阿基米德的无穷小方法

    两千多年之前，数学家阿基米德懂得用无穷小宽度的金属板来计量面积和体积。这是很了不起的数学成就。

     两千多年之后，鲁宾逊严格定义了什么是数学的无穷小。随之，无穷小微积分出现了。

    两千年，弹指一挥间。现今，我们已经进入无穷小数学时代，然而，很不幸的是，我们自己还不知道。

袁萌  陈启清 10月16日

附：阿基米德的无穷小方法

阿基米德（Archimedes） (287–212 b.c.), the greatest mathematician of antiquity（地方名）, used another procedure to determine areas and volumes. To measure an unknown gure, he imagined that it was balanced on a 2The more familiar formula A = πr2 results from the fact that π is dened by the relation C = 2πr.

lever against a known gure. To nd the area or volume of the former in terms of the latter, he determined where the fulcrum must be placed to keep the lever even. In performing his calculations, he imagined that the gures were comprised of an indenite number of laminae—very thin strips or plates. It is unclear whether Archimedes actually regarded the laminae （金属板）as having innitesimal width or breadth（无穷小宽度）. In any case, his results certainly attest to the power of his method: he discovered mensuration formulae for an entire menagerie of geometrical beasts, many of which are devilish to nd, even with modern techniques. Archimedes recognized that his method did not prove his results. Once he had applied the mechanical technique to obtain a preliminary guess, he supplemented it with a rigorous proof by exhaustion