The usual measurements being made, viz. the sides AB, BC, CD, DA, and the diagonal AC, the ground may be plotted in the usual way to scale.

D

с

b

Also the perpendiculars Dd, Bc, upon AC being drawn, and calculated on the same scale, the area of ABCD is found by Art. (253).

Then ab is drawn as a proof-line, just as in the last Ex., either within or without ABCD, and serves at once to prove the correctness, or otherwise, of the survey.

It is further to be observed, that if, from the nature of the ground, it be difficult to measure AC, that measurement may be avoided by taking ab instead, as a tie-line either within or without the plot.

Thus, proof-lines are employed not only as tests of the correctness of the work of the surveyor; but also to enable him to plot and measure areas-such as those of woods, lakes, &c.-with certainty and ease, which otherwise could not be surveyed by means of simple instruments.

270. The content of irregular fields, farms, estates, parishes, or even whole counties, when correctly planned to scale, is sometimes found by a very ingenious and simple method, as follows:

The plan being drawn upon paper, or drawing-board, of uniform thickness and texture, the portion whose area is required is cut out accurately along its boundaries with a sharp knife. Then from the same sort of paper, or drawing-board, a square is cut, which shall represent, according to the scale employed, a known area, such as an acre, or square chain, or a square mile,

&c. The two pieces of paper are then weighed in a very accurate balance, and the ratio of their weights will be that of the areas contained in them. So that, the area of the square being known to be an acre, or a square chain, or a square mile, as the case may be, the number of acres, &c. in the irregular plot is determined.

QUESTIONS AND EXERCISES L.

(1) What is the area in acres of a rectangular plot of ground, 128 yds. long, and 50 yds. wide?

Ans. 1a. 1r. 6 p. (2) State the advantage of taking the length of 22 yds. for the common Chain.

(3) When the sides of a rectangular plot are known in chains and links, how is the area obtained in acres, roods, &c.?

(4) Find the area, in acres, of a square whose side is 15 chains, 40 links. Ans. 23 a. 2r. 34 p.

(5) Shew how to find the area of a plot, of which an accurate plan has been obtained, without again going over the ground.

(6) Compute the area of a rhombus, whose base is 5 chains, 32 links, and perpendicular height is 3 chains, 7 links. Ans. 1a. 2r. 211 p. nearly. (7) Explain how the chain may be made use of for constructing a right angle.

(8) Give a brief description of the mode in which curvilinear fields are measured.

(9) Shew how, without using any offsets, a fair approximation may be made to the area of a field whose sides are not exactly straight lines.

(10) State the mode in which the observations taken in measuring a plot of ground are registered for future calculation; and write out an example of the method.

(11) When a piece of land, bounded by straight lines, is being measured, and it is not easy to traverse it,

shew how to lay down a correct plan, and thence to obtain its area.

(12) Make the largest right-angled triangle which can be constructed out of the links of two Chains fastened together; how many links are there to spare?

Ans. The sides will be 48, 64, and 80 links;

and there will be 8 links to spare.

(13) In the triangle mentioned in the last question, what multiples are its sides of those of the triangle described in Art. (266)?

Ans. 16 times as large.

(14) In measuring a surface which differs considerably from the horizontal, what deviation must be made from the mode of measuring a level surface?

(15) A field is inclined to the horizon at an angle of 121o in the direction of its length; find from the Table in (263) the length of the projection, on the horizontal plane, of a line which on the slope measures 147.5 yards. Ans. 144-00425 yds.

(16) A road rises uniformly at the rate of 300 feet per mile; what is the difference between a mile measured on the slope, and the projection of that length on the horizontal plane? And what is the angle made by the road with the horizon?

(1) Ans. 2.85 yds.; (2) Ans. Rather more than 3o.

(17) A road, a mile long, makes an angle of 5o with the horizon; what must be the length of a canal which runs parallel to the road throughout the mile?

Ans. 1753-312 yds. (18) How would the rise per mile be estimated in yards from the data in Ex. (17), by simple arithmetical calculation, without any levelling?

Ans. It is equal to (1760)2- (1753.312), in yds.

(19) On a piece of land having a uniform inclination of 50 to the horizon, a line was measured 11·73 chains long, in the direction of the inclination; required the distance, estimated longitudinally, over which the surveyor has passed. Ans. 11.676 chains.

(20) A hill is inclined to the horizon at an angle of 15° towards a river 2·5 chains broad; from its top a line is measured of 12·35 chains, in the direction of the slope; the hill rising on the other side of the stream, from its edge, is inclined at an angle of 10°, and measures to its ridge 7.5 chains; find the horizontal distance between the tops of the hills. Ans. 21.8144 chains.

(21) Describe the Cross-Staff and its use; and state the points upon which its accuracy depends.

(22) Find the areas of two pieces of land from the following notes; the measurement of the former was taken in yards, the latter in links:

MEASURING INSTRUMENTS.

Certain Measuring Instruments' have been already described, both as to their construction and use; viz. the Tape, and Chain, for measuring lengths or distances; the Protractor, for measuring angles; the Offset-Staff, for measuring short offsets; the Cross-Staff, for setting out lines at right-angles in the field; the Level, for finding the difference of level between any two given points, &c. But there are many other instruments of great value, when we come to actual work, and some of these we will proceed to describe. Especially we are required to give (which has not yet been done), such instruments as are commonly used for measuring angles out of doors; as, for instance, the angle contained by two sides of a field, considered as straight lines, and meeting at a point which is accessible; or, again, the angle in a vertical plane subtended at the eye of the observer by a lofty tower, or other building.

B

A

D

THE QUADRANT.

271. A very simple instrument may be made, as follows, for measuring, in a vertical plane, some of the more simple angles, as 15°, 30°, 45°, 60°, 75°, and 90°. Let ABCD be a square board of convenient size. Draw upon it the diagonal AC; and with centre A and radius AB describe the arc of a quadrant BED, cutting AC in E. Then ▲ BAE = L DAE = 45°. At E put 45, to shew that BE is an arc of 45°. With centre B, and radius AB, as before, set off the arc of 60°, and mark the point 60. Then from D also set off an arc of 60°, and mark that point 30, because it will determine an arc of 30° measured from B. Bisect this latter arc, and mark the point of bisection 15. Also bisect the arc D 60, and mark the point 75. The whole quadrant is now divided into 6 equal arcs

of 15°.

This square board is so fastened to a staff, about 6 feet long, with a sharp point to enter the ground, as to permit it to revolve in its own plane round a fixed axle at A; and from A a plumb-line is suspended, which serves for adjusting the vertical position both of the staff and of the board.

This is a rough instrument for measuring such angles, out of doors, as are before mentioned, and may be used effectively for certain limited purposes. When used, the face of the board ABCD is not only made vertical by means of the plumb-line, but it is turned round until it is in the same vertical plane in which the two points lie whose angular distance is required, and then the staff is fixed firmly in the ground. The observer, then, having the objects before him, whose angular distance he is to measure, places his eye at B, or D, as the case may be, and, looking along the upper edge of the board, he turns it round A, until he sees one of the