or long pickets being fixed with reference to each other, it is only necessary to measure between them, entering the distances on these lines, with the offsets to the right or left to the different short pickets marking the horizontal lines.

The water-level, already described, is adapted to this species of levelling, as the distances are always short; but in a large extent of ground, the French reflecting-level is perhaps the best instrument for tracing contour lines rapidly.

In laying down a section on paper, particularly if the ground is of gentle slope and the section of considerable length, it is usual to exaggerate the vertical heights for the purpose of rendering the undulations of the surface perceptible, which necessarily produces a distorted representation of the ground. The horizontal scale is usually made an aliquot part of the vertical, that the proportions between them may be at once obvious. Scales of 25, 50, 100 or 150 feet to one inch,* are appropriate for the latter, according to the degree of detail required in the section, and the horizontal scale may be from to of either of them; or even a less proportion if the section is of great length, and the ground generally flat, as in the figure below, plotted from the specimen of a levelling field-book in page 89.

The horizontal line, from which the vertical distances are set off, may be either on a level with one end, or some one point of the section; or a datum line may be drawn any number of feet above or below this line, exceeding the sum of all the vertical heights: this latter arrangement makes all the dimensions reduced for plotting either plus or minus, but it is doubtful if any great advantage is thereby gained.

*The plotting scales, already alluded to, are very convenient for laying down sections; and Mr. Holtzapffell's cardboard Engine-Divided Scales will be found useful where a variety of scales are often required; from their method of construction, they can be sold at the low price of nine shillings a dozen, of all descriptions in general

use.

Laying off intermediate horizontal and vertical distances, should be avoided in plotting sections; the former ought always to be measured from the commencement of the section, with as few interruptions as the length of the line will allow; and the latter from the datum line. Both horizontal and vertical distances should, particularly in a working section, have their dimensions written legibly on the drawing.

Contoured plans, from which sections are to be constructed, are generally plotted on about the same scale as special surveys of estates, &c.; that is on one of 2, 3, or 4 chains to one inch. Small portions of ground for military purposes, where the vertical distances are under five feet, may even be laid down on a scale of one chain to one inch.

Trial sections that have been run for the purpose of ascertaining the best of several routes for a railroad, canal, or other work, should invariably be all plotted on the same scale and paper, and from the same datum-line; and commencing at, and having reference to, the same points as bench-marks. By this arrangement their comparison by the eye is facilitated.

Cross or transverse sections are sometimes plotted above, and sometimes below the longitudinal section; and if only extending a few feet to the right and left, they are occasionally plotted on the line of section but if numerous, this last method causes a confused appearance in the drawing.

:

A method of combining plan and section has lately been introduced by Mr. Macneil, for the purpose of giving a popular representation of the quantity of excavation and embankment at any part of the section of a line of railway, the direction of which is shown on the outline plan of the country through which it passes by a thick black line, supposed to represent a vertical section of the rail. From the accurate section previously drawn, the heights of the embankments, and depths of excavation at the different parts of the line are transferred to this datum line on the plan; and these quantities being tinted with different colours, or, if engraved, represented the one with vertical, and the other with horizontal lines, show at a glance the general relative proportions of cutting or embankment, as in the annexed figure.

The dark line in both figures represents the surface of the railroad or embankment.

To those unaccustomed to the use of sections, this simple contrivance, by which they are rendered intelligible, is particularly useful, and has been ordered to be adopted in all plans for railways submitted to the House of Commons. Of course it is only intended to give a general idea of the quantity of work on any line of road, railroad, or canal, and to be explanatory of the report and estimate,

Numerous transverse sections are required for computing the relative proportions of embankment and excavation on any work, which operation is much facilitated. by the use of Mr. Macneil's ingenious tables, calculated upon the "Prismoidal Formula," which shows the cubic content of any prism to be equal to the area of each end +four times the middle area, multiplied by the length and divided by 6; whereas the common methods of taking half the sums of the extreme

*Of the greatest possible consequence, both for the sake of avoiding unnecessary expense, and of laying out the work to the best advantage valuable information upon which subject will be found in Mr. Macneil's work.

12

14

heights for a mean height, or of taking half the sum of the extreme areas for a mean area, are both erroneous; the first giving too large a result, and the second too little.

As this prismoidal formula is not so generally understood as is desirable, the following investigation, taken from Mr. Macneil's work, is given below :—

The figure ABCGFKDE is a prismoid, or solid figure, similar to that which is formed in excavations or embankments, in which BCDK represents the roadway, and ABCG,FKDE, parallel cross sections at each end. Then {ABCG + FKDE, +4 abeg} x = the cubic content of the solid.

=

Let BC b, the breadth at the bottom of the cutting.

CD

6

ML=h, the perpendicular depth of cutting at the higher end.
ON or MQ=h', the same depth at the lower end.
CD1, the length of the solid.

PI = r h′, the ratio of the perpendicular height of the slope, to its horizontal base, multiplied by the height, MQ or ON.

Then the content of the prism, HBCIFKDE=the area of the end, FKDE, or HBCI × C D, or = BC+PI ×MQ×CD=b+h'r×h'×l.

The pyramid AHIGE=the area of its base AHIG × its height IE, or{(b+rh) h—(b+rk') }, for the area of AHIG is equal to the difference of the areas of the ends ABCG and FKDE; and the pyramid AHEF = the area of the triangle HEF AR, or = EF × HF × × 3 } AR=b+2rh'

2

h-h'

xlxh, for AR = LQ = h—h'.

3

The content of the prismoid will therefore be equal to the sum of these three quantities: that is

b+2rh'

(b+r h')h'×l+(b+r h) h—(b+r h') h' × 3 + b + 2 r k xlx h = h2

-h'

= {6(b+rh')h'+2(b+rh) h—2(b+rh')h' +

2

3

(b+2r h' ). (h—h') } {

= {4 (b + r h') h' + 2 (b + rh) h + (b + 2 r h'). (h—h') } }

= {4 bh' + 4 r h2 + 2b h + 2 r h2 + bh-bh' + 2r hh-2 r h 2 } }

=

h—b

= { 3 b h' + 2 r h2 2 + 3 b h + 2 r h2 + 2 r h h' }

= { Q b h + b h + rh2 + r h12 + 2 b h + b h + r h2 + rh2 +2r hh2 };

=

2

{(b+r h') h' +(b+r h) h + 2 bh' + 2 b h +r h2 + 2 r hh' + r h2 }} {{ = {( b + r h') h' + (b + r h ) h + 4 ( b + r h‡h'). h + 1⁄2 )

=

2

2

Area of each end, added to four times the middle area, and the sum multiplied by the length divided by 6.