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NESBIT’S

PRACTICAL LAND-SURVEYING.

EDITED BY

WILLIAM BURNESS, F.R.A.S.

PROFESSOR OF PRACTICAL LAND-SURVEYING.

LONDON:

LONGMANS, GREEN, READER, AND DYER.

^PU0T££J

KRAKOW

k^Jechn»c^

HM353

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ADVERTISEMENT

TO

THE ELEVENTH EDITION.

The thorough revisal and enlargement of Nesbit’s Practical Land-Surveying cannot fail to continue for it the very prominent position which it has long held as a standard work. The sale of TEN LARGE EDITIONS, and the demand for the present one, is a very gratifying testimony of the patronage it has hitherto merited of land-surveyors, civil engineers, masters of seminaries, and the general public.

The subdivision of the work into Ten Parts, as in the last edition, has been continued. The First Part is elementary, and contains definitions, with a few select problems and theorems. The Second gives a description of the chain, offset-staff, compass, &c., with directions for their use. The Third and Fourth embrace the mensuration of single fields, the former with the common cross-staff and chain, the latter with the chain only. Part IV. also teaches how to survey hilly and uneven ground. The surface of the earth consists of a series of hypotenusal planes joined by curved lines, and before the dimensions of these can be plotted on to a plan, they have to be reduced to base-measure or that of a common plane; and this part gives various methods of doing so by the chain, quadrant-level, and theodolite. The Fifth, Sixth, and Seventh teach how to survey large estates, apportion land, plot, finish, transfer and reduce plans. Part VI. belongs to the higher branch of land-surveying, and is divided into four sections, the first of which is elementary, the second practical ; the third gives general directions for field operations, and the , fourth the reduction of customary to statute measure. Part VII. includes the surveying of farm-buildings, cities, towns, villages, and lands for building purposes. The Eight and Ninth teach the elements of trigonometry, and their application by examples. Railway

Surveying follows, and includes illustrated general directions for taking a preliminary survey of a projected railway. The Tenth includes Railway Engineering under seven sections, viz.:—

Section I. treats of levels and levelling, including trial, check, and cross levels; check-levelling; correction for curvature and refraction.

Section II. contains a Parliamentary survey, with plan and section; the method of keeping the level-book and of plotting the sections; also the method of laying out gradients, &c.

Section III. shows the general principles and practice of laying out railway curves, including compound, serpentine, and deviation examples, with the necessary constructions and formulas required.

Section IY. embraces railway earthworks, and gives formulas for setting out the Width of the ground, for finding the areas of the railway estate, together with auxiliary tables for finding the contents of sectional areas.

Section V. gives the methods of setting out the ground for tunnels and tunnelling, including tables of dimensions.

Section VI. is on viaducts, aqueducts, oblique railway bridges, and includes examples with dimensions.

Section VII. comprises the survey and applications of rules for the superelevation of the outside rails at railway curves.

The following is a brief summary of the principal additions and alterations made to the present edition.

Part II.—Two new articles have been written for this Part: the one on the variation of the needle of the compass; and the other, directions for correcting the compass so as to find out the true meridian.

Parts III. and IV.—In these two Parts one proposition (page 85) has been corrected, and two new ones, with a diagram, added, in order to supply the necessary instruction required in ranging and measuring lines in surveys with the chain when sight is obstructed. The methods of measuring hilly ground have been re-written; so have the articles under a Quadrant, a brief description of several additional instruments of this kind being given. General directions for surveys on the vertical plane have likewise been added in Part III.

Parts V. and VI.—Very many corrections have been made in these two divisions, paragraphs out of date suppressed, and new ones written to supply their places. General directions for eight additional surveys have been added to the latter (Part VI.), while the

statutory matter which it contained in previous editions has been transferred to the end of the work in the form of an Appendix. The two first surveys are on the horizontal plane, and embrace lands intersected by a railway. The remaining six include surveys of rivers, tidal, mountainous, and intermediate; canals, embankments, lands reclaimed from the sea, &c. They are principally on the vertical plane, and form together a large section, viz., Section III.

Part VII.—General directions for surveying and designing mansions, farm homesteads, stackyards, &c., have been added to this Part; other portions of it have been re-written.

Part VIII. has been divided into five Sections. Sect. I. consists of definitions and corollaries. Sect. II. treats of the species, constructions, ratios, resolution and solution, of triangles. Sect. III. gives the trigonometrical solution of plane triangles, right-angled, and oblique. Sect. IV. the measurement of angles, with an illustrated description of the theodolite. Sect. V. is on the trigonometrical survey of heights and distances. The first two sections have been added to this edition of the work. They have been written expressly to illustrate, by means of the several diagrams annexed, the elementary principles and practice of trigonometrical surveying: it is hoped they will prove not only instrumental in facilitating the education of private students, but also in promoting that of articled pupils in field operations with the theodolite, and in the office work of plotting the bearings. Three of the cases in Sect. III. have been re-written, and numerous corrections made in this and the remaining two sections of this Part.

Part IX. embraces examples of trigonometrical surveying; it has been divided into seven Sections. Sect. I. contains rules for finding the areas of triangles by logarithms. Sect. II. To survey a wood. Sect. III. To survey a wood or river. Sect. IV. gives the practice of taking bearings by two stations. Sect. V. To survey a town or city. Sect. VI. To range lines and take bearings in extensive surveys. Sect. VII. To survey with the circumferentor. Of these, Sect. VI. has been written for the present edition, and also the Introduction containing the new subdivision of this Part.

The section under Railway Surveying has been re-wrritten in the form of illustrated general directions for taking a preliminary survey of a portion of a projected railway. In the previous edition the illustration or plan (p. 387), and the subject-matter of the section generally, was in Part IX. under a different heading; but as

ADVERTISEMENT TO THE ELEVENTH EDITION.

viii

the field notes for the plan were wanting, and as the subject-matter was otherwise very defective, and as the former omission could not be supplied, or the latter defects corrected, without an actual survey of the ground,—conditions almost wholly impracticable,—it was considered advisable to make this change, as preliminary surveys of the kind in question are frequently demanded for reasons that will be found in the section to which we refer.

In this branch of land-surveying it is a standing rule that the measure of every line and angle shall be determined both by linear and angular instruments; and when an error or an omission is detected in any of the field books open, the error must be corrected, and the omission supplied in the field. The reason of this is, because the exceptions are few where a single line in a survey can be accurately measured with the chain; consequently, if a continuous check is not kept upon linear measure by angular, the field notes, both for plotting and setting out curve lines and inclinations or gradients, cannot be obtained with that degree of accuracy which the nature of the survey demands. If the station poles are accurately ranged, then the lines in the field close; and in plotting either plan or section, the only proof of accuracy admissible is when the lines formed by the station-points also close, for unless they do so, the subsequent operations of the survey cannot be correctly performed.

W. B.

CONTENTS.

PART I.

PAGE

Geometrical Definitions ...........

Geometrical Problems............

Prob. 1. To bisect a given line..........

2.    To bisect a given angle.........

3.    To draw a line parallel to a given line......

4.    To erect a perpendicular from a given point, near tbe middle of a

given line............

5.    To erect a perpendicular from a given point, near tbe end of a

given line............

6.    Prom a given point, to let fall a perpendicular upon a given line

7.    To construct a triangle of three given lines.....

8.    Having given the base and perpendicular, to construct a triangle . 10

9.    To describe a square whose side shall be equal to a given right line .    10

10.    To describe a rectangular parallelogram whose length and breadt]

shall be equal to two given lines......

11.    Upon a given right line to construct a rhombus ....

12.    With two given right lines, as sides, to construct a rhomboid .

13.    With a given base and two given perpendiculars, to construct

trapezoid ...........

14.    With four given sides, to construct a quadrilateral figure which ha

one right angle..........

15.    With given transverse and conjugate diameters, to construct an

ellipse...........

16.    To reduce a given trapezium to a triangle of equal area

17.    To reduce an irregular polygon of five sides to a triangle of equa

area ............

18.    To raise a perpendicular by a scale of equal parts

19.    To make a right angle by the line of chords on the plane scale .

20.    To make an acute angle........

21.    To make an obtuse angle .    .......

22.    To find the number of degrees contained in a given angle .

23.    To lay down a line making a given angle with the meridian line

Geometrical Theorems...........

10

11

11

12

12

12

13

14

14

15

15

15

16

16

PART II.

The Chain......... .

The Cross-staff........

The Offset-staff........

The Compass........

Variation of the Needle......

To find a True Meridian......

The Field-Book........

Directions to young Surveyors when in the Field, &c. Directions concerning Scales, Laying down Figures, &c.

PART III.

PAGE . 21 . 22 . 22 . 23 . 24 . 24 . 24 . 25 . 27

. 29 . 30 . 30

To Survey with the Chain and Cross......

A Table of Lineal Measures....... (.

A Table of Square Measures.......

Pjrob. 1. Square fields........

2.    Rectangular fields.......

3.    Triangular fields........

4.    Fields in the form of a trapezium ....

5.    Fields comprehended under more than four straight sides

6.    Fields comprehended under any number of crooked sides

7.    Narrow pieces of land.......

8.    Meres and woods........

9.    To find the area of segment of a circle, or any other curvilineal

figure, by means of equidistant ordinates .

10.    To find the breadth of a river

11.    Lines upon which there are impediments not obstructing the sight . 84

12.    Lines upon which impediments obstruct the sight

80

. 32 . 34 . 36 . 40 . 49 . 65 . 71

. 75 . 83

. 85

PART IV.

To Survey with the Chain only Miscellaneous Instructions .

86

. 86

. 87 . 97 . 115 . 132 . 135 . 138 . 139

. 139

Pkob. 1. Triangular fields.......

2.    Fields in the form of a trapezium

3.    Fields of more than four sides.....

4.    Meres and woods.......

5.    To measure and plan roads, rivers, canals, &c.    .

6.    To take distances by the chain and scale

7.    To erect a perpendicular by the chain

8.    From the plan of a field, and its true area, to discover the scale by

which it has been constructed .

To Measure Hilly Ground ....

Methods used by Practical Surveyors to Reduce Hypotenusal to Horizontal Lines ..............

140

141

Method I.

A table for reducing hypotenusal to horizontal lines .

141

. 142

Quadrants and their use ....

To take the altitude of a hill with the quadrant To take the altitude of a steeple .

Method II.........

To preserve the horizontal line hy elevating the chain, in ascending or descending a hill...........

PAGE

142

. 143

143

143

143

Method III........

The description and use of King’s quadrant .

Nesbit’s quadrant.....

A table by which a quadrant (invented by the author) may be constructed, answering the same purpose as King’s .

The construction of the above table ...    ...

The construction of Nesbit’s quadrant ......

The method of proving it........

The method of applying it in surveying .    .    .    .

Methods of finding the hypotenusal measure of hilly ground General directions for the vertical survey of a hill

145

. 145 . 147

. 148

148

149

151

. 151 . 152 . 156

PART V.

To Survey Farms, large Estates, or Lordships

. 159

Method I. By triangles Plan of ditto

. 159 . 160

Method II. By running lines nearly parallel to each other Plan of ditto .......

. 162 . 163

Method III. By tie-lines, &c.

Method IV. By running lines in the most advantageous manner, without regarding any particular method..........

Miscellaneous Instructions relating to running lines, putting down Stations,

ranging the Poles, measuring across Valleys, observing the Fences, &c. &c. 165 General Buies for planning large Surveys .

Directions for planning the Estate in Plate VIII.

Directions for planning the Estate in Plate X.

Drawing Paper, Pencils, &c.    ....

To compute the Contents of Estates .

The Use of the Parallel Euler, in reducing crooked fences to straight ones, in order to find the areas of fields by the method of casting .

Pbob. 1. To draw a right line from a given point so as to divide the field equally............

2.    To draw a right line from a given point so as to reduce an irregular

fieldrto a right-angled triangular one'.

3.    To reduce an irregular field to a right-angled triangle, by drawing the

equalising line through the offset fences ....

4.    To lay down a right-line offset from given dimensions, &c.

5.    To lay down a curve-line offset from given dimensions, &c.

6.    To reduce a curve-line offset area to a right-angled triangle

7.    To reduce an irregular right-line offset figure to a triangle

. 164

164

. 168

. 169

170

171

172

. 173

173

. 174

. 174 . 175 . 176 . 178 . 179

Pr.OB. 8. To reduce a right-line offset field of twelve sides to a triangle

9. To reduce a right-line offset field of eleven sides to a trapezium .

10.    To reduce a curve-line and right-line offset field to a trapezium .

11.    To divide by an equalising line a crooked eight-line offset fence

between two fields, on the principle of “ take and give ” .

12.    To divide by an equalising line a crooked curve-line offset fence

between two fields, on the principle of “ take and give ” .

A General Eule for using the Parallel Euler.....

A Book of Dimensions, Castings, and Areas, belonging to Plate VIII.

Ditto belonging to Plate X...........

To Transfer a rough Plan to a clean sheet, in order to make a finished Plan

Method I. By points..........

Method II. By tracing-paper.........

Method III. By a copying-glass .........

Method IV. By similar squares .........

Method V. By the pentagraph.........

A description of the pentagraph........

The method of using the pentagraph in copying and reducing plans .

To embellish or finish plans..........

Method I..............

To finish plans neatly with Indian ink and colours ....

How to choose Indian ink.........

Colours necessary for a Land-Surveyor ......

PAGE . 180 . 181 . 182

. 183

. 184 . 185 . 186

187

. 188

. 188

188

188

. 189

. 190

191

. 192

193

194

. 194 . 195 . 196

To finish plans highly, with Indian ink and colours

To shade and colour meadows.....

Pasture-grounds........

Corn-fields .    .......

Moors..........

Marshy ground........

Sands, rocks, and loose stones.....

Trees..........

Lakes, rivers, and the sea-shore.....

Hilly ground........

Pleasure-grounds........

Gardens    ........

The bases of buildings.......

The elevations of buildings......

Method III. .........

To finish plans highly with Indian ink only .

Shading with the pen.......

Penmanship .    .    .    .    .    .

Ornaments.........

Ornaments on Plate IX. ......

Ditto on Plate XI........

196

UtSMHHHOOOWOOaaa

. 205 . 205

205

. 205 . 206 . 206

206

Miscellaneous Instructions relating to Surveying, Planning, Casting, Valuing,

&c. &c......

A Terrier of the Survey in Plate IX.

207

PART VI.

PAGE

Directions for Apportioning Land; for Laying out, Parting off, and Dividing Land ; for Surveying Lands intersected by a Railway; Rivers, &c. &c., with six examples; and for reducing Statute Measure to Customary, and vice versa; including Scotch and Irish Measures

. 210

SECTION I.

Table for reducing acres, &c., to square links ....

Pkob. 1. To reduce acres, roods, and perches into square links .

2.    To lay out, in a square, any quantity of land proposed .

3.    Upon a given line, to make a rectangle which shall contain any pro

posed quantity of land.........

4.    To lay out any given quantity of land in a rectangle, so that one of its

sides shall be two, three, four, or any number of times as long as the other

5.    Upon a given base, to lay out a triangle that shall contain any given

number of acres, &c...........

6.    Upon a given side, or base-line, to lay out a trapezium which shall con

tain any number of acres

7.    Upon a given base, to lay out a rhombus of any content less than the

square of the base........

8.    To lay out any given quantity of land in a circle .

9.    To lay out any given quantity of land in a regular polygon .

10.    To lay out any given quantity of land in a ellipse, with a given

diameter

11.    To part from a square or rectangle any proposed quantity of ..land, by a

line parallel to one of its sides........

12.    To part from a square or rectangle any proposed quantity of land,

either in a right-angled triangle or a trapezoid, by a line drawn from any of the angles to either of the opposite sides

13.    To part from a triangle, upon the base or longest side, any proposed

quantity of land, by a line drawn from either of the angles at the base, to the opposite side........

14.    To part from a triangle any proposed quantity of land, by a line parallel

to any of its sides..........

15.    To part from a rectangle or triangle any proposed quantity of land,

upon a line on which there are offsets, when the area of those offsets, is to be considered as part of the portion to be parted off .

16.    To part from a trapezium, or any irregular polygon whatever, any pro

posed quantity of land, by a line drawn parallel to any of the sides, or by a line drawn from any of the angles, or from any assigned point in one of the sides, to any of the opposite sides .

213

. 214

214

. 215

217

.    218

.    220

. 221

222

. 224

226

227

. 228

. 231

SECTION II.

Prob. 1. To divide a square or rectangle, either equally or unequally, among any number of persons, by lines parallel to one of its sides 2. To divide a triangular field, either equally or unequally, among any number of persons, by fences made from any of its angles to the opposite side...........

. 238

PBOB. 3. To divide a triangular field, either equally or unequally, among any number of persons, by fences proceeding from any assigned point in one of its sides

4.    To divide a triangular field, either equally or unequally, among any

number of persons, by fences made parallel to one of its sides .    . 244

5.    To divide a trapezium, or any irregular polygon, either equally or un

equally, among any number of persons, by fences made in a given direction

6.    To divide a common or any quantity of land, of uniform value, among

any number of proprietors, in the proportion of their respective interests

7.    To divide a common, &c., of variable value among any number of pro

prietors, in the proportion of their respective interests

Miscellaneous Observations on Valuing Land ....

Appellations given to certain Lands.....

Directions for setting out new Roads, Sand-pits, Quarries, 'Watering-places, &c.

&c. ; and for dividing Commons and Waste Lands into Allotments .    . 255

To determine the Value of each Proprietor’s Allotment or Claim upon the Common

To set off upon the Plan each Proprietor’s Allotment or Share of the Common 257 An example of dividing the Common in Plate XII., &c.

A Book of Quantities, Qualities, Values, &c., belonging to Plate XII.

The operation of finding the Value of each Proprietor’s Share of the Common, and Directions for setting out the Allotments in the Field .

The proof of the Division.........

General Directions for apportioning Lands intersected by a Railway .

Example 1. Intersected parks and farms .    .    .

2. Intersected estates, parishes, and counties

. 242

. 246

. 249

. 251

. 252 . 253

256

. 257

. 258

. 259

260

. 261

261

. 264

SECTION III.

Directions for Surveying Rivers, Canals, Embankments, &c., in Six Surveys . 267 Example I. Directions for the survey of lands to be reclaimed from the sea

Directions for ranging station poles......

Directions for the horizontal survey.....

Directions for longitudinal and transverse sections for embankments ...........

II. Directions for the survey of a tidal river ....

1.    The breadth of a river as determined by its tributaries

2.    The breadth of a river as determined by tidal action .

3.    The velocity of a river as determined by its tidal length

4.    New channel and embankments......

A new outfall .........

Straightening a river........

5.    Methods for counteracting tidal action ....

6.    Influx of tributary rivers.......

III. General directions for horizontal survey of lands reclaimed under the first and second examples

1.    Quality of soil ....

2.    Drainage.....

3.    Embankments ....

. 267 . 268

. 271

. 273

. 274

. 275

. 275

. 277 . 278 . 278 . 279 . 280

280

. 281

282

282

PAGE

4.    Roads, fences, and farm buildings

5.    Area outside embankments Example IV. General directions for the survey of a river in a large valley

from high-water level upwards .

1.    Outfall levels of the river and its tributaries First Method op lowering Outfall

Second Method......

Third Method......

Fourth Method......

2.    General fall of the river ....

Tangential errors in levelling Errors due to refraction ....

Hypotenusal errors.....

Check-water levels .....

Longitudinal section.....

Surface level of the river ....

Plotting the dimensions ....

General rule for inclination of river .

3.    Volume of water......

4.    Area of land required for embankments

5.    Subdivision and apportioning land where the river is

straightened.........

6.    Water power, irrigation works, bleaching grounds, &c.

7.    Provision for the drainage of adjoining lands

8.    New roads, bridges, fences, drainage, and levelling of the

old channel ........

V. General directions for the survey of mountain rivers .

VI. General directions for the survey of canals, irrigation works,

and warp-lands ..........

Canals...........

Irrigation works..........

Warp-lands...........

. 284 . 284

. 285 . 286 . 286

286

287

. 288 . 288 . 289

. 291

291

. 292

292

. 294

295

. 297 . 297 . 298

. 299 . 299 . 299

300

. 301

304

304

309

310

SECTION IV.

The Method of reducing Local or Customary Measures to Statute Measure,

and vice versd............

Prob. 1. To reduce customary measure to statute measure, and vice versd

General rules..........

2. To reduce Scotch and Irish measures to English statute measure,

and vice versd...........

To estimate commons, moors, lordships, counties, or kingdoms, by the square mile, &c. .

. 311 . 311 . 311

313

. 315

PART VII.

SECTION I.

The Method of Measuring and Planning Villages, Towns, and Cities .

General Directions for taking the Dimensions of Villages, Towns, and Cities . 317

. 316

Directions for Surveying and Planning Mansions, Fa

.Note 1.—A plotting-scale divided into two chains

Answer.

PROBLEM II.

:,g<:

te

ffa-sgsitsfl;

PROBLEM X.

As the whole of A’s allotment will fall in No. 10,

sion found by trigonometry. Next, EF, taken from t

Double areas

These two plates are used to take horizontal angle

m\

3-91  

Note 2, Problem YII., where the symbols are define

Mr Basliforth says, in defence of his method of fi

39 Paternoster Row, E.C. London, April 1883.

Proctor’s Larger Star

Novels and Tales. By the

. 320 . 321 . 321 . 323 . 323 . 323 . 323 . 324 . 324 . 324

Directions for Surveying and Planning Mansions, Farm-buildings, Stackyards, &c...........

Directions for taking the Dimensions of the New Town in Plate VII.

Directions for Planning Villages, Towns, and Cities .

Drawing Pencils.........

Plotting Scales ..........

Various Plans recommended........

To clean Plans or Maps..........

India-rubber...........

Pounce or Gum Sandarach.........

Sponge .............

SECTION II.

Directions for Measuring and Planning Building-ground, and dividing it into

convenient lots for sale "......

Directions for taking the Dimensions ....

Various Examples .    ......

Directions for Planning, Dividing, Laying Out, &c. kc.

House-steads must be of various sizes, according to the respectability of the intended Buildings Different pieces of Building-ground

. 325

. 325 . 326 . 329

330

. 330

. 331

SECTION III.

Miscellaneous Questions relating to Surveying, Laying out, Parting off, and Dividing Land

PART VIII.

PLANE TRIGONOMETRY.

SECTION I.

Definitions . Corollaries

334

336

SECTION II.

Species, construction, ratios, resolution, and solution of Triangles

Right-angled Triangles.    .........

Oblique-angled Triangles.........

. 339

340

342

SECTION III.

Solution of Right-angled Plane Triangles Case I.......

349

349

II.

350

III.

Solution of Oblique-angled Plane Triangles . Case I........

. 351 . 352

352

II.

352

III.

IV.

. 354 . 355

SECTION IV.

356

The Measurement of Angles The Theodolite

PAGE

To take a Horizontal Angle with the Theodolite......

To take a Vertical Angle..........

Adjustments of the Theodolite—

I. To adjust the line of collimation ......

II. To adjust the axis of the level with that of the telescope .

III. To make the axes of the levels on the vernier plate parallel to it

. 358 . 359

The Vernier

1.    To find fractional parts of a degree by the vernier .

2.    To set the theodolite to any angle ....

. 359 . 359 . 359 . 360 . 360

360

SECTION V.

HEIGHTS AND DISTANCES.

Prob. 1. To measure a base line across a wide river

2.    To continue the direction and measurement of a given line when

buildings or other obstructions stand in the way, which can be avoided by going to the right or left .

3.    To find the distance of accessible objects when obstructions pre

vent the measurement of these distances in direct lines

4.    To find the height of a tower, standing on a horizontal plane, the base

of which is accessible ......

5.    To find the height when the base is inaccessible .

6.    To find the height of a tower standing on the side or top of an inac

cessible hill

. 360

. 363

. 364

. 364 . 365

. 366

PART IX.

TRIGONOMETRICAL SURVEYING.

SECTION I.

RULES EOR FINDING AREAS.

Preliminary Problems Prob. 1.    .

2. .

3.

4.

. 367 . 367 . 368 . 36 S . 369

SECTION II

To Survey a Wood by the Theodolite .

. 369

SECTION III

To Survey a Road or River .

. 371

SECTION IV.

The Practice of Taking Bearings by Two Stations

. 373

SECTION V.

To Survey a Town or City

. 375

SECTION VI.

To Range Lines and take Bearings in Extensive Surveys .

SECTION YII.

The Circumferentor........

To Find the Bearings of an Object by the Circumferentor . On Plotting the Bearings taken by the Circumferentor

PAGE

. 383 . 384 . 384

RAILWAY SURVEYING.

Illustrated General Directions for a Preliminary Survey of a Projected Bailway .....

To range Lines ....

Copy of a Plan of a Projected Bail way To lay down the Base Lines, and draw the Projected Bailway upon an Ordnance

or other Map.......

To range the Secondary and Minor Lines To take Bearings, and measure the First Base Line To take Bearings, and measure the second Base Line .

Details of the Survey .    ......

. 386 . 386 . 387

. 388 . 388 . 389 . 391 . 391

PART X.

RAILWAY ENGINEERING.

SECTION I.

LEVELLING.

. 392

On Levelling in General Description of the Y Level Adjustments of the Level Troughton’s improved Level Gravett’s Level Levelling Staves .

Correction for Curvature Correction for Befraction

To find the Differences of the Levels of several Points on the Earth’s Surface,

and to trace a Sectional Line of them ........

The Datum Line.....

Practical Level-Book ....

Trial Levels ......

Check Levels, and Survey of the Line .

Final Levels......

Level-Book......

Plotting the Main Section

Cross Levels......

The Application of the Correction for Curvature and Befraction in Levelling . 407 Levelling by the Theodolite .

Levelling by the Aneroid Barometer

392

. 393

394

. 394 . 395

395

. 396

397

400

400

. 401

402

402

404

. 406

406

409

409

SECTION II.

Parliamentary Plan and Section, &c.    .

The Method of Laying out Gradients .

. 410 . 411

PAGE

Tunnels.........

To determine the Rate of Inclination of a Gradient Method of determining the Heights of Roads passed over or under by a Railway, &c

412

. 412

. 413

SECTION III.

RAILWAY CURVES.

On the Use of Curves in Railways in general......

On Mechanical Railway Curves, or Curve-rulers.....

The Limit of the Radii of Railway Curves.......

Prob. 1. Case I. To determine the radius of a railway curve mechanically

II. To determine the radius of a railway curve geometrically . 417 III. To find the radius of the curve when the map is inaccurate

IY. To find the radius when the tangents cannot be prolonged to meet........

PROB. 2. To lay out a railway curve by the common method

3.    To lay out a railway curve by offsets from its tangents Case I. When the length of the curve is less than 3 of its radius

II. When the curve is any required length III. When the length of the- curve exceeds the limit adopted in Case I., but is considered too short to be divided into two parts as in Case II.

4.    To lay out the curve by two theodolites, where water, swamps,

quarries, and other obstructions prevent the use of the chain .

5.    To lay out the curve by offsets from its chord or chords, &c.

Definition of the Compound Curve .    ......

Prob. 6. To find the several Radii of the Compound Curve, &c..

Definition of the Serpentine or S Curve.......

Prob. 7. (1.) When one radius and its tangential point are given, to find the other radius and tangential point of the serpentine curve geometrically ...........

(2.) When the tangential points and one of the radii of the serpentine curve are given, to find the other radius geometrically .

PROB. 8. (1.) When the two portions of the serpentine curve have the same radius, to determine that radius geometrically, the tangential

points and their distance being given......

(2.) To find the common radius of the serpentine curve by calculation .............

PROB. 9. Tojmake a given deviation from a straight portion of a line of railway by means of three curves.......

Remarks on laying out the Curves in the four last Problems ....

. 414 . 415 . 415

. 416

. 417

418

. 418 . 420 . 420 . 422

. 424

. 424 . 426 . 427 . 428 . 429

429

. 430

431

431

432

433

SECTION IY.

RAILWAY EARTHWORKS.

On setting out the Width of Ground for a Railway

Prob. 1. To set out the width when the surface of the ground is latterly on the same level as the intended railway

. 434

Pkob. 2. To set out the width of cuttings, or embankments, when the' surface of the ground is laterally level, and at a given height above the intended railway, &c...........

3.    To set out the width of the cuttings, when the surface of the ground

is laterally sloping, &c..........

4.    To set out the widths when the surface of the ground is laterally

sloping, and when the cross-section of the works consist partly of a cutting and partly of an embankment, &c.....

5.    To find the width of cuttings when the surface is very uneven

6.    To find the quantity of land required for a projected railway, CASES

I. and II............

On Railway Cuttings in general, and Tables for finding their Contents ,

The General Earthwork Table (at the end of the book) ....

Pkob. 7. Case I. To find the contents of cuttings by the General Earthwork Table and the Auxiliary Table No. I. .

II. To find the contents of cuttings by the tables, when the depths are given in feet and decimals of feet

8.    Case I. To find the content of a cutting between two cross-sections

the areas of which, &c., are given II. When great accuracy is required, &c.

9.    To find the contents of a railway cutting, when the slopes of the two

sides are different ....

Case I. When the depths are given

II. When the sectional areas are given

10.    To apply the general table and Table No. II. to such bottom widths

and ratios of slopes as are not found in Table No. I. .

11.    To find the content of the cutting of a tunnel

An example showing the magnitude of the errors of several methods of finding the Contents of Cuttings.........

Observations important to those concerned in the Construction of Railways, on the erroneous methods of calculating the Contents of Cuttings where the surface of the ground is uneven; also on the method given in this work . 452 Investigation of the Construction and Use of the General and Auxiliary Tables 452 The Errors of the methods of finding the Contents of Cuttings by Mr. Basli-forth and others

435

436

. 438 . 440

,•441

. 443 . 444

. 444

. 446

. 448 . 449

. 449 . 449 . 450

. 450 . 450

450

455

SECTION Y.

Tunnelling.........

A Table of the Dimensions of several existing Tunnels

. 458 . 460

SECTION VI.

VIADUCTS, AQUEDUCTS, SKEW ARCHES, &C.

The Brick Viaduct at Maidenhead.....

The Viaduct over the Ouse at York.....

Aqueduct of Pont-y-Cysyllte......

Skew-bridges—On the London and Birmingham Line .

Midland Counties Railway Manchester and Birmingham Railway Dimensions of Viaducts in various Railways

. 460 . 461

461

. 462 . 462 . 463 . 463

SECTION VII.

PAGE

. 464

Super-elevation of Exterior Pail in Curves .

Tables for Peduction of Distances and Correction of Levels for Curvature, &c. . 468 Tables of Offsets for Pailway Curves, &c.

. 469

INDEX TO PRINCIPAL ENGRAVINGS, PLATES, PLANS, &c.

Plate I. The Compass.......

II. Nesbit’s Quadrant......

III.    Plan of an Estate (triangular method)

IV.    Plan of an Estate (parallel method)

V. The Pentagraph......

VI. Examples how to finish Plans with Indian ink-tures, Corn Fields, and Moors VII. Ditto—Marshy Grounds, Sands, Pocks, &c. . XII. Plan of an Inclosed Common .... VIII. Folded rough Plan of an Actual Survey .

IX. Folded finished Plan of the last (Plate VIII.)

X. Folded rough Plan of an Actual Survey XI. Folded finished Plan of Survey (Plate X.)

XIII. Folded Parliamentary Plan and Section .

Meadows, Pas-

. 23

. 147 . 160 . 163 . 191

. 197 . 199 . 259 End of work

General and Auxiliary Earthwork Tables eor Railways .

55

EXPLANATION

OF THE PBIHCIPAL

MATHEMATICAL CHARACTERS.

The sign or character = (called equality) denotes that the respective quantities between which it is placed are equal; as 4 poles = 22 yards = 1 chain = 100 links.

The sign + (called plus, or more) signifies that the numbers between which it is placed are to be added together; as 9 + 6 (read 9 plus 6) = 15. Geometrical lines are generally represented by capital letters; thus AB + CD signifies that the line CD is to be added to the line AB.

The sign - (called minus, or less) denotes that the quantity which it precedes is to be subtracted; as 15 - 6 (read 15 minus 6) = 9. In geometrical lines also, AB - CD signifies that the line CD is to be subtracted from the line AB.

The sign x denotes that the numbers between which it is placed are to be multiplied together; as 5 x 3 (read 5 multiplied by 3) =

15.

The sign -f- signifies division; as 15 -f 3 (read 15 divided by 3) =

5. Numbers placed like a vulgar fraction also denote division; the upper number being the dividend, and the lower the divisor; as y = 5.

The signs : :: : (called proportionals) denote proportionality ; as 2 : 5 :: 6 : 15, signifying that the number 2 bears the same proportion to 5, as 6 does to 15; or, in other words, as 2 is to 5, so is 6 to 15.

(called vinculum) is used to connect

The sign

several quantities together; as 9 + 3) — 61 x 2 = 12 — 6l x 2 =

6 x 2 = 12.

The sign ², placed above a quantity, represents the square of that quantity; as 5 + 3)² = 8² = 8 x 8 = 64.

The sign s, placed above a quantity, denotes the cube of that

quantity; as 9 + 3l API = 12 — 8l³ = 4³ = 4 x 4 x 4 = 64.

The sign J or placed before a quantity, denotes the square root of that quantity; as ^9 x 4 = sj36 = 6.

The sign placed before a quantity, represents the cube root of

that quantity; as y'G x 4 x 3 - ]8 = ^24 x 3 — 8l = ^72 - 8 = v"64 = 4.

DIRECTIONS TO BOOKBINDER.

Plans VIII., IX., X., XI., XIII., and Earthwork Tables at the end of the work, in the order thus given:

Fold Plans VIII. and IX. to face each other when open.

Fold Plans X. and XI. to face each other when open.

Fold Plan XIII. to face, when open, the work.

Fold Earthwork Tables to face, when open, the work.

LAND-SURVEYING.

PAET I.

DEFINITIONS, PROBLEMS, AND THEOREMS IN GEOMETRY.

Geometry originally signified the art of Measuring the Earth, or any distance or dimensions upon, or within it; but it is now used for the science of quantity, extension, or magnitude.

GEOMETRICAL DEFINITIONS.

1.    A point is considered as having neither length, breadth, nor thickness.

2.    A line has length, but is considered as having

neither breadth nor thick--

ness; as ab.

3.    Lines are either right, curved, or parallel.

4.    A right or straight line lies wholly in the same direction between its extremities, A and is the shortest distance between two points; as ab.

5.    A curved line continually changes its direction between its extremities; as AB. A’

6. Parallel lines always c remain at the same distance from each other, and, though continually produced, would a never meet; as AB and CD.

<:

7.    A surface or superficies has d length and breadth, but is considered as having no thickness; as ABCD.

8.    A superficies may be contained within one curved line, but cannot be contained within fewer than three straight lines.

9.    The area of a figure is its superficial content, or the measurement of its surface.

10.    An angle is the inclination or opening of two lines having different directions and meeting in a point, as at A, which is called the angular point; and, when three letters are used, the middle one denotes that f point.

11.    Angles are of three kinds, viz., right, acute, and obtuse.

12. A right angle is made by one right line standing perpendicularly upon another; thus, if DC be perpendicular to AB, the angles ADC and cdb are both right angles.

13.    An acute angle is less than a right angle; as CAB.

14.    The complement of an angle is what it wants to complete a right angle; as the angle DAB is the complement of the angle CAB.

15. An obtuse angle is greater than a right angle ; as BCD.

____X

16.    The supplement of an angle is what it wants of two right angles ; as the angle ACB is the supplement of the angle BCD.

17.    A triangle is a figure or superficies bounded by three right lines, and admits of three varieties; viz., equilateral, isosceles, and scalene.

18. An equilateral triangle lias all its sides equal; as ABC.

19. An isosceles triangle has only two of its sides equal; as ABC.

20. A scalene triangle lias all its sides unequal; as ABC.

21.    Triangles are also right-angled, acute-angled, and obtuse-angled.

22.    A right-angled triangle has one right angle, the side opposite to which is called the hypotenuse, the other two being termed legs, or one the perpendicular, and the other the base; thus Ac is the hypotenuse, bc the perpendicular, and AB the base.

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23. An acute-angled triangle has all its angles acute; as ABC.

24. An obtuse-angled triangle has one obtuse angle; as acb.

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25.    The longest side AB of the triangle ABC is called the base; and the line Ca falling upon it at right angles from the opposite angle C, is called a perpendicular.

26.    A figure of four sides and angles is denominated a quadrangle or quadrilateral figure.

27.    A parallelogram is a quadrilateral figure, having its opposite sides parallel and equal, and admits of four varieties; viz., the square, the rectangle, the rhombus, and the rhomboid.

28. A square is an equilateral parallelogram having all its angles right angles ; as ABCD.

29. A rectangle is a parallelogram having its opposite sides equal and all its angles right angles; as ABCD.

30. A rhombus is an equilateral parallelogram having its opposite angles equal; as ABCD.

31. A rhomboid is a parallelogram having its opposite sides and angles equal; as ABCD.

32. A trapezium is a quadrilateral figure whose opposite sides are not parallel to each other; as ABCD.

33.    A trapezoid is a quadrilateral figure having two of its opposite sides 7 parallel, and its angles, one acute, one obtuse, and two right-angled j as ad is parallel to BC, the angles A and B are right angles'; the angle D is obtuse, and c acute.

34.    A diagonal is a right line joining the two opposite angles of a quadrilateral figure; as AB.

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35.    Plane figures having more than four sides are generally called polygons, and receive their particular denominations from, the number of their sides or angles.

36.    A pentagon is a polygon of five, a hexagon of six, a heptagon of seven, an octagon of eight, a nonagon of nine, a decagon of ten, an undecagon of eleven, and a duodecagon of twelve sides.

37.    A regular polygon has all its sides and angles equal. When they are unequal the polygon is irregular.

38.    A circle is a plane figure bounded by a curved line called the circumference, which is everywhere equidistant from a certain point within it, called the centre.

39.    The circumference of every circle is supposed to be divided into .

360 equal parts called degrees;    \

each degree into 60 equal parts, called minutes; and each minute into 60 equal parts, called seconds.

40. The diameter of a circle is a right line drawn through the centre, and terminating in the circumfer- A ence on each side; as ab.

41. The radius of a circle is half the diameter, or it is a right line drawn from the centre to the cir- a! cumference; as BA.    \

42. An arc of a circle is any part of the circumference; as the arc

AB.

43.    A chord is a right line joining the extremities of an arc; as the line AB.

44.    A segment is any part of a circle bounded by an arc and its chord.

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45. A semicircle is half of a circle, or a segment cut off by the diameter; as ABC.

46. A sector is any part of a circle bounded by an arc and two radii.

47. A quadrant is the fourth part of a circle, or a sector bounded by an arc and two radii at right angles to each other; as CDB.

Corol. Hence a right angle is said to contain 90°.

Rote.—All Definitions and Rules should be committed to memory.

GEOMETRICAL PROBLEMS.

PROBLEM I.

To bisect a given Line AB.

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From A and B as centres, with any radiusgreaterthan half AB, in your compasses, describe arcs cutting each other in m and n.

Draw the line men, and it will bisect ab in c.

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PROBLEM II.

To Used a given Angle ABC.

From the point b with any radius describe the arc From A and C, with the same or any other radius, make the intersection m. Draw the line Bm, and it will bisect the angle ABC, as required.

AC.

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PROBLEM III.

To draw a Line parallel to a given Line AB at a given Distance.

From any two points, m and n, in the given line, — with the given distance as a ' radius, describe the arcs r and o. Draw CD to touch these arcs, without cutting them, and it will be parallel — to AB.

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Note.—This problem may be more readily performed by a parallel ruler.

PROBLEM IV.

To erect a Perpendicular from a given Point c, near the middle of a given Line AB.

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On each side of the point C take two equal distances cm and on; from m and n as centres, with any radius greater than cm or cn, describe two arcs cutting each other in r. Draw the line Cr, and it will be the per- -pendicular required.

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PROBLEM Y.

To erect a Perpendicular from a given Point C, near the End of a given Line AB.

From any point m, as a centre, with the radius or distance cm, describe an arc cutting the given line in C and n.

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Through n and m draw a line cutting the arc in r. Draw the line cr, and it will be the — perpendicular required.

PROBLEM VI.

From a given Point C, to let fall a Perpendiadar upon a given Line AB.

With C as a centre, and any radius a little exceeding the distance of the given line, describe an arc cutting AB in m and n. With the centres m and n, and the same or any radius exceeding half their

distance, describe arcs inter-__

secting each other in r. Draw A the line Cr, and CD will be the perpendicular required.

Note.—The last three problems may be easily performed by a square or a plotting scale.

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PROBLEM VII.

To make a Triangle with Three given Lines, any two of which must be greater than the Third (Euclid i. 22).

Let the given lines be AB = 10, AC = 8, and BC = 6 chains.

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From any scale of equal parts (which is to be understood as employed likewise in all the following problems) lay off the base AB.

With the centre A and radius AC describe an arc. With the centre B and radius bc describe another arc cutting the former in C. Draw the lines AC and bc, and the triangle will be completed.

Note.—Any trapezium may be constructed in the same manner, when the four sides and one of the diagonals are given.

PROBLEM VIII.

Having given the Base, the Perpendicular, and the Place of the Perpendicular upon the Base, to construct a Triangle.

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Let the base AB = 9, the perpendicular CD = 5, and the distance AD = 6 chains.

Make AB = 9, and ad -- 6.

At D erect the perpendicular DC, which make equal to 5. Join AC and bc, and the figure will be completed.

Note.—A trapezium may be constructed in a similar manner, when one of the diagonals, the two perpendiculars let fall thereon from the opposite angles, and the places of these perpendiculars upon the diagonal are given.

PROBLEM IX.

To describe a Square whose Side shall be equal to a given Right Line.

Let the given line ab = 4 chains.

Upon one extremity B of the given line by Problem V., erect the perpendicular BC, which make equal to AB.

With A and c as centres, and the radius ab, describe arcs cutting each other in D. Draw the lines AD and CD, and the square will be completed.

PROBLEM X.

Tc describe a Rectangular Parallelogram, whose Length and Breadth shall be equal to Tivo given Lines.

Let the length ab = 8, and the breadth bc = 4 chains.

At b erect the perpendicular BC, which make equal to

4. With A as a centre, and the radius BC, describe an arc ; and with C as a centre, and the radius AB, describe another arc, cutting the former in d. Draw the lines ad and CD, and the rectangle will be completed.

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PROBLEM XI.

Upon a given Right Line to construct a Regular Rhombus.

Let the given line AB = 4 chains.

Draw the line AB equal to

4. With A and B as centres, and the radius ab, describe arcs cutting each other in D ; then with B and D as centres, and the same radius, make the intersection c.

Draw the lines AD, DC, and BC, and the rhombus will be completed.

PROBLEM XII.

Having any Two Right Lines given, to construct a Rhomboid.

Let the given lines be AB = 7, and BC = 4 chains.

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Draw the line ab equal to 7. Take in your compasses the line BCand lay it from A to E. With A and E as centres, and the radius AE, make the intersection D. Then withB as a centre, and the same radius, describe an arc ; and with D as a centre, and the radius AB, describe another arc, cutting the former in c. Draw the lines AD, DC, and BC, and the rhomboid will be completed.

PROBLEM XIII.

Having the Base and the Two Perpendiculars given, to construct a Trapezoid.

Let the base ab = 7, and the perpendiculars bo and AD = 3 and 2 chains respectively.

Draw the base AB, equal to 7, and erect the perpendiculars, bc equal

to 3, and ad equal to 2 i-

chains. Then join DC, A

and the trapezoid will be completed.

PROBLEM XIY.

Having the Four Sides given, to construct a Quadrilateral Figure which has One Plight Angle.

Let the sides AB = 7, BC = 4, CD = 6, and DA = 3 chains; and let the angle at B be a    c

right angle.

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Draw the line AB equal to 7, and erect the perpendicular bc, equal to 4 chains.

centre, and the radius CD, describe an arc ; and with A as a centre, and the /

radius DA, describe an- ^-----j

other arc, cutting the former in D. Draw the lines CD and DA, and the figure will be completed.

PROBLEM XV.

Having the Transverse and Conjugate Diameters given, to construct an Ellipsis.

Let the transverse diameter AB = 7, and the conjugate diameter CD = 4 chains.

Draw the two diameters to bisect each other perpendicularly in the centre o. With the radius Ao, and the centre C or d, intersect AB in F and /. These points will be the foci of the ellipse. Take any point m, in the transverse diameter, and with F and / as centres

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the points G, G, g and g: thus will you have four points in the circumference of the ellipse. After this, take a second point n, in the transverse diameter, and, proceeding as before, you will determine other four points. By the same method you may determine as many more as you please ; through all of which, with a steady hand you must draw the circumference of the ellipse.

Note.—An ellipse may also be constructed as follows :—Having found the foci F and /, as before, take a thread equal in length to the transverse diameter ab, and fasten its ends, with two pins, in the points F and f; then stretch the thread to its greatest extent; and.by moving a pencil round, within the thread, keeping it always tight, you will trace out the curve of the ellipse.

The principle upon which this construction is founded may be seen in Prob. X. Part YI.

PROBLEM XVI.

To reduce a given Trapezium ABCD to a Triangle of equal area.

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Draw the diagonal db, and parallel to it draw ce, meeting ab produced in E. Draw de ; the triangle ADE is equal to the trapezium ABCD.

Note.—This and the following problem may be applied in finding the areas of trapeziums and irregular polygons, by first reducing them to triangles. .

PROBLEM XVII.

To reduce an Irregular Polygon abode, of Five Sides, to a Triangle of equal area.

Extend the side ae both ways at pleasure, and draw the diagonals ce and CA. Parallel to these diagonals draw the lines df

15

and bg ; then draw of and CG; and GCF will be the triangle required.

Note.— Any irregular polygon of more than five sides may be brought to a triangle of equal area by reducing it successively to a figure with one side less, until you bring it to a figure of three sides. Thus the trapezium abcf or gcde is equal to the polygon abode as well as the triangle gcf.

PROBLEM XVIII.

To raise a Perpendicular from any Point d, in a given Line ab, by a scale of equal parts.

Make Dm = 3; and from the points D and m with the distances 4 and 5, describe arcs intersecting each other in n.

From D, through the point n, draw the line DG, and it will be the perpendicular required.

Note.-—This problem may be performed by any other — numbers in the same proportion ; but 3, 4, and 5 are the least whole numbers that will make a right-angled triangle.

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PROBLEM XIX.

To make a Right Angle by the Line of Chords on the Plane Scale.

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Draw the unlimited line AB > then take in your compasses 60° from the line of chords, and with E A as a centre describe the arc ED. Take 90° from the same scale, and set off that extent from D to C. Draw the line AC, and CAD will be the angle required.

PROBLEM XX.

To make an Acute Angle equal to any Number of Degrees, suppose 33° 30'.

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Draw the unlimited line ab ; then take 60° in your compasses, and with A as a centre describe the arc ED.

Then set off the angle 33° 30' from D to c. Draw the line AC; and cad will be the angle required.

PROBLEM XXI.

To make an Obtuse Angle equal to any Number of Degrees, suppose 125° 30'.

Draw the unlimited line AB; then take 60° in \

your compasses, and with X A as a centre describe the /'    '

arc ed. Then set off 90° from Dto C; and from C E to G set off the excess above 90°, which is 35°

30'. Draw the line AG; and GAD will be the angle required.

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PROBLEM XXII.

To find the Number of Degrees contained in any given Angle bac.

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With the chord of 60°, and A as a centre, describe the arc mn. Take the distance mn in your compasses, and apply it to the line of chords; and it will show the number of degrees required.

Note.—Angles may be more expeditiously laid down or measured by means of a semicircle of brass called a Protractor, the arc of which is divided into 180°.

PROBLEM XXIII.

To lay down a Line making a given Angle with the Meridian, or North and South Line.

EXAMPLES.

1. Let it be required to lay down a line that ranges N.E., making an angle of 45° with the meridian line. (See The Compass, Part II)

Draw the meridian line AN; and with the sweep of 60³, taken from the line of chords in your compasses, and A as a centre, describe the arc bc.

Set off the given angle 45° from B to C; draw the line AC, and it will range N.E., as was required.

Note.—If the line had ranged N.W., the angle must have been set off on the other side of the meridian an ; and ad would have been the direction of the line.

2. Lay down a line that ranges S.W. b. W., making an angle of 26° 15' with the meridian line.

Draw the meridian line AS; and with the chord of 60° describe the arc EF.

Set off 56° 15' from e to F ; draw the line af, and it will range

S.W. b. W., as was required.

Note 1.—If the line had ranged S.E. b. E. the angle must have been set off from e to g; and ag would have been the direction of the line.

2. This problem will be found useful to young surveyors in laying down the first line, the range of which should be taken in the field by a compass.

GEOMETRICAL THEOREMS,

THE DEMONSTRATIONS OF WHICH MAY BE SEEN IN THE WORKS OF EUCLID, SIMPSON, AND EMERSON.

THEOREM I.

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If two straight lines AB and CD cut each c other in the point E, the angle AEC will be equal to the angle DEB, and CEB to AED a-(.Euclid, i. 15; Simpson, i. 3; Emerson,

i.2).    '

THEOREM II.

The greatest side of every triangle is opposite to the greatest angle (Euc. i. 18; Simp. i. 13; Em. ii. 4).

THEOREM III.

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Let the right line EF fall upon the parallel right lines AB and CD; the alternate angles AGH and GHD are equal to each other ; and the exterior angle EGB is equal to the interior and opposite upon the same side GHD ; and the two interior angles BGH and GHD upon the same side are to- c. gether equal to two right angles (Euc. i. 29 ;

Simp. i. 7; Em. i. 4).

THEOREM IV.

Let ABC be a triangle, and let one of its sides BC be produced to D ; the exterior angle ACD is equal to the two interior and opposite angles CAB and ABC ; also the three interior angles of every triangle are together equal to two right angles .

{Euc. i. 32; Simp. i. 9, 10; Em. ii. 1, 2). h

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THEOREM V.

Let the parallelograms abcd and dbce be a upon the same base BC, and between the / same parallels AE and BC; the parallelogram ABCD is equal to the parallelogram dbce / {Euc. i. 35; Simp. ii. 2; Em. iii. 6).    [/

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THEOREM VI.

Let the triangles ABC and DBCbe upon the same base, BC, and between the same parallels AD and BC ; the triangle ABC is equal to the triangle DBC (Euc. i. 37 ; Simp. ii. 2 : Em. ii. 10).

THEOREM VII.

Let ABC be a right-angled triangle, having the right angle BAC; the square of the side BC is equal to the sum of the squares of the sides AB and AC {Euc. i. 47; Simp. ii. 8 ; Em. ii. 21).

THEOREM VIII.

Let ABC be a circle, and bdc an angle at the centre, and BAC an angle at the circumference, which have the same arc BC for their base; the angle BDC is double of the angle BAC {Euc. iii. 20; Simp. iii. 10; Em. iv. 12).

THEOREM IX.

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Let ABC be a semicircle; then the angle ABC in that semicircle is a right angle (Euc. iii. 31; Simp. iii. 13; Em. vi. 14).

THEOREM X.

Let de be drawn parallel to bc, one of the sides of the triangle ABC ; then BD is to DA as CE to EA (Euc. vi. 2; Simp. iv. 12; Em. ii. 12).

THEOREM XI.

In the preceding figure, DE being parallel to bc, the triangles ABC and ADE are similar; therefore AB is to BC as AD to DE ; and AB is to AC as AD to AE (Euc. vi. 4; Simp. iv. 12 ; Em. ii. 13).

THEOREM XII.

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Let ABC be a right-angled triangle, having the right angle BAC; and from the point A let AD be drawn perpendicularly to the base BC ; the triangles abd and ADC are similar to the whole triangle ABC, and to each other. Also the perpendicular AD is a mean proportional between the segments of the base; and each of the sides is a mean proportional between b the base and its segment adjacent to that side; therefore BD is to DA as DA to DC; BC is to BA as BA to BD ; and BC is to CA as CA to CD (Euc. vi. 8; Simp. iv. 19 ; Em. vi. 17).

THEOREM XIII.

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Let ABC and ADE be similar triangles, having the angle A common to both; then the triangle ABC is to the triangle ADE as the square of BC to the square of DE. That is, similar triangles are to one another in the duplicate ratio of their homologous sides (Euc. vi. 19 ; Simp, c iv. 24 : Em. ii. 18).

THEOREM XIV.

In any triangle ABC, doable the square of a line CD, drawn from the vertex to the middle of the fyase AB, together with double the square of half the base AD or BD, is equal to the sum of the squares of the other sides AC and bc {Simp. A

ii. 11; Em. ii. 28).

THEOREM XV.

In any parallelogram abcd, the sum of the squares of the two diagonals AC and BD is equal to the sum of the squares of all the four sides of the parallelogram {Simp. ii. 12; Em. iii. 9).

THEOREM XVI.

All similar figures are in proportion to each other as the squares of their homologous sides {Simp. iv. 26 ; Em. iii. 20).

THEOREM XVII.

The circumferences of circles, and the arcs and chords of similar segments, are in proportion to each other as the radii, or diameters, of the circles {Em. iv. 8, 9).

THEOREM XVIII.

Circles are to each other as the squares of their radii, diameters, or circumferences {Em. iv. 35).

THEOREM XIX.

Similar polygons described in circles are to each other as the circles in which they are inscribed, or as the squares of the diameters of those circles {Em. iv. 36).

THEOREM XX.

All similar solids are to each other as the cubes of their like dimensions {Em. vi. 24).

PART II.

A DESCRIPTION OF THE CHAIN, CROSS-STAFF, OFF-SET STAFF, COMPASS, AND FIELD-BOOK ; ALSO DIRECTIONS TO YOUNG SURVEYORS WHEN IN THE FIELD ; &c.

THE CHAIN.

Land is commonly measured with a Chain, invented by Mr. Gunter, known by the name of ‘ Gunter’s chain.’

It is 4 poles, 22 yards, or 66 feet in length, and divided into 100 equal parts, called links; each link being 7‘92 inches. At every tenth link from each end is fixed a piece of brass with notches or points ; that at 10 links having one notch or point; at 20, two ; at 30, three ; and at 40, four points. At 50, or the middle, is a large round plain piece of brass.

The chain being thus marked, the links may be easily counted from either end; the mark at 90, 80, &c., being the same as that at 10, 20, &c. Part of the first link, at each end, is made into a large ring or bow, for the ease of holding it in the hand.

The chain should always exceed 22 yards, by an inch and half or two inches ; because, in surveying, it is almost impossible to go in a direct line, or to keep the chain perfectly stretched. Long arrows likewise keep the ends of the chains a considerable distance from the ground; the lines, consequently, will be made longer than they are in reality.

Chains, when new, are seldom a proper length; they ought always, therefore, to be examined; as should those likewise which are stretched by frequent use.

Note 1. In folding up the chain, it is most expeditious to begin at the middle, and fold it up double. When you wish to unfold it, take both the handles in your left hand, and the other part of the chain in your right; then throw it from you, taking care to keep hold of the handles. You must then adjust the links before you proceed to measure.

2. Chains which have three rings between each link are much better than those which have only two, as they are not so apt to twist.

THE CROSS-STAFF.

The Cross-Staff is an instrument used in the field by surveyors, to erect perpendiculars, and may very easily be made in the following manner.

Procure a piece of board about 6 inches square, either of sycamore, box, or mahogany.

Draw the two diagonals, and at their extremities fix four small studs or pins, which will serve as sights to direct to any object or angle.

Or, instead of studs or pins, you may saw two fine grooves at right-angles, about a quarter of an inch deep, in the board.

This being fixed upon a staff, of a convenient length for use, pointed with iron * at the bottom to enter the ground readily, the instrument is called a cross-staff.

Suppose abed, to represent a cross, and the groove ac to be directed to an object at m ; then will the groove bd point to another at n.    a

Reverse the direction of the grooves, so that bd may be in the direction of m, then if ac be in the direction of n the instrument is correct.

Note 1.—The cross must be fixed upon the top of the staff, in such a manner that it may be easily turned without moving the staff itself.

2.    The cross may be made of a circular piece of board ; you must then draw two diameters crossing each other at right angles. The fourth part of a circle will answer the purpose equally well.

3.    Great care ought to be taken in making this instrument, as its accuracy depends on the sights, or grooves, being at right angles with each other.

THE OFFSET-STAFF.

The Offset-Staff is an instrument used to measure short distances ; and may be in length 10, 12, or 15 links. It would be advisable to number the links from each end, on opposite sides, with the figures 1, 2, 3, &c., as the staff thus marked will be more convenient for use.

Note.—As the cross-staff is sometimes thought incommodious, a small pocket-cross may be so contrived as to be readily fixed, upon occasion, to the offset-staff. This may be most expeditiously accomplished by means of a hole made through the cross, admitting the top of the staff, to the eighth link, counting from the bottom or piked end ; at which place there must be attached a small shoulder, upon which the cross will rest.

THE COMPASS.

The Compass is an instrument used by surveyors, to point out the range or direction of lines, and also to show the bearings of objects. The circumference of the card of the compass contains 360°, and is divided into thirty-two equal parts, called Points, each containing 11° 15'.

Of these, the four principal (namely, East, West, North, and

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South) are called Cardinal Points; from which the names of the others are derived.

To the under-side of the card, and in the direction of its north and south lines, is attached a magnetic bar of hardened steel, called the Needle, by which the north-point is directed towards the northern part of the horizon ; and the other points, consequently, to their corresponding ones in the heavens.

The card and needle are suspended on an upright pin, called the Supporter, which is fixed in the bottom of a brass or wooden box ; and the whole is covered with a plate of glass to prevent the action of the wind upon the card.

Although the compass is divided into thirty-two points, yet surveyors reduce (them to eight, namely, the four cardinal or chief points, and the four midway between them, viz., the north-east, north-west, south-east, and south-west, which may be expressed by their initial letters, as E., W., K, S.; N.E., N.W., S.E., S.W.

Variation of the Needle.

The needle of the compass does not point due north, and the difference between the magnetic pole and the true meridian is termed the variation or declination of the needle. It is not uniform, but increases or decreases during periods of time. The angle of declination at any one time is therefore not an accurate rule for the correction of the compass at a subsequent time.

The needle is also affected by thunder-clouds, aurora borealis, and similar electrical phenomena, and also by iron and ironstone.

To find a true Meridian.

Many land-surveyors have dials in their office windows, and also a small compass; and as the shadow of the gnomon ranges due north at noon over the compass, the declination of the needle thus found gives the correction to lay down the true meridian on the plan. Others have a dial outside. Articled pupils, again, are taught in the field how to correct the compass. Thus, range two poles at noon so that their shadows shall be in a line. The declination of the needle of the pocket-compass, or of the compass of the theodolite, will give the correction. Or if the shadow of a pole is made to cross a main-line, or any other principal line driven with the chain, the angle thus formed will furnish data for laying down the meridian.

THE FIELD-BOOK.

Scarcely any two surveyors set down their field-notes exactly in the same manner. The method, however, now generally adopted, is to begin at the bottom of the page and write upward.

Each page of the book must be divided into three columns. In the middle column must be set down the distances on the chain-line at which any mark, offset, or other observation is made; and in the

right and left-hand columns respectively those marks, offsets, and observations must be entered.

The crossings of fences, rivers, &c., maybe denoted by lines drawn across the middle column, or part of the right and left-hand columns, opposite the distances on the chain-line at which they are crossed ; and the corners of fields, and other remarkable turns in the fences, to , which offsets are taken, may be denoted by lines joining or lying in the same relation to the middle column as the fences, &c., do to the chain line.

Thus a tolerably accurate representation of the fences, &c., may be sketched in the field, which will very much assist the surveyor in drawing the plan.

With respect to the characters used to denote stations, the letters of the alphabet will do very well in small surveys ; but in those of a larger extent, numeral figures must be used, and the sign + (plus) placed before each figure ; thus, + 1, or + 2, which may be read station first, or cross first; station one, or cross one, &c. Upon the plan they are generally represented by this ( 0 ) mark.

Most surveyors take the exact range of the first line, and enter it in their field-book; and from it the range of any other may be easily determined. This method I shall adopt in the following work.

The expression, R. off B, or L. off B, &c., denotes that you are to turn to the right or left hand, and measure from B, &c.

Note 1.—Many surveyors not only begin at the bottom of the field-book, but also at its right-hand side, and write toward the left, which method I generally follow myself.

2.    It is useful for a beginner to draw a rough sketch of the field, or estate which he is about to measure ; and upon it to note the stations in the same manner as they are put down in taking the survey. This will materially assist his memory in planning.

3.    The field-book, for practical use, should be made convenient for the pocket, and interleaved with blotting-paper.

4.    The field-notes should always be set down with ink, which may be carried in a bottle suspended from a button of your waistcoat. Double fountain-bottles such as are used by excise officers, are the best.

DIRECTIONS TO YOUNG SURVEYORS WHEN IN THE FIELD, &c.

In addition to the instruments already described, you must provide ten arrows, each about a foot in length, made of strong wire, and pointed at the bottom. These should be bent in a circular form

at the top, for the convenience of holding them, and a piece of red cloth should be attached to each, that they may be more conspicuous among long grass, &c.

Poles, likewise, generally called Ranging-poles, or Station-staves, will be wanted as marks, or objects of direction, each about ten feet in length, piked with iron at the bottom, and having a red or white flag at the top, that they may be better seen at a distance. Thus -equipped, and having entered the field or estate which you are about to survey, first, make yourself acquainted with its form; and then consider in what manner you must run your lines, according to the directions hereafter given in Parts III., IV., V.; after which you must proceed in the following manner.

Let your assistant or chain-leader take nine arrows in his left-hand, and one end of the chain with one arrow in his right; then, advancing toward the place directed, at the end of the chain, let him put down the arrow which he holds in his right hand. This the follower must take up with his chain-hand when he comes to it, the leader, at the same time, putting down another at the other end of the chain. In this manner he must proceed until he has put down his tenth arrow ; then, advancing a chain farther, he must set his foot upon the end of the chain, and call out “change.” The surveyor, or chain-follower, must then come up to him, if he have no offsets to take, and carefully count to him the arrows; and one being put down at the end of the chain, proceed as before, until the whole line be measured.

Each change ought to be entered in the field-book, or a mistake of 10 chains may happen, when the line is very long. The chain-follower ought to be careful that the leader always puts down his arrow perpendicularly, and in a right-line with the object of direction ; otherwise the line will be made longer than it is in reality. The follower may direct the leader by the motion of his left-hand; moving it to the right or left, as circumstances require, and always placing his eye and chain-hand directly over the arrow which is stuck in the ground. The leader likewise, as soon as he has put down his arrow, ought to fix his eye upon the object of direction, and go directty toward it. This he may easily effect by finding a tree or a bush beyond the station to which he is going, and in a straight line with it and himself.

In hilly ground, if the follower lose sight of the mark toward which he is going, he must stand over his arrowr; and the leader must move to the right or left, till he sees the follower in a direct line between himself and the mark from which they last departed.

The surveyor ought to put down at each station a small stake, called a station-stake, with the number of the station upon it; so that

any of the stations may he readily found, if there be occasion to measure the distance between two of them, as a tie or proof-line, &c.

In large surveys, there must be a cross cut in the ground, at each station, making right-angles with the chain-line; so that if the stake should be pulled up, the cross may still remain, and serve as a director.

When a survey is taken with an intent to draw a finished plan, all remarkable objects should be noted down in the field-book; as roads, stiles, gates, trees, &c.

If the surveyor can conveniently procure two assistants, the one to lead the chain and the other to follow it, it will be much to his advantage; as he will thus be left at liberty to take offsets, note down dimensions, &c., without loss of time.

He ought always to observe to whom the boundaries belong.

If the ditch be in the field he is about to measure, both it and the hedge usually belong to the adjoining field. This, however, is not always the case ; as it sometimes happens that the hedge is on the reverse side of the ditch. It is advisible, therefore, to inquire of some person resident on the spot concerning the hedges, &c.

In some places 3 feet from the roots of the quickwood are allowed for the breadth of the ditches ; in some 4, in some 5, and in some 6 ; but 4 feet, or 6 links, are commonly allowed for ditches between neighbouring estates, and 7 links for ditches adjoining roads, commons, waste lands, &c.

The ditches and fences must always be measured with the fields to which they belong, when the whole quantity of land is required; but in measuring crops of corn, turnips, &c., only so much must be measured as is or has been occupied by the corn, &c.

Upon the surveyor depends all the care of measuring, remarking, noting down, &c. It absolutely behoves him, therefore, to be not only particularly careful in his entries, and correct in his dimensions, but also extremely accurate in his constructions and calculations.

Note.—The line in which you have the misfortune to lose an arrow must be remeasured.

DIRECTIONS CONCERNING SCALES, LAYING DOWN FIGURES, &c.

Any scale of equal parts may be used in planning, or laying down figures; but that which is most convenient for use is the ivory plotting-scale, so divided on its edges that you may prick off distances by laying it upon the line.

In laying down an offset by the plotting-scale, it is best, first, to

prick off the base-line, and then upon it make a small pencil dot at every place where a perpendicular must be erected.

This being done, lay the scale across the base, so that the perpendicular line which goes across the scale, marked with oo, may coincide with the base, the edge of the scale at the same time touching one of the dots. From the dot, by the edge of the scale, draw a line (which will be perpendicular to the base), and upon it prick off the offset; or it may be pricked off without drawing a line.

Proceed thus till all the perpendiculars are erected, and then draw the fence through each of their extremities. If the fence be curved, it must be drawn by a steady hand, in the same manner as the circumference of an eclipse. (See p. 13.)

In planning, or laying down figures relating to surveying, the upper part of the paper or book used should always, if possible, represent the north. All the fences and chain-lines should first be pencilled : the first should then be drawn, and the latter dotted with ink-Great accuracy is required in the construction of figures, when the perpendiculars, &c., are to be measured by the scale. The lines should be very fine, the dots at the stations very small, and the points of the compass very sharp, in order that distances may be taken from the scale with the utmost correctness. The scale should never be smaller than two chains to an inch; for when its divisions are large, figures may be constructed with much more accuracy, and their perpendiculars, &c., measured with much greater exactness.

After having found the area of any field or estate, you may, however, lay it down by any scale that will reduce it to a more convenient size. Or you may divide the dimensions by 2, 3, 4, &c., in order to make them of a proper size to be laid down by a scale of 2, 3, or 4 chains to an inch.

.Note 1.—A plotting-scale divided into two chains to an inch on one of its edges, and four on the other, is perhaps most useful for a school-boy; but practical surveyors prefer those which have both their edges divided in the same manner, because they are more convenient in planning; and a mistake cannot be made by using one edge instead of the other.

2.    Many plotting-scales are now made of box, without the perpendicular line across them ; and a small scale, called an ‘ offset-scale,’ is used in laying down the offsets. In using these scales, the first line or division, on the plotting-scale, is made to coincide with the commencement of the base-line; and the end of the offset-scale is then brought in contact with the edge of the plotting-scale; and thus the offsets may be pricked off, without drawing the offset lines perpendicular to the base-line.

3.    An instrument called a Pricker, which may be made by putting a fine needle into a wooden haft, is used by some persons, in pricking off distances from the plotting-scale ; but a hard black-lead pencil, finely pointed, is preferable, because it does not injure the paper.

PART III.

TO SURVEY WITH THE CHAIN AND CROSS ; ALSO TO MEASURE MERES, WOODS, AND LINES UPON WHICH THERE ARE IMPEDIMENTS.

Conformably to a statute of 34 Henry VIII., an acre of land is equal to 10 square chains; that is, 10 chains in length and 1 in breadth; or 220 x 22 — 4840 square yards; or 40x4 = 160 square rods, poles, or perches.

A statute pole or perch is 16| feet in length ; but in different parts of the kingdom there were, by custom, poles of different lengths ; as 15, 18, 21 feet, &c.

The various dimensions of a piece of land are taken in lineal measure, from which its area or content is calculated in superficial measure.

Note 1. —The method of reducing statute measure to customary, and the contrary, may be seen in Part VI.

2.    By the Act 5 Geo. IV. Chap. 74, it is enacted that our present yard shall be called * The Imperial Standard Yard ; ’ that one-third part thereof shall be afoot, and the twelfth part of such foot shall be an inch ; and that the rod, pole, or perch, in length, shall be yards, the furlong 220 yards, and the mile 1760 yards.

3.    By the same statute it is enacted that all superficial measures shall be computed by the ‘ Standard Yard,’ or by certain parts or portions thereof; that the rood of land shall be 1210 square yards, and the acre of land 4840 square yards.

4.    By the Act 5 & 6 Will. IV. Chap. 63, all local or customary weights and measures are abolished ; but still it may be necessary to give the methods of reducing local or customary measures of land to statute measure, as in many old plans the contents are given in local or customary measures.

A TABLE OF LINEAL MEASURES.

Inches

Link I

7-92 =

Foot

12

1-5151 =

Yard

36

4-5454

Stat.

Perch

198

25

16-5 I 5-5 =

Chain

792

100

66

22

Fur

long

7920

1000

660

220

40

10 =

.Mile

63360

8000

5280    1760

320

80 ; 8 =

A TABLE OF SQUARE MEASURES.

Square

Inches

Square

Link

62-7264 =

Square

.Foot

144

2-2956 =

Square

Yard

1296

20-6611

Square ! Perch |

39204

625

272-25 30-25 =

Square

Chain

627264

10000

4356

484

16 =

Square

Kood

1568160

25000

10890

1210

40! 2-5 =

Square

Acre

6272640

100000

43560

4840

160 10

Square

Mile

! 4014489600164000000

27878400 3097600

102400'    6400

2560

640 =

PROBLEM I.

Square Fields.

When you enter a field, which has the appearance of a square (for few are accurately such), fix your cross-staff in a corner of it, and if

the two sides be at right angles, measure one of them, and enter its dimensions in your field-lpook. Proceed in like manner with each angle and side; and if you find all the angles right angles, and all the sides equal, the figure is a square.

To compute the Content.

Rule.—Multiply the side into itself, and the product will be the area, in square links. Cut off five places, as decimals, toward the right hand of the product, and those on the left will express the number of acres.

Reduce these decimals into roods and perches, by multiplying them successively by 4 and by 40, and cutting off five figures on the right as before, in each product.

If the dimensions be in yards, divide the square of the side by 4840, and the quotient will be acres.

Reduce the remainder, if any, into roods and perches, by multiplying it successively by 4 and by 40, as before.

Note 1.—Any person who is not in possession of a chain may take the dimensions in yards ; or, where accuracy is not required, by pacing.

2.    In measuring with the chain, it is best to set down the number of links, as 956 : where, instead of reading 956 links, read 9 chains and 56 links.

3.    The dimensions of small parcels of land, sold by the square yard, for building, &c., should be taken in feet and inches with a measuring-tape. Paving, digging, &c., should be measured in the same manner.

4.    In computing the contents of fields, it is customary, among practical surveyors, to call the remainder a perch, if it exceeds half a one; but if it be less than half a perch, it is considered as nothing.

5.    The learner should carefully work over and put down all the solutions given in this book, in order that he may better understand the different methods of calculation.

Examples.

1. What is the area in acres of the square abcd, whose side is 956 links 1

956

956

5736 4780

8604 943936 •55744

40

22⁻29760 Area 9a. Or. 22p.

2. Required the area in acres of the square whose side is 264 yards.

1056

1584

528

4840)69696(14a.

4840

21296

19360

•1936

4840)7744(lr.

4840

2904

40

484,0)11616,0(24p.

968

•1936

1936 Area 14a. Ir. 24p.

3.    If the side of a square be 1567 links, what is its area in acres 1

Ans. 24a. 2r. 9p.

4.    If the side of a square be 263 yards, what is its area in acres?

Ans. 14a. lr. 6j?.

PROBLEM II.

Rectangular Fields.

When you enter a field which has the appearance of a rectangle, try each angle, and measure each side, as before; and if you find all the angles right angles, and the opposite sides equal, the figure is a rectangle.

To compute the Content.

Rule.—Multiply the length by the breadth, and the product will be the area.

Examples.

1. What is the area of the rectangle abcd, whose length ab is 1235 links, and breadth ad, 557 links ?

1235

557

8645

6175

6175

6-87895

3-51580

40

20-63200 Area 6a. 3r. 21p.

2. Required the area of a rectangle, whose length is 235 and breadth 162 yards.

235

162

470

1410

235

484,0)3807,0(7a

3388

484)1676 (3r.

1452

to

484)8960(18^.

484

4120

3872

248 Ans. 7a. 3r. 18jp.

3. The length of a rectangular field is 1225 links, and its breadth

613 links ; required the plan and area.

Area 'la. 2r. Ip.

4. If the length of a rectangle be 135, and breadth 50 yards,

what is its area 1

Ans. 1 a. 1 r. 23p.

Rote.—As squares and rectangles seldom occur in surveying, it is more advisable to treat every field of four sides as a trapezium. (See Problem IV.) ,

PROBLEM III.

Triangular Fields.

When you have to survey a field in the form of a triangle, set up a pole at each corner, when there are no natural marks. Then measure along the base till you come to the point where you think a perpendicular will fall from the opposite angle. There plant your cross, and turn its index till the mark at each end of the base can be seen through one of the grooves. Then apply your eye to the other groove, and if you see the mark at the opposite angle, you are in the right place to measure the perpendicular; if not, move the instrument backward or forward along the line, till you can see the three marks as above directed. Enter in your field-book the distance from the end of the base to the cross, and the length of the perpendicular. Then measure the remainder of the base.

Rote. 1—Be especially careful that, in measuring the two parts of the base and the perpendicular, no confusion of arrows takes place.

2. In ranging perpendiculars by the cross, you must always proceed as above directed.

Construction.

Having the place of the perpendicular, the figure may be easily constructed, as follows. From any scale of equal parts, lay off the base ; erect the perpendicular at its proper point; draw a line from each end of the base to the end of the perpendicular, and the figure will be completed.

Rote.—Having the diagonal, the two perpendiculars, and the place of each perpendicular given, you may construct any trapezium in the same manner.

To compute the Content.

Rule.—Multiply the base and perpendicular together, divide the product by 2, and the quotient will be the area.

Or multiply half the base by the whole perpendicular, or the whole base by half the perpendicular, and the product will be the area.

Examples.

1. It is required to survey the triangular field ABC, and to find its

Measure from A toward c, and when you come to m, for instance at 935 links, try with your cross; and . if this be the point for the perpendicu-cular, measure mB = 625 links. Return and measure = 628 links, A making the whole

base= 1563 links ; then construct the figure, and find its area.

1563 base.

625 per.

~78T5 3126 9378

• 2)976875

me

4-88437

3-53748

40

21-49920 Area 4a. 3r. 21p.

2. The distance between the beginning of the base and the place of the perpendicular is 125, the perpendicular 82, and the whole base 318 yards ; what is the area of the triangle 1

318 base. 82 per.

636

2544

2)26076

4840)13038(2^.

9680

3358

4840)13432(2r.

9680

484,0)15008,0(31£>.

4 A ns. 2a. 2r. ?>\p.

3.    Measuring along the base of a triangle 862 links, I found the true place of the perpendicular, and the perpendicular itself =995 links ; the remainder of the base measured 1110 links ; what is the area of the triangle 1

Ans. 9a. 3r. lOp.

4.    Measuring along the base of the triangle field, I found the perpendicular to range at 865, and its length 645 links; the remainder of the base measured 569 links; required the plan and area.

Area 4a. 2r. 20p.

Note.—If the examples in this problem, or any of the following problems be thought too few, more may easily be supplied by the teacher sketching fields, at pleasure, with his pen, which the learner may measure by a scale. This method will be found very advantageous ; as it will give the learner a good idea in what manner he must run his lines, take his dimensions, and enter his notes, when he commences field-practice.

PROBLEM IV.

Fields in the form of a Trapezium.

A quadrilateral field, having unequal sides, may be surveyed by measuring a diagonal. This divides it into two triangles, to each of which it serves as a base.

To compute the Content.

Rule.—Multiply the sum of the two perpendiculars by the diagonal, divide the product by 2, and the quotient will be the area.

Note 1.—Always make choice of the longer diagonal, because the longer the base line of a triangle, the more obtuse is its subtending angle; and, consequently, there is the less chance to mistake, as the perpendicular will be shorter, and its place more easily and more accurately determined. After finishing the surveying, if you choose, measure the other diagonal, which will enable you to prove your work. (See Problems I. and II., Part IY.)

2.    If a field be very long, or elevated in the middle, so that you cannot see from one end to the other, it may be divided into two or more trapeziums; or you may range your lines over the hill, as directed in Part Y.

3.    When two perpendiculars cannot be taken upon either of the diagonals, such fields must be divided into two triangles by measuring a diagonal from the base of one triangle, and one side of the field for the base of the other. (See example 6.)

4.    Sometimes surveyors affect to reduce trapeziums into squares, or rectangles, by measuring all the sides, adding each two opposite sides together, and taking half their sum respectively for a mean length and breadth ; but this method leads to very erroneous results. (See Part IY., Prob. II.)

Examples.

1. It is required to survey the trapezium abcd, and find its

Measure from A toward c. Finding the perpendicular aB to rise at 473, and its length 437 links; return, and continue toward c,

1)

till you come to the place where the second perpendicular to rises. There note down its distance from A, 1128 links; measure to = 508 links; then complete the measuring of the diagonal to C, and let the whole be 1490 links.

After this, measure the diagonal bd, for a proof-line, which you will find 1152 links.

per.

945 sum 1490 diag.

85050

3780 945

2)1408050

7-04025 0-16100

40

6-44000 Area 7a. Or. Op.

2. In taking the dimensions of a trapezium, I found the first perpendicular to rise at 539, and to measure 725 links; the second at

1890, and to measure 832 links; the whole diagonal measured 2456 links j required the area of the trapezium.

Ans. 19a. Or. 19p.

Ans. 9a. 1 r. 9p.

3.    The first perpendicular of a trapezium rises at 467, and measures 545 links; the second at 1418, and measures 467 links ; required its area, the whole diagonal being 1840 links.

4.    Lay down a field, and find its area, from the following notes.

AD

1625

1252

523 C

B 639 Begin Per. on the left

636

at A

base line or diag.

range W. per. on the right

Area 9a. 1 r. 30|p.

5. Required the plan and area of a field, from the following dimensions.

AD

1744

diag.

C 545

1365

Begin

546

at A

652 B range E.

Area 10a. lr. 30/?.

6. Lay down a field, and find its area, from the following notes.

diag.

DB

1095

488

298 C

L. off D

AD

side

1358

b 532 Begin

410

at A

range E.

Answer.

Double areas.

722456 triangle ABD 326310 triangle BCD Whole area 5a. Or. 39/?.

ANOTHER METHOD.

A field of four sides may sometimes be surveyed by dividing it into two right-angled triangles, and a trapezoid.

To compute the Content.

Rule.—Multiply the sum of the two perpendiculars by their distance upon the base-line, and the product will be double the area of the trapezoid. The area of each triangle must be found as before.

Examples.

1. It is required to survey the annexed figure, and find its area.

Measure the base AD, and enter in your field-book where the two perpendiculars rise, &c., as in the following notes.

GC = 645 eb = 422 Per.

AD = 1326 AG = 952 ae= 265 Base

Triangle abe 422 per. 265 base

2110

2532

844

111830

Triangle GCD 645 per. 374 base

Trapezoid EBCG

per.

422

645

2580

4515

1067 sum 687 base

1935

7469

241230

8536

6402

733029

2)1086089

5-43044

1-72176

40

733029

241230

111830

double areas collected

28-87040 Area 5a. 1 r. 29p.

2. Required the plan and area of a field, from the following notes.

AB

1681

1015

Begin Per. on the left

432 at A base

564 d 705 c range W. per. on the right

Area 7a. Or. 1 Op.

3. Lay down a field, and find its area, from the following dimensions.

AB

1546

d 625 c 883 Begin

1146 e

564

range E.

at A

Area 8a. Or. 20\p.

PROBLEM V.

Fields comprehended under more than Four Straight Sides.

Fields having more than four sides may be surveyed by reducing them into triangles and trapeziums.

Thus, a field of five sides may be reduced into a triangle and a trapezium; of six, into two trapeziums; of seven, into two trapeziums and a triangle; of eight, into three trapeziums; &c.

The propriety of dividing fields in this manner depends entirely on the relation which the angles have to one another; it is, therefore, sometimes more accurate to divide them into triangles.

To compute the Content.

Rule.—By the rales given in the last two problems, find the double area of each triangle and trapezium contained in the figure.

Collect all the double areas into one sum, which divide by 2, and the quotient will be the whole area.

Examples.

1. Lay down a field, and find its area, from the following notes.

573

D 376

-CE

1666

1326

1000

R ofi" C

diag. 496 A

AC

1433

diag.

B 273 Begin Per. on the left

O CO

at A

diag.

range W. per. on the right

Construction.

From the notes, the figure obviously consists of five sides, and is divided into a triangle and a trapezium. Draw the base AC, which

i)

'a

make = 1433 links; at 643 links, let fall the perpendicular aB, upon which lay off 273 links; join AB and CB, and the triangle is

completed. Then, with a as a centre, and 496 links in your compasses as a radius, describe an arc; and with C as a centre, and 1326 as a radius, describe another arc, intersecting the former in b.— Through b draw the diagonal ce= 1666 links; upon which, at 573 links, erect the perpendicular cd = 376 links. Join CD, DE; and ea, and the figure will be completed.

Note.—If the learner fully comprehend the above construction, he will not find it difficult to lay down the figures belonging to the following examples; as the same process will succeed in them and all similar cases.

CO

Triangle ABC 1433 base 273 per.

4299

10031

2866

Trapezium acde

per.

872 sum 1666 diag.

5232

391209

5232

5232 872 1452752

391209 ) double areas 1452752 J collected

2)1843961

9-21980

•87920

40

35-16800 Arm 9a. Or. 3bp.

2. Lay down a field, and find its area, from the following dimensions.

I— pj

diag.

F 400

R. off E

HE

1474

diag.

D 465

465

L. off H

575 G

CH

1635

1000

diag.

910

390 I

D 235

575

r. off c

AC

1165

1000

diag.

b 263

530

400    630 I

Begin

at A

range E.S.E.

\.

/ Y\

O'

7S

\\

---------

V'

T?

Trapezium abci

Trapezium CDHi

630

263

per.

235

390

per.

893 sum 1165 diag.

625 sum 1635 diag.

4465

3125

5358

1875

893

893

Oi Cn

1040345

1021875

Trapezium DEGH

575

465

per.

Triangle efg 1150 base 400 per.

1040 sum 1474 diag.

460000

4160

7280

4160

1040

1040345 \

1021875 ( double areas 1532960 f collected 460000 )

1532960

2)4055180

20-27590

1-10360

40

4-14400 Area 20a. 1 r. ip.

3. Required the plan and area of a field, from the following dimensions.

BA

1008

diag.

e 195 Return

466

to B

AD

1345

diag.

C 415

os oo

Begin

at A

536 B range W.

Answer.

Double areas 1279095 trapezium abdc 196560 triangle AEB Whole area 7a. 1 r. 20 %p.

4. Lay down a field, and find its area, from the following notes.

diag.

DF

1940

1040

362 B

E 581

825

II. Off D

C 322

co cq

OOOI>

<; ^ i>. co

diag.

Begin

at A

665 b range W.

Answer.

Double areas 1468656 trapezium ABDC 1829420 trapezium defb Whole area 16 a. lr. 38p.

5. Draw the plan of a field, and find its area, from the following dimensions.

HK

1285

diag.

I 382

740

600

R. Off H

162 g

FH

1223

diag.

E 661

803

666

L. Off F

276 G

761

R. off D

e 409

DF

1716

1080

diag.

246 B

AD

1547

diag.

C 603

1023

5 25

Begin

at A

488 b range W.

Answer.

Double areas.

1687777 trapezium ABDC 1123980 trapezium defb 1145951 trapezium fghe 699040 trapezium hgki Whole area 23a. \r. 5p.

ANOTHER METHOD.

A field consisting of five, six, seven, or more sides may sometimes be surveyed by measuring one diagonal, and upon it erecting perpendiculars to all the opposite angles, on each side. This process will divide the whole field into right-angled triangles, and trapezoids, the areas of which must be found as before.

Examples.

1. Lay down a field, and find its area, from the following notes.

AF

1896

E 259

1342

diag.

1132

325 D

1000

c 367 b 756 Begin

763

522

at A

range E.

Triangle ABa 756 per. 522 base

Trapezoid aBCm 756

1512

1512

per.

1123 sum 241 base

367

3780

394632

1123

4492

2246

270643

Trapezoid mcEr per.

367

259

626 sum 579 base ¥634

Triangle rEF 554 base 259 per. 4986

2770

1108

4382

3130

143486

362454

3-74828

40

5688

double areas collected.

Triangle adf 1896 base 325 per. ■9480 3792

616200

394632 ] 270643 | 362454 -143486 616200 2)1787415 8-93707

29-93120

Area 8a. 3r. 30p.

2. Lay down a field, and find its area, from the following dimensions.

AK

1700

diag.

i 290

1465

1368

1055

365 H 381 G

f 144

986

794

c 250

IQ '■'ri r—i

218 E 350 d

275 b

000

Begin

at A

range W.

Answer.

Triangles and Trapezoids on the Right.

Double areas 41250 triangle ABa 228125 trapezoid «BDe 158472 trapezoid eDEm 156339 trapezoid mEGr 233498 trapezoid rGHw 121180 triangle whk 938864 sum

Triangles and Trapezoids on the Left.

Double areas 111000 triangle ACc 213548 trapezoid cCFn 207886 trapezoid nFid 68150 triangle diK 600584 sum

938864 sum brought forward 1539448 sum total

Whole area 7 a. 2 r. 31-|_p

3. It is required to lay down a field, and find its area, from the following notes.

AL

M 460

2150

diag.

1670

295 k

I 395

1530

1345

160 H

G 670 F 400

1275

880

780

465

270 e 150 d

C 405

305

000

Begin

at A

300 b range E.

Answer.

sum

Triangles and Trapezoids on the Bight. Double areas 209250 trapezoid ABDc 132300 trapezoid cDEe 242950 trapezoid £EHr 147875 trapezoid rFLKd 141600 triangle dKL 873975

Triangles and Trapezoids on the Left. Double areas 123525 triangle Acc 462875 trapezoid aCFm 422650 trapezoid mFGn 271575 trapezoid nGiw 530100 trapezoid wiML 1810725 sum

873975 sum brought down 2684700 sum total

Whole area 13a. 1 r. 27fp.

PROBLEM VI.

Fields comprehended under any Number of Crooked or Curved Sides.

When a field is bounded by crooked fences, measure a line as near to each as the angles or curves will permit; in doing which, take an offset to each corner or angle in the fence. Where the fences are curved, those offsets must be so taken that a right line drawn from the end of any one perpendicular to the end of the next, on each side, would neither exclude any part of the land to be measured, nor include any of that which is adjacent. Perpendiculars thus erected will divide the whole offset into right-angled triangles and trapezoids, the areas of which are found as before.

Note 1. — If the curves be so large that many of the offsets would be 2, 3, 4, or 5 chains long, it will be more expeditious and accurate to measure the base without taking any offsets, except such as are short, leaving stations in proper places along the base, to which, when you have obtained its length, you may return, and from them run fresh station-lines to some convenient point, or points, in the curved fence. Upon these lines take offsets as before. (See Example 3.)

2.    If any of the fences be curved inward, it is frequently most convenient to measure a line on the outside of the field, and upon it erect perpendiculars to the curved fence, which, in this case, are called insets; and the area thus included must be subtracted from the area of the whole figure. (See Example 4.)

3.    When the fences and ditches are to be measured with the field to which they belong, it is generally most practicable to fix the stations within the fences, at a little distance from the corners, and then to measure to the roots of the quick-wood, adding or subtracting 5 or 6 links, according to the custom of the place, for the breadth of the ditch. (See Example 5.)

4.    When the offsets are small, their places on the base line may be determined by laying the offset staff at right angles upon the chain ; but when large, and accuracy is required, they must be found by the cross, and measured by the chain.

5.    The base of each triangle and trapezoid, forming an offset, may be found by subtracting the distances on the chain line from each other.

6.    The following three methods of finding the area of offsets are erroneous. Some divide the sum of the offsets by their number, for a mean breadth ; others divide that sum by one more than their number, for a mean breadth ; and both multiply the whole base by the mean breadth, thus supposed to be found, for the area of the whole offset. The first of these methods generally gives the area too much ; and the second sometimes too much and sometimes too little. A third method, which is usually more accurate than either of the preceding ones, is to set down each offset twice (accountiug that one where the boundary meets the station-line) except the first and last, which are only entered once. The sum of these offsets is then multiplied by the base, the product divided by the number of offsets set down, and the quotient given as the area required.

7.    Directions for laying down offsets by a plotting-scale may be seen in Part II.

Examples.'

1. Lay down the figure of a right-line offset, and find its area, from the following notes :—

AB

n 200 m 210

1569

1249

1000

952

745

c 159 a 120

450

265

000

Begin at

range E.

in

BY THE TRUE METHOD.

120

159

Triangle Ara 265 base.

120 per. “5300"

265

31800

Trapezoid racs per.

279 sum 185 base

1395

2232

279

51615

© CD

per.

Trapezoid sceu per.

209 sum 295 base

Trapezoid ueiw 50 )

70)

120 sum 207 base

418

240

840

24840

31800' 51615 61655 24840 ' 83160 131200

double areas collected

2)384270 1 92135

3-68540

40

Trapezoid wimx

70 )

210 } Per*

Trapezoid xmnB

210

200

per.

280 sum 297 base 1960 2520

410 sum 320 base

8200

560

123

83160

131200

27-41600

Hence the true area is la. 3r 27p.

BY THE FIRST FALSE METHOD.

120

159

1569 length 134-8 breadth

50

12552

70

6276

210

4707

200

1569

2-115012

6)809

134-8 mean breadth

•461)048

18-401920

Here the area appears to be 2a. Or. 18p., which is too much by

BY THE SECOND FALSE METHOD.

120

159

1569 length 115 5 breadth

50

7845

70

7845

210

1569

200

1569

7)809

115-5 mean breadth

1-812195

3-248780

40

9-951200

Here the area appears to be 1 a. 3r. 1p., which is too little by 17p.

BY THE THIRD FALSE METHOD.

120

120

1569 length 1418 sum 12552"

159

1569

159

6276

50

1569

50

70

70

12)22-24842 1 -85403

210

210

3-41612

200

40

1418

16-64480

sum    -------------—

Here the area appears to be la. 3r. 16p., which is too little by lip.

2. Lay down a curve-line offset, and find its area, from the following notes.

AB

1012'

892

786

OO Oi ©

CO o

645

500

95

45

350

63

200

84

100

52

si ⁷

(A

52

100

CO VO ©

00 © © Ox

5200 No. 1

108

175

150

141

-rH O OO

5400

175

108

700

136

16200 No. 4

175

100

24675 No. 6

13600 No. 2

>—■ vP- ! Cs 00 Ol <| I W

45

95

140

145

80

53

133

106

7350

700

798

147

56

133

22050 No. 3

14

14098 No. 7

2U300 No. 5

i—i or to w

© eo

rH lO

OS

© I©

6360 No. 8

5200"

13600

22050

16200 double areas 20300 * collected 24675 14098

6360

2)122483

■61241

2-44964

40

Note.—When the content of a small piece of land does not amount to an acre, and if the number of figures in the area be under five, as many cyphers must be placed on the left of the area as will make it five before we can point-off the decimals, because 100,000 square links make an acre. Suppose the area to be 9625 square links; then we must make it '09625 ; and by multiplying successively by 4 and 40, we shall obtain for the area 15'4 square perches.

3. Lay down the figure of a piece of land adjoining a river, and find its area from the following notes.

EB

1350

1265

1200

1100

1000

900

800

O O xj-i o o o ^ N vfi io Ci O

(M i—i .—i

Si

700

600

500

CO

200

CO

to o

O O H H N

River

Left off e

000 and go

SW. to B

Eeturn

_o

_    I—i i—■

p O fcO tf*- Ci CO CO ®

50

82 m

CTQ

to E

River 100 links in breadth

AD

70

m 110

to o o o

oooo

CO <N o

<n <n ci o i

River

e 698

a 165

60

1410

1000

250

100

000

V/

4V

2 \fy

G_ii—e-......—

°T'

V/j    v

------

x>

Triangle bce 1950 base 698 per. 15600

1755

1170

1361100

Offsets taken on the line AD

O r—1

60

165

6000 No. 1

*—¹ LO

o<

©

O O !© T-H

r-H

6000 No. 1 33750 „ 2 18000 „ 3 11410 „ 4 69160 sum

11250

225

33750 No. 2

Offsets taken on the line CE

"I

h-»

o !o

zl

CO

CO

cr>

11410 No. 4

CO

to to

142

142

200

173

16400 No. 1

28400 No. 2

315

100

CO It— t-h

t— in

t-h t-h

154

110

31500 No. 3

CO

264

100

32700 No. 4

to

C.

Ox

t—‘

Or i-h O O

16400 No. 1 28400 „ 2 31500 „ 3 32700 „ 4 26400 „ 5 16000 „ 6 3000 „ 7 154400 sum

Ox O

160

100

OO

12$

t-

16000 No. 6

70

200

14000 No. 1

Offsets taken on the line 190 250 440

EB

100

to

CO O

44000 No. 8

223

100

to t o

Ot

O

22300 No. 2

494

100

to oa

49400 No. 9

295

244

14000 No. 1 22300 „ 2 29500 „ 3 23600 „ 4 9400 „ 5 10000 „ 6 29000 „ 7 44000 „ 8 49400 „ 9 41400 „ 10 20150 „ 11 11900 „ 12 304650 sum

100

170

29500 No. 3

414

100

-H

os

41400 No. 10

236

100

23600 No. 4

t—¹ f—1 1—¹ It*. -I

o o ©

I—I to

co co

94

1550

1361100 \ whole 69160 (double 154400 fareas 304650 / collected

100

186

2)1889310

9400 No. 5

20150 No. 11

9-44655

100

85

1-78620

100

140

40

10000 No. 6

3400

31-44800

85

100

11900 No. 12

190

290

100

29000 No. 7

Area 9a. 1 r. 31p.

4. Lay down a field, and find its area, from the following notes.

525

r. off c

perpen.

BC

^ >C ©

o to OO lO o

846

600 400 200

000

R. off B

AB

1253

1000

586

Begin at

a, the place of the perpen. range W.

Triangle ABC 1253 base 525 per.

6265

2506

6265

657825

CJ?

to o

10000

Insets taken on the line BC

Ox

OS

10000

27000

29800

15744

double

areas

collected

29800

82544

00 Ox Ox ©

185

200

984

1476

657825 triangle ABC 82544 insets 2)575281 difference

2-87640

3-50560

40

20-22400 Area 2a, 3r. 20p.

5. Draw a plan of a field, and find its area, from the following notes.

AO

1155

diag.

1000

495

CO

From

520

A gO

N.W.

To

48

the

fence

660

53

630

to A

40

500

25

380

50

200

62

From

000

D go

S.

To

the

fence

40

1012

42

950

tO D

54

600

33

400

12

260

30

150

65

000

From

ego

E.

To

the

fence

25

615

35

550

30

400

10

22

h-i

45

000

From

B go

To

the

33

1090

45

1045

56

800

40

500

48

300

30

000

i-

’r

2'i.......

•v.

-------x'

rr

kk

Answer.

Double areas 1172325'

98055 ab 32980 81254 58820 j da

Whole area 7a. Or. 34£p.

trapezium abcd

bc | offsets taken on the CD^ different lines

6. Lay down a field, and find its area, from the following dimensions.

EB

442

345

256

115

bO SO

05 rf**

000

L. off E

CE

296

200

to

CO

off c

AC

1294

B 530

1050

diag.

1000

290

485 D

R off A

DA

32

567

67

458

24

364

43

235

123

000

R. off D

fi I-H o >-H © r—I i—I

129

80 + 30

65

42

R.

® >o © o © o ,

p © r-H O © © £fc!

r—I © O

BC

E 293

584

328

R. off B

37

1000

44

900

78

750

46

85

42

000

Begin at

1)

to

/s'

“**V

\N

kM/

•A.

Answer.

Double areas 1313410 trapezium abcd 171112 triangle BEC 111401'

62395 37326 26805 33850

ab

CD | offsets taken on the

- DA [

CE | different lines EB J

Whole area 8a. 3r. 5v.

7. It is required to lay down a field, and find its area, from the following notes.

O O O CO

the

1121

1025

930

fence

to a

70

100

130

600

190

140

70

50

300

200

40

100

000

R. off D

BD

1900

diag.

1300

823 A

1000

c 780

440

R. off B

AB

1530

66

1350

73

1200

85

1100

94

1000

73

900

40

780

70

630

110

467

76

300

70

200

69

96

An&icer.

Double areas 3045700 trapezium ABCD 218592 offsets taken on the line AB 205555 ditto on the line DA Whole area 17a. 1 r. lop.

8. Required the area and plan of a field, from the following dimensions.

AD

1080

1000

950

890

820

740

650

580

535

480

420

300

220

100

50

000

o o o

so t- lO

L. off A

225

R. Off C

d 525

CA

1170

920

diag.

540 b

BO

1065

25

1005

35

946

50

870

40

830

90

780

115

715

110

80

75

55

Ox

o«o

70

65

35

to to 05 i—* CJX 05

Or O O

48

150

40

60

55

000

R.

off B

Double areas 1246050 trapezium ABCD 67710 offsets taken on the line AB 129820 ditto on the line BC 91210 ditto on the line AD Whole area 7a. 2r. 27§p.

Answer.

O OT CO © CM

Or Oi

37

225

53

170

40

130

50

50

to A

52

From the

000

fence

go K

fence

To

0j

40

the

700

645

to B

9. Lay down a field, and find its area, from the following notes.

EA

© 00 © -^ CO t* © © © GO

I—I 1—I

500

1—1 Tf 1>

450

400

350

300

250

200

to to

© © Ol

Or ©

000

R. off E

►-* t—ii i—i

tOOlCOO'-'fOMOiOSCuM

©oro©cno>e»ofco©o©o

Ot©Ox©Oi©Ot©Oi©CT(©H ~©©©©©©©©©©©H

Return

AC

B 550

1220

915

diag.

700

524 d

R. off A

DA

tel

cq

CO

875

St

490

p

q<5

R. off D

CD

CO lo

Of

SOCDCOCCS

Oi O Of o ^ o

Of Oi Oi

450

400

350

320

200

© o © eo

100

50

000

R. off C

© oo

i—i cq

© lO HO »C ©

co cq co © ©

© © © © r-l cq

EH

icocnnc©©coioo

O©t»©lO©^(M(M©00

the

1115

1075

1000

920

868

800

750

700

600

550

73

70

400

34

300

40

200

23

134

100

Begin

000

at A

range W.

Answer.

Double areas 1310280 trapezium ABCD 281750 triangle AED 116056 "I AB-i 41020 | bc

83180 [-CD offsets taken on the different lines 92400 de 53650J ea.

Whole area 9a. Sr. 22

PROBLEM VII.

Narrow Pieces of Land.

A method frequently adopted, in surveying narrow pieces of land, is to take breadths in different places, add all these breadths together, and divide their sum by their number, for a mean breadth; and this supposed mean breadth is multiplied by the length, for the area : but, as the method of finding the mean breadth is liable to mistake, this process may lead to very erroneous results.

If, for example, a piece of land taper regularly from one end to the other, you may take its breadth at each end; half the sum of these breadths will then be its mean breadth; so that you may multiply this mean breadth by the length, for the area. But if it be irregular, you must take breadths in the widest and narrowest places, or at every particular curve, noting the place of each breadth upon the chain-line. These breadths will divide it into triangles and trapezoids, which compute as before.

Note. 1.—The breadths must be taken directly across the land to be measured, and therefore, if considerable, will require the use of the cross.

2.    If a piece of land be curved, or longer on one side than on the other, by measuring along the middle you will obtain the true or mean length.

3.    When several pieces of land, of various lengths, are contiguous to each other, it will generally be most expeditious to measure only one base-line, noting the point where each piece begins and ends, perpendicularly to the line. In this case be especially careful that no confusion take place in noting down the breadths of the respective pieces.

4.    Paring, reaping, &c., both in this and the foregoing problems, should be surveyed with a slack chain, in order to obtain the measurement of the surface.

5.    It is best to take the first and last breadths of lands or ridges about half a chain from each end, and account them as the end-breadths; because, in turning, the plough usually makes some of them appear either broader or narrower than they are in reality. It may also be observed, that it is frequently necessary to take the breadths to half a link; for when the length is great, half a link in the breadth is too considerable to be neglected.

6.    If a narrow piece of land be very irregular, you may obtain its area most accurately by measuring a base-line, in a convenient position ; and upon it erecting perpendiculars to the boundaries, on each side.

7.    In surveying with the chain and cross, when the area only of any field or piece of ground is required, it is unnecessary to lay down the figure.

Examples.

1. Find the area of a tapering piece of land, whose length is 2562 links, and breadth at one end 126, and at the other 232 links.

126

232

breadths

2)358 sum 179 mean 2562 length

358

1074

895

358

4-58598

2-34392

40

13-75680 Area 4a. 2r. 14p.

2. Find the area of a piece of land, which is broadest towards the middle, from the following dimensions.

BY THE TRUE METHOD.

125

2322

2000

169

215

per.

215

1056

215

340 sum 1056 base

per.

384 sum 1266 base

169

1000

000 ‘ base.

125

per.

2040

2304

170

34

O 00

359040

384

486144

359040 ) double areas 486144 j collected

2)845184

4-22592

•90368

40

36-14720 Area, 4a. Or. 36p.

BY THE FALSE METHOD.

125

215 /-breadths

2322 length 169-6 breadth

169

3)509

169-6 mean

13932

20898

13932

2322 3-938112 3-752448

40

30-097920

Here the area appears to be 3a. 3r. 30p., which is too little by lr. 6jp.

Again, the dimensions remaining as before, suppose the piece to be narrowest towards the middle ; the area by the false method will be the same as already found.

BY THE TRUE METHOD.

215

2322

2000

169

125

per.

125

169

per.

1056

125

1000

340 sum 1056 base

294 sum 1266 base

000

base.

215

per.

2040

1764

170

1764

34

588

294

359040

372204

359040 ) double areas 372204 / collected

2)731244

3-65622

2-62488

40

24-99520

The true area is 3a. 2r. 25p.; hence the false area is too much by lr. 5p.

Lastly, the dimensions still continuing, suppose the breadth towards the middle to be greater than that at one end, and less than at the other; the false area will still be the same.

BY THE TRUE METHOD.

2322

2000

125

1056

169

1000

215 )

J69 / Per-384 sum 1056 base

169

125

per.

294 sum 1266 base

000

base.

215

per.

2304

1764

1920

1764

384

588

294

405504

372204

405504 ) double areas 372204 ) collected

2)777708

3-88854 3-55416

40

22-16640

The true area is 3a. 3r. 22p.; hence the false area is too much by 8p.

3. Draw a plan of an irregular piece of land, and find its area from the following dimensions.

AB

05 O

1325

136

1015

987

58

790

134

318

718

560

162

223

465

345

125

346

266

372

From

000

246

A go

E.

aq~

------~pB

Answer.

Double areas

361176 offsets on the right 684860 ditto on the left

2) 1046036 sum

5-23018 = 5a. Or. 36fp. the area required.

4. Find the area of five lands, from the following dimensions.

2378

185

2000

190

1700

194

1400

198

1000

200

700

195

400

189

000

185

Area 4a. 2r. 13|p.

5. Required the area of six lands, from the following notes.

3422

189

3000

2500

204

2000

1800

226

1000

800

191

000

165

Area 6a. 3r. \'2p.

6. Find the area of seven lands, from the following dimensions. Note.—In calculating the area, the half-links must be treated as decimals.

101

O CM

2900

2600

2300

2000

1900

1600

1300

1000

900

600

300

000

991

981

97*

100

7.

area,

Area 2a. 3r. 23fy>.

It is required to lay down a narrow piece of land, and find its from the following dimensions.

250

O lO o cd or

153

From

AB

1230

1050

1000

918

800

690

500

440

300

100

000

A go

o o

CD lO

300

235

108

216

130

N.

Answer.

Double areas

552890 offsets on the right 140800 ditto on the left Whole area 3a. 1 r. 35p.

8. Lay down a field, and find its area, from the following notes.

BC

-1 tfl

CO o o

O GC

521

400 300 200 100

000

R. off B

AB

1235

IQ r—I I "■»

1100

900

650

500

521 to C 430

323

245

232

350

252

268

From

200

219

240

275

000

360

A gO

N.

Answer.

Double areas

763685 offsets on the right of AB 414695 do. on the left of AB 70270 do. on the line BO Area 6a. Or. 38fp.

PROBLEM VIII.

Meres and Woods.

When you have a mere or wood to survey, by the help of your cross, fix four marks on its outside, in such a manner as to form a rectangle or square. Then measure each side of the rectangle or square, taking insets to the edge of the mere or wood, the area of which insets must be treated as directed in Note 2, Problem YI.

If the opposite sides be not found equal, or very nearly so, your marks do not form four right angles; in which case you must rectify your error.    ,

Note 1.—It is a more expeditious method to measure the four sides of a quadrilateral figure, having one right angle, paying no regard to the length of the sides. Then construct the figure by Part I. Problem XIV., and draw the longer diagonal, upon which let fall a perpendicular from each of the opposite angles. This diagonal and these perpendiculars you must measure by the scale used in plotting.

2. It sometimes happens that the mere or wood is of a triangular form; in this

case, the work may be very readily done by measuring the three sides of a triangle, taking insets as before. After which construct the triangle, and from the opposite angle let fall a perpendicular upon the base, and proceed as before.

3. By this problem you may measure fields into which you are not permitted to enter, or which contain obstructions.

Examples.

1. Let the following figure represent a mere; its area is required.

G;

gif:

iv

Hi

-:~2S

Having fixed four marks, A, B, c, and D, forming a rectangle; begin at A, and measure the line AB, taking the necessary insets, and entering them in your field-book. In the same manner proceed with the other three sides; and you will find noted the following dimensions.

DA

1100

1000

800

50

550

70

300

60

000

R. off D

CD

O O    IQQ.

Ol'OOOOCir-H©

1450

1200 1000

820

000

r. off c

BC

1100

950

110

700

70

550

142

300

100

000

R. off B

AB

1450

1200

55

1000

O bO Or CO

75

32

65

Begin at

range W.

Answer.

1595000 area of the rectangle abcd 135550 \ AB'i 183800 (

168450 j CD 95500/ DA

double areas of the insets taken on the different lines

2)583300

291650 area of the whole insets

13-03350 ditto of the mere

•13400

40

5-36000 Area 13a. Or. 5p.

2. Let the following figure represent a wood; its area is required.

li

*r

rp-'jiiLjs

7''

si

Lx.

WJS

.......

Mil

■71

!i.“j

:/ .--^

V/===^=s

Wm/&

i.

-e

Set up your cross at A, and let your assistant fix the marks B and D, so that the angle at A may be a right angle ; and measure the line AB, taking insets as before. Then fix the mark C, as most convenient; measure the other three lines, and you will find in your field-book the following notes.

DA

160

1550

1440

50

1200

1000

900

L. off D

CD

950

120

500

000

L. off c

BC

1340

1050

1000

50

700

60

400

000

L. off B

AB

1150

50

O O 10 0 05 0

100

110

160

000

Begin at

range N.

Answer.

Having constructed the figure, you will find the diagonal BD to measure 1930, the perpendicular a a 923, and c a 605 links.

Double areas

2949040 trapezium ABCD 102000 \ ab 74500 / BC 114000 T cd 83000 ; DA 373500 whole insets 2575540 wood.

insets taken on the different lines

Area 12a. 3r. 2 Op.

PROBLEM IX.

To find the Area of a Segment of a Circle, or any other Curvilineal Figure, by means of Equidistant Ordinates.

Rule.—If a right line an be divided into any even number of equal parts AC, CE, EG, &c.; and at the points of division be erected perpendicular ordinates AB,

CD, EF, &c., terminated by any curve BDF, &c.; and if A be put for the sum of the extreme or first and last ordinates, AB, NO; B for the sum I—i of the even ordinates, CD, GH, LM, &c,, viz., the second, fourth, sixth, &c.; and C for the sum of all the rest, EF, IK, &c., viz., the third, fifth, &c., or the odd ordinates, wanting the first and last; then the common distance AC, or CE, &c., of the ordinates, being multiplied by the sum arising from the addition of A, four times B, and two times C; one-third of the product will be the area

ABON, very nearly; that is,

AC, the common distance of the ordinates.

a+4 b+2 c

XD = the area, putting D =

Note.—The foregoing rule being expressed in an algebraic form, is seldom properly understood by learners; but the following one may be easily comprehended and committed to memory.

Rule.—To the sum of the first and last ordinates, add four times the sum of all the even ordinates, and twice the sum of all the odd ordinates, not including the first and last; multiply this sum by the common distance of the ordinates, divide the product by 3, and the quotient will be the area required.

Note.—The length of the base must be ascertained before you begin to take the ordinates, in order that yon may divide it into an even number of equal parts ; or you may take the dimensions without doing this, and find the areas of the pieces at the end, by the rules for triangles and trapezoids, which being added to that part of the figure computed by equidistant ordinates, will give the whole area. See the following examples.

Examples.

1. Required the plan and area of a piece of land, measured by equidistant ordinates, from the following notes.

AB

1167

1100

1000

44 EF 97 CD

900

139

800

175

700

206

600

230

500

248

400

260

300

264

200

268

Begin

100    262

000    252

at A and go W.

The first and last ordinates 252 the first ordinate AG 97 the last ordinate CD 349 sum

The even ordinates 262 second 264 fourth 248 sixth.

206 eighth 139 tenth 1119 sum

4476 four times the sum

The odd ordinates 268 third 260 fifth 230 seventh 175 ninth 933 sum

1866 twice the sum

349 the first and last ordinates 4476 four times the sum, &c. 1866 twice the sum, &c.

6691 sum total

100 the common distance

3)669100

223033 the area of the figure ACDG

Trapezoid cefd

97

Triangle EBF

67

44

44

141

268

100

268

14100

2948

Double areas 14100 trapezoid CEFD 2948 triangle EBF 2)17048 sum

8524 the area of the figure CBD 223033 ditto of the figure ACDG 2 31557 sum

1-26228

40

10-49120

Hence the area of the whole figure ACEBFDGA is 2a. 1 r. 10Jy?. nearly.

2. Lay down a piece of ground, and find its area from the following equidistant ordinates.

B(S-~

......r

-4—

00

200

230

247

250

240

216

180

130

Begin

AB

1000

900

75

800

125

700

600

500

400

300

200

100

000

at A and

tO GO

185

177

73

go E.

The first and last ordinates 130 the first ordinate 000 the last ditto 130 sum

The even ordinates 180 + 73 = 253 second 240 + 157 = 397 fourth 247 + 185 = 432 sixth 200 + 160 = 360 eighth 85 + 75 = 160 tenth 1602 sum

6408 four times the sum

The odd ordinates 216 + 125 = 341 third 250 + 177 = 427 fifth 230 + 180 = 410 seventh 150 + 125 = 275 ninth 1453 sum

2906 twice the sum

130 the first and last ordinates 6408 four times the sum, &c. 2906 twice the sum, &c.

9444 sum total

100 the common distance

3)944400

3-14800 area in square links •59200

40

23-68000 Area 3a. Or. 23£-p.

3. Required the plan and area of a piece of ground, from the following equidistant ordinates.

1200

1100

1000

250

900

246

800

700

600

500

176

400

140

300

100

200

55

100

Begin

000

’at A

range E.

Answer.

220 the first and last ordinates 4456 four times the sum, &c. 1976 twice the sum, &c.

6652 sum total

100 the common distance

3)665200

2-21733 area in square links •86932

40

34-77280 Area 2a. Or. 34fp.

s, measured by equidistant ordinates,

4. Find the area of 4 from the following notes.

182

1290

70

183

1-500

101

178

1100

115

189

1000

112

190

900

96

187

98

179

95

150

600

100

182

500

120

185

400

180

300

160

200

170

100

H CO CO co

Oh

188

000

130

Answer.

602 the first and last ordinates 7092 four times the sum, &c. 2848 twice the sum, &c.

10542 sum total

100 the common distance

3)1054200

351400

24120 trapezoid at the end

3-75520 area in square links

3-02080

40

0-83200 Area 3a. 3r. Ofp.

5. Find the area of 6 lands, by equidistant ordinates, from the following dimensions.

147

3090

153

152

3000 2700 *

hf*. Or

2400

2100

1800

144

1500

142

1200

143

900

148

600

146

145

SO o

Answer.

298 the first and last ordinates 2936 four times the sum, &c. 1174 twice the sum, &c.

4408 sum total

300 the common distance

3)1322400

4-40800

13500 trapezoid at the end 4-54300 area in square links

2-17200

40

6-88000 Area 4a. 2r. 6£p.

Answer.

743 the first and last ordinates 7900 four times the sum, &c. 3264 twice the sum, &c.

11907 sum total

100 the common distance

3)1190700

,3-96900 area in square links

3-87600

40

AB

M M bs CO O Cl

1200

180

1100

220

Or ci co be

OlSO Cl

1000

246

900

800

700

265

270

269

t- co

IQ CO OO <M

600

500

400

300

260

243

215

180

170

200

134

232

100

65

327

Begin

000

at A and

go W.

35-04000 Area 3a. 3/-. 35p.

7. Required tiie plan and area of a field from the following equidistant ordinates.

217

187

98

95

100

114

130

167

190

Begin

AB

1096

1000

900

800

700

600

500

400

300

200

100

000

at A and

oc I—■ h—1 bo

t-J Oi o

LO © to

Or Or -1 GO

© CO ©

1Q © i—i    O O

(ooohhim to

Answer.

727 tlie first and last ordinates 4384 four times the sum, &c. 1540 twice the sum, &c.

6651 sum total

100 the common distance

3)665100

221700

36288 trapezoid at the'end 2-57988 area in square links

2-31952

40

12-78080 Area 2a. 2r. 12 fp.

Rote.—Whenever the rule given in this Problem can be applied, it will be found more easy, expeditious, and accurate, in finding the areas of offsets and of narrow pieces of land, than the rules for triangles and trapezoids.

PROBLEM X.

To find the Breadth of a River.

EXAMPLE.

Let the following figure represent a river, the breadth of which is required.

JD

Fix upon anjr object B, close by the edge of the river on the side opposite to which you stand. By the help of your cross, make AD perpendicular to AB ; also make AC = CD, and erect the perpendicular DE; and when you have arrived at the point E, in a direct line with CB, the distance DE will be = ab, the breadth of the river; for by 'Theo. 1, Part I., the angle ACB = dce, and as AC = CD, and the angles A and D are right angles, it is evident that the triangles ABC and CDE are not only similar but equal.

Note 1.—The distance between a and the edge of the river must be deduced from de, when it is not convenient to fix a close by the river’s edge.

2. This Problem may also be well applied in measuring the distance of an inaccessible object; for let ac equal 8, cd equal 2, and de equal 10 chains; then, by similar triangles, as cd : de : : ac : ab; that is, as 2 : 10 : : 8 : 40 chains = ab. (See Theo. 11, Part I.)

PROBLEM XI.

Lines upon which there are Impediments not obstructing Sight.

Suppose mn to represent a deep pit or water, and A and B two objects, the direct distance of which is required.

-©

/x-

At the verge of the impediment, having fixed the mark m, in a right line with A and B, measure from A to m; and at m, by the help of your cross, erect the perpendicular ma, which measure to the outside of the interposed obstruction, as at c. Then on the other side, as at n, in a line with A and B, erect the perpendicular ne; and make nb equal to me.

Measure be, which will be equal to mn; and from n measure the distance nB ; then be added to A m and nB will give the whole distance

AB.

PROBLEM XII.

Lines upon which there are Impediments obstructing Sight.

EXAMPLE.

Suppose cdef to represent the base of a building which obstructs the sight, and through which it is necessary that a straight line should pass from an object at A to B, the direction AmB on the side A being given.

Measure from A to m; at m erect the perpendicular ma, which measure until you are clear of the impediment, as at c.

Erect the perpendicular ce, which measure until you are beyond the building, as at b.

J)

1°

■0»i

Erect the perpendicular bd; and make bn equal to me, at which point you will be in a direct line with toA. Erect the perpendicular nB, which measure • then be, added to Am and nB, will give the whole distance AB.

Note.—In the tenth and previous u c editions of the work the range am was evidently understood in this problem as given, and also in Problem VII. Part IV. p. 139, although not so expressed.

This involves an omission; for the range or direction of a line must always be known before it can be measured.

> £)-

The error is now corrected in both examples. And as impediments obstructing sight are of frequent occurrence in surveying, two propositions have been added to find the line when the range ato is not given—the one with the chain, p. 89, and the other with the theodolite and chain, Part IX.

PART IV.

THE METHOD OF SURVEYING WITH THE CHAIN ONLY; AND OF MEASURING MERES, WOODS, DISTANCES, LINES, UPON WHICH THERE ARE IMPEDIMENTS, AND HILLY GROUND.

MISCELLANEOUS INSTRUCTIONS.

The method of surveying with the chain only is adopted by most practical surveyors, and is certainly preferable to that by the chain and cross; because it is not only as accurate, but generally more expeditious.

Whatever be the form of the field or ground to be surveyed, measure as many lines as will enable you to plot it with accuracy. The plan being drawn, you may then divide the figure into trapeziums, triangles, &c., and measure the diagonals, perpendiculars, &c., with your plotting-scale.

It is better, however, to divide small pieces and single fields into trapeziums and triangles, by measuring the diagonals and bases during the survey • so that to find the area, you will have only the perpendiculars to measure with the scale.

You should also measure, in some convenient direction, a proofline to each trapezium and triangle.

Note 1.—The offsets must be treated according to the directions in Part III. Prob. VI. Or you may reduce the crooked sides to straight ones, by including as much of what does not belong to the field under your survey as you exclude of what does, in the following manner:—Apply to the crooked line in question the straight edge of a clear piece of lantern horn, so that the small parts cut off by it from the crooked figure may be equal to those which are taken in (of this equality you will presently be able to judge very correctly by a little practice); then, with a pencil, draw a line by the edge of the horn. The sides being thus successively straightened, the content may be easily found.

2.    A slender bow of cane or whalebone, strung with a silk thread, may be substituted for the horn. The thread must be applied to the crooked fence, and two marks made by which to draw a straight line.

3.    The sides may also be straightened by a parallel ruler; but the operation is generally tedious, and must be performed with the greatest care, or it will not be more correct than the foregoing method.

4.    When the three sides of a triangle are given, the area may be found as follows : From half the sum of the three sides subtract each side severally; multiply the half sum and the three remainders continually together, and the square root of the last product will be the area required. This method is too prolix, except in particular cases; the operation may, however, be considerably simplified by performing the multiplication and evolution by logarithms.

PROBLEM I.

Triangular Fields.

When you have a triangular field to survey, begin at the most convenient corner and measure each side; and while measuring any one of the sides, leave a mark in some situation on the chain-line, that the distance between it and the opposite angle being measured, may be a proof-line.

Or, leave marks upon any two of the chain-lines, and the distance between them will prove your work.

Examples.

1. It is required to construct a figure, and find its area, from the following notes.

DO

913

proof-line

return to D

CA 1029 R. off C

BC

1275

1000 R. off B

AB

Begin at

MOO

D, station for a proof-line range W.

Having constructed the figure, you will find the line DO to measure 913 links, as in the field-book; hence, you may conclude there is no error committed in taking or setting down the dimensions.

Note 1.—If your proof-line upon the plan does not agree, or nearly so, with that taken in the field, you may be assured that some error has been committed; you must, therefore, repeat the survey in order to discover it.

2. When land is level, and the lines are well driven, and not very long, you will generally find them to meet correctly.

To find the Perpendicular.

(See Part I. Prob. VI.)

Or, if you make use of a plotting-scale, lay it across the base in such a manner that any of the lines which are opposite to each other on the edges of the scale, may coincide with the base, the edge of the scale at the same time touching the opposite angle; by that edge draw a line from the base to the opposite angle; this line, or perpendicular ca, in the present case, you will find to be 878 links.

1462 base 878 per. 11696 10234 11696

2)1283636

6-41818

1-67272

40

26-90880 Area 6a. 1 r. 27p.

Computation of the Area from the Three Sides.

Here

1462 + 1275 + 1029    3766

-—= 1883, half the sum of the

Ji

Then 1883- 1462 = 421, the first remainder; 1883 — 1275 = 608, the second remainder; and 1883 - 1029 = 854, the third

three sides.

remainder; whence V1883 x 421 x 608 x 854 = V411617533376 = 641574 square links, the area equal to 6 acres, 1 rood, and 261-perches, nearly the same as before.

The log of

The same by Logarithms. 1883

421

608

854

= 3-2748503 = 2-6242821 = 2-7839036 = 2-9314579

Divide by the index of the root

The quotient is the log of 641574, the area

2)11.6144939

5-8072469

To find a Line between two Stations when the side of the field is crooked, and when the Sight on the Line sought is obstructed by a Farm Homestead or other Building.

Let ABO be the field. B and c the two stations, e the building on the crooked side, and BC the line sought.

From stations B and C range a parallel to the line sought, as in Problem VII., Part IV., placing a pole and making the angle m a right angle. Then take the inset mR, and measure from B to A, setting the proof-pole at D as in the preceding example; drive AC; next make the inset nc = mR, and the angle at n a right angle. If

1 0.Jrn

the angle n corresponds with the angle m, the line mn is parallel to BC. If not, correct station n. Measure the proof-line CD; correct station m; find mn, taking the insets and the position of the building, the areas under both having to be deducted from that of the field. Thus mn is found 1275 = to BC, the line sought.

Or, Bo may meet AC produced, the perpendicular pc forming a tieline so as to dispense with CD, when bc would become the hypotenuse to Rp and pc, and thus be found, the base Rp and perpendicular pc having been first measured with the chain. (Theorem VII., Part I.)

CA

Or t—< -5 ^ C5 03 OOiOUsSOUOSO

1252

1000

824

716

610

424

212

000

m, station for a proof-line, which goes to n, and measures 352 links.

R. off a

BC

683

536

354

o «o

229

000

R. off B

AB

48

973

76

56

n, station for a proof-line

25

ID o

Begin at

range N.E.

Computation of the Area by Offsets, &c.

t-r:

Having constructed the figure, you will find the line mn to measure 352 links, as in the field-book. You also will find the perpendicular Ba to be 528 links.

to

to Of Of Cl

1280

512

6400

00 I Of to

Cl Of

239

O f CO

162

19359

to

Cl Cl CM

2061

916

11221

113

125

565

226

113

14125

Triangle AEG 1252 base 528 per.

10016

2504

6260

661056

Offsets taken on

56

76

132

105

660

132

13860

CO Cl

124

145

620

496

124

17980

1'

CO CO

CM 'Sf

1824

912

10944

6400

19359 double 13860 17980 J collected • 10944J 68543 sum

Offsets taken on the line BC.

CD ■«#

104

182

208

832

104

18928

147

40

5880

areas

11221

14125

18928

5880

double

areas

collected

50154 sum

Offsets taken on the line CA.

212

72

55

45

1060

117

1764

1060

106

756

11660

9324

M Ol fco o I Oi - Ol

to

14840

87

186

522

696

87

16182

12402

45

69

114

108

12312

©5 t-CO CO

106

176

636

742

106

18656

661056

68543

50154

95376

whole

double areas collected

116605 14840 16182 12402

12312 I collected 18656

double

areas

9324

95376 sum

2)875129 4-3 7564

1-50256

40

20-10240 Area 4a. 1 r. 20p.

Computation of the Area by reducing the Crooked Sides to Straight Ones: generally called “ Casting.”

A A.

Having constructed the figure as before, and taken out the chainlines, draw the three dotted lines AB, BC, and CA, in such a manner that the parts included may be equal to those excluded, as nearly as your eye can judge. Then the base AC being measured, will be found = 1390 links; and the perpendicular Ba= 630 links.

1390 base 630 per. 41700

8340

2)875700

4-37850

1-51400

40

2Q-56000 Area 4a. 1 r. 20p.

Note.—Although the method of finding the area by casting (which depends entirely upon the accuracy of the eye) is occasionally adopted, it is certainly less correct than that by offsets, &c. A learner, therefore, ought to practise both until he can habitually come very near to the truth by the former.

BD

760

return to B

proof-line

DA

1035

R. off D

CD

1145

1100

1000

950

900

850

800

Cl O Q

lO T—¹

r-H r-H

750

700

600

r—I r-H

<N

O lO (M

550

500

450

180

400

159

350

142

300

165

250

173

200

161

150

126

100

65

50

000

R. off c

AC 1590

Begin at

range W.

890 A, and

A nswer>

Having constructed the figure, you will find the perpendicular Da, upon the base AC, to measure 740 links.

Double areas 1176600 triangle ACD 275770 offsets taken on the line CD Area 7a. \r. Ifp.

4. Lay down a field, and find its area, sions.

dimen-

DB

575

return to D

proof-line

CA

1320

600

It. off C

D, station for a proof-iine

BC

CiOiOMOOiOOOOiMhO

880

800

750

700

600

550

500

380

250

200

100

000

it. off B

To

the

30

930

fence

rH

875

to B

800

700

600

500

Or ot

GO

400

300

30

200

17

100

Begin

000

at A and

range N.E.

Answer.

Having constructed the figure, you will find the perpendicular Ba, upon the base AC, to measure 573 links.

Double areas 756360 triangle ABC 85060 offsets taken on the line AB 106270 ditto on the line BC Area 4a. 2r. 38».

DB

596

return to D

proof-line

CA

©©©©©lO©lO lO m rn ^ o (O

rH

1080

1000

900

r—I rH r-H

O O O O O

O O lo O ⁰⁰    >Q in

D, station for a proof-line

© © CO i—I

410

©

CO

350

300

250

CD o CO o

CO N

200

150

100

O Or

r. off c

cc oo td

750

700

650

600

520

450

380

330

270

200

150

100

K.

000 Off B.

OOiQ

H “o ^

the fence 800

755

to B

o o co

700

Brook 50 links in breadth

i—■ i—>    • i—i

to oi os ot w o On oo o o Ot

650

600

550

500

450

400

350

300

CD o

CD GO

250

200

Q O CO O d ^ cm cq 5

150

100

50

000

range N.E.

Answer.

Having laid clown the figure, you will find the perpendicular Ba, upon the base AC, to measure 587 links.

Double areas 633960 triangle ABC 110800 offsets taken on the line ab 100120 ditto on the line BC 149310 ditto on the line CA Area 4a. 3r. 35\p.

PROBLEM II.

Fields in the form of a Trapezium.

When you have a trapezium to survey, measure each side and both the diagonals, one of which will enable you to construct the figure, and the other will serve as a proof-line ; or, you may measure the longer diagonal, and a proof-line in any other direction most convenient.

FTote 1.—From various obstructions it is sometimes impossible to take either of the diagonals; in such cases measure tie lines across the angles of the field at any convenient distance (not less than two chains) from the corners. These you will find sufficient for constructing the figure, and for proofs. Or, you may take an external angle, or angles, as directed in Problem IT.

2.    When the lines, including the angle you intend to take with the chain, are of a considerable length, it will be necessary to measure more than two chains from the angular point before you take the chord-line, because a small inaccuracy in constructing the figure, when the angular distance is short, will throw the lines, when far produced, considerably out of their true position. It sometimes happens, however, in consequence of obstructions, that it is impossible to measure the chord-line at a greater distance from the angular point than one or two chains. In such cases multiply both the chord-line and angular distance by 2, 3, 4, or any larger number, as circumstances may require, and use the products resulting in laying down the figure.

3.    When the measurement of the surface is required, for reaping, &c., you must let the chain touch the sides of the lands in all places where you measure across them. If you do not measure across the lands, but along the headland, then you must add as many links to the length of the chain-line as will make it equal to one measured across the lands, parallel to and near the headland.

You may easily ascertain what number of links you ought to add by stretching the chain across the lands, and putting down an arrow at each end, after-which leave hold of one of the ends, and you will observe it recede from the arrow. The number of links by which it falls short of its former position add to each chain. Some lands you will find so low that nothing need be added to the chain-line, and some will require a link to four, three, two, or even (where the lands are very high) a link or more to one chain.

When the lands are high, if the lines measured along the headlands be not lengthened, the perpendiculars will obviously measure less than they ought to do ; consequently, the horizontal measure will be returned instead of the measure of the surface.

When the diagonal is measured with a slack chain it will give the measure of the surface; but in this case the perpendiculars will evidently be shorter than they would have been, if the diagonal had been measured with a tense chain ; consequently, the measurement will be the same, or very nearly the same, whether the diagonal be measured, with a tense or slack chain, unless the headland lines be lengthened.

4.    If two or three persons measure the same piece of land separately, or even if one person measure the same piece twice over, there will generally be a difference between the measurements; this difference, however, in small pieces, should scarcely ever exceed four or five perches.

5.    When land, crops of corn, &c., are bought and sold, the buyer and seller commonly choose each a surveyor, and in their measurements it occasionally happens that there exists a considerable difference. In this case the best method, perhaps, of adjusting a dispute is, that the two surveyors meet and jointly remeasure the land. If this fail, they or their employers, the buyer and seller jointly choose an experienced surveyor, as an umpire, and abide by his decision.

Examples.

1. It is required to construct a figure, and find its area from the following notes.

BD

diag.

return to B

AC

diag.

R. off A

DA

55 8

R. off D

P o &E O © O o

rH r-H

BC

689

R. off B

AB

1492

1000

Begin at

range W.

'••-A

Having constructed the figure, lay your scale from BtoD; and if you find it exactly 1400 links, as in the field-book, you may then measure the perpendicular Ba = 468 links, and the perpendicular T>a = 432 links; from which you will readily compute the area required.

468 |

per.

432 J 900 sum 1916 diag. 2)1724400 8-62200

2-48800

40

19-52000 Area 8a. 2r. 19/>.

BY THE FALSE METHOD.

Referred to in Note 4, Part III., Prob. IV.

1492 = ab 1626 = CD

1559

623

2)3118

4677

1559 mean length

3118

9354

9-71257

689 = bc 558 = DA

2)1247

2-85028

623 mean breadth

40

34 01120

Here the area is found to be 9a. 2r. 34p., which is too much by la. Or. lhp. ; but the more nearly a trapezium approaches to a square or rectangle the less will be the error.

2. Required the area of a field from the following notes.

DB

diag.

diag.

1365

1000

return to D

AC

1288

1000 L. of A

DA

750

L. Off D

CD

765

L. off C

BC

720

L. Off B

AB

1600

1000

Begin at

range E.

Having constructed the figure, you will find that, in consequence of the angle adb being obtuse, a perpendicular from the angle A to the diagonal db cannot be taken; you must, therefore, let fall the perpendicular Da from the angle D to the side AB, which you will find = 638 links. The perpendicular Ca will be found = 294 links.

638

Triangle ABD 1600 base per.

12800

Triangle BCD 1365 base 294 per.

5460

48

12285

96

2730

1020800

401310

1020800 ) double areas 401310 / collected 2)1422110 7-11055

•44220

40

17-68800 Area 7a. Or. 18p.

3. It is required to find the area of a field from the following notes.

BD

1236

1000

return to b

diag.

ac

1326

1000 R. off A

diag.

DA

515

400

300

200

100

000

R. off D

CD

1375

1300

1200

OO

1000

800

652

0^00

356

200

100

000

R. off c

BC

cn co co    bo

OO&MkSOO&O

664

570

483

378

300

150

100

50

000

R. off B

AB

co

725

106

500

400

335

242

50

40

000

Begi

n at

range E.

BY OFFSETS, ETC.

r-t

r;

-JS) 3»

y'e

Having constructed the figure, you will find the perpendicular D« = 512, and the perpendicular Ba = 446 links

Trapezium abcd per.

512

446

958 sum 1326 diag.

5748 1916

2874 958 1270308

Offsets taken on the line AB

242

50

12100

181

2950

100

65

18100

12100^1

32

2080

130

195

tO CO

2080

199

50

10700 double 18100 9950 I collected 10725

areas

32

9950

2950,

75

66605 sum

107

93

100

50

10700

143

75

715

1001

10725

Offsets taken on the line bo

50

181

60

25

3000

104

905

150

362

60

5200

4525

85

104

145

15600

3000

50

7250

7250

CO <M

7850

15600

double

-1 00 o<

156

2496

areas

collected

234

4525

Or

kQ

2496

40721 sum

Offsets taken on the line CD

OX O

52

5000\

84

9400

5000

136

6864

200

7696

double

50

27200

44

94

100

-4 00

Ox

27200

31800

12500

3750

areas

collected

9400

159

104210 sum

200

r-H

CS

to OS

►f*. to I ^ ox I OS

31800

75

50

6864

125

100

00 <N

12500

296

740

Ox Ox

7696

3750

Offsets taken on the line da

CO

CO r-H

O 00 ICi CM

3300

9800

double

3300

78

11500 )

100

7800

areas

collected

eo I os os t—• GO I OX 05

7800

3220

35620 sum

xo oo

r-H CM

9800

920

230

OX OS

O Ox

3220

115

100

11500

1270308]

66605 | whole 40721 ]• double areas 104210 | collected 35620J

2)1517464

>•58732

2-34928

40

13-97120 Area 7a. 2r. Up.

BY CASTING.

> r.:;

4"-

,7

:4 c

Having constructed the figure, draw the four dotted lines AB, BC, CD, and DA, in such a manner that the parts included may be equal to those excluded; then the diagonal AC will be found = 1364, and the perpendiculars Da = 636, and Ba = 476 links.

63 6

476

per.

1112 sum 1364 diag.

4448

6672

3336

1112

2)1516768

7-58384

2-33536

40

13-41440 Area 7a. 2r. \Zp.

4. It is required to find the area of a field from the following notes.

BD

1460

1000

return to B

diag.

AC

1480

1000 R. off A

diag.

DA

672

R. off D

CD

1244

1000

800

600

05 CO M O O CO Ol o s

co O

R. off C

1—1

O LO <M CO

00 1—I oo

•<1 to

650

cross the fence

O H tO OO

o o o o o o o o

83

130

100

R. off B

AB

1350

Begin at

1000

range W.

_____J

li

/■

Having constructed the figure, you will find the perpendicular Ca = 613, and the perpendicular Aa = 618 links.

Trapezium ABCD 613

per.

618

1231 sum 1460 diag. 73860

4924

1231

1797260

Insets taken on the line BC

100

130

100

83

10000

213

100

100

21300

100001

23000 1 double areas 21300 | collected 4980 j 59280 sum

130

230

CO

100

os os

4980

23000

Offsets taken on the line BC

88

112

90

85

7U20

197

100

7920'

20000 _ double areas 19700 collected 5950

CO <M CO 1—1

19700

53570 sura

200

85

100

70

976

11468

Double areas 1797260 trapezium ABCD 53570 offsets 1850830 sum

insets

153

lo

200

30600

1708

19600

OO 05

o< OO

to

© CO

CO

O Ol

6000

6000

155

200

© 00 GO <N

CO © 05

31000

19600

30600

31000

23400

11468/

double

areas

collected

-I

122068 sum

Insets taken on the line CD

117

200

23400

59280

122068

181348 sum to be deducted for the above offsets 2)1669482 whole field 8-34741

1-38964

40

15-58560 Area 8a. 1 r. 16_p.

5. It is required to find the area of a field from the following notes; neither of the diagonals having been measured in consequence of obstructions.

*»■!&

Having constructed the figure, you will find the diagonal AC = 963, and the perpendiculars Da = 257, and Ba = 316 links.

per.

257

316

573 sum 963 diag.

1719

DA

476

L. off D

CD

618

200

L. off C

r, proof-line, 17 to B

BC

443

L. off B

AB

723

192

to to

Begin at

the angle

from, mto»

from A to m, on the line AD from A tb n, on the line AB range E.

fe>-

to

3438

5157

2)551799

2-75889 3-03596

40

1-43840 Area 2a. 3r. Ip.

6. Bequired the plan and area of a field from the following notes.

BD

diag.

1437

1000

return to B

AC

939

L. off A

diag.

DA

567

L. off D

CD

712

L. off C

BC

765

L. off B

AB

1457

1000

Begin at

range E.

Answer.

Having constructed the figure, you will find one of the perpendiculars = 560, and the other =166 links; hence the area is = 5a. Or. 34^?.

7. It is required to lay down a field, and find its area, from the following notes.

BD

diag.

diag.

662

return to B

AO 533

L. off A

DA 168

L. off D

CD

439

L. off C

BC

356

L. off B

O fcO 05 . o O 05 C

o o o -<r w

62

58

range E.

Begin at

Answer.

Having constructed the figure, you find one of the perpendiculars = 168, and the other = 218 links; hence the area is = la. 2r.

8. It is required to lay down a field, and find its area, from the following notes; neither of the diagonals having been measured in consequence of obstructions.

CE 1200 1000 l. off c

AO

1094

L. off A

EA 1800 1000 L. off E

DE

837

L. off D

BD

1528

1000 860

L. off B

AB

621

Begin at

range N.

Answer.

Having constructed the figure, you will find the diagonal BE = 1927, and the perpendiculars = 580 and 637 links respectively; hence the area is = 11a. 2r. 36^.

9. The plan and area of a field are required from the following dimensions.

DB

1730

return to D

diag.

AC

1660 R. off A

O O) O O CD <N

UC5HOHCO

rH rH rH rH pH

the

fence

1580

1500

to A

1450

1400

1300

1200

1100

1000

950

900

150

210

850

800

250

700

255

630

240

550

218

117

O'

41

300

18

250

15

200

100

150

140

100

157

50

165

000

R, off D

MHM

MWOffiOMOWOMOO

the

fence

1085

920

700

600

550

500

300

450

to D

O O O HCONOJO

R. off C

OiOfflOiMOOO^COOOtCOtOOJ&WMOOlCO

the

1440

1340

1250

1200

1150

1100

1000

900

800

750

700

650

600

500

450

400

350

300

200

150

100

000

R. off B

fence

to c

_ t- CO CO °«3NOO

the

1005

930

850

fence

to B

to eo O Oi o

650

600

N lO CO oo O'! a WiO 00 b. 1CJ 50 l>

550

500

450

400

300

200

100

CO QO

50

Begin

000

at A and

go N.

Answer.

Having constructed the figure, you will find the perpendiculars xa = 810, and ca = 708 links.

Double areas

2626140 trapezium ABCD 137945 offsets taken on the line AB 167800 ditto on the line bc 157520 ditto on the line CD 395420 ditto on the line da 3484825 sum

Area 17a. lr. 27fj?.

PROBLEM III.

Fields of more than Four Sides.

When a field consists of more than four sides, divide it into triangles and trapeziums, agreeably to the directions given in Part

III., Prob. Y. Then take the dimensions of each, as directed in the last two problems.

Note.—Notwithstanding what has already been advanced with regard to taking proof-lines, you are again requested never to omit measuring such distances as will enable you to confirm every part of your survey. Some may perhaps deem this tedious and superfluous ; but the satisfaction which a surveyor finds, when his lines meet correctly, fully compensates him for his additional labour.

Examples.

1. It is required to find the area of a field from the following notes.

EB

diag.

1510

1000 R. off c

CE

1238

1000

diag.

400

R. off c

m, proof-line, goes to D, and measures 285

AC

1200

1000

R. off A

EA 383

DE

692

R. off D

CD

620

R. Off C

BC

535

R. Off B

AB

1334

Begin at

1000

range W.

©-

-r

><

"...

Having constructed the figure, you will find the perpendiculars c a = 410, A a = 330, and Da - 215 links.

Trapezium abce

per.

410

330

740 sum 1510 diag.

Triangle cde 1238 base 215 per. ~6l90 1238

7400

2476

CO

266170

1117400

1117400 ) double areas 266170 J collected 2)1383570 6*91785

3-67140

40

26-85600 Area 6a. 3r. 27p.

2. It is required to find the area of a field from the following notes.

DA

diag.

diag.

1042

return to D

CE 420

R. off C

1?

D,

Begin at

AC

768

R. off A

diag.’!

EA

58 5

R. off E

DE

518

R. Off D

CD

365

L. off C

BC

390

R. Off B

AB

1054

600

A range W.

m, proof-line, goes to c and measures 200

*•*.

X.

Having constructed the figure, you will find the perpendiculars Cff = 223, Gn = 200, and = 176 links.

Triangle ABC 1054 base 223 per

3162

2108

2108

Trapezium acde 00 )

76 |per 376 sum 1042 diag.

i—¹ to

752

1504

235042

376 391792

235042 ) double areas 391792 J collected

2)626834

3T3417

•53668

40

21 -46720 Area 3a. Or. 21p.

3. It is required to find the area of a field from the following notes.

FB

diag.

660

return to F

BE

970

L. off B

diag.

DB

diag.

268

L. off D

AD

832

R. off A

FA 285

EF

cocn^

O u> co ® ®

384

300

200

100

000

R. off E

DE

804

L. Off D

CD

O to ®

383

200

000

R. off c

BC

475

300

R. off B

m, proof-lme, goes to D, and measures 255

AB

850

Begin at

range W.

?<

s'

7~:-.

Having constructed the figure, you will find the perpendiculars Fa — 185, Fm= 185, D»-190, and Da = 216 links.

Triangle ABF 850 base 185 per.

Trapezium bdef

185

190

per.

CO CO

CO lO

id o

157250

375 sum 970 diag. 26250

3375

363750

Triangle BCD 475 base 216 per. 2850

Offsets taken on the line CD

CO CM

766

475

1915

950

19916

102600

Offsets taken on the line EF

32

53

3200

100

40

8500

double

3200

93

9300

100

3360

areas

collected

32

9300

24360 sum

53

85

84

100

40

8500

3360

157250

double areas collected

363750

102600

19916

24360

2)667876 3-33938

1-35752 40

14-30080 Area 3a. 1 r. 14p.

4 It is required to lay down a field, and find its area, from the following notes.

DB

diag.

1440

1000

return to D

AC

1078

L. off A

diag.

EA

diag.

1324

1000

return to E

DF

712

L. off D.

diag.

AD

818

L. off A

diag.

FA

755

L. off F

EF

692

L. off E

DE

754

R. off D

CD

540

L. off C

BC

1048

L. off B

AB

1360

Begin at

1000

range E.

Answer.

jgHaving constructed the figure, and divided it into two trapeziums, abcd and adef, you will find the perpendicular which falls from the angle c upon the diagonal db = 315 links, and that which falls from the angle a upon the same diagonal = 758 links.

The perpendicular which falls from the angle D upon the diagonal ea, you will find = 425 links, and that which falls from the angle F upon the same diagonal = 287 links.

Hence the area is= 12a. 1 r. 30p.

5. Required the plan and area of a field from the following notes.

EB

424

R. off E

proof-line

FE

750

L. off F

MF

400

HM

460

R. off H

227 to K, proof-line

LH

700

R. Off T.

DL

430

return to D

EL

740

p-

H •

R. Off F

cp

HE

730

400

cp

R. off H

DH

920

P-

R. off D

8?

Cp

GD

950

return to G

P-

cp

GE

580

R. off G

FG

630

R. off F

DF

540

R. off D

AD

1050

R. off A

EA

450

R. off E

DE

670

R. off D

CD

500

R. Off C

AC

780

kE

Answer.

Having constructed the figure, you will find the perpendiculars of the trapezium ACDE = 354 and 195 ; of DFGE = 404 and 340 ; of dlhf = 426 and 316; and the perpendicular of the triangle FHM = 227 links. Hence the area of the field is = 10a. 2r. 25p.

6. Required the plan and area of a field from the following dimensions.

BD

1480 to B

Return

diag. line 14

GE

1725

L. Off

diag.

line 13

CG

1295

L. off

diag. line 12

EC

935

L. off

diag.

line 11

DF

793

R. off

diag.

line 10

37

49

n o o

^ CO CO CM

60

66

62

30

l—* l—¹ fco

O oaowo

OOOOOOOGC

R.

24

OtONO CO CO CO

O O O-l CO

o ^

i—( r-H r—I rH

£

Or o Or

70

co co co

r—I r—I r-1

190

179

150

95

60

off

OOOOOOOOOOOOOOOW^¹

COOOOOOOOOOOOOOO

g“oog§S§Soo°

wNtO>OTji^cOC(5N(^HO

DC

line 8

AD

700

50

130

450

160

400

173

350

118

300

190

250

150

200

107

100

60

000

l. off

line 4

CA

r. off

1578

diag. line 3

td

60

104

72

88

100

86

05 ^ Ul C5 G5 CO OI o O O Ol Oi OJ

75

95

80

70

R. off

y OMUJW

iOOOOOOOOOOO O O O O

aoroowoo

the

line 2 fence

1820

1750

1600

1500

1400

120

1300

1200

O H r-H r—4 r-i

1100

1000

186

900

184

800

125

700

114

600

93

HP* Or

86

75

300

200

70

100

E'

Note.—The preceding figure represents the chain-line, forming the two trapeziums and the triangle, in the 6th Example.—When the learner constructs the figure, he must of course lay down the offsets from the notes, and dot all the chain-lines as before directed.

Answer.

Having constructed the figure, you will find the perpendiculars of the trapezium abcd, falling upon the diagonal CA, to measure 862 and 314 links; the perpendicular of the triangle DGC, falling upon the diagonal cg, to measure 184 links; and the perpendiculars of the trapezium cefg, falling upon the diagonal ge, to measure 513 and 300 links.

Double areas 1855728 238280 1402425 362460 133750 143250 149010 157548 200374 30696 4673521 sum

trapezium ABCD triangle dgc trapezium CEFG offsets taken on the line bc ditto on the line bc ditto on the line AD ditto on the line CE ditto on the line ef ditto on the line fg ditto on the line gd

7. Draw a plan of a small estate, consisting of four fields, and find the respective areas of the different inclosures, and the contents of the whole, from the following dimensions.

Note.—The field-notes in this Example are entered from the left towards the right; but in all the foregoing Examples, they are entered from the right towards the left. Both methods are consequently exhibited ; and, of course, the learner is at liberty, when he commences field-practice, to follow that of which he most approves.

jV

s'

n /

/\ ⁴

if

:,g<:

<£f\

---*

joy ;

J...

.«•*

■t;

LW’-i

Scale 4 Chains to an inch.

To B

>0 00 03 05 1>

O to C5 CC 05 O

662

600

500

35

© o ©

rfCIJIMHO

Go from

54

92

105

100

78

44

Go from

Go from

Go from

Oi Ci O

62

60

50

Go from

W CM OOOO'+OOO O’—'OOO'+t'-O©

To c

685

600

500

400

300

200

100

000

To D

632

600

500

400

300

250

200

100

000

To A

995

To o

615

500

400

300

200

100

45

gate

28 + 52

76

line 5

kO 00 lO t—

QO bo

« sooo

CD

diag. line 3

line 2

Go from

To

To G

Go from

Go from

32

75

55

20

Gate

28

42

40

Go from

Go from

To H 750 700 600

490

350

220

100

000

the

668

638

600

500

400

300

200

100

000

To E

965

To D

735

700

600

500

430

380

300

200

100

000

To F

593

500

400

300

200

100

000

© r-l

38

60 + 35

104

15 + 80

25

30

line 11

fence

22

24

26

42

48

45

30

18

line 10

diag. line 9

line 8

30

52

57

48

32

line 7

Go from

Go from

To

To K

Go from

Go from

54

65

63

52

30

Go from

Einis

To H

895

diag.

line 16

woooooooooo

0<MOOcr>0»00000

^|l>tOlOrt(T)tcOCO(MrHO

48

gate

32

18

28

Or GO

line 15

the

580

546

500

400

300

200

100

000

To G

908

To c

624

500

400

300

200

100

000

fence

28

co co

U) ©

to IQ o

->dH t*!

34

line 14

diag. line 13

line 12

Answer.

Having drawn the plan, you will find the perpendiculars of the different trapeziums to measure as follow, viz :—

Dm = 426, and bn = 400, in No, 1; cm = 503, „ Fn = 448 mi =515, „ Em = 498 Cn =428, „ Km = 439

3 33

3 33

No. 2;

No. 3; and No. 4.

Area of No. 1.

Double areas 821870 trapezium abcd 84786 offsets on ae 54890 ditto on bc 72968 ditto on AD ~1034514 sum

93790 insets on CD

2)940724 difference 4-70362 Area in square links

2-81448

40

32-57920 Area ia. 2r. 32\p.

Area of No. 2.

Double areas 917715 trapezium DCEF 93790 offsets on CD 55510 ditto on df 1067015 sum

60758 insets on ce 43590 ditto on ef 104348 sum 962667 difference

Area ia. 3r. lOp.

Area of No. 3.

Double areas 919804 trapezium CEGH 60758 offsets on ce 42480 ditto on eg 81310 ditto on gh 1104352 sum 54096 insets on CH 1050256 difference

Area of No. 4.

Double areas 775965 trapezium bchk 54096 offsets on CH 41024 ditto on HK 57200 ditto on KB

928285 sum 54890 insets on BC 873395 difference

Area 4a. 1 r. 18 \p. Content.

No.

A. R. P.

4    2    32£

O O 00

Sum 19    0    21

Note 1. In the last example every field is measured separately; but they are so connected by the chain-lines that no difficulty can arise to the learner in planning them. It may also be observed that no proof-lines were measured ; but they should never be omitted in practice : if they be, the surveyor cannot depend upon the accuracy of his work.

2.    If the foregoing estate be laid down upon a sheet of drawing paper, by a scale of one chain, or of two chains to an inch, a finished plan may then be made and ornamented with Indian ink, in a similar manner to Plates IX. and XI. Or the quickwood hedges may be made by a pen and Indian ink; or they may be represented by running narrow shades of colouring along the lines which form the boundaries of the fields; and each field may then be washed over with a different colour, mixed up thinly with water, and laid on with a small brush or camel’s hair pencil. (See Part V. for the method of transferring a rough plan to a clean sheet of paper, in order to make a finished plan, with proper embellishments.)

3.    In drawing the finished plan, all the out-boundaries may be considered as belonging to the fields which they respectively adjoin; that fence from b to c may be made as belonging to No. 1, that from c to D as belonging to No. 2, that from o to e as belonging to No. 3, and that from c to h as belonging to No. 4. (See a remark on the 27th page relating to fences.)

4.    The title of the finished plan of the foregoing estate may run thus :—Plan of an Estate lying in the Parish of Bradford, in the West-Riding of the County of York. *

PROBLEM IY.

Meres and Woods.

The method of measuring Meres and Woods by the Chain and Cross has already been shown in Part III. It is here proposed to survey them by the Chain only.

In this case, you must not only measure on the outside of the mere, or wood, and take insets as before directed ; but also take such external angles, or tie-lines, as will enable you to lay down the figu re.

Example.

Let the following figure represent a mere, the area of which is required :—

0 >>

00

g§§§

//

Hi

ah

............e*

Begin at+1, and measure eastward as far as+2, taking insets as you proceed; then produce the line to + 3. Return to + 2, and measure northward as far as + 4; thence run a line backward to + 3, which will tie the first and second lines. Return to + 4, continue the line to + 5, and produce it to + 6. Return to + 5, and proceed westward to + 7, the distance between which and x 6, being measured, will tie the second and third lines. Return to + 7, and continue the line to+ 8. From+ 8 proceed to + 1, and you will have obtained the following dimensions.

Note.—Here it may be observed that after the first three lines are laid down, the fourth line will exactly reach from + 8 to + 1, if the operations have been performed with correctness.

60

1625

1100

to -f-1

1000

23

800

30

600

60

96

000

From

1150

go S. to + 8

1100

1000

900

700

40

400

000

Return to

+ 5 and

56

2000

1650

1300

1000

+ 7, which is 550 from + 6 go W.

to + 6

55 0

400

40

300

+ 4, which is 490 from + 3

103

Return to

+ 2 and

1500

go 1ST. to + 3

o o *o

1200

1100

1000

850

600

400

250

000

td

*sb

<x>

Answer.

Having constructed the figure, you will find the diagonal, drawn from + 1 to + 5 = 2085, the perpendicular from + 2 upon the diagonal = 950, and that from + 8 = 890 links.

Double areas

insets taken on the different lines

Area 17a. 1 r, 35p.

3836400 trapezium made by stations 1, 2,5, and 8. 84500' 1 line 87380 i 2 53000 (3 118120 j 4 343000 whole insets 3493400 mere

PROBLEM Y.

To measure ancl plan Roads, Rivers, Canals, &c.

In measuring roads, rivers, or canals, angles or tie-lines must be taken at the different turns, in order to lay down the chain-lines * and offsets must be taken to the boundaries as you proceed, to enable you to draw the plan.

Note 1. The length of a road is generally returned either in miles, furlongs, and poles, or else in miles and yards, (See the Table, page 30.)

2. A machine called a “ Perambulator ” is sometimes used to ascertain the lengths of roads. It has a wheel of 8 feet 3 inches, or half a pole in circumference, which being made to pass over the ground, puts in motion the clockwork within ; and the distance measured is pointed out by an index on the outside. This instrument is much more expeditious for measuring the length of a road than the chain, but it is certainly less correct; for by the wheel passing over stones, sinking into holes, &c., the distance is made to appear more than it is in reality.

Examples.

1. Let the following figure represent a serpentine road, a plan of which is required.

Begin at + 1, and Qv to + 3, taking

measure offsets on both sides, as

you proceed. Return to +    2, and measure to

4, from which run a line to + 3, which will tie the first and second lines. Return to + 4, and continue the line to + 6. From + 6, proceed as before, until you arrive at + 14; and you will have obtained the following dimensions, from which a plan may be drawn.

lj

:<P

P-

..■c

'q-

III \

'Vo A⁵

•?2

.6

To + 14

58

350

60

68

200

44

150

50

Go from

100

+ 12

To +

12

+ 13 is 184 from +11

80

line 5

30

70

+ 11

86

550

33

70

300

50

200

120

135

Go from

+ 10 is 200 from + 8

cross-fence

line 4

38

LA- O

50

Go from

20

50

52

20

18

Go from

38

15

28

55

To + y

700

600

500

480

300

180

150

100

To+ 6 512

450

380

350

300

200

100

To + 3

600

480

400

350

300

at -f-1

cross-fence

84

60

65

+ 7 is 160 from + 5

line 3

70

80

+ 4 is 232 from + 3

93

line 2

cross-fence

95

80

70

line 1

Begin

2. Let the foregoing figure represent a river, a plan of which is required.

Begin at a, and measure to c, taking offsets to the river’s edge as you proceed. From c measure to d, and there take the tie or chord-line db, which will enable you to lay down the first and second lines. Continue the second line to n, and from m measure to r, at which place take the tie-line rn ; and thus proceed until you come to the end of your survey at x.

The breadth of a river, if it be everywhere nearly the same, may be taken in different places by the next problem or by Problem X. Part III.; but if it be very irregular, dimensions must be taken on both sides, as above.

When the area is required, it must he found from the plan, by dividing the river into several parts, and taking the necessary dimensions by the scale.

Rote.—Any bog, marsh, mere, or wood, whatever may be its number of sides, may be measured by this problem.

PROBLEM VI.

Talcing Distances by the Chain and Scale.

EXAMPLE.

Required the distance of an object at A, from b.

JL

First, make a station at b ; then, in a direct line with BA, set up a pole, suppose at C; measure the distance BC. Return to B, and measure in any direction, making an angle with bc, suppose to D; then set up a pole in a direct line with da, as at E. Measure the lines de and EC, and also the diagonal CD; these will enable you to construct the trapezium bced.

The lines bc and de, produced, will evidently meet at A.

Measure the line BA with the same scale by which you have constructed the trapezium, and it will be the distance required.

C.fe

|E

ci*

PROBLEM VII.

To erect a Perpendicular by the Chain, or to measure Lines upon which there are Impediments.

©R

tv A

hb-~*>—&—ep

H:

EXAMPLE.

Suppose cdef to represent the base of a building, through which it is necessary a line should pass from A, to an object at B, the direction AmB on the side A being given.

Measure from A to m; and from m, measure back to a, 40 links. Let one end of the chain be kept fast at a, and the eightieth link at m; take hold of the fiftieth link, and stretch the chain so that the two parts an and mn may be equally tight: then will mn be perpendicular to

am.

For mn will be 30, am 40, and an 50 links ; or the sides of the right-angled triangle amn will be in proportion to each other as 3, 4, and 5. (See Prob. XVIII.

Part I.)

Measure from m, upon the line mn continued, until you are clear of the impediment, as at c; then continue the line 40 links farther, to b. Find by the above process the perpendicular cd; and proceed in that direction till you are beyond the building, as at h. Again erect the perpendicular he, upon which measure till you have made hp equal to me; and you will then be in a direct line with mA. Erect the perpendicular px, which (if you have conducted the work with correctness) will be in a right line with B. Measure the distance pR; then Am, added to ch (= mp), and pR, will give the whole length of the line ab.

PROBLEM VIII.

Having the Plan of a Field, and its true Area, to find the Scale by which it has been constructed.

Rule.—By any scale whatever measure such lines as will give you the area of the figure : then say, as this area is to the square of the scale by which it was found, so is the true area to the square of the scale required.

Example.

Suppose the true area of a field, the plan of which is given, to be 9a. 1 r. 32p.; and that by a scale of 2 chains to an inch, I find the area to be 4a. Or. 32p. • required the scale by which the plan was constructed.

First, 9a. lr. 32\p. — 945000 square links; and 4a. Or. 32^?. = 420000 square links ■, then, as 420000 : 4 :: 945000 : 9. Hence it appears the plan was constructed by a scale of 3 chains to an inch.

Rote.—The principle of this process is, that the areas of similar figures are to each other as the square of their homologous sides. (Theo. XVI. Part I.)

HILLY GROUND.

The survey of a hill is taken conjointly upon the horizontal and vertical planes. Both planes are generally required to determine the area as represented upon the plan. The details of the vertical survey are shown by sections.

There are three methods practised in finding the area of a hill, viz., the horizontal, the hypotenusal, and the superficial.

The first way is that which determines the horizontal area. It accords with trigonometry on the horizontal plane, and is that which is generally practised in Land-surveying.

The second way gives the hypotenusal area, and also the angle which the hypotenusal plane makes with the horizon. Surveys of this kind are frequently ordered for plantation and other grounds, and for whole estates when they lie on the sloping sides of large hills.

The third way returns the superficial area. It is generally adopted for paring and burning, trenching, ploughing, reaping, and other agricultural works.

Methods for reducing Hypotenusal to Horizontal Lines.

Method I.

When the hill is of a regular slope, take its altitude with a theodolite, or with a quadrant; then, by a trigonometrical canon, in which the hypotenuse may be counted 100 links, determine the number of links in the base. These deducted from 100 will show the number of links by which each chain must be shortened, for the purpose of planning.

Example.

Suppose the altitude of a hill to be 16° 15', and the length of a line measured upon its surface to be 2550 links ; required the length of the line that must be used in planning.

In the right-angled triangle ABC are given the hypotenuse AC = 2550, and the angle bac = 16° 15', to determine the base AB. Or AD = 100, and the angle ead = 16° 15', to find AE.

10-00000 2 00000 9-98229

As radius

Is to the hypot. AD = 100 links So is the co-sine of the angle EAD = 16° 15'    .

1-98229

To AE= 96 links.....

Hence it appears that 4 links must be subtracted from each chain; consequently (25 x 4 + 2 =) 102 links must be taken from AC; hence AB = 2448 links, the line required.

Proof.— As, 1 : 2550 :: -96005 (the nat. co-sine of 16° 15') : 2448-1275 links = AB.

Note.—Surveyors, when serving an apprenticeship, acquire a knowledge of how to drive the chain in hilly ground ; and those who master this branch of their profession approximate very closely to the length of horizontal lines found by trigonometry.

A Table for reducing ILypotenusal to Horizontal Lines.

Different Altitudes of Hills.

Different Altitudes of Hills.

Links to be subtracted from each Chain measured upon the Surface.

Links to be subtracted from each Chain measured upon the Surface.

j-

Deg.

Min

Links.

Deg. Min.

Links.

11

14

16

18

19

21

23

24

25

27

f2oe>Sg*S2S3KS;*8o>fc

13

14

15

16

17

18

19

20

21

hhm

who

22

23

24

Note.—To construct the above table, suppose the base ab, in the preceding triangle to be = 99 '5, and the hypotenuse ac = 100 ; then by Trig, as 100 : 1: : 99’5 ⁻995, the nat. co-sine of the angle bac=5° 44'. In the same manner, the rest of the angles are obtained, by different operations, accounting the base 99 in finding the second angle, 98 in finding the third, &c.

QUADRANTS AND THEIR USE.

'As.

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/,

a¹'

I?

nn

it nisi

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&■

Quadrants differ in construction, but are similar in principle. The instrument is for taking vertical angles. In surveying, it has been nearly superseded by the semicircle of the theodolite and spirit-level, and in astronomy by the mural circle.

1.    The annexed diagram ABC represents a quadrant of the simplest mechanism. It is suspended from C either by the hand or top of a staff, the plumb-line CG being the index. The sights through which the object at D is viewed are two small holes drilled in the brass plates a and b on the radius AC.

2.    Another example is similar to the last, a telescope being substituted for the brass sights on AC.

3.    A third kind has a spirit-level on the radius bc, which is placed horizontally on CE; the quadrant or arc BA is towards D; and a telescope is affixed longitudinally on the revolving index CG, through which the object is viewed.

4.    A fourth construction has a spirit level on AC, which is placed horizontally on CE; bc is towards the object; and the telescope is now fixed tangentially upon the revolving index at the circumference of the arc, the' line of collimation thus forming a tangent.

To take the Altitude of a Hill with the Quadrant.

Upon the top of the hill fix an object, exactly as high as your eye will be from the ground, in taking the observation. At the bottom of the hill fix the quadrant staff perpendicularly to the horizon, which may be easily done by means of the plummet. Then with one eye at A, the other being closed, look through the sights, turning the quadrant until you perceive the object at D • so will the arc BG, cut off by the plumb-line CG, be the measure of the angle dce, or the altitude of the hill, in degrees, above the horizon.

To take the Altitude of a Steeple, &c., with the Quadrant.

Screw the quadrant fast to its staff, so that the plummet may hang exactly at 45° when the staff is perpendicular to the horizon. Then move the staff backward or forward (always keeping it perpendicular), until you can see the top of the object through both the sights. Measure the distance between the bottom of the staff and that of the object, which being added to the height of your eye, will give the altitude required.

Method II.

As the foregoing method of reducing hypotenusal to horizontal lines can only be applied, with accuracy, when hills are of a regular slope, surveyors, in general, elevate the chain, as they ascend or descend a hill, in order to preserve the horizontal line.

Examples.

\Zt

c —~7b

• in -

/*\

—‘c

Suppose the lines AB and BC to represent the acclivity and declivity of an irregular hill, it is required to measure them, and to preserve the horizontal line AO.

From A stretch the chain toward B, and suppose it to reach to a; the same extent, upon the base, will evidently reach from A to g ; and a perpendicular erected from g will intersect the line AB in d; hence the distance Ad, upon the hypotenuse, will make one chain upon the base. At A, stick your offset-staff into the ground, perpendicularly to the horizon, and let your assistant hold the chain, suppose at the twenty-fifth link, close to the surface of the hill, as at b ; at the same time you must elevate the end of the chain to c forming the horizontal line cb ; then move forward to b, at which place fix your staff again, as before. Let your assistant hold the fiftieth link at p, while you elevate the twenty-fifth to n, forming the horizontal line np. Again, fixing your staff at^>, elevate the fiftieth link to m, while your assistant holds the seventy-fifth at e. Lastly, put down the staff at e, and elevate the seventy-fifth link to r, while the hundredth is held by your assistant at d. There he must put down an arrow, and thus you must proceed until you arrive at B, where you will have obtained the horizontal line AD.

In descending from B to C, let your assistant hold one end of the chain at B, while you elevate, suppose, the fiftieth link to n, forming the horizontal line Bn; then fix the staff at a, perpendicularly to the horizon, and touching the chain at n. Next, let your assistant hold the fiftieth link at a, while you elevate the hundredth tom, and put down the staff at r, as before. In this manner, having arrived at C, you will have obtained the horizontal line DC, which being added to AD, will give the base or horizontal line AC, as required.

Note 1. If you wish to obtain the hypotenusal, as well as the horizontal line, divide your field-book into four columns, in one of which you must enter the number of links between a and d, &c., which being added to the horizontal will give the hypotenusal line.

2. When the ascent or descent of a hill is great, you will not be able to elevate more than 10 or 15 links of the chain at one time; for, in such cases, if you attempt to elevate 20 or 30 links, you will find that the perpendiculars ac, bn, &c. will exceed your own height before you can form the horizontal lines cb, np, &c. (See the last figure.)

Method III.

Hypotenusal lines may likewise be reduced to horizontal ones during the survey by quadrants constructed for the purpose. Of these two examples are subjoined, viz., King’s Quadrant and Nesbit’s Quadrant.

KING’S QUADRANT.

Description.

‘ The quadrant is fitted to a wooden square, which slides upon an offset-staff, and may be fixed at any height by means of a screw, which draws in the diagonal of the staff, thus embracing the four sides, and keeping the limb of the square perpendicular to the staff. The staff should be pointed with iron to prevent wear. When the staff is fixed in the ground on the station-line, the square answers the purpose of a cross staff, and may, if desired, have sights fitted to it. The quadrant is three inches radius, of brass, is furnished with a spirit-level, and is fastened to a limb of the square by means of a screw.

‘ When the several lines on the limb of the quadrant have their first division coincident with their respective index-divisions, the axis of the level is parallel to the staff.

‘ The first line next the edge of the quadrant is numbered from right to left, and is divided into 100 parts, showing the number of links in the horizontal line which are completed in 100 links on the hypotenusal line, and in proportion for any smaller number.

‘ The second, or middlemost line, shows the number of links the chain is to be drawn forward, to render the hypotenusal measure the same as the horizontal.

‘The third, or uppermost line, gives the perpendicular height, when the horizontal line is equal to 100.’

Use.

‘Lay the staff along the chain-line on the ground, so that the plane of the quadrant may be upright; then move the quadrant till the bubble stands in the middle, and on the several lines you will have—1. The horizontal length gone forward in that chain; 2. The links to be drawn forward to complete the horizontal chain; 3. The perpendicular height or descent made in going forward one horizontal chain.

‘ The first two lines are of the utmost importance in surveying land, which cannot possibly be planned with any degree of accuracy without having the horizontal line; and this is not to be obtained by any instrument in use without much loss of time to the surveyor ; whilst with this he has only to lay his staff along the ground, and set the quadrant till the bubble is in the middle of the space, which is very soon performed. And he saves by it more time in plotting his survey than he can lose in the field; for as he completes the horizontal chain as he goes forward, the offsets are always in their right places, and the field-book being kept by horizontal measure, his lines are sure to close.

‘ If the superficial content, by the hypotenusal measure, be required for any particular purpose, he has that likewise, by entering in the margin of the field-book the links drawn forward in each chain, having thus the hypotenusal and horizontal length of every line.

‘The third line, which is the perpendicular height, may be used with success in finding the height of timber. Thus, measure with a tape of 100 feet the surface of the ground from the root of the tree, and find by the second line how much the tape is to be drawn forward to complete the distance of 100 horizontal feet; and the line of perpendiculars shows how many feet the foot of the tree is above or below the place where the 100 feet distance is completed.—Then, inverting the quadrant by means of sights fixed on the staff, place the staff in such a position as to point to that part of the tree whose height you want; and sliding the quadrant -till the bubble stands level, you will have on the line of perpendiculars on the quadrant the height of that part of the tree above the level of the place where you are; to which add or substract the perpendicular height of the place from the foot of the tree, and you obtain the height required.’

T—-4

n*-

Nesbit’s Quadrant.

Plate II.

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Deg. Min. Links. Deg. Min. Links. Deg. Min. Links.

^OCO-JOSCn^COtO

SSggggggssggsss^sgggggggggojgo.o.ggg

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

The Construction of the preceding Table.

:C

A

JB

In the right-angled triangle ABC, suppose the base AB to be 100,

and the hypotenuse AC 100-5 ; then by trigonometry as 100'5 : 1 : : 100: -99502, the co-sine of the angle BAC = 5° 43'. In the same manner, the rest of the angles are obtained by different operations, accounting the hypotenuse 101 in finding the second angle, 102 in finding the third, &c.

Now from the preceding table it evidently appears that if an instrument be constructed to take the altitude of a hill at every chain, if necessary, and a line traced upon the instrument be so divided as to exhibit the number of links which the chain must be drawn forward upon the surface of the hill, to complete the horizontal chain according to the table, it may be used in surveying hilly ground.

The Method of Constructing the Quadrant, &c.

Procure a piece of soft sheet-brass, and upon it draw the lines AB and AC perpendicular to each other, and with a radius of five inches describe the quadrant BC.

Next, draw the lines DE and DF perpendicular to each other; and with four inches in your compasses for the first sweep describe the double arc EF, which divide correctly into 90 equal parts or degrees. At a proper distance, likewise, from the arc EF describe the double arc GH, and the double arc ran. Of these, the latter must be cut through the brass by a file.

You must also procure a small glass tube, nearly filled with spirit (generally called a “ spirit level ”), and a piece of sheet-brass KL, in length equal to AB, and in breadth rather exceeding the diameter of the tube, which call the “index.”

Then procure another piece of sheet-brass in the form of a semicylinder NP, large enough to admit the tube; and in it make the aperture led, in order to see the bubble.

Its edges solder to the index KL, so that the centre c may be exactly in the middle point between r and a; ra rather exceeding DE, and au being exactly equal to Dm. The end N must also be closed up by soldering a piece of brass upon it, and the end P left open in order to admit the tube.

Next, make a wooden quadrant, exactly the size of ABC, and in it a groove corresponding with the aperture run, and large enough to admit a small screw-nail, with a square head and neck, so as to run, but not to turn round, in the groove ran.

Then fix the plate abc to the wooden quadrant, by the countersunk screws, 1, 2, 3, 4, 5; taking care first to insert the screw-nail abovementioned into the aperture mn, at a small hole made for that purpose at n.

Next, fix the index kl upon the face of the quadrant by a screw-nail passing through it at a, which must enter the quadrant exactly at the centre D. The nail in the aperture mn must likewise pass through the hole at u, and upon the end of this nail must be screwed a small nut, by which the end k of the index may be made fast at any altitude.

Now, to divide the arc gh move the end K of the index toward C, until the line or edge re, which must be exactly in the centre of the index, cuts the arc EF at 8° 4', as per table; and upon the arc GH mark the first division. In the same manner, move the index until it cuts of 11° 22', and there mark the second; continuing these operations until you have made as many divisions as are necessary.— The divisions marked upon the arcs EF and GH, must then be properly cut and figured by an engraver.

Next, procure a wooden cross, RTSW, the three limbs of which must each be in length equal to AB or AC, and must form with each other three right angles, rst, tsw, and wsr.

This cross must be made to slide upon an offset-staff by means of a square or rectangular aperture through the limb RS; and if a screw be fixed in the side of the limb at n, the cross may be fastened to the staff at any convenient height, by turning the screw against the side of the staff. As it will be somewhat difficult, however, on account of the limb RS being hollow, to make a joint at s sufficiently strong to keep the limbs at right angles with each other, they may be supported by means of the brackets ab, cd, and ef. The quadrant ABO must then be fixed upon the square RST, by means of two screws passing through the bracket ab, and one through the bracket m, so that the outside of the limb SR may coincide with AB, and the outside of the limb ST with AO.

To fix the tube or spirit-level correctly in the semi-cylinder NP, screw the index fast at no altitude, and place the edge AB of the quadrant upon a level table, which you may do by laying the tube . upon it, and varying the position of the table until the bubble stands in the centre of the tube; then put the tube into the semi-cylinder np, and fix it in such a manner that the bubble may be seen at c ; after which close up the end P with brass or putty.

Note.—If the quadrant be made the same size as that in Plate II. instead of five inches radius, as before directed, it will save much trouble in dividing, as the engraver may then follow the divisions given in the plate.

The Method of Proving the Quadrant.

Let AC represent a strong plank, placed with one end against the perpendicular wall BC, and the other upon the horizontal plane AB. Lay on offset-staff, suppose of 12 links, upon AC, with one end at A and the other at m; then elevate the lower end, so that the staff An may he parallel to AB. Measure the distance mn, which suppose to he 10-5 inches ; then say, as 12 links is to 10'5 inches, so is 100 links to 87⁻5 inches, or 11 links.

Next, lay the edge AB of the quadrant upon the plank AC, and elevate the end K of the index until the bubble stands at c ; and if the index cuts off 11 links, or nearly so, upon the arc GH, the quadrant is correct.

The Method of Using the Quadrant.

Lay the staff, with the quadrant fixed to it, along the chain-line, so that the edge AB of the quadrant may come in contact with the ground; then elevate the end K of the index until the bubble stands at c ; and you will have the altitude of the hill upon the arc EF, and the number of links to be drawn forward to complete the horizontal chain upon the arc GH. If you fix the bottom of the staff into the ground upon the chain-line, the limbs ST and sw will serve as a cross by which perpendiculars may be erected.

Rote 1. In using the quadrant, care should be taken to place it upon the even part of the surface of the hill.

2.    In measuring and reducing a line upon a hill, if it happen that the end of the chain reaches exactly to the station at the end of the line, you must then deduct from the chain instead of drawing it forward. For example : if you find that the chain ought to be drawn forward 6 links, you must set down 94 instead of 100 links. Or, if the fiftieth link reach to the station, you must enter 47 instead of 50 links, &c.

3.    If you determine, by elevating the chain, and also by the quadrant, the number of links to be drawn forward upon the surface of a hill, in order to complete the horizontal chain, you will seldom find them precisely the same ; because it is almost impossible to prevent the chain from forming a curve line, or to keep the staff perpendicular to the horizon. In every case, however, the conclusions of an instrument constructed upon mathematical principles are to be preferred.

Methods for fi7iding the Hypotenusal Measure of Hilly Ground.

This is by far the most difficult part of surveying; and, though we may approach toward, we can seldom obtain the true area of hills, because their surfaces are generally so irregular that it is almost impossible to divide them into proper figures.

If the land to be surveyed lie in the form of a square, rectangle, trapezoid, trapezium, or triangle, against the side of a hill of a regular slope, take the dimensions and find the area in the same manner as if the figure lay upon a plane. But should it be required to find the area of a field (suppose in the form of a trapezium) in which there is a hill so situated as to effect the diagonal only, if the sides and diagonal be measured, and the figure laid down according to those dimensions, the perpendiculars will obviously measure less than they would have done, had the diagonal been reduced to a horizontal line; consequently, we cannot obtain the hypotenusal measure of such a field by the common method of measuring trapeziums or triangles.

In such cases, it is perhaps best, first, to measure the hill only. For this purpose, surround its base by station-staves, dividing it into an irregular polygon, each side of which must be measured. Then fix upon a convenient place near the top of the hill for a station, and between it and each station at the bottom measure a line. Thus will the whole surface be divided into triangles, the areas of which must be found by laying down each triangle separately. Or, from the three sides, you may find the area of each triangle, as already directed.

Next, measure the remainder of the field, by dividing it into proper figures. Collect all the areas together, and their sum will be the area required.

When the land to be surveyed ascends a hill on one side, occupies a plane upon the top, and descends on the other side, you must divide it into such figures as will enable you to approach as nearly as possible to the true area.

The foregoing directions may, perhaps, be found useful to a learner, but, in practice, much will always depend upon the surveyor; he ought, therefore, to be very careful, whatever be the shape or size of the hill, to divide it into such squares, rectangles, trapezoids, trapeziums, or triangles, as are most likely to give him the hypotenusal measure.

Note 1. In surveying a triangular field, of which one side passes over a hill, the other two being upon a horizontal plane of the base, it will be necessary to

divide it into two triangles, by measuring a line from some part of the fence passing over the hill to the opposite angle. Thus will two sides of each triangle be affected by the hill, the areas of which, found separately, will give the hypotenusal measure of the field.

2. After making some experiments and considering the subject very maturely, the author is of opinion that the most correct method of finding the surfaces of hills in general, is to take the dimensions in such a manner that the areas of the different figures into which the hills are divided, may be found from the lines measured in the field, without having recourse either to the scale or plan. Hence, if the figures be rectangles, their lengths and breadths must be measured in the field ; and if they be triangles, trapeziums, or trapezoids, their bases and perpendiculars must be measured in the field.

Examples.

1. The length (or hypotenusal line) of a rectangular field, lying upon the side of a hill of regular ascent, is found to be 900 links, its breadth 800 links, and the altitude of the hill 28° 21'; required the hypotenusal measure, and the length of the line that must be used in planning:

900

PART IV.

HYPOTEHUSAL AREA OF HILLS.

153

800

7-20000

•80000

40

32-00000 Area 7a. Or. 32p.

How, by the table, page 142, we find that 12 links must be deducted from each chain; hence 9 x 12 = 108, which being taken from 900, leaves 792 links, the length of the line required.

Note.—If we multiply 792 by 800, we find the product 633600 square links equal to 6a. 1r. 14p. the horizontal measure, which is less than the hypotenuse by 3r. 18p.

2. Let abcd represent a field in the form of a trapezium, lying upon the side of a hill of an irregular ascent, the sides AB and BC being upon the horizontal plane of the base; required the horizontal and hypotenusal measures from the following notes :

•I

AJ-J

no

BD

1154

12

1100

11

1000

12

900

10

800

11

700

10

600

500

400

300

200

From

100

go N.

AC

1300 R. off A

_ iffghj #i

\-

3@F

---

JO

diag.

diag. reduced for a proofdine.

DA

hhOcCMOSOSC:

990

800 700 600 500 400

o o o o

CO CM

100

r. oft D

CD

1044.

1000

900 800 700 600 500 400 300 200 100

R. off c

BC 800

R. off B

Begin

£+•

range S.W.

The Operation of finding the Horizontal Measure.

First, 700 + 1154 + 990 = 2844, the sum of the three sides, which being divided by 2, gives 1422. From this number deduct severally each side, and we obtain 722, 268, and 432, for the three remainders. Then, by multiplying the- half sum and the three remainders continually together, and extracting the square root of the product, we obtain 344768 square links, the horizontal measure of the triangle ABD.

In a similar manner, we find the horizontal measure of the triangle BCD = 405559 square links; which, added to 344768, gives 750327 square links, equal to 7a. 2r. the horizontal measure of the trapezium ABCD.

The Operation of finding the Hypotenusal Measure.

First, 1154 + 110 = 1264, the hypotenusal line BD; and 990 + 78 = 1068, the hypotenusal line da. Then, 700 + 1264

+ 1068 = 3032, the sum of the three sides, which being divided by 2, gives 1516. From this number, deduct severally each side, and we obtain 816, 252, and 448, for the three remainders. Then, proceeding as before, w~e obtain 373709 square links, the hypotenusal measure of the triangle ABD.

In a similar manner we find the hypotenusal measure of the triangle BCD = 437917 square links, making jointly 821626 square links, equal to 8a. Or. 34p. the hypotenusal measure of the trapezium ABCD, which exceeds the horizontal measure by 2r. 34y>.

General Directions for the Vertical Survey of a Hill.

Let ABC, in the diagram page 144, represent the transverse section of a hill, AC its base, and B an object at the top of the hill or section, whose altitude is represented by the perpendicular BD. The line BD may also indicate the plane of a longitudinal section.

The line AB is assumed to show the acclivity of the hill lying in the hypotenusal plane: and the crooked line BrC the declivity on the opposite side where the surface diverges from the hypotenusal plane BC, this line not being drawn.

The line AB being the base-line of the survey on that side of the hill from which the bearings are taken, it requires to be carefully measured with the chain, the same as if it lay on the horizontal plane in horizontal surveying.

The'lines AD and BD (or any other lines parallel to them that may be required) are determined by trigonometry in the following manner :—■

The angle BAD is found by taking the angle of depression at B, the two angles being equal (Theo. III. Part I.), and their complement ABD should be taken at the same time as a check angle.

The angle of elevation taken at A does not give the correct angle BAD, because the perpendiculars at A and B represent radii, and therefore are not parallel: whereas a line drawn from A so as to make with AB the complement of BAD i§ parallel to BD.

The two lines BD and AD remaining undetermined may be found by the following formulae, AB being taken as the common radius.

Any other points in AB as d may thus be determined, the triangle Adg being similar to the triangle ABD.

The details of the survey on the opposite side of the hill are more complicated, owing to the uneven surface or crooked line Brc involving oblique-angled triangles and impediments to sight between the station-poles at B and C. Thus, if the flag-staff at B cannot be seen from station C extend the base line, if practicable, as in Prob. V. Sect. V. Part VIII. until it becomes visible, at which place set a station-pole E, and take a second distance in the same direction to station E for a base line. Measure EF with the chain, and take the two adjacent angles at E and F with the quadrant or theodolite. Their sum deducted from 180° gives the other angle of the triangle EBF. BE is now found by Case I. Sect. III. Part VIII. Next take the angle of depression at B and its complement EBD, which will give the angle E sought, when BD and DE may be found by the above formulae; measure CE with the chain and deduct it from DE, which will give CD.

The hypotenuse BC and the angles which it makes with BD and DC remain undetermined, and may be found from the following formulae, BC being, in 1 and 2, the common secant sought, and the lines found, BD and DC, tangent and radius; but, in 3 and 4, it is radius and BD and DC sin and cos.

o o o o EH EH Eh EH

r! pi co

!=b Eb Ch fch P r* !=S 0 p. fL Pj Pj

o w

jri yp

m m CD CD O O

w o

g R -S .9

Ph O m ai

If the base line EF cannot be found m the horizontal plane AC, so that the station-poles E and F are seen from B, then an intermediate station-pole may be placed farther down the hill, as at r, making a triangle BCr within the vertical plane; rCE and rCF being without. Should this hypothesis fail, then a base line must be taken upon a different plane, say upon the opposite hill, and the hypotenusal line BC or rc found from the bearings taken thereon, as directed in Sect. YI. Part IX. Or the whole of the survey may be taken downwards from the top of the hill; and as the details of the operation are similar to those already given, with the exception that the offsets are taken from parallels by means of the levelling staff, they may be left for the student to work out as a simple but instructive exercise.

Of surveys involving less or more details of the above kind on the vertical plane, the following familiar examples may be enumerated.

1.    In mountainous district roads ascend from the valleys, climbing zigzag along the sides of high hills, and not unfrequently pass over the tops of low ones. Their formation often involves a vast amount of levelling, and before such work can be undertaken, a vertical as well as a horizontal survey is necessary.

2.    In many places a vertical section is necessary to show the true elevation, climate, and position of towns and other places.

3.    When a survey is ordered with a view to determine the geological character, or the agricultural and mineral value, of an estate, a vertical section is required to show the depth of the soil, and the geological strata below. Geological surveys and maps furnish an illustration.

4.    Large areas of peat-moss are to be found on hill-sides, and when the moss forms a ‘ live bog ’ a survey may be ordered on public grounds; the safety of the inhabitants farther down the hill being in danger.

5.    Quarries of slate, marble, and stone for building are often situated towards the top of hills, and surveys are required to ascertain the depth of earth upon the rock (as in Example 3), the thickness of the strata, the elevation, and probable expense of opening and working.

6.    On the Wealden and other clays, low hills often yield fine brick-earth ; hence the nature of the survey.

7.    Gravel is found drifted into low hills and flat table-lands in districts where it possesses a high value for road-making and building purposes; hence the survey.

8.    Fortifications and other military works often involve a large amount of vertical surveying.

9.    It has often been proposed to give a true plan of a country by means of miniature hills and valleys, and several exceptionary surveys of this kind have been made. The position of the hills and valleys determine the direction of the transverse and longitudinal sections.

10.    Railways, embankments, rivers, canals, &c., give rise to a larger amount of vertical surveying than any of the preceding examples. Special directions for surveys of this kind will be found in subsequent parts of the work.

PART Y.

THE METHOD OF SURVEYING AND PLANNING LARGE ESTATES.

Various methods are adopted in taking the dimensions of large estates or lordships, of which the following four examples may he given.

Method I.

Having made yourself acquainted with the form of the estate, select two suitable places, at the greatest convenient distance from each other, as grand stations ; and measure a principal base, or what is generally called a “ main-line,” from one to the other, noting every hedge, brook, or other remarkable object, as you cross or pass it • taking offsets likewise to the bends or corners of the hedges that are near you.

Next, fix upon some other suitable place, towards the outside of the estate, as a third grand station ; to which, from each extremity of the diagonal or main-line, or from two convenient points in it, lines must also be run.

These three lines being laid down, will form one large triangle; and in a similar manner, if necessary, on the other side of the diagonal or main-line a second triangle may be formed.

The survey must then be completed by forming smaller triangles on the sides of the former, and measuring such lines as will enable you to obtain the fences of each enclosure, the boundaries of rivers, roads, lakes, &c. &c., and prove the whole work.

Note 1. If the estate be of a triangular form, three lines must be run in the most convenient manner, so as to form the largest triangle possible ; after which other lines must be measured, offsets taken, &c. &c.; so that all the fences may be obtained, and the survey completed, as in Plate VIII.

2.    When an estate is divided into two triangles, it is generally best to finish one of them before you measure any of the internal lines of the other, as in Plate III. Sometimes, however, it is more convenient and expeditious to run some lines in the second triangle before you have finished the first, as in Plate X.

3.    Estates similar to those in Plates III. and VIII. are very easy to survey, as

they contain no impediments; but it is otherwise with estates like that in Plate X. where the windings of rivers, roads, and fences, make it necessary to run a great number of lines in order to obtain a correct plan of the whole estate, and the true area and position of every part.

Plan. Plate III.

Vs

■or\

y'

10

<o.

•'20''

7,'

v°

22

19o>

----o'.

V'

£8, M 6.

i«⁶;

,V

24

0- _

,'C' 13

V4

26 ?' 'K'

-6..

'.3

14.

_ -or

15^'

Scale, 8 Chains to an Inch.

4. I11 extensive surveys, where two surveyors are employed, it is best to consider the estate as divided into two distinct parts by the main-line. Each surveyor may then take a part, and make use of the base-line between them, and

measure such other lines as are necessary to complete that part of the survey which he undertakes. By this means the lines of one surveyor do not become entangled with those of the other, while the work of both is more expeditiously and correctly performed than if the two were employed on the same side of the main-line.

5.    It is sometimes advisable to divide a very large estate in the following manner : Measure a line across the estate as near to the middle as convenient; and at right angles to this line measure another line through the middle of the estate. These two lines being tied together by a connecting line measured from one to the other, will divide the estate into four parts, which may be measured separately by dividing them into triangles, and taking such dimensions as are necessary to complete the survey. This method is more especially advisable where three or four surveyors are employed in measuring a large lordship; for the first two lines being considered as out-boundaries, the estate may be measured in four separate but connected parts.

6.    The method of surveying estates by dividing them into triangles is illustrated by Plates III., VIII., and X., the last two of which are actual surveys. The field-notes belonging to them are given in an engraven Field-Book; and Plates IX. and XI. are the finished plans.

7.    No notes are given to Plate III., but the directions of the lines may be ascertained by the following particulars : The first, or main-line, leads from+ 1 to + 8, the second line from + 8 to + 10, and the third from + 10 to + 1, which three lines form the first large triangle. The fourth line extends from + 2 to +15, and the fifth from + 15 to +8, which two lines and part of the main-line form the second large triangle. The sixth line leads from + 9 to + 11; the seventh from + 20 to + 6 ; the eighth from + 7 to + 22 ; the ninth from + 21 to + 4; the tenth from+24 to +13; and the eleventh from+ 12 to+ 23; which complete the survey of the first triangle. The twelfth line extends from +5 to +17 ; the thirteenth from + 25 to the main-line south-ward of + 3 ; the fourteenth from + 1 to + 14 ; the fifteenth from+14 to+ 26; the sixteenth from+ 27 to+ 16 ; the seventeenth from + 18 to + 28 ; and the eighteenth from + 28 to + 19; which finish the whole survey.

8.    The content of the estate may be found in the following manner : Measure the lines upon the plan, and take the necessary offsets, by a scale of 8 chains to an inch, and enter the dimensions in a Field-Book. From the dimensions thus obtained, draw a plan by a scale of 2 chains to an inch ; then straighten the fences, as directed in Part IV. or Part V., and measure diagonals, perpendiculars, &c., from which compute the content of each field. The diagonals, perpendiculars, and contents may be entered in a Book of Castings, similar to those belonging to Plates VIII. and X. Any other scale will do just the same for practice.

9.    Taking the dimensions, &c., as directed in the last note, will be found of infinite service to the learner: as it will tend to make him very expert in entering the field-notes, laying down the lines, and casting the contents, which are no small acquisitions towards becoming a complete land-surveyor.

10.    In the first edition, the notes were entered from the right towards the left, in the engraven F’ield-Book ; in this edition they are entered from the left towards the right.

11.    Some surveyors represent the crossings of fences by lines drawn across the right and left hand columns of the Field-Book; and others by lines crossing the middle column. The author prefers the latter method; but every surveyor will, of course, follow that of which he most approves.

12. Many surveyors enter their notes in a book about four inches and a half in breadth, and fourteen or fifteen in length when opened ; and others prefer a book about eight or nine inches long, and seven or eight inches broad when opened. (See the description of the Field-Book, Part II.; and also the engraven Field-Book belonging to Plates VIII., X., and XII.)

Method II.

Measure a main-line as nearly to one of the out-boundaries of the estate as the curves in the hedges will permit; noting the crossings of fences, and taking offsets as before directed.

At a convenient distance measure another main-line parallel, or nearly parallel, to the first line, so that a number of fences running in that direction may be obtained ; and from any two stations in the first line measure lines to some station in the second main-line, thus forming a triangle ; so will a station in the second main-line become determined or fixed.

From the first main-line to the second, or from the second to the first, measure lines in order to obtain all the fences which run in that direction. The remainder of the fences of the enclosures contained between the first and second main-lines being obtained by running lines in the most convenient manner, you will have completed the dimensions of a portion of the estate, which may then be laid down.

Parallel or nearly parallel to the second main-line, and at a proper distance from it, measure a third ; and proceed with the internal lines as before, and you will obtain the dimensions of another portion of the estate, which may also be laid down.

Carry on the survey in a similar manner, until you finish it.

Note 1. This'method is illustrated by Plate IV. The field-notes are not given ; but the following particulars exhibit the directions of all the lines : The first mainline leads from+1 to + 6 ; the second from + 6 to + 7 ; and the third line, or second main-line, from+ 7 to+ 16. The fourth line extends from+ 16 to+1; the fifth or tie line from + 16 to + 2; the sixth from +2 to+ 14 ; the seventh from+ 17 to + 18 ; the eighth from + 12, through + 18 to + 3; the ninth from+ 4 to+ 10 ; and the tenth line leads from + 8 to + 5 ; thus all the fences between the first and second main-lines are obtained.

The eleventh line, or third main-line, leads from + 19 to+ 29 ; the twelfth from + 29 to+16; the thirteenth from+29 to+15; the fourteenth from +15 to + 28 ; the fifteenth from + 26 to+13 ; the sixteenth from+ 12 to+ 25 ; the seventeenth from + 23 to + 11 ; the eighteenth from+ 30 to+ 31 : the nineteenth from+ 9 through+31 to + 22 ; and the twentieth from+ 7 to+ 19, which completes the survey between the second and third main-lines.

The twenty-first line, or fourth main-line, extends from + 32 to + 40; the twenty-second from+ 40 to+ 29 ; the twenty-third from+ 40 to+ 28 ; the twenty-fourth from+ 28 to+ 38; the twenty-fifth from+ 37 to+27 ; the twenty-sixth from+ 24 to + 36 ; the twenty-seventh from+ 35 to+ 23 ; the twenty-eighth from + 34 to

+ 21; the twenty-ninth from + 20 to 4- 33; the thirtieth from 4- 32 to -f- 19, which finish the whole estate.

2. In order to practice the learner, a Field-Book may be formed, and the content of the estate found in the same manner as directed in Note 8, Method I.

Plan. Plate IV.

■o

■i⁹

b‘2

5 >•

203

<L²¹

.-^22

'ss

%"31

° Ki

b-Ar^

^11

30

35*|

6-

00

f-O: -. ; ib

.12

26

fV"l3

te.................

]?■

17

-6 14

38

ns

?V--

28

39?

-nr

o...

\: 16

29

Scale, 8 Chains to an Inch.

3. Some writers on surveying instruct their pupils to measure main-lines through the estate to be surveyed; and upon these, by the help of a cross, to erect perpendiculars to the opposite angles and curved fences ; and upon these perpendiculars again, if necessary, to erect other perpendiculars ; thus dividing the whole estate into right-angled triangles and trapezoids.

This method is extremely tedious, as many of the perpendiculars will be 12 or 15 chains in length, when the fields are large; and where the fences are much curved, it becomes almost impracticable, in consequence of the great number of offsets that must be taken, in order to obtain a correct plan of the estate.

Besides, when the fence to which perpendiculars must be erected is at a considerable distance from the base-line, it will be necessary for an assistant to walk along by the fence, in order to point out to the surveyor the angles and curves to which offsets ought to be taken : and if there be a crooked fence on each side of the base-line, two extra helpers will be necessary, if the surveyor intends to perform his work with expedition. Hence we see that this process of measuring not only subjects the surveyor to a great deal of extra trouble, but also to a very considerable unnecessary expense.

This method I have never followed in measuring estates ; neither have I ever seen it followed by any experienced surveyor. But where fields are rectangular, the contrary may in such cases be the rule, as shown in the opposite page 165.

Method III.

An estate of four sides may frequently be conveniently surveyed as follows : Measure four lines in such a manner that offsets or insets may be taken to the four out-boundaries of the estate; and tie the first and fourth lines together by a diagonal or tie-line measured from one to the other, at the distance of five, six, or more chains from the angular point, according to the extent of the survey; thus you will be enabled to lay down the first four lines, and also the out-boundaries of the estate.

Next proceed to obtain the internal fences by measuring lines in the most convenient manner; some of which must be run from the first to the third, or from the second to the fourth line; or in some other proper direction, so that they may become proofs and fast-lines, into which other lines may be run with propriety.

In thus proceeding, it is evident that a great deal will always depend upon the dexterity and ingenuity of the surveyor, as no directions can be given that will suit every particular case to be met with in practice.

Note.—This method of surveying an estate is exemplified by Plate XII., the field-notes of which are contained in the engraven Field-Book given with this work.

Method IV.

The method which I here intend to describe, is a compound of all the foregoing methods of surveying with the chain; for as there are never two estates to be met with which are exactly alike, sometimes one method claims the preference, and sometimes another;

PART V.

MISCELLANEOUS INSTRUCTIONS.

165

but a skilful surveyor will always adopt that by which he can take his dimensions and proofs with the greatest accuracy by the fewest lines.

If an estate be in the form of an irregular polygon of five, six, or more sides, and the fences very crooked, such an estate may generally be most easily surveyed by dividing it into triangles, as in Method I.; but if many of the fences of the different enclosures run a considerable way in the same direction, and the fields in general pretty neat trapeziums, it is commonly more eligible to proceed as directed in Method II.

Sometimes an estate varies so much in its shape, that all the methods before described may be used with propriety and advantage ; and it frequently happens that an ingenious surveyor adopts methods in particular cases entirely new to himself; care, however, must always be taken to make one line depend upon another throughout the whole survey, so that, when you come to lay it down, you may find no lines whose position are undetermined.

Note 1. Whatever method of surveying is adopted, the field-notes must be entered in a similar manner to those given in the engraven Field-Book. Some surveyors place the letter S against straight fences in the Field-Book, to distinguish them from those that are crooked ; but they may be very well denoted by drawing straight or crooked lines, as the case requires.

2. The estates given in this work as examples are not very extensive, in consequence of the serious expense that attends large plates, and the great inconvenience of folding them in books; but it may be remarked that the foregoing methods of 'surveying are applicable to estates of all sizes, even to those of many thousand acres.

MISCELLANEOUS INSTRUCTIONS.

1.    When you have an estate to survey, never begin your work too hastily. Walk over the estate, examine it minutely, and observe by which of the foregoing methods it can be most easily measured. Next determine upon that point at which it will be most convenient to begin, and never omit to take the range of the first line with a compass. If you do, it will be impossible for you to lay it down in its true position upon the plan.

2.    In measuring your main or any other chain-line, put down stations at every place to which you apprehend it may be necessary to run lines in order to complete the survey.

3.    You may sometimes put down a station, whether you see any particular use for it or not; because it may become serviceable in correcting an error should one be committed; and if it be not used, it will be immaterial.

4.    In measuring your internal lines, it will give you the least trouble to run them from one station to another, if you can make it convenient; if not, you must run them from, and continue them to some chain-line, and measure the distance upon that line to the nearest station, which may be entered in the Field-Book thus : run upon the first line, 30 links S. of + 1, &c.

5.    The place where you run upon or cross a chain-line, may he easily ascertained by setting up poles at two of the nearest stations in that line; the crossing will be at the place where you are in a direct line with these poles, which may be*represented by marks cut in the ground, pointing out the directions of the lines.

6.    In ranging the poles, there must be one fixed at the station from which you intend to depart, and another at the place toward which you direct your line, if there be no natural mark, as a tree, the corner of a house, &c. Then, in a straight line with these marks, put down poles at the distance of 4, 6, or 10 chains from each other, according as impediments may render them necessary.

7.    When you are measuring a line across a valley, you must proceed forward until you are likely to lose sight of the station to which you are going; then, let your assistant take a pole to the other side of the valley, and direct him to place it exactly in the line which you are measuring, so as to be seen from the bottom of the valley ; to this you may continue your line, and thence to the end.

8.    When the stations between which you wish to run a line are so far distant that you cannot see from one of them to the other, or when your view is obstructed by an elevation between them, you must then, accompanied by your assistant, go to the place whence you can distinctly see both ; and turning face to face, at a little distance, direct each other to the right or left, until you are both in a right line with the stations ; then, one of you putting down a pole, the line will be correctly found. If the line, however, be so long that you cannot possibly find it by the above methods, it must be ranged at random ; but, in this case, you should be extremely careful that your pole-ranger keeps one pole in a direct line with another, which he may accurately effect by always having, at least, two behind him.

9.    In measuring a line which passes over a hill, you must attend to the directions given in Part IV. in the method of measuring Hilly Ground; but you will not always find your lines to meet correctly in surveying mountainous estates.

10.    When a river runs through the estate it will be necessary to continue some of your lines across the river, in order to tie the whole survey together.

11.    Rivers, large brooks, public roads, and common sewers, should not be included in the area, but only delineated upon the plan. If, however, their areas be required, they should be given separately.

12.    Marshes, bogs, heaths, rocks, &c., belonging to the estate should be distinctly represented upon the plan; and their measurements separately returned.

13.    You will generally have an opportunity of representing some part of each hedge in your Field-Book; and you may denote on which side of the ditch the fence stands by drawing a small bush, or by specifying it in writing.

14.    In surveying estates, the crossings of fences must be taken at the outer extremities of the ditches, and not at the roots of the quickwood, because the ditch, and not the fence, is the division-line between adjoining fields; but in measuring enclosures which are separated by walls, the case is generally different, as the walls most commonly form the lines of division. It may also be observed that the ground upon which a wall stands must be measured with the field to which the fence belongs ; and as walls are generally broader at the bottom than at the top, it is necessary to attend to this circumstance in taking the dimensions.

15.    When the surveyor finds it convenient, he may put down stations at the outer extremities of the ditches; and in planning, the stations will, of course, fall upon the black lines, because they always represent the boundaries between adjoining fields. This accounts for several of the stations appearing on the black lines of the rough plans in Plates VIII., X., and XII.

16.    In taking a survey, you must enter in your Field-Book the name of each field, or of its proprietor or occupier; or you may make such remarks as will enable you to distinguish the fields from each other, &c., and after the plan is drawn, acquire from persons acquainted with the estate every necessary additional information.

17.    When hedges obstruct your sight in running the lines, it will be necessary to cut down part of their tops in order to see the poles.

18.    If it should happen that you measure a line for which you have no particular use, it will serve as an additional proof; it is evident that you had better measure too many lines than too few.

19.    In taking a survey, you ought to observe to whom the adjoining ground belongs, and specify the same upon the plan.

20.    Some of our practical surveyors use only nine arrows. When the leader has advanced ten chains, the follower goes up to him, and places his foot or offset-staff at the end of the chain, instead

GENERAL RULES FOR PLANNING LARGE SURVEYS.

The method of laying down a large survey from the Field-Book may easily be acquired by practice ; but as the least appearance of difficulty generally discourages a learner, it is presumed that the following directions may be found acceptable.

Having provided a sheet of drawing-paper of proper size, trace with a pencil a meridian, or north and south line, in such a manner that your first station may be in some convenient point in this line. Then, from your first station draw your first or main-line, making its proper angle with the meridian-line, which you may then take out with indiarubber.

Next, take separately in your compasses your second and third lines, or any two more convenient ones, forming a triangle with the main-line ; and placing one foot of your compasses in the proper centres respectively, describe arcs intersecting each other. Thus will you have three points from which to form a triangle.

In the same manner proceed with each triangle formed upon the main-line (or upon any other line), proving your work as you advance, until all the triangles are laid down; and if you find all your lines correctly meet, it will be an infallible proof of the accuracy of the work.

The chain-lines being thus laid down, next prick off the crossings of fences, and draw lines in their proper situations, from one crossing to another, to represent the straight fences.

The curved fence must be formed by laying down the offsets, as already directed.

When the whole survey is planned, all the fences must be drawn with Indian ink, the chain-lines and offsets dotted, and the stations, gates, stiles, &c., marked in their proper places; the sheet will then represent what is called a ¹ Rough Plan.’

Rote 1. When a fence represents a chain-line it must not be dotted.

2.    Practical surveyors seldom dot their chain-lines or offsets, but only mark their stations upon the plan ; but it is more satisfactory to a learner to be able to see all his chain-lines at a single view.

3.    In taking a very large survey, it is necessary that the work be laid down and proved every night; for if an error be committed, and the survey continued two or three days before it is discovered, the detection in the field will probably be attended with a great deal of trouble.

4.    In laying down large surveys it sometimes happens that one sheet of paper will not contain the whole ; in this case, two or more must be pasted together.

5.    When you have to lay down a line exceeding the length of your scale draw a line with your pencil in some convenient place upon the plan, and upon it, at two or more operations, prick off the distance in question, which you may then take in your compasses.

6.    Beam Compasses are very useful in drawing large circles. They consist of a long straight beam or bar, carrying two brass cursors ; one of them fixed at one end, the other sliding along the beam, with a screw to fasten it. To the cursors may be screwed points of any kind, as of steel, pencils, &c. To the fixed cursor is sometimes applied an adjusting or micrometer screw, by which an extent may be obtained to a very great nicety.

DIRECTIONS FOR PLANNING THE ESTATE IN PLATE VIII.

From the dimensions in the engraven Field-BooJc.

It appears by the first page of the Field-Book that the range of the first line is NNW. ; and by referring to the compass, Plate I. (page 23), we find that the angle which this line makes with the meridian line is 22° 30'.

By Prob. XXIII., Part I., lay down a line making an angle of 22° 30' with the meridian line; and by a scale of four chains to an inch prick off 2802 links from cross or station (+) 1 to + 3, and you will thus have the part of the first line.

Now, as the third line could not be run to + 1, in consequence of a large quickwood hedge intervening too far to be cut down, it was necessary to produce the first line 30 links southward, in order that the first three lines might form a triangle; consequently, the first line must be continued 30 links southward from + 1, in laying down the plan ; and this continuation completes the first line.

Take the second line, 3075 links, in your compasses, and with one foot in + 3, describe an arc; and with 3270 links, the third line in your compasses, and one foot in a point 30 links south of + 1, describe another arc, intersecting the former in + 6 ; join these three points by drawing lines from + 3 to +6, and from + 6 to the above-named point, and you will thus form the triangle 13 6.

Next, prick off stations 2, 4, 5, 7, and 8 ; and lay your plotting-scale from +2 to + 8, and if it measure 1046, as in the Field-Book, line six, you have good reason to conclude that your dimensions are thus far correctly taken and laid down.

Also, mark off + 10, try its distance from +4; likewise examine the distance from +5 to + 7 ; and if you find both these lines the same as in the Field-Book, your survey is evidently correct.

With the fourth line, 257, in your compasses, and one foot in + 1, describe an arc; and with the fifth line, 1004, as a radius, and + 2 as a centre, make another arc cutting the former in + 9 ; hence you have three points by which to form the triangle 19 2.

Lastly, complete the rough plan by pricking off, and drawing all the straight fences; laying down the off-sets; showing the gates ; numbering the fields, &c. &c., as in the Plate.

DIBECTIONS FOE PLANNING- THE ESTATE IN PLATE X.

From the dimensions in the engraven Field-Book.

We find from the fourth page of the Field-Bopk that the first line ranges W. b. N. J W., making an angle with the meridian line 84° 22^.

By Prob. XXIII., Part I., draw a line, making an angle of 84° 22^' with the meridian line ; and by a scale of four chains to an inch, prick off 5445 links from +1 to +12, and you will thus obtain the first line, upon which prick off stations 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11.

With 900 (part of the third line) in your compasses, and + 1 as a centre, describe an arc; and with 625, the fourth line, as a radius, and one foot in + 2, intersect the former arc in +22. From + 2 draw a line to +22; and from +1, through +22, draw the third line, equal to 1360, and you will obtain +23.

With the second line, 3790, in your comjDasses, and +12 as a centre, describe an arc; and with 925, part of the fifth line, as a radius, and +23 as a centre, describe another arc, cutting the former in +21. From +23, through +21, draw a line equal to 2090, and you will thus obtain the fifth line, and also stations 24, 25, and 26.

Draw a line from +12 to +21, and you will have the second line, and also stations 13, 14, 15, 16, 17, 18, 19, and 20 ; and draw another from + 26 to a point in the first line, 295 W. of + 10, and you will obtain the sixth line, and likewise stations 27 and 28.

With 2325, the twenty-sixth line, in your compasses and + 3 as a centre, describe an arc; and with 1210, part of the twenty-seventh line, as a radius, and one foot on the first line, 150 W. of + 7, describe another arc, cutting the former in + 33. Draw a line from + 3 to + 33; and from + 33, through the intersection of the first line, draw the twenty-seventh line, equal to 2040, and you will obtain + 43.

Next, with 446, the seventh line, in your compasses, and + 12 as a centre, describe an arc ; and with 2528, the eighth line, as a radius and +33 as a centre, describe another arc, cutting the former in + 29. Draw lines from + 12 to + 29, and from + 29 to + 33, and you will obtain stations 30, 31, and 32 ; and also draw the ninth line from + 32, through + 8, and + 16, to + 27, and you will have stations 34, 35, 36, and 37.

Join stations 11 and 13, and you will obtain the tenth line; 28 and 30, and you will have the eleventh line; 37 and 18, and you will obtain the twelfth line; 19 and 25, and you will have the thirteenth line ; 38 and 20, and you will obtain the fourteenth line; 20 and 39, and you will have the fifteenth line; 24 and 39, and you will obtain the sixteenth line; 39 and 19, and you will have the seventeenth line; 17 and 27, and you will obtain the eighteenth line.

From +31, through + 9, draw the twentieth line, equal to 1175, and you will obtain stations 40 and 41; join 36 and 40, and you will have the nineteenth line; and from +41, through +34, draw a line to a point in the first line 72 E. of +7, and you will obtain the twenty-first line.

Draw a line from + 35 to + 43, and you will have + 42, and the twenty-third line ; and join + 42 and + 36, and you will obtain the twenty-second line. Draw a line from + 23 to the first line, 115 E. of + 5, and you will obtain + 45, and the twenty-ninth line; from + 43 to + 45, and you will have + 44, and the twenty-fourth line ; from + 3 to + 45, and you will obtain the twenty-fifth line.

Lastly, complete the rough plan, by pricking off, and drawing all the straight fences; laying down the offsets; making the gates; forming the bases of buildings; shading the river; numbering the fields, &c. &c., as in the Plate.

Note 1. Hot-pressed drawing-paper is best for plans. Parchment and vellum are more durable, and generally used when gentlemen are desirous that the plans may be handed down to their posterity. Vellum exceeds parchment in durability ; and it may be necessary to remark, that when either of them is used for planning, it must first be rubbed with clean flannel dipped in the best Paris whiting. This operation clears its surface from grease, and facilitates the movements of the pen.

2. In damp weather paper expands, in dry weather it contracts; consequently, if a plan be drawn when the paper is in a moist state, and the content be not found till after it has become perfectly dry, the diagonals and perpendiculars will measure too little, and will of course give the area too little also, but if the plan be drawn when the paper is dry, and the area be found after it has expanded by a change in the atmosphere, the diagonals and perpendiculars will measure too much, and will consequently give the area too much likewise. Hence the necessity of having the paper in the same state of dryness when you find the area that it was in when you laid down the chain-lines, offsets, &c.

3.    The most expeditious method of laying down crooked fences is by means of an offset scale, which must be used with the plotting-scale in the following manner: Lay one edge of the plotting scale close by the base-lines, and bring the end of the offset-scale in contact with the edge of the plotting-scale, so that the edges of the scales may form a right-angle ; then by the edge of the offset-scale prick off, in its proper situation, the first offset, with a pencil finely pointed. Keep the plotting-scale firm, and slide the offset scale to the place of the next perpendicular, which prick off as before ; and thus proceed until all the offsets are finished.

4.    Properly prepared lead pencils, of different degrees of hardness, for the use of engineers, architects, land-surveyors, and artists, are always in high repute among draftsmen. For land-surveyors they should bear pointing well, so as to produce fine lines.

TO COMPUTE THE CONTENTS.

After the whole survey is laid down, practical surveyors straighten the crooked fences of each field, as directed in Part IV. ■ and then divide the fields into trapeziums and triangles, and take such dimensions by the scale as are necessary to find the separate area of each field. They then collect all the areas into one sum; afterward find the area of the whole survey, as if it were a single field, and if it appears to be equal, or nearly equal, to the sum of the separate areas, previously found, they justly infer that their survey is correct.

Note 1. Those who do not approve of finding the area by the method of casting, may make use of the offsets taken in the survey, where convenient ; and if more be wanted, they may be measured by the scale; for in measuring a number of small parts by it, some will probably be taken a little too large and others a little too small, so that, in the end, they will nearly counterbalance each other.

2.    Practical surveyors generally lay down their lines by a scale of 4 chains to an inch, when their surveys are very large; and in computing the contents, they measure the bases and diagonals by the same scale, but the perpendiculars by a scale of 2 chains to an inch; consequently, the product of the base and perpendicular of a triangle will be its area. To treat small surveys, in a similar manner, by a scale of 2 chains and of 1 chain to an inch, must, of course, be correct.

3.    When the survey is not very large, the content of each field may be set down in some convenient place upon the plan. In other cases, it may be entered within the field itself. Some gentlemen, however, prefer having the areas of their estates given in a book of particulars, containing numbers, or letters of reference, corresponding to those upon the plan.

4.    As some surveyors prefer a parallel ruler to a lantern ruler, or a bow of whalebone and silk, for reducing crooked fences to straight ones, I have in the following problems given the method of using that instrument.

5.    When there are no dimensions given in the following problems, the figures may be measured by a scale, and then laid down in the learner’s book; after which

the operations by the parallel ruler may be performed. Or, for practice, figures may be made at pleasure ; and the necessary equalising lines drawn, according to the subsequent directions.

THE USE OF THE PARALLEL RULER

IN REDUCING CROOKED FENCES TO STEAIGHT ONES, IN ORDER TO FIND THE AREAS OF FIELDS BY THE METHOD OF CASTING.

PROBLEM I.

To draw a right line AD from the point A through the line bc, so that the quantities on each side of the line AD may be equal.

Draw with your pencil a temporary line CE, at pleasure; then, your ruler being closed, lay it from C to A; hold the side that is next to you fast; open the other to B; make a mark with your pencil upon the temporary line CE, where the edge of the ruler cuts that line, as at D; draw a line from A to D, and the quantities on each side of this line will be equal; that is, the triangle ABF will be equal to the triangle CDF.

Demonstration.

Draw the line AC, and also the line BD, which is evidently parallel to AC ; then by Theo. VI., Part I., the triangle ABC is equal to the triangle ACD; take away the triangle ACF, which is common to both, and there remains the triangle ABF equal to the triangle CDF.

Note 1. The solutions of all the following problems are founded upon the foregoing demonstration.

2. If it had been required to draw the equalising line from the angle c, through the line ab, the temporary line must have been made from the angle a.

3.    All the operations must be performed with the utmost care and accuracy ; and if, at any time, the ruler be suffered to slip, the work must be repeated, or it will not be correct.

4.    When an error has been committed, it may be frequently discovered by the eye, after the equalising line is drawn.

PROBLEM II.

Let the irregular figure ABCDEA represent an offset taken in surveying a field ; it is required to draw a right line from the angle A, so as to reduce the figure to a right-angled triangle.

Produce the perpendicular BC for a temporary line.

Lay your ruler from C to E; bring it down in a parallel position to D; and make a mark upon the line BC, where the edge of the ruler intersects that line, as at m.

Lay your ruler from m to A; move it in a parallel direction to E, and make a mark upon the line BC, close by the edge of the ruler, as at F.

Draw a line from A to F; and the triangle ABF will be equal to the irregular figure ABCDEA; hence the area may be found by multiplying the base AB by half the perpendicular bf.

area

Note 1. In practical operations, the equalising and temporary lines must be made with a pencil finely pointed, and effaced with india-rubber, after the is found.

2. If perpendiculars be let fall from the angles e and d, upon the base ab, the necessary dimensions taken by a scale, and the area of the irregular figure abcdea obtained by the rules for triangles and trapezoids, it will be found equal to the area of the right-angled triangle abf : great care, however, must be used to make the lines very fine, and to take the dimensions of all the figures with the utmost accuracy.

PROBLEM III.

It is required to reduce the offset 1 2 3 4 5 to a right-angled triangle by drawing an equalising line from the fifth angle through the irregular fences.

Perpendicularly to the base, and from the first angle, draw a temporary line.

Lay your ruler from the first to the third angle, move it in a parallel position to the second angle, and mark the temporary line at number 1.

Lay your ruler from number 1 to the fourth angle, bring it down in a parallel direction to the third angle, and mark the temporary line at number 2.

Lay the ruler from number 2 to the fifth angle, move it parallel to the fourth angle, and mark the temporary line at number 3.

Draw a line from the fifth angle to number 3 ; and 5 13 will be the right-angled triangle required; hence the area of the irregular offset may be found by multiplying the base 1 5 by half the perpendicular 1 3.

PROBLEM IV.

It is required to lay down a right-line offset from, the following dimensions; to reduce it to a scalene triangle by the parallel ruler; and to find its area both by the method of offsets and casting.

300 K 100 G 200 E 150 C

West

From

AL

1500

1100

800

500

100

000

A go

Having laid down the figure, produce the side AC, at pleasure, for a temporary line.

Lay the ruler from the first angle A to the third angle E, move it parallel to the second angle C, and mark the temporary line at 1, which, in this case, is at the second angle, because AC produced is the temporary line.

II

Lay the ruler from 2 to the fifth angle K, move it parallel to the fourth angle G, and mark the temporary line at 3.

Lay the ruler from 2 to the sixth angle L, move it parallel to the fifth angle k, and mark the temporary line at 4.

Draw a line from the sixth angle L to number 4 (m), and ALM will be the scalene triangle required.

Computation of the Area by Offsets.

Here 150 x 100 = 15000, twice the area of the triangle ABC; 150 + 200 x 400 = 350 x 400 = 140000, twice the area of the trapezoid BDEC; 200 + 100 x 300 = 300 x 300 = 90000, twice the area of the trapezoid DFGE; 100 + 300 x 300 = 400 x 300 = 120000, twice the area of the trapezoid FHKG; and 400 x 300 = 120000, twice the area of the triangle hlk; then 15000 x 140000 x 90000 x 120000 x 120000 = 485000, twice the area of the whole offset; and 485000 = 2 = 242500 square links = 2a. Ir. 28p. the area required.

Computation of the Area by Casting.

From the angle M let fall the perpendicular mn, which )mu will

find to measure 323 links ; then

n.»

323 x 1500    484600 _ 24J2W)

A

links = 2A. lPv. 27-6p. the area required. It differs only four tenths of a perch from the area found by offsets.

PROBLEM V.

Lay down a curve line offset from the following dimensions: reduce it to a right-angled triangle by the parallel ruler; and find its area both by equidistant ordinates and casting.

Iv

Having laid down the figure, erect the perpendicular AP for a temporary line.

Lay the ruler from A to E, move it parallel to C, and mark the temporary line at 1.

Lay the ruler from 1 to G, move it parallel to E, and mark the temporary line at 2.

Lay the ruler from 2 to K, move it parallel to G, and mark the temporary line at 3.

Lay the ruler from 3 to M, move it parallel to K, and mark the temporary line at 4.

Lay the ruler from 4 to N, move it parallel to M, and mark the temporary line at 5.

Draw a line from N to 5 (p), and NAP will he the right-angled triangle required.

Computation of the Area by equidistant Ordinates. See Prob. IX. Part III.

Here'the sum of the first and last ordinates is nothing; (180

-f 270 + 190) X 4 = 640 X 4 = 2560, four times the sum of the even ordinates; and (250 + 260) X 2 = 510 X 2 = 7020

twice the sum of the odd ordinates; then

East

M 190 K 260 Gr 270 E 250 C 180

From

AN

1200

1000

800

600

400

200

000

A go

2560 + 1020 + 200

3580 X 200 __ ⁷¹_JOO_ ₌ 238666 square links = 2a. 1r. 21-8p. o

the area required.

Computation of the Area by Casting.

then

2a. 1r. 22p.

Measure the perpendicular AP, which you will find to be 398 links; 398 x 1200 _ 477600 = 238800 square links =

the area required; which differs only two-tenths of a perch from that found by equidistant ordinates.

PROBLEM VI.

It is required to reduce the following curve line offset area to a right-angled triangle by the parallel ruler.

Oi

Erect a perpendicular at one end of the base for a temporary line, and assume a competent number of points in the curve to denote angles.

Lay the ruler from 1 to 3, move it parallel to 2, and mark the temporary line at 1.

Lay the ruler from 1 to 4, move it parallel to 3, and mark the temporary line at 2.

Lay the ruler from 2 to 5, move it parallel to 4, and mark the temporary line at 3.

Lay the ruler from 3 to 6, move it parallel to 5, and mark the temporary line at 4.

Lay the ruler from 4 to 7, move it parallel to 6, and mark the temporary line at 5.

Lay the ruler from 5 to 8, move it parallel to 7, and mark the temporary line at 6.

Draw a line from 8 to 6, and 8 1 6 is the triangle required; hence the area of the irregular offset may be found by multiplying the base 1 8 by half the perpendicular 1 6.

Note.—When a curve-line offset area has to be reduced to a triangle by the parallel ruler, a competent number of points must be assumed in the curve to denote angles. These points must be taken at such distances from each other that a right line drawn between any two adjacent points would nearly coincide with the curve.

PROBLEM VII.

It is required to reduce the irregular figure ABCDEFGHK to a triangle by the parallel rider.

\\

Produce the base AB both ways, at pleasure, for a temporary

line.

Lay the ruler from A to H, move it parallel to K, and mark the temporary line at 1.

Lay the ruler from 1 to G, move it parallel to H, and mark the temporary line at 2.

Lay the ruler from 2 to F, move it parallel to G, and mark the temporary line at 3.

Draw a line from F to 3, and it will be a side of the required triangle.

Again, lay the ruler from B to D, move it parallel to C, and mark the temporary line at 1.

Lay the ruler from 1 to E, move it parallel to D, and mark the temporary line at 2.

Lay the ruler from 2 to F, move it parallel to E, and mark the temporary line at 3.

Draw a line from F to 3, and 3 F 3 will be the triangle required; hence the area of the irregular figure abcdefghk may be found by multiplying the base 3 3 by half the perpendicular fm.

Note.—The method of reducing fields of four or five sides to triangles of equal areas may be seen in Problems XVT. and XVII. Tart I.

PROBLEM VIII.

It is required to reduce the figure ABCDEFGHKLMN to a triangle by the parallel ruler.

•/

\\

Draw the temporary line 1 2 to touch the angle A.

Lay the ruler from A to c, move it parallel to B, and mark the temporary line at 1.

Lay the ruler from 1 to D, move it parallel to C, and mark the temporary line at 2.

Lay the ruler from 2 to E, move it parallel to D, and mark the temporary line at 3.

Draw a line from E to 3, and produce it, at pleasure, for a temporary line.

Lay the ruler from 3 to M, move it parallel to N, and mark the temporary line at a.

Lay the ruler from a to L, move it parallel to M, and mark the temporary line at n.

Lay the ruler from n to K, move it parallel to L, and mark the temporary line at m.

Draw a line from m to K, and produce it, at pleasure, for a temporary line.

Lay the ruler from K to G, move it parallel to H, and mark the temporary line at 1.

Lay the ruler from 1 to f, move it parallel to G, and mark the temporary line at 2.

Lay the ruler from 2 to E, move it parallel to F, and mark the temporary line at 3.

Draw a line from E to 3, and E 3 m will be the triangle required; hence the area of the figure ABCDEFGHKLMN may be found by multiplying the base Em by half the perpendicular 3x.

PROBLEM IX.

It is required to reduce the figure ABCDEFGHKLM to a trapezium by the parallel ruler.

J-3

-/

7; A

i) y

vr

Produce the line AB, at pleasure, for a temporary line.

Lay the ruler from A to L, move it parallel to M, and mark the temporary line at 1.

Lay the ruler from 1 to K, move it parallel to l, and mark the temporary line at 2.

Draw a line from 2 to K, and produce it, at pleasure, for a temporary line.

Lay the ruler from k to G, move it parallel to H, and mark the temporary line at 3.

Lay the ruler from 3 to F, move it parallel to G, and mark the temporary line at 4.

Lay the ruler from 4 to E, move it parallel to F, and mark the temporary line at 5.

Draw a line from 5 to e, and produce it, at pleasure, for a temporary line.

Lay the ruler from E to C, move it parallel to D, and mark the temporary line at 6.

Lay the ruler from 6 to B, move it parallel to C, and mark the temporary line at 7.

Draw a line from 7 to B, and B 7 5 2 will be the trapezium required ; hence the area of the figure abcdefghklm may be found by multiplying the diagonal B 5 by half the sum of the two perpendiculars 7 m and 2n.

PROBLEM X.

It is required to reduce the figure abcdefghklmnpr to a trapezium by the parallel ruler.

ft

/

S3 A

Continue AR for a temporary line.

Lay the ruler from A to C, move it parallel to B, and mark the temporary line at 1.

Lay the ruler from 1 to D, move it parallel to c, and mark the temporary line at 2.

Draw a line from D to 2, and produce it, at pleasure, for a temporary line.

Lay the ruler from 2 to P, move it parallel to R, and mark the temporary line at 3.

Lay the ruler from 3 to N, move it parallel to P, and mark the temporary line at 4.

Draw a line from 4 to n, and produce it, at pleasure, for a temporary line.

Lay the ruler from n to L, move it parallel to M, and mark the temporary line at 5.

Lay the ruler from 5 to K, move it parallel to L, and mark the temporary line at 6.

Lay the ruler from 6 to H, move it parallel to K, and mark the temporary line at 7.

Draw a line from 7 to H, and produce it, at pleasure, for a temporary line.

Lay the ruler from H to G, move it parallel to F, and mark the temporary line at 8.

Lay the ruler from 8 to E, move it parallel to F, and mark the temporary line at 9.

Lay the ruler from 9 to D, move it parallel to E, and mark the temporary line at T.

Draw the line DT, and D T 7 4 will he the trapezium required; hence the area of the irregular figure may be found by multiplying the diagonal D 7 by half the sum of the two perpendiculars Tm and 4 n.

PROBLEM XI.

It is required to draw an equalising line, by the parallel ruler, through the crooked fence abode, so that the two fields which it separates may be reduced to trapeziums.

tr*

71

to

■~f--------j «

Lay the ruler from A to 0, move it parallel to B, and mark the temporary line in KG at 1.

Lay the ruler from 1 to D, move it parallel to C, and mark the temporary line at 2.

Lay the ruler from 2 to E, move it parallel to D, and mark the temporary line at 3.

Draw a line from E to 3 (l), and the irregular figure ABCDEFG will he reduced to the trapezium LEFG, and the irregular figure ABCDEHK to the trapezium lehk ; hence their respective areas may be obtained by measuring diagonals and perpendiculars.

Note 1.—Sometimes the proprietors of adjoining estates agree to straighten crooked fences or brooks by giving and talcing equal quantities of land. When this is the case, you must first measure and plan the ground, then draw the equalising line as directed in the last problem, and take the distance from a to l very correctly by the scale. Measure this distance in the field from the angle a ; range the division line e l, and stake it out; and the work will be completed.

2. It will be advisable to measure, both on the plan and in the field, the parts cuts off on each side by the division-line, in order to prove the work ; for an error committed in dividing land is of serious consequence, if it be not discovered and rectified before the new fence is made.

PROBLEM XII.

It is required to draw an equalising line by the parallel ruler, so that the curved fence which separates the two fields in the following figure may be reduced to a straight fence.

Cm

to

Co

Ci

O'

Lay the ruler from 1 to 3, move it parallel to 2, and mark the temporary line in AB at 1.

Lay the ruler from 1 to 4, move it parallel to 3, and mark the temporary line at 2.

Lay the ruler from 2 to 5, move it parallel to 4, and mark the temporary line at 3.

Lay the ruler from 3 to 6, move it parallel to 5, and mark the temporary line at 4.

Lay the ruler from 4 to 7, move it parallel to 6, and mark the temporary line at 5.

Draw a line from 7 to 5, and it will reduce the figure ABCD to two trapeziums; hence their respective areas may be found by measuring diagonals and perpendiculars.

The following general rule for the parallel ruler will he found of considerable service to learners, and may be easily committed to memory.

GENERAL RULE.

1.    Lay the ruler from the first to the third angle, move it parallel to the second angle, and you will have the first mark on the temporary line.

2.    Lay the ruler from the first mark on the temporary line to the fourth angle, move it parallel to the third angle, and you will have the second mark on the temporary line.

3.    Lay the ruler from the second mark on the temporary line to the fifth angle, move it parallel to the fourth angle, and you will have the third mark on the temporary line.

4.    Lay the ruler from the third mark on the temporary line to the sixth angle, move it parallel to the fifth angle, and you will have the fourth mark on the temporary line.

5.    Lay the ruler from the fourth mark on the temporary line to the seventh angle, move it parallel to the sixth angle, and you will have the fifth mark on the temporary line.

6.    Lay the ruler from the fifth mark on the temporary line to the eighth angle, move it parallel to the seventh angle, and you will have the sixth mark on the temporary line.

7.    Lay the ruler from the sixth mark on the temporary line to the ninth angle, move it parallel to the eighth angle, and you will have the seventh mark on the temporary line.

8.    Lay the ruler from the seventh mark on the temporary line to the tenth angle, move it parallel to the ninth angle, and you will have the eighth mark on the temporary line, &c.

Note.—As the operations of the parallel ruler, in straightening crooked fences, are founded upon a mathematical truth, it is preferable to a lantern horn : but the latter may be applied with much more expedition than the former ; and if it be used by a skilful hand, its results will be sufficiently correct for general practice. (See Problems I. and II. Part IV.)

A BOOK OF DIMENSIONS, CASTINGS, AND AREAS

BELONGING TO PLATE VIII.

Names of Propietors.

No. on the plan.

Diagonals.

First Perpend.

Second Perpend.

Mr Dalton’s Close Mr Cayley’s Close Mr Whisker’s Close Mr Straker’s Close Mr Straker’s Close 'Mr Ellard’s Close

625

180

tq co    iO O

916

742

2094

1855

2197

52

123

160

328

574

247

155

481

268

05    to    ,

oo to o i ®um Perpend.

Quantity in A dec.

Quantity

in

A. B. P.

596

574

1- 12500 1 0 20

2- 73884 2 23 2-06276 2 01

13-42254 13 1 28

11- 05580 11 0 9

12- 61078 12 2 18

O GO

Whole quantity 43-01572 43 0 3

Note 1. In the first edition of this work, the content of the estate in Plate VIII. was found from a plan of 2 chains to an inch. The bases, diagonals, and perpendiculars were measured by the scale used in planning; the offsets taken in the field were used where convenient, and when those were insufficient, more were measured by the scale. In this edition, the crooked fences have been straightened by the parallel ruler, the bases and diagonals measured by a scale of 2 chains to an inch, and the perpendiculars by a scale of 1 chain to an inch ; hence the area of each triangle was found by multiplying the base by the perpendicular, and the area of each trapezium by multiplying the diagonal by the sum of the two perpendiculars.

The diagonals, perpendiculars, and areas are entered in the foregoing Book of Castings; and it may also be observed that the bases of triangles are put down in the column of diagonals, and their perpendiculars in the first column of perpendiculars.

2. In straightening crooked fences by the parallel ruler, it frequently happens that one equalising line will serve for two adjoining fields; and almost every irregular figure may be reduced either to a triangle or a trapezium. (See Problems XVI. and XYII. Part I.; and also the use of the Parallel Euler, Part Y.)

A BOOK OF DIMENSIONS, CASTINGS, AND AEEAS

BELONGING TO PLATE X.

Names of the Fields.

No. on the Plan.

Diagonals.

First Perpend.

Second Perpend.

Sum of Perpend.

Quantity in A dec.

Quantity m A. E. P.

Grime Garth . House Ing. . Sandy Field . Low Holme . Brooke Close . Low Close Marsh Close . Green Meadow Horse Pasture Cow Pasture . Calf Garth Long Meadow Biver Close . Primrose Close Bridge Ing. Shady Ing. Hare Park Long Tongue.

1166

265

95 360

OCOOO^IffiOl^WUl

<7^ CO    lO O    CO

r-H r-H r-H r—I r-H r-H r-H

1581

1098

861

1130

969

1209

846

741

962

725

1781

810

1046

1733

875

880

1092

440

196

259

215

243

336

153

161

150

180

226

cocot^c^>^ N CO CO IQ ^

r—I r-H GT r-H r—I

284

440

97 293

88

fcO OO

t—¹ hH-Oc -A

243

336

200 353 170 331 104 254 133 313 224 450 174 352

210

153

187

167

393

440

346

314

284

4-19760

6- 95640

3- 21714 2-98767 2-42950 2-35467

4- 06224 2-98638 2-45271 2-44348 2-26925 8-01450

2- 85120 4-11078

7- 62520

3- 02750 3-76320 3-10128

05 t-3 OS -J hfi M » to to M lO ft M to LO U O if-

® co o co i—i

OGOObOOCOOl—'¹ t—‘I—‘050

<Meoinooo5t-®cocq!—■ co co co co on .—ii-ncococo

7—H

O    N O

CM    i—l

Whole quantity,

67-85070

co

Note.—In the first edition of this work, the content of the estate in Plate X. was found from a plan of 2 chains to an inch, by making the crooked fences straight by a lantern horn, as directed in Part IV. In this edition, all the crooked fences have been straightened by the parallel ruler, the bases and diagonals measured by a scale of 2 chains to an inch, and the perpendiculars by a scale of 1 chain to an inch; and hence the foregoing Book of Dimensions, Castings, and Areas was formed.

TO TRANSFER A ROUGH PLAN

TO A CLEAN SHEET OF PAPER, OR TO A SKIN OF PARCHMENT OR VELLUM, IN ORDER TO MAKE A FINISHED PLAN ; ALSO TO ENLARGE OR REDUCE PLANS, ETC.

Method I.

y Points.

Having laid the fresh sheet upon a smooth table, lay the rough plan upon it, and with four small nails (or weights or books) fasten the corners of both to the table. Then, with your pricker, pierce the extremities of straight lines, and as much of the curved ones as Avill enable you to draw them on the new plan. Next separate the papers, and trace the outlines and fences with a black-lead pencil, after which draw them with a fine pen and good Indian ink.

Note.—Common ink ought never to be used in planning, because it not only sinks too deep into the paper, but generally, in process of time, becomes discoloured.

Method II.

By Tracing-Paper.

Take a sheet of writing-paper, of the same size as the rough plan, and rub one side of it with black-lead powder; then lay it upon the sheet which you intend for your new plan, with the black side downward ; upon both lay the rough plan, and fasten them all to the table, as before directed. Next run your tracer gently over all the lines upon the plan, so that the black-lead under them may be transferred to the fresh paper. They must then be drawn with Indian ink, as before directed.

Note.—This method of transferring is preferable to the former, because it does not injure the plans.

Method III.

By a Copying-Glass.

A copying-glass is a large square or rectangular piece of the best window-glass, fixed in a frame of wood, which can be raised to any angle, like a desk, the lower side resting upon a table; and a screen of blue paper may be fitted to the upper edge, and stand at right angles to it.

Place this frame at a convenient angle, against a strong light; fix the old plan and clean paper firmly together by pins, the clean paper uppermost, and on the face of the plan to be copied; lay them with the back of the old plan next the glass, namely, that part which you intend to copy first.

The light through the glass will enable you to perceive distinctly every line of the plan upon the clean paper, and you can easily trace over them with a pencil; and having finished that part which covers the glass, slide another part over it, and copy this, and thus continue till the whole be copied.

Note.—Those who have not a copying-glass may use a rectangular piece of window glass fixed in a common frame; and when copying it may be placed in an inclining position, with its top against a window, and its bottom upon the window-seat, if it be nearly level with the bottom of the window. A pane in a window is not unfrequently used for copying small drawings.

Method IV.

By Similar Squares.

The three foregoing methods of transferring or copying plans can only be applied when the rough plan is of the same size which you wish the finished one to be; but, as it may be necessary to reduce the size of the original, this may be done by similar squares.

Example.

Suppose the following inclosures to have been laid down by a scale of 2 chains to an inch; it is required to reduce them to one of 4 chains to an inch.

■*“7

By a scale of 2 chains to an inch, draw the line AB = 7 chains. At A and B erect the perpendiculars AD and BC, each of which

make = 6 chains, and join DC. Divide the lines ab and DC each into 7 equal parts, and the lines AD and BC each into 6 equal parts: join the opposite points of division, and the rectangle ABCD will be divided into 42 equal squares, the side of each being one chain.

Next, by a scale of 4 chains to an inch, draw the line ef = 7 chains. At E and F erect the perpendiculars EH and FG, each of which make = 6 chains, and join hg. Divide the lines ef and hg each into 7 equal parts, and the lines EH and FG each into 6 equal parts, join the opposite points of division, and the rectangle efgh will be divided into 42 equal squares, the sides of which will be exactly half the size of those in the rectangle abcd.

Then, with your pencil, draw within the rectangle EFGH the fences contained within the rectangle ABCD; making each fence pass through its proper situation in the corresponding squares, which may be done by observing where the lines forming the squares intersect the fences. Afterwards trace the fences with Indian ink, as before directed.

Note.—In copying or reducing a large plan by this method, you ought to number the corresponding squares in the circumscribing rectangles with the same figures, in order to prevent mistakes. These figures, as well as the lines forming the squares, should be made with a pencil, and effaced after the plan is copied.

Method V.

By the Pentagraph.

The pentagraph is an instrument for copying, reducing, and enlarging plans. Copies of surveys are also now being taken by a method termed photozincography, but from the many improvements that are annually taking place, it is seldom that a complete copy is called for, all the changes made in roads, fences, &c., requiring to be entered in the new plan. Surveyors will therefore have chiefly to

rely upon their plotting scales, proportional compasses, &c. At the same time, much may often he done by the pentagraph, so that its use should be learned.

DESCRIPTION OF THE PENTAGRAPH.

The pentagraph is generally made of wood or brass, from 12 inches to two feet in length, and consists of four flat bars or rulers, two of them long, and two short, as illustrated in the plate below. The two longer are joined at the end A by a double pivot, which is fixed to one of the rulers, and works in two small holes placed at

Plate V.

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Wash

Pat/our

house

rf

\ii,

\V

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Aik

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J8S

fiVl

kt token

the end of the other. Under the joint is an ivory castor, to support this end of the instrument. The two smaller rulers are fixed by pivots at E and h near the middle of the larger rulers, and are also joined together at the other end G.

By the construction of this instrument, the four rulers always form

a parallelogram. There is a sliding box on the longer arm, and another on the shorter arm. These boxes may be fixed at any part of the rulers, by means of their milled screws; and each of these boxes is furnished with a cylindric tube, to carry either the tracing point, pencil, or fulcrum.

The fulcrum or support K is a leaden weight; on this the whole instrument moves when in use.

To the longer instruments are sometimes placed two moveable rollers, to support the pentagraph and facilitate its motions. Their situation may be varied as occasion requires.

The graduations are placed on two of the rulers, B and D, with the proportions of \, h, &c. to -jY-, marked on them.

The pencil-holder, tracer, and fulcrum must in all cases be in a right line, so that when, they are set to any number, if a string be stretched over them, and they do not coincide with it, there is an error either in the setting or gradations.

The long tube which carries the pencil or crayon moves easily up or down in another tube; there is a string affixed to the long or inner tube, passing afterwards through the holes in three small knobs to the tracing point, where it may, if necessary, be fastened. By pulling this string, the pencil is lifted up occasionally, and thus prevented from making false or improper marks upon the copy.

THE USE OF THE PENTAGRAPH.

To reduce a plan m any of trie proportions, $, l;, \, f, dec., as marked on the two bars b and d. Suppose, for example, \ is required.

care-

Place the two sockets, at \, on the bars B and D, the fulcrum or lead weight at B, the pencil socket with the pencil at D, and the tracing point at C. Fasten down upon a smooth board or table a sheet of white paper under the pencil D, and the original map, &c., under the tracing point c, allowing yourself room enough for the various openings of the instrument. Then with a steady hand fully move the tracing point C over all the lines on the map; and the pencil at D will describe exactly the same figure as the original, but half the size. In the same manner for any other proportion, by setting the two sockets to the number of the required proportion.

The pencil-holder moves easily in the socket, to give way to any irregularity in the paper. There is a cup at the top for receiving an additional weight, either to keep down the pencil to the paper, or to increase the strength of its mark.

PAET V.

193

TO ENLARGE AND COPY PLANS.

A silken string is fastened to the pencil-holder, in order that the pencil may be drawn up off the paper, to prevent false marks when crossing the original plan, in the operation.

If the original should be so large that the instrument will not extend over it at one operation, two or three points must be marked on the original, to correspond with the same upon the copy. The fulcrum and copy may then be removed into such situations as to admit the copying of the remaining part of the original; first observing, that when the tracing point is applied to the three points marked on the original, the pencil falls on the three corresponding points upon the copy. In this manner by repeated shiftings a pen-tagraph may be made to copy an original of ever so large dimensions.

To enlarge a Plan in any of the proportions -J, | ,    <kc.

Suppose -J.

Set the two sockets at as before, and change places of the pencil and tracing point; namely, place the tracing point at D, and the pencil at C.

To copy a Plan the same size as the Original.

Place the two sockets at the fulcrum at D, and the pencil at B. In this case the lines upon the new plan will be reversed in copying.

ISTote 1. There are sometimes divisions of 100 unequal parts laid down on the bars b and D, to give any intermediate proportion, not shown by the fractional numbers.

2.    Pentagraphs of a greater length than two feet are best made of hard wood mounted in brass, with steel centres, upon the truth of which depends entirely the equable action of this useful instrument.

3.    Though I have given various methods of reducing plans, I would advise the learner, after he has found the contents from a plan of 2 chains, or of 1 chain to an inch, to draw another rough plan, of the same size which he intends his finished one to be ; and then to transfer it to a clean sheet by any of the foregoing methods. This may appear a little tedious, but it will make the learner very expert in laying down his lines, which will be found of great advantage to him when he enters upon the Practical Part of Surveying.

TO EMBELLISH A PLAN.

In order to make a neat, finished plan, some knowledge of drawing is absolutely necessary. The learner should also be a proficient in plain and ornamental penmanship; or he will not be able to finish a plan either with beauty or elegance. Every person who would excel in this art should devote all his leisure hours to copying and making out drawings, either from plans or copper-plates well executed, as nothing but practice will make a good draughtsman.

Method I.

Plans neatly finished with Indian InJc and Colours.

Having transferred the plan to a clean sheet of drawing-paper, or to a skin of parchment or vellum, by any of the foregoing methods, draw all the straight lines very finely, by the edge of a ruler, with a drawing-pen and Indian ink ; but the curved lines must be drawn by a steady hand.

Proceed next to make the representation of hedges, bushes, trees, woods, gates, stiles, bridges, the bases of buildings, &c., &c., in their proper places; running a single dotted line, in an open field, for a footpath, and a double one for a carriage-road.

Hills may be shaded with a brush or hair-pencil and Indian ink. The first wash should be weak, and the edges of the shade, particularly at the top and bottom of the hill, must be softened off with clear water, and a clean brush, kept for that purpose, at one end of the pencil-handle ; the other end being occupied by the Indian ink brush.

When the hills are very steep, and rise one above another, as those in Wales, Derbyshire, Yorkshire, Westmoreland, Cumberland, Northumberland, and Scotland, they must all be shaded according to their various inclinations; always letting one wash dry before another is laid on, and never neglecting to soften off the edges of each shade with water.

If some parts of the hills be rocky, tint them with a colour resembling stone, after they have been shaded with Indian ink and a hair-pencil in the manner exhibited in No. 2, Plate VII.

It may also be observed, that when the inclination of a hill is inconsiderable, it is never noticed by surveyors in shading or finishing their plans, and if hills be flat at the top, they are left nearly white.

The method of shading high moorish ground and hilly fields may be seen in Plates VI. and VII.; except they must not be done with lines in imitation of engraving, but with repeated washes of Indian ink.

After hills have been properly shaded with Indian ink, they may then be coloured in the manner hereafter directed for meadow, pasture, and arable land.

Lakes, rivers, brooks, &c., may also be shaded with a brush and Indian ink, pretty strongly at the edges, and softened off towards the middle; and when they are dry, they may be washed over with a light tint of Prussian blue. The shape of arrows should also be made in brooks and rivers, to show in what direction the streams run.

Meadow and pasture ground should be coloured with a transparent green, the pasture rather lighter than the meadow : arable land with various shades of fine brown, so that too many fields may not appear exactly alike; and some surveyors use both red, blue, lake, and yellow in colouring plans.

If the quickwood hedges be not made with a pen and Indian ink, in imitation of bushes, they may be represented by running narrow shades of colouring along the black lines which form the boundaries of the different enclosures.

Roads should be washed with a brownish tint, and the bases of buildings with a red one, or with Indian ink, laid on with a brush of a convenient size, as it is difficult to manage large brushes in shading small spaces.

Sands upon the sea-shore may be washed over with a mixture of brown, lake, and gamboge.

Greens of various shades may be composed of blue and yellow; a pleasing variety of brownish tints may be produced by mixing lake, red, or yellow, with a little brown ; and a shade for water may be formed of Indian ink and Prussian blue.

All the washes should be made thin, and laid on in a very neat manner, as nothing disfigures a plan or map so much as daubing on the colours too thickly.

If the estate be small, the area of each enclosure may be put down in some vacant part of the plan; but if it be large, the areas must either be entered within the fields themselves, or in a book of particulars, which may also contain any remarks that the surveyor may think necessary to make to his employer concerning the estate.

In some convenient parts of the plan, write, in various hands, with Indian ink, the title of the estate, ornamented with a compartment or device. In another vacancy introduce the scale by which the plan has been laid down, and also meridian-line, with the compass or flower-de-luce pointing to the north.

The whole may then be bordered with black lines, at a convenient distance from each other, and the space between them shaded with a hair-pencil and Indian ink. See Plates IX. and XI.

Note 1. If the learner examine a well-finished coloured map of England, or any other country, he will fully comprehend what has been said on the subject of embellishing plans.

2.    Indian ink must always be used in planning ; and as it is frequently of a very bad qualitjq it is advisable to try it before you purchase, by wetting one end of the cake, and rubbing it upon white paper. The blackest and freest is considered the best.

3.    The most convenient colours are those ready prepared in cakes, which must be used in the following manner : Dip one end of the cake in clear water, and rub a little of it upon a clean wedge-wood or earthen plate ; then mix it with water by your hair-pencil, until you have brought it to any consistency you please. Indian ink must be prepared for use in the same way.

4.    The following water-col ours may be enumerated; viz. :—

Vermilion. Prussian Blue. Prussian Green. Sap Green.

Yellow Ochre. Indian Yellow. Light Red. Lake.

Vandyke Brown.

Raw Sienna.

Burnt Sienna.

Gamboge.

By means of these colours a great variety of tints may be formed ; and a little practice will soon enable the learner to produce any shade that may be wanted for plans or maps.

5.    When the price for measuring and planning is very small, surveyors generally finish their plans neatly, but without either colours, compartments, or embellishments of any kind.

6.    Professional surveyors always enter in their field-books the day of the month and date of the year when they begin to survey an estate; and in finishing their plan, they date them accordingly, and also insert their own names, in order that gentlemen may know when and by whom their estates were surveyed.

Method II.

Plans highly finished with Indian Ink and Colours.

The foregoing method of finishing plans is very expeditious, and may suffice when the price allowed for surveying will not admit of much time being spent in making embellishments; but when a highly-finished plan is wanted, the following method must be adopted.

Meadows.

With a pen, or a very fine-pointed hair-pencil, and light Indian ink, make perpendicular and inclining strokes over the whole meadow, as represented in No. 1, Plate VI., and then wash it with a fine transparent green. The strokes must be of vai’ious lengths, but none of them should exceed the 10th part of an inch.

Pasture Ghrounds.

Pastures may be shaded with upright and sloping strokes, of various lengths, as represented in No. 2, Plate VI., and then washed over with a green somewhat inclining to yellow. None of the strokes should exceed the 20th part of an inch in length.

Corn Fields.

By the edge of a ruler, or by the hand, draw (in short dashes) fine parallel lines, at equal distances from each other, so as to give the fields the appearance of being divided into ridges and furrows, as represented in Nos. 3 and 4, Plate VI.; and then wash each field over with a different tint of brown, inclining to yellow.

PLATE YI.

2. Pasture.

1. Meadow.

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3. CW», Field.

4. C'or/i Field.

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5. High Moorish Ground.

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Moors.

With a pen, or a hair-pencil, draw the representation of a few scattered hillocks if there be any on the moor.

Draw also here and there small bushes, to represent heath, broom, whins, and such like brushwood as usually grow upon moors.

Make likewise tufts of grass, if the moor is pasturable, and then fill up all the vacant spaces with perpendicular and inclining strokes, as represented in Nos. 1 and 2, Plate VI.

If the moor be high, hilly, and rugged, with pools of Avater, caverns, roads, &c., it must be shaded with lines in imitation of engraving, as exhibited in No. 5, Plate VI.; if any parts be Avet and marshy, they must be done in the same manner as marshy ground; and if the moor contains large stones, rocks, or trees, they must not be omitted.

When you have finished shading with Indian ink, you must then colour the different parts of the moor in the same manner as they appear in nature. The parts producing herbage must be Avashed with a greenish colour, inclining to blue; the dark parts with a broAvnish tint; the lighter parts with a yellowish one, &c.; and the shrubs and bushes may be touched up with a fine lightish green.

If the moor contains whins, first Avash them with green, and then touch them up on the Avest side with yelloAv, Avhich will give them the appearance of being in blossom.

By proceeding as above directed, a variety of pleasing effects and shades will be produced, and you will be able to give your plan a very fine appearance, and make it resemble even nature itself.

Marshy Ground.

With a pen, or a fine-pointed hair-pencil, and palish Indian ink, draAv, by the hand, shortish horizontal strokes of various lengths, pretty closely to each other. Make also the representation of reeds, rushes, sedges, and strong herbage, as exhibited in No. 1, Plate VII.; wash the whole over Avith a palish green, inclining to blue; and then touch up the reeds, rushes, sedges, &c., Avith a stronger green, which soften off either towards the right or left with a lighter one, or with clear water. (See the Method of Shading Trees.)

Sands and Rocks.

Sands on the sea-shore, &c., must be represented by small dots Avith a pen and Indian ink; loose stones by figures resembling small circles and ovals, but more irregular; and rocks must be made to

1. Marshy Ground.

PLATE VII.

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appear rugged and rough, and to rise in succession, one above another, as exhibited in No. 2, Plate VII. The sands may then be washed over with a mixture of brown, lake, and gamboge; and the stones and rocks coloured with such tints as will give them the appearance of nature. Some stones and rocks are whitish, some yellowish, some greyish, others brownish, &c.; hence the propriety of always taking their real colour into consideration when we intend to give a faithful representation upon a plan.

Trees.

Trees always adorn and beautify the face of nature, and when they are neatly drawn, with a fine pen and Indian ink, they give a plan a very beautiful and pleasing appearance.

They must be made with vertical stems, neat broadish tops, shaded darker on one side than the other, and black horizontal shades at the bottom, as represented in No,. 3, Plate VII.

The lighter parts of the trees represent that side upon which the light is supposed to fall, and the horizontal shades at the bottom are intended to denote the shadows of the trees upon the ground. These shadows must always be made on the darker sides of trees, and also of every other object, where shadows are intended to be represented.

It is not material which side of a tree be left light; but we must take care to make all the trees in the same wood light on the same side; for we cannot suppose that the light can fall on the right of some trees, and on the left of others at the same time.

When a sufficient number of trees have been made to give the wood an agreeable appearance, the vacant spaces must be filled up with small bushes to represent the underwood. The whole wood should then be washed over with a lightish green ; after which the tops of the largest trees may be touched up with a darker green, and with a little brotvn or yellow, in order to produce that pleasing variety of tints which we so often behold and admire in nature.

Note 1. When the Indian ink is not perfectly dry, it will run in washing the wood with green; in order to avoid this, the green wash may be laid on before the trees and bushes are made. — This observation also points out the propriety of colouring fields before the quickwood fences are made with Indian ink.

2.    The tops of trees are formed in various ways. Sometimes they are made with jagged edges, and filled up in the middle with irregular strokes, in different directions, and some surveyors form them entirely by horizontal lines of various lengths.

3.    When trees are small and neatly made, it is unnecessary to touch them up with any colour.

4. Quickwood hedges must be made with a pen and Indian ink, in imitation of bushes, and when trees are properly introduced, they have a very good effect in the hedgerows. (See Plates IX. and XI.)

Lakes, Rivers, and the Sea-Shore.

Water must first be coloured with a fine tint of Prussian blue, and then shaded by a pen and Indian ink, with crooked or waved lines, bold near the edges, and fainter towards the middle, as exhibited in No. 4, Plate VII., which is intended to represent a mere or lake.

Rivers and brooks must also be shaded with waved lines, continued from one end to the other, as represented in Plate XI.; and the seashore in a similar manner, but much stronger and bolder than either lakes or rivers.

Note 1. Some draughtsmen do not wash with Prussian blue until they have finished shading with Indian ink, but it is much better to colour the water before it is shaded, as the ink frequently runs when a wash is laid upon it.

2. Here it may not be improper to observe that in colouring lakes, rivers, &c., with Prussian blue, the wash should be pretty strong at the edges, and softened off with water towards the middle.

Hilly Ground.

Meadow and pasture ground should first be washed with a fine green, and ploughed land with a yellowish brown, as before directed; the hills must then be shaded in lines, with a pen and Indian ink as represented in Nos. 5 and 6, Plate VII.

The sides of hills may be shaded in the manner represented in the lower part of No. 6; and when the top of a hill is level, it must be left almost without shade.

The greater the altitude of a hill, the deeper must be the shade; but the level part of a valley between two hills must be very faintly shaded.

It will add greatly to the beauty of the plan or map if all the hills be introduced in their proper places. When this is the case, and the hills are properly shaded, they form what is called a bird’s eye view ; it being supposed that the eye of the observer is elevated to some distance from the ground.

What has been said on this subject will be fully comprehended by the learner, if he carefully examine the plate to which I have already referred, and also No. 5, Plate VI., which represents a high moorish district shaded in a very neat and expressive manner.

Pleasure Grounds.

In order to draw a true plan of pleasure grounds, it is necessary to measure such lines, in taking the survey, as will enable you to lay

down correctly the shrubberies, grass-plots and fish-ponds, the bases of summer-houses and alcoves, and the turnings and windings of all the gravel-walks, &c.

The trees, bushes, bases of buildings, &c., must then be neatly made ; the fish-ponds and grass-plots properly coloured and shaded, as before directed, for lakes and meadows : and the gravel-walks washed with a fine brown inclining to yellow.

Note 1. If the mansion-house, stables, gardens, &c., be situated within the pleasure grounds, the greatest care should be taken to lay them down correctly, so that the smallest inaccuracy in the plan shall not be perceptible.

2. When pleasure grounds are surveyed and planned with adjoining estates, the same scale must, of course, be used for the whole; but when the former are measured separately, a large scale should be chosen, in order to allow sufficient room to plan . every object distinctly.

Gardens.

Gardens should be correctly and neatly planned, and finished in a tasteful and elegant manner.

The hothouses, greenhouses, grass-plots, gravel-walks, beds, &c., should all be drawn and laid out as they appear in the garden itself.

The divisions between the different beds may be made with short dashes, as represented in Nos. 3 and 4, Plate VI. ; the beds should then be lightly shaded with a pen and Indian ink; rows of bushes inserted along the sides of the walks, and at the divisions of the various beds; and trees should be shown as before directed, if there be any in the garden.

The gravel-walks must then be washed with a yellowish brown ; the grass-plots with green ; and the different beds with a light tint of yellow, red, lake,' blue, green, or any other colours, so as to produce a pleasing variety : and the trees may be touched up with a little dark green, and occasionally a brownish or yellowish tint may be used, to give them an autumnal appearance.

Note. When plans are to be finished with colours, it is not necessary to shade them so much with Indian ink as when they are finished with Indian ink only.

The Bases of Buildings.

The outlines of the bases of buildings must be made with a drawing-pen and Indian ink, bold and black on the south and east sides, or on the north and west sides ; and the spaces in the middle filled up with oblique lines, as represented in No. 7, Plate VII., which is given expressly for the purpose of making the learner fully acquainted with the method of shading the bases of buildings, drawing the plans of villages, towns, &c.

Note 1. When a proprietor wishes to have a plan of his buildings, offices, yards, &c., upon a large scale, the dimensions should be taken in feet and inches, or in 4eet and tenths, which is preferable ; because the chains and tenths of a chain, upon the plotting-scale, may then be considered as feetand tenths, and used accordingly in planning ; or, when it is more convenient, each chain may be called ten feet; consequently, each division will then become one foot. (See Note 3, Prob. I., Part III.)

2.    When it is intended to lay down buildings by a large scale, the thickness of the walls, the lengths and breadths of rooms and passages, the widths of doors and windows, the projections of fireplaces, and other necessary dimensions, should be taken, in order to produce a correct plan.

3.    After the base of a wall has been formed by parallel lines, drawn at such a distance from each other as to exhibit the wall’s thickness, the space between these lines may then be shaded by oblique lines as before directed. The doorways should be left open; the window-bottoms represented by omitting to shade them with oblique lines ; the chimney-bottoms or fireplaces exhibited by making the inside of the wall to project into the room at right angles ; and the steps of the stairs denoted by parallel lines, drawn at proper distances from each other. The insides of the rooms may either be left white or coloured at the option of the draughtsman ; and if it be thought tedious to shade the bases of the walls with oblique lines, they may be done with a brush and Indian ink.

4.    The name of every room, office, yard, &c., must be given, either within the rooms themselves, or in the margin of the plan ; and when the premises are extensive, the names of the rooms, out-offices, yards, &c., will be numerous; there will probably be the kitchen, back-kitchen, parlour, hall, breakfast-room, diningroom, drawing-room, dairy, pantry, stairs, brew-house, wash-house, coal-house, carriage-house, stables, cow-house, calf-house, hog-sty, soil-hole, barn, stable-yard, court-yard, orchard, garden, &c. What has been said on this subject will be easily comprehended by inspecting No. 2, Plate Y., which is the ground-plan of a small house, laid down by a large scale, in order to show the learner how he must proceed with plans of a similar nature.

5.    When premises are to be sold, every convenience should be pointed out on the plan, in order to promote the sale, and it will be found very advantageous to have plans of the cellars and the upper stories, and even the elevations; but this is more properly the business of an architect than that of a land surveyor. Some persons, however, will find it of considerable advantage to obtain a knowledge of both these sciences, as gentlemen frequently want not only plans of their estates, but also architectural drawings of their buildings.

The Elevations of Buildings.

In order to give a perspective view of the elevation of a building, it is necessary to be acquainted with the art of drawing in perspective; but an architectural view may be produced by taking the dimensions of the building, and laying them down by a scale of equal parts.

When it is intended to give the elevation of any buildings belonging to a farm, or the elevation of a mansion-house and offices belonging to a gentleman’s estate, the length from end to end, the perpendicular height from the ground to the eaves, the height of the gable ends, the height and breadth of the chimney-tops, the height and width of the doors and windows, their situations in the walls, and every other necessary dimension, must be measured; then these dimensions, being correctly laid down by a scale, will give an architectural view of the elevation of the building in question.

What has been advanced on this subject will be farther illustrated by referring to No. 8, Plate VII., which is an architectural view of a gentleman’s house, given for the inspection and improvement of the learner. The house itself is built with gable ends, but the roofs of both the wings are hipped at one end, which make a pleasing contrast in the elevation.

Note 1. After thekiutlines of an elevation are drawn, the common method of shading is by a brush and Indian ink, as it is generally thought too tedious to shade with strokes in imitation of engraving. The roof should be shaded pretty strongly at the ridge, and softened off towards the middle with water. It may then be washed with Prussian blue ; and if the washes, both of Indian ink and colour, be light and often repeated, a more agreeable softness will be produced than by laying on only two or three strong washes. When the roof has been thus shaded, lines may be drawn parallel to the eaves, decreasing gradually in their distance from each other towards the ridge, to represent the edges of the slates. If the house be covered with tiles,the lines must be at equal distances from each other; because tiles of different sizes are never laid upon the same house.

2.    If the front of a building project beyond the wings, it must be denoted by making its shadow fall upon one of the wings; but if the wings project beyond the front, the shade of one of them must be made to fall upon the front. (See No. S, Plate VII., where the shade of the front falls upon the right wing; if the wings had projected, the shade of the left wing would have fallen upon the front.)

3.    If a house be built of brick, it may be coloured red ; if of stone, a colour may be chosen to resemble it; and when a roof is covered with grey slates, blue slates, or red tiles, it may be coloured accordingly. Sometimes the front of a building is shaded with Indian ink, the roof tinted with blue, and the stone doorpost, window-jambs, string-courses, chimney-tops, &c., coloured so as to resemble stone. Indian ink, however, is generally used for fronts in preference to any colour, as it is considered to give buildings a much richer appearance.

4.    If there be trees about the buildings, they may be etched with a pen and Indian ink in imitation of engraving ; the ground in front should be properly shaded; the gravel-walks coloured with a light brown ; and if the elevation be bordered with black lines, as in No. 8, Plate VII., the sky may be coloured with a line blue, or shaded with Indian ink.

5.    The elevations of buildings belonging to estates that have been surveyed should be given on vacant parts of the plan as embellishments : it is very seldom indeed that they are drawTn in their true situations, because they would intercept the view of the ground-plot; and besides, they are generally laid down by a much larger scale. The mansion-house of a nobleman, well executed, on a vacant part of the plan of his estate, has a very pleasing effect, and will never fail to gratify the proprietor.

6.    It is almost superfluous to remind the young draughtsman that he should always keep his hands perfectly clean, and off his plans, so as to preserve them

from being in the least soiled in drawing them, as nothing exhibits the carelessness of a draughtsman in a more conspicuous light than seeing his work besmeared with dust, ink, or colours.

Method III.

Plans highly finished with Indian Ink.

A plan highly finished with Indian ink only has a very elegant appearance, and is considered by most persons to excel those done in colours; but the process is very tedious, and requires much time to do it neatly ; however, if the surveyor be well paid for his time, he ought to finish his plans in that manner which is most likely to give satisfaction to his employers.

Many surveyors keep plans by them, finished in various ways, as specimens, in order that gentlemen may have an opportunity of choosing in what manner they will have the plans of their estates executed.

Shading ivith the Pen.

In finishing a plan with Indian ink, a fine pen ought to be used ; and the fields should be shaded in a great variety of forms, in imitation of engraving, as exhibited in Plates VI., IX., and XI.

Some fields should be done lighter, and others darker, so as to produce a pleasing contrast of light and shade. Some may be executed in such a manner as to resemble corn-fields, as in Nos. 1 and 6, Plate IX., and 13 and 16, Plate XI., and others may be shaded like meadow and pasture, as exhibited in Nos. 1 and 2, Plate YI.

High moorish ground should be shaded as represented in No. 5, Plate YI.; and marshy grounds, sands, loose stones, rocks, trees, water, hilly fields, and the bases of buildings, as denoted in Plate YII.; and even the elevations of buildings look very elegant when they are finely shaded with lines, as No. 8 in the Plate to which we last referred.

Note. In finishing a plan with Indian ink only, it is necessary to shade it much closer and deeper than in finishing with Indian ink and colours.

Penmanship.

In making finished plans, no ornaments or embellishments will compensate for bad penmanship.

Writing, German-text, printing, and figures, are all essentially necessary for a draughtsman ; and whoever would excel in the art of planning should use his utmost endeavours to become a complete and elegant penman.

He should practise the various hands, either by copies well written or by good copper-plates, until he can make all the letters and figures

correctly and with true taste; and it will save him much trouble in making compartments and devices if he can acquire the art of flourishing and ornamenting neatly and elegantly with the pen.

Ornaments.

Any compartment or device may be chosen to fill up the vacant corners of a plan, such as the compass, scrolls of paper, wreaths or festoons of leaves and flowers, branches or sprigs of oak, palm-tree, weeping-willow, myrtle, laurel, olive, &c. ; also shields, coats of arms, columns supporting vases or urns, mathematical instruments, cattle, sheep, or whatever else may please the fancy of the draughtsman.

Ornaments on Plate IX.

In the N.W. corner is a device formed of an oak branch, leaves and acorns on the left side; and on the right side is a branch of large pointed leaves resembling sedges or sweet flags, intertwined with a string of small leaves; and both branches are united at the bottom by a bunch of riband.

In the S. W. corner is a scroll of paper, supported by a fluted column • by the side of which are some ears of corn, and at the bottom a few blades of grass and herbage.

In the N.E. corner is the sun in his meridian splendour, with a fancy device resembling an ogee cornice, fronted with reeds, and from each end of the cornice is suspended a festoon of small leaves.

In the S.E. corner is a plotting-scale, a pair of compasses, two drawing-pens, and a writing-pen, interwoven with a garland of small leaves and berries, resembling those of the myrtle.

Ornaments on Plate XI.

In the N.W. corner is a fancy device in the form of an oval, and in the N.E. corner is a rectangular device, with the exception of the arch at the top. This device is ornamented with a bunch of riband, and two festoons of small leaves and berries, hanging upon two scutcheons or shields.

In the S.W. corner is a column, at the top of wdiich is a vase encircled with leaves and flowers. On the west of the column Britannia is seated, leaning on her shield, holding a spear in her right hand, and with her left hand pointing out the science of surveying. To the east of the column are two sheep, emblems of agriculture.

The plotting-scale, drawing-pens, &c., are nearly similar to those in the last plate.

PART V.

MISCELLANEOUS INSTRUCTIONS.

207

In the S.E. corner is a parallel ruler, a plane table, a terrestrial globe, a crowing cock, and a youth seated upon a beehive, with a pair of compasses in his hand, at work upon Plate XII.

The cock is an emblem of early rising, and the beehive may be considered as an emblem of industry; and it may here be remarked, that it is impossible to attain eminence in the art of surveying without early rising, industry, and perseverance.

MISCELLANEOUS INSTRUCTIONS

Relating to Surveying, Planning, Casting, Valuing, &c.

1.    The title of a plan should set forth the name of the proprietor, and also the name of the township, hamlet, parish, and county in which the estate is situated.

2.    The names of the adjoining lordships, or the names of the proprietors of the adjoining lands, should be given on the plan, in order to point out clearly the situation of the estate and corroborate the title.

3.    All principal roads passing through the estate, from one highway to another, should be laid down, and the places to which they lead specified.

4.    All footpaths and bridle-roads should be pointed out, in order to determine the public right, and guard against encroachments.

5.    All occupation and privileged roads through adjoining estates should be noticed either on the plan or in the reference book.

6.    All ancient highways leading through the estate, although not now in use, should be particularly specified, and the names of the proprietors given, to show in whom the privilege of reopening them, if necessary, is vested.

7.    The ancient and proper names of fields should be preserved, as it generally creates confusion and mistakes when new ones are assigned without sufficient authority.

8.    It has already been observed, that the extremities of the ditches are generally the boundaries between adjoining fields ; this, however, is not always the case, as the stem of the quickwood sometimes forms the boundary; hence the necessity of obtaining an assistant who is well acquainted with all the local customs of the place.

9.    The greatest care must be taken to find the area of each field correctly; and particularly if the survey be taken for an enclosure, or to make a valuation for the land-tax, poor-rates, county-rates, and other assessments; for it is evident that if the survey be incorrect, the valuation can never be equitable, and will consequently produce nothing but disputes and dissatisfaction among the proprietors and occupiers, instead of peace, harmony, and friendship.

10.    In valuing for an assessment, great care should be taken not to over-rate the land that is of a poor quality and lies far from the means of improvement; for bad land costs the occupier as much in labour and seed as good land, and is far less productive. (See more observations on valuing land in Part VI.)

11.    In reducing a plan for portable use, care should be taken to choose a scale sufficiently large to exhibit all the irregularities in the fences, buildings, &c.

12.    Several small farms, or detached pieces of land, belonging to one jDroprietor, may be laid down upon the same sheet. They ought not, however, to be joined together, but planned as separate, estates.

13.    When one sheet of drawing-paper is too small to contain the survey, two or more must be neatly pasted together; and when those parts that have been wet with the paste are nearly dry, they may be made smooth by a warm iron. The edge of one of the sheets should be cut even, and laid nearly half-an-inch over the edge of the other sheet; and a piece of clean paper should be laid under the iron, to prevent it from soiling the plan.

14.    It has already been observed that the surveying-chain should frequently be measured. The readiest method of doing this is to drive two stakes or pins into the ground, exactly at the distance of 22 yards from each other. Professional surveyors measure their chains in this manner every morning, when they are engaged in extensive measurements. When the chain has become too long, it is better to cut a little from several of the links than to take off the rings; care, however, must be taken to keep each 10 links of an equal length, or the dimensions will be incorrect.

15.    The book of {particulars before mentioned is generally called ‘A Terrier of the Survey,’ and should contain references corresponding to those upon the plan; also the name of each field, or the name of the proprietor, or of the occupier, and the area of each field in acres, roods, and perches. If the surveyor value the estate, the terrier ought to contain the value per acre to let or for sale ; the annual value of each field to let, or the total value for sale; and also the cultivation of each field : thus will the proprietor be furnished with every necessary particular relating to his estate.

16.    The terrier may likewise contain remarks and observations on the quality of the soil; and point out the method of improving wet marshy grounds, by draining them; commons and waste lands, by enclosing them; large fields, by dividing them, &c.

17.    Some surveyors return three measurements of each field in the terrier; viz., the land in cultivation, the hedges and waste land, and the total quantity or sum of both.

18.    Arable lands should be distinguished from permanent meadows, and those which the tenant is prohibited from breaking up should be particularly noticed.

19.    In writing out a valuation book for the purpose of making assessments, all the lands and tenements in the occupation of the same tenant should be collected together, and put down on the left-hand page of the book. At the top of the page must appear the name of the tenant; and in the first and second columns respectively, the names of the pr<aprietors and the numbers on the plan. The third fourth, fifth, and sixth columns must contain the name, measurement, value per acre, and total value of each field respectively. The right-hand page may be left blank for incidental remarks, when a change of occupation takes place, or when any circumstance occurs that affects the arrangement of the book.

20.    When the valuation is high, it is frequently thought prudent to calculate the assessments from one-fourth, one-half, or three-fourths of the amount; this, however, is more properly the consideration of the occupiers than that of the land-surveyor. Sometimes the assessments are calculated from one-half or three-fourths of the valuation of the land, and from one-fourth of the valuation of the buildings.

A Terrier of the Survey in Plate IX.

CD •—'    I Nos. on the Plan.

Names of the Fields.

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Value per Acre to rent.

Total Value per Annum to rent.

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Note 1. The annual value of each field may be found from the area, and the value per acre by the Rule of Three.

2.    If one tenant occupy all the foregoing estate, his rent will be 821. 2s. 9\d. per annum ; and if the assessments be made from three-fourths of the annual value, he will be assessed at 621. 12s. Id.

3.    The terrier may be divided into any number of columns, to suit the purpose of the surveyor; and when the observations, remarks, &c., are too numerous to be contained in the columns of one page, each two opposite pages may be divided into columns, in which may be entered every necessary information relating to the estate.

4.    In extensive surveys and valuations, an alphabetical index should be annexed to the terrier or valuation-book, in order that the name of any particular proprietor or occupier may be more readily found.

PART VI.

FOR

RULES AND DIRECTIONS FOR APPORTIONING LAND ;

LAYING OUT ANY GIVEN QUANTITY OF LAND, IN ANY PROPOSED FIGURE; FOR PARTING FROM ANY FIELD OR FIGURE ANY QUANTITY OF LAND REQUIRED ; AND FOR DIVIDING A PIECE OF LAND AMONG SUNDRY CLAIMANTS IN THE PROPORTION OF THEIR RESPECTIVE CLAIMS, OR A COMMON, ETC., OF VARIABLE VALUE AMONG ANY NUMBER OF PROPRIETORS, IN THE PROPORTION OF THEIR RESPECTIVE INTERESTS.—LANDS INTERSECTED BY A RAILWAY-VERTICAL SURVEY OF EMBANKMENTS, RIVERS, CANALS, AND LANDS RECLAIMED FROM THE SEA.—ALSO, THE iMETHOD OF ' REDUCING STATUTE MEASURE TO CUSTOMARY, AND VICE VERSA.

SECTION I.

Rules and directions for laying out any given quantity of land in any proposed figure, and for parting from any field or figure any quantity of land required.

When the land to be laid out or parted off is given in acres, roods, and perches, it must first be reduced into square links; in which process the following table will be' found extremely useful.

When it is required to part off from any field or figure any quantity of land, it is generally necessary, first, to measure the whole, if the dimensions be not given.

SECTION I.

COMPUTATION TABLE.

211

A Table for Reducing Acres, Roods, and Perches into Square Links.

Acres.

Sq. Links.

Perches.

Sq. Links.

Perches.

Sq. Links.

r—I

O«DC0<lC5&(rf>.WWH

o o

<M CO

o o o o

NOOCSO

Roods.

r—I (N CO

100000 200000 300000 400000 500000 600000 700000 8OOO0O 900000 1000000 2000000 3000000 4000000 5000000 6000000 7000000 8000000 9000000 10000000 Sq. Links.

25000

50000

75000

HlMCO-^10 5Dl>oOC50TH(MCOTHCHDt>OOaJO

i—I i—I r—(i—I i—1 r—4 r—1 l—I i—It—I (M

625

1250

1875

2500

3125

3750

4375

5000

5625

6250

6875

7500

8125

8750

9375

10000

10625

11250

11875

12500

bO TO bO I—¹

CO

CO CO

CO

ox

ZD b- 00 CO CO CO

05

co

13125

13750

14375

15000

15625

16250

16875

17500

18125

18750

19375

20000

20625

21250

21875

22500

23125

23750

24375

PROBLEM I.

To reduce any number of acres, roods, and perches, into square links.

Rule.— Beduce the given quantity of land into perches, which multiply by 625, the number of square links in one perch, and the product will be the square links required. Or, find the equivalents of the acres, roods, and perches respectively in the foregoing table.

Examples.

1. Reduce 6a. 3r. 25p. into square links.

5525

2210

6630 690625 Ans.

2. Required the number of square links in 96a. 2r. 36p.

Ans. 9672500.

PROBLEM II.

To lay out, in a square, any quantity of land proposed.

Rule.—Extract the square root of the proposed area, and it will be the side of the square required.

Examples.

1. Lay out in a square 7 acres, 1 rood, and 24 perches, sq. links.

7a. = 700000 1r.= 25000 24p. = 15000

740000(860'2 links, the side of the square.

64

By the rule. A. R. P.

6 3 25

27

40

By the table, sq. links.

6 A. = 600000 3r. = 75000 25p.= 15625

690625 Ans.

Or Oi

166)1000

996

17202)40000

34404 .5596

In laying out the square in the field, let ab represent one of its sides, which make = 860*2 links. At A erect the perpendicular AD, which make = AB; and at B erect the perpendicular BO, which make also = AB. Then measure the line CD, and if you find it = 860*2 links, the work is right.

2. Required the side of a square which shall contain 15a. 2r.

32p.

A ns. 1253 links.

PROBLEM III.

Upon a given line, to 7naJce a rectangle that shall contain any proposed quantity of land.

Rule.—Divide the proposed area by the given side, and the quotient will be the other side of the rectangle.

Examples.

1. Lay out 3a. 3r. 26p. in the form of a rectangle, one side of which must be 850 links.

sq. link?.

3 A. = 300000 3r. = 75000 26p. = 16250

85,0)39125,0(460*3 links, the other side. 340 ‘

Or Cn

.T250

255

In laying out the rectangle in the field, let AB represent the given side. At A, erect the perpendicular AD, which make = 460*3 links; and at B, erect the perpendicular BC, which make = AD. Then measure the line CD, and if you find it = AB, the Avork is right.

Ans. 1265*5 links.

2. If one side of a rectangle be 525 links, required the other side, so that the figure may contain 6a. 2r. 23p.

PROBLEM IY.

To lay out any given quantity of land in a rectangle, so that one of its sides shall he two, three, four, or any number of times as long as the other.

Rule.—Divide the given area by the given number, and the square root of the quotient will be the shorter side, which multiply by the given number, and the product will be the longer side.

Examples.

1. Lay out 3a. Or. 32p. in the form of a rectangle, one of the sides of which shall be twice as long as the other.

sq. links.

3a. = 300000 32p. = 20000 2)320000

160000(400 links, the shorter side.

16__2

. .0000 800 links, the longer ditto.

Let ABCD represent the rectangle, which you must lay out according to the directions in the last problem; AD being 800 and AB 400 links.

2. A rectangle contains 7a. 2r. Op.; what are its sides, one of them being three times the length of the other 1

A ns. 1500 and 500 links.

PROBLEM Y.

Upon a given base to lay out a triangle that shall contain any given number of acres, &c.

Rule.—Divide the area by half the base, or twice the area by the whole base, and the quotient will be the perpendicular of the triangle.

Examples.

1. Lay out 3a. 2r. 16p. in the form of a triangle, the base of which must be 1200 links.

sq. links. 3A. - 300000 2r. = 50000 16p. = 10000

6,00)3600,00

600 links, the perpendicular.

Upon any part of the given base AB, suppose at D, erect the perpendicular DC, which make = 600 links; then stake out the lines AC and BC ; so will ABC be the required triangle. But if the perpendicular be erected at either end of the base, as at B, then the line AE must be staked out; and ABE will be the triangle required.

Ans. 865-2 links.

2. Required the perpendicular of a triangle, which contains 6a. 2r. 37p., its base being 1556 links.

PROBLEM VI.

To lay out a trapezium that shall contain any number of acres, dc., having one of its sides or a base line given.

Rule 1.—Divide the given area into two parts, either equal or unequal; and then, by the last problem, find the perpendicular that will lay out one of these parts in a right-angled triangle upon the given base.

You must then consider this perpendicular as one of the diagonals of the trapezium, and also the base upon which you must lay out the other triangle.

Rule 2.—Divide the given area into any two parts as before; and then find the perpendicular that will lay out one of these parts in a right-angled triangle upon the given base.

Add the square of the perpendicular thus found to the square of the given base, and the square root of the sum will be the hypotenuse. Consider this hypotenuse as one of the diagonals of the trapezium, and also the base upon which the other triangle must be laid out.

Examples.

1. Lay out 8 acres in a trapezium, upon a given side of 800 links.

BY THE FIRST RULE.

Divide the given area into 5 and 3 acres, and let the triangle upon the given side contain the greater part. oa. = 500000 square links.

8,00')! 0000,00

1250 links, the perpendicular of the first triangle, and also the base of the second.

3a. = 300000 square links.

500

125,0)60000,0(480 links, the perpendicular of the second triangle.

1000

1000

.... 0

c.

IV

a/-

In laying out the trapezium in the field, let AB represent the given side. At B, erect the perpendicular BO, which make= 1250 links. Then, upon any part of the line BO, as at D, erect the perpendicular HE, which make = 480 links. The four outlines being properly staked out, the work will be completed.

BY THE SECOND RULE.

5a. = 500000 square links.

8,00)10000,00

1250 links, the perpendicular of the first triangle.

Then, v/1250'- + SOU⁷ = V1562500 + 640000 = V2202500 = 1484 links, the hypotenuse of the first, and also the base of the second triangle.

3a. = 300000 square links.

triangle.

1484)600000(404-3 links, the perpendicular of the second 5936

. .6400

5936

•4640 4452 .188

Having laid out the triangle ABC as before directed, upon any part of the line AC, as at D, erect the perpendicular DE, which make = 404⁻3 links.

Stake all the outlines, and the work will be completed.

2. Lay out 12a. in a trapezium, upon a given side of 1400 links.

Ans. Supposing the given area divided into 7 and 5 acres : then, by the first Rule, ' the perpendicular of the first triangle is found to be 1000 links, and that of the second the same.    /

By the second Rule, the perpendicular of the first triangle is found to be 1000    /

links, the base of the second 1720 ‘5, and its perpendicular 581-2 links.

PROBLEM VII.

Upon a given base, to lay out a rhombus of any content less than the square of the base.

Rule.—Divide the content by the base, and the quotient will be the perpendicular. Then from the square of the base subtract the square of the perpendicular, and find the square root of the remainder. Upon the base, from one of its extremities, measure a line equal to this root, and at this point erect a perpendicular.

Examples.

1. Lay out in a rhombus 5a. 2r. 16p. its base being 800 links.

sq. links. 5a. = 500000 2r. = 50000 16p.= 10000

8,00)5600,00

700 links, the perpendicular.

Then, V8d0² - 700² = V⁶⁴⁰⁰⁰⁰ - 490000 = V150000 = 387-3 links, at which distance from one of the extremities of the base the perpendicular must be erected.

In laying out the rhombus in the field, let AB represent the given base. From A, on the line AB, measure 387‘3 links to D; and at D erect the perpendicular DE, which make = 700 links. At E, erect the perpendicular EC, which make = the base AB. Measure the lines CB and AE, and if you find each of them = AB, the work is right.

2. Lay out a rhombus, which shall contain 6a. 1r. 8p. upon a base measuring 900 links.

Ans. The perpendicular is found to be 700 links, and the distance at which it must be erected from one of the extremities of the base 565-7 links.

PROBLEM VIII.

To lay out any given quantity of land in a circle.

Rule 1.—If we multiply the square of the diameter of any circle by -7854, the product will be the area; consequently, if we divide the area by -7854, the quotient will be the square of the diameter.

2. Multiply the square root of the area by l-12837, and the product will be the diameter.

Examples.

1. Lay out one acre of land in a circle.

sq. links.

-7854)100000'000000(127323-65 links, the square of the

7854

diameter.

214G0

15708 .57520

54978

.25420

23562 .18580

15708 .28720

23562

.51580

47124 .44560

39270 .5290

Then, a/127323'65 = 356⁻82 links, the diameter of a circle containing one acre of land.

Or, by the second rule, VlOOOOO = 316-227; and 316-227 x 1-12837 = 356-82 links, the diameter, as before.

Aj-

In laying out the circle in the field, provide a strong cord, in length equal to the radius of the circle, which in this case will be 178‘4 links; and fixing one of its ends at B, as a centre, make the other fast to your offset-staff near its lower extremity. Then stretch the cord to A, and with the line ab describe the circle, keeping the staff perpendicular, and making a mark on the ground with its spike, by which you must stake out the circumference.

Or at proper intervals stretch the radius AB, and put down stakes in such a manner as to form the circumference.

2. Required the diameter of a circle, which will contain half acre of land.

an

Ans. 252-3 links.

PROBLEM IX.

To lay out any given quantity of land in a regular polygon.

Rule.—Divide the area of the required polygon by the area standing opposite to its name in the subjoined table, and the square root of the quotient will be the length of the side. Multiply the side thus found by the polygon’s number in the column of radii, and the product will be the radius of the circle circumscribing the required polygon.

A table of regular polygons, with their areas, and the radii of their circumscribing circles, when the side of the polygon is 1.

No. Sides.

Names.

Areas.

Kadii.

CQ^lOCONOOOSOHtM j-H T—i r—i

Triangle

Square .

Pentagon

Hexagon

Heptagon

Octagon

Nonagon

Decagon

Un decagon

Duodecagon

0-433

1-

1-72

0-577

0-707

0-851

2- 598

3- 634

4- 828 6-182 7-694 9-365

11.196

1-

1-152

O CO H CO CO ^ CO t'- Oi

CO

to Of

Examples.

Note. If the square of the side of any polygon be multiplied by the area standing opposite to its name in the preceding table, the product will be the area of the polygon.

1. Lay out 1 acre of land in a regular hexagon. 1' 0000

Here

2-598

A>.

= 38491-147; and V3849D147 = 196T91 links,

the side of the required polygon, and also the radius of the circumscribing circle, because the side of a regular hexagon \ and the radius of its circumscribing circle are always equal to each other; hence the multi-plier in the table is 1.

To lay out the hexagon in the field, draw the circumscribing circle as directed in the last problem. Then the radius ab, which is equal to the side of the

JS'B

hexagon, being applied six times, will just go round the circumference, and form the polygon required.

2. Lay out half an acre of land in a regular octagon.

Ans. The side of the required octagon is 101'76, and the radius of its circumscribing circle 132-898 links.

PROBLEM X.

To lay out any given quantity of land in an ellipsis, having one of the diameters given.

Rule.—If we multiply the rectangle of the two diameters of an ellipsis by -7854, the product will be the area; consequently, if we divide the area by ’7854, and that quotient by the given diameter, the latter quotient will be the diameter required.

Examples.

1. Lay out an ellipsis, which shall contain one acre, with a transverse diameter of 450 links.

sq. links.

*7854)100000-00000(127323-6 quotient.

7854

21460

15708 .57520

54978 .25420

23562

.18580

15708

.28720

23562

.51580

47124

.4456

Then,

127323-6

450

= 283 links, the conjugate diameter.

By Prob. XV., Part I., construct the ellipse ABCD; then by a property of the ellipse, the square of the distance of the focus from

the centre is equal to the difference of the squares of the semi-diameters : hence, we have A'225² - 14P5² = ^30602-75 = 175 links, equal Fo, or fo : and 225 - 175 = 50 links, equal AF, or b/.

Again, by another property of the ellipse, the sum of the two lines

transverse that is, Fm + fm —

drawn from the foci, A and meeting in any point in the circumference, is equal to the diameter;

AB.

Procure, therefore, a cord, and upon it make two loops, so that the distance between them may be equal to the transverse diameter ; then measure in the field the diameter AB, putting down a stake at each focus, and one at the centre o. At o erect the perpendiculars oc and oT>, making each = 141’5 links.

Put the two loops over the stakes at F, /, and stretch the cord so that the two parts Fm, fm, may be equally tight; at m put down a stake as one point in the circumference of the ellipse, and in the same manner determine as many others as you please.

But if the ellipse be very large, so that you cannot conveniently procure a cord as long as the transverse diameter, you must then erect perpendiculars, called ordinates, at every 50 links, or at every chain’s length, &c., upon that diameter, and measure the lengths of these perpendiculars by the scale.

Then measure in the field the transverse and conjugate diameters, and erect the perpendiculars in their proper places, always remembering to put down a stake at the end of each perpendicular.

2. Lay out an ellipse which shall contain 8a. 3k. 8p., one of the diameters being given equal to 800 links.

Ans. The other diameter is = 1400 links.

PROBLEM XI.

To part from a square or rectangle any proposed quantity of land by a line parallel to one of its sides.

Rule.—Divide the proposed area by the side upon which it is to be parted off, and the quotient will be the length of the other side of the figure required.

Examples.

1. From the square ABCD, containing 6a. 1r. 26p., part off 3a. by a line parallel to AB.

i)

sq. links. 6 A. = 600000 1r. = 25000 24p. = 15000

640000(800 links, the side of the square.

64

. . 0000

300000

= 375 links, the side AE or BF required.

2. From the rectangle ABCD, containing 8a. 1r. 24p., part off 2a. 1r. 32p. by a line parallel to AD = 700 links. Then, from the remainder of the rectangle, part off 2a. 3r. 25p. by a line parallel to AB.

D

Then,

800

sq. links.

2 A. = 200000 1r.= 25000 32p. = 20000 7,00)2450,00

.350 links, the .side AE or DF.

sq. links.

8 A. = 800000 lR. = 25000 24p. = 15000

7,00)8400,00

1200 links, the side AB.

350

850

the side AE. the difference = EB.

sq. links.

2 A. = 200000 3r. = 75000 25p. = 15625

850)290625(342 links, the side EG or BH. 2550

.3562

3400 .1625

1700

3. Part off 6a. 3r. 12p. from a rectangle containing 15a. by a line parallel to the longer side, the shorter being 1000 links.

Ans. The longer side of the given rectangle is 1500, and the shorter side of the rectangle required is 455 links.

PROBLEM XII.

To part from a square or rectangle any proposed quantity of land, either in a right-angled triangle or trapezoid, by a line draivn from any of the angles to either of the opposite sides.

Rule.—When the proposed area is to be parted off in a triangle, divide double this area by the base or side upon which it is to be parted off, and the quotient will be the perpendicular.

When the proposed area is to be parted off in a trapezoid, subtract it from the area of the square or rectangle, and part off the remainder in a triangle, as above directed.

Examples.

1. From ABCD, representing a square whose side is 900 links, part off a triangle which shall contain 2a. 1r. 36p. by a line drawn from the angle B to the side AD.

2a. 1r. 36p. =247500 square links.

9,00)4950,00

550 links, the perpendicular AE.

Hence ABE is the triangle required.

2. From ABCD, representing a rectangle whose length is 1265 and breadth 758 links, part off a trapezoid which shall contain 7A. 3r. 24p. by a line drawn from the angle b to the side CD.

sq. links

958870 the area of the rectangle

790000 ditto of the trapezoid

168870 difference, the area of the triangle

758)337740(445⁻5 links, the perpendicular CE.

3032 Hence abed is the trapezoid required .3454 3032

.4220

3790

•4300

3790

.510

3. From a rectangular field, whose length is 1560 and breadth 1000 links, it is required to part off a trapezoid which shall contain

12a. 3r. 12p. by a line drawn from any of the angles to the longer opposite side.

Ans. The area of the rectangle is 15a. 2r. 16p.; consequently the area of the triangle is 2a. 3r. 4p., and its perpendicular 555 links.

PROBLEM XIII.

To part from a triangle, upon the base or longest side, any proposed quantity of land by a line drawn from either of the angles at the base to the opposite side.

Rule.—Divide twice the proposed area by the base upon which it is to be parted off, and the quotient will be the perpendicular.

Or, if the proposed area be divided by half the base, the quotient will be the perpendicular.

Note.—A parallel ruler may be used with advantage in this and several of the following problems.

Examples.

1. From ABC, representing a triangle whose base AB is 1200, and sides AC and BC 1000 and 800 links respectively, part off 2a. 2r. 24p. by a line drawn from the angle B to the side AC.

2a. 2r. 24p. = 265000 square links

12,00)5300,00

441’6 links the perpendicular DE.

At A erect the perpendicular AF, which make = 441-6 links; then draw fe parallel to ab, and it will intersect the side AC in the point to which the division-fence BE must be made.

Or, by the plotting scale, erect the perpendicular de = 44L6 links, which will determine the point E.

By the scale you will find AE = 664 links; measure, therefore, in the field, 664 links from A to E ; stake out the line BE, and ABE will be the triangle required.

2. From ABC, representing a triangle whose base AB is 1300, and sides BC and AC 1100 and 900 links respectively, part off lA. 3r. 36p.

From the three sides, by Note 4, Part IV., the area of the given triangle is found = 488076 square links and 1a. Or. 36p. = 197500 square links The difference = 290576, the area of the triangle abe

1300)581152(447 links, the perpendicular de. 5200

.6115

5200 .9152 9100

.. 52

By the mode described in the last example, determine the point E, which you will find at the distance of 658 links from the angle B ; measure this distance in the field from b to e, and proceed as before.

3. From a triangular field whose sides are 1500, 1200, and 1000 links respectively, part off 3a. 2r. 16p. by a fence made from the greater angle at the base to the opposite side.

Ans. The perpendicular of the triangle required is found to be 480 links ; and it rises upon the base at the distance of 537 links from the less angle.

PROBLEM XIV.

To part from a triangle any proposed quantity of land by a line parallel to any one of its sides.

Rule.—The areas of similar triangles are to one another in the duplicate ratio of their homologous sides; hence, as the area of the triangle ABO is to the square of the side AC, or BC, so is'the area of the triangle DEC to the square of the side DC, or EC. (See Theo. XIII. Part I.)

Examples.

1. Suppose the base AB = 1200, the side AC = 1000, and the side bc = 800 links; part off 1a. 2r. 16p. by the line de parallel to ab.

ABC = 396863 square links and lA. 2r. 16p. = 160000 square links

From the three sides, by Note 4, Part IV., we find the area of the triangle.

The difference = 236863, the area of the triangle DEC.

Then, as 396863 : 1000 X 1000 :: 236863 : 596838-20; and V596838-20 - 772-5 links = DC; hence 1000 - 772-5 = 227-5 links = AD. Again, as 396863 : 800 x 800 : : 236863 : 381976-45 ; and v/381976-45 = 618 links = EC; then 800 - 618 = 182 links = BE.

Measure therefore in the field 227-5 links from A to d ; and from B to E measure 182 links; stake out the line DE, and the work will be completed.

2. From a triangular field, whose sides are 1800, 1500, and 1200 links respectively, part off 3a. 2r. 32p. by a line parallel to the shortest side.

Ans. The area of the given triangle is 892941 square links; the area of the triangle made by the line of division is 522941 square links ; and one of its sides from the angle opposite the line of division to the commencement of that line, is 1147-9, and the other 1377-4 links.

* PROBLEM XV.

To part from a rectangle or triangle any proposed quantity of land upon a line on which there are offsets, when the area of those offsets is to be considered as part of the portion to be parted off.

Rule.—Find the area of the offsets, which subtract from the portion to be parted off, and then proceed with the remainder as directed in the preceding problems.

But, in a rectangle, when there are offsets on one or both of the lines, adjoining that upon which the given quantity is to be parted off, reject these offsets, and proceed as before directed.

Then, having found the distance at which the line of division must be from that upon which the given quantity is to be parted off, find the area of the offsets contained between those lines, which area divide by the latter line, and the quotient will be the distance by Avhich the former line must be approximated to the latter.

Examples.

1. From a rectangular field whose dimensions are contained in the following notes, part off 2a. 3r. 32p., upon the chain-line ab, so that the offsets taken upon that line may be included.

DA 560

L. off D

CD 1200 1000 L. off C

BC 560

L. off B

AB

1200

1000

900

600

300

000

Begin at

H O Or OO os O § ooo

<D

bO

’b

A¹

sq. links 2a. 3r. 32p. = 295000

57000 the area of the offsets

12,00)2380,00 the difference

198 A links = ae, or BF.

Hence the irregular figure agbfe contains 2a. 3r. 32p.

2. From a rectangular field whose dimensions are contained in the following notes, part off 2a. 2r. 8p. by a line parallel to the chainline AB, so that the offsets taken upon this line, and also those upon the two adjoining lines contained between the chain-line AB and the line of division may be included.

DA

owaoo

os or Or o o o

to

Or oi O O

000

RAoff D

CD 1000 R. off C

BC

500

400

250

no o

150

000

R. off B

• rH

Cl

CD

QJ

ran ire W.

=2V

<A

sq. links 2a. 2r. 8p. = 255000

40250 the area of the offsets taken on AB 1,000)214,750 the difference

214‘750 links = Ba or Am, which we may call 215 links. Now, 215 - 150 ±= 65 = ra = cm; and, by the scale, ae

is found to measure 58, and 'mn 53 links; hence the area of the offset Bag + the area of the offset Amn =13282, which, divided by 1000, gives 13 links, the distance by which the line en must be approximated to ab. Consequently, ef is the true line of division • and the regular figure agbef contains 2a. 2r. 8p. minus the two shaded offsets.

PROBLEM XYI.

To part from a trapezium, or any irregular polygon whatever, any proposed guantity of land by a line drawn parallel to any of the sides, or by a line drawn from any of the angles, or from any assigned point in one of the sides to any of the opposite sides.

Rule 1. Having laid down the whole figure, draw a guess-line in the direction required, parting off, as nearly as can be judged, the proposed quantity; after which, by the scale, measure, with the greatest accuracy, the guess-line, and also the quantity thus parted off.

Then, if the guess-line or line of division be drawn from an angle, or from any assigned point in a side, divide the difference between the proposed quantity and the quantity parted off, by half the guessline, and the quotient will be the perpendicular to be set off, on one side, or the other, of the guess-line, accordingly as the quantity parted off is more or less than the quantity proposed. To the end of this perpendicular, from the point assigned, draw a new line of division, and it will part off the quantity required.

2. But if the guess-line be drawn parallel to any of the sides, divide the difference before mentioned by the whole guess-line, and the quotient will be the perpendicular to be set off from each end of the guess-line, on one side, or the other, as above.

Rote 1. When from a trapezium, approaching very nearly to a rectangle, it is required to part off any number of acres, &c., by a line parallel to one of its sides, it may be done as directed in Problem XI. ; and if there be offsets upon any of the lines, they must be treated as in the last problem.

In using guess-lines, it is not necessary that the learner should draw them so as to coincide in measure with those of the examples which he is performing. It will be sufficient for him to proceed in a similar manner.

Examples.

1. From a trapezium whose dimensions are contained in the following notes, part off 2a. 2r. 24p. by a line parallel to the side AB.

BD

diag.

1249

1000

return to B

B¹

f'ti H

Having laid down the figure, draw the guess-line ran parallel to AB; and from n let fall the perpendicular an; then, suppose mn = 1058 links, an will be = 230, and aA = 1052 links; therefore, m = 1114 - 1052 = 62 links.

.    sq. links

7130 the area of the triangle Ban

230 x 31 =

Then, 1055 x 230 = 242650 the area of the trapezoid Aanm and

The sum

= 249780 the area of the trapezium ABnm

2a. 2r. 24p. = 265000

AC

1112

1000 R. off A

diag.

DA

550

R. off D

CD

979

R. off C

BC

557

R. off B

AB

1114

1000

Begin at

range W.

ft

f

5 >n

15220 the difference between the quantity proposed and the quantity parted off by the guess-line; which, divided by 1058, gives 14’4 links to be set off perpendicularly from m and n toward D and C. Hence, ef is the true line of division and the trapezium aBEF contains 2a. 2r. 24p.

As A is very nearly a right angle, measure in the field 230 + 14-4 = 244*4 links from A to F. Then, upon any part of the line AB (toward b), as at e, erect the perpendicular er, which make = 244*4 links; stake out the line ErF, and the work will be completed.

2. From a trapezium whose dimensions are contained in the following notes, part off, in a triangle, lA. 3r. 12p. by a line drawn from the angle C to the side AB.

BD

diag.

diag.

1634

1000

return to B

AC

1010 L. off A

BA 620

L. off D

CD 1010 L. off C

BC 820

L. off B

AB

1300

1000

Begin at

range E.

Having laid down the figure, draw the guess-line cm, which suppose = 638 links., From m let fall the perpendicular ma, which will be = 417 links.

sq. links

Then, 410 x 417 = 170970 the area of the triangle BCm 1a. 3b. 12p. = 182500

11530 the difference between the quantity proposed and the quantity parted off by the guess-line, which divided by 319 (half the guess-line) gives 36 links, to be set off from m towards A. Hence ec is the true line of division; and the triangle BCE contains lA. 3r. 12p.

Also, AE is found = 731 links : measure, therefore, in the field 731 links from A to E; stake out the line EC, and the work will be completed.

Note.—The rules given in this problem for parting off land from irregular figures are generally adopted by practical surveyors, because they may be applied to any irregular figure whatever. Land, however, may sometimes be parted off more directly : for instance, the foregoing example may be performed by the mode followed in Problem XIII., i.e., if the given quantity, in square links, be divided by half the line bc, the quotient will be the perpendicular of the triangle bce ; then, at the distance of this perpendicular, a line drawn parallel to bc will intersect the line ab in E, the point to which the division-fence must be made.

3. From a field whose dimensions are contained in the following notes, part off 3a. 2r. 16p. toward AD, by a fence made from the side AB to the side CD, so that the fence may commence at the distance of 600 links from A.

BD

diag.

1050

return to B

AO

CO o

diag.

L. off A

DA

790

L. off D

CD

1130

1000 L. Off C

BC

640

L. off B

AB

1320

Begin at

range E.

Having constructed the figure, set off 600 links from A to E, and draw the guess-line Em, which suppose =702 links; the diagonal Am will he = 1132, the perpendicular Da = 278, and the perpendicular Ea= 318 links. Hence, the area of trapezium ADmE is found = 337336 square links; hut the quantity proposed (360000 square links) exceeds the quantity parted off by 22664 square links; this divided by 351 (half the guess-line) gives 64-5 links, to be set off from the line Em, perpendicularly towards BC.

Now, continue the line Em, and upon it erect the perpendicular nE = 64.5 links. The line FE will be the true line of division; and the trapezium ADFE contains 3a. 2r. 16p.

If it had been required to set off the perpendicular on the other side of the line Em, you must still have erected it so that its end might have touched the line CD.

Now, by the scale df is found = 553 links. Measure, therefore, in the field 600 links from A to e, and 553 from D to F; stake out the line fe, and the work will be completed.

Note.—The last example may also be performed by finding the area of the triangle ade, and subtracting it from the given quantity; then, if the remainder be divided by half the line de, the quotient will be the perpendicular of the triangle DEF.

At the distance of this perpendicular, draw a line parallel to de, and it will intersect the line cd in f, the point to which the division fence must be made.

4. From an irregular field whose dimensions are contained in the following notes, part off 2a. 3r. 20p. toward the line AE, by a fence made from the angle D to the side AB.

EB

diag.

1398

1000 R. off E

CE

1240

R. off C

diag.

m, proof-line, goes to D, and measures 324

AC

1260

1000 R. off A

diag.

EA

400

5e

200

000

R. off E

DE

600

450

300

150

000

R. off D

CD

740

R. off C

BC

550

R. off B

AB

1250

C* Or co O O Ox

1000

800

600

Co Or

400

200

000

Begin at

Having laid down the figure, draw the guess-line T>n, which suppose = 766 links; then the diagonal AD will be = 824, the perpendicular Ea = 278, and the perpendicular m= 372 links; re also will be = 228, and rn = 52 links.

‘It E

sq. links

267800 the area of the trapezium ArDE

area of the offsets taken on the different lines

16000 LJ the

12348 ) Ar {

308148 the area of the irregular figure awde

2a. 3b. 2Op. = 287500

20648 the difference between the quantity proposed and the quantity parted off by the guess-line, which divided by 383 (half the guess-line) gives 54 links, to be set off from n toward A. Hence DF is the true line of division ; and the irregular figure afde contains 2a. 3r. 20p.

Now, by the scale, Ac is found = 377 links. Measure, therefore, in the field 377 links from A to c; stake out the line DCF and the work will be completed.

Note 1. If the area of the irregular figure ade be subtracted from the given quantity, and the remainder divided by half the line ad, the quotient will be the perpendicular of the triangle adf ; the side ab being nearly straight from a to P.

Now, at the distance of this perpendicular, draw a line parallel to ad, and it will intersect the side ab in p, the point to which the division-fence must he made.

2. It is not absolutely necessary to survey and plan a whole field in order to part a portion from it, as the guess-line and portion parted off may be measured in the field; but, in my opinion, the former, in general, is a more eligible method than the latter, as you have a better opportunity of proving your work.

SECTION II.

The method of dividing a piece of land among sundry claimants, in the proportion of their respective claims, or a common, &c., of variable value, among any number of proprietors, in the proportion of their respective interests.

When land becomes the property of co-heirs, co-partners, joint purchasers, &c., it is generally divided into such shares as the co-parties are entitled to; and this cannot possibly be accurately effected without the assistance of some person who is not only well acquainted with surveying, but also with the method of apportioning land.

In this process an error is evidently much more material than one committed in surveying.—When a field, &c., is to be divided into any number of parts, equal or unequal, it is necessary, first, to ascertain its dimensions, and next to inquire of the parties concerned in what part of the property in question they wish their respective shares to lie.

PROBLEM I.

To divide a square or rectangle, either equally or unequally, among any number of persons, by lines parallel to one of its sides.

Rule.—If the parts into which the field is required to be divided be equal, divide the side which will be cut by the division-fences by the number of those parts, and the quotient will be the distance at which the division-fences must be placed from each other, and from the outsides to which they are parallel. But, if the parts be unequal, you must then part off each person’s share as directed in Section I. Problem XI.

Examples.

1. Divide the square abcd containing 5A. 2r. 20p. into three equal parts by fences parallel to the side AB.

Here 5a. 2r. 20p. = 562500 square links; and V562500 = 750 links, the side of the square. This, divided by 3, the number of parts, gives 250 links, the distance at which the first division-fence must be placed from AB, &c. Prom A and B, therefore, set off 250 links to E and E; join EF, and the rectangle ABFE will be one of the parts required.

Again, from E and F set off 250 links to G and H; join gh, and the rectangles EFGH and GHCD will be the two other parts required.

2. Divide ABCD, representing a rectangular field whose length is 1500 and breadth 800 links, among three men, A, B, and C, by fences parallel to the side AD, so that A may have 3a., B 4a., and C the remainder.

Here 3a. = 300000 square links, which divided by 800 gives 375 links = ae or DF; hence the rectangle aefd contains A’s share.

Again, 4a. = 400000 square links, which divided by 800 gives 500 links — EG or fh ; hence the rectangle eghf contains B’s share.

Now, the rectangle ABCD is found to contain 12a., consequently, the rectangle GBCH, containing 5a., is C’s share.

Note.—This and similar examples may also be performed by the following proportion : As the area of the whole rectangle is to the whole base or side cut by the division-fence, so is each person’s share of the rectangle to his share of the base.

PROBLEM II.

To divide a triangular field, either equally or unequally, among any number of persons, by fences made from any of its angles to the opposite side.

Rule.—If the parts into which the field is required to he divided he equal, divide the base or side to which the division-fences are to be made by the number of those parts, and the quotient will be each person’s share of the base. But, if the parts be unequal, say, as the area of the whole triangle is to the whole base, so is each person’s share of the triangle to his share of the base. (See Simpson’s Geom. iv. 7, and Euclid vi. 1.)

Examples.

1. Divide ABC, representing a triangular field, whose sides AB, AC, and BC are 1500, 1200, and 1000 links respectively, into three equal parts, by fences made from the angle c to the side AB.

Here AB = 1500 links, which divided by 3 (the number of parts) gives 500 links, each person’s share of the base. Erom A, therefore, set off 500 links to D, and from D 500 links to E; draw the lines CD and ce, to represent the division-fences; and the triangles ADC, DEC, and EBC, are the three equal parts required.

Here BC = 960 links, which divided by 3 (the number of parts) gives 320 links, each person’s share of the side BC. From B, therefore, set off 320 links to D, and from D set off the same distance to E; and the lines AD and AE will be the lines of division required.

3. Divide ABC, representing a triangular field whose sides AB, AC, and bc are 2200, 1700, and 1500 links respectively, among three persons A, B, and 0; so that, each person partaking of a pond at c, A may have 3a., B 4a., and C the remainder.

Having the three sides of the triangle, we find its area = 1272792 \2 square links; then, as 1272792-2 : AB == 2200 : : A’s share = 300000. square links : 518-5 links, A’s share of the base.

Again, as 1272792-2 : 2200 : : 400000 : 691 -3 links B’s share of the base.

The remainder of the base, which is 990-2 links, belongs to C ; and deducting 7a., the sum of A’s and B’s shares, from the area of the whole triangle, we find remaining for C’s share 572792 2 square links = 5a. 2r. 36p.

PROBLEM III.

From a, therefore, set off 518-5 links to d, and from D set off 691-3 links to e ; and the lines CD and ce will be the lines of division required.

To divide a triangular field either equally or unequally, among any number of persons, by fences proceeding from any assigned point in one of its sides.

Examples.

1. Divide ABC, representing a triangular field, whose sides AB, bc, and ac are 1500, 1150, and 950 links respectively, equally among three persons, by fences proceeding from a gate, 700 links distant from A on the base, leading into a lane, through which alone a road can be had to the field.

Rule.—Divide the base or side of the triangle from which the division-fences are to be run, as directed in the last problem. From the assigned point, draw a line to the opposite angle; and parallel to this line draw a line from each point of division on the base, until it intersects the opposite side. From these points of intersection draw lines to the point assigned, and they will be the lines of division required. (See Dr Hutton’s Course of Mathematics, vol. iii. chap. 7, prob. 2.)

"I)'

By the question AB = 1500 links, divided by 3, gives 500 links, each person’s share of the base. From A, therefore, set off 500 links to D, and from D set off the same extent to E. Then from the gate G draw the line GC; and, parallel to it, the lines dh and ef. From H and E draw the lines hg and FG, and they will be the fences of division required.

Now, by similar triangles (Theo. XI., part L), as AG : AG : : AD

: AH = 678-5 links; and as EG : bc : : BE : bf = 718-7 links. Measure, therefore, in the field, 678-5 links from A to H, and 718-7 from B to F; stake out the lines hg and FG, and the work will be completed.

Xote.—After the points h and f have been determined, by the parallel ruler, the lines ah and bf may be measured by the scale.

2. Divide ABC, representing a triangular field whose sides AB, AC, and BC are 1400, 1200, and 1000 links respectively, among three persons, A, B, and C, by fences proceeding from a pond which is at the distance of 600 links from A on the base, so that each person partaking of the pond, A may have 1a. 2r. 10p., B 1a. 3r. 20p., and C the remainder.

v'

Then, as 587877-5 : ab=1400 :

Having the three sides of the triangle, we find its area = 587877-5 square links.

A’s share = 156250 square links : 372 links, his share of the base.

Again, as 587877-5 : 1400 : : 187500 : 446-5 links, B’s share of the base.

From A, therefore, set off 372 links to D, and from D set off 446 -5 to E. Then from the pond P draw the line PC, and jtarallel to it the lines DG and EF. From G and F draw the lines GP and FP; and the triangle APG will contain A’s share; the trapezium PGCF, B’s share, and the triangle bpf, C’s share = 244127"5 square links = 2a. 1r. 3 Op.

By similar triangles we find AG = 744, and bf = 726-8 links; proceed, therefore, in the field, as directed in the last example.

PROBLEM IY.

To divide a triangular field, either equally or unequally, among any number of persons, by fences made parallel to one of its sides.

Rule.—By the rule given in Section I., Problem. XIY., part off the first person’s share; proceed with the remainder of the triangle, and the next person’s share in the same manner, and thus continue till the whole triangle is divided.

Examples.

1. Divide abc, representing a triangular field whose sides ab, ac, and bc, are 1500, 1200, and 1000 links respectively, into three equal parts by fences made parallel to the side BC.

The area of the given triangle is found = 598116-9 square links, which divided by 3 gives 199372-3 square links for each person’s share; consequently, 398744-6 square links is the area of the triangle AEG.

Then, as 598116-9 : 1500 x 1500 : : 398744-6 : 1500000; and Vl500000 = 1224-7 links =AE.

And as 598116-9 : 1200 x 1200 : : 398744-6 : 960000; and V960000 = 979 -8 links = AG.

Again, as 398744-6 : 12247 x 12247 :: 199372-3 : 749945-04; and V749945-04 = 865-9 links = AD.

And as 398744-6 : 979-8 x 979-8 :: 199372-3 : 480004-02; and a/480004’02 = 692-8 links = AF. Hence the triangle ABC is divided into three equal parts, as required.

2. Divide ABC, representing a triangular field whose sides AB, bc, and AC are 2200, 1700, and 1500 links respectively, among three persons, A, B, and C, by fences made parallel to the base AB, so that A may have 3a., B 4a., and C the remainder.

The area of the triangle ABC is found = 1272792-2 square links, from which taking 300000 square links (= A’s share) we leave 972792-2 square links, the area of the triangle dgc. From this taking 400000 square links ( = B’s share), we leave 572792-2 square links, the area of the triangle efc = 5a. 2r. 36p. = C’s share.

Then, as 1272792-2 : 1700 x 1700 :: 972792-2 : 2208820-46 ; and V2208820'4 = 1486-2 links = CG.

And, as 1272792-2 -1500 + 1500 :: 972792-2 : 1719669-91 ; and V1719669-91 = 1311-3 links = CD.

Again, as 972792-2 : 1486-2 x 1486*2 :: 572792-2 : 1300563-40; and a/1300563'40 = 1140-4 links = CF.

And, as 972792 : 1311-3 + 1311-3 :: 572792*2 : 1012467-60; and V1012467-60 = 1006-2 links, = ce. Hence the triangle ABC is divided into three parts, as required.

PROBLEM V.

To divide a trapezium, or an irregular polygon, equally or unequally, among any number of persons, by fences made in a given direction.

Rule.—By the rules given in Section I. Problem XVI. part ofl the first person’s share; proceed with the remainder of the figure, and the second person’s share in the same manner; and thus continue till the whole figure is divided.

Examples.

1. Divide a trapezium, whose dimensions are contained in the following notes, into three equal parts, by fences made from the side AB to the side CD.

BD

1542

1000

return to B

diag.

AC

1848

1000 R. off A

diag.

DA

915

R. off D

CD

1347

1000 R. off C

BC

885

R. off B

AB

1547

1000

Begin at

range W.

The area of the triangle ABC is found = 681942, and the area of the triangle CDA = 585949 square links; consequently the area of the trapezium ABCD = 1267891 square links, which divided by 3 gives 422630 square links for each person’s share.

Now draw the guess-line Em, which suppose = 880 links; then the diagonal EC will be found = 1028, the perpendicular ba = 387 and the perpendicular ma = 424 links; hence the area of the trapezium BCmE is found = 416854 square links, which is too little by

5776 square links. This divided by 440 (half the guess-line) gives 13 links, to be set off from m toward d ; consequently ef is the true line of division.

Again, draw the guess-line Gw, which suppose = 878 links; then will the diagonal GF = 1017, the perpendicular Ea= 430, and the perpendicular Na = 385 : hence the area of the trapezium EFwG is found = 414427 square links, which is too little by 3203 square links. This divided by 439 (half the guess-line) gives 19 links to be set off from n toward D; consequently GH is the true line of division; and the trapezium abcd is divided into three equal parts, as required.

Now, by the scale, we find BE = 450, EG = 500, CF = 508, and FH = 468 links, which distances must be measured in the field, in order to determine the situation of the division-fences.

Note.—If we subtract the area of the triangle bce from the quantity to which each person is entitled, and divide the remainder by half the line ce, the quotient will be the perpendicular of the triangle cef. By drawing a line parallel to ce, at the distance of this perpendicular, the point f may be determined.

In a similar manner may be parted off the trapezium eehg.

2. Divide a field, whose dimensions are contained in the following notes, among three persons, A, B, and O, so that each partaking of a pond at P, A may have 3a., B 4a., and C the remainder.

BD

1447

1000 R. Off B

diag.

EB

1603

1000 R. off E

diag.

CE

1300

1000 R. off C

diag.

AC

1320

1000

diag.

700

R. off A

pond

EA

750

R. off E

DE

650

R. Off D

CD

850

R. off C

BC

800

R. off B

AB

1200

1000

range W.

Begin at

From the pond P draw the line PD and also the guess-line Pm, which suppose —558 links; then will the diagonal Dm he = 1025, the perpendicular Pa = 400, and ca=195 links; hence the area of the trapezium PmCD is found = 304937 square links, which exceeds A’s share by 4937 square links. This divided by 279 (half the guess-line) gives 17‘7 links, to be set off from m toward c ; consequently pf is the true line of division, and the trapezium PFCD contains A’s share.

Again, draw the guess-line Pn, which suppose = 696 links, then the diagonal PE will be = 848, the perpendicular na = 247 and Da = 552 links; hence the area of the trapezium pmed is found = 338776 square links, which is less than B’s share by 61224 square links. This divided by 348 (half the guess-line) gives 176 links to be set off from the line Pn perpendicularly toward A; consequently, pg is the true line of division, and the trapezium PGED ontains B’s share.

Now the irregular polygon ABFPG contains C’s share, which will be found = 4a. 3r. 19p.

By the scale we find AG = 252, and bf = 545 links, which distances must be measured in the field, in order to determine the situations of the division-fences.

'QE

sA

ti/

F Li

3a

Note 1. The foregoing example may also be performed by subtracting the area of the triangle cdp from A’s share, and then laying out the remainder in the triangle cep, as before directed.

In a similar manner may be parted off B’s share.

2. The division of the last, or any other figure, may be proved by finding the area of the whole figure, which, if equal, or nearly equal, to the sum of the areas of the parts into which it has been divided, demonstrates the work to be right.

PROBLEM VI.

To divide a common, or any quantity of land, of uniform value, among any number of proprietors, in the proportion of their respective interests.

In this case the land to be divided must first be surveyed, and next the estate of each proprietor, if its quantity be unknown. Then, if it be required to make the division according to the value of each person’s estate, there must be proper persons appointed to value them, which, in this problem, we will suppose, may be done at so much per acre, uniformly, throughout each estate.

, Note 1. When the land to be divided is of uniform value nothing more is wanted than its quantity.

2. It is immaterial whether the land be valued at 5s. or 51. per acre, if the same proportion, according to the quality of the land, &c., be observed in valuing each person’s estate.

To determine each person's share.

Rule.—As the number of acres, &c., contained in the sum of the estates, is to the whole quantity of land to be divided, so is each person’s estate to his respective share. Or, as the sum of the values of all the estates is to the whole quantity of land to be divided, so is the value of each person’s estate to his respective share.

Examples.

1. Divide a common containing 56a. 2r. 16p. among three persons, A, B, and C, whose estates are 58, 96, and 128a. respectively.

Here 58 + 96 + 128 = 282, the number of acres contained in all the estates; and 56a. 2r. 16p. = 5660000 square links. Then,

A. sq. links as 282 : 5660000 :

CO <D OO lO 05 Cl

sq. links 1164113) 1926808 V = 2569078

A. R. P.

11 2 22-5) A’s 19 1 2-8y B’s 25 2 30’5) C’s

share

5659999

56 2 15’8 proof

Each person’s share thus determined, the common may easily be divided by the methods already described.

2. Three gentlemen, A, B, and C, have each an estate consisting of 300a. ; divide among them, according to the values of their estates, 75a. 3r. 32p. ; A’s estate being valued at 25s., B’s at 325., and C’s at 405. per acre per annum.

25

Here 300 x

32

40

7500 the value of A’s estate 9600 ditto of B’s [ 12000 ditto of C’s

29100 sum

And 75a. 3r. 32p. = 7595000 square links. Then

share

s. sq. links as 29100:7595000 ::

sq. links

{ 7500 ) ( 1957474 )

' 9600 Vi-J 2505567 I ( 12000 J ( 3131958 J

7594999

75 3 32 proof

Note.—It sometimes happens that two, three, or more persons join in taking a common pasture, and agree to pay in proportion to the number of cattle with

which each person depastures. In such cases, when the whole of the cattle graze an equal time, you must make use of the rule of Single Fellowship, by saying, as the whole of the cattle is to the rent of the whole pasture, so is each person’s cattle to his share of the rent. But when the cattle graze an unequal time, you must then have recourse to the rule of Double Fellowship, by saying, as the sum of the products of each person’s cattle and time is to the whole rent, so is each person’s product to his share of the rent.

PROBLEM VII.

To divide a common, dec., of variable value, among any number of proprietors, in proportion to their respective interests.

In a work of this kind the quantity of every different quality is required, not only of the land to be divided, but also of each proprietor’s estate; consequently the surveyor, accompanied by the persons appointed to value, generally called * Commissioners,’ must examine each person’s estate, and also the common, previously to the survey being taken.

In doing this they must stake out lines between the different qualities of the soil; and, in surveying, these lines (called by surveyors ‘ quality-lines ’) must be considered as boundaries, and represented in the field-book, and upon the plan, by small dashes.

By way of distinction there ought to be two stakes put down at each angle formed by the quality-lines; and also marks cut in the ground, pointing in the direction of these lines, so that if the stakes should be pulled up, these marks may serve as directors.

When the survey is finished and laid down, every different quality represented upon the plan must be successively numbered, 1, 2, 3, &c. The surveyor must then require the commissioners to put the different valuations upon the land; and, in doing this, he must accompany them with the plan, in order that both he and they may know the ground corresponding with each number. Surveyors generally use letters to represent the different values of land :

Thus a may denote 1 shilling

99

99

99

99

99

99

99

99

99

99

99

99

99

99

99

99

99

and x „

»

„ 20

„ 30 shillings

By putting three of these letters together, and adding their separate values, the value per acre per annum may be set down as high as sixty shillings; and, by adding more letters, it may be carried to any height required. By the use of these letters the confusion arising from a multiplicity of figures is avoided.

The land being valued, you must then proceed to find the quantity contained under each number on the plan, and also its value.

In doing this, it is unnecessary to bring the decimals into roods and perches, or to retain more of them than the three next the acres, as the operation is thus considerably simplified.

If the fourth figure in the decimals be 5, or greater, add 1 to the third : that is, if the content be 3⁻54585, set down 3,54G.

When the content does not amount to an acre, and the number of figures is under five, add as many ciphers to the left as will complete that number: that is, if the content be ⁻8626, set down ‘086. Then, multiply the acres and decimals, contained in each number, by the valuation per acre put upon the respective numbers, and the product will be the value in shillings and decimals. (See the note on page 54.)

MISCELLANEOUS OBSERVATIONS ON VALUING LAND.

1.    Proprietors require to appoint judicious commissioners to value for an enclosure. They should not only be well acquainted with the quality of the soil, but should also be able to judge how far every part of the common is capable of being improved after it has been inclosed, or they will not be able to put a just valuation upon it.

2.    In valuing, not only the quality of the land, but also its situation, must be attended to ; for, if one part of the land to be divided lies in a valley (not subject to be flooded), near a proprietor’s messuage, and another part upon a hill, at the distance of two or three miles, it is evident, allowing the land to be all of the same quality, that the former situation is much more desirable than the latter; because it is nearer the homestead, and consequently better situated for agricultural purposes.

3.    The manner in which the climate and seasons may operate upon the produce of the ground, in consequence of its local situation, should always be taken into consideration. If one field lies towards the south, and another towards the north, and both be of the same quality, the field that faces the south is more valuable than the other, as the crops on the former will not only be brought to a

greater degree of perfection by the influence of the sun, but will also be ready for harvesting sooner, and consequently may be brought to an earlier and frequently to a better market.

4.    In valuing a common for an inclosure, the improvements that may be made by fencing, draining, and cultivation, should never be overlooked. If one person should have an allotment awarded to him in the best part of the common, but where no improvement can be made, and another person’s allotment, of equal value, be laid out in the worst part of the common, but where much improvement may easily be made by cultivation, it is manifest that the latter allotment will, in a few years, be more valuable than the former. Besides, as quantity is always given to compensate for any deficiency in quality, the proprietor who has his common-right laid out in the worst part of the ground, will not only receive more land than the other, but will soon be able, by a trifling expense in cultivation, to make it worth more per acre.

5.    In valuing either old inclosed lands or commons, the distance of the ground from good springs of water should be regarded. In many parts of England, and particularly upon the Wolds in Yorkshire, the occupiers of land frequently suffer great inconvenience in driving their cattle a considerable distance to watering-places • and the cattle themselves are sometimes much injured in droughty summers, for want of a regular supply of wholesome water. Hence a farm that is well watered is worth more to rent than another farm of equal quantity and quality, but destitute of water.

6.    The distance of farms, common-rights, &c., from market-towns is also of considerable importance ; because land always increases in value as it approaches the vicinity of large towns. Besides, as the means of tillage abounds in such places, improvements may be made at less expense than where the land lies at the distance of many miles. It may also be remarked that the occupiers of the former can always find a ready market for the produce of their land, while the occupiers of the latter are under the necessity of being at a considerable expense in transporting their goods to market, and in procuring the various articles that are indispensably necessary for the use of their families.

Appellations given to certain lands.

1. Moors are large, uncultivated tracts of ground, generally overgrown with furze, broom, heath, and other small shrubs, as Bumbles-moor in Yorkshire, and Blackstone-edge, partly in Yorkshire and partly in Lancashire.

2.    A Fell is a large open portion of land, generally less overrun with shrubs than a moor, as Gateshead Fell in the county of Durham.

3.    A Heath is an open ground, abounding with the plant called heath, or any other shrubs, as Hounslow Heath in Middlesex.

4.    "Wolds are high open grounds, as the Wolds in Yorkshire and Lincolnshire.

5.    Downs are fine open pasture grounds, as the Downs in Kent, Sussex, and Surrey.

6.    Fens are low wet tracts of ground, as the Fens in Lincolnshire.

7 Marshes are low swampy grounds, and when adjoining the sea or the sides of rivers, they are mostly excellent pastures; as the Marshes in the counties of Durham and York, contiguous to the river Tees; those in the counties of York and Lincoln, contiguous to the river Humber; and the rich marsh of Romney, in the county of Kent, adjoining the straits of Dover.

8.    Mosses are black, turfy, boggy moors, as Ashton Moss, and many others in Lancashire.

9.    Forests are wild uncultivated tracts of ground, generally abounding with trees, as Sherwood Forest, in Nottinghamshire, and the New Forest, and that of East Bere, in Hampshire.

10.    Ings are large open meadows, generally situated on low, level grounds. Fields and tracts of land known by the local name of “ The Ings,” abound in almost every county of England.

11.    Holmes are hilly, fenny, or level grounds, adjoining to, or encompassed by rivulets or brooks. Many rich and fertile pasture grounds in this country are known under the local appellation of “ the Holmes.”

12.    Open fields are unenclosed lands, generally divided into furlongs by mere forms, and occupied by different tenants.

Some furlongs are usually in corn, some in meadow, and others in pasture; and the cattle and sheep which depasture are tended by shepherds. Large tracts of land upon the Wolds in Yorkshire are cultivated in this manner.

13.    A Furlong of land is used in some old books to express the eighth part of an acre; hence 20 perches, or 605 square yards, make a furlong.

The term is also used to denote any number of lands adjoining each other in open fields, and running in the same direction from one head-land to another, and known by some particular name, in order to distinguish the different parts of the field from each other.

14.    Mereforms are narrow pieces of swarth, dividing lands or furlongs in open fields from each other.

15.    An Ox-gang or Ax-gate of land is usually taken for 15 acres, being as much land as it is supposed one ox can plough in a year.

In Scotland 13 acres are denominated an Ox-gang, and in some places the term is used to denote as much land as will summer one

ox.

This word is corruptly called OsTcen in Lincolnshire and some other counties.

16. A Hide of land, sometimes met with in old books, was such a quantity as might be cultivated, in the compass of a year, with one plough, having meadow and pasture sufficient to feed the cattle belonging thereto. The term was also frequently used to denote as much land as would maintain a family.

Some writers make the hide to contain 60, some 80, some 100, and others 120 acres.

Sir William Dugdale, the antiquarian, says, that a Barony, in former ages, was a certain portion of land held immediately of the king, and contained not less than 40 hides, or 3840 acres—a statement that gives 96 acres to a hide.

Directions for setting out new roads, sand-pits, quarries, watering-places, dec., and for dividing commons and waste lands into allotments.

1.    Before commons and waste lands are divided and allotted, new roads must be set out upon them, in the most convenient and advantageous manner. They should, whenever it is practicable, be set out in such directions as to form right angles, or as nearly right angles as possible, at the places where they meet or intersect each other, or come in contact with ancient highways. They should not be less than thirty feet in breadth, and set out in right lines; be cause straight roads not only look better than crooked ones, but also occupy less ground.

2.    All old roads leading over commons or waste lands about to be inclosed, may be stopped or diverted at the discretion of the commissioners; and such old roads must be surveyed and allotted as part of the common or waste lands.

3.    Certain portions of commons should always be set out for sand or gravel pits, and for quarries, if the commons contain either sand, gravel, or stone. The portions of ground thus set out are considered

as public property, from which every person who receives a common-right may take materials for building houses, making fences, and repairing roads.

4.    If there be any good springs of water on commons, they must either be left uninclosed for the public watering-places, or the water must be conveyed to more convenient situations, by means of drains, or channels, and troughs or reservoirs made for its reception.

5.    In some places the lord of the manor claims one-twelfth, in some one-sixteenth, in others only one-twentieth, of all commons and waste lands ,• whatever be his claim, however, it must be set out before any other allotment, after its value has been ascertained from the quantity and value of the whole common. Besides this allotment, the lord of the manor will, of course, be entitled to his proportional share of the remainder of the waste lands, in the same manner as any other proprietor.

6.    When it can be done, it is very desirable to ascertain the value of any other claim, and to set out, for the proprietor of the same, an allotment of equivalent value ; thus will the whole place become free, and the occupiers of lands be exempt from what they may generally deem an unpleasant tax upon their industry, but which may, nevertheless, be as justly due to the claimants as the rent of a farm is to the landlord.

7.    If the clerk’s salary arise from the lands, which is the case in some places, a common-right may also be set out in lieu of it; and if another can be obtained as a small endowment for a town’s school, the inhabitants will not have cause to repent, if they be judicious in the choice of a master.

8.    After the roads, sand-pits, quarries, watering-places, manorial rights, &c., have been set out, the remainder of the common or waste lands must be equitably divided (quantity, quality, and situation of place being regarded), among the owners and proprietors of messuages, cottages, lands, tenements, and hereditaments situated in the township or place where the inclosure is to be made and executed.

Note—The first step towards inclosing wet, marshy grounds, is to have them well drained, for without this be done, every attempt at improvement will be vain.

To determine the value of each proprietor’s allotment, or claim upon the

common.

In doing this the value only can be used; for, if we make use of the quantity, in allotting land of different qualities, the proprietor

who has his allotment in land of the best quality will obviously receive more than his just right, while those whose allotments fall in land of inferior quality lose part of their property. Hence, you must say, as the value of the whole estates is to the value of the common, or land to be divided, so is the value of each person’s estate to the value of his allotment, or claim upon the common.

To set off, upon the plan, each proprietor's allotment or share of the common.

When you find that a proprietor’s allotment falls in that part of the common which is of uniform quality, you may easily determine the quantity to which he is entitled, by saying, as the value put upon the number in which his allotment falls, is to 1 acre, so is the value of his claim to the quantity of land which his allotment must contain. Then set off the allotment upon the plan by some of the methods already described.

But it commonly happens that a proprietor’s allotment falls in different numbers. In such a case you must draw a guess-line or lines, and measure separately, by the scale, the pieces cut off belonging to the different numbers : then multiply the different quantities by their respective values; and if the sum of the products be equal to the value of the claim in question, the guess-line or lines are right; if not, they must be altered, until they part off the exact portion. After each proprietor’s allotment is set off upon the plan, if you find the quantity and value of all the allotments equal to the quantity and value of the whole common, the division is right.

Example.

Lay down a plan from the engraven field-book belonging to Plate XII., and divide the common among the three proprietors, A, B, and C, according to the different qualities of their estates and of the common.

Xote 1.—The learner should lay down the plan, from the field-notes, by a scale of two chains to an inch, and find the areas of all the fields from his own dimensions, as directed in Part Y. The diagonals and perpendiculars from which the above areas were found, are not given, as this would have rendered the work too easy to exercise the genius of the student; he may, however, retain his own dimensions, and enter them in “A Book of Dimensions, Castings, Quantities, Qualities, and Values, adapted to Plate XII.” (See page 187.)

A Booh of Quantities, Qualities, Values, d-c.

Belonging to Plate XII.

A’s Estate.

No. on the Plan.

Quantity.

A. Dec.

Quality.

Value Shil. Dec.

I—i CM CO

Total

7-565

7-609

7-301

22-475

§ §1?

378-250

304-360

277-438

960-048

B’s Estate.

Total I

7-858

7- 892

8- 223

23-973

290-746

260-436

328-920

880-102

C s Estate.

Total

O 1—J CM CO

7-819 7 078 7*481

22-378

xh

xso

xs

297-122

424-680

374-050

1095-852

The value of the whole Estates

2936-002

The Common.

10-061

4-680

4- 446

5- 995

301-830

159-120

168-948

167-860

Total I 25-182

797-758

2. If the learner should not be able to find such dimensions as will make his agree exactly with those given in the foregoing Book of Quantities, it will be

areas

a matter of no consequence, provided the difference be not too considerable; and any difference in the areas will also produce a difference in the values, all the numbers in his book will differ from the given numbers. This, however, will tend much to his improvement, as he will be under the necessity of making all his own calculations, both in finding the areas and values of the different fields, and also in dividing the common and proving the division.

as

The operation of finding the value of each proprietor's share of the common; and directions for setting out the allotments in the field.

s.

s.

As 2936': 797-758 :

s.

s.

960 048 )

, 880-102 V . : { 1095-852)

( 260-860 ) A’s )

: 1 239-137 IB’s lvalue ( 297-761 J C’s J 797 758 proof.

Plate XII.

01

12

xh

13

11

vx

A Plan from the Engraven Field Book.

7,

4}-

North.

xh

O.v

sx

ox

OX

xso

Scale, 8 chains to an inch.

sx

xg

xh

As the whole of A’s allotment will fall in No. 10, say, as 30 : 1 :: 260-860 : 8 695 acres, the quantity of land which A’s allotment must contain.

Prom 10-061 take 8-695, and we have 1-366, the remainder of No. 10, in value = 40-980, which will form part of B's allotment. Then, from 239-137 take 40-980, and there remains 198-157; consequently, B must have land equivalent to this value from Nos. 11 and 12.

The remainder of these Nos. and the whole of No. 13 will he C’s allotment, which you must measure, &c., as a proof.

In setting off the allotments upon the plan we find that one end of. the division-fence between the allotments of A and B falls at the distance of 827 links from + 8 toward + 1, and the other end at the same distance from + 7 toward + 2. We find, likewise, that one end of the division-fence between the allotments of B and C fall at the distance of 1465 links from+8 toward+ 1, and the other end at the distance of 1478 links from+ 7 toward + 2. Measure, therefore, these distances in the field, stake out the division-fences, and the work will be completed.

Note.—The fences of old inclosures are generally very crooked ; but the fences of new inclosures are always set out in straight lines, when it is practicable.

THE PROOF OF THE DIVISION.

A’s Allotment

No. on the Plan.

Quantity A. Dec.

Quantity

A. E. P.

Quality.

Value Shil. Dec.

Part of 10

8-695

8    2    31

|    260-850

B’s Allotment

Part of 10

11

12

1- 366

2- 580 2-910

8-g.s

40-980

87-720

110-580

Total

6-856

6    3    16

239-280

C’s Allotment

Part of 11

12

Whole of 13

2-100

1-536

5-995

xd

xh

sh

71-400

58-368

167-860

Total |    9-631

9 2 52

297-628

Sum total |    25-182

25 0 29

797-758

GENERAL DIRECTIONS FOR APPORTIONING LAND INTERSECTED BY A RAILWAY.

On referring to Plate XIII. at the end of the work, it will he seen how railways, by intersecting fields, necessitate new lines of fences. Thus, in the parishes of Westbrook and Winston, there is not a single field through which the line passes but what requires to be altered, either by land being added to or taken from it.

Estates, parishes, and counties are severed in a similar way to farms, lands being taken from one that require to be added to another before they can be used in the most profitable manner: while the railway itself has become an intervening estate, containing a large area of land, now' in the occupation of a new proprietary.

In a survey of this kind the present and future value of the land is that which principally engrosses the surveyor’s attention. In other words, the survey is more a work of valuation than measurement, and as directions for the latter have already been given, those now offered will chiefly refer to the former.

The valuation of severed lands may, for the sake of practical instruction, be illustrated under two examples. First, intersected parks and farms; second, intersected estates, parishes, and counties.

Example I.

Intersected Parks and Farms.

In this example we shall suppose the railway to pass through a large estate, severing the game-preserves and other lands in the natural possession of their proprietor, and also those held on lease by tenants, cutting off portions from fields in the occupation of one farmer so small, or detached, as to render it necessary to add them to fields on the opposite side of the line belonging to other tenants. It may also be assumed that the severed lands are farmed under different systems of husbandry, some being under permanent meadow, others under aration.

As a general rule, it may be given that railways will ultimately determine the direction of fences and the form of fields, just as roads, rivers, and boundaries of estates have hitherto done, similarly to what is exemplified in the parishes of Westbrook and Winston, as illustrated Plate XIII.

The doctrine thus taught obviously involves the resubdivision of land on both sides of the railway as the most profitable practice. To this rule there are but few exceptions. It implies that if the lands are allowed to remain as they are, in a severed state, a permanent loss will be sustained; but, as in the parallel example of a road, present losses will be more than compensated by future gains, provided a proper subdivision is effected; consequently, as the proper subdivision of the land may be presumed to be the object of the survey, the two alternatives call for special consideration at its commencement.

1.    If lands intersected by a railway are allowed to remain in a severed state, or if they are improperly laid out into fields, their value per acre for agricultural purposes will be permanently decreased.

2.    If lands intersected by a railway are properly laid out into rectangular fields of the required size, their value per acre will be permanently increased.

3.    Capital invested according to the first rule will not return interest. The investment is a dead loss, and ought, therefore, to be borne exclusively by the landlord, because the tenant, on his part, under such circumstances, has annually to bear an increase of expense and decrease of profit, as a permanent loss per acre.

4.    Capital invested in accordance with the second rule will return interest, and if the investment is judiciously made, the amount of interest realised will, in the vast majority of cases, be highly remunerating to the landlord, while the improvements effected ought at the same time to afford equal encouragement to tenant’s capital.

The agricultural losses sustained under the first rule may be accounted for by reference to Plate XIII. Thus, if we suppose, what is too commonly the practice, that the triangular portion of field No. 5, lying between the railway and the open drain in the parish of Westbrook, is added to field No. 6, by the removal of the fence that is cut by the line, and that the triangular portion of No. 6, lying between the railway and the road, is added to No. 5, then, under such data, a permanent loss per acre will be sustained of a twofold character, apart from the dead loss borne in removing old fences and making new ones; and also apart from the greater area of land under fences.

The twofold loss thus sustained is occasioned by an increase of expense and a decrease of produce, as formerly stated, the first arising from working acute-angled and shapeless fields, which will be greater under steam-culture than it has hitherto been under horse-culture; and the second from the greater length of headlands.

Another source of loss arises from roads crossing the line, as in the case of the farm-road leading up to Hill Farm, which is raised eleven inches; but in many instances the inclination amounts to

several feet, so that in hay and corn harvest two horses are required to do what formerly was done with one, while a boy has to attend to the gates at these two busy and expensive periods of labour.

Permanent losses of this kind are first experienced by tenants, but ultimately by landowners. In the case of the home-farm the latter bears the whole loss.

There are other losses which have to be valued that are not of a permanent character, and which, therefore, may have to be solely borne by the tenants whose lands are severed. Thus, an inferior quality of soil, in a high state of fertility by artificial means, taken from a tenant at the commencement of his lease, may be worth more annually for a short term to his neighbour who gets it than a better quality of land in an exhausted state from mismanagement received in exchange. Instances might be quoted where differences amount to more than the rent of the land during the remainder of a lease ; and with the progress of science and an increasing amount of capital invested in farming, such differences are of greater importance than they were formerly.

For agricultural purposes a rectangle is the most profitable form of a field. This may be given as an established formula—that which is practically embodied in the second and fourth rules above.

Any deviation from the rectangle, occasioned by the inclination of the land or other cause, will be attended with a corresponding permanent loss; and the amount of this loss will be greater under steam-culture, as now performed by a wire rope, than it has hitherto been under horses. In other words, the best system of cultivation is that for which provision should be made by the surveyor.

As the railway determines one boundary fence, the direction of the parallel and perpendicular fences may also be considered as ultimately fixed. Thus, the two limit of deviation lines may be taken to represent the first two parallels, one on either side of the railway, as shown in Plate XIII.

In the case of the park, woods, game-preserves, pleasure-grounds, and home-farm, in the natural possession of their owner, present losses are generally greater, and future profits more distant, owing to the larger amount of ornamental work destroyed by the railway, and the corresponding increase of capital required for new improvements of this description. But, granting this, a railway may be made far less objectionable in a park than a common road, while in very many parks and demesnes it may be the base-line of much-needed improvements, alike for ornament, pleasure, game, and profit.

When an old plan of the estate is furnished, as is generally the case, a copy of the park may be taken by tracing-paper, and the railway laid down upon it. This will enable the surveyor to determine the new design, and apportion the grounds.

In a work of this kind every case must be its own rule. When the line crosses a river, road, or valley, at a distance from the mansion, it may have to be left open for ornament, but generally it will require to be covered. This may be done by planting forest timber, ornamental trees, and shrubbery favourable to game, to compensate for other grounds taken into the home-farm to square off arable fields. In the case of permanent meadows and other grass lands, if angles and straight lines are avoided, the more shapeless the areas are the better, provided the outlines are natural. Although game require retirement, yet pure air, light and sunshine, with natural food, are essentially necessary to health and the fine normal flavour of their flesh, and if these are secured for them, they will soon become familiar to the most noisy traffic on the line, and even prefer it to the interior of close, dark, and unhealthy forests and game-preserves.

Lands thus laid out for timber and game-preserves on both sides • of a railway should communicate with each other by bridges under or over the line, as shown in the section Plate XIII. at No. 1 and No. 2 ; and where a choice exists, the latter, generally speaking, will give more satisfaction than the former. When a railway crosses a river, a brook, or even a drain, a footpath on one side will form a very inviting communication between game preserves.

Example II.

Intersected Estates, Parishes, and Counties.

A railway may be considered a permanent landed estate, in the possession of a separate and distinct proprietary, and, therefore, a suitable boundary between farms, estates, parishes, and counties.

The practical rule involved in this can be profitably carried out in all cases where a railway intervenes between farms, estates, parishes, and counties, in such a manner that the land taken from one can be equitably given in exchange for what has been cut off from another, the work of apportioning being made in accordance with the old familiar maxim of “take and give,” as shown in Prob. XI. and XII., Part Y.

It may also be profitably carried out in cases where the small differences of value between portions of severed land can be settled,

either by money or by an exchange of land in some other place. Thus, A may receive from C land to compensate for what B has received from A, while C may receive from B wrhat will balance the triune account.

Exchanges of this kind enhance the value of land in several ways. It does so to railway companies and the public as follows. Thus, if timely notice were given the former that the lands intersected by their line would be exchanged, and that their estate would be made a permanent boundary, they would often be enabled to obviate bridges under and over the line, and crossings on a level with the rails; consequently, capital would be economised in its formation, and expenses in working it afterwards, while in the outset they could afford to give a higher price for the land.* The safety of the public, again, would be better cared for, more especially at the crossings to fields and farms on a level with the rails, cattle being very liable to get upon the line at such places, gates to be left open, and things to be dropped from carts and left upon the rails.

In exchanges between landowners of portions of their estates present and future values require to be considered under the following three heads :—1. The agricultural value of the land; 2. The mineral resources of the land; 3. The adoption of land for building purposes, and others of a kindred character.

1.    In examples of land exclusively adapted for agricultural purposes to present values must he added facilities for future improvement. Thus, according to the old adage, “ Lay clay on sand and you buy land.” If, therefore, an estate possesses within itself facilities for improvement by the mixing of soils, and which may not have existed prior to the formation of the railway, the materials having to be conveyed along the line or part of them, such must be carefully taken into account. On the other hand, some sandy, clayey, or mossy lands may be worth more to another estate than to the one to which they belong, because the former possesses such facilities while the latter does not.

2.    Railways have considerably affected the mineral value of land. Generally an advance has been realised, but there are examples to the contrary, so that every individual case must furnish its own rule for valuation.

3.    In the immediate vicinity of towns, or railway stations, or places on the line where a new railway station might advantageously be made, lands adapted for building purposes always form a prominent topic in a survey accompanied with a valuation and plan of exchange.

When the growth of the population of a place is rapid under the prospect of a continued state of prosperity, lands for building may increase in value.

Again, when a toAvn is overcrowded by narrow streets and antiquated houses, there is every prospect of suburban lands increasing in value, if the trade of the place is flourishing.

The extension of railway accommodation is also thinning out many of our large towns, and as the inhabitants are being spread over a large area, market-gardens, nursery-grounds, dairies, and various manufactories are being removed farther inland. In all such cases there is an increasing demand for land a short distance from town, with a corresponding increase of value.

Rectangular areas of land are of more value for building purposes than triangular and shapeless pieces. This is of growing importance to landowners and tenants for long terms, more so than it was some time past. The increase of value is partly owing to the architectural reformation that old streets and buildings are now undergoing; but principally to the general progress of science demanding a greater economy of time and labour. Thus, when a street crosses another at right angles, the traffic in the one has only to wait until that of the other moves across the breadth of the street in which the stoppage takes place. But when streets cross each other diagonally the crossings increase in length, so that the expense of time and labour may be twice and even thrice that of the rectangular system. And this, too, is not the worst feature of diagonal crossings, for horses cannot turn round the acute angles without stopping both the up and down traffic, Avhile in turning at obtuse angles the opposite error is experienced, viz., too great a velocity, and tear and wear of streets and vehicles. To these practical objections must be added other three, viz., the greater area of land consumed both by streets and houses; the increase of expense in their formation and annual repair; and the reduction in the accommodation which they afford for business ; and when the student sums up the objections on both sides, he will then be able to appreciate the difference between the rectangular and diagonal systems of laying off land for building purposes, and how much two proprietors of building lands would gain by exchanging triangular and shapeless pieces of land situated on opposite sides of a railway, so as to enable each to square off his property, and thus give it the highest value.

SECTION III.

GENERAL DIRECTIONS EOR TAKING SURVEYS OF RIVERS, CANALS, EMBANKMENTS, AND LANDS RECLAIMED FROM THE SEA.

The directions in this section will be given under the following six examples, viz. :—

Example I.—Directions for the survey of lands to be reclaimed from the sea.

Example II.—Directions for the survey of a tidal river.

Example III.—Directions for the horizontal survey of lands reclaimed under Ex. I. and II.

Example IY.—Directions for the survey of a river in a large valley from high water level upwards.

Example V.—Directions for the survey of mountain rivers.

Example VI.—Directions for the survey of canals, irrigation-works, and warp-lands.

EXAMPLE I.

DIRECTIONS FOR THE SURVEY OF LANDS TO BE RECLAIMED FROM

THE SEA.

Before land is reclaimed from the estuary of a tidal river on one or both banks, by means of embankments, a survey of it is usually made. The object of the survey is to enable the owner or owners to ascertain the area that can be recovered, the dimensions of the embankments required, and whether the land, when reclaimed, will return sufficient interest to justify the investment of capital necessary to complete the work, and lay out the land for agricultural purposes.

Such a survey necessarily includes a plan and section of the embankments and mouth of the river. The land to be reclaimed may be presumed to extend to both sides, but as the two areas are similar in character, directions for one will be sufficient.

The embankments are thus what are usually termed the sea embankments, in contradistinction from those of the next example that extend inland along both sides of the tidal part of the river.

as

Parallel examples are to be found at most of our large rivers, such those that flow into the Wash, from which large areas of fen land have been reclaimed and added to the counties of Lincoln, Huntingdon, and Cambridge. Holland presents examples of still greater magnitude. In this country, in our colonies, and other parts of the world, a vast amount of land may yet be profitably reclaimed.

In some places surveys are easily taken both for plan and section. Thus, where the land is only under water at spring tides, it will be covered with herbage, so that a complete survey, horizontal and vertical, may be taken at first, as no difficulty stands in the way of determining the position of the embankments, fences, water- courses, roads, buildings, &c., required in laying out the land reclaimed for agricultural purposes. In others the work embraces au indefinite amount of sectional detail, owing to the nature of the subsoil, and the difficulty of determining the position and magnitude of the embankments. The latter will be the most instructive example to adopt. The more advisable course in examples where the position of the embankment cannot be determined at once, is to include two areas of land by means of two boundary lines, the one area greater than will probably be reclaimed, and the other less.

The boundary line of the largest area may be termed the maximum-extreme, or the exterior line, or outer line; and the boundary line of the lesser area the minimum-extreme, or the interior line, or inner line.

A preliminary horizontal survey for the plan will thus be necessary. It will embrace the details of the area between the outer and inner boundary lines, but only a" general outline of the lands inside and outside.

The survey for the sections should precede that for the plan, as the position of the two boundary lines cannot be determined until the former work is concluded. This arises from the treacherous

nature of the subsoil, which, in the vast majority of cases, cannot be judged as to quality from mere superficial appearances. The boundary lines have consequently to be carefully examined before the several stations are finally fixed upon for the maximum and minimum extremes between which the position of the embankment is presumed to run. In doing so, to avoid, if possible, going over the line a second time, the distances of the ground line below the surface level of the sea at high water should be taken and entered into the field-book, which will form part of the details of the sectional survey.

Directions for ranging station-poles.

The first station-poles that require to be set are those at the mouth of the river, two on each side, one for each boundary-line.

The outfall or mouth of the river is often the most difficult point in the survey to determine. In some cases, for example, it is more advisable to make a new outfall than to continue the old one; consequently this is the first point that requires to be settled, and the following general rules may be given for that purpose:—

1.    W}ien the channel of the river at low water contracts at its mouth, when the subsoil is of such a quality as to permit of an embankment being easily thrown across this narrow place from one side to the other, and when the old bed of the river above this expands into basins, it may then be advisable to change the outfall.

2.    When the land to be reclaimed extends to a great distance on each side of the river, and when an embankment can be thrown across the channel, the river may be divided, and its waters discharged into the sea by two outfalls, one on each side. This may be done partly to intercept the water from tributary streams and drains, and partly to throw it into the sea at a higher level.

3.    It may be advisable to change the outfall in order to economise capital invested in the formation of the embankment.

4.    The'outfall maybe changed to afford increased facilities for inland navigation.

5.    The outfall may be changed to facilitate the accumulation of mud, and the formation of new lands for subsequent reclamation from the ocean.

Having fixed upon the place for the station-poles at the mouth of the river, which we shall suppose is not to be changed, two on each side, a transverse section of the channel and embankments between them should be taken and entered into the field-book. The ranging of the other station-poles of the survey may then be proceeded with, at the different angles or points of the two boundary lines, together with those of the seaward and landward stations.

If the ground cannot be conveniently driven with a chain, which is often the case, the poles should be ranged purposely for the three areas of land, being ascertained by trigonometry; the first comprising all included by the survey beyond the outer boundary line; the second that within the inner line; and the third area, the land lying between the two boundary lines.

One reason for ranging two boundary lines, is to furnish the necessary data for obtaining an estimate of any line which may be determined upon between them for the embankment. So far as the work of the surveyor is concerned, no more difficulty is experienced in giving a section of the embankment at the exterior boundary than at the interior one. In point of fact the two lines are two sections, either of which may be adopted, or any third line between them. But were only one longitudinal section of the embankment given, it might be otherwise with those who have to carry out into practice the proposition of reclaiming the land from

the ocean. But when they have the alternative left of turning to the right hand or left, between two extremes sufficiently far apart to meet all the exigencies of the case, the practical solution of the problem as to the position of the embankment can then be determined satisfactorily.

When the boundary lines are covered with a sufficient depth of water to float a flat-bottomed boat or coble, the station-poles may often be more conveniently ranged by such means, during the flow and ebb of the tide, than at low water. And when thus ranged the levels of both boundary lines may be taken at high water by the same means, and the ground, longitudinally and transversely, between them also examined by boring into the subsoil under the water over the edge of the boat. Between the seaward and landward stations the surveying staff can thus employ their time without interruption until the station-poles are ranged, the lines examined, and the levels are taken.

The station-poles in the two boundary lines are ranged purposely to obtain the levels of the transverse and longitudinal sections, as well as the boundary lines themselves, and care should be taken that the line of the intended embankment lies between them, so that when it is made, the exterior line shall fall without, upon the sea side, and the interior one shall fall within, upon the opposite or land side.

The area between the boundary lines should be carefully examined, so as to determine if a sound foundation for the embankment exists between them.

The station-poles in each boundary line should be numbered, and the two lines distinguished from each other by different letters of the alphabet, and this requires to be done both upon the plan and sections. Thus, the station-poles of the exterior line may be numbered la, 2a, 3a, &c., and those of the interior line lb, 2b, 3b, &c. And as the survey commences at the mouth of the river, it will be more convenient for the consecutive numbers to run from that point, the two stations on each side; there being la and lb on the left bank, and lc, Id on the right bank.

The number of station-poles in each boundary line require to be equal, and the corresponding numbers should be opposite each other. Thus la should be opposite lb, 5a opposite 5b, and so on for the other numbers in each line.

The object of this order is to get the data for the transverse sections. Thus between la and lb there is a vertical plane that must be surveyed and shown upon the plan by a section, the ground line being represented by the bottom line, and the surface or high-

water level at the top of the station-poles being shown by the upper line.

The number of transverse sections will depend upon the nature of the ground. If the surface is a regular plane, one for each straight line will be sufficient, but if undulating, one may be required for every concave and convex part.

The transverse section has a twofold purpose to serve. First, It shows the slope or angle of depression seawards which is required both for the embankment and the drainage of the land. Second, The levels being thus given at the two extremes (the outer and inner boundary lines), and shown upon the transverse section, the height of the surface level at high water above any point in the ground line between them can be easily ascertained from the scale upon which it is drawn.

Two longitudinal sections are required, one for each boundary line. They show the top and bottom levels at each station-pole along both lines, and the numbers and letters of the alphabet that distinguish them from each other should be carefully attended to, so as to obviate error in copying from the working plan and section.

The levels are taken at high water by placing a mark upon the station-poles at the highest point to which the tide rises for the surface level, and by measuring its distance from the ground to obtain the bottom level; and the marks thus placed upon the station-poles should be carefully examined a second time at high water, in order to ascertain if the levels are correctly taken, before the base line is ranged and the theodolite placed for taking the bearings. This is usually termed “ checking the levels.” It should be done between flow and ebb, when the weather is calm and the surface of the water “ smooth ” and free from ripple, and when right they will all lie in one plane.

Directions for the Horizontal Survey.

As the horizontal plane intersects the vertical ones at the several pairs of station-poles, the points of intersection and lines which they form are consequently common to both surveys. In all cases, therefore, when the vertical lines cannot be measured with the chain, this part of the vertical survey may be profitably supplied from data taken under the horizontal survey. Such, according to hypothesis, is the case in the present example.

The base line should range parallel to the general direction of the boundary lines, or as nearly so as practicable. Level ground and easily gone over should be chosen, in order to secure the greatest accuracy in measuring with the chain. This is necessary, as the true length of the lines between the station-poles in the two boundary-lines will depend upon that of the base line. In the field the significant practical rule, the lest of three, is generally given, i.e,, measure the base line three times and take the average.

The bearings are taken as in ordinary surveys with the theodolite, and directions for such will be found in Sect. YI. Part IX. and p. 385.

The stations and lines which they form should be shown upon the plan, as represented in the plan of the Preliminary Survey for a projected Railway, Part X. Sect. I. And the two boundary lines may be represented in a similar way to what the two lines indicating the Limits of Deviation of the Parliamentary Survey are in Plate XIII.

If the Surveyor undertakes to lay down a line representing the centre of the embankment, it may be determined as in Sect. I. Part X., and shown between the exterior and interior boundary lines upon the plan, as the line of railway is represented in Plate XIII. between the two lines indicating the Limits of Deviation.

The object of thus showing the working details of the survey, both horizontal and vertical, is to enable engineers, contractors, or their draughtsmen to take off working plans more readily for the execution of the work of embanking, &c., and for this purpose doubles of the plans and sections are generally ordered when surveys are extensive.

Where several landowners are interested they may order an approximate estimate of their respective shares of the land to be reclaimed, which can easily be taken and shown upon the plan, by rules previously given in the work. But the final work of apportioning shares should be left to a more complete survey when the embankments are finished, and when the lands recovered from the sea and the river can be laid out into fields, and the lines accurately measured with the chain. (See Ex. III.)

The position of sluices, syphons, or pumping stations, and existing embankments, require to be taken and shown upon the plan, and also the position of watercourses necessary for the drainage of adjoining lands that lie above the level of the top of the embankments.

The general rule for the latter is, that no water should be allowed to descend from a higher level into the watercourses of the lands to be reclaimed that can otherwise be drained off to the sea or river.

Directions for Longitudinal and Transverse Sections for Embankments.

When the survey includes longitudinal and transverse sections of the embankment, it is more accurately and easily done to show them upon a separate plan by themselves, than to crowd them into the preliminary one, along with those previously given.

This part of the survey includes the .ground from whence materials are to be obtained for the formation of the embankment; and the quantity of materials will depend upon the dimensions of the embankment, quality of soil, and exposure to the sea.

The exceptions are few where the foundation of an embankment can be safely laid upon the surface of the ground to be reclaimed. Even in those cases where it has to be run through a shallow arm of the sea, according to the plan pursued in many cases in Holland, and also in this country, a proper foundation should afterwards be dug out inside of sufficient breadth to contain a body of clay that will prevent the water outside from finding its way by any porous strata to the interior, and thus ultimately endangering the safety of the whole area of land reclaimed.

As a general rule, sea embankments should be formed of inorganic materials, as clay or clayey gravel, and the nearest ground containing such, if at command, is generally that included in the survey. With the means of railway conveyance now at command, contractors enjoy many facilities for getting an ample supply of such materials that were not within the reach of their predecessors, so that if the surveyor finds a choice, the best quality may be preferable, although situated at a greater distance.

The principal reasons why vegetable and other organic matter should be excluded from the interior of embankments are as follows :—

1.    All organic matter of the kind in question is subject to decomposition. If, therefore, an embankment, or any portion of it extending from the outside to the inside, is formed of materials that contain a large percentage by bulk of such, this percentage may in the course of time be dissipated in the atmosphere, and the inorganic portion reduced to an open porous body approaching in character as if formed of pure sand or gravel, through which the sea-water will ooze, and eventually flow.

2.    Embankments which contain a large percentage of vegetable matter are liable to destruction from vermin of a very multifarious and indefinable character, as rats, moles, mice, worms, and innumerable lands of insects, some of which are so small as to be invisible

to the naked eye. This motley brood live upon each other and the decaying vegetable matter as their normal food : the organic portion of the embankment is thus literally consumed, while the inorganic part that remains is so burrowed as to resemble in some measure a sponge, through which the sea water will ooze, eventually affecting a breach, as in the previous case.

3. A third reason may consist of a combination of the first and second, and in practice this is perhaps what is generally experienced, the depredation of insects and other animals, together with the action of their excreta and putrefying remains, hastening the chemical changes that destroy the stability of the embankment.

When old embankments that have to remain are included in the survey, they should be carefully examined in order to have destructive causes of the above kinds, and the extent of injury done, faithfully delineated. For this purpose a good microscope is essentially necessary; and if it is discovered that the embankment is giving way, it may be advisable to recommend the employment of men to whose professional labours these branches of science belong. The amount or area of damage done will, however, require to be examined by the surveyor, and shown upon the plan and section of the old embankment.

EXAMPLE II.

DIRECTIONS FOR THE SURVEY OF A TIDAL RIVER.

The present survey may be presumed to contemplate improvements extending from the mouth of a tidal river upwards to the highest point where the tide flows. The field operations lie between Examples I. and IV., and are chiefly sectional in character, embracing longitudinal and transverse sections of the channel and its embankments.

Many of the directions given in the last example relative to the ranging of the station poles, the taking of levels at high water, and the survey of the horizontal and vertical planes for the preliminary plan and sections, are applicable to the present, and therefore need not be repeated.

The chief points requiring notice are as follows :—First, The breadth of the river as determined by its tributaries. Second, The breadth of the river as determined by tidal action. Third, The velocity of the river as determined by its tidal length. Fourth, New channel and embankments. Fifth, Methods for counteracting tidal action. Sixth, Influx of tributary rivers.

1. The breadth of a river as determined by its tributaries.

The increase required in the breadth of a river to contain the waters of tributary streams, so as to maintain its velocity at low water, and prevent damming back above the point of junction, is directly as the volumes of water which the tributaries discharge.

If, for example, a tributary river discharges a volume of water equal to that flowing in the tidal river, then the breadth of channel required by the latter below the point of junction will be double that above where the two rivers unite. Or the breadth of the united rivers, as it requires to be shown upon the plan and transverse section, is equal to the sum of the breadths of the two rivers before their junction, their velocities and volumes of water being equal.

Had the tidal river contained twice the volume of water it received from its tributary, then the increase of breadth required would have been one-third; and for any other increase of volume, greater or less, a corresponding increase in the breadth of the channel at low water is necessary, to preserve the natural velocity of the tidal river below the point of junction, and to prevent the damming back process above this point (allowing no increase of breadth for friction, which must always be determined on the spot).

In applying the above data either to. the laying out of a new channel for the tidal part of a river, or in straightening or effecting any improvement of the old one, it must be borne in mind that an increase in the depth will not compensate for breadth. The rule is founded upon an established law that cannot be violated without consequential losses being sustained, for a greater velocity effected by an increased altitude of the surface-level of the water,, involves the damming back process above the point where this superelevation of the surface-water takes place; and when the increase of depth is effected by digging a deeper channel, the practice is abortive, amounting to a standing pool below the natural inclination of the bed of the river, so that no increase of velocity is obtained to convey the extra volume of water received from the tributary.

2. The breadth of a river as determined by tidal action.

The action of a tidal river upon the sides or slopes of its channel and embankments, and also upon its bottom, is directly as its depth, the quality of the ground being equal or uniform.

This rule is based upon the established data at sea, viz., the deeper the ocean the higher the waves and the greater the force with which they strike the vessel or lash the shore, and the heavier the amount of damage done during a storm.

On a sandy, gravelly, clayey, or muddy sea-shore, the counteracting force of the return water down the inclined slope formed by the flowing and ebbing tide is a safer defence to the slope than the more solid materials of its structure.

Under differences of geological strata the coarser the quality of the gravel or sand, the more obtuse is the angle of the slope of the sea-beach, and vice versd.

The same practical data will be found exemplified in beds of rivers that have been scooped out by the natural action of the water.

The above data may be seen instructively illustrated on a sea-beach during a storm, or channel of a river during heavy floods; and both examples should be examined on shores of different qualities of soil, to obtain the invaluable information they afford.

The length of the transverse section representing the breadth of a tidal river at its mouth will therefore be as the height of the high-water level line above the ground line or bottom of the channel; but inversely as the angle of depression which the ground line makes with the horizon or high-water level.

As the extra depth of the river above its mouth produced by tidal action gradually decreases until it ceases to exist, it consequently follows that the extra breadth required for tidal action will also decrease in a corresponding manner, and finally cease to exist at the point where the tide ceases to flow.

This gradual decrease of breadth requires to be carefully ascertained in the survey and delineated upon the plan, and also shown by the transverse sections of the river, at its mouth and at the junctions of the tributary streams and upper end of the survey where tidal action terminates.

The general doctrines taught in the preceding directions have reference to the uniform retarded and accelerated velocity of the river during the flow and ebb of the tide, so essentially necessary to the safety of its channel and embankments. Thus during the flow of the tide the velocity of the river will be retarded, and the direction of the current at its mouth and upwards gradually reversed, owing to the damming back process that takes place by the rising of the surface-level of the ocean above that of the water in the river. But when the ebb commences the river will again begin to flow to the ocean with an accelerated velocity. Now, if the above rules have been properly carried out, then the retarding and accelerating processes will be so uniform as hardly to be perceptible to the eye of the observer, the downward current of the river being gradually counteracted by the upward flow of the tide, while the

elevated surface of the dammed-back water will during the ebb as gradually fall to the lower-water level.

If, however, the river is confined between high embankments at its mouth, and if the channel upwards is of unequal breadth, being narrow in some places, and spread out into basins of great width in others, so as to form inland seas or lakes during high water, then “ races ” (as they are sometimes termed on old plans and in old deeds) will be found in the narrow parts between them. The force of the stream jin such places is not easily calculated, or embankments defended from the impetuosity of the current during spring-tides, when a more than ordinary rapid fall of the ocean in front is attended by a heavy inland flood behind, or vice versd, when an extra high tide is accompanied with a low level of the river, so that there is little or no counterbalancing action to the upward current, by the damming process of the downward one.

3. The velocity of the river as determined by its tidal length.

The velocity of a tidal river at low water is inversely as the length of the tidal part; or, the shorter the distance between the mouth of a tidal river and the highest point to which the tide reaches, the greater the velocity of the river at low water; and, vice versd, the greater the distance the less the velocity.

By straightening a meandering tidal river its velocity at low water will be increased, because its course will be shortened.

The practical data upon which the above two rules are founded may be given thus. If we assume the difference of the depth of the river at its mouth between low water and high water to be 12 feet, then this is the fall of the whole tidal portion of the river included in the survey. If we next assume that the length of the tidal part is twelve miles, that the ground is of equal quality and the inclination uniform throughout its length, it will give a fall of one foot per mile. But if the length is twenty-four miles, and the other conditions the same, then the fall per mile will only be six inches ; consequently the velocity in the latter example will only be that due to six inches, or one-half the former.

The formation of bars in the beds of rivers is produced by a decrease of velocity, the sand and gravel being deposited when the force of the current is so reduced as not to be sufficient to carry them farther. Bars across the mouths of rivers are formed in the same way, or jointly by the action of the sea.

The extra depth of the tidal part of a river which gradually increases towards the sea, when the bottom soil is of uniform quality, is produced, partly by the accelerated velocity of the water during

the ebbing of the tide ; partly by the scooping-out process of ojiposing currents during the flow of the tide; and partly by the action of the waves towards the sea in stormy weather.

If the ground forming the bed of the river is not composed of materials of uniform quality, the inclination of the bottom will not be uniform throughout its length.

Bars formed in the bed of the river require to be shown both by longitudinal and transverse sections ; inequalities in the bottom, produced by different gradients, only in the former, unless special circumstances call for a transverse representation. Bars beyond the mouth of the river seawards are usually shown upon the plan, the depth of Avater over them being represented by figures. All bars, both at sea and in the channel of the river, if dry at low water, require to be surveyed and shown upon the plan as dry land; where rock crops out in the bottom, it should be indicated both on plan and section, and also any special diversity of soil, as a change from clay to sand.

4. New channel and embankments.

A new channel for a river may be occasioned either by changing its outfall, or straightening the course of the old one, and both (it may be presumed) are included in the survey.

A new outfall.

In surveying ground for a new outfall, the inclination of the bottom of the sea and the quality of the ground require, in the vast majority of cases, to be carefully examined for a considerable distance out seawards, before the depth and breadth of the mouth of the new channel can be determined, according to the directions given under the second head of this example. The exposure of the new outfall also requires to be attended to ; for if the one side of the' channel is more exposed than the other, a greater slope must be given both to it and the embankment, so as to make the return action of the water down the inclined plane defend it, as formerly shown, during a heavy storm.

In a few exceptional cases the new outfall made be made over a rocky bottom at various levels between high and low water, in which the breadth of the river from its mouth upwards will be determined accordingly. Thus, if the new channel is made to intercept the water from the elevated grounds on one side of the low lands, and to discharge itself over a rocky bottom on a level with high water, then the new river will cease to be a tidalfone and to be subject to tidal action; consequently its breadth will be determined by its

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inclination and volume of fresh water, as in the next example. If, however, the rock is on a level with low water, then the breadth of the river will depend upon the nature of the ground forming the sides of the channel, and upon the exposure of its mouth to the waves of the sea during storms; and for intermediate levels the rock will defend the channel from tidal action according to its height, tidal action being inversely as the height of the rocky bottom above low water level.

Straightening a river.

In the language of practice, straightening a river more frequently means reducing the number of curves and right lines than the formation of one straight channel.

In every survey, therefore, the direction of a new channel for a river has to be determined on the spot.

If the object of a new channel is to effect a more efficient state of drainage, then the nearer its direction approaches a right line, the shorter will be the whole length of its course, and the greater the fall gained at low water for any given distance.

By such means the length of the tidal portion of some rivers in the United Kingdom may be reduced to one-half—of others to one-third, and of one or two exceptions to one-fourth of their present length. In such cases the increase of fall gained, and also the area of land reclaimed from the river, may be easily calculated.

The directions for the capacity of the straight portion of a new channel are the same as those given for the survey of the longitudinal and transverse sections of the old.

If a new channel is to be opened between two bends of the old so as to cut off one intervening bend, the best line of direction will be that of tangents to the two curves.

In straightening a river according to this rule the radius of any curve may be easily increased, so as to change the angularity of the current when it strikes the concave bank at too acute an angle. This may be done, for example, by extending the tangential direction of the new channel outwards, the required distance from both the old bends, so that the two station poles at the two extremities of the straight portion of the new channel shall be in the two curves at the two points where the two radii touch the new line of channel, so as to form with it right angles.

on

In the field this is done by going backwards until the whole of the curve on one hand and new channel on the other shall appear the river side of the station poles ; and the centre of the circle at each curve is the point where two lines ranging from station poles

at the two extremities of the curve intersect each other, the one line forming a right angle with the new channel, upwards or downwards as the case may he, and the other a right angle with the direction of the old channel, downwards or upwards as the case may be.

The less the radius of any curve, the less must be the inclination of the slope of the channel and embankment on the concave side, to defend it from the current, and, vice versd, on the opposite or convex side to maintain the capacity of the river required by tidal action.

A transverse section of a curve of a river is thus different from that of the straight portion of the channel.

5. Methods for counteracting tidal action.

One method for obviating the destructive action of tides in tidal rivers has already been given, viz., by changing the outfall to a rocky bottom.

A second method is by a tide-lock, or several tide-locks, if the volume of water is large. Such may be constructed either at the mouth of the river, or in the first narrow place above it, if preferable.

A third method may consist of a tide-lock for shipping at some distant point communicating with the river by means of a canal, the channel of the river itself being defended by a series of waterfalls from the high to the low water level, constructed on the principle of a salmon ladder, and which might serve that purpose.

A fourth method consists of floodgates or sluices, in number according to the size of the river, across its mouth or some more convenient place above it.

These several methods are now in operation, and hence call for surveys.

The details of the practice are similar to those of canal-surveying, and will be given under Example YI.

6. Influx of tributary rivers.

The survey of a tidal river generally includes that of its tributaries to a distance as far up from their mouths as contemplated improvements extend.

Where lands have been reclaimed, or are proposed to be so, at the mouth of a tidal river, as in the example, the first two tributaries surveyed are those that previously discharged, or now discharge, the drainage waters of the more elevated grounds through the unreclaimed land into the sea, which require to be diverted from their

old channels and turned into the river by two new courses—one on each side the main stream.

Where old embankments exist at the inland boundary of the newly-reclaimed land, they may probably be turned to profitable account by forming one of the embankments of the first tributary, if the upland drainage-water cannot be intercepted at a higher level.

In those cases where no land has previously been reclaimed, and where embankments do not exist, the high-water levels of spring tides often determine the course of the two first tributaries for intercepting the drain age-water of the elevated grounds, and discharging it into the main river by opposite channels.

These are questions that always require to be determined on the spot by taking the levels of high water at the necessary station poles, and the other station-levels on more elevated ground. They are seldom surrounded with much practical difficulty; and, as the formation of the two first intercepting tributaries form the initiatory work of the contractor in the reclaiming of the lands lying between them and the ocean, this division of the survey should be the first finished, as the plans or copies of them may be demanded before the others.

The waters'of a tributary and main river should unite with equal velocities, so as not to form bars across either channel, or in any way disturb the uniform flow of the principal stream.

Where a tidal river runs through a narrow valley, tributaries very frequently enter at too great a velocity; but such can easily be reduced by means of a waterfall, or series of waterfalls, as in the case of mountain streams in Example V.

The longitudinal and transverse sections of tributary rivers are similar to those of the principal, so that the directions given relative to the latter are applicable to the former.

EXAMPLE III.

GENERAL DIRECTIONS FOR THE HORIZONTAL SURVEY OF LANDS RECLAIMED UNDER THE FIRST AND SECOND EXAMPLES.

When a large area of land has been recovered from the sea and a tidal river by embankments, it requires to be surveyed and laid out into fields and farms for agricultural purposes. And if it belongs to different proprietors, as is frequently, if not generally, the case, when the reclaimed lands extend to both sides of the river, the whole will have to be apportioned according to the several proprietary rights involved.

In its general character the survey resembles that of a common proposed to be apportioned under the Inclosure Act, for which directions have already been given; but in the following details it differs in many respects—viz., 1. Quality of soil; 2. Drainage; 3. v Embankments ; 4. Roads, fences, and farm buildings; and 5. Area outside embankments.

1. Quality of soil.

The agricultural value of the land cannot be separately investigated from its proprietorship. This arises from the difficulty of distinguishing at times between different qualities of soil, especially that reclaimed from the sea, before it has yielded crops of any kind, so as to afford practical evidence of its productive value per acre; consequently, surveys of this kind, when marches are not well defined, often call for the highest degree of professional talent to make the most of the land, and to do justice to the landowners and tenants interested. On the other hand, when the line of subdivision between properties is well defined, and when the land is of uniform quality, the problems of apportioning and laying it out into farms and fields are of easy solution.

2. Drainage.

In laying off the principal drains for the drainage of the land they should, as far as practicable, form boundary lines between estates and farms. They are usually divided into main and tributary. The former are the largest; they carry the waters of the latter directly to the sluices or pumping-stations at the embankments ; and the latter, sometimes designated cross-drains from their position, discharge the water they collect from the fields and farm ditches into the former. When the land is comparatively level, the water in the small ditches between the headlands of fields very frequently flow to both sides from the middle—a difference in the depth being all the fall that can be given them. In such cases the watershed should be shown upon the plan. In other cases the water-shed may be at one side or between two fields—the water drained from the one field running to the right, and that from the other flowing to the left. In other cases, again, the field ditches are cut to a more uniform depth for the purpose of bottom irrigation, as shown under Example VI.

Straight lines and rectangular fields form the rule, one that also applies to main drains, estates, and farms. The tributary drains, however, where they enter the main drain, should have a gentle curve in the direction of the main current, so as the waters of

SECTION III.

the former shall not retard the velocity of those of the latter in times of flood, or silt up the channel when the stream is low, and its force reduced to a minimum.

The laying off of the main and tributary drains necessarily involves a large amount of levelling, which is now done with instruments, either spirit-levels or draining-levels; and, as the inclination is often less than 1 in 5000, the work requires to be carefully performed, otherwise flood-water will prove its inaccuracy.

3. Embankments.

The lines for the main drains and other subdivisions having been determined, and station poles ranged for measuring them with the chain, the ranging of station poles along the top of the embankments from one extremity of the survey to the other, may be proceeded with as the next step in field operations. The base-line of the previous survey with the theodolite may be continued; but if the lines along the embankments form a series of right lines of considerable length, with but few curves to break their angular connexion, it may be advisable to adopt one of them for a base-line —or the whole series for a series of base-lines—as right lines along the top of embankments can always be driven with greater accuracy than those across broken and uneven ground. The true position or range of the right lines on both sides of a curve can be easily determined by special check-lines and the other bearings of the survey.

The preliminary survey was made exclusively with the theodolite, the measuring of the base-line with the chain excepted; the present is taken both with the chain and theodolite, the distance between the station poles now placed upon the top of the embankment, as well as at the base, being carefully measured with the chain, as also the length and angle of the slopes and total breadth of the base : all deviations from the former dimensions or position of the embankment made during its formation require to be measured, as also the inside drains running parallel to it, and junction of the main drains therewith. The true position of sluices, pumping stations, &c., require to be shown.

The directions for taking the bearings in this survey are similar to those given in the preceding, and also those for laying down the true position of the embankments and river upon the plan.

The lines and offsets to subdivisions, &c., are measured with the chain and offset-staff, and the measurements entered in the field-book as in ordinary surveys.

4. Roads, Farms, and Farm-buildings.

Public roads, and private ones leading from them to farm homesteads, require to be between fences, but farm roads from the homestead to the fields may run alongside fences, being only fenced on one side.

In laying off land for public roads in a level district of newly-reclaimed fen or marsh land, the two chief points for consideration are as follows :—First, that without increasing their length they shall serve as much as possible the private purposes of the individual farms through which they pass; Second, that they shall form the shortest or the nearest and most direct route to church, markets, railway-stations, &c.

Sometimes the direction of roads and fences is determined by dry grounds rising above the general level of lands recovered from rhe sea and tidal rivers, and offering inviting sites for farm-buildings. As such are in many cases invaluable, they may not only determine the ground for the homestead, and the direction of roads and fences, but also the size of farms; consequently such are preliminary questions that ought to be settled with landowners and tenants before boundary and subdivision lines are finally determined.

Live fences have sometimes an open ditch on each side, in other cases only a ditch on one side, while in a few exceptionary examples as yet the lands are thoroughly drained and cultivated close up to the bottom of the hedges, the hedgerows being regularly hoed and kept as clean and free from weeds as drilled green crops of beans. Such, therefore, are also preliminary topics that, like the above, should be determined before the lines are finally ranged for being driven with the chain, more especially if the plan is to be drawn to a large scale, and everything shown in a highly-finished style.

The directions for laying off ground for farm-buildings, including labourers’ cottages, water for the same, and for stock in the fields, are similar to those given in a subsequent place, Part VII.

Lands reclaimed from the river are generally added to the adjoining farms, the fences running directly to the river. This may give rise to alterations in the cross-fences, and the rule for determining the position of both subdivisions will be the size of the farm and fields in every individual example.

5. Area outside Embankments.

new

The extent of the survey outside the embankments will depend much upon whether land is accumulating, or the contrary. If deposit is being formed at every tide, landowners have an

interest in ascertaining the direction of this new formation, in order to watch its progress and preserve the rights which they or their posterity may possess to reclaim it from the sea at some future period. A survey, therefore, may be. presumed to determine the proprietary rights of those who employ the surveyor. It may extend to a considerable distance beyond what dry land appears above low water, and the depths of ground under low water level are taking by soundings, and shown upon the plan in their proper places. Where the nature of the ground will not permit of station poles being set, as is frequently the case, buoys require to be anchored for taking the bearings. Buoys may thus be in deep water and station poles on dry ground, and the bearings are taken partly with the theodolite from the top of the embankments, or any other land station, and partly with the compass—a mariner’s compass being preferable in a boat at sea to the pocket one of the land-surveyor or the compass of the theodolite. The position of the buoys at sea, and land-stations from which the bearings of the former are taken, should be shown upon the plan, so as to enable landowners at any time to take the depth of the sea at low water at any desired point, in order to ascertain whether deposit was making progress or not.

EXAMPLE IY.

GENERAL DIRECTIONS FOR THE SURVEY OF A RIVER IN A LARGE VALLEY FROM HIGH-WATER LEVEL UPWARDS.

The survey under this example is presumed to contemplate improvements in straightening and embanking a large river immediately above high-water level, either in terms of 24 & 25 Viet. c. 133, or by the mutual agreement of the proprietors interested. It may be presumed either to commence at that point of the river where Example II. terminated, and to end where the next example begins, or to commence at the ocean. Field operations will comprise horizontal data, as shown upon the plan, and vertical data, as represented by longitudinal and transverse sections. But as rules for the former have already been given, the directions will chiefly apply to the latter, the vertical survey under the following eight heads :—1. Outfall levels of the river and its tributaries; 2. General fall of the river; 3. Volume of water; 4. Area of land required for embankments; 5. Subdivision and apportioning of land where the river is straightened; 6. Water power, irrigation works, bleaching grounds, &c.; 7. Provision for the drainage of adjoining lands;

1. Outfall levels of the river and its tributaries.

The beds of rivers at high water outfalls may frequently be lowered so as to drain more efficiently at low water the lands lying on both banks upwards. The outfall-level is, therefore, the first that has to be surveyed.

There are four methods by which the surface level of the outfall of a river may be lowered; viz., first, by widening the channel; second, by deepening the channel; third, by straightening the channel j and fourth, by increasing the velocity of the current.

FIRST METHOD.

Rivers gradually increase in breadth from their sources to their mouths, according to the influx of their tributaries, the width being thus inversely as the depth at any section.

The natural breadth of a river is, by the above rule, determined by its tributaries and the quality of the soil of which the bed is formed ; consequently, if the natural breadth of the outfall is doubled, the depth will be reduced one-half, and the surface level thereby lowered in the same ratio (making no allowance in breadth for extra friction). Thus, if the present depth is 6 feet and the width 300, then by increasing the width to 600 feet, the surface level of the outfall will be lowered three feet for drainage, the velocity in both cases being the same.

In practice, the rule will be found generally to apply thus : the river at the outfall, or towards this point, will be found to be narrowed by various obstructions, thereby damming back the water and raising its surface level; consequently, if such obstructions are removed, it follows that the river will be widened, and its surface level lowered.

The narrowing causes that require special notice are rock; the central piers of bridges; tide-locks and sluices; rocky channels; quays and harbourage for shipping • boats and barges moored or plying in the river ; trees deposited; piles driven in for fencing and other purposes.

When obstructions of the above kind can be removed the increase of breadth gained requires to be shown upon the plan ; and the same data applied to the outfalls of tributaries.

SECOND METHOD.

When the present breadth of the outfall is sufficient, but the

bottom of the channel so elevated by rocks, bars, or the like, that it can be lowered, such would lower the surface level at low water, and thus improve the drainage during a large period of the ebb and flow of the tide. This would allow the tide to flow farther up the river, thereby subjecting it to tidal action, for which provision will require to be made as previously directed (Example II.), but if half the period of the ebb and flow can be gained for drainage, the advantage may do more than compensate for all expenses.

The formation of new bars across the outfall, in examples where old ones have been removed, can be prevented by reducing the current of the river to a uniform velocity. Thus, the cause of the deposit of sand and gravel being a greater velocity and force of current at some part in the river above the outfall than at the outfall itself, it follows that if the river is deepened at that part, and widened, if necessary, so as to bring the bottom and sides of the channel to a uniform inclination, then the formation of a new bar could not take place; for the velocity and force of the current at the latter section would be equal to its velocity and force at the former section; consequently, no depositing materials would exist to be thrown down, the scooping out and depositing causes being both obviated.

The details of the survey under this method lie chiefly on the vertical plane in the longitudinal direction of the river, but two transverse sections will be required, one at the outfall, and the other at that part of the river above proposed to be deepened.

THIRD METHOD.

The increase of fall gained by straightening a river has already been shown in Example II. Thus, if by straightening a river its length can be reduced to one-half, the fall per mile will be doubled, and the surface level lowered by such means directly as the velocity is increased. Hence the gain for drainage.

In practice, the rule may be carried out in two ways. First, By straightening the river and removing the outfall farther down the tidal channel, which would improve the drainage at low water. Second, The old tidal bed may be preserved, and two tributaries, one on each side, taken farther down by two straight channels to improve the drainage. An extra fall of several feet may frequently be gained by such means. Under the second way the surface level of the outfall in the old bed will be lowered directly as the amount of tributary water removed.

Under this method the details of the survey are chiefly on the horizontal plane.

FOURTH METHOD.

It is manifest that if the velocity of a river at its outfall can he doubled, the depth will he reduced one-half, and the surface level lowered in the same ratio, as the quantity of ^water discharged in both cases would be equal in equal times.

In practice, the rule is seldom applicable to examples of the kind under survey unless where stagnation of the water in the river takes place at the outfall, not from narrowing cases, as in the first method, but from the reverse, too great a breadth, the object of the survey being to obtain the necessary sectional data for confining the river within a proper breadth of channel at its outfall, and thus obtain a sufficient velocity to prevent deposit of every kind.

This extra breadth of outfall may be effected bjr one of two causes either by a rapid above having a sufficient force to scoop out a broad pool at the outfall, and thus form a bar on the other side; partly by the joint action of the tide; or else by too small a velocity, the growth of reed, and accumulation of muddy deposit. Both examples are more frequently met with when the outfall is at the sea, than when it is far inland at the point where the ordinary tide ceases to flow.

In the case of a rapid above the outfall and a bar below, the rule is to deepen the channel at both places, so as to bring the bottom to a uniform inclination, as under the second method, and to reduce the width of the outfall section to its proper dimensions.

In the case of a deposit of mud the narrowing of the channel at the outfall, and the confining of the river in its proper rectilinear direction, will increase the velocity, clear out the bed and lower the surface level of the outfall at low water ; but if the growth of reed has been allowed to commence, and the plants to take deep root, they may require to be cut: but the execution of works of this kind do not belong to the survey: at the same time, the surveyor may be called upon to report as to their magnitude and how they are to be done.

In both these examples the details of the survey are similar to those of the second method.

The above methods of lowering the surface level of the outfall of the principal river are applicable to the outfall levels of its tributaries ; but when the general inclination of the bed of the former is altered, that will also affect the outfall levels of the latter.

2. General fall of the river.

The position and level of the outfall section having been taken

the next branch of field operations is to find the whole fall of the river between the lower and upper ends of the survey, in order to determine its general inclination, and how much additional depth can be gained for drainage and water-power purposes. Technically the operation is termed levelling, an illustrated description of which will be found in Sect. I. Part X. as applicable to railway surveying. The work is done by instruments of various kinds, but as levels thus taken are subject to errors that require to be corrected, it is usual in river surveying to check them by water levels, all our large valleys affording ample opportunity for taking “ check levels ” of this kind. In this and in several other respects, the survey differs widely from that of a railway.

The errors that require to be corrected are three in number : first, tangential errors; second, errors due to refraction; and third, hypotenusal errors.

Tangential Errors.

There are three errors of this kind that require notice. First, deviation from the arc as to curvature, or errors in curvature; second, deviation from the. arc as to length, or errors in length ; and third, deviation from the horizontal level, or errors in the practice of taking the horizontal levels.

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First. If BE, in the annexed diagram, represents an arc of 60°, then ab is radius, bd tangent or the horizontal level, and AD secant. In the direction of the secant AD the extent of deviation is ED, which in this case is equal to radius ; but if a perpendicular is-dropped from BD to E, parallel to AB, the extent of error will then be equal to the versed sine, or half the radius AB, when the arc is 60°. The former is the extent of error usually adopted, viz., the difference between the radius and the secant of the arc; so that, taking the diameter of the earth at 7912 miles, the correction for 60° would be 3956 miles, i.e., the semi-diameter.

If we suppose the length of the tangent one mile, then, according to Theor. VII. Part I, ed is equal to V(ab" + bd²) - ab ; the extent of error would, therefore, be V(3956² + l²) — 3956 = ‘000128 + of a mile, or 8 inches. Por two miles V(3956² + 2²) - 3956 = ’000505 + of a mile or 32-028 inches

Second. The next error consists in taking the length of the tangent for the length of the arc, or rather a rough guess for it, as the line measured in practice makes an angle of elevation with (bd) the horizontal level. Thus in the diagram if bf is an arc of 45°, then BC is tangent and equal to radius or 3956 miles ; whereas bf is only 3107‘04 miles, according to data subsequently given under Example VI. The error in length for 45° is, therefore, 848⁻9 miles. For 60° the tangent bd is 6301-6 miles; the arc BE 4142-7 miles, giving 2158-8 miles as the tangential error in length.

Third. There are two methods of taking horizontal levels, the one with a backsight from the station, at which the level is placed to the levelling staff; and the other with a backsight and a foresight to two levelling staves. The former practice may be termed the single tangent method, and the latter the double tangent method.

The single tangent method of taking levels is the correct one. It is subject to the two preceding tangential errors, and also to those arising from refraction and measurement with the chain. As the dead levels of canals will furnish a more instructive illustration of the practice than the surface levels of a river, it will be illustrated under Example VI. But general directions will be given under the longitudinal section of this example.

The double tangent method of levelling, although it has received • the sanction of long use as being sufficiently correct for general purposes, is nevertheless erroneous; consequently the single tangent method has to be adopted when a greater degree of accuracy is required, or when obstructions prevent the level from being placed half-way between the two levelling staves. An illustrated description of the practice will be given under Railway Surveying, Sect. I. Part X., as it is better adapted for roads and railways than for surveys of rivers, canals, waterworks, and the like. It will, however, be advisable in this section to point out to the student the errors that attend the double tangent method of levelling.

The double tangent method is based on assumptions contrary to fact and geometrical rule, as the following three examples will show.

Ex. 1. The practice in the case of an acclivity or a declivity assumes that the two secants of the two arcs are equal, while it proves the contrary, the readings from the two levelling staves indicating a difference. Thus, as the two levelling staves represent two “ normals,” or portions of two radii, it follows that when they indicate a difference, as they always do on an inclined plane, the two radii, with their respective arcs, tangents, and secants, are also different, which is the contrary of the assumption that they are equal.

Ex. 2. If an exceptionary position of the level and levelling staves is assumed, viz., that the readings from the latter are equal, thereby proving the two tangents, secants, radii, and arcs equal, as when the ground is a horizontal level, or when the level is placed on the top of a hill, and the two levelling staves on the two declivities downwards—or when it is placed in the bottom of a concave surface or valley, and the two levelling staves on the two opposite acclivities upwards—then another assumption (which will be pointed out under the next example), that the back-secant is different from the foresecant, and that the two thereby correct for curvature, is contrary to the previous one that they are equal.

Ex. 3. The practice assumes that the level is equidistant from the two levelling staves between which it is placed; (see diag. Sect. I. Part X.), that the latter represent two secants ; the former a common radius; the two lines of sight two tangents; and that as the two tangential deviations are the one minus and the other plus, they, therefore, correct each other. Now these assumptions, passing over the errors already pointed out, can only, under the most favourable circumstances of the case, be received as very distant approximations, obtained contrary to geometrical rule, for the half-way position of the level cannot be determined until the whole distance is measured with the chain, until the horizontal levels are taken, and the true lengths of the two arcs and two tangents ascertained, which cannot be done by the practice itself. Hence the reason why surveyors are obliged to adopt the other method.

Errors due to refraction.

Directions for correcting errors arising from refraction on the horizontal plane will be found in Sect. I. Part X.

On the vertical plane the correction for refraction is different. Thus, when the telescope is levelled in a horizontal direction, the amount of refraction is 33'; but when elevated to 45° it is only 1', or more correctly 57", being three seconds less than the thirty-third part of what it is at the horizon ; while directed to the zenith it is nil, a star or bird passing over the field of the telescope appearing then in its true position.

Ilypotemsal Errors.

Errors of this kind arise from the undulating and uneven surface of the ground, which requires very great skill in driving the chain to take the distances between the level and levelling staff with that degree of accuracy which the nature of the survey demands. In horizontal surveying the correctness of the length of the lines measured with the chain is tested both by tie lines and the trigonometrical data obtained from the bearings taken with the theodolite; but in a vertical survey there are fewer opportunities of testing the geometrical truthfulness of the work.

The line measured with the chain in the vertical plane is the hypotenuse of a right-angled triangle, represented by ci in the diagram to Example VI., the triangle there being ixc, and the errors that require to be corrected are' the deviations from that line. In practice the rule is to place the level and levelling staff in the same hypotenusal plane. This frequently gives rise to short backsight distances at one operation, when the greatest degree of accuracy is ordered. And when a concave or convex surface intervenes between the level and levelling staff, a second levelling staff, and a third if required, is, or are, placed between upon the highest and lowest grounds; the object of the intervening levelling staves being to take the offsets from the line of collimation or sight, so as to determine the curvature of the surface (the line measured), and the true length of the hypotenusal line sought. (Statipn poles will answer for intermediate staves.) If, in measuring the hypotenuse on the horizontal plane the leader was to place his arrows sometimes a few feet on one side of the line, and sometimes a few feet on the other, the driver would be aware that the measurement thus found would exceed the true length of the hypotenuse sought, just as the two sides of a parallelogram do that of the diagonal. And the same practical rule applies to similar deviations from the hypotenuse on the vertical plane.

Check-water levels.

In valleys check-water levels can frequently be taken by the water levels of canals and mills; or they may be taken by damming back the water in side drains, or even in the river itself under survey. Examples of this latter kind might be quoted where the dead level for several miles was ascertained in less time than it could otherwise have been. Thus, a dam was made in the evening, and next morning stakes were driven in to the water’s edge, which finished the work. In extensive surveys, where the investment of large capitals is involved, check-water levelling should never be lost sight of when at command ; more especially if the inclination of the river is small, and consequently every inch of fall gained for drainage invaluable.

Longitudinal section.

In comparatively level valleys the inclination of rivers is frequently so nearly uniform that a longitudinal section is not ordered, transverse sections at particular parts being considered sufficient. In other cases* longitudinal sections may be required for those parts of the river that are straightened; but when extensive

improvements throughout the survey are contemplated, a section of the whole length may be necessary.

If we suppose that the diagram, Prob. V. Part IV., represents a portion of a river that is proposed to be straightened by a new rectilinear channel; that a and x are two stations in the survey; and that the new channel is to be on the same side of these two stations as the old one, it will give the student an illustration of an example of the latter kind requiring a longitudinal section for the whole length.

The horizontal survey may be taken as there directed; a plan drawn according to the dimensions given in the field-book; the course of the new channel represented by two straight parallel lines, and the severed lands apportioned.

Under such conditions the dimensions sought are the lengths of the old and new channels, the fall between station x and station a, and the surface levels of the old river, when such are necessary.

If + a is visible from + x, or if +x and + a are seen from a third station beyond + n, so as to form two straight lines xn and net, then the line ax sought may be found as follows :—

Measure an and nx with the chain, and take the included angle with the theodolite, which will give ax by Case II. Part VIII.

If + a is visible from + x, the angle of depression would give the angles of the right-angled vertical triangle of which ax is hypotenuse, so that the fall may be got by trigonometry, the correction for curvature being the radius of the earth at x minus the versed sine of the arc between that radius and the radius of the earth at a, similar to what is shown in the diagram, Example VI., the corresponding right-angled triangle there being icx, and the correction ix-xz — zi, the fall between i and c. Or it may be got by various formulae given under Example VI.

If the levelling staff when placed at + a is sufficiently high to reach the line of collimation or horizontal level at + x, the fall will thus be determined at once, the correction for curvature being now secant minus radius.

But if the levelling staff has to be placed at + n and + a, or any other number of intermediate stations, then the sum of the falls is the total fall, and the corrections for curvature and refraction have to be made at each fall, and not for the sum as a whole.

In a similar manner if three intermediate station-poles are ranged in a line between + a and + x, one in cm, the second in mr, and the third in the line opposite, it will intersect these three lines (which belong to the horizontal survey) so as to form four triangles.

In each of these two sides and the included angle are determined by the horizontal survey, all the given angles being proved by tie lines. The four remaining sides or distances can therefore be got by trigonometry, when their sum will give the total distance ax.

The length and fall of the new channel thus found will give its inclination, and also the fall of the old one, but not the length and inclination of the latter.

The length of the old channel is determined by the horizontal survey. It should be carefully measured with the chain, and will be found in the above example nearly one-third of its length longer than the new one; i.e., the length of the new channel is only two-thirds that of the old one, so that there is a gain on the fall of one mile in three for improved drainage.

Surface levels of the river.

If the velocity of the river is uniform, its length and fall will give its inclination; or if its inclination is uniform, its fall and length will determine its velocity. But such conditions are never realised in a meandering river, the bends invariably proving the flow to be both accelerated and retarded; for a decrease of velocity at a bend involves an increase above, while as rivers descend toward their outfalls, their general velocities decrease. Hence the practice of taking the surface levels of the stream in order to determine how far it deviates from the hypotenusal plane of its inclination.

Although the course of the river is crooked, its longitudinal section may be represented as straight. Thus in the annexed diagram if ab is the length of the old channel, ED the length of the new one, and bc or EE the fall without correction for curvature, then the triangles ABC and EBC will be two right-angled triangles • similar to cix in the diagram Example YI. The two normals in

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the two cases are BC and EE in the above, and ix and yc in that of Ex. YI. The hypotenusal plane of the old channel is ab, and that of the new one eb.

The correction for curvature is generally so small as with difficulty to be represented on the section. At the same time, it must not be overlooked in plotting the dimensions, for reasons subsequently given.

If BC is found by trigonometry, then EC is the sine of the lesser

arc of which Er (not shown) is radius, r being the centre of the great circle where the two normals BO and fe meet; Cn is the versed sine of the arc En, and represents the correction for curvature. The fall is thus Bn; but if the fall has been taken from the levelling staff ef, according to a former hypothesis, then the correction for curvature is sec. — rad. or fe - Fro - Em = Bn, the fall as before.

The correction for curvature on ab is greater than that on EB, although both lie between two stations on the survey, because the old channel traverses a curved surface. But the difference is less than it would be were the channel straight. The actual difference is, therefore, very small in the majority of examples. It requires however, to be attended to, for reasons which will be seen in the next paragraph, and in practice it is generally approximated on what is termed “ the safe side.”

If it is intended to give the new channel the same inclination as the old one (which is a general rule, the inclination being rather less than greater, so as not to wash away soil), then draw the line ed a distance below the parallel to AB through E equal to the difference between the corrections for curvature of AB and EB, which will give the fall mi of the new channel, and bd the fall gained for drainage.

At any place between + a and + x the fall for drainage may be got by an intermediate normal at that place or a vertical line nearly parallel to bd between the two lines EB and ED.

The surface levels of the stream may at some places rise above AB, and at others fall below it. Thus, if no obstructions from piers of bridges, fords, and the like exist, then the surface water-line will be concave, falling below AB; but if side and bottom obstructions are numerous, the water-line will be convex, rising above ab. At acute bends of the river as at + 6 (page 136) a damming back process, with an elevated surface and retarded velocity, will be found. This involves a greater velocity at the narrow part above + 7, so that the former level may be above AB and the latter below it. These require surface level stations, one or two at each bend and narrow place.

Plotting the dimensions.

As the field operations of the vertical survey commence at station x, so do the office ones of plotting the dimensions. Thus draw the normal BC indefinite in the direction of C; the tangent bf indefinite in the direction of G; the diagonal be according to dimensions found, making with BF the angle of depression also found, and the sine EC at an angle equal to fbe, If the work is thus far correct EC will cut BC in C, making a right angle at C, and BC equal to the dimen-

sion found by trigonometry. Next, EF, taken from the field-book, will cut BG in F, making the angle at E greater than a right angle, and the angle at F less than a right angle. Produce CE, in the direction EA indefinite, and with the measured length of the old channel from B cut CA in A. Draw AB and EB; also ED, nearly parallel to AB, as before directed. Mark off the surface level stations +1, +2, &c., on BA, commencing at B. The deviation of the water-line from BA may then be plotted from the readings of the levelling-staff at +1, +2, &c., the operation being similar to plotting a crooked fence on a horizontal survey from offsets taken from a line driven with the chain.

In the preceding example, from the ’r shortness of the distance between + x and + a it is assumed that the latter station is seen from the former, and that the height EF is taken from the levelling staff at one operation : but when the survey extends over ten, twenty, or forty miles, this cannot be done.

In a comparatively level valley, forty miles, and even the length of a degree, or seventy miles, may be included between two normals as BC and FE, and two parallels BF and CE, the former, BF, a tangent to the greater radius Br and arc Bm, and the latter CE, a sine to the lesser radius Er and arc En.

In this case the distance EB, between the two extreme stations E and B, would be determined from the horizontal survey as before and so would BA. But besides BE and BA, there would now be a series of diagonals corresponding to the number of intervening levelling-staff stations. In most examples the position of the series of diagonals would be above the common diagonals BA and BE; in many places, however, they may be below, but in all their position and length can be easily determined, and plotted with the section according to constructive geometry, the series of diagonals being similar to the sides of a polygon on the horizontal plane, and the surface levels of the stream may be accurately drawn therefrom.

In more extensive surveys the longitudinal section should be divided into sectors, as in the diagram Example VI., which contains three, purposely drawn to show the principles of plotting.

The vertical offsets with the levelling-staff being taken above the diagonal AB, and not above the sine AC, the practice of laying down the surface levels from the latter, or from a datum line below it, is not correct, because AG is greater than bc ; consequently the offsets above are greater than those below.

General rule for the inclination of the channel.

The angle of depression of the channel of a river should be that which will give to the flowing current a force sufficient to keep it clear of sediment in times of drought, when the surface level of the stream is low, but not such as will wash away its sides and bottom during periods of flood water, when the surface level is high.

If the fall and inclination are less than what is given in the rule, deposit will be formed, and the channel of the river gradually silted up. On the other hand, if it is greater, the force of the stream will wash away first the lighter materials of which the channel is formed, thus undermining the heavier, so that its proper shape will be destroyed, and its course made liable to change at every storm or heavy fall of rain.

In practice it is difficult to make provision for the two extremes, the minimum and maximum volumes of water, as sediment cannot be avoided during low water without injury being sustained in times of heavy floods. Practical rules are sometimes advanced relative to the proper fall per mile that will obviate both objections; but such can never be safely taught, unless upon the spot, and even then with much caution and allowance for differences in the quality of soil; for as such rules are deduced from the actual velocity of other rivers, because they are considered favourable .examples, it follows that if there exists the slightest difference between the two soils of which the channels are composed, either silting or washing away will be the inevitable result, as shown in Example II.

When a new channel is proposed, as in the above example, the advisable plan for both the surveyor and drainage engineer is to make ample provision for flood waters, so as to obviate ruinous tear and wear. To effect this, all differences arising from corrections for curvature, refraction, &c., should be thrown upon the proper or safe side, so that the fall and inclination shall be rather less than calculated ; and the practical reason for this is, that the small deposit of silt that takes place at low water will be removed during floods if the true form and direction of the channel are preserved.

In the generality of cases the old channel will be the best example as a practical rule for the new one, the soils opposite being for the most part equal, and the lowest velocity of the old river may be taken for the general inclination of the latter.

3. Volume of water.

The volume of water during the highest floods is that for which provision has to be made by the survey, and this has to be ascer-

tained by measurement. In the same manner the transverse sectional capacity of a new channel may be determined from that of the old one. A diagram of a semi-transverse section will be given under the next head, Embankments.

4. Area of land required for embankments.

The areas of land required for the two embankments, including the two forelands, have generally to be computed separately from the area occupied by the channel between them and from the two areas outside : the several areas being shown upon the plan.

For the sake of illustration, let the annexed diagram represent a semi-transverse section of the river at station x, page 136. Then x will show the land outside at + .r, a the top of the embankment, l the foreland between the embankment and edge of the channel, c the bottom of the channel, Bn the depth of the fall (see previous diagram), cc' the horizontal plane of the sine EC, and 1, 2, 3, 4, and 5, the vertical offsets.

/*/

n¹

The inside capacity being uniform throughout the bottom c, the slopes of the channel and embankment, and the level of the foreland, will be similar at other sections; but the offsets will decrease towards A, and the surface level at x may be higher or lower.

By improved drainage rain-water is now much faster removed from land than formerly, and a corresponding provision should be made for such, by higher embankments and a greater breadth of foreland between the bottom of the inside slopes and edge of the channel of the river.

The foreland should be parallel to the top of the embankment and surface of the water, and not much above the ordinary level of the latter during the winter season.

The bends or curves of rivers should be of the greatest possible radius, and the slopes on the concave side opposed to the current at bends should be greater, or of a less angle, than where the river is

straight; and the top of the embankment whose concave slope faces the current, should be higher than the opposite convex one.

A greater area of land is required for the embankments and channel at curves than where the river is straight; and the additional breadth, both for the embankments and channel, should, if practicable, be taken from the opposite or convex bank of the river.

5. Subdivision and apportioning of land where the river is straightened.

The method of apportioning land on the equitable principle of “ take and give ” is similar in this example to that described when land is intersected by a railway, Sect. II.

6. Water power, irrigation works, bleaching grounds, dec.

If levels are properly attended to, the straightening and deepening of rivers will give an increase of water-power for mills, proportionally to the increase of fall gained, but it will reduce facilities for irrigation, bleaching, and like purposes.

When the bottom of the channel is lowered, water-wheels require to be lowered to the same extent, and also the watercourses.

The two watercourses may also require to be of greater length, especially the one above the mill, in order to draw oft the water from a higher level, so as to avoid an intercepting wire across the stream, which is always objectionable.

Such alterations involve a large amount of both field and office work, but, although the greatest degree of accuracy is required in taking the levels, the operation is not generally attended with much difficulty. If the water-power is large and of great value, longitudinal and transverse sections of the mill-courses may be included in the survey, but when it is small, the details may all be shown upon the plan, the fall being indicated by figures.

When water is taken from the river for irrigation, breweries, bleaching, &c., by gravitation, the lowering of the channel will render it necessary to draw off the water from a higher level, which will increase the length of the Avatercourse. Thus if the channel is loAvered one foot, and if the fall per mile is one foot, then the increase of length for the watercourse will be one mile.

7. Provision for the drainage of adjoining lands.

All surface and spring water gathered above the level of the top of the embankments is generally discharged into the river between embankments, and what is collected below that level is discharged by sluices and similar contrivances.

As the water from elevated grounds may frequently be discharged into the river between embankments at a much higher level than what is requisite to drain the lower grounds, the side drains for the latter may have to run under the former, and be discharged into the main channel at a point farther down, in order to gain the proper fall.

The levels for such tributaries are generally included in the survey of the principal river; and when the works are large, sections of them may be required, but when small the vertical details may be shown upon the plan.

8. New roads, bridges, fences, drainage, and levelling of the old channel.

If, in straightening a river, the new channel shall cross a road, a new bridge will be required; and as old bridges are frequently not well situated, the old road may also have to be taken up, and a new one made. For similar reasons old fences will often have to be removed and new ones formed.

A new bridge, in such cases, is generally considered a favourite place for a transverse section; and if a new road is also required, the prolongation of the transverse level of the roadway on both sides of the new channel will form the surface-level of the longitudinal section of the road.

The foundation of the piers of bridges should be of such a depth as to be defended by a pool in a similar way to what is shown in the next example of mountain streams—the safety-line being considerably above the base-line of the masonry, or top of the piles, where such are used.

The superficial area of the old channel will have to be surveyed, and sometimes its capacity also measured, in order to ascertain the quantity of materials required to fill it up. This latter measurement is usually given in cubic yards. The expense of filling up the old course with the materials excavated from the new one, may likewise have to be estimated before the lands can be valued and apportioned.

When a tributary discharges itself into the river at the part straightened, it may in some cases be allowed to enter the new channel by the old one ; or a new course may have to be surveyed and laid off for it, between its old mouth and the new river.

When the old river, or a portion of it, is to be filled up, the course of the Avhole, or the part, as the case may be, will have to be shown on the plan by dotted lines; but where the whole or the part is to be left open, such will require to be represented upon the plan accordingly.

"When the old channel is to be filled up, and when a large covered sewer is placed in the bottom of the old channel for the purpose of drainage, its course may have to be represented upon the plan by a dotted line, its mouth where it enters the new channel being shown on the longitudinal section, when such is ordered.

The ranging of the poles for the horizontal survey, the measuring of the lines with the chain, the taking of the bearings with the theodolite, and the other minor details, are similar to what will be found under Prob. V. Part IV., and subsequently Part IX. Section VI.

EXAMPLE V.

GENERAL DIRECTIONS FOR THE SURVEY OF MOUNTAIN RIVERS.

That which chiefly distinguishes the rivers of mountainous districts from those of a comparatively level open country is their greater fall and velocity, and the peculiar phenomena to which such give rise, as the formation of pools, rapids, waterfalls, cascades, &c.

Pools and rapids are occasioned partly by the scooping-out process of the water, and partly by the formation of bars across the river. In all cases where the fall is great, the water, by its continuous scooping-out action, either cuts down to the rock, or sweeps before it, in heavy floods, stones and gravel to a less inclination, where such materials accumulate in a bar, forming a rapid below and a pool above. In this manner bar after bar rises in the channel, until the whole bed of the river, from its source to its confluence with the ocean or large river, in the campaign valley, is formed of an alternating series of pools and rapids, differing in length and depth according to the nature of the ground and force of the current.

In those cases where the water washes away the soil in the natural formation of a channel, and flows upon a rocky bottom, waterfalls and foaming cataracts are formed of various magnitudes.

In other examples rivers flow into and out of lakes, in channels consisting of pools and rapids, or of rocky bottoms.

The pools and rapids formed in gravelly soils are a natural protection to the bed of the river and the adjoining lands on both banks. The water, when it leaves the pool at the top of the rapid, begins to descend with a minimum velocity and force, which gradually increase to a maximum; but the accelerated velocity and force acquired in its descent is lost in the pool below. As the river thus pursues its course it leaves the pools above it at a minimum velocity, but enters those below at a maximum.

The depth and form of the pools at the bottom depend upon the force of the current and the nature of the subsoil. If water is gently decanted into a full vessel, the influx current will only penetrate a short distance into the stagnant fluid below; but if the decanter is elevated, and the volume and force of the water increased, it will sink to a greater depth. The river as it flows down the gradient forming the rapid into the pool below, is governed by the same hydrostatic law. When small it will penetrate to a short distance below the surface, but when swollen it will sink to a greater depth. If large stones are embedded in the subsoil, they may deflect the current, and thereby prevent the bottom being scooped out to so great a depth as otherwise it would have been. But where bottom soils are equal, the depth of the pool will be directly as the altitude and force of the swollen current during the greatest floods.

The elevation of a bar above the general inclination of the river will, like the depth of a pool, depend upon the force of the current and the nature of the accumulated debris of which it is formed. If the ground is full of large stones, consisting of fragments of rock, they will accumulate during the natural formation of the channel by the flowing stream, and thus form a more elevated, acute, and permanent ridge across the current than when the stones are of a less size and rounded by attrition. But when the bar is composed of finer gravel, it will be “ more flat,” will give to the rapid a greater length, and be much more “ liable to shift ” in times of heavy floods than the previous example.

The depths and water-level lengths of the pools; the depths and hypotenusal lengths of the rapids from smooth water to smooth water, and the heights and lengths of the bars, require to be ascertained in the survey, and shown upon the longitudinal section; the breadths and depths are represented upon the transverse section; and the breadth, horizontal length, and direction of the river are shown upon the plan.

Pools, rapids, waterfalls, and cataracts are frequently known to anglers and others by their names; and these may require to be shown both upon the plan and sections.

It is seldom found advisable to change the course of a large river in a hilly country; on the contrary, the difficulty experienced is to confine them within the channels they have naturally scooped out for themselves; for unless the pools are of sufficient depth, and the bars formed with proper materials, it is impossible to prevent harm in storms of more than ordinary magnitude and duration.

When a survey is undertaken for the express purpose of supplying the necessary data required for embanking a river, so as to

prevent it from overflowing its banks, and for deepening the pools and improving the rapids, so as to obviate the shifting of the channel, the height and position of the embankments, and the materials of which the bar is formed, are the first matters of inquiry that engage the attention of the surveyor.

If the river has previously been allowed to overflow its banks during heavy floods, the confining of it within embankments will raise its surface level and increase its velocity and force.

The increase in the altitude of the river, at the top of the rapid, will be found in the majority of cases to endanger the stability of the bar, and hence the safety of the channel below. This arises from the increased velocity and force being produced by artificial means, whereas the defence is natural, the bar being formed by the river when flowing at a less depth.

Sometimes the stability of the bar and rapid may be increased by artificial means sufficient to counteract a greater force of current. This is done either by throwing in heavy stones in the smooth water immediately above the rapids whose bars are liable to be washed away, or by driving in piles and wattling them with small timber.

As the breadth and depth of the river at the top of the rapid during a heavy flood are determined by the height and position of the embankments, they should be placed as far from the river’s edge on both sides as the circumstances of the case will permit, the depth being inversely as the breadth.

When a river has left its bed, the same data are applicable in the formation of new bars and embankments in order to direct it back into its natural channel. Wherever the surface of the water is raised on the top of a rapid, the stability of the bar must be increased, and also the depth of the pool below; and this has to be done for several pools and rapids down the river, so as to restore the deranged balance of things to a state of equilibrium, similar to what is found in natural examples of stability.

When arable, meadow, or pasture lands have been washed away or covered with gravel, they require to be measured, so as to enable the landowner to settle questions of value with his tenants. In such cases a section to show the subsoil and drift is not unfrequently ordered.

When the course of a river has to be changed, a pool of considerable depth and length is best for the purpose, more especially if the bend is quick. A rapid should in such cases be avoided, if possible, as the concave bank and embankment are difficult to protect at the top of the rapid when the river is much swollen.

A ferry is generally at tlie narrowest and smoothest part of a pool, and a ford at the first smooth water immediately above a rapid; both, require to be shown upon the plan, and also the ferryman’s house and boat.

The piers of bridges should be protected by pools of sufficient depth to prevent harm to their foundations during the greatest flood.

Where the foundations of the piers of old bridges are so high as to form rapids, such data should be carefully shown upon the sections, and special attention drawn in any report that may accompany the survey relative to details of tear and wear, &c., that cannot be otherwise represented.

Lakes are sounded and the depths shown upon the plan, as in the case of the ocean.

In surveying waterfalls and cataracts the chief points that require attention are the peculiar geological strata of the rocks that form the bed of the river, the fracture and angles thus formed, and the wearing away of the bottom by the continuous action of the water. Such involves a large amount of work, but it is simple and easily performed by those who have a taste for drawing.

EXAMPLE VI.

GENERAL DIRECTIONS FOR THE SURVEY OF CANALS, IRRIGATION-WORKS, AND WARP-LANDS.

Surveys under this example are either of works that are finished or else to furnish data whereby others may be executed at some future period.

Canals.

In Canal-Surveying the levels are necessarily “ water-levels ” or “ dead-levels,” as they are sometimes termed in contradistinction from horizontal levels on which water flows.

When the canal is formed and in operation, the details of the vertical survey for the sections are of the simplest kind, as the levels can then be taken from the water, the normals from the height of the locks, and the lengths and breadths by measurement with the chain.

The details of the horizontal survey for determining the direction of the line of the canal, the area of land which it occupies, including the position of basins, quays, &c., and the severance of property which it has effected, are similar to what has already been given, and to the corresponding data in Railway-Surveying, which will be found under that head, p. 385.

If the survey is made purposely to determine the levels and other data for a canal intended to be made, its details are more diversified and differ in many respects from those of a railway.

Under such conditions there are two levels given ; viz., the-lowest water level, and the highest one—the object of the canal being to raise the navigation from the former to the latter by means of a series of locks and dead levels, as from c to b in the annexed diagram.

The first step in the survey is to find the difference and distance between the two given levels. This may be done either by the common levels and chain, as in the case of the river (Example IV.), or by the theodolite and chain. We shall adopt the latter, as it will illustrate a survey on the vertical plane conducted on the principles of plane trigonometry. The difference between this practice and that with the level will also be so pointed out as to be easily understood, the theodolite being in point of fact a spirit level.

The object of taking the levels is to find the height of the upper level above the lower one, which determines the number of locks on the line, and that of measuring the hypotenusal distances with the chain to find data for ascertaining the true lengths of the dead

levels or arcs that determine the length of the canal within the survey.

The survey may be represented as lying between two arcs of two concentric circles on the vertical plane.

If, in the foregoing diagram (drawn on an exaggerated scale purposely to illustrate details which otherwise could not have been shown and made intelligible to the private student), we suppose abr to represent a section of the vertical plane, then ab and cd will be the two arcs between which the survey lies ; the former showing the upper water level line, and the latter the lower one.

If we further suppose four locks on the line, such data will give three intervening arcs; and if we again assume that the three arcs subtend equal angles, then ce in the diagram will show the first lock, im the second, no the third, and ub the fourth : ei will show the first intervening arc, mn the second, and on the third.

The lines sought are therefore bd, ei, mn, and ou on the vertical plane.

The lines measured are the hypotenusal distances, and normals or vertical-offsets, the former lying between the theodolite and levelling-staff by the chain, and the latter by the levelling-staff.

The angles measured with the theodolite are those which each hypotenusal line makes with its horizontal level or tangent and sine, and the normals or portions of the two radii between which it lies.

In the diagram the hypotenusal line between the first two radii is the diagonal ci, which represents the first line measured with the chain. The first normal taken from the levelling-staff is represented by cy. The other diagonals, in and nu, may be drawn in a similar way; also their two normals at i and n; their two sines from i and n, and their two tangents from n and u—the construction of the second and third sector being similar to that of the first.

In each sector there are three right-angled plane triangles, and two oblique-angled triangles, with certain lines and angles, measured to find other lines and angles, by rules given in geometry and trigonometry. The triangles of the first sector only are completed, those of the other two being left for exercises to the student; and the three sectors have been made equiangular at the centre r, for the express purpose of illustrating some of the most remarkable properties of the circle, those properties most commonly met with in the practice of surveying. Two of the corresponding triangles of the three sectors are equiangular, but not equilateral, the sides increasing in length as the arcs and radii increase in length. The other three triangles are neither equiangular nor equilateral.

The three right-angled triangles in the first sector are (1) cxi, (2) cxr, and (3) yir: and the two oblique-angled triangles are (1) cyi, and (2) icr.

The two equianglar triangles are yir and cxr.

They are equiangular, first, because the two angles at i and x are right angles, and the angles at y and c each the complement of r, which is common to both; second, because yi is parallel to cx, the angle iyr is equal to the angle xcr (Theor. III. Part I.); and as the angle at r is common, and the remaining angles right angles, the triangles are therefore equiangular.

The angles of the remaining three triangles are dissimilar.

The two chief lines sought in this sector are the normal ce, representing the height of the first lock, and the arc ei, representing the length of the first level of the canal.

If we suppose the surveying-staff to be divided into two companies, one with the theodolite and levelling-staff, and the other with the chain, and that the station-poles along the canal-line are ranged, those shown on the vertical plane being c, i, n, and b, then the field operations will be performed and the measurements entered in the three field-books that would be open under such an hypothesis, as follows:—

At station c the angle yd is measured with the theodolite, the vertical bearing being taken from y to i. (Before removing the instrument the horizontal bearings are also taken ; but as they are similar to those of the railway survey, p. 385, the directions in this section are confined to the survey of the vertical plane.) The bearings are next taken at station i ; first the angle of depression yic, which gives at the same time icx. The measurement of the angles with the theodolite, and normals with the levelling-staff, now proceed together, cy representing the first one that is found by the latter instrument. When the angle of depression yic is accurately taken and entered in the field-book, the signal is given to those in charge of the levelling-staff, who enter the height cy in their field-book, which measurement includes the height of the theodolite above the ground at station i. This latter has therefore to be taken at each angle of depression, and afterwards deducted from the former, to obtain the true normal found. The theodolite is then turned round, and the vertical bearing min taken before the station-pole is replaced. It is next removed to station n, when the levelling-staff is taken to station i.

During the above operations, the chain may be employed in measuring the side lines of the horizontal survey; but when those working the levelling-staff replace station-pole i, and proceed to n, then the diagonal or base line ci may be driven.

This concludes the field operations for the first level ei or sector eri. Those of the other two sectors mm and oru are performed in a similar manner.

Erom the two angles thus measured, viz., yci and yic, the other angles of the five triangles may he determined. Thus 180° — (yci + yic) gives cyi and xcr; yic— icx; 90° — icx—cix; 90° — xcr — r; and icx -f xcr = icr.

From the two sides found, viz., ci and cy, and the above angles, the other sides of the triangles may now be determined by the following formulae :—

R : cos. icx : : R : sin. icx : : R : tan. xcr : : rx: R R : tan. r

ci : cx ci : ix cx: xr r : rc

To find cx To find ix To find xr To find rc To find yi To find yr

: : cos.

: : rx + xi : yi R : sec. r : ; rx + xi : yr

The two lines yi and ir may also be obtained by formulas for oblique-angled triangles, Part VIII. Section II., and the secants yr and cr by Theorem VII. Part I., as shown in Example IV. The line ce is now got by subtracting cr from ir; and ey representing the correction for curvature of the horizontal level iy from the arc ei, is obtained by subtracting ir from yr, or the radius of the arc from its secant.

The arc ei, representing the length of the first level of the canal, remains to be determined.

The difference between radius and secant is found with geometrical accuracy in several ways, but the true length of a curve or of an arc of a circle, as ei, is only approximated. Thus, it is less than its tangent, but greater than its chord, as will be seen from the diagram. Taking the diameter, for example, to be 1, the circumference of a circle has been calculated to be 3-14159; now 3T41 would lie within the circle, -while 3T42 would lie without. Between these two lesser and greater differences, another one, viz.,

3-141592265358979 lies; still it is upon the inner side of the curve : and in this manner the decimal has been extended to upwards of two hundred places, and might be carried to as many more. But for all practical purposes it may be closed at 3T416 without, or 3T415 within, the former being generally adopted. From such data the following rules have been deduced :—

To find the circumference, multiply the diameter by 3T416.

To find an arc, multiply -0174533 by radius, and the number of degrees which the arc contains.

Irrigation-works.

Irrigation-works may be considered with a view to their survey under the following three heads :—1. Bottom irrigation ; 2. Surface irrigation; 3. The modern system of applying liquid manure to land.

1.    Extensive examples of the first kind are to he found in our West India colonies, and many other parts of the world. The works consist principally of parallel canals and ditches, intersecting comparatively level land.

The nature of the survey is therefore similar to that under Example III., the canals and ditches being used for the twofold purpose of surface drainage in the rainy season and bottom irrigation during the intervening periods of drought.

2.    Surface irrigation, the second example, is principally practised for the growth of rice and grass. The water is generally conveyed to the fields in both cases by gravitation, either from canals or rivers, naturally or artificially formed, but it is differently applied afterwards, paddy fields consisting of a series of dead levels, while water-meadows lie at various inclinations.

In surveys of this kind, field operations are simple, but multitudinous in character, the levels required for distributing the water being exceedingly numerous; and as they are represented upon the plan, they furnish a corresponding amount of office work.

3.    The third practice of applying water to land is by forcing it through pipes, either by gravitation, or by steam, or other power. It may first be thrown, by means of pumping apparatus, into tanks or cisterns, situated on elevated ground, and from thence be distributed throughout the fields by gravitation; or it may be forced directly to the land. The pipes and hydrants are ramified throughout the fields in such a manner that each of the latter has a certain area of land assigned to it, so that the liquid may be showered equally over it by means of a hose of the proper length screwed on to the hydrant. When one area is finished, the liquid is turned off, the hose screwed on to another hydrant, and thus the work proceeds until the several hydrantal areas are gone over.

It does not belong to this work to describe the details of liquid manuring land on this plan. Enough has been said to show the student who may not be acquainted with the practice the nature of the levels that have to be taken, and the survey required in laying down the pipes and hydrants.

The vertical survey closely resembles that of water-works for supplying towns, castles, farm-homesteads, and fields, with water.

It embraces two levels, a lower one at the river or fountain-head, or liquid manure tank, and a higher one at the reservoir, or its equivalent, the height to which the liquid has to be forced. The work of taking the levels is therefore similar to that in canal surveying.

With regard to the horizontal survey, a hose screwed on to a hydrant or mouth of a stopcock would distribute water over a circular area of land. The areas, however, into which fields must be subdivided are necessarily square areas; consequently each square area has to be inscribed, as it were, within the circular area which the length of the hose and jet will cover. As gases are very liable to be disengaged from drainage water and liquid manure, and to collect in the bends of pipes, where they pass over elevated ground, hydrants should be placed at such, so as to draw off the gases which there collect.

Warping land.

When flowing water holding organic and inorganic matter in suspension is turned into an area of land inclosed by embankments, and then allowed to stagnate, such matter is deposited, and termed “ warp,” and the process “ warping.”

The annual overflowing of the Nile in Egypt furnishes a practical example on a large scale. Warping has also been extensively practised in this country, the warp being chiefly obtained from tidal rivers, some of which have been termed “ muddy to excess,” a depth of from six to twelve inches of warp being left by them upon the land in a single summer season.

The survey is similar in character to that for embanking land when the work is not attended with any difficulty. The height and breadth of the embankments at the base are generally shown by figures upon the plan, in the same manner as the area of land inclosed for warping. As the depth of the warp will be directly as that of the water, the depths of the surface of the land at the lowest and highest places below the top of the embankment may have to be taken and shown upon the plan, as different depths of the ocean are directed to be represented in Example III. Such bottom levels are ■ easily found by setting station-poles at the places, and raising flags or marks upon them until such marks appear in a line or in the same plane with the top of the embankment levels.

SECTION IY.

THE METHOD OF REDUCING LOCAL OR CUSTOMARY MEASURES TO

STATUTE MEASURE, AND VICE VERSA; ALSO, THE METHOD OF

REDUCING SCOTCH AND IRISH MEASURES TO STATUTE MEASURE,

AND VICE VERSA.

It lias been already observed that formerly, by custom, the perch varied in different parts of England, and with it consequently the acre also varied in proportion.

In Devonshire and part of Somersetshire 15, in Cornwall 18, in Lancashire 21, and in Cheshire and Staffordshire 24 feet were accounted a perch.

In the common field-lands of Wiltshire, and in some other counties, there was a customary measure of a different nature, viz., of 120 instead of 160 statute perches to an acre; consequently 30 perches of statute measure made 1 rood of customary, or 3 statute roods made 1 customary acre, or 30 statute perches made 1 rood, and 4 such roods made 1 acre, customary measure.

In some places, an acre of this measure was called a day-work, or a day’s work of land.

We may also observe that the customary measures of Scotland and Ireland differed very greatly from the English statute measure.

PROBLEM I.

To reduce customary measure to statute measure, or statute measure to customary measure.

GENERAL RULES.

RULE I. To reduce customary measure to statute measure.

Multiply the number of perches, customary measure, by the square feet in a square perch, customary measure; divide the product by the square feet in a square perch, statute measure, and the quotient will be the answer in square perches, which reduce to roods and acres by dividing by 40 and by 4 in the usual manner.

RULE II. As the square yards in an acre, statute measure, are to the square yards in an acre, customary measure ; so are any number of acres and decimals, customary measure, to their equivalent in acres and decimals, statute measure. Then reduce the decimals to roods and perches by multiplying by 4 and by 40 in the usual manner.

Hote 1. The roods and perches in the given quantity must he reduced to decimals of an acre before stating the question.

2.    By reversing either of the preceding rules, we can reduce customary measure to statute measure.

3.    It is scarcely necessary to remark that the length of any perch multiplied by itself will give the number of square feet in a square perch of the same measure; hence we have 16'5x 16⁻5=:272-25, the statute perch; 15x15 = 225, the Devonshire and Somersetshire perch ; 18 x 18 = 324, the Cornwall perch ; 21 x 21 = 441, the Lancashire perch; and 24x24 = 576, the Cheshire and Staffordshire perch.

4.    It may also be observed that 4840 square yards make 1 statute acre; 4000 made 1 Devonshire or Somersetshire acre; 5760 made 1 Cornwall acre; 7840 made 1 Lancashire acre; and 10240 square yards made 1 acre of the customary measure of Cheshire or Staffordshire. Also, 3630 square yards made 1 acre of the customary measure of Wiltshire. .

5.    When it was intended to find the area of an estate }n customary measure only, it was generally thought most convenient to take the dimensions by a chain properly adapted for that purpose. The Devonshire and Somerset chain w-as 60 feet; the Cornwall chain, 72 feet; the Lancashire chain, 84 feet; and the Cheshire and Staffordshire chain, 96 feet in length. Each of these chains was divided into 100 equal links, in the same manner as the statute-chain; consequently the customary measure was found by the same rules as the statute measure.

6.    It may also be observed that the Devonshire and Somersetshire link was 7'2 inches; the Cornwall link 8'64 inches; the Lancashire link lO’OS inches ; and the Cheshire and Staffordshire link was 11"52 inches in length.

Examples.

Ex. 1. In 32 acres 2 roods and 20 perches, Devonshire and Somersetshire customary measure, how many acres, &c., statute measure 1

By Rule I. Here 32a. 2r. 20p. — 5220 customary perches; and 5220 x 225 = 1174500, the square feet in 5220 customary perches; then, 1174500-00 = 272-25 = 4314 statute perches = 26a. 3r. 34y>., statute measure.

By Rule II. Here 32a. 2r. 20p. — 32-625 acres; then, as 4840 yds. : 4000 yds. : : 32-625a. : 26-96281a. = 26a. 3?-. 34y>., statute measure, the same as by Rule I.

Proof. As 4000 yds. : 4840 yds. : : 26-96281a. : 32⁻625a. = 32a. 2r. 2Op., customary measure, the same as given in the question; and obtained here by reversing Rule II.

Ex. 2. In 45 acres 3 roods and 15 perches, Wiltshire customary measure, how much statute measure 1

By Rule II. Here 45a. 3r. 15p. = 45-875 customary acres; then, as 4840 yds. : 3630 yds. : : 45-875a. : 34-40625a. = 34a. 1 r. 15^., statute measure.

Proof. As 3630 yds. : 4840 yds. : : 34-40625a. : 45⁻875a. = 45a. 3r. 1 bp. customary measure, the same as given in the question.

Note.—In solving this and every similar question, the learner must recollect that 30 perches make 1 rood Wiltshire customary measure. (See the Introduction to this Section.)

Examples for practice.

1.    Reduce 47a. 3r. 20p. Cornwall customary measure, to statute

measure.

Am. 56a. 3r. 36p.

Ans. 36a. lr. 10-3p.

3. Reduce 127a. Ir. 2Gp. Cheshire and Staffordshire customary

Ans. 269a. 2r. 1-01 p.

2.    Reduce 22a. 1?\ 27p. Lancashire customary measure, to statute measure.

measure, to statute measure.

Am. 40a. lr. Op.

4.    How many statute acres are contained in 53a. 2r. 20p. of the customary measure of Wiltshire ?

5.    A gentleman’s estate near Manchester, according to an ancient plan and survey, contains 345a. 2r. 30p. Lancashire customary measure, what is its content in statute measure 1

Am. 559a. 3r. 30^.

6. A gentleman’s estate near Chester contains 436a. 3r. 36^. Cheshire customary measure, and is let at £8, 10s. 6d. per acre per annum; what would be an equivalent rent per acre if the estate were let by the statute acre 1

Am. £4, 0s. Id. per acre, statute measure.

PROBLEM II.

To reduce Scotch and Irish land measures to English statute measure ; and vice versd.

The Scotch chain, by which land was formerly measured in Scotland, was divided into 100 links, in the same manner as the English statute chain; but the Scotch link was 8⁻88 inches in length; consequently the length of the chain was 888 inches, or 74 feet English measure.

The Scotch fall or perch was 18-5 feet in length; and hence the square fall or perch contained 342⁻25 square feet; and 40 falls or perches made 1 rood, and 4 roods made 1 acre. Also, 160 falls or perches made 1 acre; and consequently the Scotch acre contained 6084| square yards English measure.

The Irish chain was also divided into 100 links, each link being 10'08 inches in length; and consequently the length of the chain was 1008 inches, or 84 feet English measure.

The Irish perch was 21 feet in length, and hence the square ■ perch contained 441 square feet, and 40 perches made 1 rood, and 4 roods made 1 acre. Also, 160 perches made 1 acre; and con-

sequently the Irish acre contained 7840 square yards, English measure.

Note 1. The general rules given in the first problem may be applied to the reduction of both Scotch and Irish measures. Or, multiply Scotch acres by 1*25712, and Irish acres by 1*61933, and the respective products will be English statute acres. Also, if English statute acres be divided by these numbers, the respective quotients will be Scotch and Irish acres.

2.    As both the Scotch and Irish chains were divided into 100 links, in the same manner as the English chain, it is manifest that the rules given in this work for finding the areas of different figures, and for laying-out, parting-off, and dividing land, were equally applicable in all cases of surveying, whether the dimensions were taken with the English, Scotch, or Irish chain.

3.    The Scotch acre contained 1244-J square yards, and the Irish acre 3000 square yards more than the English statute acre. It also appears that the Irish measure was the same as the Lancashire customary measure.

4.    The Scotch mile was 1973L and the Irish mile 2240 English yards ; and - hence it appears that the Scotch mile was 213|-, and the Irish mile 480 yards

more than the English mile.

Examples.

Ex. 1. Reduce 36a. lr. lOp. Scotch measure to English statute measure.

By Rule I. Prob. I. Here, 36a. lr. lOp. = 5810 Scotch perches; and 5810 X 342*25 = 1988472*5 square feet in 5810 perches; then 1988472*5 -f 272*25 = 7303*8 perches = 45a. 2r. 23*8p. statute measure.

By Note 1. Prob. II. Here 36a. lr. lOp. = 36*3125a., and 36*3125 X 1*25712 = 45*64917a. = 45a. 2r. 23*8p. statute measure.

Ans. 58a. dr. lip.

Ex. 2. Reduce 36a. lr. lOp. Irish measure to English statute measure.

Ex. 3. The length of a rectangular field measured by the English statute chain is 1435 links, and its breadth 923 links; required the area of the field in English, Scotch, and Irish measures.

Ans. 13a. Or. 39p. English measure; 10a. 2r. 5fp. Scotch measure ; and 8a. Or. 28|p. Irish measure.

REMARK.

The preceding problems and examples will be found useful, not only in reducing customary measures to English statute measure, but also in conveying information to our young surveyors relating to ancient measures, which will soon become only matters of history.

ESTIMATING LAND BY THE MILE.

The Method of Making a Bough Calculation of the Number of Acres contained in a Common, Moor, Lordshijp, County, or Kingdom.

Endeavour to ascertain in miles, as nearly as you can, either by your own observations or from the information of others, the mean length and breadth of the land to be estimated • then multiply the length by the breadth, and the product will be the area in square miles. Multiply this area by 640,. the number of acres in a square mile, and the product thus obtained will be the area in acres, according to this method of calculating.

Note 1. The mean length and breadth of a county or a kingdom may be found from a map, in the following manner:—Measure several lengths by the scale of miles upon the map; add them together ; and divide their sum by their number for a mean length. A mean breadth may be obtained by a similar process.

2.    The foregoing method of finding the area of counties and kingdoms must, of course, be liable to considerable inaccuracy, not only as regards the method of taking the dimensions, but also as respects the correctness of the map and scale ; for it is evident that if these be not truly delineated, the dimensions can never be obtained to any degree of accuracy.

3.    When you have a correct map and scale of a county or a kingdom, its content may be found to a considerable degree of accuracy by the following method :— Divide the map into triangles and trapeziums in the most convenient manner, and straighten the crooked shores or coasts, either with a lantern horn, as directed in Part IY., or by the parallel ruler, as directed in Part V. Measure the bases, diagonals, and perpendiculars correctly, by the scale of miles belonging to the map ; find the area of each figure separately ; and the sum of these areas will be the whole area required.

Examples.

miles.

3-75

1. Suppose the mean length of a common or moor be estimated at 3f miles, and its mean breadth at 2\ miles, what is the area in acres, according to this estimation 1

2-25

GO lO C

8-4375 miles.

640

3375000

506250

5400-0000 acres. A ns. 5400 acres.

2.    If the mean length of a lordship be estimated at 4^ miles, and its mean breadth at miles; what is the content in miles and acres ?

Ans. 10-625 miles and 6800 acres.

3.    The mean length of a county, found from a map, is 63 miles, and its mean breadth 42 miles; what is its area in miles and acres 1

Ans. 2646 miles and 1693440 acres.

4.    According to Parliamentary Papers 1861, the content of Ireland is computed at 32A18 square miles; what is its area in acres'?

Ans. 20,811520 acres.

5.    According to Parliamentary Papers, the content of Scotland is computed at 31‘324 square miles; required its area in acres.

Ans. 20-047360 acres.

Ans. 37-324160 acres.

6. According to Parliamentary Papers 1861, the extent of England and Wales is computed at 58,319 square miles; what is the area in acres 1

Note.—The county maps of the Ordnance Survey may be taken as examples for exercises.

PART VII.

THE METHOD OF MEASURING AND PLANNING MANSIONS, HOMESTEADS, YILLAGES, TOWNS, AND CITIES; DIRECTIONS FOR MEASURING AND PLANNING BUILDING GROUND, AND DIVIDING IT INTO CONVENIENT LOTS FOR SALE; AND MISCELLANEOUS QUESTIONS RELATING TO SURVEYING, LAYING-OUT, PARTING-OFF, AND DIVIDING LAND.

SECTION I.

THE METHOD OF MEASURING AND PLANNING VILLAGES, TOWNS, AND CITIES.

Villages, towns, or cities, mansions, and farm-homesteads, present themselves in almost every extensive survey, and are generally measured and planned with the adjoining or surrounding lands. The method of taking and laying down the dimensions of such places, and finishing the plans, will be given in the following sections.

Besides the plans of towns and cities are so essentially necessary for the purposes of commercial and general reference, that surveyors are not unfrequently employed in forming correct drawings

of the same, in order to have them engraved and published in copperplates.

Without this art, we could not obtain the iclmography of towns and cities ; neither could we have any just idea of the shape, extent, and direction of the streets ; the size and number of the public buildings; the local conveniences enjoyed by the inhabitants, &c., of those places which circumstances will not permit us to visit.

Directions for taking the Dimensions of Villages, Towns, and Cities.

The dimensions of villages, towns, and cities may generally be obtained by the chain only, as the streets are usually wide enough to admit of angles or tie-lines being taken at the meetings or intersections, in the same manner as directed in Problems IV. and V., Part IV. In these Problems the methods of measuring meres, woods, roads, rivers, and canals, are illustrated; and as the learner becomes completely master of this branch of surveying, any difficulties that present themselves in measuring will be easily surmounted.

It sometimes happens that the tie-lines cannot be measured at a greater distance from the angular points than 30 or 40 links. In such cases the tie-lines must be taken to a quarter of a link, and both them and the angular distances must be multiplied by 2, 3, 4, or any larger numbers, as circumstances may require; and the products used in laying down the chain-lines. (See Problem II. Part. IV.)

The notes taken in measuring towns and cities must be entered precisely in the same manner as in surveying estates; and in measuring along the streets offsets must be taken to the houses on both sides of the chain-line, and particularly to every corner and projection; even the small projections of bow-windows must not be omitted.

Sketches of the bases of the buildings, particularly the corners and projections, must be made in the margin of the note-book, in order to assist the surveyor in drawing a correct plan.

All public buildings, such as churches, prisons, castles, courthouses, market-places, halls, colleges, mansion-houses, &c., must be distinctly noticed; and the range of the first line should be taken with the compass, in order that the draughtsman may be able to lay down every street in its true direction.

Note 1. In measuring along the streets, all the offsets to the buildings must be taken at right angles to the chain-lines. The bases of the buildings and all the projections must be sketched as you proceed ; and the breadths of the buildings, the lengths and breadths of the projections, &c., must be correctly measured and entered opposite to those parts of the sketch to which they respectively belong. The sign + (plus) is usually placed between the breadth of a building, at its

perpendicular distance from the chain-line. The method of sketching the bases of buildings, and entering the notes, is exemplified in pages 4, 10, and 12, of the engraved field-book, to which the learner is referred. (See also Plate VII. No. 7, page 199.)

2.    When a town and the surrounding or adjoining lands are both to be measured and planned together, the dimensions must be taken with Gunter’s chain; and the lines measured along the streets must be properly connected with those measured in surveying the adjoining estates; but if the plan of a town only is required, it is more convenient to take the dimensions with a chain of 50 feet in length, divided into 50 links, and an offset-staff of 10 feet in length.

3.    As station staves cannot always be fixed in the streets, in consequence of the pavement, they must either be set in pedestals made for that purpose, or two or more assistants must each hold a staff in those places that are pointed out by the surveyor.

4.    Sometimes it is most convenient to measure external or main-lines on the outside of the town, as in surveying a mere or wood (Pro|b. IV. Part IV.); and in running such lines, stations must be left at the end of the streets, as you pass them, in order that lines may be run from one station to another in measuring the streets.

5.    In some situations, and under certain circumstances, it is more eligible to measure the first line along one of the principal streets, and to intersect this line by another, measured along some other principal street, nearly at right angles with the former ; then these two lines being tied together by a connecting line, measured in the most convenient manner, will divide the town into four parts, each of which may be measured separately by running lines in the most advantageous manner.

6.    In putting down stations at the ends of the streets, &c., the number of the station may be made upon the wall of the opposite building (if there be one) with red or white chalk, in such a situation that an offset may be taken, at right angles to the building, from the station marked upon the wall to the station on the chain-line. This offset being entered in the book, and again measured from the station on the wall, at right angles to the building, will give you the station on the chain-line whenever you may want to find it.

7.    When the foregoing method cannot be adopted, in consequence of not being able to take a right-angled offset from any building to the station which you wish to fix, then two lines may be measured from the station to the corners, or to any other parts of two adjoining buildings ; and the intersection of these lines, when measured from the buildings, will give the station required.

8.    After all the principal streets have been measured, then proceed to the smaller and intermediate streets ; and lastly to the lanes, alleys, courts, yards, and every other part which it may be thought necessary to represent upon the plan.

9. When any of the streets are so narrow as not to admit of tie-lines being taken with the chain, the angles which the chain-lines make with each other, at the meetings or intersections of the streets, must be taken in degrees and minutes by a theodolite; and in planning, they must be laid down as directed in Problems XX. and XXI. Part I.

10. What has been advanced on this subject will, no doubt, be acceptable to learners ; but as towns are built after such a variety of plans, and consequently vary so much in their forms, no directions can be given that will be applicable

to every particular case to be met with in practice. A great deal will always depend upon the skill and judgment of the suiweyor, who should, after duly examining every part of the town, endeavour to run his lines in the most advantageous manner.

Examples.

Let it be required to measure the New Town, No. 7, Plate TIL

In order to follow the method described in Note 4, we shall begin at the south-west corner, as in Problem IV. Part IT. ; although the survey would be conducted precisely in the same manner if we began at any other corner.

1st Line. Put down + 1, at the SW. corner, and proceed towards the SE. corner; taking offsets to the buildings, wherever it is necessary, and sketching their bases in the margin of the notebook. At the end of High Street, put down + 2; at Queen Street, + 3; at Low Street, + 4; at the SE. corner, + 5 ; and produce the line at pleasure to + 6.

2nd Line. From + 5 proceed towards the NE. corner, but when you arrive at the end of York Street, put down + 7, and thence run a tie-line to + 6. From + 7 proceed with the main-line, and at King Street put down + 8 ; at George Street, + 9 ; at the NE. corner, + 10; and continue the line to + 11.

3rd Line. From + 10 go towards the NW. corner; but when you come to the end of Low Street put down + 12, from which run a tie-line to + 11. Proceed from + 12, and at the end of Queen Street put down + 13; at High Street, + 14; and at the NW. corner, + 15.

4th Line. From+15 proceed towards the SW. corner; and at the end of George Street put down + 16 ; at King Street, + 17; at York Street, + 18; and continuing the line to + 1, you will have circumscribed the town with four main-lines into which the lines measured along the streets must be run.

Rote.—After the first three lines are laid down, it is evident that the fourth line will serve as a check, and will reach exactly from + 15 to + 1, if all the operations have been conducted with accuracy.

5th Line. From +18, through York Street, to + 7.

6th Line. From + 8, along King Street, to +17.

7th Line. From +16, through George Street, to + 9.

8th Line. From +12, along Low Street, to +4.

9th Line. From + 3, through Queen Street, to +13.

10/A Line. From +14, along High Street, to + 2; thus the survey of the town is completed.

Rote 1. The chain-lines and stations do not appear upon the plan, as they could not have been conveniently entered without increasing its size; the

learner will, however, find no difficulty in making a similar plan, two or three times as large ; drawing the chain-lines, and putting down the stations in their proper places. Or he may take the dimensions of the given plan with a small scale, enter them in a note-book, and then draw a rough plan by a larger scale, and after that a finished one, which will be an exercise that will tend much to his improvement.

2.    The survey of this town might have been carried on according to the directions given in Note 5, by measuring a line through King Street, and another through Queen Street; and then connecting these two lines together by tie-lines taken at the point of intersection.

3.    Here it will be proper to observe that in taking an angle with the chain or theodolite, at the intersection or meeting of two lines, either the external or internal angle may be taken, as circumstances may make it most convenient; but it should always be remembered, that neither very acute, nor very obtuse angles should be measured, if it can be avoided, as both are liable to errors in laying down. Those angles which approach nearest to right angles should always be preferred as being most correct.

4.    By way of proof it is an excellent plan to take both the angles. If they be taken by the chain, you will have a check-line by the scale; and if taken by the theodolite, their sum should be 180 degrees ; and you will also have a proof in planning, in consequence of having measured an angle and its supplement. (See Definition 16, and Problems KX. and XXI.)

Directions for Surveying and Planning Mansions, Farm-buildings, Stack-yards, &c.

Landed estates, generally speaking, have mansions for their proprietors, and homesteads for tenants, and these occupy a prominent place on every plan.

In a survey, one of two ways may be ordered. First, a new site for a homestead or mansion may be proposed, involving the resubdivision of the land by new fences and roads; second, such improvements may be finished before the surveyor is employed.

Under the first proposition, the chief points that engage attention are, first, the form of position of buildings and fields; second, the site of the homestead and labourers’ cottages ; third, water for cattle, irrigation, and machinery; and fourth, roads : under the second proposition, surveying and planning only are involved.

1. A modern farm-homestead may be considered a manufactory consisting of a rectangular block of buildings, requiring for a site an area of land of a similar form, and in size according to the size of the form and number of cattle kept.

"When the cattle-yards are roofed over, the position of the houses is of little moment, but when open they should have a southern exposure.

Kectanglar fields are to be preferred, and fences should run north and south, east and west. But this position, although the most

favourable, is not always practicable, so that every individual case must be its own rule.

When the front of the houses does not run parallel to the fence opposite, there will be a triangular piece of land on each side to appropriate to some useful purpose. To throw such pieces into the adjoining fields would, in many cases, destroy their appearance and make them a perpetual eyesore, and -what is worse, more expensive to labour. Small paddocks for calves, or for liquid manuring and soiling, if properly managed, are always the best paying land on the farm, and highly ornamental if well laid out; so that by such means every inch of land about the homestead can be turned to the best advantage.

2.    The farm homestead and labourers’ cottages should be centrally situated, but in practice the third and fourth points generally determine the sites of both upon a farm.

3.    Water for the homestead, the cottages and fields being invaluable, is the first thing that should engage attention. The means of supply, be it pump or otherwise, should never be omitted, and when the water is conveyed from a distance, the fountain-head and direction of the pipes should also be indicated upon the plan. When a sufficient supply can be had for irrigation, or for water power, the same rules apply to water-courses, mill-dams, &c.

4.    Roads and tramways for traction engines should be laid out on the same principles as railroads relative to levels. In many cases, therefore, their direction will not only determine the site of the homestead, but also the range of the fences.

When the farm has only to be surveyed and planned, it is usual to range one of the lines close past the homestead, so that the ground plan of the buildings can be laid down from it, as in the preceding example of a new town, No. 7, Plate YII. Part Y.

The home farm of the landowner can be surveyed and planned as above, but improvements in connexion with the mansion, &c., involve much ornamental work, for which no general directions can be given. When the surrounding grounds are extensive and thickly wooded, some difficulty may be experienced in ranging lines and driving them with the chain, but the bearings can always be taken from some eminence by the theodolite, according to Section VI., Part IX., and the details easily worked out with the chain and plotting scales afterwards.

Directions for planning villages, towns, and cities.

All the main-lines must first be laid down, and the stations upon them marked off. The lines measured along the streets must then be

drawn, and the stations upon them denoted. The bases of the buildings must next be laid down from the offsets, so as to form the streets, and shaded as directed in Part V., and exhibited in Plate VII. The rough plan must then be transferred to a clean sheet, by some of the methods described in Part V., in order to make a finished plan.

The bases of all public buildings, such as churches, castles, prisons, session-houses, market-places, infirmaries, hospitals, mansion-houses, monuments, &c., should be delineated upon the plan with the utmost correctness; and most surveyors draw the bases of the columns which support the roofs of market-crosses, the galleries of churches, &c., as exhibited in the plate to which we last referred.

The streets are usually left white; but some draughtsmen prefer colouring the causeways with a tint of blue to distinguish them from the carriage-roads, which are generally washed with a yellowish brown.

The grass-plots in gardens, public squares, &c., whether they be rectangles, rhombuses, circles, ovals or regular polygons, should be correctly delineated upon the plan; then shaded with. Indian ink and washed with green, in the same manner as pasture grounds; and trees, water, pleasure-grounds, gardens, gravel-walks, &c., must be shaded and coloured as directed in Part V.

The name of the village, town, or city should be given in conspicuous characters, in some vacant part of the plan or map; and the names of all the streets, public squares, churches, colleges, halls, prisons, castles, court-houses, mansion-houses, market-places, lanes, alleys, courts, yards, &c., must be entered in their respective situations in the manner exhibited in Plate VII.

Hote 1. If the dimensions be taken and laid down in feet, a scale of feet must be given ; if in yards, a scale of yards must be given ; if in chains and links, a scale of chains and links must be given; and if the town or city be very large, a scale of miles and furlongs may be given upon the plan, for the purpose of measuring distances; and as 220 yards make a furlong, the distance of one place from another may be easily obtained in miles and yards.

2.    Any remarks or explanations that it may be thought necessary to give, may be entered in some vacant corner of the plan.

3.    All plans, ornaments, &c., should first be drawn in pencil; and it will tend much to the improvement of the learner, if he form all his printing, German text, and large-hand letters by the pencil also, and then finish them with Indian ink.

4.    In forming letters, ornaments, &c., with the pencil, the lines and strokes should be made as fine as possible, as the ink frequently runs upon the lead when the pencil has been used too freely ; hence the necessity of applying Indian rubber after the outlines have been finely drawn with Indian ink, in order to remove the lead which is not covered by the ink, before we proceed to finish the letters, ornaments, &c.

5.    If the pupil does not succeed well in his first attempt with the pencil, the letters, ornaments, &c., must be effaced with Indian rubber; and he must repeat the process until he can form all the letters, devices, &c., correctly.

6.    Properly prepared lead pencils are essential for fine drawing, and will be found to answer well in making letters, ornaments, &c., when they are of a middling degree of hardness; because the marks made by them may be easily effaced.

7.    After practice has made the learner a proficient in penmanship, he will be able to print text, and write more expeditiously without the use of the pencil.

8.    Here it may not be improper to caution the learner against a very common fault of young draughtsmen ; namely, that of making their lines and letters too strong, both with pencil and ink.

9.    Engraven plans of towns, &c., as copy-books, may now be had in almost every county town in the kingdom, and if the learner can get possession of a plan of a place with which he is personally familiar, the example will be none the less interesting and instructive. And besides the ground plans of towns delineating the bases of houses, the areas of streets, parks, &c., those of large farmeries, with their outbuildings, gardens, and small paddocks adjoining, covering altogether an area of several acres of land, are not unfrequently called for on a larger scale than is exemplified on the general plan or atlas of the estate. This is more especially the case as to the ground plans and isometrical views of the mansion, gardens, pleasure-grounds, and buildings of the home farm of the proprietor. Engraven plans of many of these, and designs for an indefinite number more, are now accessible, and may be studied with advantage, and even transferred to drawing paper on a different scale. Nor is it advisable that the learner should exclusively confine his source of information to engraven plans, for many (especially farmers’ and land-surveyors’ sons) may very profitably get for an exercise to give in to their teachers a ground plan and view of their parents’ homesteads or residences. Boys are generally fond of drawing, so that the more rudimentary period of education passes over smoothly, but when they advance to what may be termed usefulness, the measuring of the garden and the laying it down with mathematical accuracy upon a base line, is often surrounded with greater difficulty; but they should ever bear in mind that it is upon this latter that their professional eminence in surveying is to depend.

TO CLEAN PLANS OR MAPS.

It has already been intimated to the young draughtsman, that every precaution should be taken to keep plans and maps clean in executing them ; but notwithstanding the greatest possible care, they will generally be somewhat soiled, both upon the face and back (perhaps in consequence of misfortunes), either by dust, ink, or colours; hence it is necessary to clean them before they are delivered.

To dean plans or maps that are soiled with dust, Indian inlc, or colours.

Take a sharp penknife, with a roundish point, and scrape those parts gently which are besmeared with ink or colours, until you

efface the blots; then use clean Indian rubber freely to those places^ that are soiled with dust; and lastly, rub the whole map well with white bread, taking care to pare the bread as it accumulates the dust.

Note 1. Indian rubber is made from the juice of a large tree, which grows in South America, Java, and India. The juice is obtained by making incisions through the bark of the tree. From the wounds thus formed the juice flows abundantly. It is usually brought to Europe in the form of pear-shaped bottles, which are made by spreading the juice over moulds of claj'. These are dried in various ways according to the season. This done, the clay in the inside is then picked out by instruments.

2. When Indian rubber has become foul by frequent use, it may be cleaned by washing it in lukewarm water and soap.

To clean plans or maps that are blotted with common ink.

If the blots be light, they may be scraped out with a penknife, or effaced by rubbing them repeatedly with clean paper wet in water or saliva; but when they are deep, acid or salt of lemons must be used in the following manner : Dissolve a small portion of the acid in hot water, and with a clean hair-pencil, dipped in the solution, ■wash the blots until they are removed.

Note 1. Recent blots are easily obliterated; but when they are old, and very deep, it will be found necessary to let the paper dry, and repeat the wash several times. Salt of lemons is sold in small boxes by druggists.

2.    When you have to write upon those places from which the blots have been

emoved, the paper will bear the ink better if you rub a little pounce upon it with

clean paper, and then smooth it with your folder, or with the haft of your penknife. Sometimes the paper will bear the ink pretty well, after a blot has been removed, by merely smoothing it as above directed.

3.    Pounce or gum sandarach is a resinous gum which exudes from the juniper-tree, and is of a pale yellowish colour. It is imported in small pieces or tears, about the size of peas; and when reduced to powder, and passed through a fine sieve, it is used for rubbing upon writing paper, in places where any blot or writing has been scraped out.

4.    Sometimes a spurious kind of pounce is made by reducing resin (the juice of the Scotch fir) to a powder ; but this should never be used, as it is too gummy. It may be known from genuine pounce by being of a darker yellow, and having a stronger odour.

5.    Large blots of Indian ink, common ink, or colours may be removed from maps or strong drawing paper by washing them repeatedly with a sponge and clear water, always taking care to squeeze and wash the sponge as it absorbs the ink or colours; repeating the process until the blots disappear.

6.    Sponge is a kind of marine animal production, a species of the class polypi, found adhering to rocks, shells, &c., chiefly in the Mediterranean Sea. The best sponges are those which are white, light, and have the holes small, and near to each other.

7. Sponge is much used by landscape painters, in washing their pieces with water, after they have finished shading them with neutral tint. This process removes the superabounding shade, discharges the clouds, produces a remarkable softness, and prepares the pieces for taking the colours with ease and freedom.

SECTION II.

DIRECTIONS FOR MEASURING AND PLANNING BUILDING-GROUND, AND DIVIDING IT INTO CONVENIENT LOTS FOR SALE.

Directions for measuring building-ground.

Land lying in the vicinity of large towns is frequently sold by the square yard, for building-ground; and as it always bears a high price when the situation is eligible, it is of the greatest importance, both to the buyer and seller, to ascertain its contents with the utmost accuracy.

In order to accomplish this desirable object, the dimensions should be very correctly taken with a measuring-tape divided into yards, tenths, and hundredths ; or with a tape divided into feet and tenths, or feet and inches.

When the dimensions are taken in feet and inches, the inches must be reduced to the decimal parts of a foot: and the area found from such dimensions must be divided by 9, to bring it into square yards.

Whatever be the shape of the ground to be measured, it must be divided into such squares, rectangles, trapezoids, trapeziums, or triangles as will give the true content of the whole : and if the side be crooked, offsets must be taken, as directed in Problem YI. Part III.

Narrow pieces of building-ground must be measured by Problem VII.; and if they be very irregular, their areas may be correctly found by the method of equidistant ordinates described in Problem IX., Part III.

Note 1. As a measuring tape is not so convenient in taking the dimensions of land as a chain, it is more eligible to use the latter when the land to be measured is extensive; the greatest care, however, must be used in order to obtain the dimensions correctly, which should be taken to a quarter of a link.

2. The chain must be completely stretched, and held at the bottom of the arrows in measuring; and if it be an inch or two overlong, an allowance must be made in the dimensions ; thus, if a line 650 links be measured by a chain that is 2J inches above 66 feet, we shall have 6£ x 2£ = 16£ inches = 2 links nearly ; hence the true length of the line will be 652 links.

3.    The above method may also be adopted in measuring' land, when it is found necessary to correct the dimensions taken by a chain that exceeds the proper length.

4.    As 4840 square yards make 1 acre, 1210 square yards, 1 rood, and 30£ square yards, 1 perch, we can easily reduce acres, roods, and perches to square yards, in the following manner : Multiply 4840 by the number of acres ; 1210 by the number of roods ; and 30‘25 by the number of perches ; then the sum of these three products will be the square yards required.

5.    When the area is in square links, divide it by 20‘6611, the number of square links in a square yard; and the quotient will be the area in square yards. (See the Table of Square Measures in' Part III).

6.    Building-ground is generally sold in small parcels. Sometimes, however, it is sold by whole fields together, which are afterwards divided by the buyer, and retailed out in small lots.

Examples.

1. The length of a rectangular piece of building-ground is 65-8 yards, and its breadth 32-6 yards; what is its area in square yards, and its value at 5s. 9d. per square yard 1

yds.

65-8

32-6 3948 1316

1974

2145-08 area

yd.

s.

yds.

.£

s.

d.

As 1    : 5-75    : : 2145-08

21 the value.

2.    The length of a rectangle measures 85-36,and its breadth 43-28 yards; what is its area in square yards, and its value at 6s. 3d. per square yard ?

Ans. The area is 1694-3808 square yards; and the value of the land 1154/. 9s. 10$d.

3.    The parallel sides of a piece of ground in the form of a trapezoid measure 84-63 and 72-78 yards, and the perpendicular distance between them 56-59 yards; what is its area in square yards?

Ans. 4453-91595 square yards.

4.    The diagonal of a trapezium measures 236-5 feet, one of the perpendiculars 189-3 feet, and the other 127-9 feet; what is its in square yards, and its value at 1/. 6s. 6d. per square yard ?

Ans. The area is 4167-655 square yards; and the value of the ground 5522/. 2s. lO^d.

5.    The base of a triangle measures 369-9 feet, and the perpendicular 234-7 feet; what is its area in square yards, and its value at 2s. 6d. per square yard ?

616    14

area

6.    The three sides of a triangle measure 362 feet 3 inches, 316 feet 6 inches, and 284 feet 9 inches respectively; what is its area in square yards.

8$d.

Ans. By Note 4, Part IV., you will find the area to be 4810 square yards.

7.    Draw a plan of an irregular piece of land, and find its area in square yards, from the following dimensions, taken in feet.

AB

1286

145-6

247-6

1015

987

59-8

790

317-6

136-5

720

560

164-2

223-5

465

346

124-8

345-2 372-4 Begin at

268

000

245-3

and go West.

Answer.

Double areas

359147"3 offsets on the right 676164-8 ditto on the left

2)1035312-1 sum 9)517546-05 area in square feet 57517-33 ditto in square yards.

8. Required the plan of a j)iece of building-ground, and also its area in square yards, from the following equidistant ordinates, taken in feet.

AB

255-8 250-6 246-3 235-4 221-2 201-9 176-8 168-5 157-7 146-0 139-1 Begin at

1000

CO

700

600

500

400

300

200

100

000

and range East.

Answer.

394⁻9 the first and last ordinates 4009-6 four times the sum, &c. 1604-0 twice the sum, &c.

6008-5 sum total

100 the common distance

3)600850-0

9/200283-3 area in square feet 22253-7 ditto in square yards.

9. Required the plan of a piece of ground, and likewise its area in square yards, from the following dimensions, taken in feet.

BC

to Ot © Oo M Oi OOiOMOII^QMO OC©-boh©i6bALi©

846

co

700

600

500

400

300

200

100

000

R. off B

r-H H

Begin at

o bO i>

^ co o oi td ©ow

525-6 c and go West.

Answer.

Double areas

658576-8 triangle ABC 81307-2 offsets on bc

2)739884-0 9)369942-0 area in square feet

41104-6 ditto in square yards.

10. Required the plan of a portion of building-ground, and also its area in square yards, from the following dimensions, taken by Gunter’s chain ; likewise its value, supposing it to have been sold by auction at 14s. 9d. per square yard.

AB

221

1175

25i

1100

454

1000

40 o

900

54f

800

68J

700

704

600

65|

60i

551

300

200

100

Begin at

000

and go West.

Note.—In calculating tlie area, the quarter-links must be treated as decimals.

270

474| A

C 4S2J

BD

1235

1075

E. Off B

Answer.

Double areas

1181895'00 trapezium abcd 112881 *25 offsets on ab

2)1294776-25 sum

647388*125 area in square links.

By note 5, we have 647388-125 ~ 20-6611 = 31333 67, the area in square yards; then, as 1yd.: 14-755. : : 31333-67yds. :462171-63255. = 23108/. II5. 7|c/., the value required.

Directions for planning building-ground, and dividing it into convenient

lots for sale.

Building-ground may be laid down by a plotting-scale, whether the dimensions be taken in yards or in feet, by calling each chain upon the scale one yard, or one foot, as the case requires ; and the intermediate divisions will evidently be tenths of a yard, or tenths of a foot.

If the plot of ground be small, the scale made choice of should be pretty large, so as to exhibit every part of the ground distinctly.

After the plan has been drawn, the ground must be judiciously divided and laid out, by making streets at a proper distance from each other : and then subdivided into convenient parcels or lots for

sale, according to the situation of the place and the size of the houses for which the ground is best adapted.

Main or principal streets should be much wider than it is necessary to make back or intermediate streets; and the size of the house-steads adjoining main streets should exceed the size of those adjoining back streets.

When practicable, streets should be laid out in straight lines; and if possible on the rectangular plan shown Plate VII. Part V.

Streets are laid out of very different breadths, from 15 to 90 or 100 feet; but when ground is of great value, the breadth of the streets becomes an object of consideration, whether the ground occupied by them be given by the seller, or purchased by the buyer of the adjoining lots.

Building-ground may sometimes be very elegantly laid out in a square or a rectangle. When this is the case, the houses are built on the margin or outside of the square ; and in the middle is left an open area, which is generally ornamented with grass-plots, gravel-walks, trees, &c.

If the ground will admit, it is very desirable for each house to have a garden laid out in the front; which must, of course, be sold with the house-stead. The open area may be divided into as many equal parts as there are house-steads in the square ; one part may be sold with each house-stead •, and the respective purchasers may occupy the whole as joint property.

Ground laid out in this manner generally fetches a good price, as most persons think it more pleasant and healthful living in squares than in streets.

After everything has been properly and judiciously arranged upon the plan, such dimensions must be taken by the scale as will enable the surveyor to stake out all the streets, squares, lots, house-steads, &c., in the field. This being done, the ground may then be considered as ready for inspection and sale.

Note 1. House-steads must be laid out of different sizes, according to the respectability of the intended buildings.

2. A plan of ground or buildings, intended for sale, is generally left for inspection at the office of the surveyor or solicitor employed on the occasion, from the time of advertising to the time of sale. Also, the special conditions of the sale, not specified in the advertisement, may commonly be known at those offices, or by applying to the proprietor, or to his agent, previously to the day of sale.

3. Building-ground is generally sold by auction ; and if it be divided into small lots, it will tend much to promote the sale, as many persons may be desirous of purchasing a single house-stead, who would not find it convenient to purchase a lot containing two or three house-steads.

4.    The price of building-ground varies from sixpence to upwards of twro guineas per square yard, according to the eligibility of the situation.

5.    The method of laying out building-ground so as to form straight streets at right angles to each other is exemplified in the Plan of a new Town, Plate VII.

SECTION III.

MISCELLANEOUS QUESTIONS RELATING TO SURVEYING, LAYING OUT, PARTING OFF, AND DIVIDING LAND.

1.    The base of the largest Egyptian pyramid is a square whose side is 693 feet; how many acres of ground does it cover?

Ans. llci. Or. ip.

A ns. 25 yards.

2.    Eequired the side of a square garden that cost 3/. 18s. \\d. trenching, at 1 \d. per square yard.

Ans. 8a. 2r. 17^.

4.    The area of a rectangular field is 14a. 2r. 1 \p.; what is its length, its breadth being 925 links ?

5.    A rectangular allotment upon a common cost 78/. Is. 10\d. digging and levelling, at 11. 10s. per acre; what will be the expense of fencing it half round, at 5s. 6d. per rood; its length being 1225 links.

3.    Eequired the area of a rectangle whose length is 1275 and breadth 675 links.

Ans. 1575 links.

Ans. 17/. 18s. 8d.

6. Measuring along the base of a field in the form of a rhomboid, I found the perpendicular to rise at 678, and its length 1264 links; the remainder of the base measured 2435 links; what is the area of the field ?

Ans. 39a. 1 r. lofqo.

7. A grass-plot, in a gentleman’s pleasure-ground, cost 3/. 14s. Id. making, at id. per square yard; what is the length of the base, the perpendicular being 40 feet, and the figure a rhqmbus ?

Ans. 50 feet.

8.    What is the area of a triangular field, the base of which measures 3568 links, the perpendicular 1589 links, and the distance between one end of the base and the place of the jierpendicular 1495 links ?

Ans 28a. lr. 15|p.

9.    After measuring along the base of a triangle 895 links, I found the place of the perpendicular and the perpendicular itself =994 links, the whole base measured 1958 links; what is the area of the

triangle ?

Ans. 9a. 2r. 37p.

10.    The area of a triangle is 6 acres, 2 roods, and 8 perches, and its perpendicular measures 826 links ; what will be the expense of making a ditch the whole length of its base, at 2s. 6d. per rood ?

Ans. 61. 4s. 7|d.

11.    What is the area of a triangle whose 3 sides measure 15, 20, and 25 chains respectively ?

A ns. 15 acres.

Ans. 561YS446 feet.

12.    Required the area of a grass-plot in the form of an equilateral triangle, whose side is 36 feet?

13.    What is the area of a triangular field whose 3 sides measure

2564, 2345, and 2139 links?

Ans. 23 a. 2 r. 0 £p.

Ans. 843/. 7s. 8d.

14.    The three sides of a trianglar fish-pond measure 293, 239, and 185 yards ; what did the ground which it occupies cost, at 185/. per acre ?

15.    How many square yards of paving are there in a trapezium whose diagonal is found to measure 126 feet 3 inches, and perpendiculars 58 feet 6 inches and 65 feet 9 inches 1

Ans. 871’47569 yards.

16. In taking the dimensions of a trapezium, I found the first perpendicular to rise at 568, and to measure 835 links; the second at 1865, and to measure 915 links; the whole diagonal measured 2543 links ; what is the area of the trapezium ?

Ans. 22a. In Op.

17. Lay down a trapezium, and find its area from the following dimensions; namely, the side AB measures 345, BC 156, CD 323, DA 192, and the diagonal AC 438 feet.

Ans. 52330-33406 feet.

18. What is the area of a trapezoid whose parallel sides measure 25 and 18 feet; and the perpendicular distance between them, 38 feet ?

Ans. 1197 feet.

19.    The parallel sides of a piece of ground measure 856 and 684 links, and their perpendicular distance 985 links ; what is its area?

Ans. 7a. 2r. 13\p.

20.    If the parallel sides of a garden be 65 feet 6 inches and 49 feet 3 inches, and their perpendicular distance 56 feet 9 inches; what did it cost at 325/. 10s. per acre ?

Ans. 24/. 6s. 7\d.

21. It is required to lay down a pentangular field, and find its annual value at 21. 5s. per acre, the first side measuring 926, the second 536, the third 835, the fourth 628, and the fifth 587 links; and the diagonal from the first angle to the third 1194, and that from the third to the fifth 1223 links ?

Ans. 181. 10s. 7\d.

22. The diameters of an elliptical piece of ground are 330 and 220 feet; and how many quicks will plant the fence forming the circumference, supposing them to be set 5 inches asunder ?

Ans. 2073.

23. Given the lengths of 7 equidistant ordinates of an irregular

Ans. 1704 feet.

piece of ground, as follows; 15, 19, 20, 23, 25, 30, and 33 feet; and the length of the base 7 2 feet; require the plan and area.

A ns. 48-072 yards.

24.    What must be the length of a chord which will strike the circumference of a circular plantation that shall contain just an acre and a half of ground ]

25.    The annual rent of a triangular field is 43/. 15s., its base measures 25, and perpendicular 14 chains; what is it let for per

acre]

A ns. 21. 10s.

26.    The transverse diameter of the ellipse in Grosvenor Square measures 840 links, and the conjugate 612, within the Avail; the wall is 14 inches thick; what quantity of ground does it inclose, and how much does it occupy ] A ns. The wall incloses 4a. Or. 6p., and

occupies 1760-531 square feet.

27.    Two sides of an obtuse-angled triangle are 5 and 10 chains; what must be the length of the third side that the triangle may contain just two acres of ground ]

Ans. 8-06225 or 13-60147 chains.

Ans. 134-3538.

28.    What is the area of an isosceles triangle inscribed in a circle A\rhose diameter is 24 ; the angle included by the equal sides of the triangle being 30 degrees ]

29.    The side AB of a triangular field is 40, BC 30, and CA 25 chains; required the sides of a triangle parted off by a division-fence made parallel to AB, and proceeding from a point in CA at the distance of 9 chains from the angle A.

Ans. 16, 19-2, and 25⁻6 chains.

30. A field in the form of a right angled triangle is to be divided between two persons, by a fence made from the right angle, meeting the hypotenuse perpendicularly, at the distance of 880 links from one end; required the area of each person’s share, the length of the division-fence being 660 links]

Ans. 2 a. 3 r. 24J p. and I a. 2 r. 21 \p.

31.    It is required to part from a triangular field whose 3 sides measure 1200, 1000, and 800 links respectively, 1 acre, 2 roods, and 16 perches, by a line parallel to the longest side.

Ans. The sides of the remaining triangle are 927, 77 2f and 618 links respectively.

Ans. The division fence is 22-8035 chains, and it divides the base into two parts, whose lengths are 17-0087 and 12-9913 chains respectively.

32.    The base of a field, in the form of a trapezoid, is 30, and the two perpendiculars are 28 and 16 chains respectively; it is required to divide it equally between tAvo persons, by a fence parallel to the perpendiculars.

33. A gentleman a garden had,

Five score feet long and four score broad; A walk of equal width half round He made, that took up half the ground ; Ye skilful in geometry,

Tell us how wide the walk must be.

Ans. 25-96876 feet.

Note 1. If the sum of the two diameters of an ellipse be multiplied by 1-5708, the product will be the circumference, exact enough for most practical purposes. (See Question 22.)

2. All the foregoing questions are taken from the author’s Treatise on Practical Mensuration; consequently, their solutions may be found in the Key to that work.

PART VIII.

PLANE TRIGONOMETRY.

Plane Trigonometry teaches how to determine certain unknown parts of a triangle from having the other parts given. The method of resolution and solution is based upon the relation that exists between the sides and angles of triangles. Thus, in the right-angled triangle ABE (fig. 1), if the sides AB and AE, together with the included angle EAB, are given, the side subtending the -'given angle and the angles adjacent to it may then be found.

SECTION I.

DEFINITIONS.

1.    The circumference HBKE of a circle is divided into 360°, each degree into 60', and each minute into 60".

2.    An arc is a part of the circumference lying between two radii, and is the measure of the angle which the latter form at the centre. Thus AB is the measure of the angle ACB (by construction = 60°).

3.    When the two diameters HK and eb divide the circle into four equal parts, HCB, BCK, KCE, and ECH, they are termed quadrants of the circle.

4.    A right angle is a quadrant, and contains 90°.

5.    The complement of an arc is what it wants of 90°. Thus ah and AB are the complements of each other.

6.    The supplement of an arc is what it wants of 180°. and AE are the supplements of each other.

Thus AB

7.    The sine AS of the arc AB that measures an acute angle ACB is a right line drawn from the extremity A of the radius CA perpendicular to the radius cb that bounds the other extremity of the arc at B, and is parallel to bd.

Pi g. 1.

i)

Y,

II

S.

8.    The cosine of an arc that measures an acute angle is the sine of its complement. Thus AG is the cosine of AB, and as the cosine of HA.

9.    The versed sine of an arc that measures an acute angle is that segment of the radius which lies between the sine and the arc. Thus SB is the versed sine of ab, and gh the versed sine of ha.

10.    The coversed sine of an arc is the versed sine of its complement. Thus gh is the coversed sine of ab, and SB the coversed sine of HA.

11.    The sine EN of the arc AE that measures an obtuse angle as ACE, is a right line drawn from the extremity E of the radius CE perpendicular to the opposite radius AC produced to N, and is parallel to the tangent AL from the other extremity of the arc at A.

12.    The tangent bd of the arc AB that measures the acute angle ACB is a perpendicular drawn from the extremity b of the radius cb that meets the other radius CA produced to D.

13.    The cotangent of an arc that measures an acute angle is the tangent of its complement. Thus Hi is the cotangent of AB, and BD the cotangent of HA.

14.    The tangent AL of the arc AE, which measures the obtuse angle ACE, is a perpendicular drawn from the extremity A of the radius CA, that meets the other radius EC produced to L, and that is parallel to the sine EN, drawn from the opposite extremity of the arc.

15.    The secant CD of the arc AB that measures an acute angle, as ACB, is the radius AC produced to D that limits the tangent BD, drawn perpendicularly from the extremity of the other radius at B.

16.    The cosecant of an arc is the secant of its complement. Thus Cl is the cosecant of AB, and CD the cosecant of HA.

17.    The secant CL of the arc AE that measures an obtuse angle, as ACE, is that portion of the right line el consisting of EC produced to L, that lies between the centre C and the extremity L of the tangent AL, that makes with AC, the other radius, an angle equal to its supplement, and with AC produced an angle lcm = ACE.

18.    The versed sine ES of the arc AE that measures the obtuse angle ACE is that portion of the diameter EB that lies between the sine AS and the other extremity of the arc at E.

Corollaries from the above definitions and diagram.

1.    The sine of an arc is equal to half the chord of double that arc, AS = ^ar.

2.    The sine AS of 60° bisects the radius cb in s, CS = SB.

3.    The sine of 90° = radius.

4.    The cosine of 60° = its versed sine, AG = SB.

5.    The chord of 60° and the tangent of 45° are each equal to radius, AB = AC = HF.

6.    The sine, tangent, and secant of 45° are respectively equal to the cosine, cotangent, and cosecant of 45°.

7.    The tangent of 60° is equal to the chord of 120°, BD = AR.

8.    The secant of 60° = diameter, CD = EB.

9.    The secant of 45° bisects the chord of 90°, and is equal to it, ce = EK.

10.    The square of the diameter is equal to twice the square of the chord of 90° or secant of 45°, eb² = ek² + bk².

11.    The square of the diameter is equal to the square of the chord of 120° added to the square of the radius, eb² = ae² + ab².

12.    The square of the chord of 90° is equal to twice the square of the radius, CF² = CH² + HF².

13.    The sum of the three angles of a triangle is equal to 180°.

14.    If one of the acute angles of a right-angled triangle be given, the other acute angle is its complement, so that all the angles are

given. And if any two angles of a triangle = 90°, tlien the remaining angle = 90°, and the triangle is right-angled.

15.    If one of the angles of an oblique-angled triangle be given, then that angle subtracted from 180° gives the sum of the other two angles; and if two angles are given, their sum subtracted from 180° gives the remaining angle.

16.    If one of the equal angles of an isosceles triangle be found, then the other two angles are determined, because the sum of the two equal angles adjacent to the base subtracted from 180° gives the remaining angle at the vertex. And if the two angles at the base are equal, the sides that subtend them are equal, and the triangle is isosceles.

17.    The two acute-angled triangles ABO and AER are each equilateral and equiangular, and therefore similar to each 'other, and each angle is 60°.

18.    The three right-angled triangles ASB, asc, and CGA are similar and equal to each other; ANE = ESA; also DBC = LAC = MEC.

19.    A triangle standing upon the chord of an arc with two sides radii is isosceles.

20.    In the resolution of a right-angled triangle, as CAL, if C is taken for the centre of the circle, and CA for radius, then AL is tangent and CL secant. If L is taken for the centre of the circle, and LA for radius, then AC is tangent, and LC secant. And if L is taken for the centre, and lc for radius, then CA will be sine and AL = cosine.

21.    To every arc. measuring either an acute or obtuse angle two sines, two tangents, and two secants can, and often require to be drawn.

22.    In an arc subtending an acute angle the two sines and two tangents cross each other within the angle, so that the centre of the circle and the two points of intersection, the one where the two sines cross, and the other where the two tangents cross, are in a right line that bisects the arc, its chord, and the angle it measures.

23.    Each of the two radii, which includes an acute angle at the centre, forms a portion of the two secants to an arc; and when the angle is 60° the portion of each that is inside the arc is equal to the remaining part outside the arc. Thus CB = bl, so that the tangent AL of 60° diverges from the arc AB, a distance equal to radius.

24.    In an arc subtending an obtuse angle the two sines from its opposite extremities fall without the arc perpendicularly upon the two secants, now portions of other radii than those which include the angle. The two tangents are limited by the two secants, ranging from the centre at an angle equal to the supplement of the arc, and

in the opposite direction of the two radii that include the arc or angle. The secant therefore contains no part of the radii that include the angle, but the whole right line formed by each radius and its secant together is greater than the secant of its supplement by radius. Thus el = cl + eg.

25.    In arcs subtending obtuse angles the parallelism of sines with tangents, and vice versd, lies on the opposite sides of the centre, the tangent being on one side, and its parallel sine on the other. Thus AS is parallel to EM, and EN to AL.

26.    In arcs subtending acute angles the parallelism of the sines and tangents of an arc lies within the angle upon the same side of the centre. Thus AS is parallel to BD, and BY to AL.

27.    The sines, tangents, and secants of an arc are equal in magnitude or length to the sines, tangents, and secants of the supplement of that arc; but their position is different, so that a sine, tangent, or secant of the one cannot properly be termed the sine, tangent, or secant of the other, while the versed sine of an obtuse angle is greater than the versed sine of its supplement. -

28.    It is the direction and position of the two sides of a triangle that determine the measure of the angle they include, and vice versd the measure of the angle they include that fixes the position of any two sides.

29.    When a right line is drawn from the obtuse angle of an oblique-angled triangle, perpendicular to the base or side subtending the obtuse angle, it resolves the oblique-angled triangle into two right-angled triangles; the perpendicular thus drawn becomes the sine to each of the two angles adjacent to the base; the sides that include the obtuse angle are radii; and the segments of the base are each equal to the cosine of its respective adjacent angle. Thus in the obtuse triangle EAL, AS is sine to E and L, ES = cos. e, and LS = cos. L, and ASE and asl are each right-angled triangles.

Note 1. The sine, tangent, and secant of an are, are in some books defined to be the sine, tangent, and secant of the supplement of that arc. This, however, is not correct: the two arcs measure two angles of a different species, the one being acute and the other obtuse, according to their geometrical definitions (see Def. 12, 13, 15, 18, 19, 20, 22, 23, and 24, Part I.); consequently each has a separate and independent existence distinct from the other. Thus bd, although defined to be the tangent of ae, is not its tangent; neither is em the tangent of ab. Again bv is not the sine of ae, nor en the sine of ab ; much less is es the versed sine of ae, viz., es also the versed sine of ab ; and lastly cd is not the secant of ae, nor cm the secant of ab. Hence the conclusion of dissimilarity as to position (cor. 27).

2. The diagram is drawn purposely to illustrate the dissimilarity of position that exists between the sines, tangents, secants, and versed sines of angles and their supplements, and also to show some of the properties of triangles formed by lines in and about the circle that are of daily use in taking bearings and tie lines in surveying. It may also be added here that figs. 2 and 3 of the next section will still further show how absolutely necessary it is to attend to such distinctive characteristics as the position of lines, and the species of angles they subtend, in order to secure accuracy in the trigonometrical resolution of the triangles into which a survey may be divided, and to avoid ambiguity and error in plotting the same.

SECTION II.

ON SPECIES, CONSTRUCTION, RATIOS, RESOLUTION, AND SOLUTION, OF

TRIANGLES.

1.    Under Geometry, Part I., six kinds of plain triangles are defined, viz., (1) right-angled; (2) equilateral; (3) isosceles; (4) scalene; (5) acute-angled; and (6) obtuse-angled.

2.    The angles of triangles are divided into three kinds, and defined (1) right angles; (2) acute angles; and (3) obtuse angles.

3.    A plane triangle consists of six parts, viz., three sides and three angles; and if three of these, including one side, are given, the others, technically termed the sought parts, may be ascertained from the ratio or relation that subsists between them and the given parts.

4.    In Plane Trigonometry triangles are divided into two classes, right-angled triangles and oblique-angled triangles. (The latter thus includes the five species of triangles that are not right-angled.;

5.    In a trigonometrical survey the ground is divided into triangles. Thus any three station-poles not in a right line form a plane triangle.

6.    Fields, farms, and estates may be divided and measured witli

5 + 10, + 5 + 10 ;ain the estate (Plate

CO I

+ <1

the chain, either on the triangular method, as illustrated Plate III. Part V., or on the parallel method (Plate IY. Part V.), and yet both may be surveyed by the theodolite as if divided into triangles independently of those of the former survey. Thus the survey (Plate III.) is divided into two large triangles +1 +8 +10, and +    + 8 + 15 ; or into four triangles + 1

+ 8 + 5 + 15, and +15 + 5 + 1.

IV.) may be divided into two triangles + 1 + 6 + 32, and + 1 + 32 + 40. Or the same station-poles shown on the plan may be used in taking the bearings, so as to give a much larger number of triangles in the trigonometrical survey. Thus each of the quadrilateral areas may be taken as forming two triangles.

7.    In large surveys, where each of the interior triangles is bounded

by other three triangles, if the sides of one are determined, that will give one side in each of the adjacent triangles.

8.    In trigonometrical surveying a side of one of the triangles is measured with the chain. This is termed the main-line or base-line. The other two sides may then be determined from the relation they bear to it (the measured line), and to the angles they subtend; and as a side of each of the adjacent triangles is thus found (by last article, 7), the sides of all the other triangles in the survey may therefore be ascertained from trigonometrical data.

9.    The first triangle surveyed and plotted is the one that contains the base-line; and as this side is measured with the chain and the two adjacent angles with the theodolite, such data will give the other angle and species of triangle either by cor. 14, 15, or 16, Sect. I.

Eight-angled triangles.

10.    The field-book of the trigonometrical survey supplies only part of the dimensions requisite for plotting. Technically these are termed the “ bearings,” or “ the given sides and angles,” so that the remaining parts are found by calculation, or construction, or both, as illustrated in the following four examples.

Ex. 1. Let the base line be eb in the triangle AEB (fig. 1): let B = 60° and E 30°, then (by cor. 14) the triangle is right-angled and scalene. The side eb and angles B and e are given to find BA and ea : A is found by cor. 14.

Ex. 2. If the angles at the base are each 45° then ekb would represent the triangle; and as the angles E and B are equal, so are the sides BK and EK. The triangle is therefore both isosceles and right-angled, with a side and three angles given to find the remaining two sides.

Ex. 3. If one of the angles at the base is a right angle, then ASE would represent the triangle ; ES the base line measured with the chain; E and S the angles taken with the theodolite ; EA and SA the sides sought, and A the remaining angle, determined by co?. 14. Sect. I.

Ex. 4. In the interior triangles of a survey the two sides ES and SA, or ea and As, or ea and ES, and the angle A = 90° may be found, when the remaining side EA and two acute angles E and A adjacent will be the parts sought: or ES and the angles E and A; or AS and the angles E and A, as the case may be.

11.    The large triangles on the plans Part V. Plates III. and IV. may be considered copies of those in the field, so that plotting the first two bearings is simply a work of copying (Problems XIX. XX. and

XXI. Part I.), and when the two hearings or angles at the base are given the first triangle may be formed (cor. 28). Thus draw the base-line EB; next EA indefinite, making E = 30° (Problem XX. Part I.); BA in the same manner, making B = 60°: the two indefinite lines will, if the work is correctly .performed, intersect each other at A, and form the triangle EAB right-angled at A. The lines EA and BA may then be measured by the scale on which EB has been laid down upon the plan, or they may be found by trigonometry. Thus, if EB is radius, as in the first two examples, then BA would be sine and EA cosine. The ratio would therefore in these two examples be that of radius to the sine and cosine of the angle E = 30° In the third example ES is radius, SA tangent, and EA secant, consequently the ratio is that of radius to the tangent and secant of E = 30°. In the former case, either of the sides about the right angle may be sine or cosine ; he., if the one is sine the other is cosine, and in the latter, either side about the right angle may be radius or tangent ; he., if the one is radius the other is tangent.

12. The trigonometrical solution of the four examples would therefore be effected by the following formulas:—

Ex. 1. To find BA To find EA Ex. 2. To find bk To find EK Ex. 3. To find EA ., To find SA Ex. 4. To find EA To find E To find ES To find E To find SE To find A

33

33

33

33

33

33

formula 1    Rad.    : sin. E : :    EB    :    BA

2    Rad.    : cos. E : :    EB    :    EA

3    Rad.    : sin. E : :    EB    :    BK

4    Rad.    : sin. B : :    BE    :    EK = BK

5    Rad.    : sec.    E    :    :    ES    : EA

12    AS    : SE : :    rad.    :    tan. A

33

33

33

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an

Note.—Under right-angled triangles it is common to consider the right angle as understood to be one of the parts given, although not so expressed. This however, is objectionable, for when articled pupils enter the field with such formulas stereotyped as it were upon their memories they are worse than useless to them, as they frequently lead to much confusion. In taking the bearings, right angles have to be found in the same way as acute and obtuse ones ; and it may be further observed that unless the two acute angles —90° as a proof check, right-gled triangles are treated as oblique ones ; and as a general rule this is the more advisable one to follow in the trigonometrical survey. Thus the right-angled triangle-eab (Fig. 1) is resolved into two right-angled triangles by drawing the • perpendicular as.

Oblique-angled triangles.

13. When three parts of an oblique-angled triangle are given to find the remaining parts, it requires to be resolved into two or more right-angled triangles. This is done by drawing perpendiculars from the angles to the sides that subtend them (by cor. 29), or to the sides that subtend them produced. Thus AS (fig. 1) resolves the three triangles eab (right-angled), EAL (isosceles), and CAB (equilateral) each into two right-angled triangles.

Ex. 1. If the angles at the base are each 60° then the remaining angle is 60° and the sides opposite the given angles each equal to the base (cor. 17). In this case resolution is unnecessary.

Ex. 2. If the angles at the base are equal, but greater or less than 60°, then the triangle is isosceles, and a perpendicular from the vertex to the base will resolve it into two equal right-angled triangles. In this case half the base may be taken for radius, when the side opposite the right angle would be secant of the given angle. Thus let EAL (fig. 1) be the triangle, EL the base ; E and L each 30°; then ES would be radius, and EA found by formula 5 under article 12 -} AL and A by cor. 16.

Ex. 3. If the angles at the base are acute and unequal, the one being 45° and the other 60°, then the remaining angle opposite the base or given side is 75° and the triangle acute and scalene.

Fig. 2.

Let ABC (fig. 2) be the triangle, AB the base, A and B the angles taken with the theodolite, AC and BC the

sides sought, C being found by cor. 15. R Draw the perpendiculars AD, bf, and CE, which resolve AB into six right-angled triangles.

The sides therefore are to one another as the sines

In the solution of this example each of the angles A, B, and C is common to two right-angled triangles; their complements therefore are equal, or the remaining acute angle of each of the two triangles is equal each to each, so that the triangles are similar, each perpendicular being the sine of the two angles adjacent to the two segments of the base. The sides opposite the equal angles are thus proportional, and the ratio is that of a greater to a less magnitude, the given side being the greatest, or the given side being the radius of the greatest of the three circles now involved.

Again, to find cb we have C the centre

of the angles opposite. Thus AB : AC : : sin. c : sin. B. To find AC, therefore, we have sin. C : rad. : : AD : AC ; sin. B : rad. : : AD : AB. Then inversely rad. : sin. B : : AB : AD; and by ex o’quo inversely sin. C : sin. B : : ab : AC.

with cb radius, and bf sine; and. A the centre with AB as radius, and BF sin. Then sin. C : rad. : : BF : CB; sin. A : rad. : : BF : AB; by inversely rad. : sin. A : : AB : BF; and by ex aequo inversely sin. C : sin. A : : AB : CB.

Ex. 4. If the angles adjacent to the base-line are unequal and their sine less than 90°, then the triangle is obtuse-angled and scalene, the obtuse angle being opposite the base.

Fig. 3.

Let abc (fig. 3) be the triangle, ab the base-line, A and B the angles adjacent whose sum is 60° ;

C = 120° (by cor. 15). Draw the perpendiculars, AD on BC produced, BF on AC produced, and a CE on AB, the base. In this case because AC and BC are each less than AB, the sines AD and BF of the angles A and B when AB is radius, falls without the triangle; and because C is obtuse AD is the sine of C = 120°, when CA is radius; and BF the sine of C=120° when cb is radius. But when A and B are the centres, and AC and BC the radii, then CE is the common sine; and the segments of the base the cosines of A and B. The sides AC and BC may thus be obtained by several methods of solution. Thus ab : AC : : sin. C : sin. B. Or rad. : sin. B : : BA : AD; and as the angle ACD is equal to BCF = 60°, then by making AD radius, rad. : sec. A : : AD : AC. BF and BC may be found in a similar manner. The area of the triangle ABC is obtained from CE as sine to A and B, AC and BC being respectively the two radii.

Ex. 5. If one of the angles at the base is obtuse, say 120°, and the other less than 30°, so that the acute angles found by cor. 15 are unequal, then the triangle is obtuse and scalene, as in Example 4, the greater side being opposite the greater angle, and the less one opposite the less angle.

Let ABC (fig. 3) be the triangle, AC the base-line, A and C the given angles, then B is also given (cor. 15). Draw the perpendicular CE, which will give the trigonometrical data for finding the segments AE and EB of the base AB, and also BC. Thus by making AC radius, CE is sine and AE cosine ; CE may therefore be found by formula 1, art. 12, and AE by formula 2. Again, by making BE radius, EC would be tangent, and BC secant of the angle B, so that by formulas 6 and 11 conversely be and BC would be found. Thus tan. b : rad. : : EC : BE; and tan. B : sec. B : : EC : BC. AE + EB = AB. The two sides AB and BC may also be found as by Examples 3 and 4.

Ex. 6. If one of the angles at the base is obtuse, and the other the greatest of the two acute angles, as B (fig. 3), then BC would be the base-line. The solution would be the same as in the last case (5), but as BC is the less side, the ratio is now that of a less magnitude to a greater.

14.    The preceding examples have reference to the triangle on the base-line, when that side only and the adjacent angles, and hence by cor. 15 all the angles, are given. More frequently in practice the three sides of the first triangle are carefully measured with the chain before the trigonometrical survey begins. This is done purposely that the one survey may be a proof check on the accuracy of the other.

15.    In the other and interior triangles the parts found by previous cases (articles 7 and 8) are more diversified. Thus two sides and an angle opposite one of them may be found from two previous cases, or two sides and the angle they include, or two sides and two angles (i.e., all the angles, cor. 15), and in some central and exceptionary cases the three sides only.

16.    In examples of this kind the construction, resolution, and solution, are more simple, provided the species and ratios are properly attended to in the trigonometrical survey.

Ex. 7. Given the two lesser sides and the angle opposite the least of them, to find the other side and two angles of an acute-angled scalene triangle.

Let abc (fig. 2) be the triangle, AC and BC the given sides, A the angle = 45°. Then AB is the side sought, and C and B the angles. Draw CE and bf, which resolve the triangle into four right-angled triangles. In this case the triangle is scalene and acute-angled, and the ratio that of a less magnitude to a greater, because the lesser of the two given sides is opposite the given and smallest angle. The solution would therefore be to find B, BC : CA : : sin. A : sin. B, which also gives c (by cor. 15). Then sin. A : sin. C : : BC : BA.

Ex. 8. Given the two lesser sides and the angle opposite the greatest of them, to find the other side, and two angles of an acute-angled scalene triangle.

Let ABC (fig. 2) be the triangle, bc and AC the given sides, B the angle = 60°. Then AB is the side sought, and A and C the angles (any one of which will determine the other by cor. 15). Draw CE and bf, which resolve the triangle into four right-angled triangles. . In this case the species and ratio are similar to those of the last example (7), as also the resolution. The construction and solution are also the same in principle, the only difference being that the greatest of the two given sides, instead of the less, is the last line in the

construction; the following formula will give the angles sought; as CA : EC : : sin. B : sin. A.

Ex. 9. Given the two greater sides, and the angle opposite the greatest of them, to find the other side and two angles of an acute-angled scalene triangle.

Ex. 10. Given the greatest and least side, and the angle opposite the latter, to find the remaining side and angles of an acute-angled scalene triangle.

In both these examples, ABC (fig. 2) may represent the triangle, so that the species, construction, resolution, and solution are similar, but the ratio is from a greater to a less magnitude.

Ex. 11. Given the sides about the obtuse angle, and the less of the two acute angles, to find the other side and two angles of an obtuse-angled scalene triangle.

Let AC and BC be the two sides, and A the angle given of the triangle ABC (fig. 3). Then AB is the side sought, and B and C the angles. Draw CE, which gives two right-angled triangles, AEC and BEC.

In this example the sides AC and BC are by hypothesis already drawn upon the plan, and their extents or lengths known, as also the bearing of AB from c, which determines the direction and position of AB, B and C (cor. 28). ■ But before AB is drawn, its length, and also the angles C and B, require to be found, as a check upon accuracy, the bearings having to be corrected for obliquity and refraction. Hence the object of the solution. To find CD, rad. : sin. A : : AC : CE ; rad. : cos. A : : AC : AE. To find B, BC : CE : : rad. : sin. B; rad. : cos. B : : BC : BE\ AExEB = AB; 180°-A + B = C (cor. 15).

Ex. 12. Given the two greater sides, and the angle opposite the lesser of them, to find the other side, and two angles of an obtuse-angled scalene triangle.

Ex. 13. Given the greatest and least side, and the obtuse angle, to find the other side and angles of an obtuse-angled scalene triangle.

In both these cases ABC (fig. 3) may represent the triangle. In the former case, CE will resolve the triangle, as in Example 11, but in the latter (13), BF or AD is requisite, as in previous resolutions, Ex. 4, 5, and 6. The two solutions would therefore be accordingly, the ratios being from a greater to a less magnitude.

Ex. 14. Given the two greater sides, and the angle they include, to find the other side and two angles of an acute-angled scalene triangle.

Let AB and AC (fig. 2) be the two given sides, and A the angle of the triangle ABC; then bc and the adjacent angles B and C are the parts sought.

In this case, and in all examples of this kind, the given angle determines the position of the given sides that include it (by cor. 28), so that the construction and plotting is simple, and the parts sought easily measured by the scale and protractor.

The ratio for the trigonometrical solution is equally definite. Thus, as the given sides are opposite the unknown angles, and the unknown sides opposite the given angle, it follows that c must be greater than B, and BC less than AC, the least of the two given sides. The ratio, therefore, is that-of a greater to a less magnitude.

Again, as 180° — A = B + C, it follows that half the sum of the unknown angles is given. Now as half their difference added to half their sum gives the greater (c), and as half their difference taken from half their sum gives the less (b), it is usual to find half the difference. Thus :—•

AB + AC : AB - AC : : tan. \ (c + b) : tan. J (c- b).

The angles, however, may be otherwise found by resolving the triangle as in fig. 2. Then C is common to two right-angled triangles ADC and BFC, and B to other two. Either, therefore, can be determined in the process of finding BC. Thus, if AC is radius and A the centre, then CE is sin. A, and AE cos. A; consequently, as AB is given, AB - cos. A = EB. To find B, make BE radius and EC tangent. The formula for solution would then be BE : EC : : rad. : tan. B ■, and to find BC, rad. : sec. B : : BE : BC; 180° - A + B = c (cor. 15).

Exercise 1. Given AC and CB, and the included angle C (fig. 2); required the formulas for finding AB, and the angles A and B.

2. Given CB and BA, and the included angle B (fig. 2); required the formulas for finding AC and the adjacent angles A and C.

Ex. 15. Given the two greater sides, and the angle they include to find the other side and two angles of an obtuse-angled scalene triangle.

Let AB and AC be the two given sides, and A the angle of the triangle ABC (fig. 3). Then BC is the side, and B and C the angles sought. Draw CE. The solution is similar to that of Example 14, B and BC being given in this case instead of A and AC in the previous one.

Exercise 1. Given AC and CB and the included angle c (fig. 3); required the formulas for finding the greater side AB and the acute angles A and B.

2. Given CB and BA and the included angle B (fig. 3); required the formulas for finding AC, A and C.

Ex. 16. Given the three sides to find the angles of a scalene triangle.

Let ABC, either fig. 2 or 3, be the triangle. From c, the angle opposite the greater side, draw the perpendicular CE. In this example, according to hypothesis, the triangle is constructed by the plotting of the adjacent |;hree triangles, and from the solutions of them the three sides AB, AC, and BC, were previously found. Such being the case, the perpendicular CE is drawn purposely for giving the content or area, and its length is easily found by the scale. Very generally this is the practice followed when none of the angles are taken in the field. There are, however, exceptions, for which provision requires to be made, i.e., to find the angle A or B, of which CE is sine, AC or BC being radius. In the two right-angled triangles AEC and EEC, the right angles at E and the two sides opposite are given. There is thus only two parts given in each; a part is therefore wanting to effect the solution. This may be found by several propositions, from whence general rules are deduced. One will suffice for an example. Thus, to find the segments of the base AE and EB whose sum AB is given, AB : AC + BC : : AC - BC : AE - eb. Then half the difference of the segments, added to half their sum, gives the greater AE, and half their difference taken from half their sum, gives the less EB. Having thus found AE and EB, they are the cosines of A and B, AC and BC being the radii. To find A, AC : AE : : rad. : cos. A. To rad. : sin. A. : : AC : C

B, BC : BE : : rad. : cos. B. To find CE,

17. In the construction of all scalene triangles, whether acute-angled, obtuse-angled, or right-angled, the arcs of the lesser radii that measure the angles they include intersect in two points the lines that unite them with the longer radii. Thus, in the right-angled triangle EAB (fig. 1), with centre A and radius AB, the greater side is intersected in two points B and C. In fig. 2, AC, with centre A, intersects BC in C and n; BC, with centre B, intersects AC in C and m; CB, with centre C, cuts AB in B and o. And had AOC been the - triangle, then oA with centre o would have cut AC in two points. In the obtuse-angled triangle ABC (fig. 3), CB, with centre C, intersects AB in B and x. And when the base of an isosceles triangle is less than one of the sides, and only one side, the base and vertical angle given the species of the triangle being unknown, then the base line may cut the unknown side in two points.

30. These examples show that in the construction of triangles, and hence in plotting the bearings of the trigonometrical survey, when two sides of a triangle and one angle only are given, then, either the species of the triangle and ratios of the sides must be known, or else the given angle requires to be the one that is included by the two given sides, so as to determine their position (cor. 28), and hence the position of the other line between their extremities, and of the adjacent angles thus formed, otherwise the construction will be what has been termed ambiguous, i.e., one of two triangles, as pointed out in Euclid’s Data, Prop. XLVII., Case 3.

31.    If, however, the species of the triangle, and ratios of the sides, and the position of the lesser given radius determined by the given angle (cor. 28) are known, such data will also determine the position of the intersecting radius or side that cuts the indefinite and first side drawn at the proper length. Tims, in fig. 1, because the triangle is right-angled at A, that fixes the direction and position of the lesser radius AB. But had the triangle been in species an obtuse-angled isosceles one, then the position of the lesser side or radius would have been AC, and the triangle ACE. In like manner ABC (fig. 2), being in species an acute-angled scalene triangle, that determines the position of the two lesser radii. But in the obtuse-angled scalene triangle ABC (fig. 3), CxA is also an obtuse-angled scalene triangle, so that species alone is not enough in this class of triangles, the ratios being also necessary. In other words, it is the position of CB, and the relation of the greater side to the other two sides, that determines the construction of the triangle.

32.    In plane trigonometry triangles exist, and hence their species and also the ratios of the sides and angles exist. If, therefore, a side and two angles are given, or two sides and an angle, or any three parts including a side (art. 3), then the trigonometrical relation that subsists between them and the unknown parts determines the latter. According to the general definition given of trigonometry, it is this relation that supplies the mathematician and surveyor with data for finding the unknown parts. This is consequently the given hypothesis in every proposition in plane trigonometrj^. The sides and angles sought are therefore as definite and free from ambiguity as those given, otherwise they could not be determined.

33.    The ambiguity referred to under (30) is, it will thus be seen (32), exclusively confined to constructive geometry. Applied to land-surveying it is wholly confined to the work of plotting the bearings or copying the triangles of the field on to the plan.

34.    In the trigonometrical survey the species and ratios of the triangles should be carefully determined in the field; as this, practically speaking, is the principal part of the work, or the principal function of plane trigonometry. If this is done, no ambiguity will be experienced in plotting the bearings. (See Sect. VI. Part IX.)

SECTION III.

PLANE TRIANGLES.

The solution of plane triangles may be reduced to four cases, in which the given parts are the following:—

Case I. A side and three angles.

Case II. Two sides and an angle opposite one of them.

Case III. Two sides and their included angle.

Case IY. The three sides.

SOLUTION OF RIGHT-ANGLED PLANE TRIANGLES.

Case I.

Given the angles and hypotenuse or a side, to determine the triangle.

Rule.—As sine of any angle : side opposite : : sine of any other angle : its opposite side.

1. Given ¹ the hypotenuse AC = 720 links, and the angle A = 53° 8'; to find the sides ab and bc, and the angle C.

By construction.

Draw the line AB, of any length make the angle A = 53° 8', draw the hypotenuse AC = 720 links; from C let fall the perpendicular CB ; then ABC is the triangle required : and AB, being measured, will be found = 432 links, and bc = 576 links; also the angle C will be found = 36° 52'.

To find BC. As sine z B = 90°

: hyp. AC :: sine A = 53° 8'

By calculation.

The angle c = 90° - 53° 8' = 36° 52'.

To find ab.

10 00000 As sine b = 90°

: hyp. ac

:: sine ^ C = 36° 52'

10-00000

= 720 links 2-85733 2-90311

= 720 links 2-85733 9-77812

12-76044

1000000

12-63545

10-00000

: side bc = 576 links 2-76044

:side AB = 432 links 2-63545

Note.—The logarithmic operation is performed by adding together the second and third terms of the above proportions, and subtracting the first from the respec-

tive sums, the remainders being the fourth terms : which operations must be understood to be performed, in all the following examples, it being unnecessary to put down the work at length as above.

2. The side AB is = 4320 links, and the angle A = 53° 8'; required the other parts.

By construction.

Make AB = 4320 links, at B erect the indefinite per-<> pendicular BC, make the angle A = 53° 8', and draw AC, meeting BC in C; then AC will measure 7200 links, BC = 5760 links, and the angle C = 36° 52'.

By calculation.

The angle c = 90° - 53° 8' = 36° 52'.

To find BC.

As sin. ^c = 36° 52'

: side AB

:: sin. ZA = 53° 8'

: side bc

3.    In a right-angled triangle are given the hypotenuse = 98 chains and one of the angles = 56° 48', to find the sides and area.

A ns. Sides 82 and 53-66, area 22a. Or. Ip.

4.    One side of a right-angled triangle is 2350 links, and its opposite angle 50°; required the other side and hypotenuse.

A ns. Side — 1977 links hyp. = 3067-5 links.

To find AC. As sin. ZG = 36°52'

: side AB

: : sin. B = 90° : side AC

9-77812 = 4320 links 3-63548 10-00000 = 7200 links 3-85736

9-77812 = 4320 links 3-63548

9-90311 = 5760 links 3-76047

Case II.

Given the hypotenuse, a side, and the right angle, to determine the triangle.

Rule.—As hypotenuse : radius : : given side : to sine of opposite angle.

1. The hypotenuse AC is 24 chains, and the side AB = 14 chains 40 links; required the other side and angles.

'o

By construction.

Make the side AB = 14-40 chains, erect the indefi-o nite perpendicular BC; with centre A and radius AC = 24 chains, describe an arc cutting BC in c; join AC; then BC will be found =19 chains 20 links, and the - angles A and C respectively 53° 8' and 36° 52'.

By calculation.

To find the angle C. As hyp. aC = 24 : sin. A B = 90°

: : side AB = 14-40 : sin. ^ c = 36° 52'

1-38021

10-00000

1-15836

9-77815

To find bc. As sin. B = 90°

: hyp. AC = 24 : : sin. Za = 53° S'

: side BC = 19-20

10-00000

1-38021

9-90311

1-28332

Hence angle A — 90° - 36³ 52' = 53° 8'.

Note.—The side BC may he found by Euc. I. 47, thus,

Vac² — AB² = jy/24² - 19-4² = V368-64 = 19 chains 20 links, the same as above.

2.    The hypotenuse of a right-angled triangle is 960 links, and one of the sides 768 links; required the other side and angles.

Ans. Side 576 links, angles 53° 8' ancl 36° 52'.

3.    In a right-angled triangle are given the hypotenuse = 3350 links, and a side —2631 links; required the other side and area.

Ans. Side 2074 links, area 27a. lr. op.

Case III.

Given the right angle and the adjacent sides to find the other parts of the triangle.

Rule.—As a side : radius : : the other side : tangent of angle adjacent to the first side.

As the sine of this angle : rad. :: second side : hypotenuse.

1. The sides of a right-angled triangle are AB 2880 and BC 3840 links; required the hypotenuse AC and angies A and C.

'o

By construction.

Make AB = 2880, perpendicular to which make BC = 3840 links; join AC; then AC will be found = 4800 links, angle A = 53° 8', and angle c = 36° 52'.

By calculation.

To find angle A. As side AB = 2880 : rad.

: : side BC = 3840 : tan.

To find AC. As sin. A =53° 8'

: rad.

: : side BC= 3840 : hyp. AC = 4800 Angle c = 90° - 53° 8'= 36° 52'.

3-45939 10-00000 3-58433 A = 53° 8'1    0-12494

9-90311

10-00000

3*58433

3-68122

3. The sides of a right-angled triangle are 6250 and 1250 links required the hypotenuse, angles, and area.

A ns. Hyp. 6374 links, angles 78° 41' and 11° 19', area 39a. Or. 10p.

OBLIQUE-ANGLED PLANE TRIANGLES. Case I. and II.

Given two sides and an angle opposite one of them, or given two angles and a side opposite one of them, to determine the triangle.

Rule.—As a side : sine of its opposite angle f : any other side : sine of its opposite angle.

And conversely, As sine of an angle : opposite side : : sine of any other angle : to its opposite side.

1. In an obtuse-angled scalene triangle are given the angle A = 32° 15', the angle B = 114° 24', and the side AB = 98 chains, to find the sides AC and BC, and the other angle.

\\

'A

ts

By construction.

Make AB = 98 chains, draw BC, making the angle B = 114° 24', also draw AC making the angle A = 32° 15', and meeting BC in C ; then AC and BC, being measured, will be found respectively 162'34 and 9542, and the angle C = 33° 21'.

By calculation.

The angle c = 180° - (114° 24' + 32° 15) = 33° 21'.

Since the angle B is greater than 90° , its sine cannot be found in the tables, but by taking it from 180°, its supplement 65° 36' is obtained, the sine of which is also equal to the sine of the angle B.

To find AC. As sin. C = 33° 21'

: side AB = 98 sin. b= 114° 24'

: side AC= 162-34

To find BC.

9-74017

1- 99123 9-95937

2- 21043

9-74017

1-99123

9-72723

1-97829

As sin. c = 33° 21 : side AB = 98 :: sin. A = 32° 15'

: side BC - 95T2

2. In an acute-angled scalene triangle are given AB = 60 chains, BC = 9542 chains, and the angle C — 33° 21'; required the greater side AC, and the angles A and B.

By construction.

Make bc = 95-12 chains, draw AC, making the angle c = 33° 21'; with the side AB = 60, as radius, and centre B, describe an arc cutting AC in A; then ABC is the triangle required.

AC — 108'87 chains, angle A = 60° 38', Ji. angle B = 86° 1'.

By calculation.

To find angle A.

O

As AB = 60 : sin. c = 33° 21' : : BC = 95-12 : sin. A = 60° 38'

1-77815

9-74017

1-97827

9-94029

To find AC. As sin. C = 33° 21'

: AB = 60 : : sin. B = 86° 1'

: AC = 108-87

9-74017

1- 77815 9.99895

2- 03693

The sum of the angles C and A subtracted from 180°, leaves the angle B = 86° 1'.

3. In the obtuse-angled scalene triangle ABC are given the two greater sides BC = 95-12 chains, BA = 60 chains, and the angle c opposite the lesser of them = 33° 21' ; required the side AC and the angles A and B.

By construction.

Make BC = 95-12 chains; draw CA indefinite, making the angle C = 33° 21'; then with BA = 60 chains as radius, and centre B, de- B scribe an arc cutting CA in A; draw ba ; then ABC is the triangle required. AC = 50 03 chains; A = 119° 22'; and B = 27° 17'.

To find A.

As BA = 60 : sin. C = 33° 21'

: : BC = 95-12 : sin. A = 119° 22'

By calculation.

1-77815

9-74017

1-97827

9-94029

To find AC. As sin. c = 33° 21'

: BA = 60 : : sin. B = 27° 17'

: AC = 50-03

9-74017

1-77815

9-66124

1-69922

= 180° - (119° 22' + 33° 21')

= 27° 17'

4. In a plane triangle AB=98, and BC = 95-12 chains, and angle C = 33° 21'; required the other parts.

5. In a plane triangle are given two angles 79° 23' and 54° 22', and a side 1250 links opposite the first angle, to find the other parts.

Ans. Sides 1033-6 and 918-7 links.

Case III.

Given two sides and their included angle to find the rest.

Rule.—As the sum of the given sides : their difference : : co-tangent of half the included angle : tangent of half the difference of the required angles. This angle, added to half the complement of the included angle, gives the greater required angle, and subtracted gives the lesser. The other side is then found by Case I.

1. Given the side AB = 9800 links, the side BC = 9512, and their included angle B= 114° 24', to find the other side and angles.

By construction.

.qT

t>

'sx

By construction.

Make BC = 95-12 chains, draw ac indefinite, making the angle C = 33° 21'; N with AB as radius and centre B describe \ the arc act, cutting AC in A; then ABC is ’*, the triangle required.

° ' chains; A = 32°l5'; and B = 114° 24'.

AC = 162-34

N/

By calculation.

To find angle A.

As ab = 98 : sin. C = 33° 21' : : BC = 95-12 : sin. A = 32° 15'

To find AC. As sin. c = 33° 21'

: AB= 98

: : sin. b = 114° 24'

: side AC == 162-34 Angle B -180° - (a + c) = 114° 24'.

1-99123

9-74017

1-97827

9-72721

9-74017

1- 99123 9-95937

2- 21043

yj Make AB = 9800 links, and the angle ° b = 114° 24'; draw BC, which make = 9512 / links : then ABC is the triangle required. The angles A and c measure 32° 15' and 33° 21', and AC 16234 links.

By calculation.

To find AC. As sin. c = 33° 21'

: ab = 9800 : : sin. B = 114° 24'

: AC = 16234

9-74017

3- 99123 9-95937

4- 21043

To find the angles.

4-28583

2-45939

9-80919

7-98275

As AB + bc = 19312 : AB - BC = 288 : :cot. {B = 57° 12'

: tan. i (c - a) 0° 33' comp. |b = 32° 48'

sum 33° 21' = ^/c cliff. 32° 15' = ^A

2.    Given the two sides 103 and 126 chains, and their contained angle 56° 30', to determine the triangle.

A ns. The angles 72° 20' and 51° 10', the side 110*3 chains.

3.    Two sides of a triangle are 34500 links and 17407, and their included angle 37° 20'; required the other angles and side.

Ans. The angles 27° 4' and 115° 36', the side 23200 links.

Case IY.

Given the three sides to find the angles.

Rule I. From half the sum of the three sides subtract the side opposite the angle sought, and the logarithms of the half sum and remainder, and increase the index of the sum by 20; from the sum thus increased, subtract the sum of the logarithms of the sides containing the angle sought; the remainder, divided by 2, is the log. cosine of half the angle sought.

Rule II. From half the sum of the three sides subtract each of the sides containing the angle sought, add the logarithms of the two remainders, and increase the index of the sum by 20; also from half the sum of the three sides subtract the side opposite the angle sought, and add the logarithms of the half sum and remainder; then half the difference between these logarithmic sums is the log. tangent of half the angle sought.

The remaining angles may be found by Case I.

Nots.—Rule II. is to be preferred when the angle sought is very small or near 180°.

1. Given the three sides ac = 98, ab = 162-34, and BC = 95 12 chains, to find the angles of the triangle.

By construction.

C = 114° 24'.

Draw the longest side AB = 162-34 chains; with AO = 98 as radius, and centre A, describe an arc; with BO = 95-12 as radius, and centre B, describe another arc cutting the former in c ; draw AO and BC; then \ ABO is the triangle required. The ■2A angles being measured, will be ³ found A = 32° 15', B = 33° 21', and

By calculation. (Rule I.)

■J(ab+ac+bc)

|(ab+ac+bc)-bc— 82-61 1‘91703 1-99123

sum

= 177-73 2-24976

ac = 98 ab = 162-34 2-21043

sum + 20 = 24T6679

= 4-20166

2)19-96513

9-98257

cos. iZA=16° 7' 30"

.    ^/a= 32° 15' 0"

(Rule II.)

£(ab+ac+bc)-ab= 15-39 1-18724 J(ab + ac+bc)-ac= 79-73 1-90162 |(ab+ao+bc)

|(ab+ac+bc)-bc= 82-61 1-91703

sum-e20 = 23-08886

= 177-73 2-24976

sum

= 416679

2)18-92207

9-46104

tan. L/a=16° 7' 30"

^a = 32° 15' 0"

The remaining angles may be found by Case I.

2. In a plane triangle are given the three sides 7000, 10400, and 14202 links to find the angles.

Ans. 27° 59', 44° 12', and 107° 49'.

Ans. 57° 27', 57° 59', and 64° 34'.

3. When the sides of the triangle are 2253, 2240, and 2400 links, what are the angles 1

SECTION IV.

THE MEASUREMENT OF ANGLES.

The measurement of horizontal and vertical angles is usually taken with a theodolite.

THE THEODOLITE.

Q1(q

Two of the principal parts of a theodolite are the circular brass plates A and B, which turn one upon the other horizontally on the vertical axis C. Th circumference of the lower plate B is divided ™ into 360°, and these again into half degrees.

[</

At the extremities of a diameter of the upper plate A are fixed two verniers for reading off the minutes in the divisions of the plate b.

Ky

N’.°

re

O'

J>—cT

This plate has its clamp-screw H, and its adjust- E ing - screw I, which {jh? clamp and adjust the (r| whole instrument. To the plate A are fixed two spirit-levels, d, d, at right angles to each other; and a compass J concentric with it. It has also its clamp and adjusting screw (not shown in the figure).

¹

The given parts of a triangle are generally marked with a dash ( — ), and the required parts with a circle (o), as in the figures annexed to the constructions.

²

The sides adjacent the right angle of a right-angled triangle 2683 and 4100 links; required the hypotenuse and angles.

Arts. Hyp. 4900 links, angles 56° 48' and 33° 12'.

dDt

These two plates are used to take horizontal angles and lines.

\%

The vertical semicircle M turns on an axis supported by the frame KL, resting on the plate A; its axis passes perpendicularly through the common axis of the instrument; its motion is therefore perpendicular to the horizontal plates. It is graduated like the plate B, and by means of a vernier, fixed above the compass, the clivisons can be read off to minutes. It has its clamp-screw o, and its adjusting screw P, and is used for taking vertical angles and lines.

The telescope is placed above the vertical circle in two receptacles, called Y’s from their shape, and secured above by two clips i, i. It has a long spirit-level fixed beneath and parallel to it. The object-

glass is adjusted to the sight by the screw Q and the eyeglass by moving it backwards and forwards with the hand.

Between what are called the two parallel plates F and G are fixed two pairs of conjugate screws, b, b, &c., by means of which the upper one F can always be made horizontal. These two plates are connected by the ball and socket D, and to the lower one G are fixed the three legs by which the whole instrument is supported.

To take a horizontal angle with the theodolite.

The bubbles in the two levels d, d, by the proper opening of the three legs of the theodolite, should be made central, and the plummet, suspended by a hook under the body of the instrument, should be made to hang above the station at which the angle is to be taken : this done, then unclamp the whole instrument by means of the screw H, the other motions being kept clamped. Set one of the two levels on the horizontal plate A over one of the pairs of opposite screws b, b, and the other level will be over the other pair, because the pairs of screws as well as levels are conjugate, i.e., at right angles to each other. If both the bubbles in the levels be not in the centre, loosen one of the conjugate screws, and tighten the corresponding one till the levels are accurately adjusted. Loosen and tighten the other pair if required, till the same result is obtained with respect to the other level. If the last operation throw the former adjustment out, repeat the adjustments on each pair of screws till both shall be level.

Next clamp the whole instrument • then unclamp the vernier plate A; set the broad arrow of the vernier to 360°, or zero, on the plate B, and reclamp the plate A. This must be carefully done by the microscope E and the adjusting-screw I.

Again unclamp the whole instrument, and turn it to the left of the two stations, between which the angle is to be taken, till the intersection of the cross wires in the telescope cut the flag or other object in the station as accurately as can be done with the hand ; then clamp the screw H, and by slowly turning the screw I the greatest accuracy may be obtained.

Now unclamp the upper plate A, and turn it round till the cross wires in the telescope cut the object in the second station; then clamp and adjust with the two screws attached to the plate A, till having obtained perfect accuracy, read off the angle by means of the vernier with the glass E : in the same way, read off the angle with the other vernier, and the mean of the two will be the correct angle.

To take a vertical angle.

The horizontal plates being set level, as already explained, bring the bubble of the telescope level to the centre of its tube, observing at the same time whether the zero point of the vertical circle coincides with that of the vernier by the microscope N. These points being found to coincide, raise or depress the telescope till the cross wires cut the required object: then clamp with the screw O, and adjust with the screw P, and having obtained perfect accuracy, read off the angle, which will be an angle of depression, if the broad arrow be between the zero of the vertical circle and the object-glass of the telescope; and an angle of elevation, if beyond them.

V

ADJUSTMENTS.

1. To adjust the line of collimation.

Direct the telescope to some well-defined object at a great distance, adjust it till the intersection of the wires cut it accurately, then turn the telescope on its axis, and observe whether the intersection of the wires still continues to cut it during a whole revolution. If it does, it is in adjustment; if it does not, the line of collimation, or optical axis of the telescope, is not in the line joining the centres of the eye and object-glasses.

To correct this error, turn the telescope on its axis, and by means of the four conjugate screws, m, &c., correct for half the error, alternately loosing one screw and tightening its opposite one.

2. To adjust the axis of the level with that of the telescope.

Make the telescope perfectly level by the tangent screw P; then reverse the telescope in the y’s : if the level remains the same, it is in adjustment; if not, correct for half the error by the screw/at the end of the level, and for the other half by the screw p. Replace the telescope in its former position, and correct again, if necesary.

3. To make the axes of the levels on the vernier plate parallel to it.

Set one of the levels over one pair of the conjugate screws, b, b, &c., then the other level will be over the other pair; make both the bubbles come to their respective centres, turn the instrument half round, and if the bubbles deviate from their centres, correct half the error by the small screws on the levels, and the other half by the screws b, b, &c. Repeat this operation till the bubbles are central in every position throughout a whole revolution of the instrument.

THE VERNIER.

The vernier is a contrivance for measuring the fractional parts of the half-degrees on the lower plate of the theodolite. The space occupied by 29 or 31 of these half-degrees is used to form the scale of the vernier, and divided into 30 parts; whence it is obvious that the difference of one division on the vernier and one on the lower plate will be ^th in excess or defect of the |° on the lower plate, or a single minute; and the difference between 2, 3, 4, &c., divisions on the two plates will be 2, 3, 4, &c., minutes. Hence the following methods of finding the minutes corresponding to a fractional part of a degree on any given angle, &c.

1. To find the minutes in the fractional part of a degree of a given angle by the vernier.

If the broad arrow of the vernier is between a full degree and a half-degree, say between 40° and 40^°, then the angle is 40° and as many minutes as are shown by the number of the first divisionline of the vernier that coincides with a division on the lower plate, reading forward as the degrees are numbered. If the broad arrow be between 40^° and 41°, then the angle is 40° 30'plus the number of minutes shown by the first coincidence of the divisions of the vernier and those on the lower plate. If the coincidence take place at the 13th division on the vernier, then the angle in the former case is 40° 30', and in the latter 40° 30' + 13' = 40° 43'.

2. To set the theodolite to an angle of any number of degrees ancl minutes.

If the required angle be 48° 28', bring the vernier till the broad arrow shall be in such a jmsition within the half-degree above 48°, that the division numbered 28 on the vernier shall coincide with the same numbered line on the lower plate; the theodolite is then set to the required angle.

SECTION Y.

TRIGONOMETRICAL SURVEY OF HEIGHTS AND DISTANCES.

PROBLEM I.

To measure that part of a base line which crosses a river.

Let DBA be the direction of the line, which has been measured to the edge b of the river; it is required to find the width ba of the river.

1. Having set up flags at any convenient points B and A (supposing sight not to be obstructed), range and measure the line BC perpendicular to BA, and let bc = 576 links, then take with the theodolite the angle ACB, which suppose to be 53° 8'; required the distance BA, and from thence the width of the river ba, the distance Bb being 36 links, and aA 43 links.

a;

.    9-77812

.    2-76042

.    9-90311

. 2-88541

sin. A = cos. c = 53° 8' : side bo = 576 .

: : sin. C = 53° 8' .

: BA = 768

ba = 768 - (36 x 43) = 689 links, the breadth of the river.

The distance ba may be found by construction. See solution of right-angled plane triangles, Case 1, Ex. 2.

Note.—If the angle acb had been 45°, the angle a would also have been 45°, and therefore ba = bc. In this case no calculation or construction would be required. A cross-staff, having also sights placed at this angle, is sometimes used for thus expeditiously solving this problem, when the nature of the ground admits of the line bo being measured of a sufficient length for the purpose.

2. The distance BA may be found independently of trigonometry thus. Having measured the perpendicular bc = 576 links, as in the last example, range and measure a line CD perpendicular to CA, and meeting the base-line DBA in D; measure db = 432 links : then (Euc, VI. 8)

As BD : BC : : BC : BA, that is, 432 : 576 : : 576 : 768 = BA, Btf (576)²

or BA

432

768 links.

3. Should obstructions prevent the measurement of the perpendicular CD, let the line CD = 576 be measured as before; at any convenient distance bh = (suppose) 200 links on the base-line, range and measure HI perpendicular to DBA, till the point I shall be in a direct line with the flags at A and C, and let Hi = 726 links; then

4. Should impediments prevent the lines BC, HI, being measured perpendicular to HBA, measure BC to make any angle with HA, and at a convenient distance BH measure Hi parallel to BC, which may be done by erecting two equal perpendiculars to BC, the point I being in a right line with A and C as before.

B/fi

AC

The rule for finding BA will be the same as in the last example ; the calculation in both cases may be readily performed on the ground.

5. Lastly, if the nature of the ground be such that none of the four preceding methods can be adopted, neither to the right nor to the left of the base-line, measure BC in the most convenient direction, and take the angles ABC, ACB with the theodolite; then by taking the sum of these angles from 180°, the angle A will become known, whence, by Case I. of oblique-angled triangles :

As sin. ^ A : BC : : sin. AT C : BA.

This problem may also be solved by construction.

Examples for practice.

1.    BC-308 links (fig. to first method) is measured perpendicular to BA, and CD is ranged perpendicular to CA, meeting DA in D, the distance BD being 400 links, and the flags at B and A being at the water’s edge ; required the distance BA.

Ans. By the second method ba is found — 237 links nearly.

2.    Given BC = 400 (fig. to first method), HI = 520, and BH = 180 links, to find ba.

Ans. By the third method ba is found = 600 links.

3. Given BC = 8 chains, and the angles at B and C respectively 30° and 45° ; required the distance BA, and the perpendicular width AD of the river.

As HI - BC : BH : : BC : BA,* that is, 726 - 576 : 200 : : 576 : 768 links = BA, BH x BC 200 x 576 HI - BC

or BA =

150

= 768 links.

Ans. ba = 5S5 links, and ad = 292 links.

* This rule is derived from similar triangles ; for hi : bc : : ha : ba, and hi -bc : ha - ba : : bc : ba, i.e., hi - bc : hb : : bc : ba.

PEOBLEM II.

To continue the direction and measurement of a given line when buildings or other obstructions stand in the way, which can be avoided by going to right or left.

V,//,

1. At any convenient point A in the given line aACc, which is obstructed by the buildings H, take an angle aAB - 120° (thus making the angle CAB = 60°); range and measure the line ab, till, by taking an angle b = 60° the line BC will clear the obstructions at H ; then measure BC = AB, and take the angle BCc = 120°. Then Cc is in the range of the given line aACc, and AC = AB or BC, the triangle ABC being thus made equilateral 3 hence the measurement of the line aA.Cc may proceed from C, after having added the distance thus found to the measure at A.

2.    If the nature of the ground be such as not to admit line AB to make with aA an angle aAB = 120°, take any other angle, and suppose it to be 132°, which taken from 180° leaves the angle CAB = 48°; measure AB,'till by taking an angle at B = 84° = twice the complement of the angle cab, the line BC may clear the buildings; then measure BC = BA, taking the angle CCB also =132°, and continue the line in the direction cc. The triangle ABC is thus made isosceles and a perpendicular let fall from B to H on AC will divide ABC into two equal right-angled triangles.

Hence as rad. or sin. 90° : side AB : : sin. /_ B, or 42° : AH =

AC, whence AC becomes known.

3.    If it be convenient to take the angle aAB = 135°, the angle B = 90°, ab = BC = (suppose) 800 links, and lastly the angle ccb = 135°, thus making each of the angles CAB, ACB = 45°, the triangle being right-angled at B, as well as isosceles. Hence, by Euc. I. 47,

AC = ab x 2 = 800 X 1-414 = 1131 links.

Note.—By any of these methods the distance ac between any two given points A and c may be found, when sight is not obstructed, as when h is a lake or projection of a sea-coast, intervening between a and c, that cannot be measured across with the chain.

are

PROBLEM III.

To find the distances of three accessible and visible objects, when obstructions prevent the measurement of these distances in direct lines.

■ef

i.

LetA,B, and D be the three objects, thedistances of which, AB, AD, and BD, cannot be measured directly, on account of the obstruction shown in the figure. Set up a flag at C in the direction AB, take the angle ACD, measure the distance CD, and take the angles ADC and ADB. Then in the triangle ACD Are given the angles ACD and ADC, and their included side CD, to find AD. In the triangle ADB are given the angles ADB and DAB = ACD + ADC, and their included side AD, to find AB and BD.

Ex. Let CD = 1962 links, and the angle C = 57a, CDA = 14°, and ADB = 41° 30'; required the distances AD, DB, and AB.

The angle dac = 180° - (57° + 14) = 109°,

The angle dab = 57° + 14° = 71°.

As sin. 109° : CD = 1962 : : sin. 57° : ad = 1740 links.

The angle b = 180 - (71° + 41° 30') = 67° 30'.

As sin. 67° 30' : ad = 1740 : : sin. 41° 30' : ab = 1248 links,

: : sin. 71°

: BD = 178.1 links.

By construction.

Draw the indefinite line cb, make the angle c = 57; draw CD, which make = 1962 links, lay off the angles cda = 14°, CDB = 14° + 41° 30' = 55° 30', and draw the lines da and DB meeting CB in A and B, then the distances AD, AB, and BD will be found as above.

PROBLEM IV.

To find the height of a tower or any other object, standing on an horizontal plane, the base of the tower, dec., being accessible.

Let CBbe a tower, standing on an horizontal plane. Measure any convenient distance, AB, in a direct line from the tower; fix the theodolite at A and take the vertical angle Cab, the line ab being in the level of centre of the vertical arc of the instrument. Then in the right-angled triangle acb are given ab = ab,

and the angle Cab, to find Cb ; to which add Aa = B& = height of the centre of the instrument, and the sum will be the height of the tower.

Ex. Let AB = 200 feet, the angle cab = 46° 30', and the height of the centre of the instrument 5 feet; required the height of the tower.

As rad. = sin. 90°: tan. Cab = 46° 30' : : 200 — ab : cb — 210-76

A a = B 6 =    5

The height of the tower BC

= 215-76 feet.

PROBLEM V.

To find the height of a tower or any other object, standing on an horizontal plane, the base of the tower, &c., being inaccessible.

Let cb be a spire or steeple inaccessible on every side, on account of trees, a portion AD of the ground line AB being horizontal, and passing through the bottom of the spire. Take the angle cdb = 51°

30', measure the distance AD = 75 feet, and take the angle A =

26° 30'. Required the height of the spire and the distance of the first •= station from the centre of its base. A

Subtract the angle A from the angle CDB, the remainder 25° = angle A CD. Then in the oblique-angled triangle ADC, ad and all the angles are given to find DC = 79-18; and in the right-angled triangle dbc, the hypotenuse DC and angle bdg are given to find BC = 61-97, and DB equal 49-29 feet.

Or BC may be found by a single operation by the formula BC = AD x sin. A x sin. D rad. x sin. (d - a)’ and of the sines of the angles A and D ; from the sum subtract the log. sin. (d-a) with its index increased by 10, and the remainder is the log. of the height bc. Thus,

Log. ad sin. 51° 30' . sin. 26° 30' .

C.M

I!

ft

that is, add together the logarithms of ad,

.    1-87506

.    9-89354

.    9-64953

21-41813 . 19-62595 .    1-79218

sin. 25° + 10 Log. bc = 61-97 .

To this result the height of the instrument should be added.

Examples for practice on the two last problems.

1.    At the distance of 45 feet from the bottom of a steeple, on an horizontal plane, the angle of elevation is 48° 12', and the height of the instrument 5 feet. Required the height of the steeple.

A ns. 55 feet 4 inches.

2.    The angles taken at two stations A and D (see last fig.) on an horizontal plane passing through the base of an inaccessible tower CB, are 28° 34' = CAB, 50° 9' = CDB, the distance of stations being 30 yards = AD; required the height of the tower, and its distance from the station D, the height of the instrument being 5J feet.

A ns. Height of tower 3P69 yards ; distance from D = 24*99.

PROBLEM VI.

To find the height of a tower standing on the side or top of an inaccessible hill.

Let CE be a tower on the inaccessible hill e, adb the horizontal plane, the point B being in the prolongation of the line CE; adb the level of the centre of the instrument, with which the angles a. cdb, Eclb were taken, and found to be respectively 44°, 67° 50'

IK

and 51°. Required the heights of the tower and hill, the height of the instrument being 5 feet, and ad = 134 yards.

II

I?!

ssifl

Subtract the angle a from cdb, the remainder 23°    50' = acd:

hence all the angles of the triangle acd are given, and the side ad, to find C d = 230-36 yards.

In the triangle cdE all the angles are given, viz. cdc = 67° 50'-51°= 16° 50', dCE = 90°- 67° 50'= 22° 10'. Hence the angle CKcl — 141°, and CE = 106 yards, the height of the tower.

In the right-angled triangle cdb, the angle cdb and DC are given to find cb = 213-34 yards: hence = 5e &C — CE = 107*34 yards, to which add the height of the instrument = 5 feet= L66 yards, and there results eb = 109-01 yards, the height of the hill.

Examples for practice.

1. The angles of elevation of a castle standing on the top of a rock are gad = 40°, cdb = 63° 20', and ODE = 14° 30', and the distance of the stations A and D, which are in a line with the castle, is 40 yards. Required the height of the castle.

A ns. 24’69 yards.

2. When the angles cad, cdb, and cde are respectively 30° 30', 59° 5', and 13° 52', and the distance AD on the horizontal plane = 52 yards, what is the height of tower ?

PART IX.

TRIGONOMETRICAL SURVEY.

Section I. Rules for finding the areas of triangles, &c., by logarithms.

Section II. To survey a wood.

Section III. To survey a road or river.

Section IV. The practice of taking bearings at two stations. Section V. To survey a town or city.

Section VI. To range lines and take bearings in extensive surveys.

Section VII. To survey with the circumferentor.

SECTION I.

RULES FOR FINDING THE AREAS OF TRIANGLES, ETC., BY LOGARITHMS.

PROBLEM I.

To find the area of a triangle when two sides and their included angle

are given.

To the logarithms of the two sides in links add the log. sine of the included angle, and from the sum subtract 15-30103, and the remainder is the log. of the area in acres and decimals of an acre.

Ex. The two sides of a triangle are 960 and 576 links, and their included angle 53° 8'; required the area.

Log. 960 .-. 2-98227 576 . . 2-76042 Log. sin. 53° 8' . . 9-90311

15-64580 15-30103 Log. 2-212 . . . 0-34487

0-848

40

A.

P.

33

32-920 Ans. 2

PEOBLEM II.

To find the area of a triangle when two angles and their included side

are given.

From the sum of twice the log. of the given side in links and the log. sines of the given angles, subtract the log. sine of the sum of the given angles increased by 15-30103, and the remainder is the area in acres.

Ex. Two angles of a triangle are 59° 46' and 60° 14', and their included side 1000 links; required the area.

Log. 1000 . . 3-00000 6-00000

Log. sin. 59° 46' . . 9-93665 60° 14' . . 9-93855

25-87520

Log. sin. (59° 46' + 60° 14') - 120° . 9-93753

15-30103

. 25-23856

Log. 4-332 • 0-63664

1-328

40

13-120 Ans.

►p*- >

R. P.

1 13

PEOBLEM III.

To find the area of a triangle when the three sides are given.

From the half sum of the three sides subtract each side separately, add together the logarithms of the half sum and the three remainders in links, and divide the sum by 2, diminishing the index by 5, and the result is the log. of the area in acres.

Ans. 102a. Or. 16y>.

PROBLEM IV.

To find the area of a trapezium when the tivo diagonals and the angle made by their intersection are given.

The rule for the area is the same in this as in Prob. I., the diagonals being used as sides.

SECTION II.

TO SURVEY A WOOD.

The following figure represents a wood, the plan and area of which are required.

O"

.....-v

'■'O

Fix station-flags so that the lines compassing the wood shall be as near it as possible, that the offsets may be conveniently taken, and. that the stations, at the same time, are on ground proper for

2 A

to + C

from +E

50

O O CO OOOi^

from + d

58° 20' to +A

to + B

896

675

241° 40'

60

to 4- E go SE.

from + A

80° 31'

to + C go W.

to 4-E

1150

from + o

47° 48'

4-D

1000

to 4- E

go NE.

50

900

to 4-D

00

550

00

950

100

300

120

500

110

160

000

188

110

from + b

111° 41'

000 •

to + D go S.

Begin at

go N.

Explanation of the, method of taking the angles, &c.

From the preceding field-notes it will be seen that AB is first measured, as a base line for the plan. At B the angle ABC is then taken, which is 80° 31', thus giving the direction of the next line bc.

In taking this angle, the theodolite is fixed at B; the two zeros on the horizontal plates being clamped together, the telescope is directed to A, and the whole instrument clamped; the upper plate is then unclamped, and the telescope turned round to c, and the angle ABC found to be 80° 31'. In a similar way the angles BCD, cde, are found respectively 111⁰ 41' and 47° 48', each of the three lines BC, CD, de turning respectively to the left of the line that preceded it. But in taking the angle DEA, it is found to be 241° 40', which being greater than 180°, that is, greater than the semicircle adb by the arc bc; therefore, the line EA turns to the right. Lastly, the angle EAB is found = 58° 20', the line AB turning to the left.

It thus appears that the magnitude of the angle determines whether the new line turns to the right or    the left of the    old    one,    that is, the

new line turns to the left of the old    one if the angle is    less    than 180°,

and to the right if greater than    180°, provided    the    zero    of the instrument be directed towards the beginning of the    old    line; therefore

the bearings of the lines bc, CD, de, AE, though put down by way of check, may be omitted in the field-notes.

The remainder of the field-notes are similar to those given in the preceding parts of this work, and therefore require no further explanation.

On plotting and proving the work.

Draw the line AB in the given direction, and of the given length, 1150 links, as a base line; place the centre of the protractor at B with its straight edge close against AB, and from the end towards A mark off 80|°, then through B and the protractor-mark draw EC = 1340 links. Proceed similarly with the lines CD, CE, and the angles C, D. At e the angle is 241f°: the next line ea must therefore turn to the right, and the angle to be marked off at E must be 360° -241§° = 118J°; or, if the protractor be a circular one, the whole angle may be marked off at once. At A, the commencement of the base line, the angle is 58J°, which is a check angle to prove the accuracy of the previous work.

Moreover, since the sum of all the interior angles of any polygon is equal to twice as many right angles as the figure has sides, lessened by four; as the given figure has five sides, the sum of all its interior angles must be 2 x 5 - 4 = 6 right angles = 6 x 90° = 540°, that is,

80° 31' 111⁰ 41' 47° 48' 241° 40' 58° 20'

SECTION III.

TO SURVEY A ROAD OR RIVER.

371

540° O' proof, as respects the angles.

The angles may, therefore, be considered as having been accurately taken. But the proof of the plotting being correct, is when the work is found to close, that is, when the line ea exactly reaches to the starting-point A, or so very near to it that the error is immaterial.

Having plotted the work, the area is found, by the methods already given in the preceding parts of this work, to be 9a. 1 r. Tip.

A large pond, mere, or lake is surveyed and mapped precisely in the same manner as in the case of the wood just given.

SECTION IIL

TO SURVEY A ROAD OR RIVER.

1. The figure Problem V. Part IV., represents a road, the map of which is required from the following notes; and as the angles at

to + 14

Or 05 ^1

60

<M 7-1

44

554

to fence

© © CM

512

450

380

from + 1 from + 9

234° 20' 276° 8' + 12

80

to + 14 ch to + 14.

20

70

18

300

80

30

752

from + 1

100

108° 20'

93

70

720

to fence + 12

to + 6

86

650

76

560

33

38

bo M

OO

55

Begin at

© © © © © i-i

to fence + 2

130

from + 6

300    50

165

135

58° 20'

36

cr. fence, to + 12.

70

go N. §E.

38

cr. fence.

40

84

52

60

50

180    65

100

from + 2    297° 13' to + 9.

Explanatory remarks, &c.

Station-flags being set up at 1 and 2, and the line measured, a flag is then set up at 6, and the angle 2 taken, which being less than 180°, shows that the line 2 6 is on the left of 1 2. Having measured the line 2 6, the angle 2 6 9 is next taken, which, being 297° 13', or greater than 180°, shows that the line 6 9 turns to the right. Similar observations will apply to the remaining lines. But at station 12, besides the angle to determine the position of the line 12 14, a check angle is taken between an imaginary line, drawn from 12 to 1 and the line 12 14, which is 234° 20'; therefore, after the work is plotted, the protractor will show this angle if the work be right.

If the station-flag at 1 cannot be seen from the station at 12, a flag may be fixed in any other given point in the line 1 2, or in its prolongation, for the purpose of taking the check angle; and if the road is very long, and has several bends, a proportionate number of check angles ought to be taken.

The method of plotting the preceding work will be sufficiently obvious from the -example given in Section II. of the survey of a wood.

section⁻ iv. TAKING BEARINGS AT TWO STATIONS. 373

2. If the figure just referred to, i.e., Problem Y. Part IV., represent a river, it may be surveyed in the following way.

Measure a system of lines, ac, cm, &c., to x, similar to those used in the last example, and take the angles between them, but omitting ab, nr, &c., which being tie-lines are not required.

If the breadth of the river is very unequal, a similar system of lines must be used on the other side of it: but if its width is everywhere nearly equal, its breadth may be found by any of the methods given in Prob. I., Heights and Distances, or by Prob. X. Part III.

The map of the river being thus obtained, the area, if required, may be readily found by one or other of the methods already given.

SECTION IV.

THE PRACTICE OF TAKING BEARINGS AT TWO STATIONS.

Let F, G, H, E, D, C, be six points, or stations, the positions of which, and the distances FG, GH, HE, ED, DC, CF, are to be determined. Measure a base line AB on convenient and level ground, and at A and B angles being taken between AB and the points F, G, H, E, D, C, their positions and distances will be determined by the intersection of the lines that include these angles, as shown in the figure.

The lines FG, GH, &c., being thus determined, may form new base lines for other portions of the survey.

Note.—It is not necessary that the stations a and b should be in the lines eh, ce, as in the figure, these positions being only assumed as being convenient for checking the work, which can be better done by taking angles at any given point o in ab.

The base ab, and the several angles taken at the stations A and B, are shown in the following field-notes, also the angles taken at o, a station in ab, as check angles.

to -b C to -b D to -b E to +H to +G to "b E

From + b

352° 28' 322° 37' 287° 14' 107° 14' 48° 18' 16° 56'

From + A

341° 11' 292° 18' 221° 14' 167° 44' 99° 59' 44° 17'

to +H to +G to +F to + c to +D to -b E

4323

2220

Begin at

go W.

From + A

348° 28' 328° 38' 310° 56' 130° 56' 48° 48' 23° 40'

to + H to +G to +F to -b C to -b D to -b E

B.

v-

■x

.........-r

-V'

X.

VXS

'E

I)

On laying down and proving the work

Draw the "base line ab in the given direction, and of the given length, 4323 links, making the station O thereon, for the check or proof angles. With the protractor, applied to the line AB, and its centre at A, lay off the angle BAF =16° 56', reckoning from B. Move its centre to B, and lay off the angle abf= 360° - 310° 56' = 49° 4r, reckoning from A, and the intersection of the lines AF, BF, will give the point F. The position of the point G may be determined in the same way, by the intersection of AG, BG; as also the points H, E, D, 0; recollecting that when the angles exceed 180° to take them from 360°, and to plot the remainder from right to left, looking from the centre of the instrument along the base line.

If the work be correct, the lines HE, cf will respectively pass through the stations A, B ; this, however, does not check the points D and G : therefore join OD, and apply the protractor to AB with its centre at 0, and if it show the angle AOD to be 99? 59', the point D is in its right position. In the same way, if the angle AOG=360C -292° 18' = 67° 42', the point G is in its correct position; and similarly with respect to the points E, c, F, H.

By calculation.

In the triangle abf are given the side ab, and its adjacent angles BAF = 16° 56', and abf = 360° — 310° 56' — 49° 4', from which

AF and bf may be found. Again, in the triangle abg, are given AB, and the angles BAG = 48° 18', and abg = 360° - 328° 38' = 31° 22', from which AG may be found. Now in the triangle afg, the two sides AF and AG, being determined by the two preceding operations, may be considered as given, and their included angle FAG is also given in the field-notes, being = 48° 18' - 16° 56' = 31° 22', from which fg may be found. And similarly gh, he, &c., may be found.

2576,

ii

FC =

By either of the preceding methods are found CD de = 2102, eh = 2625, hg = 1977, gf = 1995, a 2071 links.

TWO EXAMPLES FOR PRACTICE BY TWO STATIONS.

It is required to find the positions and distances of the points D, E, F and g, from the field-notes No. 1, also from the field-notes No. 2.

No. 2.

No. 1.

From 4- A

331° 2' 314° 0' 70° 42' 27Q 50' at + b

to +G to +F to + E to +D

From + a

336Q 0' 321° 30' 68° 40' 31° 30'

at 4- B

to + G to +F to +E to + D

From + B

22Q

OlOlOO

to 4- E to 4-D to 4-G to 4-F

From 4-B

at 4- A

319° 26' 381° 40' 50° 14' 24° 10' at 4- A

to 4- E to 4~ D to 4- G to +F

to +B 8000

to + B 15421

4000

Begin at +A

Begin

6000 at -1- A

SECTION V.

TO SURVEY A TOWN OR CITY.

Let the following figure represent a portion of a town or city, the map of which is required.

Begin the survey at the meeting of three or more principal streets, and select a station, as A, where the longest prospects can be obtained along all the streets diverging from it, in order to get the longest main or station lines. Having fixed the theodolite at that station, take the direction of one of the principal streets as a base line, as AB,

which must be directed to some well-defined object, as to the right or left side of a door or window, or a projecting corner of a house, or to an angle, or upright post of railing, or any other mark that can be well remembered; a description of which should be also put down in the field-book.

Angles between this base line and the other lines, diverging from station A must then be taken, the objects to which these lines are directed being well defined, in the manner already explained with respect to the base line.

The lines thus set out must next be measured with the chain, taking offsets to all projecting corners of buildings, bends, or openings ;'and

26 a

■V :

90

ft

■Jp^Z

-o/i.

-A-

m\

mi

If !

li

■■•O.......

fv

■ow--o-r-

I IK

yriii .

,/P

......i

(if

to all public buildings, as cliurches, markets, &c., also to statues, obelisks, &c., leaving stations at the ends of the streets to the right and left of the line, and taking the angles of their several directions. The same process is to be repeated on the other lines till the whole is finished.

Thus, fix the theodolite at A, and take the bearings of all the streets meeting there; then measure AB, taking offsets to the openings, corners, &c., of buildings, leaving stations at m, n, o, opposite the directions of the streets on the right. If the survey is for a railway, for which some of the buildings are required to be taken down, the entries should be made at a sufficient distance

section vi. TO RANGE LINES AND TAKE BEARINGS. 377

apart to admit sketches of the buildings, gardens, yards, &c., belonging to the different proprietors, with the dimensions put upon them, as shown at a, b, and c; so that the several parts may be accurately mapped, and their areas determined, for the purpose of their valuation. The same should be done in parish maps, where the valuations of the different properties are required for the purpose of rating, &c.

The base line AB being measured, fix the theodolite at B, and take the angles of the several streets diverging from that point, then measure BC, taking the offsets, and making stations opposite the directions of the streets to the right and left, as at p and C. Make C the third station, at which take the angles and proceed to D, as before. From thence proceed in a similar manner to E, from which return to the first station A, and in measuring EA leave a station at t, from which the station m in AB may be joined by the line tm, and the station p in BC by the line tp, crossing the line joining the stations D and m in v, stations in all these lines having been left opposite the streets and lanes, to the right and left of them, as at q, r, s, w, x.

If the interior angles of the five-sided figure ABODE have been correctly taken, their sum will be equal to 2x5 — 4 = 6 right angles, as shown in Sect. I. Also the line tm determines the position of AE, independent of the angles; in the same manner, the lines T>m, tp, determine the positions of the line BC, CD, and de. But lines can seldom be obtained in large towns thus effectually, to tie the main lines independently of the angles, this result being referable to the line tm, which is actually a tie-line; therefore, when the peculiar position of the street, in which the line tm is drawn, is wanting, the angles and the closing of the work must be altogether relied on, as they constitute a sufficient proof of themselves.

SECTION VI.

TO RANGE LINES AND TAKE BEARINGS IN EXTENSIVE SURVEYS.

Before a single line can be measured with the chain or the angle which one line makes with another, taken with the theodolite, the three station poles that determine the triangle must be set, and as it is the direction of the lines that determines the measure of the angles, and, on the contrary, the measure of the angles that gives the direction of the line (cor. 28), it follows that the slightest deviation in either will mar the trigonometrical relation that exists between them, so that the subsequent work of plotting cannot be performed with geometrical accuracy. In Constructive Geometry triangles have to be made, but in Trigonometry they are presumed to exist, and their sides and angles to have a definite relation to each other.

In every survey the first work is to set the poles at the principal stations, and thus determine the triangles of the trigonometrical survey, and to fix upon the main line, which should be measured with the chain before the taking of the bearings with the theodolite commences. This is done by the surveyor going over the ground, and either setting the poles or marking the best places for the stations.

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Between the principal stations intervening poles require to be ranged afterwards, before the lines can be accurately measured with the chain. When the lines are long, this is generally done with the aid of a telescope only; but when sight is obstructed the theodolite may be necessary. Thus in the annexed diagram the stations A and B are not seen from each other; consequently, before the two intervening station poles x and y can be ranged so as to enable those in charge of the chain to measure the line, the angles A and B must first be determined, which is done in taking the bearings. An ex-e7 ample will best illus-/ trate this.

Ex. To range and measure a line between two stations when sight is obstructed by a town, village, farm-homestead, plantation, or other impediment.

Let A and B be the two stations, CDEF the obstruction to sight, and AB the line sought.

Take any third station g visible from A

Then measure AG and gb with the chain, and take the angle

and b.

G with the theodolite. ' Having thus found two sides and the in-

section VI. TO RANGE LINES AND TAKE BEARINGS. 379

eluded angle, the other two angles and side may be got by Case III. Part VIII. Sect. III.

When the line AB has to be measured with the chain as is generally the case, find the angles A and B as before, and then either range + x and +y with the theodolite, or by Problem XXII. Part I. This done, the line may be measured with the chain by Problem XII. Part III.

In extensive surveys the bearings should be kept well in advance, in order not to give any hindrance to that division of the staff in charge of both chain and field-book on their arrival at stations such as A and B in the above example. And besides expediting the work, it is also in favour of accuracy ; for when the principal lines, with the exception of the base line, are first determined by trigonometry, it serves as a check upon the measurement with the chain.

The principal stations should also be so organised that the bearings taken at them shall be a proof-check upon each other.

This is done by making each flagstaff visible from-as many of the other principal stations on the survey as possible, so that the total area is thus twice taken, and the position of things several times determined.

In surveying thickly-wooded parks, game-preserves, deer-forests, and large plantations covering hundreds and even thousands of acres, it is often necessary to place station poles in any open grounds most convenient, and to take the bearings of such from other stations on elevated places at a distance. Such auxiliary stations are not unfrequently upon the estates of adjoining proprietors, whose leave must necessarily be obtained before field operations commence. Sometimes the smoke that issues from the chimneys of the shooting-box of the sportsman, or of the cottage of his gamekeeper, is the only thing that indicates their respective sites. In such cases a flagstaff may have to be fixed on the top of a tree. In other examples the chimney-tops may be visible from some elevated ground, and thus serve the purpose of flagstaffs.

On mountainous estates—as in some parts of Wales, north of England, and greater part of Scotland—a station pole on the top of a hill can seldom be seen from any part of the valley below, owing to the height and curvature of the former, and narrowness of the latter. In such cases the principal station poles will require to be placed at proper distances along the brows of the two opposite mountains, although only one of them may be included in the survey. The two sets of station poles on the opposite hills should be visible from as many of the stations on the open grounds of the valley as possible. Towards the top of the mountain being surveyed, other

station poles may then be placed, visible from those on its brow farther down.

There are three principal reasons for placing station poles on opposite mountain-sides, as above directed, that require to be noticed.

First. The line between two flagstaffs, one on the brow of each hill, right and left, forms an office or plotting check, the bearings of the two being taken from a station in the valley below, and also the bearings of the stations in the valley from the two elevated ones above on each side.

Second. Lines along the brow of a hill can frequently be measured with the chain when those in the valley cannot, owing to lakes, rivers, or growing timber, being in the way.

Third. The bearings of stations on the opposite mountain, both above and below its brow, can be taken, and the various conspicuous objects thus laid down upon the plan in their true position—such as a lake, a waterfall, meanderings of a river, bridges, rocks, roads, residences, particular outstanding trees, &c.

In a campaign country, the station poles in the valleys may, with few exceptions, be seen from those on the tops of hills; but owing to the greater number of the latter, it is often advisable to continue the direction of lines rectilineally over them. This may be done by means of poles, but more easily by the theodolite placed on the top of the hill. In preliminary surveys taken with the telescope for the purpose of determining the best places for the stations of the trigonometrical survey, straight lines may thus be produced over hills, by simply laying down the offset-staff in the proper direction both ways, and then by directing the telescope first from the one end of the offset-staff, and then from the other, along its length. An approximation of the bearings may at the same time be taken by a protractor and the shadow of a staff, or by a pocket compass.

The work of taking the bearings commences first at the one end of the main line and then at the other. Thus, if we suppose BH in the diagram a main line and that at + b the bearings of o from H are first taken; then in going along the line to H, the angle AGH would be taken in passing, which would also give agb. At H the angle bho would be taken. The telescope would then be turned back again to B, and next round from B to A, in order to find the angle gha, which, with AGH previously found, would give GAH.

The line BH, and consequently BG and GH, having been previously measured with the chain, the necessary trigonometrical data are now given to find the remaining parts of the three angles BOH, EGA, and agh. Thus BH and the three angles are given

section VI. TO RANGE LINES AND TAKE BEARINGS. 381

to find BO and HO; GH and the three angles to find AG and AH; and AG and gb and the included angle AGB to find AB and the angles ABG and BAG.

In a similar manner all the sides and angles of the several triangles into which the survey may "be divided are found. At every station there are always two lines ranging from it, more frequently three and four; and sometimes when the flagstaff occupies a central position, as at o in the diagram Section IV., a great many. And as the work proceeds, the readings from the compass and measure of the angles, horizontal and vertical, from the theodolite, are carefully recorded in the field-book in their respective columns ■ one for the horizontal angles, one for the vertical when taken, one for the compass, and the two outside spaces for remarks, &c., as usual.

In large surveys, the calculations and plottings should never be allowed to fall behind, as many things are liable to escape the memory that cannot be entered in the field-hooks. And besides this, it is necessary that the triangles he plotted to test their own accuracy, as well as that of the measurement with the chain.

With the exception of the main line, the chain should follow the theodolite. This is not an absolute rule, hut one of expediency, which will he found for the most part advisable to observe.

When the details within the large triangles are left to he worked up in the office, those in charge of the field-book cannot he too careful to avoid omissions as to the proximate measure of angles not taken by the theodolite, and the range of lines not driven by the chain, that determine the species of the triangles and the ratios of their sides, otherwise they may fall into what is termed “ the ambiguous case” already noticed (Section II. Part VIII.) In other words, when in the field they should take a note of the trigonometrical relationship of the sides and angles of the triangles to each other, more especially as to whether the sines are those of acute or obtuse angles, and whether the diameter includes that portion of it in which the centre of the circle lies (Art. 34, Part VIII. Section II.)

Omissions of the above kind are only experienced in plotting, which proves that they belong to Constructive Geometry, and not to Trigonometry or the mensuration of superficies, as formerly shown. To beginners, anxious to get quickly over the ground, such omissions are by no means unfrequent misfortunes. It must be borne in mind, however, that no ambiguity exists in the field, for there lines and angles bear a closer resemblance to the definition given them than do those of the office and class-room. If therefore, in driving the line AC (fig. 2, Section II. Part VIII.), it is omitted to enter • in the field-book that the sine of c is within the triangle, and in driving

AC (fig. 3) that the sine of C is that of an obtuse angle, and consequently falls upon the diameter BCD, produced without the triangle, then ambiguity in the office is the natural concomitant of such an omission. And this, too, does not represent the full amount of negligence in the field, for in driving either of the two lines from C, the staff are proceeding to a definite point of the compass, one too far asunder from that of the other line to be mistaken for it.

The professional objection of surveyors' to treating the ambiguous case as a proposition, either in trigonometry or the mensuration of superficies, is perhaps the most convincing proof that it belongs exclusively to Constructive Geometry. Thus, if admitted into the former, a triangular field would contain either the one or the other of two measures, either the area within the triangle ABC (Example II. Part VIII.), or that within ABC (Example 3); consequently two plans would be required, a greater and a less, for which payment would assuredly be received for neither. Hence the practical conclusion.

In field operations it is always preferable to determine either the three angles and one side or two sides of a triangle, or else two sides and the included angle. But this cannot always be complied with, and the following general rules will assist beginners in making provision for the exception, as by Article 34, Section II. Part VIII.

1.    When the sum of the adjacent or opposite angles is equal to a right angle, the other angle is also a right angle.

2.    When the sum of the two angles adjacent to any one of the given sides is less than a right angle, the other angle opposite that side is obtuse.

3.    When the sum of the two angles adjacent to any one of the two given sides is greater than a right angle, the other angle opposite that side is acute.

4.    When the angle included by the two given sides is obtuse, then the other two angles of the triangle are acute.

5.    When the complement of the given angle is greater than the angle included by the two given sides, the other remaining angle is obtuse.

6.    When the complement of the given angle is less than the angle included by the two given sides, the remaining angle is acute.

7.    The nearer the angle opposite the given angle is to a right angle, the greater is the liability to error, in taking approximate estimates of angles.

SECTION VII.

TO SURVEY WITH THE CIRCUMFERENTOR.

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The circumferentor is sometimes used in surveying. It consists of a flat bar of brass bb, about fifteen inches long, with sights CC at its opposite ends, and two narrow slits b, c for taking observations ; in the middle of the bar is a circular brass box A, containing a magnetic needle, which, as usual, is covered with glass. The ends of the needle play over a brass circle g, which f. © is divided into 360° in such a U; manner that the two quadrants are at right angles to the line drawn through the sights.

The instrument is usually •supported on a tripod E, and can be turned in any direction by means of its socket-joint.

When the magnetic needle is properly balanced, and moves freely in its horizontal position, it will retain its position of magnetic north, while the sights are turned from one station to another; consequently the number of degrees which the angle contains can be read off. The great length of the magnetic needle increases the accuracy of the circumferentor, for, if it were short, and consequently the graduations on the brass circle proportionately small, the angle could not be read off with sufficient accuracy.

The instrument is chiefly used in surveying mines, coal-pits, woods, &c.

The circumferentor is sometimes provided with a spirit-level F, with adjusting screws a, a, &c., a tangent screw m, a vernier, &c., in which case it may be made to answer most of the purposes of a theodolite.

The needle should not be suffered to play longer than necessary, but be lifted off its centre, otherwise the delicate point on which it turns would soon be destroyed. The instrument usually has the east and west marked contrary to their true positions, in order that by the reading of the needle the actual direction of the line is shown.

To find the hearing of an object by the circumferentor.

Place the circumferentor over the station, turn N. sight to the object, and looking through the S. sight adjust with the tangent-screw till the hair in the N. sight exactly cuts the object—this part of the operation being the same as in the theodolite. When the needle has perfectly settled, read off the degrees to which its N. end points, from the N. or S. line of the compass-box, accordingly as the N. or S. end of the needle is in the N. or S. part of the compass-box; the angle thus read off is called the bearing of the object.

If the needle stand between two degrees, as between 40° and 41°, turn the instrument gently till it stand exactly at 40°, and clamp it; detach the sights, and bring them with the vernier, with which they are connected, carefully back to the object: the number of minutes to be added to the angle will be shown by the first coincidence of a division on the vernier with one on the horizontal plate ; if the coincidence take place at the 27 division, the angle will be 40° 27'.

The method of taking an angle with this instrument without using the needle, is the same as with the theodolite.

On plotting the bearings taken by the circumferentor, dec.

In laying down the bearings, in cases where the angles are taken from the magnetic meridian of the circumferentor, that meridian is first drawn on the plan, and from a point in it the bearing of the first line from it is laid off with the protractor; the length of the line is next laid off, and through its extremity another meridian is drawn parallel to the former; the second bearing, and the length of the second line, are then laid off in the same manner; and so on till the work be completed, and the variation of the compass being known, the true meridian may then be drawn on the plan, that the work may have its proper position on the finished plan.

The above method is the same as that of laying down a wood, or road survey, by the theodolite, except in laying off the angles.

When the angles are taken with the circumferentor without reference to the magnetic needle, the method of laying down the work is the same as in theodolite surveying.

Note. In coal-mines, when a pit or shaft is required to be sunk from the surface to a given point in the works below, it is usual to take the several bearings and distances with this instrument, from the bottom of an existing shaft along the passages in the mine, to the required point; and then to repeat the same operation * on the surface of the ground, without plotting the work on paper; by which means complete accuracy is frequently obtained.

RAILWAY SURVEYING.

A complete survey for an intended railway contains six problems, three of which are field operations, and the remainder office work. They are as follows :—

Prob. I. To range the station poles that form the chain and other

lines.

Prob. II. To measure with the chain the chain lines ; with the theodolite the angles which they form, also the angles of depression, to obtain data for the reduction of the hypotenusal planes of the earth’s surface to a common plane; and with levelling instruments to take the levels of stations.

The field notes under this problem are in three field-books; the first for the chain, the second for the theodolite, and the third for the levelling instruments.

Prob. III. Given the field notes, to plot the plan and sections.

Prob. IY. To apportion the land required for the intended railway.

Prob. Y. To find the field notes required for setting out the railway estate and the line, including the diversion of roads, &c.

Prob. YI. To set out the railway estate and line, including roads, &c., whose diversion is necessary.

In a preliminary survey, and in a parliamentary survey, and also in surveying a railway which passes through an estate being surveyed, only a portion of the above field and office operations are required.

In the following directions for a preliminary survey, the lines are ranged for laying down the railway from a series of connected baselines. It is seldom, however, that lines thus ranged can be measured by the chain with sufficient accuracy, and when such is the case, even ground adjacent must be selected for base-lines, so as to obtain the true lengths of the former by trigonometry.

Another method is to range the right lines of the projected railway, and either to join them by the conveying tangents from the two tangential station poles, or else by the chord-line that lies between them (the two tangential stations), and then to take the bearings of these stations from the adjacent base-stations, so as to find their reduced distances or base-lengths required for plotting; and when an adjacent base-line is not perfectly level, its hypotenusal length must be reduced to base-measure, otherwise the field notes, both for plotting and setting out the railway, will be erroneous. (See Advertisement, p. viii, under Railway Surveying.)

2 B

Illustrated general directions for a preliminary survey of a portion of a projected railway.

Landowners (resident and absentee), the executors of minors, mortgagees, and capitalists, often order plans purposely to ascertain how far projected railways will interfere with their respective interests involved, and whether. they would be justified in farther supporting such projects.

Plans of this kind are of the simplest character, consistent with the purpose they are intended to serve. The several resident agents in charge of estates proposed to be intersected watch such railway movements closely, and never fail to apply to their employers for instructions how to act. They are often by profession land-surveyors themselves, and, when such, they send to their absent employers a sketch of the projected line on tracing-paper, taken from the plans of their estates; or when otherwise, they employ their surveyors to do so for them, who may give a general sketch of the whole intended line upon an ordnance map.

Again, several parties may join and order a survey and plan of the whole district through which the projected railway is intended to pass, or only of that portion of it in which they are more immediately interested.

The annexed figure, already referred to, may be presumed to represent a plan of this latter kind. It is, in point of fact, a copy of a plan of a portion of a district surveyed for the purpose of the projected railway rst, the base-line AC being ranged so as to avoid Lynch Wood. This line terminates at + C, partly on account of its diverging too far from the line of the railway, and partly to avoid the River Ouse beyond + C. The next base-line commences at + B in AC, and runs close to the railway at T. Beyond + D (on the complete survey of the whole line) it continues the same direction till its deviation requires another base-line.

To range the base-lines,

on a

The initiatory step in such a survey is to range the base-lines. This is done by going over the ground and carefully examining it from points where the projected railway may be presumed to be level with the surface. Such points are generally determined by the amount of excavation, on the one hand, that is required to fill up the hollows on the other. An experienced eye will approximate very closely to such data. Having come to a definite conclusion on these points, and the curvature to which they may give rise, the flagstaff's are then set so as to avoid obstructions, and be visible from each

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other. Such fiagstaffs then constitute the principal stations in the survey, the lines which they make being base-lines.

The several base-lines are denominated first, second, third, &c. The numbers run from the point where the survey commences, and each base-line should embrace as great a length of the projected railway as is consistent with details. In the figure there are only two base-lines in the survey, AC being the first base-line and bd the second basedine.

To lay down the base-lines, and draw the projected railway upon an ordnance or other map.

Having fixed flagstaff's at the principal stations A, C, B, and D, the next work is to find out the points to represent those stations upon an ordnance map (or any other map thought 23referable). This done, the projected line is then drawn upon the map with a steady hand, its direction being subject to any deviation which the details of the survey may suggest.

To range the secondary and minor lines.

Before placing the station poles that determine the secondary and minor lines, and that enable the surveyor to take bearings, he must first make himself thoroughly acquainted with the boundaries of estates, counties, and parishes, in order to ascertain how far the projected line will interfere with private and public interests. This is a work often surrounded Avith much difficulty, interest being not unfrequently concealed at this stage, and when doubts exist an increased number of station poles may be advisable so as to get fences, roads, and other objects faithfully delineated, purposely to avoid objections to the accuracy of the plan.

seen

The ranging of the secondary and minor lines on each side of the base-lines, and the setting of the station poles, are similar in railway surveying to what they are in an ordinary survey of several adjoining estates. For the details of the work, we may, therefore, refer back to previous sections. For such data, ordnance maps are seldom or never to be relied upon. The bearings of old church steeples and like prominent objects may be correct, but it is for the most part otherwise with the areas and positions of fields, and the innumerable little things upon which interested parties place a value—things that require to be shown upon the plan—some of which may not have existed at the time of the ordnance survey ; consequently all details should be carefully laid down from an actual survey. In the example, the different stations on each side of the base-lines, and the secondary and minor lines which they make, will be

upon the drawing or plan, and the plotting data they involve understood.

The distance on either side of the projected railway that requires to he surveyed is invariably specified in the order or agreement. It generally considerably exceeds the parliamentary requirements subsequently given and illustrated in Plate XIII., by “ limits of deviation,” and may'extend from 30 chains to a mile. Thus in the figure it extends to Lynch Church on the one side, and to Linton Church on the other, both of which serve as stations.

The reason of this extra breadth, as compared with the parliamentary limits, is to make ample provision for deviation, and to include the interests of all parties that may be effected, in order to induce them to come forward, and either support or oppose the projected railway as circumstances may direct.

To take hearings and measure the first hase-line AC.

When the surveying staff employed is large, it may be divided as in land-surveying; one party or division taking the theodolite, and the other the chain. But when it is small, the taking of the bearings and measuring with the chain proceed together, an extra labourer being employed to carry the theodolite when the latter work is being prosecuted between stations. This conjunct method is that which the present example illustrates, the chain and theodolite going together, the staff being undivided.

The work commences at station A, and in a complete survey of the whole length of a projected railway, the bearings of the first baseline should be taken from a meridian, as in land-surveying. But when a portion of such a survey is only called for, the bearing of the first base-line may be taken from that of the base-line immediately preceding it on that side, provided station A is in it. If the contrary is the case, as it generally is, two or three other methods may be adopted. First, the base-line AC may be produced to the auxiliary station a, and the bearings at station A taken from it. Second, the bearings at A may be taken from C. Third, the bearings at A may be taken from station 1, the line y 1 thus serving as a meridian.

The first and second of these methods are identically the same, although differing in the details indicated. In point of fact, the line CA.a is only a portion of a base-line which the learner may presume has either been ranged from a meridian or from another base-line, for had the plan represented the survey of the whole length of the projected railway, then the line ca would have extended beyond a in the direction A a until its divergence from the railway called for

another base-line ; and we may further observe that, in a similar manner, the base bd would have proceeded till its deviation called for a base-line on that side, which may be taken as the third base-line, to range from station 15, in the same manner as the second base-line ranges from B on the first base-line.

From these observations, the learner will readily perceive how the total number of base-lines in a complete survey of the whole length of a projected railway may be ranged from a meridian, and their positions accurately delineated upon a plan. He will also be able to comprehend the proper relation that exists between' the survey of a part and of the whole.

According to the above data, the range of the base-line AC with the meridian may be considered as having been previously taken. The bearings at A, therefore, commence either from the auxiliary station a, by the first method, or + 1 by the third method. The former is the more accurate of the two methods, as it gives the whole of the details required for rectilinear and angular proof; for if the reading at c is 180°, the line CA« is correct, and if the sum of the angles at A is 360, the whole of the bearings are true, as will be seen below. The readings of the bearings may be entered in the field-book thus:—

From + a in AC produced

231° 40'

180°

90° 40' at + A

to + 2 to + c to + 1

Begin

The several angles at A are now found and read thus :— aAl lAC

ca2 2ay yAa

90° 40'

89° 20'

180

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51° 40'

39°

89° 20'

180

Proof 360

360

XII. Part III.

The bearings at A being taken, the work of measuring with the chain commences from that station to station 3. At + 3 the instrument is then placed, and the bearings from + A to + 1, from + 1 to + 4, at the north-eastern boundary of Lynch Wood, from + 4 to + c, and from + c to + Lynch Church, are each taken and entered in the field-book, as were the bearings at + A. The part of the base-line between + 3 and + 5 is next driven, the River Ouse being crossed by the bridge on the road from Hurst to Linton, that portion of the line across the river being measured according to Problem At + 5 the bearings are taken from + 3 to + 6

to get the southern boundary of Lynch Wood, the northern boundary of Linton Mere, and the boundary between the two parishes shown by dotted lines, from + 6 to Linton Church, from Linton Church to + C, and from + C to Lynch Church; also from Lynch Church to + 3 as a check upon the last angle, so as to determine accurately the position of the church. When the ground is undulating, checks of this kind should never be omitted. The measurement now .proceeds to + 7, where an angle is taken to + 8, the line passing over that from + 5 to Linton Church, a minor station pole being placed at the crossing of the two lines. The remaining portion of the first base-line is then driven, viz., from +7 to + C, the distance across the River Ouse being measured by any of the methods given under Problem I. Part VIII. At C the bearings to Linton Church, and to stations 9 and 10, are taken, which conclude the work on the first base-fine.

To take bearings and measure the second base-line bd.

The staff now go back to B, and from that station take the bearing from + C to + D, which determines the range of BD. The first portion of the line, viz., from +B to +11, is then measured, the distance across the River Ouse being found, as before, where the first base-line crosses near Linton. The measuring of the remaining portions of the line, and the taking of the bearings at the several stations upon it, may advantageously be left for the learner to conclude as an exercise.

Details of the survey.

The measuring of the secondary and minor fines, the taking of the offsets, and the whole of the remaining portion of the work, with the exception of levelling, and reporting on the probable amount of damages along the projected fine, is so exactly similar to land-surveying, that the more advisable course in an educational work of this kind is to leave it to be performed by the student. Such, when taken in detail, forms very suitable exercises, and by this time he must be considered qualified to go through the Avhole successfully.

With regard to levelling and reporting, all that is generally demanded of the surveyor, relative to the former, in a preliminary survey, is to take some points in the projected railway presumed to be on a level, or in the same horizontal plane with the surface of the ground, or nearly so, the difference between water-level fines and horizontal being nominal, and from these stations to take the angles of elevation and depression, so as to be able to give an approximate estimate of how much land it will occupy. On the severance of fields, the diversion of roads, watercourses, and other encroachments of this kind, the surveyor may report. But land-owners and their agents, when once they get the plans, generally go over the ground along with their tenants and satisfy themselves in such points, and for the most part they are by no means strangers during the survey at the several station-flags.

PART X.

RAILWAY ENGINEERING.

SECTION I.

LEVELLING.

Levelling is the art of representing the inequalities of the upper boundary of any section of the earth’s surface, and of determining the relative heights of any number of points in that boundary, above or below a line equidistant, at every point, from the earth’s centre. This line is termed a level line,¹ and is that which water assumes when at rest.

The operation of levelling may be performed by various instruments, depending on the action of gravity: the most simple of which are those formed with a 'plumb line, such as the ordinary mason’s level; but these are only suited for very limited operations.

The fluid or water-level consists principally of a glass tube, bent like the letter u, open at both ends, and partly filled with water ; the surfaces of which, in the two arms of the tube, will stand at the same level by the action of gravity; and a line of sight taken over these two surfaces is therefore a level line. This instrument has been much improved, but, in all its modifications, it is found inconvenient in practice.

There are also the reflecting and Cambrian levels, which are useful only for special operations of small extent, such as military engineering, &c.

(1.) The instrument used for levelling by civil engineers is called a spirit-level, or simply, a level.

DESCRIPTION OF THE Y LEVEL.

(2.) This instrument consists of a portion of a theodolite brought to its utmost practical perfection, and constructed for levelling

purposes only; its chief parts are the telescope AA with the spirit-level DD beneath it, and two parallel plates with four conjugate screws K, K, &c., between them; to the lower of these plates are fixed the legs that support the instrument; and on the upper one rests a pillar that supports a strong brass plate GG, in which is fixed the compass H. This plate supports the telescope by means of two pillars, F, f, called y’s from their forked shape, and

--CZ3--

Ki

Ul

u ill umilil

from which the instrument has its name. The telescope is provided with a screw c, to adjust the object-glass, and four other screws to adjust the cross wires within it. The spirit-level is adjusted by the screw e, having a joint at the other end. The instrument is clamped by the screw I, and brought accurately into the required line by the tangent screw L.

ADJUSTMENTS.

(3.) There are three adjustments of the level; firstly, that of the line of collimation of the telescope ; secondly, that of the axis of the level of the axis of the telescope; thirdly, that to make the telescope always at right angles to the vertical axis of the instrument.

The two first adjustments are the same as those for the theodolite, which see : the last is as follows.

Having fixed the instrument on the ground as nearly level as can be judged by the eye, set the telescope over either pair of conjugate screws K, K, &c. ; by means of which bring the bubble to the middle of the level tube; turn the telescope over the other pair of conjugate screws, and correct the level; turn it back to the other pair, and correct again, if necessary. Then reverse the direction of the tele-

scope over either or both pairs of screws, alternately backwards and forwards ; and if the bubble be found to stand in the middle in all directions, the instrument is then in adjustment; if not, the error arises from the axis of the level not being perpendicular to the vertical axis of the instrument. To correct this error, raise or depress the movable support of the telescope by the milled-headed screw beneath, till the bubble be brought half-way to the middle of the tube; then correct for the other half by the conjugate screws; repeating the correction till perfect accuracy is obtained.

Note.—This adjustment is the same in all levels.

The other adjustments are either different in different levels, or not required from the nature of their construction.

troughton’s IMPROVED LEVEL.

(4.) This level is preferred by many, on account of its adjustments not being so liable, after they are once perfected, to derangement. The spirit-level is placed above the telescope, and over it the compass-box. In most instruments the telescope shows the object

TG2T

TEJ

inverted. The spirit-level is firmly fixed in its cell, and the adjustment of its axis with that of the telescope, or line of collimation, is given by two screws near the eye-glass.

gravatt’s level.

(5.) This level is another modification of the preceding one; it is usually called the Dumpy, from the comparative shortness of its telescope. It has an object-glass of larger aperture, and shorter focal length; its spirit-level, as in the preceding one, is fixed above the telescope; it has also a small mirror so fixed on a hinge that the position of the air-bubble can be seen by the observer while he reads off the staff.

Many varieties of levels are used by engineers; but the differences in construction are generally only slight. It is important to have one made by a good maker, or much trouble may be caused in its use. The telescope should be short and of large diameter, and it should always reverse the image, by which much light is gained. The reversal of the figures is only a little perplexing at first; it is very soon learnt, and no one who has been accustomed to a reversing level would willingly change to an upright one.

LEVELLING STAVES.

(6.) These staves are generally from twelve to fourteen feet in length, but made to slide telescopically into about one-third of this length for portability. They are divided into feet, and again into tenths of a foot; and, lastly, subdivided into hundredths of a foot. The feet are marked by large figures—the tenths by smaller ones.

There are several varieties of staves sold by instrument-makers.

CORRECTION EOR CURVATURE.

X)

(7.) Let bdeh be a line of sight as given by a level, i.e., a horizontal line ; BCFG an arc of a great circle of the earth, and A its centre. It will be seen from the figure that the heights DC,

/G

EF, HG, of the apparent level BH, above the true level,.continually increase from the point B. The height DC of the apparent level above the true, is nearly equal to the square of the distance bd, divided by twice the earth’s radius ab, that

-rt)2

is, DC = -—    Similarly EF =

be2 JAB

&c. j therefore the corrections DC, ef, &c., are always proportional to the squares of the distances bd, be, &c., seeing that 2ab is

a constant quantity. By substituting the value of in miles,

the rules for finding the correc-

BD²

2ab’

Suppose the diameter of the earth to be 7958 miles, and that the distance BC = 1 mile, then the correction for curvature DC = BC²

of a mile = 7’962 inches = nearly 8 inches. If the distance be = 2 miles, then the correction ef = be² -i 2ab = tsts a m^e = 31’848 inches.

chains, &c., in the formula DC = tions are derived.

2ab = 1» * 7958 = T 1

When the distance is in miles.

Rule I.—Take of the square of the distance in miles for the correction in feet.

When the distance is in chains.

Rule II.—Take of the square of the distance in chains, from which cut off two places for decimals, for the correction in inches.

When the distance is in yards.

Rule III.—Multiply the square of the distance in yards by *00000257, for the correction in inches.

CORRECTION FOR REFRACTION.

(8.) The refraction varies according to the state of the atmosphere, but it may generally be taken at \ of the correction for curvature as an average; and since refraction makes objects appear higher than they really are, the correction for it must be deducted from that for curvature.

Examples.

1. Required the correction for curvature and refraction, when the distance of the object is 2^ miles.

Here (2J)* x |=(?)²x

cq iio

27

= — = 3-375 feet, cor. for curvature

cor. for refraction cor. required.

2. Required the correction for curvature and refraction, when the distance is 40 chains.

■it of which = ⁻482 2-893

Here 40² q- 800 = 2*

\ of which = -285 T7l5

inches, correction for curvature cor. for refraction cor. required.

3. Required the correction, as in the last examples, when the distance of the object is 1000 yards.

Here 105 0² x -00 0 0 0 257 = 2-57 inches, cor. for curvature

of which = -367 2-203

cor. for refraction cor. required.

Note.—These corrections, it will be seen, are very small at short distances, and are frequently obviated in practical levelling. See note to Art. 11.

THE PRACTICE OF LEVELLING.

(9.) To find the differences of the levels of several points on the earth's surface, and to trace a sectional line of them.

Let A, b, c, D, E, be the points on the earth’s surface, a sectional line of which is required to be traced. Set up the levelling staves

MtLj

perpendicularly at A and B; between which fix the level L on the ground; make the spirit-level horizontal, and turn the object-glass of the telescope to the back-staff A; then the staff A is cut at a by the horizontal line of sight aa' of the telescope, through which the observer can read off the height A a. Now turn the object-glass of the telescope to the staff B, the horizontal line of sight aa! cutting it at a', and read off the height Ba : then the difference between the heights B a', A a, is the number of feet that B is lower than A, because the line aa' is horizontal. Now let the staff A be removed to C, the staff B remaining in the same position, excepting that its graduated side must now be turned towards C, and having removed the level L, place it between B and C, and the spirit-level being again made horizontal, the line of sight, in this case, being hV, read off the heights Bb, Cb' the difference of which is the number of feet that C is lower than B, as before.

In this manner, by alternately moving the levelling staves, placing the level between them, and reading off the heights below the line of sight or collimation, the operation is conducted to F.

Now, as there is a continued fall from A to D, the sum of the differences of the readings at B and A, at C and B, and at D and c, will give the actual fall vertically from A to D; in the same way the rise from D to F is estimated : if, therefore, the fall from A to D be greater than the rise from d to F, the difference of these two will be the actual fall from af, or the distance in feet, estimated vertically, that the station A is higher than the station F.

EXAMPLE.

Feet Feet f 3 *50 )

Let the readings on the staves A and B be <    >•

f 5-04 )

17-39/

/ 4-04 )

17-00/ _

Total fall from A to D . . 9-38 sum.

on B and C

on C and D

4-07 diff.

2-35 „

2-96 „

Feet Feet

The reading on the staves D and E | 5.93 |

/ 6-81 )

1 4-86 /

Total rise from D to F . . . 2-12 sum. Actual fall from A to F . . . 7*26 difF.

on E and f

0-17 diff.

1-95 „

The difference of the sums of the back and front readings of the staves, or “ back and fore sights,” as they are termed, will also give the difference between A and F: thus—

Back Sight Fore Sight

7-57

3-50

5-04

7-39

4-04

7-00

6-10

7-93

6-81

7-86

25-49

32-75 sum of fore sights 25-49 sum of back sights 7*26 diff. the same as before.

But in order to draw the section to show the undulations of the ground between A and F, the horizontal distances of the several points B, c, D, E, and F, must be measured from the first station a j this is usually done with the chain during the operation of levelling. These distances with the back and fore sights may be arranged in a Field-Book of the following form, which, though not the practical form, may probably be better understood by the student.

FIELD-BOOK.

Back Sight. Fore Sight. Fall.

Rise.

Reduced Levels below Station A.

Distances.

05 <35 OK W OOHOOCl M O    O

NC50COCD LO co o cp CO

t-

4-07

2-35

2-96

0-17

1-95

4-07 at B

6- 42 „ c 9-38 „ d 9-21 „ e

7- 26 ,, f

Chains.

6-00

12-00

18-00

24-00

30-00

25-49    32-75

* 25-49

7-26 diff. the same as the last of the reduced levels

In the above Field-Book it will be seen that the differences 4-07 and 2*35 in the column marked Fall are added together, making 6-42 for the fall at c, in the column of reduced levels : to this sum the next difference 2-96 is added for the fall at D. The other differences, in the column marked Rise, are successively subtracted from the last sum 9-38 for the falls at E and E: the latter of which, agreeing with the other difference, shows the accuracy of the castings. The last column shows the horizontal distances of the several points B, C, &c., from A.

td|

6-42

'o'

9-38

>r*

7/ 05

45-93

43-58

40-62

40-79'

Em

I Ant*

Datum

In plotting from the above columns of reduced levels and distances. the former are always taken from a larger scale than the latter,

otherwise the variations in the ground would,[in a great many cases, be scarcely perceptible; the horizontal scale used in plotting the following section is 10 chains to an inch, and the vertical one for the reduced levels 25 feet to an inch, which latter scale has been thus chosen, that the section may be the same as that in the figure to Article 9, the vertical scale practically used being commonly 50 or 100 feet to an inch.

Having drawn the horizontal line A/, set off the distances Ah = 6 chains, Ac = 12 chains, &c., from the column of distances, on a scale of 10 chains to an inch; then set off the vertical lines b~B, cc, &c., respectively, 4-07 feet, 6-42 feet, &c., from the column of reduced levels, on a scale of 25 feet to an inch, and through the points A, B, C, &c., draw the line ABCDEF, which is the section required.

THE DATUM LINE.

(10.) It is found inconvenient in practice to plot a section generally by the method already given, as the reduced levels, in extensive operations, would alternately rise and fall above and below Af (see last figure), and thus produce much confusion: therefore, a line hi, called a datum line, is assumed at 50, 100, or 200 feet, &c., below the first station A, so that it may be always below the sectional line ABCDEF. To or from this assumed distance of the datum line the rise or fall is respectively added or subtracted, and the next rise or fall added to or subtracted from the sum or difference, as in the following Level-Book; thus giving a series of vertical heights to be set off, always upwards from the datum line, through the upper extremities of which the section line is to be drawn.

PRACTICAL LEVEL-BOOK. Datum line 50 feet below station A.

Back Sight. Fore Sight. Rise. Fall.

Reduced

Levels.

Dist.

Remarks.

3-50

5-04

4⁻04

6-10

6-81

1> O CO CD io co 9 9 co lO ^

i to to

to 03 6 i os or

25-49

32-75    2-12    9-38

25-49 7 "2 6 differences = 7 "2 6

2-12

50-00 D. 45-93 43-58 40-62 40-79 42-74

50-00 /

6 00 B. M. at A. 12-00 18-00 24-00

30-00

7-26

diff. between last red. level and datum

Having set off the distances on the datum line hi, draw the vertical lines HA, mB, nc, oD, pis,, and IF, setting off their heights as in the column of reduced levels, and through the points A, B, &c., draw the required section ABCDEF.

(11.) It will he seen that the operation of levelling is extremely simple, as are also the castings of the Level-Book, and the plotting of the section. Great care is, however, necessary in reading off the staves, which ought, at the same time, to be held in a perfectly upright position ; the castings in the Level-Boolc ought also to be carefully checked by taking the respective differences of the sums of the back and fore sights, and of the sums of the rises and falls, also of the datum number and the last reduced level; which three differences must agree, as shown in the preceding Level-Book, otherwise errors will be found in extensive operations to creep in imperceptibly.

Note.—Calculations to find the corrections for curvature and refraction .are sometimes necessary in levelling, but as these would be extremely tedious, especially in extensive operations, they may be obviated by fixing the level always halfway between the staves, as nearly as the eye can judge, and thus the errors in both become equal and opposite, and consequently correct each other.

The general practice of levelling being now laid down in a brief manner, more details shall next be given.

TRIAL-LEVELS.

(12.) The general direction or route of a proposed line of railway between two places being determined upon, the line is marked out on such a map or plan of the district or country as is most convenient. (For this purpose the ordnance maps are the best, if the line be in England or Ireland.)

The levels are then taken, as near as possible to the direction thus marked out on the map ; if any deviations are made from the projected line, they are confined to the intervals between where it crosses turnpikes and other public roads, the relative heights of the points where it crosses these roads being always determined; also marks, usually called bench marks (abbreviated B. M.), are made in or near them, these marks being described in the Level-Book for future reference. Cross-levels are also frequently taken, especially on the roads, to show the inclination of the ground to the right and left of the line, so that the engineer may know how to improve the line, and prepare it for further operations.

If the line, thus obtained, be thought not satisfactory, other directions, embracing either the whole, or only part of the distance between the two places, are tried in the same manner, and that direction which includes the fewest engineering difficulties is adopted.

CHECK-LEVELS AND SURVEY OF THE LINE.

(13.) Check-levels are sometimes taken to ascertain the accuracy of the trial-levels, if any doubt exist in this respect, and more especially if the line be a detached one; i.e., not connected at its commencement or termination with existing lines, nor anywhere intersecting them. This, however, can rarely be the case in these times, when so many railways exist, the sections of which are published, and from and to which almost all new lines proceed. Hence the elevations of the commencement or termination of almost all new operations are so well known, not only with respect to these points, but also to other intermediate points of intersection with existing lines, as to obviate the use of check-levels, provided the trial-levels have been taken by a careful leveller.

(14.) The trial-levels being considered satisfactory, the survey of the surface is immediately commenced to the width of from 5 to 10 or 15 chains on each side of the projected line, accordingly as the property of individuals may appear to be affected, or the engineering difficulties or requirements of the line may demand. This survey must include every enclosure, building, road, &c., within the above-named limits; it is usually on a scale of from 5 to 10 chains to an inch. It is in every respect the same as in the preliminary survey, Section I., to which reference is given for the details of the work.

FINAL LEVELS.

(15.) The point of commencement of the proposed railway being determined upon by the previous operations, set one of the levelling staves on that point and call it the first back station, and select a point as near this station as possible, the height and position of which must be entered in the Level-Book as a mark of reference for future operations. This is called a bench mark; it is marked B. M. in the column of remarks in the Level-Book; it may be either between the first back station and the second, or between the second and third, accordingly as convenient objects for the purpose present themselves. The most proper places for bench marks are prominent stones on buildings, or other fixed stones, hinges of gates, milestones, notches near the roots of trees, &c., the positions of which must be described in the Level-Book. Fix the level firmly in the ground about half-way between the first and second stations, and make the axis of the spirit-level horizontal in all positions; read off the staff at the first back station; this reading is placed in the first column of the Level-Book, marked* Back Sights. If neither the bench mark nor any other intermediate point requiring notice, occur between the two stations, turn the telescope to the second station and read off the height; this reading is placed in the third column of the Level-Book, marked Fore Sights. Bemove the first staff to the bench mark, or intermediate station (if one occur), having at the time removed the instrument; read off the back sight and enter it in the Level-Book, as before; then read off the staff on the bench mark, and enter the reading in the column marked Inter. ; lastly, read off the forestaff, as before : thus continuing and taking as many bench marks, and other intermediate sights, as may be thought necessary; the chaining of the line proceeding at the same time, the results of which are placed in the column of distances. It will readily be seen that four assistants will be required by tlie leveller, i.e., two for the staves and two for the chaining. (For method of keeping Level-Book, see next page.)

From this Level-Book the method of proceeding, to any extent, may be sufficiently seen, with the help of the following remarks on the method of obtaining the reduced levels, at the same time.

Note.—That wheresoever the back or fore sight exceeds the usual height of the levelling-staff, that staff must be understood to have been placed on the top of a gate or rail-post, and its height afterwards measured, and added to the reading of the levelling-staff.

(16.) Bemarhs on obtaining the reduced levels, &c., in the following Level-Boolc, <&c.

The method of casting the Level-Book referred to, is the same, with respect to the back and fore sights, as that given in Art. 10, page 353. These castings must be first done, and the intermediate ones, in the column marked Inter., afterwards.

an

The datum line is assumed 100 feet below the first station or commencement of the levels : this number is placed first in the column of reduced levels. The back sight is 2'20 feet, and its corresponding fore sight 13'40, which shows that the ground falls; the difference of these 11 -20, is put in the column marked Fall, which, being subtracted from the datum 100 feet leaves *88-80, which is put under the datum, in the column marked Reduced Levels, being marked with asterisk or cross, and shows the height of the forestaff above the datum line. The same remarks will apply to the back and fore sights 1'80 and 10T0, and to 7'30 and 1'80, observing that in the latter case the ground rises, and the difference must be put in the column marked Rise, and added to the next preceding reduced level that is marked thus *.

LEVEL-BOOK.

The projected railway from the Grand Junction Railway to A biergelly. (See Section, Plate XIII.)

Back

Sights.

Inter.

Fore

Sights.

Rise.

Fall.

Reduced

Levels.

Distances in Chains.

Remarks.

Datum

Ft.

100.00

On

Rail

2.20

13.40

11.20

*88.80

0.45

1.80

8.36

80.44

0.80

Below level of rails of tlie Grand Junction Railway, at its j uric-, tion therewith.

On side of railway embankment.

B. M. on stone

7.30

7.70

4.80

4.60

10.10

2.50

2.70

8.30

0.40

*80.50

80.10

83.00

83.20

3.00

5.00

7.00

10.00

opposite Sheaf Inn, at foot of railway embankment.

1.80

5.50

*86.00

11.10

Edge of drain (3 ft. deep).

Occupation road to Elm farm andWing-field cottages.

At foot of perpendi-

bank^-f

12.00

10.00

5.00

14.00

12.20

0.00

2.00

o o o o o o o o CO o c4 c4 rH c<i

88.00

93.00

*98.00

99.80

*110.00

11.13

15.00

17.00

20.00

23.86

cular 2 ft. high.

At top of ditto.

7.86

3.80

2.20

Or rfx

bb

Oh Oh

114 06 115.66

24.20

Centre of open occupation road to Hill farm.

On Bank top.

B. M. on flat square stone at corner of Hill farm barn, 1 ch. to left.

7.20

5.00

0.00

2.00

px tO^q

to to bo

O O Oh

*117.86

120.06

52.36

29.30

29.30

*123.06)

100.005

26.50

28.00

32.00

Footpath.

23.06

23.06

The difference between the first back sight 1 -80, and the first intermediate one 10 T 6, is 8‘36, which shows that the ground falls; it is therefore put in the column of falls, under 11 -SO^ancl subtracted from the next preceding number in the column of reduced levels, marked thus'", and the remainder 80⁻44 is then put in that column. In like

level -BO OK—continued.

Back

Sights.

Inter.

Fore

Sights.

Rise. Fall.

Reduced Distances Levels. in Chains.

Remarks.

Ft.

9.83

13.50    8.50

8.89

10.60    4.70

1.40

1.70

CO CJi to or o

*0 b M b b

0 0 0 0 4**

*123.06

*124.00

129.00

*136.10

142.00

145.00

32.60

33.00

35.00

37.00

39.00

6.70

3.90

140.00    39.40

2.60 8.00

*144.10    40.00

1.05    9.15

1.00 6.00

15.65

8.10

14.60

5.00

In plantation.

Ditto.

Ditto.

15 links over plantation hedge, on edge! of bank of road to Westwick.

Centre of road to West wick.

On further bank of ditto. B. M. on square stone in line.

0.40    3.90

13.50

12.50

1.60 11.60

17.40

3.50

5.00 14.00

14.50

17.00

10.00

12.90

9.00

*117.00

113.50

*100.00

90.00

*87.10

78.10

47.50

48.00

49.00

50.20

51.15

51.65

Crosses footpath to Westwick.

Edge of steep brow.

0.00    5.00

*92.10    52.00

12.00

14.00

15.00    8.50

0.10

2.50

Bottom of brow.

Edge of brook.

Centre of do. (water 2 ft. deep).

Further bank of brook.

10.00

0.00

1.70

15.00    5.80

14.20

12.00 8.00

0.40

0.60

135.18

80.24

54.94

1.00

H H O lO CO C5 TJ? CO t-H HH rH    HH rH

*104.00

*115.50

122.00

*130.50

*138.80

53.50

55.45

56.10

56.95

58.75

80.24

148.00 *153.40 *167.00 171.00 *178.00 ^ 123.061 54.94

59.80

60.65

61.60

62.25

63.40

Footpath to farmhouse.

Centre of public road to Winston, road 50 links wide.

manner all intermediate differences are added to, or subtracted from, the next preceding reduced level number marked thus *, accordingly as they stand in the column marked Rise or Fall.

The proof of the accuracy of the work is by taking the differences of the sums of the back and fore sights, and of the first and last

reduced levels, which differences must be equal; thus, in the latter portion of the Level-Book, the sums of the back and fore sights are 135*18 and 80*24, the difference of which is 54*94; and the first and last reduced levels are 123*06 and 178*00, the difference of which is also 54*94, which shows the work to be right.

(17.) As a means of checking the accuracy of trial-levels it has sometimes been recommended to employ two levellers, with separate instruments placed at a short distance from one another, to read from the same staves at the same time, and frequently to compare their reduced levels, correcting the errors immediately, if any occur; this would completely obviate the numerous mistakes that arise, in extensive operations, through wrong readings, wrong entries in the Level-Book, &c. ; since the method of proving the Level-Book by differences only proves that the reduced levels are correct with respect to the back and fore sights entered in the book, and not the accuracy of the actual levels.

PLOTTING THE MAIN SECTION.

(18.) First draw the datum line, as in Plate XIII.; and from a scale of 5 chains to an inch lay off the    f, and 1 mile distances,

repeating them after every successive mile, and marking the miles 1, 2, 3, &c. Now lay off on the datum line, from the column of distances in the Level-Book, the several distances up to the first mile ; do the same between the second and third line, between the third and fourth, &c., the distances always beginning afresh from the end of every mile in the Level-Book. Draw lines perpendicular to the datum line at the end of every mile, and parallel to these perpendiculars draw lines in pencil of the height given in the column of reduced levels, and at their corresponding distances, from a scale of 50 feet to an inch; and through the upper extremities of these lines draw the section, carefully noting the roads, paths, rivers, brooks, woods, &c., as given in the column of remarks in the Level-Books, which, it will be seen, are copied in the plate.

CROSS LEVELS.

(19.) Cross levels are taken principally on roads crossed by the line of railway, to show the position of the surface of the ground, partly with a view, if possible, of improving the main line, and partly to show the nature of the approaches of the cross roads to the bridges, where required, and the quantity in respect to depth and length of cutting, or of height and length of embankment required where the main line is crossed either over or under by bridges, or on the line of the rails.

The heights in the cross-section are usually taken at every chain’s length, to the distance of 10 chains on each side of the line, as will be seen in the following notes; remembering always to take the levels from the left to the right of the onward direction of the main line, otherwise serious errors might arise by plotting the section in the -wrong direction.

The cross-section is usually plotted on the same scale as the main section; some engineers adopt a larger one. The student can have no difficulty in plotting from the following notes. (See Plate XIII., Cross-Section No. 3.)

Cross levels No. 3 on road from Winston to Mold.

Remarks.

Back Sights Inter. Fore Sights Distance.

1-13

0-00

1-86

1-00

2-89

2-00

6-70

400

8-83

5-00

102

7-70

10-64

6-00

8-00

2-83

10-76

9-00

6-60

7-31

10-00 10 31

Line crosses. Reduced level 104 feet

0-61

0-48

0-65

OO CC CD

ip

kQ

CO

1-63

6-70

8-06

11-00

1300

15-00

17-00

19- 00

20- 00

(20.) THE APPLICATION OF THE CORRECTION FOR CURVATURE AND REFRACTION IN LEVELLING.

Ill the preceding examples on levelling, the application of the correction for curvature and refraction has been avoided, by assuming that the levelling instrument was always placed in observing at or near the middle point between the staves; thus making the errors arising from these causes correct each other. (See note to Art. 11.) But in a case where a very long sight is taken in one direction, as, for instance, the fore sight, the back sight at the same time being a very short one, these corrections must be strictly attended to, as in the following

Example.—In taking levels for a projected railway, a swamp of 148 chains in width occurred in the line; the level was placed at the edge of the swamp, the back-staff was placed one chain behind the instrument, on the level of the sw'amp, the reading thereon being

5⁻56 feet; the level being reversed, was found to cut a notch in a post, standing nearly in the line, on the opposite edge of the swamp, and on going round the swamp to the post, and measuring the height of the notch thereon, it was found to be 7⁻36 feet above the level of the swamp; required the correction for curvature and refraction.

By Art. 8, Rule II. (148)²-4-800 = 27‘38 inches = cor. for cur.

¹

What is here termed a level line, is, in reality, a great circle of the earth, a small arc of which, on account of the great magnitude of its radius, may be considered as a straight and level line.

deduct

for ref.

3-91 -

Correction for curvature and ref. = 23A7 = 1*95 feet.

Therefore 7'36- 1 -95 = 5*41 feet, is the corrected fore sight; the hack sight, being at so short a distance, needs no correction : hence the difference of the levels of the edges of the swamp is 5'56 — 5-41 = 0'15 feet.

Note.—From the above example it will be seen that had the correction not been made, the error in the levels would have been nearly 2 feet.

(21.) When several assistants are employed on the same section, a uniform system should be strictly adopted, and the superintendent of the work should fix upon the bench marks, occasionally checking the work of the others. When a section has been thus taken in several parts at the same time, after putting the parts carefully together, a common datum line must be assumed for them all, and the result checked by adding or subtracting the differences of the reduced levels at the points of junction, accordingly as they rise or fall.

Intermediate sights, in ordinary levelling, need only be taken to tenths of feet, as it would be a waste of time to attempt greater accuracy, excepting where the levels are taken to bench marks, in which the same accuracy should be observed as in taking back and fore sights. The height of the optical axis of the level, when on the line, or the level of the line of section, may always be put down as an intermediate sight, if required. It will here be proper to observe that it is not necessary that the levelling instrument should be placed directly on the sectional line, while making observations therewith, but in any convenient position either to the right or left of it.

In taking levels through towns, the operation frequently proceeds in a zigzag direction, such as the streets may present, the length of the required sectional line being determined from a map of the town, and the several heights of the points in the section obtained from the reduced levels, corresponding to the points in the streets where the sectional line crosses them. In a similar manner the elevation of the extreme points of the estates of proprietors and occupiers, who are hostile to engineering operations, are obtained by going round

Note.—All the methods of levelling, and of laying down sections therefrom, given in the preceding articles of this section, will apply to canals, roads, sewers, drains, &c., with the same facility as if they had been written for these purposes.

LEVELLING BY THE THEODOLITE.

(22.) When the levels are required to be taken over very high and rapidly rising summits, on the acclivities and declivities of which it is found very difficult to fix the levelling instrument, or over steep and almost perpendicular cliffs, where it cannot be fixed at all, the operation would be best performed by the theodolite, which must be set perfectly level, both with respect to the spirit-levels on their vernier plate, and that which is attached to the telescope ; and the angles of elevation and depression of the required points, both before and behind the instrument, respectively, taken by means of the vertical arc; the distances on the slope of the observed points from the instrument being, at the same time, measured as correctly as circumstances may permit, from which data the perpendicular elevation and depression of the points, as well as their horizontal distances from the instrument, may be found by the rules of right-angled plane trigonometry. These operations may be repeated for any number of stations, recollecting to make the necessary corrections for curvature and refraction, and to take into account the heights of the instrument and of the objects placed in the observed points. Operations of this kind should be performed when the atmosphere is settled, otherwise the refraction will be found so extremely variable and deceptive as to produce considerable error.

LEVELLING BY THE ANEROID BAROMETER.

It is often very useful, in going over a line of country, to be able to form a general approximate idea of the levels or altitudes of the ground, without going to the trouble of levelling it in the usual way. For this purpose the aneroid barometer (which is now made small enough for the pocket) is admirably adapted, as it will indicate differences of altitude to a few feet with tolerable precision. Full directions for the use of the instrument are usually sold with it.

It must, however, be borne in mind that the pressure of the atmosphere, even in the same place, is constantly varying, and that, therefore, unless this source of error is provided against, the indications of the barometer may be exceedingly delusive. If accurate indications are desired, a duplicate instrument, remaining stationary

in one place, should, be constantly watched and registered during the survey, and its indications carefully compared with that carried about over the ground.

SECTION II.

PARLIAMENTARY PLAN AND SECTION, ETC.

(23.) When application is made to Parliament for authority to make a railway, it is necessary to prepare a survey of the intended line, showing both its horizontal and vertical position; the documents exhibiting these features being called the “ Parliamentary Plan and Section.” Copies of these and some other documents have to be deposited, in due form, at certain places, in compliance with certain rules established by Parliament for the purpose.

These rules are called the “ Standing Orders of the Houses of Lords and Commons on Private Bills.” They contain many directions not only for the preparation of the plans and sections, but also for the laying out of the lines, all which must be carefully adhered to by the engineer and surveyor, or he will risk the rejection of his Bill. The standing orders are altered and amended from time to time, and it is therefore highly necessary that any engineer who contemplates going to Parliament, should provide himself with a copy of the latest edition, which is sold by Parliamentary stationers at a moderate price.

According to the standing orders of 1863, the Plan must be drawn to a scale of not less than four inches to a mile, and where any buildings are included within the limits of deviation, enlarged plans must be given to a scale not less than \ of an inch to 100 feet.

The Section must be drawn to the same horizo7ital scale as the plan, and to a vertical scale of not less than 1 inch to every 100 feet. The line of the railway shown is to correspond with the upper surface of the rails.

A model plan and section usually accompany the published book of standing orders.

Method of preparing a plan and section of a railway, as required by the standing orders of the House of Commons, preparatory to obtaining an Act of Parliament for its construction.

Plate XIII. is a portion of a plan and section, with cross-sections &c., of a railway prepared for the above-named purpose. The plan or map of a portion of the country through which it passes occupies the lower portion of the plate, the proposed centre line of the railway being marked thereon by a strong black line, and by dots where

it passes through a tunnel. The fields, &c., in the parish where the railway commences are numbered consecutively to the boundary of the next parish, where the numbers commence afresh, and so on through the successive parishes or townships; the numbers referring to corresponding numbers in a- book of reference, in which are descriptions of the several properties, with the names of owners and occupiers. On each side of the centre line, and parallel to it, at the distance of 100 yards, or 454 links from it, are dotted lines called the limits of deviation: within the space included by these dotted lines, the engineer, on being empowered to construct the railway, is allowed to deviate from the line, as projected on the map, should he think it advisable for the sake of improving the line, or of avoiding expensive severance, &c.

The main section, with its accompanying cross-sections, occupies the upper portions of the plate (these sections are the same as those referred to in some of the preceding articles), to prepare which, for Parliamentary as well as practical purposes, it will be first necessary to explain the method of laying out gradients.

THE METHOD OE LAVING OUT GRADIENTS.

(24.) The gradient of any portion of a railway means the inclination of the surface of the rails with respect to a horizontal line. A level line of railway would, doubtless, be preferable to any other; but the unevenness of the earth’s surface puts this out of the question in by far the greatest number of railways. It is well to make the gradients as flat as possible, consistent with economy in construction of the works; they were formerly limited to about 1 in 264, but owing to the improvements in the tractive power, they now frequently reach 1 in 100, beyond which they should not go except in very difficult cases. Parliamentary committees exercise a rigid investigation into the necessity of steep gradients in lines brought before them.

The usual practical method of laying out railway gradients is by applying one end of an extended silken thread to the commencement of the railway on the section, the other end being so applied that the extended thread may cut the curved boundary of the section or profile of the earth’s surface, so as to leave an equal portion of space both above and below the thread, as nearly as can be judged by the eye, in order that the cuttings from the spaces or parts of the section above the thread, may produce materials sufficient to fill up the spaces or parts below the thread, or form the embankments of the railway. The position of the thread being thought satisfactory for the purposes required, its extremities are

marked, and a line ruled in the place occupied by it for the first gradient. The second, third, &c., gradients are laid out in a similar manner to the end of the section. See Art. 26 and the two following notes.

Note 1.—The excavations and embankments of a railway are made about 2 feet lower than the level of the rails, thus giving a line parallel thereto, called the balance or formation line; the 2 feet filled up with gravel, to form the road and the beds for the sleepers of the rails.

2. If the position of the first gradient, though favourable in itself, cause the following gradient, or gradients, to be less favourable, with respect to the quantity of cuttings and embankments, it is advisable to alter the position of the first gradient to one less favourable, provided that the compound results of cuttings and embankments on the several successive gradients, as now altered, is more favourable than in the preceding case. In this manner, it is requisite to change the positions of the several gradients repeatedly till the minimum, or least possible quantity of cuttings and embankments, shall be required in the construction of the railway, keeping in view the required limit in the ascent and descent of the several gradients; the difficulty of making the excavations, throughout the whole length of the line, being supposed, at the same time, to be nearly equal. But where the geological character of the country through which the railway passes, differs considerably, presenting for excavation strata varying throughout the length of the line from loose sand to hard rock, and vice versa, the facility or difficulty of making the excavations must be carefully considered in laying out the gradients; larger excavations being advisable where they can be easily made—and smaller where with difficulty. The least ‘possible expense ivill be incurred in the construction of a railway by talcing into account all these circumstances.

(25.) If anywhere in the section the excavations reach 60 feet in depth, and afterwards increase rapidly in depth, it is a more economical method of proceeding to make a subterraneous passage, called a tunnel, through the deep part, than to cut the whole open to the surface of the ground, which in many cases would be next to impossible. Tunnels are cut to the width and depth of 30 feet, for railways on the narrow gauge; if on the broad gauge, to the width and depth of 36 and 32 feet respectively, the width and depth being less in both cases, if the material to be cut be hard rock. The diminished quantity of cuttings, where tunnels occur, must be taken into account in laying out the gradients.

(26.) To determine the rate of inclination of a gradient.

AE, in Plate XIII., is the first gradient on the railway section, the cuttings or excavations above it being considered to be equal to the requirements of the embankments below it; at its commencement A, its height AO is 100 feet; and at its termination B, its height EC is 130 feet above the datum line; thus giving a rise of 130 - 100

= 30 feet; the horizontal length oc of the gradient is 65‘GO chains = 4329 f6 feet. There is, therefore, a vertical rise of 30 feet in a horizontal distance of 4329-6 : hence 30 : 4329-6 : : 1 : 144-32; or, in round numbers, a rise of 1 in 144, which is called the inclination of the gradient ab, and is thus noted on the section, Inclination 1 in 144.

The rule for finding the rate of inclination of a gradient may be thus briefly enunciated:—

Multiply the horizontal length of the gradient in chains by 66, and divide the product by the difference of the heights of the gradient at its extremities, above the datum line, and the quotient is the horizontal distance to a rise of 1 foot, or the rate of inclination required.

(27.) The excavations at B being about 60 feet deep, and continuing to increase rapidly, a tunnel BD, to the length of 462 yards, is introduced on the next following gradient; the tunnel having passed beyond the summit, till its depth at D again becomes 60 feet; its height, as shown on the section, is 21 feet, being the height to which the cuttings are reduced by the ballasting below and the arch above. The gradient, of which the tunnel forms a part, is assumed to extend beyond the limits of the plate ; its position, therefore, is not determined by the excavations and embankments shown thereon, but in conjunction with those beyond its limits.

(28.) Method of determining the heights of the several roads passed over or under by the railway, and whether they should be raised or lowered to give sufficient height for viaducts or bridges; or to be raised or lowered to be passed on the level of the railway.

When a road is passed over by the railway, the usual height allowed from the surface of the road to that of the rails is 18 or 20 feet; and when the road is passed under by the railway, the height allowed is 18 or 19 feet.

The heights or depths of the roads, rivers, &c., above or below the rails, or gradients, is sometimes found by measuring them carefully by the vertical scale; but they may be found more accurately by the following method:—r

The distance of the middle of the first road from the commencement of the section is 10 chains, its height above the datum lines is 83-20 feet, the horizontal length OC of the first gradient is 65-60 chains, and the gradient rises 130-100 = 30 feet. There is, therefore, given the length and rise of the gradient to find its rise at any other given point; which may be done by similar triangles ; thus—

As lior. length of gradient 65-60 ch. : 30 ft. : : 10 ch. the dist. of road : 4’57 ft., the rise of the gradient at a ; this added to 100 ft., the height of the gradient at A, gives 104-57 ft., the height of the point a above the datum line; but the height of the road above the same line is 83-20 feet, therefore 104-57-83-20 = 21-37 ft. = 21 ft. 4 in. is the height of the rails at a above the road; which, being above 20 ft., shows that the level of the road may remain unaltered, the height of the bridge required for the railway being 18 ft.; thus leaving 3 ft. 4 in. for the thickness of the arch and ballasting, and its span being taken 24 ft., as being sufficient to allow carriages to pass one another on the road below the line.

The distance of the occupation road per Level-Book, is 24⁻06 ch., and the reduced level is 110 ft., therefore 65 60 ch. : 30 ft. : : 24-06 ch.

: 10-91 ft., hence

100 + 10'91 = 110-91 ft. height of gradient at road 110-00 ft. do. of road

0‘91 ft. 11 inches, the height the road must be raised to be passed over on level of rails. In raising or lowering roads for this purpose, a rise or fall of 1 foot in 20 is required.

In the same manner, the road from Westbrook to Hurst is found to be 22 feet above the gradient; the hedge of the brook that forms the boundary of the parishes of Westbrook and Winston, 36 ft. 5 in. below the gradient; and the road next following, 40 ft. 5 in. above it.

SECTION III.

RAILWAY CURVES.

(1.) On the use of curves in railways in general.

The use of curves in railways is absolutely necessary on account of the natural unevenness of the earth’s surface, it being desirable to attain the nearest possible practical level by avoiding hills, crags, mountains, &c., and by winding round them by means of curves. Curves are also equally necesssary in avoiding other natural and artificial obstructions, in many cases not materially affecting the level of the line, as lakes, swamps, the windings of sea-coasts, and of rivers; also cities, towns, villages, parks, pleasure-grounds, &c. In this manner a great saving is effected in the expense of extensive excavations, embankments, tunnels, viaducts, &c., as well as the expense of the severance of valuable property, which would otherwise be required. Besides, it is frequently desirable to make a

winding railway, in order that it may embrace in its route some important city, town, harbour, &c., or make a junction with another railway. Straight lines in railways are, however, much to he preferred to curves, and are therefore adopted as far as possible; they are first set out as bases for further operations; and every curve is subordinate to two straight portions of the line, which are tangents to it at its commencement, and termination. The curves adopted in practice are always arcs of circles. Sometimes two, three, or more consecutive circular arcs, having a common tangent at their point or points of junction, are joined together, as in the case of the compound curve; and sometimes two circular arcs are connected, having their convexities turned different ways, with a common tangent at their point of junction, as in the case of the serpentine, or S curve.

]STote.—Other curves beside the circular arc might sometimes with advantage be adopted as the curves of railways; but their construction is attended with difficulty.

(2.) On mechanical railway curves, or curve-rulers.

Mechanical railway curves, sometimes called curve-rulers, are a series of segments of circles made of hard wood, as box or mahogany, or strong pasteboard, with their radii in inches marked on them. These curves commonly begin with a radius of 2| inches, and terminate with one of 160 inches, or’upwards; the smallest radii usually increased by \ inch up to 10 inches; then by inches up to 20 inches; afterwards by 2 inches up to 80, including all numbers ending in 5; lastly, by 5 inches, till near the end of the series, when the radii increase by 10 inches.

The annexed figure represents the railway curve-ruler of 16 inches radius. If the scale of plan, to which this curve-ruler is applied be 5 chains to an inch, it will represent a curve or circular arc of 16 x 5 = 80 chains = 1 mile radius. If the scale of the plan be 12 chains to an inch, it will represent an arc of 16x12 = 192 chains = 2 miles and 34 furlongs radius; and similarly for other scales.

(3.) The limit of the radii of railway curves.

As a general rule the curves of a railway should be kept of as large a radius as possible consistently with the economical laying-out of the line. One mile radius or upwards is quite unobjectionable ; but half a mile, or even less, will be sanctioned by Parliament

in difficult cases, particularly near large stations, where the speed is usually slackened. In all sharp curves the outer rail requires to be raised above the level of the inner one, to counteract the effect of the centrifugal force. (See Formulae in Section VII.)

PROBLEM I.

. Case I.

To determine the radius of a railway curve mechanically, that is, by the curve-rulers ; the positions of the straight or tangential portions of the railway being given.

This problem admits of an indefinite number of solutions, but the curve adopted is generally that which falls on ground presenting the fewest natural or artificial obstructions, provided its radius be not too great. (See Art. 3.)

Let AB, CD be two straight portions of a railway, the positions of which are given by an accurate plan; and which, when prolonged, meet at T. Apply several of the series of curve-rulers (see Art. 2) to touch the line AB, without cutting it, at the points B, b', b", &c., and also to touch CD, in like manner, at the points C, c', c", &c. ; and let BC, Be', Be", &c., be the curves thus obtained. Then

whichsoever of these curves may appear, on a careful consideration, to be least obstructed by hills, lakes, rivers, build-\ ings, &c., must be D adopted as part of the line of railway, provided its radius be equal to, or exceed, the required limit. The radius is determined by multiplying the number on the curve-ruler by the number of chains per inch on the scale of the plan.

Example. If the curve BC be drawn by the curve-ruler, numbered 22, and be adopted as part of the line, and the scale of the plan be 5 chains to an inch, required the radius of the curve.

22 x 5 = 110 chains = 1 mile 3 fur., the radius required.

Note.—By this method the radii of railway curves are commonly found in practice; but since a circular arc of large radius apparently coincides with its

C"

R‘.

sC'

cizz-.zzizziZ.~~

A/

tangent for a considerable distance, it is difficult, in this manner, to determine the exact starting-points of the curve; besides, if the plan be not strictly accurate, as is too often the case, the positions of the tangents cannot be said to be given, and therefore this method should only be employed in roughly determining the radius and starting-points of the curve, so that it may fall, in its progress, on the most favourable ground.

Case II.

To determine the radius of a railway curve geometrically, the starting-point B being given.

Prolong AB, DC till they meet at T; bisect the angle ATD by the line TO; and at B draw BO perpendicular to AT, meeting TO in 0. Then o is the centre, and BO the required radius of the curve BC.

Or,

Make TC — tb ; and draw the perpendiculars BO, CO, meeting in 0. Then BO, or CO, is the radius of the curve.

In the same manner, the other radii c'o', c"o", &c., of the curves b'c', b''c", &c., may be found.

Case III.

To find the radius of the curve by taking dimensions on the ground, the map being known to be inaccurate, and the starting-point B being given.

Range the tangents AB, DC till they meet at T; measure BT ; then measure on TD till TC = BT; also measure BC. Then bo = ^ BC. BT -f V BT² - £ BC² ¹ = radius required.

Ex. Let BT = 82-50 chains, and BC = 132-00 chains. Then BO = \ (132 x 82-5) - V(82-5)² - i (132)² = 5445 -f 49-5 = 110 chains, the radius required.

Or by Trigonometry.

Range the tangents AB, DC till they meet at t, as before, then measure BT, and take the angle BTC, of which is the BTO; then in the triangle BTO, right-angled at B, are given BT and the </bto to find BO; whence by trigonometry,

cot. ,/bto : rad. : : TC : BO, the radius required.

Ex. Let bt = 82-50 chains, and the angle BTC = 106° 16'.

9-87501

10-00000

Then cot. J a BTC = cot. a BTO, 53° 8'

: rad. or sin. 90°    .....

1- 91645

2- 04144

: : BT = 82-50 chains, .    .    .    .

: BO = 110 chains, the required radius .

Case IY.

To find the radius of the curve, when the plan is known to be inaccurate, and the tangents AB, DC cannot be prolonged to meet at t on account of obstructions, the starting-point b of the curve being given.

From b measure a line BD, on the most convenient ground, to meet CD at D, taking the angles tbd, tdb ; the sum of which taken from 180° gives the Z BTD, whence by trigonometry, sin. Z BTD : BD : : sin. TDB : BT.

Now in the right-angled triangle bto are given BT and the angle BTO = \ Z DTD, to find the radius BO, which may be done as in Case III.

PROBLEM II.

To lay out on the ground a railway curve by the common method.

Let AB, CD be two straight or tangential portions of a railway, the points BC being required to be joined by a circular curve BC, to which AB, CD shall be tangents at the points B and C ; and let BO be the radius of the curve, which is supposed to be determined by one or

r.

3.

to

other of the cases in Problem I. accordingly as the map is found to he accurate or inaccurate. Put the radius BO = r; measure on the tangent ab prolonged the distance Bpx = 1 chain, as is usual in

practice ; set off pLq₂ =

1*

at right angles to Bp₁ • then qx is the

first point in tlie curve. Through. B^ measure the right line Bqxp₂

= twice Bpx — 2 chains, and set off p₂q₂ =    = twice pyqx at right

angles to Bqxp₂; then q₂ is the second point in the curve, repeating the last operation till the curve reach the point c. Lastly, q₃Cp5 being measured = 2 chains the last offset j?₅d will be found = the first offset pxqu or half the preceding one ptC, if the work be right. See the following notes and example.

Note 1. Only a small part of the operation for a curve of great length is shown in the figure ; but as the whole of the work, except the first and last offsets, is alike, to show more would be unnecessary.

2. The values of — for all radii, from 15 chains to 540, are given in Table No. 2 r

4, at the end of the work.

Ex. Let bo = r = 80 chains = 1 mile: then — = _L_ of a chain,

2r 160

which being multiplied by 792, the number of inches in 1 chain, gives 1 x 792

160

= 4-95 inches, the first offset pxqx5 whence 4-95 x 2 = 9-9 inches, the second offset p₂q₂.

Or by Table No. 4, opposite 80, in the column marked Radius, &c., stands 4'95 inches, in the column marked Offsets, &c., which is the value of ptqx, as above.

3. By this method the greater part of British, as well as foreign, railway curves have been laid out, having been invented by Mr. T. Baker, C.E., at a very early period, when the Stockton and Darlington Railway was laid out. It was eagerly adopted by railway surveyors, as it involves very little calculation, and does not require the use of an angular instrument. It is, however, defective in practice, on account of its requiring the coupling together of so very many short lines, as small errors will unavoidably creep in and multiply (and more especially so if the ground be rough), so that the curve has frequently to be retraced several times

* Demonstration.—Complete the semicircle bcq, prolong bo to Q, and join Then because Bp*, Bp₂, b<7₂, are always, in practice, so very small,

S'iQj BS's-

compared with bq that they nearly coincide with the curve, and consequently B/q is approximately = b<?i = q₂p₂ = qxq₂ = ^bq.₂; . *. by the nature of the circle the As BqiQ, Bjpjgq, Bp₂q-₂, are similar, and

bq : Bq₂ : : Bq₂ : qx

₌ B?i* ₌ B£]² ₌ 52 BQ BQ !_.

BQ

also :    : : bq₂ : p₂q.₂ = b^j x bq₂

bo = ^BQ being put — r, and Bpi = bq₂ = &c. = 5 ; and when 6=1 chain, as in practice, Piqi — and P-2& — Psqs — &c. = Q- A A.

before it can be got right, especially if it be a long one. This defect in the above method induced the author to prepare three other and more accurate methods, which are given in the following Problems.

4. When the curve shall have been laid out with sufficient accuracy, the rods or quills that mark the ends of the offsets, qgiz, &c., must be taken out and their places supplied by strong wooden stumps, about 16 or 18 inches in length, and 1J inch square, each end of the curve being marked with three stumps, or one large one with a cross or some other conspicuous mark on it. The straight portions of the line must also be marked with stumps at the end of every chain.

PROBLEM III.

To lay out a railway curve by ordinates or offsets from its tangents, no material obstructions being supposed to exist on the convex side of the curve, to prevent the use of the chain.

Case I.

When the length of the curve is less than of its radius.

Let BC be the curve, AB, CD, tangents thereto, which are ranged till they meet at D, and BO the radius of the curve, which must be correctly determined by one or other of the cases in Prob. I.

if

,/'2

q-2

q-i

PI

V‘

a/

i)

according to circumstances. Measure on BT the usual distance B\px = 1 chain, and lay off the offset pgp = i- (r being the radius of

ZiT

the curve as before), then measure plp.i = 1 chain, and lay off

PA = 2r-1 2² 3² 4²

2²

The successive offsets at the end of every chain being

2r’ 2r’ 2r’ 2r’ <^C‘,V or’    amounts to the same thing, the

2d, 3d, 4th, &c., offsets are respectively 2², 3², 4², &c., times the first offset, or 4, 9, 16, &c., times the first one.

Having laid out the offsets in this manner, till the last one, which suppose to be p₄q₄ is either within, or very little more than, a chain from T, make p'₄T =p₄T, and lay out the same offsets in an inverted order, on TC, as were laid out on BT, that is, beginning with the greatest first and ending with the least.

1 x 792

Ex. Let the radius of the curve be 80 chains, then p₄q₄ —

2x 80

= 4-95 inches; whence p₂q₂ = 4 x 4⁻95 = 19²8 inches, p₃q₃ — 9 x 4'95 = 44 55 inches = 3 ft. 8⁻55 in., &c., or the value of the first offset may be taken from the table at the end of the book.

To make the distances of the stumps equal.

As it can rarely happen in practice that the distance q₄q₄ will be = 1 chain, when this is the case it will be better to set out the curve from C as was done from B; then, if the distance q₄q'₄ be less than 1 chain, suppose m links less, lay the chain from q'₄ to q'₃, and at the

m (1 — to)

then s will be the

2r

distance q'₄t = m, lay off the offsets ts =

point for the stump, instead of q'₄, thus making q'₄s = 1 chain. This operation must be repeated between q₃ and q₂, q₂ and q₄, &c., the last point to be changed necessarily falling on the tangent CD. If the distance q₄q'₄ be greater than 1 chain by m links, lay the chain

(    1²\ l² 2 2    (    2²\    2²

RAILWAY ENGINEERING, from c/₄ in the direction of q₄, and at the distance of q₄ v = m links, lay off the offset vu = (jLt-P.)— , then u will be the place for the stump;

422

PARI X.

next lay the chain from q₃ in the direction of u, and repeat the previous operation : this must be done to the end of the curve, the last mark to be changed falling, in this case, within the curve. The stumps may now be put down as pointed out in Note 4, Prob. II.

Ex. 1. When the distance q₄q₄, is less than 1 chain by 40 links,

the radius of the curve being 80 chains.

2r

Here „ 1 (? ~ »>) « ₌

•60 x -40 x 792

(1 - -40) x -40

Too

Ex. 2. When the distance q₄q'₄ is more than 1 chain by 40 links,

_(1+ ro)m_(l + '40) x ‘40 160

ch. =

160

= 1T88 in. = 14 inch nearly.

and rad. = 80 chains. Here vu

2r

ch. x

(1-40 x -40) x 792

160

- 2‘742 in. = 2f inches nearly.

Case II.

When the curve is any required length.

As it is inconvenient in practice to have the offsets greatly to exceed two chains in length; if, therefore, the curve be a long one, the offsets may be confined within proper limits by dividing the curve

PI

p~.

*qi

T p’3

V3\

pi

pi

pi

Pi Tt

>2

‘li

(pi

q,i

'I

qi

B.

into two or more parts, and introducing one or more additional tangents. In the annexed figure the curve BC is divided into two parts in o', at which point the tangent TcV is drawn, meeting the tangents bt, t'c in t and t', the tangents bt, to' being each taken =

Bp + \p₃q3³ wherein By>₃ is taken the nearest whole number of chains to |bo, and p₃q₃ is the offset corresponding to Bp₃. By thus taking the length of BT, tc', the last offset on bt and the first one on TC will meet at q₃, the middle point of bc', and therefore the distances of the ends of the offsets that form the curve will all he sufficiently near to equality, i.e., to one chain. Also by trigonometry, BT : BO : : rad. : tan. l bto — \ l etc' ; whence the l BTC' becomes known, and consequently the direction of the new tangent tcV is also known. Having, therefore, determined bt and L BTC', the offsets may be laid out to p₃q₃; and having made Tp₃ — Tp'₃, the same offsets may be laid out in an inverted order on tc' ; the order of the offsets being a second time inverted on c't', and a third time on t'c. If the distance of the middle offsets q₂qf₂ be less or greater than 1 chain, that distance, and consequently the following ones, must be made equal to 1 chain, as in the preceding case.

2bo

Ex. Let bo = 240 ch., then bp₃ — |bo — 30 ch., p₃q₃

3Q2

= 1-875 ch. : whence bt. = tc' = Bp₃ + \p₃q₃ = 30 +

480

|1-875    30-2344 ch. = 30-23| ch. nearly : and by trig.

As 30-2344 (bt) log. : 240 (BO).    .

: : rad. .

.    1-48050

.    2-38021

: tan. L BTC' = 82° 49'

. 10-00000 .    10-89971

L BTO = 165° 38', whence the position of the tangent tc't' becomes known; it will be unnecessary in this case to find l tt'c.

curve

part c'c of curve, if required, is the same as in Case I.

By table 4, the first offset on bt, viz., pxqx = L65 in., from which the other offsets may be found by multiplying by 4, 9, 16, &c., which may be laid out as in the last case, p₃q₃ being the 30 th offset from B, and meeting the corresponding offset p'₃q₃ at q₃, the middle of the bc'. The method of making the distances equal on the second

Note.—If the curve be a long one, it may be divided, in this manner, into 3, 4, or 5 parts, according to its length.

Case III.

When the length of the curve exceeds the limit adopted in Case I., but is considered too short to be divided into two parts, as in Case II.

Let the curve be one of 80 chains radius, and between 32 and 38 chains in length, then to avoid the trouble of adding another tangent, the offsets beyond the 8th must be calculated from the following formulae:—

Mi = r -

m* = r -

&c. = &c.

- 9²

- 10

If the radius of the curve be 160 chains, and its length about 50 or 60 chains, the offsets must be calculated, as above, beyond the 12    2² 32

2r 2 r 2 r

the true lengths of the offsets within the limits assigned, but beyond these limits the errors of the offsets begin gradually to augment, till they become too considerable to be overlooked. See Demonstration to Case I.

15th. The formula

&c., is a very near approximation to

PROBLEM IV.

To lay out the curve, by two theodolites, where water, swamps, quarries, or other obstructions prevent the use of the chain.

i;

vD

\/

The position of the tangents AB, DC to the curve at its extremities B and C, its radius OB or OC. and the l BOO = supplement of the angle made by the tangents when prolonged, being determined; join

SECTION III.

RAILWAY CURVES.

425

Take the a qxCB = an arc whose sine is —, wherein d = bqx — qxq₂ — &c., and r = OB, or, because d is, in practice, usually required to be = 1 chain, take a gfCB = arc whose sine is

The several angles may then be arranged as below :—

1

= arc to sin. —

- ?iCB

A 2.CB = 2 A (^CB, A q₃CB = 3 A (frCB, A q&B =4 A ^iCB, &c. =

&C.

2r*

and l qxBC = a BOG -

A q₂BG =

L q₃BC =

A g^BC =

&C.

tOH tCH. fcC;M bob-⁴

A BOC -

A BOC -A BOC -

&c.

L qx CB,

L 2iCB, L qxGB, L (^CB,

Therefore if theodolites be fixed at B and c, and the angles g^CB, gxBC be taken at the same time, the intersection of Bqx and C^ will give the point qx. In the same manner by taking the angles q₂CB, q₂BC, the intersection of Bq₂ and cq₂ will give the point q₂, &c.

Ex. Let the radius OB = 80 chains, and the angle between the tangents AB, DC, when prolonged to meet, be 160°; then its supplement = A BOC = 80°, and a BOC — 40°. Also a qxCB = arc to sine

= 0° 2Y 29", whence a^bc = \ aboc-

b (⁼

•00625

160 ~

LqxCB = 40° - 0D 21' 29" = 39° 38' 31"; Lq₂CB - 2 LqxCB = 2x0° 2T 29"= 0° 42' 58" and Lq₂BC = \ A BOC - 2 a qxGB = 40° - 0° 42' 58"= 39° 17' 2", &c. arranged for same time.

The angles would be best use as follows, the opposite ones to be taken at the

^<^03 = arc to sin.-00625 = 0° 21'29", xh^BC = ^ ^boc - Z.qxcb = 39° 38'31"; ^A^₂cb = 2 Afg^CB •Afg^CB = 3 AqiCB Afg₄cB = 4 Z 2iCB &c. =

This list of angles must be continued till n a g^CB can no longer be taken from \ a BOC, and one angle in each column being taken at the same time, as LqxGB at C and LqxBG at B, will give the point qx: and so on for the other points q₂, q₃, &c. See the following notes.

Note 1.—Where the obstructions are such as to prevent the stumps being put down at the consecutive points qx, g₂, q₃, &c., it would be best to take every fourth angle, thus obtaining the points g₄, qa, q₁₂, &c., which will be 4 chains apart on the curve, the intermediate points being left to be put in when the 'work of the line has progressed so far as to present a better opportunity. Or, if it

= 0° 42'58", Afg₂BC = £ aTboc- 2 A^glCB = 39°l7' 2"; = 1° 4' 27", ^qsBC = l ATboo- 3 Z?iCB = 38°55'33"; = 1°25'56", ^qiRG = \ AfBOC— 4 AfgiCB = 38° 34' 4";

&c.

&c.

&c.

&c.

should be thought preferable to avoid taking a multiplicity of angles, every fourth angle may be taken in any case, as this method is equally available whether obstructions exist or not.

Demonst. to Prob. IV.—The angles Bt^c, Bq₂c, B23C, &c., being in the same segment, are well known to be equal and constant, and also equal to half the supplement of ^boc, and consequently the sums of the angles of the triangles Bq^, B^c, &c., adjacent to bc are equal to \ ,/boc, whence Z.q\BG = ^ zIbog- Z.qiCB, Z.q₂bc = % /.boc- Z-%gb= (because equal angles stand on equal arcs) J ^!boc-2 <rLq₂CB, &c. Also join oq₂ and put BO = r, and Bgq = 5, then it is well known that j «CBOqi= ^.qicb, and because the sides bo, q₂o of the triangle bo^i are

; or, by taking rad. = 1, and 5 = 1 chain,

equal, sin. J PlBoqi=sm. ^dq₁CB =-

1 . 1

as required in practice, sin. *dqiCB = or    = arc to sine

•rad. x 5

2~r

Q. E. D.

PROBLEM Y.

To lay out a railway curve by means of ordinates or offsets from its chord or chords no material obstructions being supposed to exist on the concave side of the curve to prevent the use of the chain.

Let bc'c be a portion, or the whole, of a curve of a railway • at, t't' and CD tangents to the curve at B, c', and C; O the centre, OB the radius, bc' and c'c chords of the curve; which chords must not exceed 40 chains, if the radius be 80 or 120 chains, but they may be 60

•g'

A3

■s

5....

//•"

•Q

chains, if the radius exceed 120 (this limitation is necessary to prevent the offsets pgqv p₂q₂, &c., being too long, as was observed with respect to the offsets from the tangents, in Prob. III.); and let the radius q₃o bisect the curve and chord in q₃ and ps respectively. Then from the right-angled triangle Bpso, in which bo and Am =

Jec' are given, the L BOy>₃ — l tbc may be found, which determines the position of the chord bc' ; also op₃ may be found from the same triangle being = V OB² - bp²₃ — V ob² - d B(/² Put OB = r, op₃ = s, and the J chord Bp₃ = n chains, then

qxPi- V r² - (n - l)² - s*

Mh = V f²-(n- 2)²-s,

&c. =    &c.

isVi= V r² - (n - 3)² - s - r - s.

q₃pz being supposed to be the offsets at the middle part of the curve ec', not the third offset as shown in the figure, it being impossible to draw all the offsets without confusing the figure, or making it unnecessarily large. After reaching the middle point q₃ of the curve, the same offsets are repeated in an inverted order till the curve shall have been set out to c'. The same operation may be repeated as often as necessary, till the whole curve be completed, observing to make the l bc'c = 2 complement of l tbc', which has been already found. See the following notes :—

Note 1. When bc' is the whole curve, and its chord Bp₃c includes a fractional part of a chain, the distance of the offsets on each side of the middle of the curve •will be less than one chain; therefore that distance, and consequently the following ones, must he made equal, as was shown with respect to the distances in Problem III.

2.    If the last chord, which suppose to be c'c, be less than the preceding chord or chords, the oc'c, must be found, and added to the <^obc' or oc'b, which will give the ^Tbc'c, showing the direction of the chord c'c.

3.    This method of laying out the curve is seldom used, on account of the calculations it involves. It may, however, be used with advantage where a winding river or cliff is close to the convex side of the curve, or protrudes in some places a little through the curve; thus preventing the use of any other method, except that in Problem IV., which requires two theodolites.

Definition of the compound curve. (See figures to Prob. VI.)

The compound curve bcc'C', joining the tangents AB, DC", is composed of three circular arcs BC, CC', c'o", having common normals, oc, o'c' at their points of junction c, c'; and therefore common tangents at the same points, the radii of the three portions of the curve being respectively ob — OC, o'c — o'c', and o"c' = o"c". This kind of curve is adopted where the line is required to pass through

* Demonstration.—Draw oq parallel to bc', giQ _Lto oq, and join gqo ; and let n — number of chains in %bc' or p₃c', and c'y>i=l chain. Then p₃p₁ = oq = n- 1; and (Euc. I. 47) 2iQ= V r²-(ra-l)²; whence qip\ — V r²-(n¹ - l)²-y>iQ : but y>iQ=y>₃o = s, STPi = V r² - [n - l)² - s. Similarly, q-₂p₃ is founds Vrs-(na—2)s-8, &c. Q. E. D.

given points, as C and c', to avoid obstructions, or where a principal station or terminus is at or near c"; in the latter case the radius o"c" may, if required, be less than 80 chains.

The compound curve may consist of two, three, or more portions of different arcs; thus the curve bcc' consists of two portions, be, cc\

PROBLEM YI.

1. To find the several radii of the compound curve mechanically.

Let AB, DC" be the tangential portions of the line, which are required to be joined by a curve passing through the points C, c/, the point c" not being given. Select a curve-ruler such that, being applied to touch AB at B, it may also pass through c; if this curve do not pass through c', but through some other point E, another curve-ruler of less radius, in this case, must be selected, and such that it may touch the arc BC at 0 without cutting it or its prolongation towards E, and also pass through the point c : if this curve cut the prolongation c"t' of the tangent dc", another curve-ruler of less radius than the last one must be selected, and such that it may touch curve CC' at c' and the tangent dcC at c": thus completing the curve BCC'o". The radii OC, o'c', o"c" may be determined from the curve-rulers, as in Case I. Prob. I.

2. To find the radius co' of the compound curve bcc' geometrically, the starting point b and the radius bo being given.

(f

C':

h. y

- - c\

o'V.....c....../

,h

v'

A*

1°'

From the given point b in the tangent ab draw the given radius bo jl to ab; and draw the curve to some point c, where it is found

convenient to change the radius : draw the radius oc, and from thereto draw d', meeting the tangent dt in t'; make t'd = t'c, and from c' draw do' j_ to td meeting co, prolonged if necessary in o'; then o' is the centre of the arc cd of the curve, conformable to the nature of tangents.

The method of constructing the curve, when it consists of three or more parts, is sufficiently obvious.

3. One of the two radii of the compound curve and its starting and dosing points being given, to find the other radius.

Let ab, c'd, be the tangents, b and d the starting and closing points of the curve. Draw the perpendiculars bo = dh — given radius to the tangents; join oh, and bisect it in /; draw fo to oh, meeting dh prolonged in o'; join o'o, and prolong it till oc — dh : then the points o, o' are the centres of the arcs be cc, which constitute the compound curve, o'c-o'd being the radius required.*

"Note.—In the compound curve bcc'c", where the radius c"o", which is to be found, is less than the preceding radius co, the J.c"h is made = co ; ho' is joined and bisected in F ; and fo" drawn -L to ho', meeting c"h in o", which is the centre of the arc e'e", &c.

Definition of the serpentine or S curve. (See figure to Prob. VII.)

The serpentine curve BGC is used in railways, when obstructions or some other cause render its adoption preferable; it consists of two circular arcs, having their convex sides turned in opposite directions, like the letter S, whence it is sometimes called the S curve; the two portions BG, GO of the curve having a common normal OGo' at their point of junction G, and therefore a common tangent at the same point. This curve affords the most easy means of joining two parallel, or nearly parallel, portions of a line of railway.

PROBLEM VII.

1. JVhen one radius and its tangential point are given, to find the other radius and tangential point of the serpentine curve geometrically.

From the given tangential point c draw the given radius co j_ to the tangent CD, and draw the curve CG to some point G where it is found convenient that it should have its point of contrary flexure;

Q. E. D.

* Demonstration.—Since of = fh, and fo' is J_ to oh, oo'=o'h; also bo was made = c'A = oc; .'. o'c—o'c' and the normal o'oc is common to the arcs of the curve.

through OG draw the normal OGO'; from G draw GT j_ to ogo', to meet the tangent AT; make TB = TG; and draw BO' to AT, meeting

\b

T'

O'

OGO' in o'; then o is the centre, and o'b = OG is the radius of the curve BG, as is evident from the nature of tangents.

Note.—The radius o'b and tangential point b may be found mechanically, i.e., by the curve-rulers, as in Problems I. and VI.

2. When the tangential points and one of the radii of the serpentine curve are given, to find the other radius geometrically.

Erom the given tangential points c and B draw CO, BH, respectively j_ to the tangents CD and BA, and equal to the given radius ; join OH, and bisect it in F; draw fo' j_ to oh, meeting fib prolonged in o', and join o, o'; making o'g = o'b; then o' is the centre, and o'b - o'g is the radius of the portion BG of the curve, as required.⁴ •

Rote 1. The radius bo' may be found mechanically as in the preceding case.

2. The radius o'b may be found from the following formula, wherein 5 — bc, r = given radius oc, a = Zl tbc, and a'= zl t'ce.

• See investigation and fig. to Problem VIII.

o'b =

5 (5- 2r sin. a')

2^5sin. a + 2rsin.'- f¹ ^

Ex. Let 5 = 200 chains, r = 110 chains, a — 415°, and a' = 20°; the sin. a =

45° —20°

•70711, sin. a' = ‘34202,

•21644 ; and by the formula o'b =

— = 12° 30', the sine of which is

_200 (200-220 x ‘34202)    _

2 (200 x ‘70711 x 220 x dn644\~

= 82‘22 chains.

200 x__124‘7556 2 x 151‘7282

It may thus be readily ascertained whether the required radius o'b is greater, equal to, or less than 80 chains; if less, the given radius r ought either to be diminished, or the distance BC of the tangential points ought to be increased, according to circumstances, in order that o'b may be of the required length, assuming that the portion bg of the curve is not near a terminus or principal station.

PROBLEM VIII.

1. When the two portions of the serpentine curve have the same radius, to determine that radius geometrically, the tangential points and their distance being given.

Let AB, DC be the tangents, B and C the tangential points, and BC the given distance. Draw Bo = Co' respectively j_ to AB, DC, and of any convenient length; through o, parallel to BC, draw og

o'

indefinitely; with the compasses apply o'o" - 2co' = 2bo ; through C, o" draw co"o, meeting bo prolonged in o; and through o, parallel to o''o', draw 00, meeting Co' prolonged in o; then 0 and o' are the centres, and OB and o'c are the equal radii of the serpentine curve BGC, the common normal of the portions BG, GC of the curve being ogo' = 2bo = 2co'.⁵

2. To find the comwm radius of the two portions of the serpentine curve by calculation, the same things being given as in the preceding case, and the angles tbc, t'cb.

Put BC = <3, BO = Co' = r, L TBC = a, and L t'cb = a'; then ___<3

r ~ sin. a + sin. a + 2 sin. of arc to cos. (cos. a + cos. «')

Ex. Let 3 = 200 chains, a = 27°, a! = 50° : then from a table of nat. sines sin. a =-45399, its cos. =‘89101 ; sin. a =-76604, its = ■64279: whence |- (cos. a + cos. a) — \ (‘89101 +‘64279) = -76690 = cos. and 2 sin. of arc to cos. ‘76690 is 1‘28334 ; therefore 200 200 ^ , .

COS.

•45399 H- -76604 + 1-28334 ° 250337 ='⁹'89 chams> 0r n^ 80 chains, the radius required.

Note.—The method of forming the serpentine curve with a common radius is much to be preferred to any other, when the nature of the ground will admit of its being done; and more especially so, when the data, as in the preceding example, will only just give a common radius of 80 chains : for, if the radius of one of the portions of the curve had been taken greater than 80 chains, the other radius would have necessarily been less than 80 chains.

The Continental engineers, in laying out serpentine curves, usually place, if possible, a short straight piece of line between the two members of the curve, to ease the transfer of the train from one direction of curvature to the other.

PROBLEM IX.

To make a given deviation HQ from a straight portion of a line of railway ahd by means of three curves, bg, gqg', g'c, having their radii OB, o'q, o''c, all equal, in order that the lateral works of the line may avoid the building, or other obstruction b, which is close to the centre of the straight portion of the line.

1. Construction. — From the given point H draw HQ = given deviation to AD; on QH prolonged, take Qo' = OP = given radius ; with the compasses apply PB = QP = twice given radius; draw

given, whence the comp, of Ao'qo = ^oo'q is known, and may be thus expressed, sin. ^foo'Q = sin. of arc to cos. \ (cos. a+cos. a'). Whence oq = pp'=oo’x sin. /oo'q = 2rxsin. of arc to cos. J (cos. a+cos. a'), S = BC = BP + pp'+p'c=r sin. a + r sin. a' + 2r sin. of arc to cos. \ (cos. a + cos. a'), from which

Q. E. I.

r~sTnTa + sin. a'+2 sin. of arc to cos. £ (cos. a + cos. a')'

When the radii are unequal, and one of them, as r = o'c is given, and the other = bo is required; then o'o = R + r, bp = r sin. a, po = p'q = b cos. a, o'q=e cos. a + r cos. a', andoQ = PP'= V⁰'⁰²⁻⁰^²— V(k+p)²- (e- cos. a + r cos. a')'², S=B sin. a a'+ ^/(b + r)¹—(r cos. a + r cos. a')². By transposing and squaring, and

+r cos.

remembering that sin.² + cos.²=l, &c., there results 5³-25 (r sin. a + r sin. a')-

— a!

__ d —

2rr (1 — cos. a cos. a — sin. a sin. a') = 2Rr (1 — cos.a — a') = 4ej- sin. -    ^

, whence

5 (5-2r sin. a')

2 (f+2r sin.²^-^—)

This formula is used for finding the value of bo',

¹

Demonstration.—In the A bth, th = Vbt2 ~ bh2 — VB'f2 - i bc², and by the similar As bht, bho, bt²-^ bc² (=th) : bt : : | bc ( = bh) : bo = | bc : BT-f-Vbt - £ bc². Q. E. D.

²

Demonstration.—Draw the radius q₃o, and q₃a J_ to ob. Put OB = og³=r, and q₃a—p-iB — ^; then oa— \/r² - S’², and bu=p₃q₃ — ob - oa = r — vV²S². Now, if S be taken successively = 1², 2², 3², &c., chains, the values of piqi, p₂q₂, Psqs, &c., will be respectively r— sfr² -1², r- sir²- 2², r— ²/r²— 3², &c. But as l², 2², 3², &c., are very small compared with r\ within the limitation assigned to Case I. of this Problem, the values of sjr² —1², sjr²- 2², &c., may be taken re-1²    2²

respectively r — — r— —, &c., without material error, whence p>iqi = r-2r    2r

&c. But when the length of the

curve exceeds \ of its radius, only 5 or 6 of the offsets ought to be taken in this manner, and the remainder, say, commencing with ought to be taken =r-Ur² — 7², &c. ; although in a curve of 80 chains radius, the 10th offset by the contracted method exceeds the same offset by the correct method by only \ of a link out of 62§ links. This.difference, however, becomes gradually greater as the distance on the tangent approaches the 20th chain.

³

Demonstration.—By the similar triangles obt, q₃p₃T:, ob : By)3 + y>₃T (bt) : : p₃q₃ ; y>₃t ; but since B/>₃ = i-OB, or the nearest whole number to it, and since pjs must always be very small compared with b^₃, it may be rejected; .³. ob : -Iob (= very nearly to bt) : : p₃q₃ : p₃t = —p₃q₃, wherein n may be taken any whole number, 8 being the most convenient, as in the example.

⁴

The demonstration in this case is similar to the one given to Problem VI.

⁵

Demonstration.—Draw o"b parallel to ob; then by similar triangles, co': o'o" = 2co' :: co' : 2co'=oo', co" : co : : co' = bo = So" : co'

. \ co" : co :: co'=bo = bo” : bo = co'. Q. E. D.

+ Investigation.—Draw po, p'o'J_ to bc; through o draw oq parallel to bc, meeting o'p' prolonged in q : then oo'= 2r, bp = r sin. a, op = p'q = r cos. a, cp’—r sin. a', o'p'=r cos. a', o'q = o'p' + p'q=:?' (cos. a + cos. a') : whence, from the right-angled triangle oqo', sin. ^ o'oq \ (cos. a + cos. a'), which is. therefore

Note 2, Problem YII., where the symbols are defined. Q. F. 1.

BO to AD ; through o' draw o'go parallel to pb, meeting OB m 0; make CH = HB; and join CQ, QB, the latter cutting 00' in G, and the former cutting o'oo"( which is similarly drawn to 00') in g' : then B and C are the starting and closing points of the curve, of which the separate portions are BG, GQG', g'c, d and the chords BG, GQ, QG', g'c are all equal.¹

Q'\

a.....

>G'

ry

2. Calculation.—Take bh = HC = VQH (4BO - QH), and BG = GQ = QG' = g'c =

VBO . QH, which chords of the arcs bg, GQ, &c., thus become known; and, since the common radius BO is given, the construction of the curve is obvious.t

Ex. Let the given deviation QH = 2 chains, and the common radius BO = 85 chains; then BH = HC = ^2(340.- 2) = V676 = 26 chains, and BG = GQ = &c. = \^2 x 85 = ¹/l70 = 13-04 chains.

REMARKS ON LAYING OUT THE CURVES IN THE POUR LAST PROBLEMS.

Having in the four last problems given various methods of determining the radii and common normals, indicating the positions of the tangent points of the parts of the compound, serpentine, and deviation curves, the method of laying out the curves themselves by Problems II., III., IV., or V., according to circumstances, will be readily seen, recollecting that when junction-points of curves of different radii occur, as cc', first fig. to Prob. VI., to commence the operation afresh, by using the radii and tangents of the respective portions of the curve.

SECTION IY.

RAILWAY EARTH WORKS.

On setting out the width of ground for a railway.

After the centre stumps of the railway have been put down, which, as before observed, are usually at the distance of one chain apart, the line must next be carefully levelled, and the number of the stumps entered in the Level-Book, in a vertical column; and opposite each number, in a second column, the depth of the cuttings or embankments (see Level-Book, page 441); and in a third column, the computed or horizontal half-width of the surface cuttings as found by Problems I. and II. following; the depths of the cuttings and embankments being estimated from the balance line, which is 2 feet below the line of the rails or gradients, the 2 feet being filled up with gravel to form the way, and the beds for the sleepers of the rails.

The side stumps are next to be put clown, which must be placed two on each side of every centre stump, in a direction perpendicular to the length of the line; or, if the line be curved, in a direction perpendicular to the tangent to the curve at the centre stump : the two interior stumps, i.e., those next to the centre one, to mark the width of the cuttings, and the two exterior ones to mark the ditches of the side fences. The distances of every two of the interior stumps are to be entered in the Level-Book, opposite the number of the centre stump, in the main section, in order to ascertain the quantity of cuttings for the contractor; and the distances of every two of exterior stumps from the centre one are to be similarly entered, to ascertain the quantity of land which will be required from each proprietor for the works of the line; which last may be calculated accurately by the method of equidistant ordinates. (See Problem YI.)

PROBLEM I.

To set out the width when the surface of the ground is laterally on the same level as the intended railway.

From the centre stump, perpendicular to the direction of the line, set out half the bottom-width for the cutting, to which add the width of the side-fence, putting down a stump at each distance; then repeat the operation on the other side of the line.

Ex. If the bottom-width of the railway be 30 feet, and the breadth of one of the side fences be 12 feet, required the widths for cutting and for fences.

30 -i* 2 = 15 ft. = dist. of side-stump from centre for cutting.

15 +12 = 27 ft. = dist. of side-stump from centre for fence.

Therefore, 15 x 2 = 30 27 x 2 = 54

= whole width for cutting.

and

= whole width for fences.

PROBLEM II.

To set out the width of cuttings or embankments, when the su/rface of the ground is laterally level, and at a given height above the level of the intended railway, the ratio of the slopes being given.

Let ABCD be a r. cross-section of the cuttings, RS the horizontal surface of the ground, AB the bottom width, AC, BD the slopes, Mm = Aa = bb the perpendicular depth, and M the middle stump. Multiply the given depth Mm by the ratio of the slopes, to which add half the bottom-width Am, or UM: set out this distance from M to c for the half-width of the cuttings, to which add the width of the side fence for the whole half-width, repeating the same operation on the other side of M.

Ex. If the bottom-width AB or ab be 30 feet, the depth of the cuttings 28 feet, and the ratio of the slopes 1J : 1, and the width of one of the side fences 12 feet, required the width of the cuttings, and of the land for the works of the railway.

= 42 + 15 = 57 feet = MC = md = width of

~2

(28 x 1§) + ³⁰

cuttings; and 57 + 12 = 69 feet = MR = MS = | width of land. The doubles of which are the whole widths.

If io = bottom-width = ab, a = depth of cuttings = Mm, / = width of one of the side-fences = RC, and the ratio of the slopes r : 1. Then

1: r :: a : aC = ar, hence ar + J iv

= MC = MD, ar + \ w + / = MR = MS, and 2 {ar + \ w + /) -w + 2ar + 2/= RS. Construction.—Draw ab = given width = 30 feet, perpendicular to which, at its middle point m, draw mM = given depth = 28 feet; through M parallel to AB draw RMS; through A and B draw A a, b5 parallel to Mm; make aC = to = 1£ times Mm, join AC, BD, and make

RC, ds = width of one of the side-fences = 12 feet. Then abdc is a cross section of the cuttings, CD the surface-width thereof, and RS the whole surface required for the railway.

By inverting the last figure, so that ABDC may represent the cross-section of an embankment, it will be readily seen that the same method will apply, for setting out its half-widths, MC, MR, as that just given for the cuttings; as the dimensions are the same in both cases.

Note 1. The ratio of the slopes is the proportion of the horizontal line ca to the depth Aa or Mm. Thus, when the ratio is 1 : 1, ca = Aa: when the ratio is 2 :1, ca—^Aa, &c. This ratio depends on the nature of the material through which the cuttings are made; if it is close-jointed hard rock, the ratio is £ or 4 to 1 ; if soft, or loose-jointed rock, or strong clay, the ratio is 1 or 1J to 1 ; if springy ground, or loose sand, the ratio is 2 or 2J to 1.

2. The computed half-widths, in the third column of the Level-Book (following Problem Y.), are found by this Problem.

PROBLEM III.

To set out the width of the cuttings, when the surface of the ground is laterally sloping, the height of the centre stump above the level of the intended railway, the ratio of the slopes, and the lateral fall or rise of the ground in a given horizontal distance being given.

C    ......

D'

Let abdc be a cross-section of the cuttings, RS the sloping surface of the ground, AB the bottom-width, Mm the depth of the cuttings, M the middle stump, AC, BD the slopes, and c'd' a horizontal line passing through M, MC⁷ = MD' being the computed half-widths.

Having set the level so that by turning the telescope two or three chains’ length of the line may be seen, if possible, on both sides of it, place a levelling-stalf at M, and another at q, and observe the level readings on them, the difference of which will be pq; measure with the tape-line in feet the distance Mq on the slope and the horizontal distance mp; take the computed half-width from the Level-Book, or find it by Problem II., and multiply it by the distance Mg' on the slope, and reserve the product. Add and subtract the product of the difference of the staff-readings, and the ratio of the slopes, to and from the horizontal distance Mp, and reserve the sum and difference ; divide the reserved product by the reserved sum for the corrected

half-width MC, and by the reserved difference for the corrected halfwidth MD.

Ex. Let the depth of the cuttings at the centre-stump M be 20 feet, the bottom width AB = 30 feet, the distance on the slope = 25 feet, the distance Mp on the level = 24 feet, the difference of the stave readings pq = 7 feet, and the ratio of the slopes 14 : 1 ; required the corrected half-widths MC, MD.

30

20 x 1| + — =45 feet = computed half-width.

'A

25

1125 reserved product.

Cl r—*

7 x n = oi

reserved sum 34|)1125(32,6 feet = cor. \ width MC reserved diff. 13|)1125(83-33 feet = cor. £ width MD.

Put MC' = h, pq = h, mq = s, M.p = l, MC = x, MD = x', and the ratio of the slopes r : 1.

Then x = hs

l + rh

— MC*

and af =

I - rh

= MD.

If the numbers in the preceding example be substituted in these formulae, the work will stand thus :

45 x 25

l + rh 24 x (7 x 1£)    34|

45 x 25

24 - (7 x 1£) “ T3f ~ to each of which values of the corrected half-width, the width of

MC =

MD =

bs

hs

b — rh

1125    2250

69

1125    2250 = 83.33f

27

* Demonstration.—Draw cc X to c'd'; then by similar triangles, s : h : : x : — = , and mc = c'm ~ c'c—b-^¹⁰⁰; again, by similar triangles

cc, hence c'c =

rhx

.iife

(bs-rh.) ₌ hx whence x-

Is

bs

l + rh

= mc ; and in a

similar manner is found x' —

bs

I - rh

= md. Q. E. D.

Cor. When the difference of the level readings is so large that the horizontal distance l cannot be conveniently measured, the value of l = V ss - h² must be substituted, thus giving

MC -

bs

Vs² - A² + rh

, and md =

bs

V S² - A² - rK

one of the side-fences must be added for the whole breadth of land required.

Note 1. When the difference of level-readings is small, l may be taken = s without material error.

2. A little practice will make this method of finding the corrected half-"widths very easy. It is evidently much superior to the method of finding the same things by the blundering approximations often in use, especially where the difference of the levels of c and d is very considerable, and the surface of the ground has a regular slope, which often happens to be the case, or very nearly so; or, if the surface be a little uneven, a skilful engineer, by applying the tape-line in the general direction of the surface of the ground, will be readily able to correct the difference pq of the level-readings, so as to adapt it to his purpose. If, however, the surface of the ground be very uneven, which sometimes happens to be the case, it will be best to use the method of approximation, as in Problem Y.

Construction.—Draw the cross-section abd'c' as in the preceding Problem; make Mp = given horizontal distance; draw pq perpendicular to c'd' and equal to the difference of the level-readings; and through draw RS, cutting AC' in C and BD' prolonged in D : then ABDC is the required cross-section of the cuttings, RC, DS being the breadths of the side fences.

By reversing the cross-section, it will be seen that the preceding calculation and construction will also apply to an embankment; but in this case the lesser distance mc must be measured up the slope, and the greater md down it.

PROBLEM IY.

To set out the widths when the surface of the ground is laterally sloping, and when the cross section of the works consists partly of a cutting and partly of an embankment, the things given being the same as in the preceding Problem.

Let acpbd be a cross-section of the works of a railway, consisting

p. ...a.. c'

i aJA

O'

partly of the cutting BPD, and partly of the embankment APC; AB the bottom-width ; M the centre stump, and Mm the depth of the cutting; RS the sloping surface of the ground, &c. When the embanked part, apc, is less than half the bottom-width Am, proceed to find, as in Problems II. and III. the computed half-width MD/, and from thence the corrected half-width md. But to find the corrected half-width MC (which consists

partly of an embankment), multiply the distance md (just found) by the difference of the bottom-width and the estimated half-width, and divide the product by the estimated half-width, and the quotient is the corrected half-width MG.*

Ex. Let the bottom-width ab = 30 ft., the depth Mm=3 ft., the ratio of the slopes 1| : 1, and the difference of level-reading 7 ft., at the distances of 25 and 24 ft. from the centre-stump on the surface-slope, and on the level respectively; required the corrected half-widths MD and MC.

OA

By Problem II. 3 x 1| + — = 19* ft. = estimated half-width md',

By Problem III. —— = ^1^ = 36'll ft. — corrected half-24-(7 x li) 27

width MD.

19*

By Problem IV. —_...L-    = 19 *44 ft. = corrected half-

width MC.

If the same symbols be used for the given parts in this example, as in Problem III., and w = bottom-width: then

= 36-11 ft. = corrected half-width.

MD =

Is _ 19* x 25 l-rh 24 -(7xli)

MC

= -yG9|)25 = 19-44 ft. = corrected half-width. l-rh 24-(7xlJ)

Note.—This operation for finding mc, it will be seen, is different from the preceding one. See Demonstration.

Construction. The operation for this Problem is the same as that for Problem III., except that AC is drawn parallel to bd.

By reversing the cross-section, it will be readily seen that the same calculation and construction will apply in the case of apc being a cutting, and pbd an embankment, observing that the corrected half-width for the cutting is the shorter distance, and vice versd.

P * Demonstration. Draw a c J_ to c'd' prolonged, and let pq, which is J_ to pc'o', be the difference of level-leadings at m and q; then, if a — iim, uc = Am — ^w— $ bottom width, the other symbols being the same as in the Demonstration to Problem III., ar~co'and c'ai=Me - cc'=\w - ar ; whence, by the Demonstration

above referred to, mc =    : p>ut & — ^w + ar, or ar—b-\w; this value

l-rh

being substituted in that of mc, gives⁸- - -

l — rh

x which is the rule given in words at length. Q. E. D. b

= MC. Also

s (to — b) l-rh ’

= MC =

bs

l -rh

PROBLEM Y.

To find the width of the cuttings when the surface of the ground is laterally very uneven.

C' c

PY!

It~p c

Let abdmc be a cross-section of the cuttings, Mm the depth, c'm = MD' = estimated half-width, CDM the uneven surface of the ground, &c. The solution of this Problem will be best effected by giving an example in numbers.

Let ab = 30 ft., Mm = 30 ft., and the ratio of the side-slopes as 1|- : 1; then mc' = MD'= (30 x 1|) + (1 x 30) = 60 ft. Measure from M horizontally the distance Me? =60 ft.; the point d is directly above d'. Place levelling-staves at M and d, and observe the difference of the readings at M and d, which, in this base, is 7 ft.; whence 7xll = 10| ft. = approximate distance dD, and Md + dD = D = 60 + 10| = 70J ft. Now place a levelling-staff D, and observe the reading, which is found to be 7'8 ft. greater than that at M, or 0'8 ft. greater than that at d; whence 0⁻8 x l£= P2 ft., and consequently 70|- x 1*2 = 71‘7 ft., which is a still nearer approximation to the true distance MD or Mj, it being measured horizontally. The operation for finding Me, or MC measured horizontally, is the same as the preceding, excepting that the product of the staff-readings is subtracted from the estimated half-width, &c. In this manner the distance Me = horizontal distance MC is found to be 52’6 feet.

Note 1. The widths ec, ds, of the side-fences, must be added to the above results for the whole width.

2.    If the difference of the staff-readings at m and d be very large, it will require three or four approximations similar to those given in the preceding example to find the true corrected half-width.

Construction.—Take the levels of the several undulations of the surface CMD, making c'md' the datum-line, and draw the cross-section ABDMC by the methods already given.

By reversing the cross-section ABDMC, its application to an embankment is obvious, observing also to reverse the distances MC, md, as previously noticed.

3.    The truth of the method of approximation used in this example is too obvious to require a demonstration.

Level-Book.

No. of Stump.

Depth of Cuttings or Embankments.

Computed

'Half-width.

Corrected Half-widths for Edge of Cutting or Foot of Embankment.

North.

South.

Whole Widths including Fences, each Nine Feet in Width.

Feet.

Embankt. Cutting.

216

217

218

219

220

221

222

223

30-00

3-00

28-00

19- 68

20- 00 16-08 30-00 32-18

Feet.

60-00

19-50

57-00

44- 42

45- 00 39-12 60-00 63-27

Feet.

52-60

19-44

57-95

44-42

32-60

39-72

54-24

63-00

Feet.

71-70

36-11

56-08

44-42

83-33

5912

68-05

63-62

Feet.

142-30 73-55 133-55 106-82 133 93 116-84 140-29 144-62

Note.—The depths of cuttings or embankments in the 2d column of the preceding Level-Book are found by calculation, or by carefully measuring them from the section by the vertical scale, but the latter method is not sufficiently correct. The computed half-widths in the 3d column are found by Problems I. and II. The corrected half-tvidths in columns 4th and 5th, by the five preceding Problems, according to the nature of the cuttings or embankments.

PROBLEM VI.

To find the quantity of land required for a projected railway.

Case I.

In p/reparing the preliminary estimates for a projected railway, the quantity of land required for the purpose is usually found without paying any regard to the lateral inclination of the ground, by taking a considerable length of the section at once, especially if the surface thereof have a regular rise or fall, and by measuring the depth of the ends of such length with the vertical scale.

Rule.—Find the surface-widths, fences included, at each end of the given length, by Problems I. and II., add them together, multiply the sum by the length in chains, and divide the product by 1320 for the area in acres.

Ex. Let the length of the sectional surface be 18 chains, and the depths at the ends 22 and 38 feet, the bottom-width of the railway 33 feet, and the ratio of the slopes 11 to 1; required the area of the surface, the width of the side-fences being 9 feet each.

By Prob. II. w + 2ar + 2!/= 33 + (3 x 22) + (2 x 9) = 117 w + 2br + 2/= 33 + (3 x 38) + (2 x 9) = 165

282 18

1320

A. R. P.

12 507-6 11 42-3

3-84545 = 3 3 15

or by putting l = given length, and taking half the sum of the widths, there will result (tv + ra + b + 2f)l

18

660

= (33 + 1| x 22 + 38 + 2 x 9)-^ = 3-84545 acres.

Case II.

When the exact quantity of land for the railway is required.

Rule.—Take the whole widths at the end of every chain, from the 6th column of the Level-Book, for the several widths; add continually together the first and last widths, and twice the sum of all the intermediate widths, and divide the whole sum by 1320 for the area in acres.

Ex. Required the area corresponding to the several widths in the Level-Book at the end of Problem Y.

73-55

133-03

106-82

133-93

116-84

140-29 704-46

1408-92 = twice sum of intermediate widths 142-30 = first width 144-62 = last width

1320

12 169-584 11 14-132

1-28473 = 1A. lR. 5|p. =area required.

Note.—It is very common in practice to find the areas of the quantities of land required from the several proprietors, by actual measurement from the two chain maps, made for the use of the contractors, after the several widths have been laid down thereon : copies being taken, at the same time, from the maps, on tracing-paper, showing the position and quantity of land required from each proprietor.

ON RAILWAY CUTTINGS IN GENERAL, AND TABLES FOR FINDING THEIR

CONTENTS.

In preparing the preliminary estimates for a railway, the contents of the cuttings are usually found by tables for the purpose, the surface of the ground in the several cross-sections being assumed to be on a level with the centre of the line. But when power has been granted for constructing the line, the cross-sections are carefully taken at the end of every prominent variation of the surface of the ground, or, if consistent with accuracy, at the end of every one, two, or three chains in length; and the several cross-sections are then plotted on a large scale, which may be done by the methods given in the preceding Problems. Their areas may then be found by actual measurement • or they may be reduced, where the surface of the ground is uneven, to horizontal sections. The contents are found, either by taking the mean of every two succeeding sections, ivliich method is very erroneous where the areas of the sections differ greatly, or by finding the contents from the tables, by using the mean depths of the several sections, which method is correct; but the mean depth used in this method cannot be accurately found in many cases without considerable calculation. Some use a mean of the mean depths as the bases for a mean area, which method is also very inaccurate, especially where the areas of the extreme sections differ greatly. The magnitude of the errors in both cases will be pointed out in the investigations at the end of these Problems.

On Earthwork Tables.

Tables for this purpose have been published by Sir John M‘Neill, Mr Bidder, Mr Bashforth, Messrs Sibley and Rutherford, and others; all of which are well adapted for finding the contents of • cuttings, assuming the surface of the ground to be laterally level with respect to the direction of the cutting. But none of these tables are accompanied with directions for finding the contents from sectional areas, he., from the areas of working drawings, excepting Mr Bash-forth’s tables; but his mathematical investigation of the rule for using them in finding the contents from working drawings, where the surface of the ground is laterally sloping or uneven, is founded on a false assumption, and therefore his results are erroneous, and especially so where the sectional areas differ considerably. (See the investigations at the end of these Problems.)

THE GENERAL EARTHWORK TABLE.

(At the end of the booh.)

This Table, with the help of the Auxiliary Earthwork Tables, Nos. 1 and 2, on the same sheet, possesses the advantage of being general for all varieties of slopes and bottom widths in common use, as well as for decimal parts of feet in the depths. It may also, with a very trifling preliminary calculation, be made to extend to every variety of bottom width and ratio of slope that can occur, if even the slopes of the two sides differ in the same cutting : and with the help of a table of square roots it will apply, with all attainable mathematical accuracy, to cuttings where the surface of the ground is uneven. The investigation of the method of forming the Tables and using them, will be given at the end of these Problems. The contents in the General Table, and those in Table No. 2, are calculated to the nearest unit for one chain in length, and checked by differences; the side-slopes being assumed to be extended till they intersect. The auxiliary Table No. 1 gives the depths of the intersections of the side slopes below the balance-line, and the corresponding number of cubic yards to be deducted from the contents for each chain in length.

PROBLEM VII.

Case I.

To find the contents of cuttings by the General Earthwork Table, and the Auxiliary Table No. 1, at the end of the book

Let ABbdcG be a cutting, ab = ab = bottom width on the formation level, MM/ and mm' the perpendicular depths at the middle of the two ends of the cutting; AC, bd, ac, bd, the side-slopes, which, being prolonged two and two will meet at the points N and n; also MM' and mm', being prolonged, will meet at the same points. The distance m'n = ran in feet and decimals is given in the Auxiliary Earthwork Table No. 1, for all bottom widths and ratios of slopes in common use, at which distance a line must be ruled on the section, parallel to the balance-line, or at the same distance + 2 feet from the line of the rails, in which latter case the balance-line need not be drawn. From the line thus ruled, the depths of the cutting must be measured to adapt them to the General Earthwork Table; or a mark might be made on the vertical scale with Indian ink (which is easily washed off) at the same distance, which mark might then be applied to the line of the

rails in measuring off the depths. For measuring the depths of embankments, the line must be ruled at the same distance—2 feet above the line of the rails. When the several quantities of a cutting or embankment have been taken from the Table, and their sum multiplied by the ratio of the slopes, the cubic yards to be deducted for each chain in length, for the particular bottom width and ratio of slopes, must be taken from Table No. 1, and multiplied by the whole length of the cutting, and the product, being subtracted from the result obtained from the General Table, will give the content of the cutting in cubic yards.

The method of using the Tables will best appear from the following examples.

Ex. 1. Let the several depths of a railway cutting to the intersection of the slopes, at the end of every chain, be as in the annexed table, the bottom width 30 feet, and the ratio of the slopes 1| to 1; required the content of the cutting in cubic yards.

Note.—By the Table No. 1 the depth to be added to the depths of the cutting, for bottom width 30 ft. and ratio of slopes 1 ^ to 1, is 10 ft., therefore the line from which the depths in the annexed table are measured is 10 + 2 =12 ft. below the line of the rails.

The corresponding number of cubic yards to be subtracted is carried to two places of decimals, or, if the nearest whole number had been taken, the quantity would have been in excess or defect by several cubic yards, when the cutting is of a considerable length.

Ex. 2. Let the several depths to the intersection of the slopes and their distances be as in the annexed table, the bottom width 30 feet, and the ratio of the slopes 1J to 1; required the content of the cutting.

Note.—When any of the distances is greater or less than 1 chain, the corresponding quantity must be multiplied by that distance; as in the cases of the distances between the depths 45 and 50, and between 30 and 10, the former distance being 3 chains and the latter 60 links.

Dist. in Chains.

1-00

2-00

3-00

4-00

5-00

6-00

9-00

10-00

11-00

11-60

Dist. in Chains.

Depths

Qts. per Table.

1-00

2-00

3-00

4-00

5-00

6-00

10

29

32

33

39

35

10

1003

2276

2582

3175

3350

1365

Bor slope 1 to 1 . . 13751 ^ to 1 . * 6875-5 l|to 1 .. 20626-5 = 2200

Subtract

366-67x6

Content in cubic yds.

= 18426-5

Depths

10

20

16

25

32

39

45

50

40

30

10

Dist. greater than 1 Chain.

3 x 5520

•60 x 1059

For slopes 1 to 1 .

\ to 1 .    .

ljtol . . 366-67x 11-60 . Content in cubic yards

99

99

Qts. per Table.

570

795

1044

1996

3091

4319

16560

4971

3015

635-4

.    36996-4

.    18498-2

.    55494"6

= 4253-4 .    51241-2

Ex. 3. The depth of a cutting to the intersection of the slopes and their distances in feet are as in the annexed table; the bottom width is 36 feet, and the ratio of the slopes 2 to 1; required the content of the cutting in cubic yards.

Note.—When the distances are given in feet, the quantities from the General Table must be multiplied by their respective distances; also the quantity from Table Ho. 1 must be multiplied by the whole distance, and the final result divided by 66, as in the annexed example. See Demonstration at the end of these Problems.

Dist. in Feet.

Depths

Qts. x by Length.

Products.

o O CO CO OCON

H H H

39

621000

664752

419400

For slopes 1 to 1 .    . . .    1705152

99

2 to 1 .    .    .    .    3410304

396 x 278 = 110088 66) 3300216 50003

Content in cubic yards .    .

Case II.

To find the contents of cuttings by the Tables, when the depths are given in feet and decimals of feet.

Rule.—Let any two succeeding depths be denoted by a and b, and let the decimal parts of the depths be respectively denoted by a' and V; find the quantity corresponding to a and b from the General Table, as in the former case; then

2a+ b

-jq-} or its nearest whole number, and the decimal a’ will show the

or its nearest whole number, and the decimal V will show the

number of cubic yards to be added in Table No. 2, and 2b + a

~To~’

cubic yards to be added in the same Table.

Ex. 1. Let the depths to the intersection of the slopes be 61-6 and 39'4 feet, their distance 1 chain, the bottom width 36 feet, and the ratio of slopes 2 to 1; required the content of the cutting in cubic yards.

Put 61 =a, its decimal ⁻6 = «' 39 = b, its decimal A = b'.

For slopes 1 to 1

6334

Then the depths a and b, per General Table, give 2 a + b

6210

10

= 16T, or its nearest whole No. 16 and {a') -6 per

78

Table No. 2 .

2& -f- Qj

10

— 13-9, or its nearest whole No. 14 and (b) ⁻4 per )

46

Table No. 2 .

. 12668

For slopes 2 to 1    »

By Table No. 1, for bottom width 36 feet and ratio of slopes 2 to 1, No. of cubic yds. to be deducted

Content in cubic yards

0/» i    t

Note 1. Care must be taken to use the decimal a' with —V, and the decimal

396

12272

10

b' with

2 b -j- ct

, in finding the quantities in Table No. 2, as a mistake might easily

be made in this matter, which would lead to an erroneous result.

2. If any two succeeding depths be nearly equal, and the sum of their decimal parts be together equal to 1 foot, or nearly so, by adding 1 foot to the lesser depth, and rejecting the decimal in the larger depth, and using the depths thus altered as whole numbers, a result sufficiently correct will be obtained ; as in the following example.

Ex. 2. Let the depths of a cutting be 5029 and 48*7 feet, their distance 1 chain, the bottom width 33 feet, and the ratio of the slopes 1| to 1; required the content.

5990 for 1 to 1

By adding the decimal -29, in the larger depth, to 48-7, the depths may be called 50 and 49, for which, by the General Table, the content is

From which deduct, from Table No. 1

2995 for J to 1 8985 for 1J to 1 443-67

854U33 cubic yards.

Note 1. The results of all the examples in the two preceding cases only differ by a very small fraction of a cubic yard from the true contents obtained by calculation from formula (1), page 456. The method of finding the contents to decimals or tenths of a foot in the depths has been particularly discussed, on account of its utility in finding the contents for actual contract-work from the working drawings, in which, as great accuracy is required, the contents should be found to two places of decimals, or to xJ-j-ths of a foot in the depths, as in Case II. of the following Problem.

2. When one or both of the given depths exceed the limits of the table, find the content corresponding to half the two depths, and four times the result will be the content required.

PROBLEM VIII.

Case I.

To find the content of a cutting between two cross-sections, the areas of which, in the intersection of the slopes, the length, the bottom width, and the ratio of the slopes, are given.

Rule.—Find the square roots of the given areas either by a table of square roots, or by actual extraction ; with these roots, as depths, proceed to find the content from the General Table, as in Problem VII., from which deduct the quantity corresponding to the given bottom width and ratio of slopes from Table No. 1, and multiply the remainder by the length for the content.

Note.—If the length be given in feet, multiply the content found by the above rule by the feet, and divide by 66 for the content.

Ex. 1. Let the areas of the two ends of a cutting be 5141 and 1444 square feet, the bottom width 30 feet, the length L60 chains and the ratio of the slopes 1| to 1; required the content of the cutting in cubic yards.

i)

Let the annexed figures be the two cross-sections, the side slopes CA, DB; ca, db, a being prolonged till they meet at N and n, the area CND = 5141, the area cnd = 1444; and since the bottom width AB = ab, and the ratio of the slopes are given, the solidity of the prism, the ends of which are ABN = abn, is given in Table No. 1, and is to be deducted from the content found by the General Table as in Problem VII.

The square roots of 5141 and 1444 are By General Table, for 71 and 38 (71 x 2)+ 38

71-7 and 38 7483

By Table No. 2, for

Yq =18 and-7

103

Content to intersection of slopes By Table No. 1, for bottom width 30, and slopes l|to 1 366-67. Content for 1 chain in length

7586

7220-33

1-60

11552-528 cubic yards.

Content for 1-60 chains in length

Case II.

When great accuracy is required, especially in the measurement of contract work, the second decimals, or -j^ths, of feet, must be considered in the calculation by taking for them -j-^th of their responding quantities in Table No. 2.

Ex. Let the areas of the several cross-sections to the intersection of the slopes, and their distances, be as in the annexed table, the bottom width 36 feet, and the ratio of the slopes 1J to 1; required the content of the cutting in cubic yards.

Dist in Chains.

Areas in Sq. ft.

V Areas.

Product.

1-00

2 00

4-20

6 00

8-00

9 00

2161

3759

5141

4100

4221

3136

2727

56

fcO    O ■<! Co Or

fcO “<1 CO I—i O

11271 x 2-20 10170 x 1-80 8959 x 2-00

From Table No. 1 for bottom width 36 ft. and slopes 1^ to 1, 528 x 9'00 Content in cubic yards.....

cor-

Contents.

7147

10832

24796

18306

17918

7160

86159

4752

81407

PROBLEM IX.

To find the content of a railway cutting when the slopes of the two sides

are different.

Case I.

When the depths are given.

Rule.—Find the content by the tables, as in the two preceding Problems, multiply the result by half the sum of the side slopes, and from the product deduct half the sum of the cubic yards, corresponding to the given bottom width and the ratios of the slopes of the two sides, multiplied by the whole length of the cutting, and the remainder is the content in cubic yards.

Ex. Let the depths and their distances be as in the annexed table, the bottom width 36 feet, and the ratios of the slopes of the two sides 1 to 1 and 1|- to 1 • required the content of the cutting.

Note.—Here the half sum of the first terms of the ratios of the slopes, viz., ^ (1 + 1J) = 1J is the number by which the sum of the quantities, per tables, is to be multiplied, or | added, as in the operation : and half the sum of the cubic yards, for the given bottom width and ratios of the two slopes, is multiplied by the whole length of the cutting, and deducted, agreeably to the rule.

Dist. in Chains.

Depths.

Qts. per Tables.

1-00

2-00

61

52

44'46

7820

5473

Content .

15296 25

Case II.

When the sectional areas are given.

Rule.—Find the content from the tables as in Problem VIII. and deduct for the quantity below the cutting, as in Case I. of this Problem.

Note.—This rule seems too obvious to require an example.

PROBLEM X.

To apply the General Table and Table No. 2 to such bottom widths and ratios of slopes as are not found in Table No. 1.

Put w = bottom width, and r to 1 the ratio of slopes.

Then-^- = feet to be added to depth of cutting, or distance from

bottom of cutting to the intersection of slopes. ll?c²

And ----- = cubic yards to be deducted for each chain in length.

Ex. If the bottom width be 28 feet, and the ratio of slopes 1^ to 1; then iv = 2r = 28 ~ (2 x 1^) = 111 = 11*2 feet = distance from bottom of cutting to intersection of slopes, and 11 w² -f 18r = 11 x 28²-fl8 x 1^ = 383’29 cubic yards to be deducted for each chain in length.

PROBLEM XI.

To find the content of the cutting of a tunnel.

1.    When the length is given in yards, and the width and height in

feet.

mean

Rule.—Multiply continually together the length, width, and height, and divide the product by 9.

2.    When the length is given in chains, and the width and height in

feet.

Rule—Multiply the continued product of the length, width, and mean height by 22, and divide by 9.

Ex. The length of cutting of a tunnel is 1053 yards, its width 30 feet, and mean height 32 feet; required the content.

= 11232 cubic yards.

The following example shows the magnitude of the errors of several methods, sometimes used, for calculating the contents of cuttings.

Ex. The areas of the two cross-sections of a cutting, to the

1053 x 30 x 32

intersection of the slopes, are 10324 and 400 square feet, their distance 4 chains, the bottom width 36 feet, and the ratio of the slopes 1 to 1; required the content of the cutting by the General Table, &c., and by the erroneous methods sometimes used.

Note.—The great difference of the sectional areas in this example shows very prominently the errors of several methods of finding the contents of cuttings : at the same time it is proper to remark that similar differences in the sectional areas very frequently occur in practice, and that the erroneous methods give the contents very near the truth only when these areas are nearly equal.

10324 = 101-607 V400 = 20 Deduction from Tab. No. 1 Content for 1 chain in length

| Content by General Tab.

&c. 10394

792

9602

.    38408 cub. yds.

True content for 4 chains in length

By Mr Bashforth’s Method.

(1.) By Table No. 1 the depth below the formation-level to the meeting of the slopes is 18 ft.; hence the area of the triangle below it is ^ (36 x 18) = 324 square feet, which deducted from the given sectional areas, gives 10000 and 76 sq. ft. for the sectional areas, as used by Mr Bashforth.

| Content by Gen. Tab., &c. 8920-4

Whence V 10000_ = 100

V 76 =    8-718

.    35681-6 c.y.

2726-4

Content for 4 chains in length

Error in defect, compared with the true content given above, being above 7-| per cent,, by Mr B.’s method.

(2.) By taking a mean of the areas for a mean section.

00

(10000 + 76) x _ x 4 = 49260-4 cubic yards; which exceeds

the true content by 10852-4 cubic yards, being above 22 per cent, in excess.

(3.) By taking a mean depth the error in defect is just half of the preceding error, or nearly 11 per cent.

OBSERVATIONS IMPORTANT TO THOSE CONCERNED IN THE CONSTRUCTION OF RAILWAYS, ON THE ERRONEOUS METHODS OF CALCULATING THE CONTENTS OF CUTTINGS, WHERE THE SURFACE OF THE GROUND IS UNEVEN; ALSO ON THE METHOD GIVEN IN THIS WORK.

The magnitude of the errors of the methods (1), (2), and (3), in the last example, for calculating the contents of cuttings, where the surface of the ground is uneven, from sectional areas, is strikingly apparent; and since many tables are not accompanied by any directions for their practical application in this particular case, one or other of the last two of these defective methods is still frequently used by engineers and contractors ; thus causing continual disputes concerning the contents in consequence of the different parties using irreconcilable methods, some preferring one and some the other, as giving, in their respective judgments, the true content.

It is therefore the interest of those concerned in the construction of railways to adopt the method given in this work for finding the contents of cuttings, where the surface of the ground is laterally sloping, &c., as no other method combining all attainable mathematical accuracy has, to my knowledge, been yet published : those already published referring only to cuttings where the surface of the ground is either level, or the sections thereof assumed to be reduced to a level, which, in many cases, is a work of great labour to perform accurately. See the following investigations.

INVESTIGATION OF THE CONSTRUCTION AND USE OF THE GENERAL AND AUXILIARY TABLES, AND OF THE ERRORS OF OTHER METHODS.

Let ABDC, abdc, be vertical cross-sections of a railway cutting, the

At

-------------

"li

surface widths CD, cd, being horizontal, AC ca, BDdb, the side slopes, AB = ab the bottom width. Prolong the planes of the side slopes downwards and to the left till they meet in NraV, and also meet the prolongation of the plane CDdc in cv, dv. Because AB = ab, the cross-sections are J. to VN. Let fall the ± NM, nm on CD, cd respectively, bisecting them in M and m, and also bisecting AB, ab in m' and m. Put MN = a, mn = b, Nn = Aa = Bb = l, AB = ab = w, all in feet, and the ratio of the slopes, i.e., cm : mn : : r : 1. Then

A CDN = a²r, A cdn = b²r. By similar figures a : b : : NV : NV -1, whence NV =

a-b

Pyramid vcnd = |nvx a²r.

.....Ycdn - §(nv — l)b²r; whence

Frustum CDNncd = J NV xaV-| (nv -1) b²r = Jr (a² - b²nv + b²l) — (by substituting the value of NV)

JrZ(a² + ab+ b²) cubic feet

. (1)

=.^-(a*+ db + b²) cubic yards

In the General Table r=l, and / = 1 chain = 66 feet; and .*. the solidity

22

• (2)

S = —(a² + ah + b²) cubic yards

wherein a and b have all integral values from 0 to 72.

Auxiliary Table No. 1.

r : 1 : : J w{ = am') : Ni' = ~

whence A ABN = A abn = J- AB x nm' = —

.*. Prism ABXnab = sq. ft. =    = cubic yds.

4r H 108r    J

.    (3)

• (4)

.    (5)

From (3) and (5) the depths to be added, and the cubic yards to be deducted in Table No. 1, are calculated, by taking l — 66 feet, and w and r all the values most commonly used in practice.

1 1•

-yg- = cubic yards to be deducted, Table No. 1.

= cubic yards to be deducted for the length L.

Let 2 = sum of all the solidities, S, s', &c. (per General Table), the

Hw²

length of which is L, and % = from (6), then 2r = sum of solidities for slopes r to 1, from which deduct kxl, and there results 2r - jcl = cubic yards in the whole cutting .

Whence the method of finding the contents of cutting in Problem VII.

vM

’ ' 108r

11 w²l

" 18r

• (6)

• (7)

Cor 1. The content of a cutting having only two given depths is (sr - x)l cubic yards.

Cor. 2. If the ratio of the slopes of the two sides of a cutting be r to 1 and p to 1, and Jc, x the corresponding cubic yards to be deducted. (Table No. 1.)

\ { 2 (r + p) - l(& + x)} cubic yards.

Let in (2) a and b, hitherto supposed to be integral numbers, have the increments or decimals a and /3, so that a and b become respectively a + a and b + (3, which being substituted for a and b, neglecting the squares and product of a and /3, as being comparatively small, - there results:

22

22

22

~{a² + ab + b²)+ ~{2a + b)a + — (2b + a)l3 .

By subtracting (2) from (9), there results the sum of (2a + b)a and ~(25 + a)/3, or

22/25 + a\

V 10 )

• (9)

27

27

/2a + b\

WO—)

10a and —

10/S

. (10)

which are the increments of the formula (2) arising from the addition of the decimal parts a and /3 to the depths a and b, from which Table No. 2 has been calculated, the quantities 2a+b, and 2b + a being divided by 10, to prevent a too great extension of the Table ; and the corresponding factors a and (3 multiplied by the same number, that the product might retain their original value, the decimal points being still affixed to the values of a and /3 in the horizontal line at the top of the Table.

In consulting Table No. 2 the nearest whole numbers to —^

are taken as the small errors, thus resulting, will usually

and

2b + a

~w

balance one another in a long cutting, and can never in any case amount to much.

Let the plane VCD revolve on the side vc, so that the lines CD, cd may incline from their hitherto assumed horizontal position, thus making the side slopes unequal; and let the area CDN = A, and the area cdn = B, the other symbols being the same as before; then

V A : V B : : VN : VN - whence VN =

l Va

Pyramid VCDN = |.VNx a, Pyramid Vcdn = l (vn-1)xb, whence the frustum CDismcd = s = ^vnxa-^ (vn - Z) x b = J (a-bx vn + b l),

. (11)

or s = ^ l (a + B + \ra + b), in cubic feet, and taking l — 66 feet 2

= - (A + B + V A x v b) cubic yards

Therefore the Table must be consulted for depths J A and a/ B, the ratio of the slopes being included; for if a and b be the mean depths for the areas A and B respectively, the slopes being r to 1, then 99

a²r = A, b²r = b and

27

22r

(A 4- B + A X gj B^ = - (cfi +

27

ab + b²).

Therefore in reality the table is consulted for the depths a r —

A, and b J r= V b.

. (12)

If 2' = sum of all the solidities S, s', &c., for depths V A and J B, &c., and length L; then by subtracting xL, as in (7), there results 2' — xL cubic yards in the whole cutting

Whence the Eule in Problem VIII.

.    (13).

Cor. 1. The content of a cutting, having only two given sectional areas, is (s - x) L cubic yards

Cor. 2. Formulae (11), (12), and (13) will evidently hold, if, instead of the straight lines CD, cd, being the surface edges of the cross-sections cdn, edn, of the cutting, these lines be the chords of similar curves forming the surface edges of the cross-sections, and the similarly situated points in the two curves be joined by right lines (which prolonged will all meet in the vertex V), thus forming the surface of the cutting, a necessary condition in taking cross-sections ; or the small inequalities of the earth’s surface between the cross-sections must be balanced, as nearly as can be judged by the eye, so that the surface may fulfil this condition, in order that all attainable mathematical accuracy may be arrived at in finding the contents.

THE ERRORS OF OTHER METHODS OF FINDING THE CONTENTS OF RAILWAY CUTTINGS.

Mr Bashforth, in his investigations for finding the contents of railway cuttings, where the surface of the ground is level, or assumed to be so, includes the prism ABNnba, and afterwards deducts it, which is mathematically accurate. But in finding the contents from sectional areas, where the surface of the ground is laterally sloping,

or uneven, lie altogether leaves out the above-named prism; for he says (Art. 15, p. 11 of his work), “ The only way to proceed in such a case is—find the areas of the cross-sections in square feet, take out of Barlow’s Tables the square root of each, and treat these square roots precisely as if they had been measured heights, excepting that there will be nothing to be deducted for the prism, and no multiplication for the slopes, these having been already accounted for in finding the areas.”

Let a' and b' be the sectional areas CDBA, cdba respectively, as used by Mr B., and Z = n» = length of the cutting (fig. page 452), then according to his method, the content of the cutting is

.    (14)

gj (a'+ Va' x b' + b')

which is erroneous in every case, except where the areas represented by a' and b' are similar and equal. For when the equal areas ABN, abn are omitted, the areas represented by a' and b' are dissimilar,, and .-. the proportion */a' : Vb' :: NY : nv, on which Mr B.’s rule is founded, does not hold, it being true only in respect to the areas CDN, cdn, because the solids VCDN, vcdn are similar. Moreover, if the plane VCD revolve on the side CD so as to bring the lines cd, ab, to coincidence, or almost to coincidence, in the points b and d, the dissimilarity of the planes cdba, cdba will become more strikingly evident, whereas with the addition of the As, ABN, abn, they still remain similar.

Let AABN = abn — a, then a' + a and b' + a are the sectional areas corresponding to A and B in formula (11) of my investigation, whence by that formula the true content, including the prism abnnba, is

(a' + b'+ 2a + Va'

b' + a) cubic yards. But the solidity of the prism is ~ cubic yards. Whence the content of the prismoid or cutting is (a' + b' + 2a + Va' + a X B'

8l

+ a — 3a), or

(a' + b'+ Va'

B' + a — a)?

from which subtract the content according to Mr B. (14), and there results.

81

vV+axB' + a- Va' x b' - a) == error in defect.

Let a' - and b' = 36a, which is a case that may occur in practice, then

al

gy (vA’ + a x b' + a — */a' x b' — K) = gT (2 */135 — 3 — 1) = gy x 2-8007 = error in defect by Mr B.’s method.

+ 3 + 36) =

al

and g-y (A'+ \Ja! x b' + b') = g.

Wm

x 39|- =

whole content by Mr B.’s method.

al

al

.-. whole content : error in defect:: gj x 39^ : gy x 2-8007

2-8007

the error in defect

391 :

100 : 7-g-, or 7| per cent.

(15)

See Ex. at the end of Problem X., where the error of Mr Bash-forth’s method is shown by giving actual areas.

Note.—The errors of this method are not so prominent where the sectional areas approach near to equality, as in the case of the Burnley cross-sections, in the Ex. Art. 20, page 13, of Mr B.’s work: his method, however, is erroneous to a greater or lesser extent, in every case, except where the sectional areas are equal and similar; for they cannot be similar without being equal, while the bottom width remains the same.

¹

Demonstration.—Because qp, Boare parallel, as are also o'o, pb; o'p = bo = (by const.) qo' = given radius, and o'o = pb = 2qo' = 2go' ; .\ go' = go = qo =bo, which are similarly proved to be = g'o' = g'o¹' = co". Q. E. D.

\ Demonstration.—Draw oa perpendicular to BQ, bisecting it in a ; then by similar triangles bo : Ba = Jbq :: bq : qh, whence bq² = 4bo . qh, or bg² = |b^² = bo . QH, or bg = a/bo . qh. Also bh² = hc² = bq² - qh² = 4bo . qh - qh² =■ qh (4bo - qh), or bh — hc ⁼ a/qh (4bo - qh)7 Q. E. D.

Mr Basliforth says, in defence of his method of finding the contents of cuttings from sectional areas, in the “ Mechanics’ Magazine” for Sept. 11, 1847, p. 249, “ In the case of contract estimates, numerous cross-sections ought to be taken; and it matters little whether we start from the intersection of the slopes or the formation-level.” The above-noticed error (15) shows that it “ matters” so much as 7} per cent, in defect. Besides, there is no need of taking “ numerous cross-sections,” as Mr B. recommends, especially where the surface of the ground is laterally sloping like a geometrical plane, or curved like a conical surface, one or other of which cases very frequently occurs, so that cross-sections, taken at a considerable distance from one another to the intersection of the slopes, may be considered as similar, or so very nearly so as not to induce any important mathematical error, which conditions may be easily determined by the eye. Moreover, the expense and trouble of taking “ numerous cross-sections,” plotting them, and finding their areas, are very considerable ; and, therefore, ought to be avoided, together with all erroneous methods, such as Mr B.’s, of finding the contents of cuttings, as his

“numerous cross-sections” only tend to diminish the errors, without wholly getting rid of them.

The error of the method of finding contents hy mean areas.

Let A and B be the areas of two cross-sections of a cutting to the intersection of the slopes, and l its length : then the mean area is \ (A + b), and the content in cubic yards is l x J (A + b) = l x f (A + b); from which subtract the true content, equation (11), p. 455, and there results

st ^ {f (a + b) - (a + b + VAB)} = l (JA - a/b) = error in excess.

Which error is very great when the areas A and B differ considerably. See (2), p. 451.

The error of the method of finding the content hy mean depth.

Let a and h be the depths of two cross-sections to the intersection of the slopes, the ratio of which is r to 1, and 1 = length of the cutting; then |(a + h) = mean depth, -Jr(a + 5)² = mean area, and the content in cubic yards = -fT l x \r {a + If¹ - rl x f (a+ 5)²;

which subtract from equation (1), p. 411, and there results

sy rl (a² + ah + h² - f (a + 5)²} = rl (a - hf = error in

defect:

Which error is very considerable, when the depths a and h differ greatly.

Note.—The errors of these methods, as here shown, are the same as if the areas and depths had only extended to the formation-level, since a common quantity, i.e., the prisfti below the formation-level, is here included, and afterwards excluded, by taking the differences.

SECTION y.

TUNNELLING.

(1.) Previous to setting out the earthwork of a tunnel, the levelling operation must be repeated with great care, and should also be checked hy the method given in Art. (22), Section I., especially if the tunnel pass under a very high summit: for, if the section be incorrect, the gradient or gradients, on which the tunnel is formed, will not meet at the points shown thereon, and thus embarrass the mining operation.

(2.) If the tunnel is formed on one gradient, as bd, Plate XIII.,

the gradient must incline to one of the extremities of the tunnel, as at D, in order to discharge the water generated therein. Strong poles or masts must be firmly fixed on the surface, in the intended direction of the tunnel, of which one must be on the summit of the hill; at which place a temporary observatory is frequently erected, especially if the summit be a very high one, and the tunnel a very long one. Shafts must be sunk at the distance of four or five chains from one another, in the direction of the poles (and observatory, if there be one), in order to ventilate the tunnel, as well as to check the accuracy of the work as it proceeds. If the tunnel be a long one, it would be preferable, if convenient, to form it on two gradients, inclining to its opposite extremities to liberate the water, and thus to aid the mining operation, which is commonly commenced at both ends of the tunnel at the same time.

(3.) When it is necessary to have a curve in the direction of a part or of the whole of the tunnel, that direction must be carefully laid down on the surface, by the methods given in Section III., making allowance for acclivities and declivities, poles or masts being fixed therein, as pointed out in Art. (2), that the shafts may be sunk so as to meet the mining operations of the tunnel, as well as to check their accuracy in point of direction, and this will be the more especially necessary in the curved part of the tunnel.

(4.) The mining operation of the tunnel should commence when the depth of the cuttings at each end is about 60 feet. The width and depth of the excavation of a tunnel, on the narrow gauge, should be about 30 feet each, and must be dug 5 or 6 feet below the intended line of the rails, to give space for the inverted arch and the ballasting, excepting where the excavation is made through rock sufficiently hard to form the side-walls of the tunnel, in which case 22 or 24 feet in width, and about 26 feet in height, will be sufficient, the excavation in this case being terminated below by the balanceline, or formation-level. The depth and width of the excavation for a tunnel on the broad X gauge must, in both cases, be proportionately larger.    £7

The annexed figure is a cross-section of □ the masonry of a tunnel, which, of course, is 3 such as is required where the tunnel is made B ^ through loose earth, only the arch above being required when made through hard rock.

Note.—The remarks at Articles (25) and (27), Section II., refer only to the projection of tunnels on Parliamentary maps.

A TABLE OF THE DIMENSIONS OF SEVERAL EXISTING TUNNELS.

Names of Railways and of Tunnels.

Length

in

Yards.

Width

in

Feet.

Height

in

Feet.

Shafts.

Primrose Hill l Weedon (Kilsby .    .    .

Cromford and High Peak—Buxton Great Western—The Box Tunnel . Manchester and Leeds

London and Bir- j mingham

1120

418

2398

580

3227

22

22

10

North Midland

London and Brighton j

Sheffield and Manchester Chester and Holyhead

South-Western

| Littleborough .

Claycross .    .

Melford .    .    .

Merstham Balcombe j Five tunnels )

{ Total length J Three tunnels Total length f Seven tunnels )

| Total length j

2869

1806

836

1780

1192

6245

2130

1998

eq

CO    i—i I—I

HM (M (M

13

SECTION VI.

VIADUCTS, AQUEDUCTS, SKEW ARCHES, ETC.

What are here given on these subjects are necessarily only outlines, referring to some few existing structures, without entering into the details of specifications, working drawings, &c., which are foreign to the nature of this work, and properly belong to the department of the architect.

Railway Viaducts.—It would be impossible to enumerate the whole of the viaducts constructed since the introduction of railways. Structures in timber, brick, stone, and iron, of various designs, have been erected; and in some cases there is a novelty of principle accompanied with great boldness of execution.

The Brick Viaduct at Maidenhead, constructed by Mr. Brunei for the Great Western Railway, is one of the best examples in that material : it is composed of two elliptical arches spanning the • Thames, each 128 feet, with a versed sine of 24^ feet; the pier between the two arches is 30 feet in width. The arch in the middle is, feet in thickness, which gradually increases towards the abutments. Besides these two grand brick arches, on each bank of the

river are four others ; those on the abutments span 21 feet each, and the six others 28 feet each.

The Viaduct over the Ouse, near York, consists of three arches, each 66 feet span; the piers are 10 feet in thickness, and the width of

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the arches 28-jt feet on the soffite, and their thickness at the keystone 3|- feet, the voussoirs gradually increasing towards the springing.

This is a specimen of a stone viaduct combining great strength and elegance in its construction.

Aqueduct of Pont-y Cysyllte is 4 miles from Chirk, over the river Dee; its length is 1007 feet, and the river is 127 feet below the water-level of the canal carried over it.

To construct an aqueduct upon the usual principles, with piers and arches above 100 feet in height, and of a sufficient breadth and strength to afford room for a puddled water-way, would have been not only expensive but extremely hazardous. Telford, who had already carried the Shrewsbury Canal by a cast-iron trough 16 feet above the level of the ground, formed the idea of doing the same in the present instance, which was approved and finally adopted. The foundation on which the piers are erected is a hard sandstone ; their height above low water in the river is 121 feet: at the bottom they are 20 feet by 12, at the top 13 feet by 7-J. For a height of 70 feet above

the foundations, they are built solid, and the remaining 50 feet hollow, the walls being only 2 feet in thickness, with one cross inner wall; by this means the centre of gravity of the piers is thrown lower, and the masonry economised. The width of the water-way is 12 feet; of which the towing-path covers 4| feet, leaving 7J feet for the boat: as the towing-path is supported by iron pillars, the water fluctuates, and recedes freely as the boat passes.

There are 18 of these stone piers, besides the abutments; the span of the arches is 45 feet, and their rise above the springing 7|- feet, and the total expense of its construction was 47,018/.

This aqueduct almost rivals the works of a similar kind left us by the ancient Romans : the introduction of iron, however, for a watercourse, is a novelty with which they were unacquainted : in this instance it has proved admirably well fitted for the purpose to which it is applied : had the channel been constructed with stone or brick at this great elevation, it would have been less secure, as there would have been a constant danger of leakage.

SKEW OR OBLIQUE BRIDGES.

One of the skew bridges on the London and Birmingham Line, a fine erection of this kind, is 23| feet in height from the surface of the road below to that of the rails, making an angle of 32° with the road: the direct span of the arch is 21 feet, and oblique span 40. The arch, which is 2\ feet thick, is the segment of a cylinder, the internal radius of which is 12J feet, and the versed sine 5f feet. The angle of which the coursing joints of the sofiite cross the axis of the cylinder is 53° 25', and the joints of the face of the arch all converge to a point 32J feet below the axis of the cylinder, and 45 feet below the crown of the arch.

er

The Midland Counties Railway possesses another variety of skewed brick bridge, the span of which is 42|- feet, and versed sine 11 feet;

the arch consists of six ribs, 2 bricks in depth, and 4 feet in thickness, the total breadth of the bridge, measured at right angles to the face, being 24 feet.

Manchester and Birmingham Railway.—In Fairfield Street is a skew bridge, the oblique span of which is 128f feet, with a versed sine of 12 feet; the width from face to face is 31 feet.

Six ribs of iron abut on as many independent walls, and project before each other 13 feet; an ornamental stone parapet and cornice crown this viaduct, presenting a novel and agreeable appearance.

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This bridge, executed after the designs of Mr G. W. Buck, is remarkable for its acute angle, which is 24J°. The weight of the ironwork employed on the six ribs was 540 tons, and the whole was admirably secured together; the remainder of the viaduct is formed with brick arches of 45 feet span.

DIMENSIONS OF VIADUCTS IN VARIOUS RAILWAYS.

In the Grand Junction Railway, at Vale Royal, is a viaduct of stone over the river Weaver, in which are five arches, each 63 feet in span and 60 in height; the length of the viaduct is 456 feet. Where this railway crosses the Mersey and Irwell Canal there is a viaduct of stone having 12 arches ; the two in the centre are 75 feet in span, and the remainder from 40 feet to 12^.

In the Newcastle and Carlisle Railway, near Brampton, is a viaduct, which crosses the public road and the river Gelt, at the height of 80 feet above the bed of the river, over which it is carried in an oblique direction. The arches, which are three in number, are 33 feet in direct span, and are built at an angle of 45°.

In the Birmingham and Derby Junction Line, between Kingsbury and Tam-worth, over the Anker, is a viaduct of 18 arches, each of 30 feet span, and one oblique arch of 60 feet span; its height above the river is 23 feet, and the cost was 18,000Z. In the same line, between Tam worth and Burton-on-Trent, is a viaduct | mile in length, built upon above 1000 piles, driven 15 feet below the bed of the river.

In the Newcastle and Shields Railway there is a viaduct over the Ouseburn of 9 arches, two of which at the ends are of stone, and the others are of timber, resting on stone piers ; the three central arches have each a span of 116 feet, and two others 110 ; the total length of the viaduct is 750 feet, and its height above the water 180 feet. In the same line, the viaduct at Willingdon Dean has 7 timber

arches, five of which span 120 feet each, and the two exterior each 115 feet; the whole length is 1050 feet, and the height 82 feet.

In the Taff Vale Rail-way, near Quaker’s Yard, is a viaduct crossing the TafF, the length of which is 600 feet, and the height above the river 100 feet, having 6 arches. In the same line, at the conflux of the Rhondda and Taflf, is a viaduct having an arch of 100 feet span, and 60 feet in height.

In the London and Brighton Railway, across the valley of the Ouse, is a viaduct 1437 feet in length, and the height varies from 40 to 96 feet; it is formed of 37 brick arches of 30 feet span.

The London and Greenwich Railway is a continuous viaduct of more than 1000 brick arches, each 18 feet span, 22 feet in height, and 25 feet in width. It is 3f miles in length, and cost 266,3221. per mile.

The extension of the South-Western Railway through the metropolis, from the Nine Elms to Waterloo Bridge, is a continuous viaduct, like the last-mentioned one, in which are several strong and elegant oblique iron arches for the purpose of crossing some of the principal streets.

The student’s attention should also be directed to the many magnificent railway bridges and viaducts of iron which have been erected of late years; such as the Britannia and Conway Bridges in North Wales; the High Level Bridge at Newcastle-on-Tyne; and the great Victoria Bridge in Canada by Mr Robert Stephenson; the Windsor, Chepstow, and Saltash bridges by Mr Brunei; the new Charing Cross Bridge by Mr Hawkshaw ; and the Viaduct at Crumlin in Monmouthshire, by Messrs. Kennard. Descriptions of these will be found in engineering works.

I shall conclude this section by recommending to those who wish for scientific and practical information of the first order on this subject, “ The Theory, Practice, and Architecture of Bridges,” by J. Hann and others; “ Practical and Theoretical Essay on Oblique Bridges,” by G. W. Buck, M. Inst. C.E. ; “ Treatise on the Equilibrium of Arches,” by Joseph Gwilt, architect, F.S.A.; Cresy’s “ Encyclopaedia; ” and a “ Practical Treatise on the Construction of Oblique Arches,” by J. Hart.

SECTION VII.

SUPERELEVATION OF EXTERIOR RAIL IN CURVES.

The superelevation of the exterior rail in curves, the radii of which are within certain limits, is absolutely necessary to counteract the centrifugal force caused by the velocity of the train, since all moving bodies have a tendency to continue their motion in a direct line. From this cause the carriages of a railway train are driven towards the exterior rail, and would finally be thrown off the rails,

sect. vii. SUPERELEVATION OF EXTERIOR RAIL.

465

were it not for the conical inclination of the tire and the flanges of the wheels.

Let w=weight of the moving body or train, v=its velocity per second, r = radius of the curve, and g = force of gravity at the earth’s surface; then, by Dynamics, the centrifugal force

(1)

, wv2 / = ——

When R = 1 mile=5280 feet, v=velocity = 60 miles per hour = 88 feet per second, and g=32|- feet, then

/=

= 2¹2-w;

W x 882 32i x 5280

that is, the centrifugal force that urges the moving body to leave the curve, in this case, is of its weight.

This force is slightly counteracted by the conical inclination of the tire of the wheels, each pair of which are firmly fixed on the axle that turns with them. This inclination of the tire, together with the lateral play of the flanges of i an inch on each side, and the centrifugal force impelling the carriages of the train, when moving in a curve, towards the exterior rail, enlarge the diameter of the exterior wheel, and diminish that of the interior, thus causing the train to roll on conical surfaces, which necessarily produces a centripetal force, the centre of which force is the vertex of the cone, of which the increased and diminished diameters of each pair of wheels are sections.

Let d be the outer diameter of the wheels, d the increment and consequently the decrement that the diameters of the exterior and interior wheels respectively receive, through the joint action of the centrifugal force and the inclination of the tire: then under these circumstances the respective diameters of the exterior and interior wheels will be

d x d and d-d

also if R' be the radius of a circle which the centre of the carriage would describe in consequence of the inclination of the tire of the wheels, and b the breadth of the road or gauge : then r' + ■§•& and r' — \b are the radii which would be respectively described by the exterior and interior wheels; and by similar triangles

whence

d + d : d - d : :n' + U :K' -\b, did:: 2r' : b, and

n-bd 2

Or, if 1 = inclination of the tire, and A the deviation of the

wheels, the increment or decrement <5 = A—; and, by substitution,

■ (2)

„ bdn R =--

4A

Also W and v representing the weight and' velocity of the train, as in (1), and g the force of gravity, the centripetal force corresponding to the radius r' will be

f=

wv²

gu'

• (3)

Or, by substituting the value of r' from (2)

■ (4)

wv² 4 A

bdn

Since the forces / and f (1) and (3), act in contrary directions, they will hold each other in equilibrium when they become equal, and the train will cease to have a tendency to leave the curve; this takes place when

wv² _ wv2 gn gpJ

or r = r'

wv² _ wv

Also from (1) and (4)

whence

• (5)

. bdn :

A —- .

4r

which is the deviation required to produce the equilibrum between the centripetal and centrifugal forces of the train. Therefore since r = r', i.e., the vertex of the imaginary cone of which each pair of wheels are sections will coincide with the centre of the curve, there will in consequence be no dragging of either of the wheels on the rail.

In practice, however, it is safer to neglect the effect of the coning of the tires, and to calculate the superelevation of the rail as if the tires were cylindrical.

The rule for this is as follows :—

Let R = radius of curve in chains.

b = breadth of gauge of the line in feet; or more properly, the distance from centre to centre of rails.

V = velocity of the train in miles per hour. x = superelevation to be given to the outside rail in inches. Then

bv²

82*5 r

For the narrow gauge, where b = 4 ft. 11 in. = 4*91,

V² n

nearly-

Example.—On a curve \ mile (= 40 chains) radius it is expected that trains will travel at 40 miles an hour. What ought to be the superelevation of the outside rail %

= 2*35 inches. Answer.

40² _1600 17 x 40    680

TABLES FOE SEDUCTION OF DISTANCES AND COEEECTION OF LEVELS FOE CUEVATUEE, ETC.

No. 3.

No. 1.    No. 2.

Reduction for each Difference between apparent and Difference between apparent and True Chainforthefoilowing True Level for distances in Chains. Level for distances in Miles. Angles of elevation. Correction in Decimals of Feet. Correction in Feet and Decimals.

Angles.

Reduction in Links.

Distances in Chains.

For

Curvature.

For

Refraction.

For Curvature and

Refraction.

Distances in Miles.

For

Curvature.

For'

Refraction.

For Curvature and

Refraction.

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•0008

•0010

•0011

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•0016

•0018

•0020

•0021

•0023

•0025

•0027

•0029

•0031

•0033

•0036

•0038

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•0043

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•0048

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■0054

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16-675    2-382    14-293

20177    2-882    17-295

24012    3-430    20*582

28181    4-026    24-155

32-683    4-669    28’014

37-519    5-360    32-159

42-688    6-098    36-590

48-191    6-884    41-307

54-027    7-718    46‘309

60-197    8-600    51-597

66-700    9 529    57’171

71-520 10-217    61-303

80-687 11-527    69-160

88167 12-594    75‘573

96-000 13-714    82-286

104-167 14-881    89-286

112-667 16-095    96‘572

121-500 17-357 104-143 130-667 18 666 112001 140167 20-024 120'143 150-000 21-428 128'572 160-167 22-881 137-286 170-667 24-381 146-286 181-500 25928 155'572 192-667 27-524 165 143 204-167 29-167 175'000 216-000 30-857 185-143 228-167 32-581 195-586 240-667 34-381 206-286 253-500 36-214 217'286 266-667 38-075 228’592 294-000 42-000 252-000 322-667 46-095 276’572 352-667 50-381 302-286 384-000 54-857 329-143 416-667 58-095 358’572 450-667 64-381 386'286 486-000 69-428 416-572 522-667 74-667 448'000 560-667 80-095 480-572 600-000 85-714 514-286

TABLES OF OFFSETS FOR RAILWAY CURVES, ETC.

No. 4.

Offsets or Ordinates at the end of the first Chain from tangent point of Railway Curves.

No. 5.

Horizontal Distances to an Unit’s Height to the following Angles of Elevation.

Radius of Curve in Chains.

Offsets in Inches and Decimals.

Radius of Curve in Chains.

Offsets in Inches and Decimals.

Radius of Curve in Chains.

Offsets in Inches and Decimals.

Radius of Curve in Chains.

Offsets in Inches and Decimals.

Angle of Elevation.

Hor. Distance to 'one

Perpendicular.

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20    25

29

40

35

10

15

23

32

30    39 5

40    50! 6ll 6

52

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42    51

66

59

55

74

68

64

84

79

IS 74    88 7

89 103 120

105

121

138

156

95

81    90 101 113

123 139 157 177 198 [

1431 160 178 199 221 i 244i H

99

117

108 120 133 147

138

128

160

149

140

172

162

186

154

177

169

164

201

193

218

183 196 211! 2271 245

209

235

222

250

238

266

255

284

274

187

211

237

296 42 323 352

13

257

279

303

328

355

353

382

384

414

265 286 309 333 359 3871 416 447

^ 0^ CO lb. H

14

479

15

514

5501

315

304

294

336

TS 264 279 297

310 328 348 369 201 3261 343 362 382 404

325

358

392

316 340

348

381

372

365

416

407

398

392

428 453

442

433

470

426

462

499

420

455

492

530

451

486

524

563

480 508 539 570 604

CO O C?l M OO

«o —t to to

Vx Cr\

os

CO

516

553

592

633

551

589

629

671

587

627

626

17

667

710

666

707

750

675 714 754 795

706

18

21 359

22

394

377

413

397

433

418

23

24

431

469

451

490

472

512

455

495

535

441

479

519

561

465

504

545

491

531

519

25

509 530 553 578 604

587

573

616

560

602

640

548

590

633

678

579 612 646 682 719 759

622

666

655

711

700

690

746

736

727

783

773

765

821

812

861

805

853

903

631 661 692 724 759 794 832 871 912 954

799

847

895

946

co o to

99]

9981044

26

27

551

594

573

28

29

639

617

662

685

710

596

641

688

736

622

667

715

763

649

695

743

793

30 733 759 785 814 844

31 783

32

33

834

809

677 707

724

773

824

755

739

805

856

788

838

891

876 909 944

772 807

34

887

861

837

35

942

915

970

890

866

944

920

897

1001

975

952

930

964

1032

1008

1066

985

1042

1020

1101

A GENERAL TABLE FOR THE CALCULATION OF EARTH-WORK FOR RAILWAYS, &c.

AUXILIARY EARTH-WORK TABLES.

No. 1.

No. 2.

749 7921 19 793 837 882l_20 838 8831 9301 9781 ²⁴

886 931

979

1027 1078 ²²

935

981

1029

1079

1130

1183

23

985 1032 1081 1132 1184 1238 1293 24 1037 1085 1135 1186 1239 1294 1350 14081 25 1091 1140 1190 1243 1296 1352 1409 1467 15281 26

1529 1590 1652 27 1591 1653 1717 1782 28 1656 1718 1783 1849 1916 1722 1785 1850 1917 1986 20561 30 981 1019 1059 1101 1144 1189 1235 1283 1333 1384 1437 1492 1548 1606 1665 1727 1789 1854 1920 1987 2057 2127'2200l 31

822

873

858

844

910

895

882

926

964

948

935

922

988

975

964

1030

1018

1007

1073

1062

1052

1118

1107

1098

1155

1146 11961247 130013551411

11641212

1203

1262

1254

1313

1306

1366

1360

1421

1415

1477

1472

1469

1531

15351595

1003

1044

1086

1130

1176

1223

1272

1322

1375

1428

1484

1542

1599

1660

100010371076

1078

1137

1057

1115

1095

1176

1155

1135

1216

1117

1195

1177

1159 1203

1238

1220

1257 1300

1282

1265

1249

1345

1327

1311

1392

1296

1375

1359

1440

1345

1423

1409

1489

1396

1474

1460

1541

(1448

1526

1513

1594

1502

1580

1568

1557

1648

1635

1624

161416731733

1692

1682

1751

1741

1705

1762

1822

1811

1802

1873

1883

1946

1795 1859 1924:1991 2059 2129 2201 2274 2349 1865 1929 1996|2063 2132 2203 2276 2350 2426 1937 2002 2069.2137 2207 2279 2352 2427 2504 2010 2076 2144 2213 2284 2356 2431 2506 2584

32

2503

2582

2663

998 1027 1058 1091 1125 1161 1199 1238 1278 1321 1365 1410 1458 1507 1557 1609 1663 1718 1775 1834 1894 1956 2020 2085 2152 2220 2290 2362 2435 2510 2587 2665 2745

33

2662 34 2743 2826| 35 2827 2910129941 36

36    1056 1086 1118 1151 1186 12231261 1301 1343 1386 1431 147711525 1575 1626 1679 1734 179018481907 1969 2031 2096 2162 2229 2299 2369 2442 2516 2592 2669 2748 2829 2911 2995 3081 3168 37

37    1115 11461179 1213 1249    1287 1326 13661409 1453 1498    1546 1595 1645 1697 1751 1806    186319221982 2044 2108 2173 2240 2308 2378 2450 2523 2598 2675 2753 2833 2915 2998 3082 3169 3257 3346 38

38    1177 1208 1242 1277 1313    1352 1392 14331476 1521 1568    1616 1665 1717 177018241881    1938 1998 2059 2122 2168 2252 2320 2389 2460 2532 2607 2682 2760 2839 2919 3002 3086 3171 3258 3347 3438 3530 39

39    1239 1272 1306 1342 1379    1419 1459 15021546 1591 1639    1687 1738 179018441899 1956    2015 2075 2137 2201 2266 2333 2401 2471 2543 2616 2691 2768 2846 2926 3007 3091 3175 3262 3350 3439 3531 3623 37181    ⁴⁰

40    1304 1337 1372 1409 1447    1487 1529 1572 1617 1663 1711    1761 1812 1865 1920 1976 2034    2093 2154 2217 2281 2347 2415 2484 2555(2628 2702 2778 2855 2934 3015 3097 3181 3267 3354 3443 3533 3625 3719 3814l39lll ⁴⁴

41 1370 1404 1440

A Table of Feet to be added to the Depths of Cuttings, and the Number of Cubic Yards to be deducted from the Content of each Chain in Length, for the most common Bottom Widths and Ratios of Slopes.

A Table of Cubic Yards to be added for the Decimal Parts of a Foot, to be used with the general Table for the Calculation of Earthwork, when required.

Ratio of Slopes.

ssssssss

~«o» HC« M(M(Nhh

Bottom Width 27 feet.

Bottom Width 30 feet.

Bottom Width 33 feet.

Bottom Width 36 feet.

Bottom Width 39 feet.

rv(2a+b) or

*(2 b+a)

•1    *2    *3    *4    5    *6    *7    *8 *9

Feet to be added to Depths.

Cubic Yards to be deducted.

Feet to be added to Depths.

Cubic yards to be deducted.

Feet to be added to Depths.

Cubic Yards to be deducted.

4.50    148.50    5.00    183.33    5.50    221.83

5.40    178.20    6.00    220.00    6.60    266.20

6.75    222.75    7.50    275.00    8.25    332.75

9.00    297.00    10.00    366.67    11.00    443.67

13.50    445.50    15.00    550.00    16.50    665.50

18.00    594.00    20.00    733.33    22.00    887.33

27.00    891.00    30.00    1100.00    33.00    1331.00

54.00    1782.00    60.00    2200.00    66.00    2662.00

M M bO M M

to OS ^ 00 to to OS

o o o o 0 © to o | added to Depths. 00000000

Feet to be

Cubic Yards to be deducted.

Feet to be added to Depths.

Cubic Yards to be deducted.

264.00

316.80

396.00

528.00

792.00

1056.00

1584.00

3168.00

6.50    309.83

7.80    371.80

9.75    464.75

13.00    619.67

19.50    929.50

26.00    1239.33

39.00    1859.00

78.00    3718.00

OOOMOSOu^MWM

HOOOOSOS^ibWUlHO

OOOMMOStHibWtOtOH

0O C* CO to

WMMMMMMHMM

*UO)OSCtiMMtOHHO

MMCOtObOtOtOtOtOtOM«->HMMH

ibMHCOQOOSibWHOOOOSClMMO

1477

1643

1599

1516

1689

1736 1785 1836 1888 1942 1997 2054 2113 2173 2235 2299 2364 2431 2499 2569 2641 2714 2789 2866 2944 3024 3105 3188 3273 3359 3447 3537 3628 3721 3816 3912 4010 4109 42

1912 1965 2020 2076 2134 2193 2255 2317 2382 2448 2515 2585 2655    2728    2802 2878 2955 3034 3115 3197 3281 3367 3454 3543 3633 3725 3819 3914 4011 4110 4210 4312 43

1991 2044 2100 2157 2215 2276 2338 2401 2466 2533 2602 2672 2743    2817    2892 2968 3047 3126 3208 3291 3376 3462 3550 3640 3731 3824 3918 4015 4112 4212 4313 4415 4520    44

2070 2125 2181 2239 2299 2360 2422 2487 2553 2620 2690 2761 2833    2907    2983 3060 3139 3220 3302 3386 3472 3559 3648 3738 3830 3924 4019 4116 4215 4315 4417 4521 4626    47321    45

45 1650 1687 1727 1767 1810 1854 1899 1947 1995 2046 2098 2152 2207 2264 2323 2383 2445 2509 2574 2641 2709 2779 2851 2924    2999    3076 3154 3234 3315 3399 3483 3570 3658 3747 3839 3931 4026 4122 4220 4319 4420 4523 4627 4733    4841149501 '46

42

43

1437

44

1507

1472

1542

1509

1577

1614

1580

1547

1652

1619

1587

1692

1660

1734

*C ONN r—< t—■I r—I r—t

1672

1746

1717

1822

1792

1763

1868

1839

1811

1888

1861

19161966

1938

2017

46 1724 1762 1802 1844    1887    1932    1978    2026    2076 2127    2180 2235 2291(2349    2409    2470 2532 2597 2663 2730 2800    2871 2943 3017 3093 3170 3249 3330 3412 3496 3582 3669 3758 3848    3940 4034    4129 4226 4325    4425 4527 4631 4736 4842 4951 5061    5172 47

1800 1839 1880 1922    1966    2012    2059    2108    2158 2211    2264 2320 2377 2435    2496    2558 2621 2686 2753 2822 2892    2963 3037 3112 3188 3267 3346 3428 3511 3596 3682 3770 3860 3951    4044 4138    4235 4332 4432    4533 4635 4740 4846 4953 5062 5173    5286 5400 48

2712 2778 2845 2915 2985    3058 3132 3208 3285 3364 3445 8527 3611 3697 3784 3873 3963 4055    4149 4244    4341 4440 4540    4642 4745 4851 4957 5066 5176 5287    5401 5515 5632    49

2801 2871 2939 3009 3081    3154 3229 3306 3384 3464 3545 3628 3713 3799 3887 3977 4068 4161    4256 4352    4450 4549 4650    4753 4857 4963 5071 5180 5291 5403    5517 5633 5750 58691 50

50    2037 2079 2122 2167 2213 2261 2311 2362 2415 2470 2526 2584 2643 2704 2767 2831 2897 2965 3034 3105 3178 3252 3328 3405 3484 3565 3647 3731 3817 3904 3993 4083 4175 4269 4364 4461 4560 4660 4762 4865 4971 5077 5185 5295 5407 5520 5635 5752 5870 5990l6llll 51

51    2119 2162 2206 2251 2299 2347 2398 2450 2504 2559 2616 2675 2735 2797 2861 2926 2993 3061 3131 3203 3276 3351 3428 3506 3586 3667 3751(3835 3922 4010 4099 4191 4283 4378 4474 4572 4671 4772 4875 4979 5085 5193 5302 5413 5525 5639 5755 5872 5991 6112 6234 6358 *2

......— —    —- —— — —.............. 5095 5202 5310 5420 5532 5645 5760 5876 5995 6114 6236 6359 6483 6610 53

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INDEX.

A bbey b* Overton's English Church History 14

Abney's Photography..............

Acton's Modern Cookery...........

Alpine Club Map of Switzerland

Guide (The).................

Amos's Jurisprudence ..............

-Primer of the Constitution............ 5

-50 Years of English Constitution    5

Anderson's Strength of Materials Armstrong's Organic Chemistry Arnold's (Dr.) Lectures on Modern History 2

-Miscellaneous Works

-—Sermons ................

-(T.) English Literature ....

10

20

17

10

10

IS

15

Poetry and Prose ...    6

Arnott's Elements of Physics

Atelier (The) du Lys .........

Atherstone Priory...............

Autumn Holidays of a Country Parson ...    7

dyre's Treasury of Bible Knowledge

18

18

20

Bacon's Essays, by Whaiely

--Life and Letters, by Spedding ...    5

--Promus, edited by Mrs. Pott

-Works.............................

Bagehot's Biographical Studies.........

--Economic Studies ............

---Literary Studies...............

Bailey's Festus, a Poem..................

Bain's James Mill and J. S. Mill......

-Mental and Moral Science ...

-on the Senses and Intellect ...

--Emotions and Will...............

Baker's Two Works on Ceylon.........

Balts Alpine Guides .....................

Ball's Elements of Astronomy .........

Barry on Railway Appliances .........

-& Braniwell on Railways, &c..

Bauerman s Mineralogy...................

Beaconsfield's (Lord) Novels and Tales 17 & 18 ---—----Speeches

21

18

17

10

10

13

10 & 11

20

10

14

19

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Froude's Caesar

Gairdner's Houses of Lancaster and York 3 Ganot's Elementary Physics

-Natural Philosophy

--Puritan Resolution ...............

-Thirty Years’ War ...............

--(Mrs.) French Revolution ......

----StruggleagainstAbsolute

10

18

18

18

13

10

10

io&ii

Beaconsfield!s{Lord) Wit and Wisdom

Becker's Charicles and Gallus............

Beesly's Gracchi, Marius, and Sulla...

Bingham's Bonaparte Marriages......

Black's Treatise on Brewing ............

Blackley's German-English Dictionary

Bloxam & Huntington’s Metals ......

Bolland and Lang's Aristotle’s Politics

Boultbee on 39 Articles.....................

--'s History of the English Church... 14

Bourne's Works on the Steam Engine...i3&i4 Bowdler's Family Shakespeare

Brabourne's Fairy-Land.........

--Higgledy-Piggledy

-Voyage in the ‘ Sunbeam ’...

Bray's Elements of Morality............

Browne's Exposition of the 39 Articles

Brownings Modern England .........

Buckle's History of Civilisation.........

Buckton's Food and Home Cookery...

- Health in the House.........

Bulls Hints to Mothers..................

- Maternal Management of Children. 21

Burgomaster’s Family (The)

Cabinet Lawyer..................

Calvert's Wife’s Manual......

Capes's Age of the Antonines.

-Early Roman Empire

Carlyles Reminiscences.......

Commonplace Philosopher........

Conder's Handbook to the Bible Conington's Translation of Virgil’s TEneid 19 ---Prose Translation of Virgil’s

Poems.............................................

Contanseau's Two French Dictionaries ...

Conybeare and Howson’s St. Paul............

Cotta on Rocks, by Lawrence ...............

Counsel and Comfort from a City Pulpit... 7 Cox's (G. W.) Athenian Empire ...

--Crusades...............

---Greeks and Persians

Creighton!s Age of Elizabeth.........

Cresy's Encyclopaedia of Civil Engineering 14 Critical Essays of a Country Parson Culley's Handbook of Telegraphy...

Curteis's Macedonian Empire ......

Davidson's New Testament ........

Dead Shot (The) .......................

De Caisne and Le Maout's Botany De Morgan!s (Mrs.) Memoir of her Husband 4

De Tocqueville's Democracy in America... 4 Dewes's Life and Letters of St. Paul

Dixon's Rural Bird Life...............

Doyles English in America....'.........

Dresser's Arts of Japan ................

Dun's American Farming and Food

Eastlake's Hints on Household Taste

- Foreign    Picture Galleries...,

Edmonds's Elementary Botany..........

Ellicott's Scripture Commentaries.....

--Lectures    on Life of Christ .

Elsa and her Vulture ......................

Epochs of Ancient History................

--English History .............

- Modern History .............

Evans's Bronze Implements .............

EwalcFs Antiquities of Israel.............

-Apostolic    Age ...................

--Christ and His Times..........

-History of Israel ................

Fairbairn's Information for Engineers.

--Mills and Millwork..........

Farrar's Language and Languages ....

Fitzwygram on Horses ...................

Francis's Fishing Book ...................

Freeman's Historical Geography .......

---United States...................

-English in Ireland -History of England

-Short Studies.........

-Thomas Carlyle......

Monarchy .....................

Glazebrook's Physical Optics

Goethe's Faust, by Birds......

■--by Selss ......

---by Webb ...

Goodeve's Mechanism ........

Gore’s Electro-Metallurgy...

Gospel (The) for the Nineteenth Century .    16

Grant's Ethics of Aristotle...............

Graver Thoughts of a Country Parson Graves's Life of Sir W. Hamilton ...

Greville's Journal...........................

Griffin!s Algebra and Trigonometry...

Grove on Correlation of Physical Forces... 9 Gwilt's Encyclopaedia of Architecture...... 13

Hale's Fall of the Stuarts........................ 3

Halliwell-Phillipps's Outlines of Shakespeare’s Life ....................................

Hartwig's Works on Popular Natural

History, &c.......................

Hassall's Climate of San Remo.

Haughton's Physical Geography

11

17

16

19

19

11

11

13

17

18

Jago's Inorganic Chemistry ............

"Jameson's Sacred and Legendary Art Jenkins Electricity and Magnetism... Jerrold! s Life of Napole Johnson's Normans in I

21

IS

IS

IS

IS

IS

20

12

11

11

16

Macalister's Vertebrate Animals Macaulay's (Lord) Essays.......

13

21

21

21

17

18

16

.... x6 ...w    16

20

10

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19

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17

11

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16

Heer's Primeval World of Switzerland

Helmholtzs Scientific Lectures.........

Hobart's Medical Language of St. Luke... 6

Herschels Outlines of Astronomy.....

Hope's Hester, a Novel .................

Hopkins's Christ the Consoler ........

Horses and Roads .......................

Hewitt's Visits to Remarkable Places Hullah's History of Modern Music

- Transition Period ...........

Hulme's Art-Instruction in England Humes Philosophical Works...........

/hue's Rome to its Capture by the Gauls... 3

-History of Rome

In the Olden Time .....

Ingelow's Poems ........

12 12 10

on

Europe......

- Patentee’s Manual.........

Johnston's Geographical Dictionary

Jukes's New Man........................

-Second Death ................

■    Types of Genesis ............

Kalisch's Bible Studies .................

- Commentary on the Bible.

--Path and Goal................

Keary's Outlines of Primitive Belief.

Kellers Lake Dwellings of Switzerland.... 11 Kerls Metallurgy, by Crookes and Rbhrig. 14

Landscapes, Churches, &c Latham's Handbook of English Language 7 Lecky's History of England

--European Morals

— -Rationalism ......

-Leaders of Public Opinion...

Leisure Hours in Town ...............

Leslies Political and Moral Philosophy ...    6

Lessons of Middle Age .......

Lewes's History of Philosophy

Lewis on Authority ..............

Liddell and Scott's Greek-English Lexicons 8 Lindley and Moore's Treasury of Botany ... 20

Lloyd's Magnetism ........................

- Wave-Theory of Light.........

Longman's (F. W.) Chess Openings...

— ------Frederic the Great

--German Dictionary ...    7

■■■ ■■■ ■    (W.) Edward the Third

-Lectures on History of England    2

—--St. Paul's Cathedral

Loudon's Encyclopaedia of Agriculture ... 14

■ -----Gardening... 11 & 14

-Plants

Lubbock's Origin of Civilisation ...

Ludlow's American War of Independence 3 Lyra Germanica

10

History of England ...    1

Lays, Illus. Edits. ...12 & 18 —— Cheap Edition..* 18

Macaulay s (Lord) Life and Letters

----Miscellaneous Writings 6

-----Selected Writings......5 & 6

--Speeches

--Works

MacCullaglis Tracts ....................

McCarthy's Epoch of Reform ........

McCulloch's Dictionary of Commerce

Macfarren on Musical Harmony.....

Macleod's Economical Philosophy.....

---Economics for Beginners ..

---Elements of Banking........

---Elements of Economics....,

--- Theory and Practice of Banking 21

Macnamara’s Himalayan Districts .

Mademoiselle Mori............,.........

Mahaffy's Classical Greek Literature Manning's Mtssion of the Holy Ghost ... 16

Marshman s Life of Havelock ........

Martineau's Christian Life..............

--Hours of Thought........

-Hymns.......................

Maunder's Popular Treasuries.........

Maxwells Theory of Heat..............

May's History of Democracy...........

-History of England ..............

Melville's (Whyte) Novels and Tales

Mendelssohn's Letters ....................

Merivale's Fall of the Roman Republic ...

- General    History of Rome ......

- Roman    Triumvirates...............

- Romans    under the Empire......

Merrifiehds Arithmetic and Mensuration... 10 Messer Agnolo’s Household Miles on Horse’s Foot and Horse Shoeing 19

-on Horse's Teeth and Stables.........

Mill (J.) on the Mind...........................

Mills (J. S.) Autobiography..................

-Dissertations & Discussions 5

---Essays on Religion.....

---Hamilton’s Philosophy

--Liberty ....................

-Political Economy .....

---Representative Government 4

--Subjection of Women

--System of Logic ......

---Unsettled Questions

-Utilitarianism .........

Millard's Grammar of Elocution...

Miller s Elements of Chemistry ...

---Inorganic    Chemistry .......

---Wintering    in the Riviera...,

Milner's Country Pleasures ..........

Mitchell's Manual of Assaying......

Modern Novelist’s Library............

Monck's Logic ...........................

Monsells Spiritual Songs...............

Moore's Irish Melodies, Illustrated Edition 12

--Lalla Rookh, Illustrated Edition.. 12

Morris' s Age of Anne Mosley's Reminiscences of Oriel College... 3 Muller's Chips from a German Workshop. 7

--Lectures on India .....

-Origin &c. of Religion

-Science of Language .,

-Science of Religion ....

-Selected Essays ........

Nelson on the Moon............

Nevile's Horses and Riding

New Testament (The) Illustrated. Newman's Apologia pro Vita Sua.

Nicols's Puzzle of Life................

Northcott's Lathes & Turning ....

12

ii

13

Oliphant's In Trust..............................

Orsi's Fifty Years’ Recollections............

Our Little Life, by A. K. H. B.............

Overton's Life, &c. of Law.....................

Owen's (R.) Comparative Anatomy and

Physiology of Vertebrate Animals.........

---Experimental Physiology ... io

- (J.) Evenings with the Skeptics ...    6

17

io

Smith’s Rome and Carthage............

-(J. Shipwreck of St. Paul ...

Southey's Poetical Works ...............

--& Bowles's Correspondence

Stanley's Familiar History of Birds ...

Steel on Diseases of the Ox ............

Stephen's Ecclesiastical Biography.....

Stonehenge, Dog and Greyhound......

Stulls's Early Plantagenets ............

Sunday Afternoons, by A. K. H.B. ...

Supernatural Religion.....................

Swinlurne's Picture Logic ..............

Perry's Greek and Roman Sculpture ......    12

Payen's Industrial Chemistry........

Pewtner's Comprehensive Specifier

Piesse's Art of Perfumery ...........

Pole’s Game of Whist.................

Powells Early England..............

Preece & Sivewright’sTelegraphy..

Present-Day Thoughts.................

Proctor's Astronomical Works .....

-Scientific Essays...........

Public Schools Atlases ..............

13

20

14

20

10

.....8&9

11

Quain’s Dictionary of Medicine

21

Rawlins on’s Ancient Egypt .........

-Sassanians ..............

Recreations of a Country Parson ...

Reeve's Cookery and Housekeeping Reynolds's Experimental Chemistry Rich's Dictionary of Antiquities ...

Rivers's Orchard House...............

- Rose Amateur’s Guide......

Rolinson’s Arden, a Novel............

Rogers's Eclipse of Faith and its Defence 15

Roget's English Thesaurus.....................

Ronalds' Fly-Fisher’s Entomology .........

Rowley's Rise of the People ..................

-Settlement of the Constitution ...

Rutley’s Study of Rocks

N Cl tx

Tancock’s England during the Wars,

1765-1820 .......................................

Taylor's History of India .....................

-Ancient and Modem History ...    3

- (Jeremy) Works, edited by Eden 16

Text-Books of Science...........

Thomi's Botany ..................

Thomson's Laws of Thought ..

Thorpe's Quantitative Analysis Thorpe and Muir's Qualitative Analysis ... 10

Three in Norway................................. 16

Thudichum s Annals of Chemical Medicine 12 Tilden s Chemical Philosophy

--Practical Chemistry..

Trevelyan s Life of Fox........

Trollope's Warden and Barchester Towers 18 Twiss's Law of Nations in Time of War... 5 Tyndall's (Professor) Scientific Works... 9&10

10

10

10

10

12

20

12

11

11

17

19

IO&II

Sandars's Justinian’s Institutes......

Sankey's Sparta and Thebes .........

Seaside Musings ........................

Scott's Farm Valuer.....................

-Rents and Purchases.........

Seelohm's Oxford Reformers of 1498

--Protestant Revolution ...

Sennett's Marine Steam Engine......

Seth & Haldanes Philosophical Essays ...    6

Sewells Passing Thoughts on Religion ...    16

-Preparation for Communion

-Private Devotions.............

-Stories and Tales .............

Seymour’s Hebrew Psalter.............

Shelley's Workshop Appliances.......

Short's Church History .................

Simcox's Classical Latin Literature..

Skobeleff and the Slavonic Cause ..

Smith!s (Sydney) Wit and Wisdom ..

-(Dr. R. A.) Air and Rain .....

--(R. B.) Carthage & the Carthaginians    2

21

21

13

16

16

18

16

10

14

Unawares ....................................

Unwin's Machine Design ...............

Ure’s Arts, Manufactures, and Mines

Ville on Artificial Manures.

Walker on Whist.................................

Walpole's History of England ...............

Warlurton's Edward the Third ............

Watson's Geometery ...........................

Watts's Dictionary of Chemistry............

Well's Celestial Objects........................

Weld's Sacred Palmlands .....................

Wellingtons Life, by Gleig ..................

Whately's English Synonymes...............

-Logic and Rhetoric...............

Whites Four Gospels in Greek...............

-and Riddle's Latin Dictionaries ...

Wilcocks's Sea-Fisherman.....................

Williams's Aristotle’s Ethics..................

Willich's Popular Tables .....................

Wilson's Studies of Modern Mind .........

Wilt's Myths of Hellas, translated by

Younghusland.................................

Wood's Works on Natural History.........

Yonge’s English-Greek Lexicons Youatt on the Dog and Horse ..

Zeiler's Greek Philosophy

Spottiswoode &> Co. Printers, New-street Square, London.

18

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