396).īeeley Philip and Scriba, Christoph. J’ay de la peine a m’imaginer qu’il le croye luy mesme, et il me paroit plus vraysemblable qu’il s’est voulu sauver dans l’embaras et dans l’obscurité” (in Huygens 1888–1950, Vol. Again, Huygens wrote in similar tones to Robert Moray, in March 1669: “la derniere response de Monsieur Gregory sur le suject de la Quadrature, ou il n’a rien fait qui vaille, et je voudrois bien scavoir s’il y a aucun des geometres par de la qui prenne pour des demonstrations ce qu’il donne pour telles. Gregory” to which Huygens refers is obviously the last letter published in the Transactions in February 1669 (Gregory 1668c). The infinite therefore becomes one more element in the mathematical calculations of these authors, which in seventeenth century mathematics opened up a world of possibilities in the series and in their relations with infinitesimal calculus.We can glean this conclusion from Huygens’ remarks in a letter to Oldenburg, from March 30, 1669: “Il me semble par la response de Monsieur Gregory qu’il s’est trouvé fort embarassé de mes derniers instances, car au lieu d’y respondre pertinemment, il ne cherche qu’a embrouiller tellement la dispute, et la rendre si obscure que personne n’y comprendra dorenavant rien” (in Huygens 1888–1950, Vol. Harmonic triangle has an open visual structure in which the number of terms arranged in this way can be made infinite. On the other hand, at the same time, Leibniz defines the harmonic triangle from the study on harmonic series, analyses its properties, and uses it to perform the summations of infi-nite series through one procedure called by him “sums of all the differences”. We show that, on the one hand, Mengoli uses triangular tables as a tool of calculus, and uses the harmonic triangle to perform quadratures through one procedure called by him “homology”. In this article we analyze and compare the independent treatment of harmonic triangle by Mengoli and Leibniz in their works, referring to their sources, their aims, and their uses. Pietro Mengoli (1627-1686), rather at the same time, used the harmonic triangle as a triangular table to perform quadratures and also the interpola-ted harmonic triangle to calculate the quadrature of the circle. Leibniz studied it in many different texts throughout his life. The harmonic triangle was defined by Gottfried Wilhelm Leibniz (1646-1716) in 1673, and its definition was related to the successive differences of the harmonic series. Research on the manuscripts of James Gregorie and David Gregory shows that David's Geometria practica is actually James's, and that David's optical book heavily borrows from James's optical manuscript." Placing the contributions of Newton and Gregorie in the context of 17th-century discussions on indivisibles, l argue that Gregorie and Newton were idiosyncratic in their rejection of indivisibles. The last chapter studies and translates an unpublished mathematical manuscript featuring results similar to those in section 1 of Newton's Principia. It appears that Gregorie's work is a substantial counter-example to the standard thesis that geometry and algebra were opposed forces in 17th-century mathematics. The third chapter studies Gregorie's work on "Taylor" expansions and his analytical method of tangents, which has passed unnoticed so far. The new optical science sought experimental confirmation for its basic notions and results and thus provided a direct methodological antecedent to Newton's Principia. I argue that Gregorie, Barrow, and Newton produced a methodological revolution in geometrical optics. A major center of interest is the origins of the notion of geometrical optical image, which are shown to have been influenced by the philosophical empiricism. The second chapter studies Gregorie's contributions to optics, including a description of a hitherto unpublished manuscript. It is argued that John Collins, representing the world of practical mathematicians, was a source of motivations for some of Gregorie's mathematical discoveries, and that Gregorie's attempts to publicize his contributions failed because of institutional practices characteristic of the early Royal Society. Gregorie's correspondence with Newton is studied. Evidence on Gregorie's life within 17th-century Scottish universities, his involvement in setting up the St Andrews observatory, his activities in the early 1670's as leader of a Scottish network of mathematical virtuosi, and his juvenile astrological concerns is here produced for the first time. The first chapter contains a narrative of Gregorie's life and works. "As a general conclusion this dissertation suggests that the mathematical and optical contributions of James Gregorie, Isaac Barrow and Isaac Newton are more closely related to one another than it is usually acknowledged.
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