Prob. VI. To lay out a rhomboid of given area, the lengths of its

sides being given

Prob. VII. To lay out a regular polygon of given area

Prob. VIII. To lay out a circle of given area

Prob. IX. To lay out an ellipse of given area

Prob. X. To divide any proposed quantity of land from a triangle,

by a line parallel to one of its sides

·

Prob. XI. To lay off any given quantity of land from an irregular

field or common

Prob. XII. To divide from a field or common, bounded by straight

fences, any given quantity of land, by a line parallel to one of

the straight fences

SECTION II. ON DIVIDING LAND OF EITHER EQUAL OR VARIABLE

VALUE AMONG VARIOUS CLAIMANTS, IN PROPORTION TO

THEIR SEVERAL CLAIMS.

Prob. I. To divide a rectangular piece of land of equal value through-

out, either equally or unequally, among any given number of

claimants, by fences parallel to one of its sides

Prob. II. To divide a triangle of equal value throughout, either

equally or unequally, among several claimants, who shall all

have the use of the same watering-place, situated at one of the

angles of the field

Prob. III. To divide a triangular field of equal value throughout,

either equally or unequally, among sundry claimants, by fences

running from a given point in one of its sides

Prob. IV. To divide an irregular field with any number of sides,

among sundry claimants, so that they may all have the use of

a pond situated at a given point within the field

Prob. V. To set out from a field or common of variable value, a

quantity of land, that shall have a given value, by a straight

fence in a given direction.-Cases I. to IV. (the last Case may
be solved by Prob. VI., Chap. VI., Part II.)

Prob. VI. Case I. To divide a common of uniform value, among

any number of proprietors, in proportion to the values of their

respective estates

Case II. To divide a common of variable value, among

any number of claimants, in proportion to the values of their

respective estates

Roads, quarries, watering-places, &c., required to be set out previous

to inclosing commons, wastes, &c.

PAGE

104

105

106

107

108

108

110

112

118

114

115

117

122

The method of dividing and allotting the remaining part of the com-

Principles and Practice of Levelling

133

To draw a sectional line on the earth's surface, the Levels of which have

been taken

135

On Railway Curves in General

Prob. II. To find the Tangential Angle for a cord of one chain, the

radius being given

CHAPTER II.

THE METHOD OF LAYING OUT RAILWAY CURVES, TURNOUTS, CROSSINGS, ETC.

Prob. I. To determine the Radius of a Railway Curve

162

Prob. III. To find the length of a Tangent, the radius and angle of

intersection being given

163

Prob. IV. To find the Angle of Intersection when it happens to fall in

a sheet of water, a wood, or a pile of buildings -

Prob. V. To find the length of the Curve, the angle of intersection and

the tangential angle being given

Prob. VI. To lay out a Curve on the Ground with the Chain only, no

obstruction being on the convex side of the Curve

-

163

163

164

Prob. VII. To lay out the Curve by offsets from its chord or chords,

when obstructions occur on the convex side of the Curve

Prob. VIII. To find the length of the Chord of a Curve

Prob. IX. To find the second Radius of a compound Curve, the oth

radius, tangential points, and angles being given

PAGK

166

168

168

169

Prob. X. To find the second Radius of a compound Curve, the other
radius, tangential points, and angle of intersection being given
Prob. XI. To determine the reversing point and the common Radius of a
serpentine or reversed Curve, the tangent points, &c., being given
Prob. XII. To find the Chords and second Radius of a serpentine or
reversed Curve, the tangential points and first radius being given -
Prob. XIII. To find the common Radius of a serpentine or reversed
Curve, the tangential points being given

Prob. XIV. To find the common Radius of a reversed Curve, the length

of the common tangent and the angles of intersection being given 171

Prob. XV. To unite the Tangent Points by a parabolic vertical Curve,

the rate of inclination of gradients, number of stations, and tan-

gent points being given

Turnouts and Crossings

Prob. XVI. To find the Frog Angle and Chord of a Turnout, the radius

of a centre line being given

·

Prob. XVII. To find the Radius of the centre line of a Turnout, the

frog angle being given -

170

172

174

174

Prob. XVIII. To find the position of a given Frog practically on the

ground

Prob. XIX. To make a given deviation from a straight line of Railway

by three Curves, &c.

Examples of expensive severance of property by improperly setting cut

Railway curves

The practice of Engineers in the adoption of Curves in various Railways

Coreless Expenditure in the Construction of Railways from the non-adop-

tion of Curves, from improperly laying out Gradients, Exactions of

Landowners, &c.

179

CHAPTER III.

ON SETTING OUT THE SURFACE WIDTHS OF RAILWAYS.

Prob. I. To set out the width of a Railway Cutting, when the surface

of the ground is laterally level, and at a given height above the

formation level, the ratio of the slopes being given

Prob. II. The same things being given as in the last Problem, to set

out the width of the Cutting, when the ground is laterally sloping,

the lateral fall of the ground in a given horizontal distance being

also given

Prob. III. The same things being given as in Prob. II., to set out the

width, when it consists partly of a Cutting, and partly of an Em-

bankment

Prob. IV. To find the surface width of a Cutting, where the ground is

very uneven

Prob. V. To calculate the quantity of land for a projected Railway

Prob. VI. To find the exact quantity of land for a Railway

181

182

184

THE METHODS OF FINDING THE CONTENTS OF RAILWAY CUTTINGS.

Tables for finding the contents of ditto

Prob. I. To find the contents of Railway Cuttings when the surface lines

of the several sections are level

192

194

196

198

198

199

199

Prob. II. To find the contents from sectional areas. Cases I. to III.

Prob. III. To adapt the General Table to all widths of formation level

and ratios of slope

Prob. IV. To find the content of a Cutting when each of the sides have

two different ratios of slope

Prob. V. To find the content when the ratio of the slopes of the two
sides are different. Cases I. and II.

Prob. VI. To find the quantity of the Cutting of a Tunnel

Problems and Formula of utility in Land and Engineering Surveying

original and select, for the exercise of students

Centrifugal force of trains in Railway Curves

Formula for the super-elevation of the exterior rail

Plan of part of the Borough of Liverpool, with contour lines

Addenda to methods for laying out Curves

On the Division of Land

Correction of Levels for Parallax and Refraction

219

PREVIOUS to commencing the various subjects of Land and Engineering Surveying, it will be necessary to give a clear view of Practical Geometry, which is especially requisite for those who are unacquainted with this branch, as well as those parts of the Mathematics which are equivalent to it.

PRACTICAL GEOMETRY.

DEFINITIONS.

1. A point has no dimensions, neither length, breadth, nor thickness.

2. A line has length only, as A.

3. A surface or plane has length and

breadth, as B.

A

B

4. A right or straight line lies wholly in the same direction, as A B.

5. Parallel lines are always at the same distance from each other, and never meet when prolonged, as A B and C D.

6. An angle is formed by the meeting of two lines, as A C, CB. It is called the angle ACB, the letter at the angular point C being read in the middle.

7. A right angle is formed by one right line standing erect or perpendicular to another; thus, ABC is a right angle, as is also ABE.

8. An acute angle is less than a right angle, as DB C.

A

A

E

B

B

-D

9. An obtuse angle is greater than a right angle, as D'BE.