The innovation as a product of the tradition: Between Apollonius and Descartes a curve theory by Grégoire de Saint-Vincent (Q2702312)

The author examines Grégoire St. Vincent's \textit{Opus geometricum} (Antwerp, 1647; essential part goes back before 1625), huge and not well-organized work, skillfully pointing out its characteristics as a work envisaging new mathematics such as integral calculus and Descartes' analytic geometry. The significance of this work lies not in its subject (quadrature of the circle) but the many innovative ideas and techniques it contains. The most remarkable is `functionalization' of conic sections, especially of hyperbola. Their properties are fully exploited to manipulate ratios and proportions, so that the curves become rather instrumental for handling the magnitudes. This is a step ahead from Apollonius' treatment of conics, though conic sections are not yet reduced to quantitative relations between ordinate and abscissa as in Descartes. Besides, Grégoire's work contains innovative ideas such as an attempt to generalize the method of exhaustion (involving necessarily a generalization of the concept of curved line) and recourse to converging geometric progression. However, Grégoire's mathematical language remained that of classical theory of proportion, which was insufficient to describe the results and develop the potential consequences of his innovation. The author convincingly concludes that Grégoire's innovation was captive to tradition.NEWLINENEWLINEFor the entire collection see [Zbl 0954.00008].