学微积，讲历史，不做迷路人

  目前，国内微积分教材，讲微积分发展史，一般讲到牛顿、莱布尼兹为止，其实相差十万八千里，甚胡

是胡说八道。

  请见本文附件。

袁萌  陈启清  12月9日

附件：

Epllogue（无穷小微积分课程的结束语）

  How does the infinitesimal calculus as developed in this book relate to the traditional (or e, 3) calculus? To get the proper perspective we shall sketch the history of the calculus. Many problems involving slopes, areas, and volumes, which we would today call calculus problems, were solved by the ancient Greek mathematicians. The greatest of them was Archimedes (287-212 B.C.). Archimedes anticipated both the infinitesimal and the   （ε,δ）approach to calculus. He sometimes discovered his results by reasoning with infinitesimals, but always published his proofs using the "method of exhaustion," which is similar to the （ε,δ）  approach. Calculus problems became important in the early 1600's with the development of physics and astronomy. The basic rules for differentiation and integration were discovered in that period by informal reasoning with infinitesimals. Kepler, Galileo, Fermat, and Barrow were among the contributors. In the 1660's and 1670's Sir Isaac Newton and Gottfried Wilhelm Leibniz independently "invented" the calculus. They took the major step of recognizing the importance of a collection of isolated results and organizing them into a whole. Newton, at different times, described the derivative of y (which he called the "fluxion" of y) in three different ways, roughly

(1) The ratio of an infinitesimal change in y to an infinitesimal change in x. (The infinitesimal method.) (2) The limit of the ratio of the change in y to the change in x, l'ly/ l'lx, as l'lx approaches zero. (The limit method.) (3) The velocity of y where x denotes time. (The velocity method.)

In his later writings Newton sought to avoid infinitesimals and emphasized the methods (2) and (3). Leibniz rather consistently favored the infinitesimal method but believed (correctly) that the same results could be obtained using only real numbers. He regarded the infinitesimals as "ideal" numbers like the imaginary numbers. To justify them he proposed his law of continuity: "In any supposed transition, ending in any terminus, it is permissible to institute a general reasoning, in which the terminus may

EPILOGUE 903

also be included."1 This "law" is far too imprecise by present standards. But it was a remarkable forerunner of the Transfer Principle on which modern infinitesimal calculus is based. Leibniz was on the right track, but 300 years too soon! The notation developed by Leibniz is still in general use today, even though it was meant to suggest the infinitesimal method: dyjdx for the derivative (to suggest an infinitesimal change in y divided by an infinitesimal change in x), and s~ f(x) dx for the integral (to suggest the sum of infinitely many infinitesimal quantities f(x) dx). All three approaches had serious inconsistencies which were criticized most effectively by Bishop Berkeley in 1734. However, a precise treatment of the calculus was beyond the state of the art at the time, and the three intuitive descriptions (1H3) of the derivative competed with each other for the next two hundred years. Until sometime after 1820, the infinitesimal method (1) of Leibniz was dominant on the European continent, because of its intuitive appeal and the convenience of the Leibniz notation. In England the velocity method (3) predominated; it also has intuitive appeal but cannot be made rigorous. In 1821 A. L. Cauchy published a forerunner of the modern treatment of the calculus based on the limit method (2). He defined the integral as well as the derivative in terms of limits, namely

f

b b f(x) dx = lim If(x) Llx. a Ax-o+ a