TANGENT, in geometry, is defined, in general, to be a right line, which touches any arch of a curve, in such a manner, that no right line can be drawn between the right line and the arch, or within the angle that is formed by them. The tan gent of an arch is a right line drawn per pendicularly from the end of a diameter, passing to one extremity of the arch, and terminated by a right line drawn from the centre through the other end of the arch, and called the secant. And the co-tan gent of an arch is the tangent of the com plement of that arch. The tangent of a curve is a right line, which only touches the curve in one point, but does not cut it.
In order to illustrate the method of drawing tangents to curves, let A C G, Plate XIV. Miscel. fig. 10, be a curve of any kind, and C the given point from whence the tangent is to be drawn. Then conceive a right line, en g, to be carried along uniformly, parallel to itself, from A towards Q; and let, at the same time, a point, p, so more in that line, at to de scribe the given curve, A C G ; also let in en, or C n, express the fluxion of A ?n, or the velocity wherewith the line, in g, is carried; and let n S express the corres ponding fluxion of m p, in the position nt C g, or the velocity of the point, p, in the line, in g moreover, through the point, C, let the right line, S F, be drawn, meeting the axis of the curve, A Q, in P.
Now it is evident, that if the motion of p, along the line in g, was to become equable at C, the point, p, would be at S, when the line itself had got into the po sition in S g ; because, by the hypothesis, C n and n S express the distances that might be described by the two uniforin motions in the same time. And if w s g be assumed to represent any other posi tion of that line, and s the contemporary position of the point,p, still supposing an equable velocity ofp ; then the distances, C v, and v s, gone over in the same time by the two motions, will always be to each other as the velocities, or as C a to a S. Therefore, since C v: v s C : n 5, (which is a known property of similar tri angles), the point, s, will always fall in the right line; F C S, (fig. 11) : whence it ap pears, that if the motion of the point, p, along the line, at g, was to become uni form at C, that point would then move in the right line, C S, instead of the curve line, C G. Now, seeing the motion of p, in the description of curves, must either be an accelerated or retarded one ; let it be first considered as an accelerated one, in which case, the arch, C G, will fall wholly above the right line, C D, as in fig. 10 ; because the distance of the point, p, from the axis A Q, at the end of any given time, is greater than it would be if the acceleration was to cease at C ; and if the acceleration had ceased at C, the point, p, would have been always found in the said right line, F S. But if the mo
tion of the point, p, be a retarded one, it will appear, by arguing in the same man ner, that the arch C will fall wholly below the right line, C D, as in fig. 11.
This being the case, let the line in g, and the point p, along that line, be now supposed to move back again, towards A and m, in the same manner they proceed ed from thence ; then, since the velocity of p did before increase, it must now, on the contrary, decrease ; and therefore, as p, at the end of a given time, after repass ing the point, C, is not so near to A Q, as it would have been, had the velocity continued the same as at C, the arch, CA (as well as C G) must fall wholly above the right line, F C D: and by the same method of arguing, the arch, C h, in the second case, will tall wholly below F C D. Therefore F C D, in both cases, isa tan gent to the curve at the point, C ; whence the triangles, F m C, and C n being similar, it appears that the subtangent, m F, is always a fourth proportional to n S, the fluxion of the ordinate, C a, the fluxion of the absciss, and C tn, the ordinate ; that is, Sn:nG an C ; m F. Hence, if the absciss, A in x, and the ordinate an we shall have m by means of which general expression, and the equation expressing the relation between s and g, the ratio of the fluxions, a': and j will be found, and from thence the length of the sub-tangent, in F, as in the following examples.
1. To draw a right line C T, a tangent to is given circle, (fig. 12) B C A, in a given point, C. Let C S be. perpendicu lar to the diameter, A B, and put A B = eieB S =-. x, and S C y. Then, by the property of the circle, y. (= C S.) = B S XAS(=xxa—x) =ax-50; whereof the fluxion being taken, in or der to determine the ratio of a:: and y, we get 2yy =al:— 2 x a7:; consequently y a — 2 x which multiplied by y, gives y = the sub-tan ya x gent, $ T. Whence, 0, being supposed the centre, we have S (reri a — x): C S (= y) C S (= y) : S T ; which is also Ibund to be the case from other princi ples.
2. To draw a tangent to any given point, C, (fig. 13) of the conical parabola, A C G. If the haus rectum of the curve he denoted by a, the ordinate M C, by y, and its corresponding absciss, A M, by x ; then the known equation, expressing the relation of x and y, being a s = y=, we have, in this case, the fluxion a .1: =2 y y ; 2 y y x whence a. ---=—; and consequently, —re_— y a 2 2 a x 2 x = F. Therefore a a the sub-tangent is just the double of its corresponding absciss, A M. And so for finding the tangents of other species of curves..