极限思想外文翻译.pdf.doc

BSHM Bulletin, 2014 Did Weierstrass’s differential calculus have a limit-avoiding character? His definition of a limit in style MICHIYO NAKANE Nihon University Research Institute of Science Technology, Japan In the 1820s, Cauchy founded his calculus on his original limit concept and developed his the-ory by using inequalities, but he did not apply these inequalities consistently to all parts of his theory. In contrast, Weierstrass consistently developed his 1861 lectures on differential calculus in terms of epsilonics. His lectures were not based on Cauchy’s limit and are distin-guished by their limit-avoiding character. Dugac’s partial publication of the 1861 lectures makes these differences clear. But in the unpublished portions of the lectures, Weierstrass actu-ally defined his limit in terms ofinequalities. Weierstrass’s limit was a prototype of the modern limit but did not serve as a foundation of his calculus theory. For this reason, he did not provide the basic structure for the modern e
d style analysis. Thus it was Dini’s 1878 text-book that introduced the definition of a limit in terms of inequalities. Introduction Augustin Louis Cauchy and Karl Weierstrass were two of the most important mathematicians associated with the formalization of analysis on the basis of the e
d doctrine. In the 1820s, Cauchy was the first to give comprehensive statements of mathematical analysis that were based from the outset on a reasonably clear definition of the limit concept (Edwards 1979, 310). He introduced various definitions and theories that involved his limit concept. His expressions were mainly verbal, but they could be understood in terms of inequalities: given an e, find n or d (Grabiner 1981, 7). As we show later, Cauchy actually paraphrased his limit concept in terms of e, d, and n0 inequalities, in his more complicated proofs. But it was Weierstrass’s 1861 lectures which used the technique in all proofs and also in his defi-nition (Lutzen€ 2003, 185–186). We