‎"CAYLEY, ARTHUR.‎
‎Sur quelques propriétés des déterminants gauches.‎

‎Berlin, Georg Reimer, 1846. 4to. In ""Journal für die reine und angewandte Mathematik, 32. Band, 2 Heft, 1846"". In the original printed wrappers, without backstrip. Front wrapper with a brown mark to top left corner, otherwise fine and clean. [Cayley:] Pp. 119-23. [Entire issue: Pp. 93-180, (2)].‎

Reference : 45097

‎First printing of Cayley's paper on Rotation matrix or skew-symmetric matrix (a square matrix A whose transpose is also its negative).""By 1846 Cayley had made use of four dimensions in the enunciation of specifically synthetic geometrical theorems, suggesting methods later developed by Veronese (C. M. P., I, no. 50 [1846], 317-328). Long afterward Cayley laid down in general terms, without symbolism, the elements of the subject of ""hyperspace"" (cf his use of the terms ""hyperelliptic theta functions,"" ""hyperdetermtnant,"" and so on) in his ""Memoir on Abstract Geometry"" (C. M. P., VI, no. 413 [1870], 456-469), showing that he was conscious of the metaphysical issues raised by his ideas in the minds of his followers but that as a mathematician he was no more their slave then than when remarking in his paper of 1846 (published in French): ""We may in effect argue as follows, without having recourse to any metaphysical idea as to the possibility of space of four dimensions (all this may be translated into purely analytic language)."""" (DSB)‎