two of these equations, we deduce from them

sin 0,002, cos 0,002, sin 0,004, cos 0,004, &c.,

and taking a 0o,001, b = 0o,002, a = 0a,002, b = 0,003, &c., we obtain by means of the last two equations

sin 0,003, cos 0o,003, sin 0,005, cos 0,005, &c.

It will be perceived from what has been said, how a set of trigonometrical tables may be formed. There are other methods, more convenient for calculating the sines of any arcs whatever, by means of converging series, which are deduced from the equations of art. 11. They may be found in the introduction to my Treatise on the Differential and Integral Calculus.

21. To render the calculation more easy, the custom has been for a long time to use logarithms instead of the values of the sines, cosines, tangents, and cotangents; and in most tables the latter are not to be found. The questions, therefore, which present themselves, are of the following nature.

1. An arc being given, to find the logarithm of its sine, that of its cosine, tangent, or cotangent.

2. The logarithm of the sine, that of the cosine, tangent, or cotan» gent of an arc being known, to find this arc.

In solving these questions, regard must be had to the particular disposition of the tables, that are used, as they are not all alike, and each is usually accompanied with the necessary directions. I shall omit, therefore, giving the instruction, that the student may want on this subject. I will merely mention the tables of Callet, as the best for the ancient division, and those of Borda, or those of Hobert and Ideler for the new. 22 Trigonometrical tables extend only to the fourth part of a circle; but they give, notwithstanding, the sines and cosines, the tangents and cotangents, for all arcs however great. This I shall now proceed to show, by tracing the progress of the trigonometrical lines with respect to the different degrees of magnitude, which an arc of a circle is capable of assuming.

In order to comprehend fully what I am about to offer, we must first understand the continuity, which prevails among the

different results obtained from the same algebraic expression, or from the same geometrical construction, and which consists in this, that each value, which the expression in question assumes, is preceded and followed by values, which differ as little as we please, from the first, and that in describing a line, each point is preceded and followed by points, which are immediately contiguous. Fig. 10. This being supposed, if we conceive the radius MC (fig. 10), at first coinciding with AC, to turn about the point C, as upon a pivot, this radius will form, successively, with AC, all possible angles; and the point M, situated at the extremity, will pass over all the points of the circumference of the circle ABA'B'A, or, which is the same thing, will describe it. By following with attention this motion, we see in the first place, that at the point A, where the arc is nothing, the sine is also nothing, and the cosine does not differ from the radius AC. When the radius CM moves off from AC, the sine PM increases, as the point M, which I shall call the describing point, advances toward B, and when it has reached it, PM becomes equal to CB, or to radius. Under the same circumstances, the cosine PC diminishes continually and becomes nothing, when the point M is in B; the angle ACB is then a right angle, and the arc AB. The point M being continued beyond B, the sine decreases and the cosine, which falls now upon the diameter AB on the side of the point C opposite to that in which it was before, increases. This is evident. from the figure; P'M', the sine of ABM', is less than BC, the sine of AB, and CP', the cosine of the first of these arcs, exceeds the cosine of the second, which is nothing. It may be remarked, that P'M' and CP' are, respectively, the sine and cosine of the arc A'M', counted from A', and the supplement of ABM'; whence it follows, that an obtuse angle has the same sine and the same cosine, as its supplement.

When the point M has arrived at A', the sine is nothing, as at the point A, and the cosine is again equal to radius. At the point A' the arc ABA' is equal to the semicircumference ; the angle ACM has attained its greatest magnitude, but there is nothing to prevent the radius CM and the describing point, being continued below the diameter AA'. The sine, which then becomes P"M", falls also below the diameter, and increases, according as the point M" approaches to B', while the cosine. GP" diminishes. At the point B', where the arc ABA'B' is 3

of the circumference, or, the sine is equal to the radius CB' and the cosine is nothing. Lastly, from B' to A the sine B""M"", constantly below AA', diminishes continually, and the cosine CP", which is now on the same side of C, that it was in the first quadrant AB, increases and becomes equal to radius in A. At this point the sine is nothing; the describing point has com pleted a revolution, but we may suppose it to begin another, and by considering, as a single arc, the whole course passed over by this point from the commencement of its motion, we have arcs that exceed a circumference and which have the same sines, cosines, tangents, cotangents, as those which are described in the first revolution. These considerations lead to consequences, that are of the greatest importance in analysis, and which I have developed in my treatise on the Differential and Integral Calculus. 23. It may be well now to see how the algebraic expressions for the sine and cosine correspond with the different circumstances, which we have been considering. In order to this, I make, in the first place, a = in the equations

There are two things to be attended to in these expressions, namely, their absolute value, and the sign with which it is 5 effected.

This value is verified by the figure; for AB being, if we take the arc BM' for b, the arc AM' will be +b; but P'M' being the sine of A'M', as well as of AM', will be the cosine of BM' or of b; while CP' will be the sine of b.

As to the sign, which affects cos (+b), it signifies, that if we regard, as positive, the sine and cosine of an arc less than the fourth of the circumference, the cosine of a greater arc will be negative, while its sine will be positive. If we make b, we have cos л = 1, sin л = 0.

Again, if we suppose, that in the equations (A), α = π, we shall obtain, according to what precedes,

Fig. 4.

The absolute value of these formulas may be verified, as easily as that of the preceding; the sign shows, that every arc comprehended between л and, has its sine and cosine negative; and when b = л, we have

Lastly, when a = 7, the equations (A) are reduced by means of the values just found to

from which it follows, that every arc comprehended between 3 and л, or 2 л, has its cosine positive and its sine negative. The results, then, to which we have arrived, are

1. That from the point A to the point A', at which the arc ABA', the sines are positive;

2. That from the point A to the point A, at which the arc AВA'В'A 2л, that is, from л to 27, the sines are negative; 3. That from the point A to the point B, at which the arc AB, the cosines are positive;

4. That from the point B to the point B', at which the arc ABA'B' = 1⁄2 ï, that is, from to, the cosines are negative;

5. Lastly, that from the point B' to the point A, at which the arc ABA'B'A = 27, that is, from л to 2 л, the cosines are positive.

It will be readily observed, that the sines change their sign, when they pass below the diameter AA', and the cosines, when they pass from one side to the other of the point C, or according as they fall on this side or that of the diameter BB' perpendicular to AA'.

By attending to these things we shall be able to extend the formulas of art. 11 to all possible magnitudes of the arcs AM and MN (fig. 4); and the values deduced from these formulas will agree with those which are derived from the construction and reasoning employed in the article referred to, if we apply them immediately to the proposed arcs. The application of these formulas would be a useful exercise for the learner.

24. By following the course of the tangents we find, that they Fig. 10. increase continually from the point A (fig. 10), to the point B, at which the arc AM becomes equal to . At this point the secant NC, coinciding with CB, is parallel to the tangent AN, and therefore no longer meets it; so that the arc AB has not,

properly speaking, a trigonometrical tangent. We say, nevertheless, that its tangent is infinite; but the meaning of this expression is, that by taking the point M sufficiently near to the point B, we may make the tangent AN greater than any assignable quantity. It is in this manner, that we show the truth of the equation tang a = which gives for tang a a value, so

sin a
cos a'

much the greater, as cos a becomes smaller, or as we approach nearer to the point B.

When a = 0,5 it becomes cos a = sin a (20), and, consequently, tang 0,5 = 1.

This may be shown also by the triangle CAN (fig. 9), which Fig. 9. becomes isosceles in this case, since the angle ACN, being equal to half a right angle, is necessarily equal to the angle ANC; the tangent AN is then equal to radius (Geom. 48).

When the arc AM (fig. 10), is greater than, the radius Fig. 10, CM will no longer meet the line AN above the diameter but below it. The true tangent AN' is equal, as may be easily shown, to A'n', the tangent of the arc A'M', the supplement of AM', but it lies in an opposite direction. In the third quarter of the circle the tangent which has nothing at the point A', returns above the diameter AA', and AN is the tangent of AA'M". The radius becomes again parallel to AN at the point B', and the tangent is infinite; beyond this point it falls below the diameter, and the arc AA'M"" has AN' for its tangent.

25. I proceed now to point out what results from the algebraic expression, tang a =

sin a

cos a

to

It is evident, that its value will be positive in all those cases, where the sine and cosine have the same sign, or from 0 to π, and from л to; it will, consequently, be negative from л, and from 2 to 2л; whence it follows, that for the tangents, as well as for the sines and cosines, a change of sign corresponds. to a change of situation; we find likewise, that the cotangents are positive from 0 to π, from л to л; and negative from to л, and from л to 2л.

26. We sometimes meet with negative arcs in a calculation, the sines and cosines of which may be determined by the formulas of art. 11. As the expression sin (a - b) changes its sign, when we change a into b, and b into a, it is manifest, that