ε-δ論法による微積分学の形成におけるCauchyとWeierstrassの寄与

抄録

This paper clarifies Cauchy's and Weierstrass's contributions to the construction of differential calculus represented in terms of epsilonics. In the eighteenth century the limit concept had a geometrical image that is typically represented in >indefinitely approaching to a fixed value>. In 1820s Cauchy described this concept in terms of inequalities and defined the limit. Since his new calculus theory was based on this concept, he could transform previous results from calculus to his new theory developed only by algebraic techniques. He also defined his original concept of infinitesimals based on the limit concept. The relations between the infinitesimals and infinitely large numbers or infinitesimally small changes can be represented in term of epsilon-delta inequalities. Although Cauchy occasionally used the term of infinitesimals in the usual sense, he substantially developed his calculus theory in epsilonics using his infinitesimals. Weierstrass noted the differential calculus needs to apply neither Cauchy's limit nor infinitesimals, but the relations that involve them. Neither isolated limits nor infinitesimals can be written in terms of epsilon-delta inequalities, but their relations can. Weierstrass began his 1861 lectures on the differential calculus by defining the fundamental concepts in terms of epsilon-delta inequalities. His original limit concept was also defined in terms of these, without any geometrical image. In contrast to Cauchy, Weierstrass's theory was pure algebraic and had no geometrical background. Although both mathematicians basically developed their differential calculus in epsilonics, the essential difference between their approaches lies in this point.