TY - JOUR

T1 - Almost equal

T2 - The method of adequality from diophantus to fermat and beyond

AU - Katz, Mikhail G.

AU - Schaps, David M.

AU - Shnider, Steven

N1 - cited By 18

PY - 2013/9

Y1 - 2013/9

N2 - We analyze some of the main approaches in the literature to the method of 'adequality' with which Fermat approached the problems of the calculus, as well as its source in the (Greek Source) of Diophantus, and propose a novel reading thereof. Adequality is a crucial step in Fermat's method of finding maxima, minima, tangents, and solving other problems that a modern mathematician would solve using infinitesimal calculus. The method is presented in a series of short articles in Fermat's collected works (Tannery and Henry 1981, pp. 133-172). We show that at least some of the manifestations of adequality amount to variational techniques exploiting a small, or infinitesimal, variation e. Fermat's treatment of geometric and physical applications suggests that an aspect of approximation is inherent in adequality, as well as an aspect of smallness on the part of e. We question the relevance to understanding Fermat of 19th century dictionary definitions of (Greek Source) and adaequare, cited by Breger, and take issue with his interpretation of adequality, including his novel reading of Diophantus, and his hypothesis concerning alleged tampering with Fermat's texts by Carcavy. We argue that Fermat relied on Bachet's reading of Diophantus. Diophantus coined the term for mathematical purposes and used it to refer to the way in which 1321/711 is approximately equal to 11/6. Bachet performed a semantic calque in passing from parisoō to adaequo. We note the similar role of, respectively, adequality and the Transcendental Law of Homogeneity in the work of, respectively, Fermat (1896) and Leibniz (1858) on the problem of maxima and minima.

AB - We analyze some of the main approaches in the literature to the method of 'adequality' with which Fermat approached the problems of the calculus, as well as its source in the (Greek Source) of Diophantus, and propose a novel reading thereof. Adequality is a crucial step in Fermat's method of finding maxima, minima, tangents, and solving other problems that a modern mathematician would solve using infinitesimal calculus. The method is presented in a series of short articles in Fermat's collected works (Tannery and Henry 1981, pp. 133-172). We show that at least some of the manifestations of adequality amount to variational techniques exploiting a small, or infinitesimal, variation e. Fermat's treatment of geometric and physical applications suggests that an aspect of approximation is inherent in adequality, as well as an aspect of smallness on the part of e. We question the relevance to understanding Fermat of 19th century dictionary definitions of (Greek Source) and adaequare, cited by Breger, and take issue with his interpretation of adequality, including his novel reading of Diophantus, and his hypothesis concerning alleged tampering with Fermat's texts by Carcavy. We argue that Fermat relied on Bachet's reading of Diophantus. Diophantus coined the term for mathematical purposes and used it to refer to the way in which 1321/711 is approximately equal to 11/6. Bachet performed a semantic calque in passing from parisoō to adaequo. We note the similar role of, respectively, adequality and the Transcendental Law of Homogeneity in the work of, respectively, Fermat (1896) and Leibniz (1858) on the problem of maxima and minima.

UR - http://www.scopus.com/inward/record.url?scp=84878368095&partnerID=8YFLogxK

U2 - 10.1162/POSC_a_00101

DO - 10.1162/POSC_a_00101

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SN - 1063-6145

VL - 21

SP - 283

EP - 324

JO - Perspectives on Science

JF - Perspectives on Science

IS - 3

ER -