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branch more immediately respects navigation, surveying, and what is commonly called altimetry and . longimetry, or heights and distances, if indeed this must be distinguished from furveying.
Accessible lines are measured by applying to them some certain measure, as an inch, a foot, &c. a number of times; but inaccessible lines must be measured by taking angles, or by some such-like method, drawn from the principles of geometry.
When instruments 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 similar triangles, without any regard to the quantities of the angles.
Angles of elevation, or of depression, are usually taken either with a theodolite, or with a quadrant, divided into degrees and minutes, and furnished with a plummet suspended from the center, and two fights fixed perpendicularly upon one of the radii. To take an Angle of Altitude and Depression with the
Quadrant. Let A be any ob
A* ject, as the top of a tower, hill, or other eminence; or the sun, 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.
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 shut, you perceive
the object A through the two fights E, F; then will the arc gh of the quadrant, cut off by the plumb line BH, be the measure of the angle ABC required.
The angle ABC of depression of any object A is taken in the faine manner, except that here the eye is applied to the center, and the measure of the angle is the arc
The observations with the quadrant, necessary to determine the heights and distances of objects, will be sufficiently apparent from the manner in which the following examples are proposed; and the solution may easily be given, by any one who understands plane trigonometry.
The construction of the figures to the following examples, are omitted; but they are to be constructed as in the problems of trigonometry.
Having measured A 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 10'0000000 Tot.2 D47° 30'10'0379475 SO DE 100
TO CE 109:13 2'0379475
AC I 1413
Or, without the Logarithms. As 1 (rad.): 1.0913085 (tan. of 47° 30'):: 100 : 100 X 1.0913085 = 109-13085 = ce.
By the calculation the height ce is found cqual to 109:13, to which DB (equal to EA equal to 5 feet the height of the instrument) being added, gives ac equal to 114:13, the whole height.
Note. If you go off to such a distance from the bottom, as that the angle of elevation shall be 45°, then will the height be equal to the distance with the height of the center of the instrument added.
EX AMPLE 11.
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,Asr: 10934702 (tan. 62°40') :: 18: 1934702 X 18 = 34-824636
Then ✓ 182 + 34•824636* = ✓ 1536-755272 = 39'2014 = AC. Or as 1 : 2.1778594 (secant of 62° 40') :: 18:39*204692 = AC.
Ans. The height BC = 34:82, and the length of the ladder AC = 39'2.
From the top of a fhip’s mast, which was 80 feet above the water, the angle of depression of another ship’s hull, at a distance, upon the water, is 20°; what is their distance?
As 1 : 2*7474774 (tan. 70°):: 80: 2.7473774'X 80 = 219*790192 feet = Ac the distance required.
EX A M P LE JV.
What is the perpendicular height of a hill whose 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°
From the top of a tower, whose height was 120 feet, I took the angle of depression of two trees which lay in a direct line, upon the same horizontal plane, with the bottom of the tower, viz. that of the nearer 57°, and that of the farther 25°1 : what is the distance between the two trees, and the distance of each from the bottom of the tower?
As I : 6494076 (tan. L BAC=33') :: 120:•6494076x 120 = 77-928912 feet = BC the distance from the bottom of the tower to the nearest tree.
Andas I • 20965436 (tan. . BAD = 64°):: 120: 2.0965436 X 120 = 251.585232 feet = ED the distance of the farther tree.
Therefore, BD - BC = 2510585232 — 770928912 = 173•65632 feet = co the distance between the two trees.
An obelisk standing 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 same direction 60 feet farther, the fame angle was 23° 45', the height of the instrument being five feet : what was the height of the obelisk ?