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SACCHERI S FIRST WORK ON GEOMETRY A PRECURSOR TO HIS EUCLIDES VINDICATUS . First edition, exceptionally rare, of Saccheri s first published work on geometry, a precursor to his famous treatise on non-Euclidean geometry, Euclides ab omni naevo vindicatus (1733). "Saccheri proceeded in his youthful work with a severa methodo , aiming towards the construction of a geometry solidly based on its axioms and certain in its conclusions … for him the problem is that of deepening the meaning and the rigour of the logical structure of Euclidean geometry and of the demonstrative logic which allows mathematical truths to be linked with absolute rigour to the postulates.The frequent and systematic use of the reductio ad impossibile in the demonstrative procedures used in the Quaesita geometrica shows that the Jesuit mathematician, already in 1693, was above all interested in putting to the test those logical instruments which he would use with such skill later" (Brigaglia & Nastasi, p. 37, our translation). The Quaesita elegantly solves, using synthetic Euclidean methods, some geometric problems that had been proposed by the brilliant, but now practically unknown, Sicilian mathematician Count Ruggero Ventimiglia, which he had proposed in a work published in 1692. The problems themselves are less important than the methods Saccheri uses to solve them, methods which show Saccheri to be a significant figure in the history of projective geometry. "The solutions of the first two problems gave Saccheri, assisted by his friend and colleague Giovanni Ceva, the opportunity to rediscover, by a systematic employment of the properties of harmonic and anharmonic ratio, many important theorems of Desargues, La Hire, and others … The recent edition, moreover, of Newton s manuscript work on Geometria , and Whiteside s careful notes upon it [Papers, Vol. VII, Ch. 2] have given us a new perspective on the refound interest in classical geometry during the later seventeenth century … We have now verified to our satisfaction not only that the development of Ceva s and Saccheri s mathematical thinking derived in an organic way from the study of the pole-polar properties of conics in the Italian tradition, but that they were also greatly influenced (in their search for unifying theoretical principles) by the ideas of René Descartes, even while refusing to follow his preference for algebraic technicalities" (ibid., p. 7). Saccheri was certainly familiar with Cartesian methods. His solutions of the 4th and 6th problems strongly suggest that he first treated these problems analytically and then converted the solutions into synthetic form. Again, in the 3rd problem, Saccheri gives the sought for hyperbola without explanation, and only verifies a posteriori that it satisfies the required properties. "But this is even more evident from his correspondence, in which Saccheri, for example, studies in a completely analytical way a non-trivial sextic … in many Jesuit colleges, particularly Spanish ones (and Pavia was then Spanish), geometry was taught according to analytical methods" (ibid., p. 31). A second edition was published in the following year under the title Sphinx Geometra, seu quaesita geometrica proposita, et solute, to which Ventimiglia s original questions were added. OCLC lists 6 copies worldwide (Bayerische Staatsbibliothek (2 copies), BnF, Biblioteca Nazionale Centrale, Rome (2 copies, from the Jesuits Collegio Romano and Casa Professa), University of Turin (Giuseppe Peano s copy)) no copy in the US. No other copy on RBH. "In 1685 Saccheri (1667-1733) entered the Jesuit novitiate in Genoa and after two years taught at the Jesuit college in that city until 1690. Sent to Milan, he studied philosophy and theology at the Jesuit College of the Brera, and in March 1694 he was ordained a priest at Como. In the same year he was sent to teach philosophy first at Turin and, in 1697, at the Jesuit College of Pavia. In 1699 he began teaching philosophy at th.
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