method of measuring the parts required by scale and compaffes.

For, a radius being taken at pleasure, and the lengths of the fines, verfed fines, tangents, and fecants of all arcs from 1 to 90°, calculated in proportion to it, and ranged in order in tables, or marked on a scale, affords a canon for calculating or measuring the angles and fides of any triangle; one fide, and other two of its parts, being given.

21. The three angles of any plain triangle are equal to two right angles, or 180°.

COR. I. If one angle of a plain triangle be given, the fum of the other two angles may be found, by fubtracting the given angle from 180°,

2. If two angles of a plain triangle are given, the third may be found, by fubtracting the fum of the two given angles from 180o.

3. The two acute angles of any right angled plain triangle, are equal to one right angle, or 90°.

4. If one of the acute angles of a right angled plain triangle be given, the other is found by fubtracting the given angle from 90°.

22. In any right angled plain triangle ABC, Fig. 27. the line AC oppofite to the right angle, is called the Hypothenufe; and AB and BC, which contain the right angle, are called Sides, or Legs.

23. Plain triangles are either right angled, or oblique angled; and each kind hath several cases. 24. Similar triangles are those which have their angles equal, each to its correspondent angle.

The

25. The fides about the equal angles of fimilar triangles are proportional. Eucl. B. 6. prop. 4.

THEOREM. If from any point in the fide of a plain triangle, a right line be drawn parallel to one of the other fides, it will cut off a triangle fimilar to the whole. Fig. A. plate 2.

Let ABC be a triangle, and from any point E in the fide, AB draw EF parallel to BC; I fay the triangle AEF is fimilar to the triangle ABC.

For, fince the line AB falls upon the two parallels EF and BC, the angle AEF is equal to the angle ABC; and, for the fame reason, the angle AFE is equal to the angle ACB; also the angle at A is common to the two triangles AEF, ABC; and therefore the triangle AEF is fimilar to the triangle ABC. In like manner, if FD be drawn parallel to BC, it may be de monftrated that the triangle FDC is fimilar to the triangle ABC. Hence, and from the 4th prop. of Euclid's 6th book, we have the following proportions, viz.

26. By Radius, in Trigonometry, is understood, the radius of the tables of fines and tangents, commonly called the Trigonometrical Canon; and is generally supposed to consist of 100000000 equal parts.

RIGHT ANGLED TRIANGLES.

CASE I. Fig. 27.

In the right angled plain triangle ABC, suppose the two fides AB and BC were given, and it were required to find the angles at A and C.

If the given fides are equal between themselves, each of the acute angles is half a right angle, or 45°; but, if not, the angles may be found by this proportion:

As one of the given fides

Is to the other,

So is the radius

To the tangent of the angle adjacent to the firft fide. For, produce AB and AC to D and E, so that AD may be equal to the radius. About the center A, with the radii AB and AD, describe the arcs Be and DF, whofe tangents are BC and DE; each of thefe arcs is the measure of the angle at A, and the triangles ABC and ADE are fimilar: Now, fince AB, BC, and AD are known; DE, or the tangent of the angle at A may be found thus, as AB: BC :: AD: DE. That is,

As AB: BC :: Rad. : Tan. A. Hence,

As any fide of a right angled plain triangle AB, Is to the other fide containing the right angle BC, So is the radius

To the tangent of the angle adjacent to the first fide A. In the fame manner it appears, that

CB

CB: BA:: Rad. : Tan. C; and when either angle is found, fubtract it from 90°, and the remainder is the other angle.

In the right angled plain triangle ABC, fuppofe the hypothenufe AC and the fide AB were given, and it were required to find the angles at A and C. As the hypothenuse AC

Is to the fide AB,

So is the radius

To the fine of the angle at C.

Make CE equal to the radius ; and about the center C defcribe the two arcs An, Em, whofe fines are AB, ED; either of which is the fine of the angle C, according to the radius affumed: And because the triangles ABC, ECD are fimilar, and CA, AB, and CE, are known; ED, or the fine of the angle at C, may be found thus,

As CA: AB::CE: ED; that is,

As CA: AB:: Rad.: Sine C.

In the fame manner it may be demonftrated, that AC: CB:: Rad. : Sine A.

As the hypothenuse

Is to any fide,

So is the radius

Therefore,

To the fine of the angle oppofite to that fide.

Hence we have rules for finding any part of a right angled plain triangle, the hypothenufe, or a fide, and any other two of its fix parts, being given.

C A SE III. Fig. 27.

In the right angled triangle ABC, suppose the fide AB, and the angles at A and C were given, to find the fide BC, and the hypothenufe AC.

1. To find the fide BC.

Since by Cafe 1. AB

Therefore

Rad.

BC :: Rad.: Tan. A,
Tan.A:: AB : BC.

2. To find the hypothenuse AC.

Since by Cafe 2. AC: AB :: Rad. : Sine C,
Therefore Sine C: Rad. :: AB : AC.

In the right angled triangle ABC, suppose the hy. pothenuse AC, and the angles at A and C were given, and it were required to find the fides AB and BC.

1. To find AB.

Since by Cafe 2. AC: AB :: Rad.: Sine C;
Therefore Rad.: Sine C:: AC : AB.

2. To find BC.

By the fame Case, Rad. : Sine A:: AC: CB,

EXAMPLES in right angled Triangles.

CASE 1. Fig. 27. In the right angled triangle ABC, there is given the fide AB=429, and BC=316, to find the angles at A and C.