Paris: Bachelier, 1822.
1st Edition. Hardcover. 4to - over 9¾ - 42" tall. Very Good. Item #003552
4to (263 x 210 mm). xlvi, 426 pp., including half-title, 12 folding engraved plates, errata leaf bound in as facsimile. Later half buckram and marbled boards, gilt-lettered red morocco spine label (minor wear to extremities). Brown stain at head of title possibly from removed signature, text only very little browned, minor scattered foxing throughout, half title and title little dust soiled, dampstaining to blank gutter of first preliminary pages, a few annotations and diagrams in pencil. Provenance: L. Givebin (signature on half title and title); Collection of Peter and Margarethe Braune. Still very good, wide-margined copy. ----
DSB XI, p. 77-80; Gascoigne 214. FIRST EDITION of the first book on projective geometry. Poncelet, a pupil of Monge, "during his years in the prisons of Russia, meditated on the real cause of the power which algebraical analysis possessed, on the reason why geometry proper was deprived of it, and what might be done to give it a similar generality. . . . He was thus led to the enunciation of his celebrated and much-criticised principle or law of continuity. Analytical geometry, by substituting an algebraical expression for a geometrical figure, could apply to it all the artifices of abstract analysis" (Merz). In this book Poncelet announced his discovery of the principle of projection and the principle of figures. The Traité "was the first book wholly devoted to projective geometry, a new discipline that was to experience wide success during the nineteenth century. In this domain Poncelet considered himself the successor to Desargues, Blaise Pascal, and Maclaurin and the continuator of the work of Monge and his disciples. Concerned to endow pure geometry with the generality it lacked and to assure its independence vis-à-vis algebraic analysis, Poncelet systematically introduced elements at infinity and imaginary elements, thus constructing the space employed in complex projective geometry. Basing his efforts on the principle of continuity and the notion of ideal chords, he also made extensive use of central projections and profitably utilized other types of transformations ... The distinction Poncelet made between projective and metric properties prefigured the appearances of the modern concept of structure. Among the many original results presented in the Traité are those stating that in complex projective space two nondegenerate conics are of the same nature and have four common points (a finding that led to the discovery of cyclic points, imaginary points at infinity common to all the cirlces of a plane), and that all quadrics possess (real or imaginary) systems of generatrices. The decisive influence that Traité des propriétés projectives des figures exercised on the development of projective geometry ... is brought to light by most commentators, particularly by E. Kötter, who made the most complete analysis of it. . . Of the later memoirs, the most striking is devoted to the theory of reciprocal polars, which in Poncelet's hands became an extremely fruitful instrument of discovery, although he did not perceive the more general character of the principle of duality, which was pointed out shortly afterward by Gergonne, Plücker, Möbius, and Chasles. Although it was prematurely interrupted, Poncelet's geometric work marks the first major step toward the elaboration of the fundamental theories of modern geometry." (DSB) - Visit our website to see more images!
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