Euler in the History of the Calculus

by Corry Shores
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Up to the beginning of the 18^th century, calculus methods were considered as means to solve geometrical problems, and bases for these methods were sought in geometry rather than arithmetic. This however changes in the beginning half of the 18^th century.

Although most of Leonhard Euler’s predecessors considered differential calculus to be bound-up with geometry, he rendered caculus into a “formal theory of functions which had no need to revert to diagrams or geometrical conceptions” (Boyer 243b). He was “the first mathematician to give prominence to the function concept and to make a systematic study and classification of all the elementary functions, together with their differentials and integrals” (243c).

Functions for Euler were not quantities conceived as depending on variables, that is, “as an analytic expression in constants and variables which could be represented by simple symbols” (243cd). Rather, functionality for him was more a matter of formal representation than of conceptual recognition of a relationship (243d).

Euler did not consider the infinite and infinitesimal as holding some deep mystery; “an infinitely small or evanescent quantity he held to be simply one which will be zero” (244a). The infinitesimal for him was not “a constant quantity less than any assignable magnitude;” for, Euler rejected any sort of mathematical atomism or monadology (244b). Any number less than a given quantity must necessarily be zero, he argued.

Although he admitted the existence of an infinite number of infinitesimals, as found in the differentials of higher orders, these he held were all zero. Leibniz had at one point suggested that the differentials could be regarded as qualitative zeros, which nevertheless retained by the law of continuity the character of the relationships of the finite quantities from which they were derived. Euler, in conformity with his formalistic view, held less philosophically that the zeros represented by differentials were to be distinguished through the recognized fact that the ratio 0/0 could, in a sense, represent any ratio of finite numbers, n/1. Thus for Euler the calculus was simply the determination of the ratio of evanescent increments – a heuristic procedure for finding the value of the expression 0/0. (244c.d)

Euler’s formalist approach to calculus freed it from all “geometrical fetters. It also made more acceptable the arithmetic interpretation which was later to clarify the calculus through the limit concept which Euler himself neglected” (246b).

Boyer, Carl B. The History of the Calculus and its Conceptual Development. New York: Dover Publications, 1949.