By F. Zabeeh
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Additional info for Avicenna's Treatise on Logic, Part One of Danesh-Name Alai: A concise philosophical Encyclopaedia and Autobiography (Pt. 1)
5 Jean-Victor Poncelet (1788–1867) is the first to have viewed projective geometry as an independent branch of geometry, defined by its specific methods, its distinctive results and its particular problems. 6 Poncelet focused on a type of transformation called ‘perspectivities’. 1 belong to the O-pencil. Let P, P′ be two distinct planes not containing O. A set of P′-points is said to be in perspective (from O) to a set of P-points if and only if each point of the former set is related to a point of the latter by a line of the O-pencil.
His new definitions paved the way for a characterization of projective geometry in terms of incidence. Alas, his programme did not come to fruition. Klein showed that some key projective theorems (Pappus’s and the fundamental theorems) could not be reached by Von Staudt’s method. To understand the depth of the problem, let us recall that the fundamental theorem is needed to prove that the two rival notions of homography (as the product of perspectivities and as transformations which preserve cross-ratios) were coextensive.
There is absolutely no doubt that Russell took the second account to be the most fundamental one. For him, Pieri, not Pasch, should be regarded as the true founder of projective geometry (PoM, p. 421): The true founder of non-quantitative Geometry is Von Staudt. It was he who introduced the definition of a harmonic range by means of the quadrilateral construction, and who rendered it possible, by repetitions of this construction, to give projective definitions of all rational anharmonic ratios ...