Methods for finding 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 affect 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 the 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
800

7.20000

4

·80000

40

32-00000 Area 7a. Or. 32p.

Now, 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:

=

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 99078= 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, we 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. 34p.

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 is parallel to BD.

The two lines BD and AD remaining undetermined may be found by the following formulæ, 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 involv