1.8 莱布尼兹的天才发明

1.8莱布尼兹的天才发明

   从数学发展的历史长河中来看，在我们国内发表此文（1.8莱布尼兹的天才发明）具有标志性意义。

  注：法国数学家柯西（分析数学奠基人）竟然也是莱布尼兹的信徒。

  请见本文附件。

  感叹国内无穷小“痴迷者”太少了。

袁萌  陈启清  10月10日

附件：

1.8 Innitesimals in the 17th to the 19th century

There can be no doubt that in the 1670’s, some 1900 years after

Archimedes lived, innitesimals were conceived by Leibniz. Moreover, he formulated their main properties, and many contemporary mathematicians as well as mathematicians after him, among them Euler and Cauchy, were able to successfully work with them. But the theory of the innitesimals lacked a rigorous basis, and during some 200 years all trials to improve this situation were in vein, so that at last one gave up, the more so because in the 1870’s Weierstrass came up with a rigorous theory of limits and continuity, which became the basis of what now is known as classical analysis, and where there was and is no need to consider innitesimals any more.

It is quite interesting to see how Euler [2] shows the well-known product formula for the sine function. He begins his proof with the equality, 2·sinh x = (1 + x/n)n −(1−x/n)n, valid for – in Eulers’s own words – ‘innitely large values’ of n. Obviously, this is only true up to an innitesimal. Then the right-hand side is treated as if n were

34

a classical natural number. Th