in the construction of tables of natural sines, tangents, &c. To glance, for a moment, at this application, in passing, suppose it were required to find the natural sines to each of the 10 first minutes of the quadrant.

The radius of a circle being 1, the semi-circumference is known to be 3.14159265358979. This being divided successively by 180 and 60, or at once by 10800, gives 0002908882086657, for the arc of 1 minute. Of so small an arc the sine, chord, and arc, differ almost imperceptibly from the ratio of equality; so that the first ten of the preceding figures, that is, 0002908882 may be regarded as the sine of 1'; and in fact the sine given in the tables which run to seven places of figures is ⚫0002909. By chap. i. art. 19, we have, for any arc, cos = √(1-sin2). This theorem gives, in the present case, cos l'′ = 9999999577. Then, by prop. 12, cor. 2, (A), of this chapter, we shall have

2 cos l'x sin l'
2 cos l'x sin 2'
2 cos l' x
2 cos l' x

sin 0'sin 2′ = •0005817764

- sin l'

=

sin 3'
sin 4'

- sin 2'

sin 3'
sin 4'
&c.

2 cos l'x sin 5'

&c.

sin 3'0008726646 sin 4' = ⚫0011635526 sin 5'0014544407 sin 6' = •0017453284 &c.

Thus may the work be continued to any extent, the whole difficulty consisting in the multiplication of each successive result by the quantity 2 cos 1'=1.9999999154. Or, the sines of 1' and 2' being determined, the work might be continued by the last proposition, thus:

sin l': sin 2'- sin l' :: sin 2' + sin l': sin 3′
sin 2': sin 3′-
sin 3': sin 4'

sin l' :: sin 3′ + sin 1': sin 4'

sin 1':: sin 4' + sin 1': sin 5'

sin l' :: sin 5′+ sin l' : sin 6'

In like manner, the computer might proceed for the sines of degrees, &c. thus:

sin 2° sin 3°

sin 1°: sin 2° - sin 1°:: sin 2° + sin 1°: sin 3°
sin 1°:: sin 3° + sin 1: sin 4°
sin 1° :: sin 4° + sin 1° : sin 5°
&c.

To check and verify operations like these, the proportions should be changed at certain stages. Thus, sin 1°; sin 3° sin 2°:: sin 3° + sin 2°: sin 5° sin 1° sin 4° sin 3°: sin 4° + sin 3°: sin 7° sin 4° : sin 7°

sin 3°:: sin 70+ sin 3°: sin 10o. The coincidence of the results thus obtained, with the analogous results in the preceding operations, will manifestly establish the correctness of both.

The sines and cosines of the degrees and minutes up to 30°, being determined by these or other processes (some of which will be indicated in the 4th chapter), they may be continued thus:

sin 30° 1' = cos l'

sin 30° 2' = cos 2'

sin 30° 3′ cos 3'

sin 29° 59′

And these being continued to 60°, the cosines also become known to 60°; because

The sines and cosines from 60° to 90°, are deduced from those between 0° and 30°. For

sin 60° 1' = cos 29° 59'

sin 60° 2′ cos 29° 58'

&c. &c.

&c.

21. The sines and cosines being found, the versed sines are determined by subtracting the cosines from radius in arcs less than 90°, and by adding the cosines to radius in arcs greater than 90°.

22. The tangents may be found from the sines and

by the theorem for the tangents of the sums and differences of arcs. For, when the radius is unity, the tangent of 45° is also unity, and tan (A + B) will be denoted thus:

And this, again, may be still further simplified in practice.

23. The secants may readily be found from the tangents by addition. For (prop. 3) sec A tan A + tan comp A. Or, for the odd minutes of the quadrant the

secants may

1

be found from the expression sec = COS

Other methods for all the trigonometrical lines are deduced from the expressions for the sines, tangents, &c. of multiple arcs; but this is not the place to explain them, even if it were requisite to introduce them at large into a cursory outline.

CHAPTER III.

Solutions of the several Cases of Plane Triangles. 1. THERE being in every plane triangle, six things, namely, three sides and three angles, of which some three are given to determine the other three; and the combinations that can be formed out of six quantities,

6.5.4

taken three and three being = or 20; it might at

1.2.3

first sight be imagined that 20 distinct rules would be required in this branch of trigonometry. But though the varieties of data are in truth thus numerous, it will

soon appear, that the number of cases which require distinct rules, are very few. Let A, B, C, denote the angles of a plane triangle, and a, b, c, the sides respectively opposite; then the twenty varieties of data will be these, viz.

(3) abc... (1) ɑba

(1) aca
(2) bca

(1) аbв (2) abc
(2) асв (1) acc
(1) bcв

(4) ABC... (1) ABа
(1) Aca

...

(1) bcc

(*) acb ....... (1) ACC
(1) Bcb

(1) BCC.

Here the varieties which are marked by the figure 1, have this in common, that a side and its opposite angle are two of the given parts; and, if it be considered that when two angles of a plane triangle are known, the third is, in fact, given, because it is the supplement of their sum, it will appear that those varieties which are marked with an asterisk fall under this first case: so that the varieties comprehended in this case are fifteen. Of the remaining five, three, indicated by the number 2, have this in common, that two sides and an included angle are given; they, therefore, constitute a second case. The example marked 3, has three sides given, but no angle; this makes a third case: and the remaining variety in which the three angles are given, but no side, would make a fourth case, were it not (see chap. i. 6) that for want of a side among the data, the problem thus expressed is unlimited. Our twenty varieties, therefore, only furnish three distinct cases, to the solution of which we shall now proceed.

CASE I.

2. When a side and its opposite angle are among the given parts.

Here the solution may be obtained by means of chap. ii. prop. 14, where it is demonstrated that the sides of plane triangles are respectively proportional to the sines of their opposite angles.

C

In practice if a side be required, begin the proportion with a sine, and say,

As the sine of the given angle,

To its opposite side;

So is the sine of either of the other angles,
To its opposite side.

If an angle be required, begin the proportion with a side, and say,

As one of the given sides,

Is to the sine of its opposite angle;
So is the other given side,

To the sine of its opposite angle.

The third angle becomes known, by taking the sum

of the two former from 180°.

Note 1. Since sines are lines, there can be no impropriety in comparing them with the sides of triangles; and the rule is better remembered by young mathematicians, than when the sines and sides are compared each to each.

Note 2. It is usually, though not always, best to work the proportions in trigonometry by means of the logarithms, taking the logarithm of the first term from the sum of the logarithms of the second and third, to obtain the logarithm of the fourth term. Or, adding the arithmetical complement of the logarithm of the first term to the logarithms of the other two, to obtain that of the fourth.

Note 3. It is an excellent plan to accustom the pupil to draw (previously to his commencing the computations in plane trigonometry) not a rough, but a neat and accurate, sketch of the triangle proposed, from the given data, by means of scale, protractor, and compasses. Such construction will enable him at once to trace the peculiarities of the problem, and to detect its ambiguities, if there be any. It will also, if accurately performed, upon a scale of moderate size, give the sides to within their 200th part, and the angles true to within half a degree.