Methods for finding the hypothenusal 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 hypothenusal 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 hypothenusal

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 hypothenusal 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.

Several very experienced Land-Surveyors with whom the Author is acquainted, perfectly agree with him on this subject.

EXAMPLES.

1. The length (or hypothenusal 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 hypothenusal 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 × 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 hypothenusal by 3R. 18P.

2. Let A B C D represent a field in the form of a trapezium, lying upon the side of a hill of an irregular ascent, the sides A B and B C being upon the horizontal plane of the base; required the horizontal and hypothenusal measures, from the following notes.

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 A B D.

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 A B C D.

The Operation of finding the hypothenusal Measure.

First, 1154 + 110 = 1264, the hypothenusal line BD; and 99078 1068, the hypothenusal line D A. 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 hypothenusal measure of the triangle A B D.

In a similar manner we find the hypothenusal measure of the triangle B C D = 437917 square links, making jointly 821626 square links, equal to 8a. Or. 34p. the hypothenusal measure of the trapezium A B C D, which exceeds the horizontal measure by 2r. 34p.

NOTE 1. -If you lay down the trapezium by the horizontal and hypothenusal lines respectively, and measure the perpendiculars by the scale, you will find the areas the same as those resulting from the foregoing operations.

2. From these examples, it appears that the difference between the horizontal and hypothenusal measures of billy fields, is often very considerable, and is deserving of particular notice. For instance; suppose the field, in the last example, to have been sown with wheat, and the owner to have sold the crop at the rate of 121. per acre; the reapers have a claim upon the buyer for the hypothenusal measure; but if he makes his payment to the seller, by the same admeasurement, he will receive 81. 11s. more than his due.

Practical Surveyors, however, in general, (as before observed,) return the horizontal measure, in surveying estates; whence few farmers, comparatively speaking, are charged for more; and ought not, therefore, when they sell a crop of corn, &c. to expect pay for the hypothenusal measure.

REMARKS.

Since the publication of the first edition of this Work, the Author has consulted several eminent Land-Surveyors, and also Commissioners for Inclosures, in very extensive practice, in the WestRiding of Yorkshire, and in Cumberland, and Westmoreland, places noted for their hills; and they, without one exception, inform him that the horizontal measure of hilly ground is always returned, both by them, and by every practical Surveyor with whom they are acquainted.

Some late writers on Surveying contend very strenuously for the hypothenusal measure of hills; but the Author and his friends have no hesitation in saying, that those writers are very deficient in practical knowledge.

If we consider the earth as a perfect sphere whose diameter is 7957 miles, it is not necessary to take its curvature into consideration in surveying single Fields, Farms, or Lordships; for it is evident that the quantity of land, even in the County of York, would form such a small spherical segment, that its convex surface would exceed the area of its base extremely little.