As the base, or sum of the segments,
Is to the sum of the other two sides;
So is the difference of those sides,

To the diff. of the segments of the base.

Then take half this difference of the segments, and add it to the half sum, or the half base, for the greater segment; and subtract the same for the less segment.

Hence, in each of the two right-angled triangles, there will be known two sides, and the right angle opposite to one of them; consequently the other angles will be found by the first theorem.

Demonstr. By theor. 35, Geom. the rectangle of the sum and difference of the two sides, is equal to the rectangle of the sum and difference of the two segments. Therefore, by forming the sides of these rectangles into a proportion by theor. 76, Geometry, it will appear that the sums and differences are proportional as in this theorem.

Draw the base AB = 345 by a scale of equal parts. With radius 232, and centre A, describe an arc; and with radius 174, and centre B, describe another arc, cutting the former in c. Join AC, BC, and it is done.

Then, by measuring the angles, they will be found to be nearly as follows, viz.

ZA 27, 4 B 37°, and c 115o.

2. Arithmetically.

Having let fall the perpendicular cp, it will be,
As the base AB AC + BC: AC BC: AP

-

that is, as 345 : 406′07 :: 57·93 : 68∙18 = AP its half is

BP,

34.09

Then, in the triangle APC, right-angled at P,

Again, in the triangle BPC, right angled at P,

So that all the three angles are as follow, viz.

the A 27°4'; the B 37° 20'; the 4 c 115° 36′.

3. Instrumentally.

In the first proportion.-Extend the compasses from 345 to 406, on the line of numbers; then that extent reaches, on the same line, from 58 to 68.2 nearly, which is the difference of the segments of the base.

In the second proportion.-Extend from 232 to 2061⁄2, on the line of numbers; then that extent reaches, on the sines, from 90° to 63°.

In the third proportion.-Extend from 174 to 138; then that extent reaches from 90° to 52° on the sines.

The three foregoing theorems include all the cases of plane triangles, both right-angled and oblique. But there are other theorems suited to some particular forms of triangles, which are sometimes more expeditious in their use than the general ones; one of which, as the case for which it serves so frequently occurs, may be here taken, as follows:

THEOREM IV.

When a Triangle is Right-angled; any of the unknown parts may be found by the following proportions: viz.

As radius.

Is to either leg of the triangle;

So is tang. of its adjacent angle,

To its opposite leg;

And so is secant of the same angle,
To the hypothenuse.

E

E

Demonstr. AB being the given leg, in the right-angled triangle ABC; with the centre A, and any assumed radius AD, describe an arc DE, and draw DF perpendicular to AB, or parallel to BC. Then it is evident, from the definitions, that DF is the tangent, and AF the secant of the arc DE, or of the angle A which is measured by that arc, to the radius AD. Then, because of the parallels BC, DF, it will be,

as AD: AB :: DF: BC and :: AF AC, which is the same as the theorem is in words.

Note. The radius is equal, either to the sine of 90°, or the tangent of 45°; and is expressed by 1, in a table of natural sines, or by 10 in the log. sines.

Make AB

1. Geometrically.

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 BC 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 rights angled 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

B D L

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

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 square root of 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.

C

B

G

H

round