000			03507cam a2200421 4500500
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041			_afre
042			_adc
100	1	0	_aDhombres, Jean _eauthor
245	0	0	_aThe indefinite in Pascal’s Treatises on roulette; his ontology, sources and posterity
260			_c2023.
500			_a28
520			_aThe expression “indefinite number” for the division into equal parts of a curve like a roulette is frequent in the book by Pascal, the Lettres de A. Dettonville of 1658–1659, accompanied explicitly by a hypothesis, a “given” which is the knowledge of the ratio from the perimeter of a circle to its diameter. The rectification of a circle opens the path to possible rectifications, if not all rectifications. Exegetes have not seriously exercised themselves on the meaning of this indefinite which has its source in part in Roberval as we will show; it was for Leibniz the subject of an initiation story, that went too far by certain commentators. Even by going through a few comments on the Pascalian corpus, I force words by linking to the expression of indefinite integral, certainly absent from Pascal, but it is worth to play this game. The latter finds its source in the logarithm as expressed by Grégoire de Saint-Vincent. This connection makes it possible to consider the summation as a function, the value for example of the length of any arc of roulette found by the very young Christopher Wren after the first publicity given to the Pascal’s challenge, and which forced him to modify the questions he asked. It therefore required much more than the knowledge of the number π, and a knowledge which does not imply anything numerical, but a classification of the things available to do mathematics and to go as far as that of the length of any arc of a circle. It is precisely a function, without the name, where the indefinite is that of the variable, as the indefinite of equal divisions is in the number of these. The roulette is indeed the curve pretext for Pascal’s challenge and his successful work goes far beyond it. Fixing myself on the word indefinite, looking for sources, I question Pascal at the heart of his device of hypotheses necessary for the pursuit of mathematics. It went so far from the indefinite number of a division reduced to a simple trick of straightening the indivisibles so to enter with curvilinear integrals in the new terrain of integration that provides equivalences and formulae, but without any algorithm for it.
690			_amethod of indivisibles
690			_acurvilinear integral
690			_aGilles Personne de Roberval
690			_ahistory of mathematics
690			_aLettres de A. Dettonville
690			_asine function
690			_aindefinite division
690			_aindefinite integral
690			_aHonoré Fabri
690			_ainfinite
690			_aBlaise Pascal
690			_amethod of indivisibles
690			_acurvilinear integral
690			_aGilles Personne de Roberval
690			_ahistory of mathematics
690			_asine function
690			_aLettres de A. Dettonville
690			_aindefinite division
690			_aindefinite integral
690			_aHonoré Fabri
690			_ainfinite
690			_aBlaise Pascal
786	0		_nRevue d’histoire des sciences | Volume 76 | 2 | 2023-06-27 | p. 303-340 | 0151-4105
856	4	1	_uhttps://shs.cairn.info/journal-revue-d-histoire-des-sciences-2023-2-page-303?lang=en&redirect-ssocas=7080
999			_c564214 _d564214