eye of the student, serves admirably to guard him against the admission of error; and is, therefore, well fitted for adoption at the commencement of this branch of science. But of late years, another method, first introduced by Euler, has been generally employed by the continental mathematicians, and very frequently by those of England. It is analytical. The nature and mutual relation of the lineo-angular quantities, sines, tangents, &c. being defined by a few obvious and simple equations, every other theorem and formula that is likely to be of use, is deduced with great facility by the mere reduction and transformation of the original equations. This method serves greatly to shorten almost all trigonometrical investigations, except a few which lie at the foundation of the science: and admitting of an extension to which the geometrical method cannot lay claim, at the same time that it proceeds to some of its most important results with great rapidity; no treatise on trigonometry can be complete that does not assign to this manner of handling the subject a considerable place. 2. Previously to the student's entrance upon this department of enquiry, it will be expedient for him to trace the mutations which the principal intermediary or lineo-angular quantities undergo, when they relate to arcs found in different quadrants of the same circle. To this end let him draw afresh and lay before him, the diagram given at chap. i. art. 18, and first trace the mutations of the sines and the cosines; keeping in mind this general principle, that every variable algebraic quantity changes its sign after it becomes 0. 3. Let the arc be supposed to commence at A, and to increase in the direction A BEA'B. As the arc augments through the first quadrant AE, the sine augments till it becomes equal to ce, the radius; passing from 90° to 180° through the quadrant EA', the sine diminishes, but is to be regarded as positive till the arc becomes AEA' or 180°. In that state of the arc the value of the sine is obviously 0. Passing on to the third quadrant, as when the arc is AEA'B', the sine D'B' is directed contrarily to what it was in the first two quadrants, and is then to be regarded as negative. In this state it continues to increase through the third quadrant, at the end E' of which it is again equal to radius. From thence the negative value of the sine diminishes, till at the end A of the fourth arc, it again passes through zero, and the sine becomes positive in the fifth quadrant; as it obviously ought to do, since the fifth quadrant is coincident with the first. In the 1st and 2d quadrants, then, the sines are +, In the 3d and 4th, they are —. If the arc were taken negatively, its sine would, for like reasons, be negative in the first half, positive in the second half, of the circle. 4. The cosine of an arc is equal to radius at the point A, where the arc is evanescent. From thence it diminishes while the arc increases through the first quadrant; at E the cosine is 0. Then it changes its sign and continues negative through the arc EA'E', that is, from 90° to 270°. At Ef the cosine is again 0; and, of course, from E' to A, through the fourth quadrant, it is again positive. The rule is the same for a negative arc. 5. With regard to the signs of the tangents, it is evi dent, since tan = . (chap. i. 19), that when the signs of the sine and cosine are both alike, whether positive or negative, the sign of the tangent is positive; and when those signs are unlike, the sign of the tangent is negative. The tangent, therefore, is positive in the first quadrant, the sine and cosine being then both positive: it is negative in the second quadrant, the cosine being then negative though the sine remains positive; it becomes positive again in the third quadrant, the sine and cosine being then both negative: finally, it is negative in the fourth quadrant, since the sine is then negative and the cosine positive. The tangent, therefore, changes its sign at the end of every quadrant, where it passes alternately through nothing and infinity; these being, indeed, the algebraic indications of the changes of Sign. "if the arc is regarded as negative, the rule for the tangents becomes inverted; the tangent being then negative in the first and third, positive in the second and jourth arcs. 6. For the cotangents the rule of the signs is the same as for the tangents: this is evident, because 7. So again, the rule for the secants is the same as for - - l the cosines; because sec = cos' 8. The rule for the cosecants is, also, the same as for - l the sines; because cosec = sin' 9. Proceeding in this way, a second, a third, &c. time over the circumference of the circle, like muta tions would occur. The results of the whole may be thus tabulated:— 10. Let it be remembered that when the tangent passes through infinity, the secant does, and for the same reason, viz. because they in that case could only limit each other at an infinite distance. The principal changes, then, in point of magnitude, may be easily traced, and tabulated as below; where the sign oo denotes infinity. . . . .90°. ... 180°. . . .270°.... 360°) . . . . R. . . . . 0 . . . . ~ R . . . . O .... - Co . . . . SeC . . . . . . . Co . . . . - R. . . . . -oo . . . . COS . . . . ... 0 . . . . -R .... 0 . . . . cot . . . . ... O . . . . -oo . . . . 0 . . . COSeC . . co . . . . R. . . . . - Co . . . . R. . . . . 11. These particulars premised, the properties and relations of the sines, tangents, &c. of combined and multiple arcs, as well as rules for the solutions of triangles, may be investigated analytically. i. ABc, in the annexed figure, be C any plane triangle, c the vertical angle, & No Sin .... tan . . . . cD a perpendicular let fall from it upon the base or base produced, and let a, b, and c, denote the sides respectively opposite to the angles A, B, and c. Then, since Ac = b, AD is the cosine of A to that radius; consequently, when radius is unity we have a D = b cos A. In like manner BD = a cos B. Therefore, AD -- BD = AB = c = a cos B + b cos A. If one of the angles, as A, were obtuse, the result would, notwithstanding, be the same; because, while on the one hand cos A would be negative, AD, lying on the contrary side of A to what it does in the figure, must be deducted from BD to leave AB, and a negative quantity subtracted, is equivalent to a positive quantity added. By letting fall perpendiculars from the angles A and B, upon the opposite sides, or their continuations, precisely analogous results will be obtained. They may be placed together, thus:– a = b cos c + c cos B = a COS G + C COSA c = a cos B + b cos A 12. If BC, or a, be regarded as radius, CD will be the sine of the angle B to that radius; therefore, to the radius unity, CD will be = a sin B. So again, for a like reason, cd = b sin A. Consequently, a sin B = 5 sin A, rt sin A or H = H. In a similar manner, we may obtain b sin a ax sin A b sin n - - = ..., and F = H. Or, changing the denominac st in C C Si in C tors, the relations of all the six quantities may be thus expressed, sin A sin B sin c al ==== +...... (2) These equations are, manifestly, of similar import with chap. ii. prop. 14. 13. From the last equations we have, sin A sin B a = c + .... b = c : , slin C - st in C Substituting these values of a and b for them in the equation c = a cos p + b cos A, and multiplying the - in c ... . equation so transformed, by **, it will become sin c = sin A cos B + sín B cos A, Now, since in every plane triangle, the sum of the three angles is equal to two right angles, A + b = suplement of C ; and, since an angle and its supplement o the same sine, it follows that sin (A + B) = sin c; whence sin (A + B) = sin A cos B + sin B cos A. 14. If in this equation B be regarded as subtractive, then will sin B obviously be subtractive also ; but cos B will not change its sign, because it will still continue to be estimated in the same direction on the same radius The equation will, therefore, become sin (A – B) = sin A cos B – sin B cos A. 15. Conceive B' to be the complement of B, and 3 O to be the quarter of the circumference, or the measure of a right angle: then will B" = }O – B, sin B" = cos B, and cos B" = sin B. But, by the preceding article, sin (A – B') = sin A cos B" – sin B' cos A. Substituting for |