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GREEK GEOMETRY ? A CRUCIAL INFLUENCE ON DESCARTES. First edition of Pappus' Collection, translated with commentary by Federico Commandino, a princely copy from the notable collection of the great Papal family and patrons of learning, the Piccolomini, Dukes of Amalfi, thence by marriage to the German nobleman von Troilo. The Collection is ?by far the most important of Pappus? works ? without it, much of the geometrical achievement of his predecessors would have been lost forever ? The Collection deals with the whole body of Greek geometry, mostly in the form of commentaries on texts which it is assumed the reader has to hand. It reproduces known solutions to problems in geometry; but it also frequently gives Pappus? own solutions, or improvements and extensions to existing solutions. Thus Pappus handles the problem of inscribing five regular solids in a sphere in a way quite different from Euclid; gives a broader generalization than Euclid to the famous Pythagorean theorem, and provides a demonstration of squaring the circle which is quite different from the method of Archimedes (who used a spiral) or that of Nicomedes (who used the conchoid). Perhaps the most interesting part of the Collection, measured by its influence on modern mathematics, is Book VII, which is concerned with the problems of determining the locus with respect to three, four, five, six or more than six lines. Pappus? work in this field was called ?Pappus? problem? by Ren? Descartes, who demonstrated that the difficulties which Pappus was unable to overcome could be got round by the use of his new algebraic symbols. Pappus thus came to play an important, if minor, role in the founding of Cartesian analytical geometry. And it is another mark of his originality and skill that he spent much time working on the problem of drawing a circle in such a way that it will touch three given circles, a problem sophisticated enough to engage the interest, centuries later, of both Fran?ois Vi?te and Isaac Newton. For his own originality, even if his chief importance is as the preserver of Greek scientific knowledge, Pappus stands (with Diophantus) as the last of the long and distinguished line of Alexandrian mathematicians? (Hutchinson Dictionary of Scientific Biography). ?He formally defined analysis and synthesis as they are still commonly applied in the solution of geometrical riders. Pappus stumbled upon the projective invariance of the cross-ratio of four collinear points and other related results reclaimed by modern projective geometry; and he gave the first recorded statement of the focus-directrix property of the three conic sections. He formulated the ?centrobaric? theorems, frequently attributed to Paul Guldin (1577-1643), for calculating the volume and surface generated by a plane figure rotating about an axis in its own plane. He discussed theoretical mechanics, the equilibrium of a heavy body on an inclined plane, the use of the mechanical powers, and the construction of mechanical toys? (Biographical Dictionary of Scientists). Provenance: Ex libris inscription of Princess Maria Piccolomini and signature of Count Franz Gottfried von Troilo on title; shelfmark on front free endpaper. Pappus of Alexandria (c.? 290?? c.? 350 AD) was the most important mathematical author writing in Greek during the later Roman Empire. Other than that he was born at Alexandria in Egypt and that his career coincided with the first three decades of the 4th century AD, little is known about his life. ?In the silver age of Greek mathematics Pappus stands out as an accomplished and versatile geometer. His treatise known as the Synagoge or Collection is a chief, and sometimes the only, source for our knowledge of his predecessors? achievements. The Collection is in eight books, perhaps originally in twelve, of which the first and part of the second are missing ? The several books of the Collection many well have been written as separate treatises at different dates and later brought toge.
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