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Gauss, Grunert, and Mobius glanced Post Date: Mon, 28 Jul 2008 12:32:13 +0000 It now became his ambition to secure a mathematical chair at a university, but in this he never succeeded. In 1844 appeared his great classical work, the Lineale Ausdehnungslelire, which was full of new and strange matter, and so general, abstract, and out of fashion in its mode of exposition, that it could hardly have had less influence on European mathematics during its first twenty years, had it been published in China. Gauss, Grunert, and Mobius glanced over it, praised it, but complained of the strange terminology and its " philosophische Allgemeinheit. Autor of the post: Undefined Need we mar vel if Post Date: Mon, 28 Jul 2008 12:20:16 +0000 " Eight years afterwards, Bretschneider of Gotha was said to be the only man who had read it through. An article in Crelle s Journal, in which Grassmann eclipsed the geometers of that time by constructing, with, aid of Ms method, geometrically any algebraic curve, remained again unnoticed. Need we mar vel if G-rassinann turned Ms attention to other subjects, to Schleiermacher s pMlosophy, to polities, to philology ? Still, articles by him continued to appear in Crelle s Journal, and in 1862 came out the second part of Ms Ausdehnungslehre. Autor of the post: Undefined At the age of fifty-three Post Date: Mon, 28 Jul 2008 12:00:43 +0000 It was intended to show better than the first part the broad scope of the Ausdehnungslehre, by considering not only geo metric applications, but by treating also of algebraic functions, infinite series, and the differential and integral calculus. But the second part was no more appreciated than the first. At the age of fifty-three, this wonderful man, with heavy heart, gave up mathematics, and directed his energies to the study of Sans krit, achieving in philology results which were better appreci ated, and which vie in splendour with those in mathematics. Autor of the post: Undefined Grass mann developed the idea Post Date: Mon, 28 Jul 2008 11:45:12 +0000 Common to the Ausdehnungslehre and to quaternions are geo metric addition, the function of two vectors represented in qua ternions by Saft and Fa3, and the linear vector functions. The quaternion is peculiar to Hamilton, while with Grassmann we find in addition to the algebra of vectors a geometrical algebra of wide application, and resembling Mobius s Barycentrische Calculj in which the point is the fundamental element. Grass mann developed the idea of the "external product," the "inter nal product," and the " open product. Autor of the post: Undefined Only in recent years Post Date: Mon, 28 Jul 2008 11:33:14 +0000 " The last we now call a matrix. His Ausdehnungslehre has very great extension, having no limitation to any particular number of dimen sions. Only in recent years has the wonderful richness of his discoveries begun to be appreciated. Autor of the post: Undefined Discoveries of less value Post Date: Mon, 28 Jul 2008 11:18:41 +0000 A second edition of the Ausdehnungslehre of 1844 was printed in 1877. Peirce gave a representation of Grassmann s system in the logical notation, and Hyde of the University of Cin cinnati wrote the first text-book on Grassmann s calculus in the English language. Discoveries of less value, which in part covered those of Grassmann and Hamilton, were made by Saint- Venant (1797- 1886), who described the multiplication of vectors, and the addition of vectors and oriented areas; by Cauchy; whose "clefs algebriques " were units subject to combinatorial mul tiplication, and were applied by the author to the theory of elimination in the same way as had been done earlier by Grassmann; by Justus Bellavitis (1803-1880), who published in 1835 and 1837 in the Annali delle Scienze his calculus of sequipollences. Autor of the post: Undefined Hankel, then decent in Leipzig Post Date: Mon, 28 Jul 2008 11:04:01 +0000 Bellavitis, for many years professor at Padua, was a self-taught mathematician of much power, who in his thirty-eighth year laid down a city office in his native place, Bassano, that he might give his time to science. The first impression of G-rassmann s ideas is marked in the writings of Hermann Hankel (1839-1873), who published in 1867 his Vorlesungen uber die Complexen Zahlen. Hankel, then decent in Leipzig, had been in correspondence with Grassraann. Autor of the post: Undefined Hankel was a close student Post Date: Mon, 28 Jul 2008 10:44:45 +0000 The "alternate numbers" of Hankel are sub ject to his law of combinatorial multiplication. In consider ing the foundations of algebra Hankel affirms the principle of the permanence of formal laws previously enunciated incom pletely by Peacock. Hankel was a close student of mathe matical history, and left behind an unfinished work thereon. Autor of the post: Undefined Schlegel was at one time Post Date: Mon, 28 Jul 2008 10:30:55 +0000 Before his death he was professor at Tubingen. His Com- plexe Zahlen was at first little read, and we must turn to Victor ScMegel of Hagen as the successful interpreter of Grass mann. Schlegel was at one time a young colleague of Grass- maun at the Marienstifts-Gyrrmasiuin in Stettin. Autor of the post: Undefined Benjamin Peirce (1809-1880) was born Post Date: Mon, 28 Jul 2008 10:16:37 +0000 Encouraged by Clebsch, Schlegel wrote a System der Baumlehre which explained the essential conceptions and operations of the Ausdehnungslehre. Multiple algebra was powerfully advanced by Peirce, whose theory is not geometrical, as are those of Hamilton and Grass- mann. Benjamin Peirce (1809-1880) was born at Salem, Mass, and graduated at Harvard College, having as undergraduate carried the study of mathematics far beyond the limits of the college course. 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