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nydus/A History of MathematicsPublic

This text examines the transition from the Middle Ages to the Modern era, highlighting how the fall of Constantinople and the invention of the printing press catalyzed a revival of classical learning. It traces the shift toward scientific inquiry through the rise of pure mathematics and astronomy, detailing the intellectual struggle against established scholastic and ecclesiastical authority.

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Table of Contents

Algebra

could be reached in a shorter way by some new ideas advanced in his father's books, and he proceeded to elaborate this abridged method, and to apply it in the study of tides. He was thus led to a new geometric analysis. In 1840 he had made considerable progress in its development, but a new book of Schleiermacher drew him again to theology. In 1842 he resumed mathematical research, and becoming thoroughly convinced of the importance of his new analysis, decided to devote himself to it. 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 Ausdehnungslehre, 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

Möbius glanced over it, praised it, but complained of the

strange terminology and its "philosophische Allgemeinheit." 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 his method, geometrically any algebraic curve, remained again unnoticed. Need we marvel if Grassmann turned his attention to other subjects,–-to Schleiermacher's philosophy, to politics, to philology? Still, articles by him continued to appear in Crelle's Journal, and in 1862 came out the second part of his Ausdehnungslehre. It

was intended to show better than the first part the broad scope of the Ausdehnungslehre, by considering not only geometric 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 Sanskrit, achieving in philology results which were better appreciated, and which vie in splendour with those in mathematics.

Common to the Ausdehnungslehre and to quaternions are geometric addition, the function of two vectors represented in quaternions by Sαβ and Vαβ, 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 Möbius's Barycentrische

Calcul, in which the point is the fundamental element. Grassmann developed the idea of the "external product," the "internal product," and the "open product." The last we now call a matrix. His Ausdehnungslehre has very great extension, having no limitation to any particular number of dimensions. Only in recent years has the wonderful richness of his discoveries begun to be appreciated. A second edition of the Ausdehnungslehre of 1844 was printed in 1877. C. S. Peirce gave a representation of Grassmann's system in the

logical notation, and E. W. Hyde of the University of Cincinnati

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