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branch more immediately refpects navigation, furveying, and what is commonly called altimetry and longimetry, or heights and distances, if indeed this must be diftinguifhed from furveying.

Acceffible lines are measured by applying to them fome certain meafure, as an inch, a foot, &c. a number of times; but inacceffible lines must be measured by taking angles, or by fome fuch-like method, drawn from the principles of geometry.

When inftruments are used for taking the quantities of the angles in degrees, the lines are then calculated by trigonometry: in the other methods the lines are calculated from the principle of fimilar triangles, without any regard to the quantities of the angles.

Angles of elevation, or of depreffion, are usually taken either with a theodolite, or with a quadrant, divided into degrees and minutes, and furnished with a plummet fufpended from the center, and two fights fixed perpendicularly upon one of the radii.

To take an Angle of Altitude and Depreffion with the Quadrant.

Let A be any object, as the top of a tower, hill, or other eminence; or the fun, moon, or a star; and let it be required to find the measure of the angle ABC which a line drawn from the object

makes with the horizontal line BC.

A*

Fix the center of the quadrant in the angular point, and move it round there as a center till with one eye at D, the other being fhut, you perceive

the

the object A through the two fights E, F; then will the arc GH of the quadrant, cut off by the plumb, line Bн, be the meafure of the angle ABC required. The angle ABC of depreffion of any object A is taken in the fame manner, except that here the eye is applied to the center, and the measure of the angle is the arc

GH.

The obfervations with the quadrant, neceffary to determine the heights and diftances of objects, will be fufficiently apparent from the manner in which the following examples are propofed; and the folution may eafily be given, by any one who underftands plane trigonometry.

The conftruction of the figures to the following examples, are omitted; but they are to be conftructed as in the problems of trigonometry.

EXAMPLE I.

Having measured AB equal to 100 feet from the bottom of a tower, in a direct line on a horizontal plane, I then took the angle CDE of elevation of the top, and found it to be 47° 30', the center of the quadrant being fixed five feet above the ground: required the height of the tower.

As radius

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Tot. D47°30′10·0379475

So DE 100

To CE 109.13 2.0379475

Add AE 5

AC 114 13

A

Or,

Or, without the Logarithms.

As 1 (rad.): 10913085 (tan. of 47° 30′):: 100 : 100 X 1.0913085 = 109.13085 = CE.

By the calculation the height CE is found equal to 109.13, to which DB (equal to EA equal to 5 feet the height of the inftrument) being added, gives AC equal to 114.13, the whole height.

Note. If you go off to fuch a distance from the bottom, as that the angle of elevation fhall be 45°, then will the height be equal to the distance with the height of the center of the inftrument added.

EXAMPLE II.

From the edge of a ditch of 18 feet wide, furrounding a fort, I took the angle of elevation of the top of the wall, and found it 62° 40′ required the height of the wall, and the length of a ladder neceffary to reach from my station to the top of it.

First, As1: 1934702 (tan. 62°40′)

::18: 1934702 X 1834 824636

= BC.

Then 182 + 34°8246362 = ✓1536.75527239'2014

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AC. Or as 1 2 1778594 (fecant of 62° 40) 18:39204692 AC. :: Ans. The height BC3482, and the length of the ladder AC 39'2.

EXAMPLE III.

A

B

From the top of a fhip's maft, which was 80 feet above the water, the angle of depreffion of another ship's hull, at a distance, upon the water, is 20°; what is their distance?

As

B

A

As 1:2·7474774 (tan. 70°) :: 80:27473774'× 80 =219.790192 feet AC the distance required.

EXAMPLE IV.

What is the perpendicular height of a hill whofe angle of elevation, taken at the bottom of it, was 46°; and 100 yards farther off, on a level with the bottom of it, the angle was 31°;

LC 46°

fubtract

B

LD 31

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From the top of a tower, whofe height was 120 feet, I took the angle of depreffion of two trees which lay in a direct line, upon the fame horizontal plane, with the bottom of the tower, viz. that of the nearer 57°, and that of the farther 25°: what is the distance between the two trees, and the distance of each from the bottom of the tower?

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:: 120: 6494076X 12077928912 feet BC the diftance from the bot

tom of the tower to

the nearest tree.

B

Andas 1 · 2·0965436 (tan. ▲ BAD = 64°1):: 120: 2.0965436 X 120 251.585232 feet BD the diftance of the farther tree.

Therefore, BD-BC251.585232-77928913 = 173.65632 feet CD the diftance between the

two trees.

EXAMPLE VI.

An obelifk ftanding on the top of a declivity, I measured from its bottom a distance of 40 feet, and then took the angle formed by the plane and a line drawn to the top 41°; and going on in the fame direction 60 feet farther, the fame angle was 23° 45', the height of the inftrument being five feet: what was the height of the obelisk?

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