‎"CAYLEY, ARTHUR. - THE THEORY OF FORMS (QUANTICS) - A NEW ASPECT OF NON-EUCLIDEAN GEOMETRY.‎
‎A Sixth Memoir upon Quantics. Received November 18, 1858, - Read anuary 6, 1859.‎

‎(London, Richard Taylor and William Francis, 1859). 4to. No wrappers as extracted from ""Philosophical Transactions"" Vol. 149 - Part I. Pp. 61-90. Clean and fine.‎

Reference : 42296

‎First appearance of this pathbreaking paper in which Cayley unites 'Metrical Geometry' and 'Projectice Geometry' by introducing ""imaginary"" elements to metrical properties.""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 Euclideangeometry and the common non-Euclidean gemoetries into one comprehensive theory.""(Bell in ""Men of Mathematics"").In non-Euclidean geometry prepared the way for Klein's splendid discovery that the geometry of Euclid and the non-Euclidean geometries of Lobatchewsky and Riemann are, all threee, merely different aspects of a more general kind of geometry which includes them as special cases..Dealing with the relations between metrical and projective geometry Klein remarks (In ""Entwicklung der Mathematik"", Teil I p. 148): ""Vor allem kommt für uns sein (Cayley's) berühmtes 'A Sixth Memoir upon Quantics"" im betrachtt. Quantioc heisst soviwel ""Form"", d.h. homogenes Polynom von zwei, drei oder mehr Variablen, wonach man binäre, tertiäre usw. Formen unterscheidet...""‎