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162 equal parts, and the angle A = 53° 7′ 48′′; then raise the perpendicular BC, meeting AC in c. So shall AC measure 270, and вC 216.
Extend the compasses from 45° to 53°, on the tangents. Then that extent will reach from 162 to 216 on the line of numbers.
Note. There is sometimes given another method for rightangled triangles, which is this:
ABC being such a triangle, make one leg AB radius; that is, with centre A, and distance AB, describe an arc BF. Then it is evident that the other leg BC represents the tangent, and the hypothenuse AC the secant, of the arc BF, or of the angle A.
In like manner, if the leg BC be made radius; then the other leg AB will re
present the tangent, and the hypothenuse AC the secant, of the arc BG or angle c.
But if the hypothenuse be made radius; then each leg will represent the sine of its opposite angle; namely, the leg AB the sine of the arc AE or angle c, and the leg BC the sine of the arc CD or angle A.
Then the general rule for all these cases is this, namely, that the sides of the triangle bear to each other the same proportion as the parts which they represent.
And this is called, Making every side radius.
Note 2. When there are given two sides of a right-angled triangle, to find the third side; this is to be found by the property of the squares of the sides, in theorem 34, Geom. viz. that the square of the hypothenuse, or longest side, is equal to both the squares of the two other sides together. Therefore, to find the longest side, add the squares of the two shorter sides together, and extract the root of square that sum; but to find one of the shorter sides, subtract the one square from the other, and extract the root of the remainder.
OF HEIGHTS AND DISTANCES, &c.
BY the mensuration and protraction of lines and angles, are determined the lengths, heights, depths, and distances of bodies or objects.
Accessible lines are measured by applying to them some certain measure a number of times, as an inch, or a foot, or yard. But inaccessible lines must be measured by taking angles, or by such-like method, drawn from the principles of geometry.
When instruments are used for taking the magnitude 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, or some other geometrical property, without regard to the measure 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 centre, and two open sights fixed on one of the radii, or else with telescopic sights.
To take an Angle of Altitude and Depression with the Quadrant.
Let A be any object, as the sun, moon, or a star, or the top of a tower, or hill, or other eminence: and let it be required to find the measure of the angle ABC, which a line drawn from the object makes above the horizontal line BC.
Place the centre of the quadrant in the angular point, and move it VOL. II.
round there as a centre, till with one eye at D, the other being shut, you perceive the object a through the sights; then will the arc GH of the quadrant, cut off by the plumbline BH, be the measure of the angle ABC as required.
The angle ABC of depression of any object A, below the horizontal line BC, is taken in the same manner; except that here the eye is applied to the centre, and the measure of the angle is the arc GH, on the other side of the plumb-line.
The following examples are to be constructed and calculated by the foregoing methods, treated of in Trigonometry.
Having measured a distance of 200 feet, in a direct horizontal line, from the bottom of a steeple, the angle of elevation of its top, taken at that distance, was found to be 47° 30'; hence it is required to find the height of the steeple.
Draw an indefinite line; on which set off Ac=200 equal parts, for the measured distance. Erect the indefinite perpendicular AB; and draw CB so as to make the angle c = 47° 30', the angle of elevation; and it is done. Then AB, measured on the scale of equal parts, is nearly 2181
What was the perpendicular height of a cloud, or of a balloon, when its angles of elevation were 35° and 64°, as taken by two observers, at the same time, both on the same side of it, and in the same vertical plane; the distance between them being half a mile or 880 yards. And what was its distance from the said two observers?
Draw an indefinite ground line, on which set off the given distance AB 880; then A and B are the places of the observers. Make the angle A 35°, and the angle" 64°; then the intersection of the lines at c will be the place of the balloon: whence the perpendicular CD, being let fall, will be its perpendicular height. Then by measurement are found the distances and height nearly as follow, viz. AC 1631, BC 1041, DC 936.
Having to find the height of an obelisk standing on the top of a declivity, I first measured from its bottom a distance of 40 feet, and there found the angle, formed by the oblique plane and a line imagined to go to the top of the obelisk, 41°; but after measuring on in the same direction 60 feet farther, the like angle was only 23° 45'. What then was the height of the obelisk?
Draw an indefinite line for the sloping plane or declivity, in which assume any point A for the bottom of the obelisk, from which set off the distance Ac=40, and again CD=60 equal parts. Then make the angle c=41°, and the angle D= 23° 45'; and the point B where the two lines meet will be the top of the obelisk. Therefore AB, joined, will be its height.
Wanting to know the distance between two inaccessible trees, or other objects, from the top of a tower 120 feet high, which lay in the same right line with the two objects, I took the angles formed by the perpendicular wall and lines conceived to be drawn from the top of the tower to the bottom of each tree, and found them to be 33° and 64°. What then may be the distance between the two objects?