Page:Encyclopædia Britannica, Ninth Edition, v. 13.djvu/16

6 INFINITESIMAL CALCULUS The method of Cavalieri was severely criticized "by some of his contemporaries, more especially by Guldin. They alleged that, since a line has no breadth, no number of right lines, however great, when taken together, could make up a plane area. This objection was answered by Cavalieri ; but the reply was stated in Pascal, the clearest form by Pascal, who observed (letter to M. do Carcavi, 1658) that the method of indivisibles possessed all the rigour of that of exhaustions, from which it differed only in the manner of expression, and that, when we con ceive an area as a sum of a system of parallel ordinates, we mean in reality an indefinite number of rectangles under the several ordinates, and the small equal portions into which we conceive the common perpendicular to these ordinates to be divided. This passage is remarkable as was well observed by Carnot as it shows that the notion of mathematical infinity, as now employed, was not strange to the geometers of that time ; for it is clear that Pascal employed the word " indefinite " in the same signification as we now attach to the word " infinite," and that he called " small" that which is now called "infinitely small," also that he neglected these small quantities in comparison with finite quantities thus he regarded as simple rectangles the small portions of the area of the curve comprised between two consecutive ordinates, neglecting the small triangles which have for their bases the differences of these ordinates. Carnot adds that no person attempted to reproach Pascal with want of rigour in his demonstrations. Pascal applied the method of Cavalieri with eminent success to the investigation of properties of the cycloid, and other problems. His researches, according to D Alembert, closely approach to the integral calculus, and form the con nexion between the methods of Archimedes and of Newton. Wallis. The most important application, however, of Cavalieri s method was that of Wallis, Savilian professor at Oxford, who, in 1G55, gave an admirable specimen of this method in his Arithmetical, Infinitorum, sive nova Methodus in- quirendi in Curvilineariim Quadraturam. Pursuing Cava lieri s views, he reduced the problem of finding the areas of a large number of curves, and the volumes of solids of revolution, to the summation of the powers of the terms of arithmetical series, consisting of an infinite number of terms, or rather to the determination of the ratio of the arithmetical mean of all such powers of the terms to the like power of the last term. For example, in the series of square numbers 0, 1, 4, 9, 16, &c., the ratio of the mean to the last is, for the first ,, three terms, + 1 + 4 4 + 4 + 4 = 15 + T > f r ^ JTJ-; for the first four, + 1+4 + 9 + 16 0+1+4+9 9+9+9+9 J . = 1 4. _i 16 + 16 + 16 + 16 + 16 aT; in like manner the next fraction is ^ + - 3 V- Hence Wallis noticed that the fractions approach nearer and nearer to ^ ; and, as the denominators in the fractions -jV, Jg-, ^ -i form an arithmetical series, with a common difference 6,. it follows that, when the number of terms is indefinitely increased, the resulting fraction becomes ultimately ^. Wallis applied the same method to the series 0, I 3 , 2 3 , 3 3 , &c., and found without difficulty that the aforesaid ratio is ^ in this case ; and so generally. He also introduced into analysis the notation of fractional indices instead of radicals, and extended his method of summation to series proceeding by fractional powers of the natural numbers 1, 2, 3, &c. Wallis was enabled by these principles to obtain the areas of many curves, and the volumes of solids which had not been ^previously found. _ He also, by aid of this method, combined with the principles of " interpolation," arrived at bis well-known expression for TT, viz. : K _2^ 4j6 6-8 f ^3-3 5-fl 77 Again, in his treatise De Curv. rectif. (1G59), Wallis showed that certain curves were capable of being " recti fied," or that straight lines might be found to which they were exactly equal, a remark which was Very soon verified by a young English mathematician William Neil, who, by Wallis s method, obtained in 1660 the length of any arc of a semicubical parabola. This is the first curve that was rectified. The cycloid is the second ; its rectification was effected by Sir C. Wren (Phil. Trans., 1673). Tlio methods we have thus far considered were more especially precursory to the integral calculus, having mainly refer ence to the quadrature of curves and cubature of solids. We now propose to consider the question of tangents to curves, in which the differential calculus may be said to have originated. The great discovery of Descartes in his application of Des- algebra to geometry (1637) imparted to the latter science cartes the character of abstraction and generality which distin guishes modern from ancient geometry. By it the study of curves was brought under the domain of analysis, and in- steadof investigation being restricted to particularproperties of a few isolated curves, as it had been hitherto, general views and methods applicable to all curves were introduced. Hence the general problem of drawing tangents to curved lines started immediately into prominence. It was found necessary to depart from the definition of tan gents given by the ancient geometers, and to consider them in other points of view. A tangent, accordingly, came to be regarded either (1) as a secant of which the points of intersection became coincident ; or (2) as the prolongation of the element of the curve, regarded as a polygon of an infinite number of sides ; or (3) as the direc tion of the resultant motion by which the curve may be described. The first view was that of Descartes and Fermat ; the second was introduced by Barrow, who thus simplified the method of Fermat ; and the third was that of Roberval. Descartes s method of drawing a tangent consisted in supposing a circle (whose centre he placed on the axis of x) to cut the curve in two points ; then, if the radius of the circle be supposed to decrease, its centre remaining fixed, so that the points of section approach nearer and nearer and finally coincide, the circle will touch the curve ; thus, by aid of the equation of the curve, the problem was reduced to one of finding the condition of equal roots in an equation. This method is remarkable as being the first general process of applying analysis to the problem of tangents ; at the same time it is only capable of practical application in a small number of simple cases. Many years subsequently (Act. Erud. Lips., 1691) John Bernoulli extended Descartes s method with success to the problem of finding the centre of curvature and the equation of the evolute of an algebraic curve. In his application he supposed the centre of a circle taken on the normal to a curve, and the centre to vary until three of the points of intersection of the circle with the curve became coincident, i.e., so that the resulting equation should have three equal roots. Thus, for example, he showed, without difficulty, that the evolute of a parabola was a semicubical parabola. He also remarked that, when four roots coincide, the centre of curvature becomes a cusp on the evolute. It should also be noticed that we owe to Descartes the general method of drawing a tangent to a roulette. This was given by him in a letter to Mersenne (Aug. 23, 1638), from which we take the following extracts : " I have been very glad to see the questions which you say that the geometers, even M. Roberval, whom you esteem the prin cipal of them, confess that they cannot solve ; for in investi gating them I may discover whether my analysis is better than theirs. The first of these questions is that of drawing