3 Dec 2008

Carnot in the History of the Calculus

by Corry Shores
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L.N.M Carnot wanted to make rigidly precise the theory of the calculus, and in particular the notion of the infinitesimal. But

in his selection of the unifying principle, however, he made a most deplorable choice. He concluded that "the true metaphysical principles of the Infinitesimal Analysis . . . are nevertheless . . . the principles of the compensation of errors," as Berkeley and Lagrange had suggested. (257-259)

Carnot reverts to Leibniz' ideas, and held that we may be certain that quantities are "rigorously equal" if we can prove that "their difference cannot be a 'quantité designée." Carnot further followed Leibniz in saying that we may substitute one quantity for another if there is only an infinitesimal difference between them. He also claimed that "the method of infinitesimals is nothing more than that of exhaustion reduced to an algorithm" (258a).

Carnot also followed Leibniz' law of continuity, holding that we may view the infinitesimal analysis according to two points of view: 1) by taking the infinitesimals as "quantités effectives," or 2) by taking them as "quantités absolument nulles."

In the first case, he felt that the calculus was to be explained upon the basis of a compensation of errors: "imperfect equations" were to be made "perfectly exact" by the simple expedient of eliminating the quantities whose presence occasioned the errors. (258b)

In the second case, calculus is an "art" of comparing vanishing quantities so to determine the relationships between them.

Vanishing quantities for Carnot were not null in themselves, but were rather assigned null values by Leibniz' law of continuity (285d).

Carnot assessed that all the calculus methods (and their precursors) throughout history were based on the method of exhaustion reduced to a "convenient algorithm" (259b).

However, Carnot did not develop much past Leibniz, because he considered derivatives in terms of equations rather than as functions, and he was more concerned with applying the method than with the logical reasoning involved (259d).

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

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