A Sixth Memoir upon Quantics, extracted from The Philosophical Transactions of the Royal Society, Vol. 149, Part I, 1859, pp. 61-90

Cambridge: Royal Society, 1859. 1st Edition. FIRST EDITION (extracted from the 1859 Philosophical Transactions) OF THE PAPER IN WHICH CAYLEY "LAID THE FOUNDATIONS OF NON-EUCLIDEAN GEOMETRY" & "DEMONSTRATED THAT EUCLIDEAN GEOMETRY WAS PART OF PROJECTIVE GEOMETRY RATHER THAN THE CONVERSE" (Pierpont, "Cayley's Definition of Non-Euclidean Geometry," AJM 53, 1, 1931, 117; Princeton Companion to Mathematics, 772). Cayley's ‘Sixth Memoir upon Quantics' "contains the first analytical treatment of a projective metric from the standpoint of the algebraic theory of invariants" (Biagioli, Space, Number, and Geometry, 123).

Beginning with an introductory memoir in 1854, Cayley "composed a series of ten ‘Memoirs on Quantics,' the last published in 1878, which for mathematicians at large constituted a brilliant and influential account of the theory as he and others were developing it" (DSB, III, 165). [Note that we offer all ten memoirs together in a separate listing]. Quantics is "a term [Cayley] coined for algebraic forms, now referred to as multilinear homogeneous algebraic forms" (Princeton).

The ‘Sixth Memoir' offered here is the best known of the ten. In it, and in order to unite metrical geometry and projective geometry, Cayley introduced metrics into projective geometry by means of a fixed, embedded conic called the absolute; in other words, he introduced an ‘imaginary' element to metrical properties. "Hitherto, affine and projective geometry had been regarded as special cases of metric geometry. Cayley showed how it was possible to interpret all as special cases of projective geometry" (DSB 165).

In this memoir, Cayley "set himself the task of "establishing the notion of distance upon purely descriptive principle" [showing] that, with the ordinary idea of distance, it can be rendered projective by reference to the circular points and a line at infinity -- and that the same is true of angles (Bell, Men of Mathematics). "The fundamental notions in metrical geometry are the distance between two points and the angle between two lines. Replacing the concept of distance by another, also involving "imaginary" elements, Cayley provided the means for unifying Euclidean geometry and the common non-Euclidean geometries into one comprehensive theory" (Bell).

Within Cayley's ‘new' definition of distance as the inverse sine of a certain function of the coordinates, the properties usually known as metrical become projective properties (having reference to a certain conic called by Cayley the Absolute). Cayley proves that, when the Absolute is an imaginary conic, the Geometry so obtained for two dimensions is spherical Geometry" (Russell, An Essay on the Foundations of Geometry).

"The full significance of Cayley's ideas was not appreciated until 1871 when Klein showed how it was possible to identify Cayley's generalized theory of metrical geometry with the non-Euclidean geometries of Lobachevski, Bolyai, and Riemann. When Cayley's Absolute is real, his distance function is that of the ‘hyperbolic' geometry; when imaginary, the formulas reduce to those of Riemann's ‘elliptic' geometry. A degenerate conic gives rise to the familiar Euclidean geometry. Whereas during the first half of the century geometry had seemed to be becoming increasingly fragmented, Cayley and Klein, through the medium of these ideas, apparently succeeded for a time in providing geometers with a unified view of their subject. Thus, although the so-called Cayley-Klein metric is now seldom taught, to their contemporaries it was of great importance" (DSB 165). CONDITION & DETAILS: Cambridge: The Royal Society, 1859. Extracted from The Philosophical Transactions, Vol. 149, Part I, 1859, pp. 61-90. Clean, bright, and in near fine condition.