In attempting to give an introductory sketch of pro jective geometry we have started from the point of view of ele mentary Euclidean geometry and freely made use of conceptions which do not belong in projective geometry as the subject is finally defined with the aid of the group idea. This is probably the course which will be generally regarded as advisable in first approaching the subject. But since projective geometry is a collection of very general and significant propositions from which it is possible to specialize in a variety of directions, it is an attractive idea to make projective geometry the starting point of the logical formulation of geometry as a whole and arrive at the various more special branches by a process of specialization. The still more general branches will of course continue to be reached by the process of generalization. The problem of stating the axioms in purely pro jective terms and deriving the theorems from this foundation by purely logical processes has engaged the attention of several mathematicians in recent years and is quite fully developed in the last two books cited in the bibliography below.
The conception of points at infinity goes back to G. Desargues (cf. his Brouillon-projet, 1639, in Oeuvres de Desar gues, 1864), and many of the individual conceptions of projective geometry can be traced back to remote antiquity; but it may be said to appear first as a definite branch of science in 1822 in the Traite des proprietes projectives des figures of J. V. Poncelet. The
development of the science was participated in by nearly all the geometers of the nineteenth century, notably by Carnot, Brian chon, Gergonne, Chasles, Mobius, Monge, Steiner, Plucker, Clif ford, Cremona, H. J. S. Smith and H. Wiener. The clear separation of projective from metric properties dates from the publication of the Geometrie der Lage of K. G. Ch. von Staudt (Nurnberg, 1847) and his Beitriige zur Geometrie der Lage (Nurnberg, 1857). The formulation of the group-theoretic classi fication of geometries is due to F. Klein. The study of the axioms and the logical organization of the science as a separate entity has been participated in by many mathematicians during the last forty years, notably by M. Pieri.
P. Field, Projective geometry (1923) D. N. Lehmer, An elementary course in synthetic projective geometry (1917) ; G. B. Mathews, Projective geometry (1914) . Advanced: H. F. Baker, Principles of geometry (1922) ; L. Cremona, Elements of projective geometry (1885) ; F. Enriques, Vorlesungen caber Projektive Geometrie (Leipzig, 1913) ; J. L. S. Hatton, The principles of projective geometry (1913) ; M. Pieri, Geometria projettiva (Turin, 1891) ; T. Reye, Die Geometrie der Lage (5th ed., Leipzig, 1909) ; F. Severi, Geometria projettiva (Padua, 1921) ; 0. Veblen and J. W. Young, Pro jective geometry (igio) ; A. N. Whitehead, The axioms of projective geometry (1906). (0. V.)