Bruxelles: M. Hayez, 1837. First edition. The second part of this work, written in response to a question posed by the Royal Academy of Brussels in 1829, is an important memoir in which Chasles casts projective geometry as the study of linear transformations that preserve the cross ratio. The long introduction constitutes one of the most important histories of geometry.
“The Apercu Historique was inspired by the question posed by the Royal Academy of Brussels in 1829: a philosophical examination of the different methods in modern geometry, particularly the method of reciprocal polars. Chasles submitted a memoir on the principles of duality and homography. He argued that the principle of duality, like that of homography, is based on the general theory of transformations of figures, particularly transformations in which the cross ratio is preserved, of which the reciprocal polar transformation is an example. The work was crowned in 1830, and the Academy ordered it published. Chasles requested permission to expand the historical introduction and to add a series of mathematical and historical notes, giving the result of recent researches. His books and almost all of his many memoirs are elaborations of points originally discussed in these notes.” (DSB).
In ‘A History of Geometrical Methods’ Coolidge writes: “The greatest contribution which Chasles made to projective geometry, and it certainly was great, was in the study of the cross ration, whether of collinear points or of concurrent lines, or yet of coaxel planes. We this in Notes IX and X of [his Apercu]. He discussed the six cross ratios of four elements of one of these forms, and this gives him a deeper insight into the meaning of an involution. In Note XV we find the theorem, which was later to play a vital role in the work of Steiner, that four points of a conic determine with any fifth point of that curve four lines having the same cross ration. ... Another fundamental idea developed by Chasles was homography, or the general linear transformation in the plane or in space. Here a point corresponds to a point, and points of a plane or line to points of a plane or line. Central projection and homology discussed by Poncelet are special cases of this. He added the invariance of cross ratios as part of his definition, not realizing that this invariance could be proved, ... I think we should give him credit for being the first to give a proper definition of these important transformations”.
The two works published as volume XI of the 'Mémoires couronnés par l'académie royale des sciences et belles-lettres de Bruxelles'. 4to (278 x 210 mm), contemporary calf, some light fosing toi titel pages, otherwise fine, pp ,572;573-851 .