On the Historical Development of Infinitesimal Mathematics

1. LEIBNIZ AND L'HOSPITAL. Exactly 300 years ago there appeared in Paris the first book on differential calculus, Analyse des infiniment petits, by Marquis Guillaume F.A. L'Hospital (1661-1704). The book was based on materials supplied by Johann Bernoulli (1667-1748). Johann and his older brother Jakob (1654-1705) were the first 4'comrades-in-arms" of Gottfried Wilhelm Leibniz (1646-1716), one of the two discoverers of the calculus. In 1684 Leibniz published for the first time a few simple applications of the differential calculus, without making any attempt to provide clear justifications for them. But we find such attempts in his letters to judicious contemporaries. For example, on 30 March 1690, Leibniz wrote to John Wallis (1616-1703): CCIt is useful to regard quantities as infinitely small, so that, when their ratio is sought, they are omitted, rather than viewed as 0, when they turn up next to quantities that are incomparably larger. If we have x + dx, then dx is omitted. Similarly, we cannot let xdx and (X)2 stand next to one another. Thus if we have to differentiate xy, we write (x + dx)(y + dy) xy = x dy + y dac + dxdy. But here dxdy should be omitted, for it is incomparably smaller than xdy +ydx. Hence, in each particular case, the error is smaller than any finite quantit,v." (Leibniz, Math. Schriften IV, 63; our translation. D.L., A.S.) Leibniz proceeds pragmatically. He states a rule which gives a correct result in Cevery special case.' At first sight, it all looks like facts based on experience, but theoretical justifications are in the offing. Leibniz seems to add to the system of quantities apparetntly measured in terms of real numbers ideal elements dx, dy, . . . such that a (positive);dx is smaller than every (positive) real quantity. Then the algebraic rules of computation are applied in the usual manner. The worked examples show that at the end of a computation the differentials are replaced by 0s. (In the last step we are, to some extent, anticipating history, for it was soon noticed that the stipulation pertaining to the omitting of a quantity against an incomparably greater one can be problematic. A case in point is (x + CtX)2-x2 = 2x * dx + (£¢)2 for x = 0t). This recipe can be used for algebraic expressions. As early as 1684 Leibniz obtained the root formula