Page:Encyclopædia Britannica, Ninth Edition, v. 13.djvu/17

INFINITESIMAL CALCULUS tangents to curves described by a roulette motion. My solution is as follows. If a rectilinear polygon be conceived to roll on a right line, the curve described by any one of its points will be composed of a number of arcs of circles, and the tangent at any point on one of these arcs is perpendicular to the line drawn from the point to that in which the polygon is in contact with the base, when describ ing the arc. Consequently, if we consider a rolling curve as a polygon of an infinite number of sides, we see clearly that the roulette traced by any point must possess the same property; that is to say, the tangent at any of its points is perpendicular to the right line connecting it with the point of contact of the rolling curve and its base." In this we perceive that Descartes gave a genuine and most important application of the infinitesimal method. Again, Descartes first introduced the method of indeter minate coefficients into analysis, a principle, as was ably shown by Carnot, which is of itself sufficient to establish, by ordinary algebra, the fundamental principles of the in finitesimal calculus. Fermat. r -^ ne method of Fermat for drawing tangents was based on his method of maxima and minima. This latter was founded, as already observed, on a principle of Kepler s, viz., that, whenever a magnitude attains a maximum or mini mum, its increment or diminution, for a very small change in the variable on which it depends, becomes evanescent. Accordingly, to determine the maximum or minimum of any function of x, Fermat substituted x + c instead of x, and equated the two consecutive values of the function ; then, removing the common terms, and dividing by e, he made e = 0, and obtained an equation for determining the maximum or minimum value. Thus, adopting the modern notation, let y =f(x), and y x =f(x + e), then f(x + e) -/(#) = 0. Dividing by e, c)-f(x) = 0; hence / (x) = 0. Thus the roots of the derived equation, f (x) = Q, correspond to the maximum or minimum values of f(x). Consequently we see that Fermat s rule agrees with that of the differential calculus, and in fact is the method of the calculus as applied to such cases. 1 In consequence of Fermat s both having introduced the conception of an infinitely small difference, and also having arrived at the principle of the calculus for determin ing maxima and minima, it was maintained by Laplace, Lagrange, Fourier, and other eminent French mathema ticians that Fermat ought to be regarded as the first in ventor of the differential calculus. In reply to this we need but introduce the remark of their distinguished country man Poisson, "that this calculus consists in a system of rules proper for finding the differentials of all functions, rather than in the use which may be made of these infinitely small variations in the solution of one or two isolated pro blems" (Mem. de I Acad. des Sci., 1831). Fermat seems to have given no general demonstration of his method, but contented himself with giving particular applications of it to some pro blems of maxima and minima, as well as to finding the tan gents to and the centres of gravity of a few curves. Fermat applied his method to drawing a tangent, as fol- lows : 1 Fermat was in possession of his method in the year 1629, as appears from a statement in one of his letters to Itoberval, although it was not made public until this correspondence was printed by M. Ilerigone in his Cursus Mothematicus (1644). Suppose CD (fig. 1) the ordinate, and CF the tangent at the point C in a curve, meeting the axis AB in F ; from E, a near point on CF, draw an ordinate EG ; then CD FC IIC -___ > ___ , if the curve be concave to the axis, JJr Lrr Irr and , if the curve be convex. CD PF ^ GF" Hence, in cither case, the ratio of the ordinate CP to the sub- tangent PF is a maximum or a minimum relatively to the ratio for a near ordinate II G to GF, the abscissa measured from F, the foot of the tangent. Accordingly, if CP = ?/, IIG^^, and PF = , we have, by the method of maxima and minima, 1L = JLL t te It is easily seen that this method furnishes the ordinary value for the subtangent, as obtained by infinitesimals ; for, denoting by x, y the coordinates of C, let < = FD, ~DG = dx, yi = y + dy, and we have y y "T cty , ct-x t t + dx dy from which the subtangent t can be obtained. The method of Fermat was improved and extended by an Italian, Cardinal Ricci, in his Geometrica cxer- citatio (166G). Ricci was the first who showed that, if (a - x) m x n is a maximum, we must have no, iii + n This he easily established when m and n are integers, from the principle that if a magnitude bo divided into r equal parts, their continued product is greater than that obtained by dividing it into r parts in any other manner The following application, as given by him, to the curve y m -=*px", m>n, will help to illustrate this method of drawing tangents. To draw the tangent at C (fig. 1) take AF : AD = m - n : n, and join FC ; then FC touches the curve at C. For the product AF m ""AD" is a maximum by the preceding lemma; hence the product AF m ~"AG" is not a maximum for the line FG ; consequently AF "-"AD" > AF B> -"AG ~~FDi~ FG" but, from the equation of the curve, f^-f^ , FP CD CD CD or EG > Gil i.e., the point E falls outside the curve. In like manner it can be shown that any other point on CF lies outside the curve, and con sequently CF touches the curve at C. Barrow, Newton s predecessor in the Lucasian chair of Barrow. mathematics at Cambridge, simplified and extended the method of Fermat, and advanced a step further in the development of the infinitesimal method, by the introduc tion of two infinitesimals instead of one in the problem of drawing a tangent. His method was as follows : Let a; y be the coordinates of a point P on a curve (fig. 2), and take Q an adjacent point ; let e = PR = MN be the incre ment of x, and a = QR the increment of y ; then, sub stituting x + e for x, and y + a for y, in the equation of the curve, subtracting the equation of the curve for the original values, and rejecting all terms of the second and higher degrees in a and e, he obtained the limiting value of a : e, or of PM : MT, thus determining the value of the sub- tangent. The triangle PQR, which has for its sides the elements of the curve, of the abscissa, and of the ordinate, has been called Barrow s differential triangle. The elements which Barrow represented by a and e Leibnitz subsequently styled dy and dx, the differentials of the ordinate and abscissa of the point on the curve. Thus