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The acceptance of the theory of proportion in the sixteenth and seventeenth centuries - Barrow’s reaction to the analytic mathematics. (English) Zbl 0602.01008

”In any case, seventeenth century mathematicians were in a deep forest of conflict, and historians are left to speculate on the shape of the whole forest itself.” With this sentence, the author closes a study which contains more than the title reveals. What is at stake is the problem of the relation between the ”old” and the ”new” mathematics in the 17th century - the ”new” one being the symbolic algebra (or analytical method) of Viète, Descartes, and their successors, the ”old” one the classical Greek geometry with its rigorous synthetic proofs. In the author’s words: ”The main concerns of Descartes, Wallis and Leibniz, the heuristic analysts according to our distinction, were to make new effective mathematical tools in order to solve difficult problems. On the other side, Barrow’s emphasis seems to have been to provide a strict basis for contemporary mathematics. This paper will show that there were some important confrontations between the two philosophical points of view, and will indicate how these confrontations came to be solved.” Sasaki’s reason for taking Barrow’s treatment of the theory of proportion (and in particular definition 5 of book V of Euclid’s ”Elements”, i.e. Eudoxos’ definition on the equalities of ratios which holds for incommensurable quantities as well) is that ratios and proportion until the 17th century played as central a rôle in mathematics as later (beginning in particular with Leibniz) the concepts of equation and function. - Besides Barrow’s ”Lectiones mathematicae”, the views of Wallis and his opponent, those of the forerunners Tartaglia and Ramus, and interpretations by later Historians are also discussed in this penetrating study. Not mentioned, however, is the irony of history that at the very moment when ratios in mathematics seemed to have become obsolete as entities of their own standing, they returned in another guise: as ratios of vanishing quantities, in connection with infinitesimal problems.
Reviewer: C.J.Scriba

MSC:

01A45 History of mathematics in the 17th century
01A40 History of mathematics in the 15th and 16th centuries, Renaissance

Biographic References:

Barrow, I.
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