From I. and II. we have, writing b, B for a, A, and

41.

To shew that in a triangle ABC, in which C is a right angle, A and a are of the same affection, as are also B and b.

Now since b is less than π, sin b is positive; therefore tan a and tan A must be of the same sign. And because is the limit both of a and of A, these angles must be both greater or both less than a right angle; that is, A and a are of the same affection.

Similarly, from sin a =

tan b tan B'

it appears that B and b

are of the same affection.

42. If in a right-angled triangle, an angle and the side opposite to it be the only given quantities, the triangle cannot be determined.

For if the circles AB and AC intersect again in A', and C be a right angle, it is evident that ACB and A'CB have the angles A, A'

B

equal, and CB the side opposite to these angles is the same in both triangles. It is therefore ambiguous whether ABC or ABC be the triangle sought.

This ambiguity will also appear, if it be attempted to determine the triangle by Napier's rules.

tell from the equation

For we cannot

whether we are to take the angle AC, or its supplement A'C.

Α'

43. The solutions of the other cases of a right-angled triangle from two given parts are not ambiguous, if attention be paid to the two following principles,

(1). The greater side is opposite to the greater angle.

(2). An angle and the side opposite to it are of the same affection.

For example: Let c and A be given, to find a, B, b.

And since a and A are of the same affection, the greater or lesser angle which satisfies this equation is to be taken for a,

according as A is greater or less than

П

2

Again, sin

( 1 −c),

or cos c,

= cot A. cot B;

tan B = cot A. sec c.

And B is 90° as the second member of the equation is positive or negative; that is, as A and c are of like or unlike affection.

Again, sin

π

C or cos c, = cos a . cos b ;

cos b = cos c. sec a.

And bis 90°, according as the second member of the equation is positive or negative, that is, as a and c are of like or unlike affection.

44. In selecting a formula attention must be paid to the principles laid down in Appendix 11. to Pl. Trig. The following formulæ may be used with advantage, when the side or angle required is small, or nearly equal to one or to two right angles.

By Napier's rules, cos c = cot A. cot B, whence c cannot be accurately determined, if it be either a very small angle or nearly equal to two right angles.

When c and A are given, if a be nearly a right angle, it cannot be accurately determined from its sine. In this case cot B = cos c. tan A determines B, and a may be found from

45. DEF. A triangle is called quadrantal, if any one of its sides be a quadrant.

A quadrantal triangle ABC, of which the side c is a quadrant, may be solved by Napier's rules, if the qua

drantal side be neglected, and

be taken for the circular parts.

π 2

π

П

a

b, A, B, C

2

Let A'B'C' be the polar triangle. It may be solved

by Napier's rules, because C'(= π − c = 7) is a right angle ;

T

A Collection of Examples is added at the end of this Treatise.

CHAPTER IV.

ON THE SOLUTION OF OBLIQUE-ANGLED TRIANGLES.

46. LET the three sides be given, (a, b, c).

The angles may be determined from one of the formulæ (xvi.), (xvii.), (xviii.), (xix.).

47. Let the three angles be given, (A, B, C).

The sides may be determined from one of the formulæ (xx.), (xxi.), (xxii.), (xxiii.).

48. Let two sides and the included angle be given, (a, C, b).

By Napier's first and second analogies, (xii.) and (xiii.),

And A and B being known, c is found from