## History of Modern Mathematics (Google eBook) |

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Abel Abelian functions Abelian integrals algebraic algebraic curves American Mathematical Society analytic geometry applied bibliography Bibliotheca Mathematica binary Brioschi Bulletin calculus Cantor Cauchy Cayley Cayley's Chasles Clebsch complex numbers contributions contributors convergence Crelle cubic Dedekind determinants Dirichlet discovery Eisenstein elliptic functions Euler Evanston Lectures formula foundation fundamental theorem Galois Gauss gave Geschichte der Mathematik Grassmann groups H. J. S. Smith Halphen Hermite Hesse integrals investigation Jacobi Ji.oo Klein Kronecker Kummer labors Lagrange Laplace Legendre Leipzig Liouville Mathe mathematicians matical memoir mentioned Mobius modern mathematics Monge nineteenth century noteworthy partial differential equations Picard plane curves Poincare Poncelet principle problem projective geometry prominent published Quaternions quintic quintic equation recent Riemann roots Schwarz Steiner Sylvester synthetic geometry ternary ternary forms theory of Abelian theory of elliptic theory of forms theory of functions theory of numbers tions transcendent numbers treatises trigonometry Weierstrass writers York Mathematical Society

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Page 46 - Abelian functions is an extension of Clebsch's scheme. In this extension, as in the general theory of Abelian functions, Klein has been a leader. With the development of the theory of Abelian functions is connected a long list of names, including those of Schottky, Humbert. C. Neumann, Fricke, Konigsberger, Prym, Schwarz, Painleve', Hurwitz, Brioschi, Borchardt, Cayley, Forsyth, and Rosenhain, besides others already mentioned. Returning to the theory of elliptic functions, Jacobi (1827) began by...

Page 66 - Bartels went to Kasan in 1807, and Lobachevsky was his pupil. The latter's lecture notes show that Bartels never mentioned the subject of the fifth postulate to him, so that his investigations, begun even before 1823, were made on his own motion and his results were wholly original. Early in 1826 he sent forth the principles of his famous doctrine of parallels, based on the assumption that through a given point more than one line can be drawn which shall never meet a given line coplanar with it....

Page 37 - The theory of singular solutions of ordinary and partial differential equations has been a subject of research from the time of Leibniz, but only since the middle of the present century has it received especial attention. A valuable but littleknown work on the subject is that of Houtain (1854). Darboux (from 1873) has been a leader in the theory, and in the geometric interpretation of these solutions he has opened a field which has been worked by various writers, notably Casorati and Cayley. To the...

Page 53 - Clairaut's (1731), in which, at the age of sixteen, he solved with rare elegance many of the problems relating to curves of double curvature. Euler (1760) laid the foundations for the analytic theory of curvature of surfaces, attempting the classification of those of the second degree as the ancients had classified curves of the second order. Monge, Hachette, and other members of that school entered into the study of surfaces with great zeal. Monge introduced the notion of families of surfaces, and...

Page iii - EDITORS' PREFACE. THE volume called Higher Mathematics, the third edition of which was published in 1900, contained eleven chapters by eleven authors, each chapter being independent of the others, but all supposing the reader to have at least a mathematical training equivalent to that given in classical and engineering colleges. The publication of that volume was discontinued in 1906, and the chapters have since been issued in separate Monographs, they being generally enlarged by additional articles...

Page iii - ... Monographs, they being generally enlarged by additional articles or appendices which either amplify the former presentation or record recent advances. This plan of publication was arranged in order to meet the demand of teachers and the convenience of classes, and it was also thought that it would prove advantageous to readers in special lines of mathematical literature. It is the intention of the publishers and editors to add other monographs to the series from time to time, if the demand seems...

Page 27 - But about the time of Jacobi's closing memoirs, Sylvester (1839) and Cayley began their great work, a work which it is impossible to briefly summarize, but which represents the development of the theory to the present time. The study of special forms of determinants has been the natural result of the completion of the general theory. Axisymmetric determinants have been studied by Lebesgue, Hesse, and Sylvester ; per-symmetric determinants by Sylvester and Hankel ; circulants by Catalan, Spottiswoode,...

Page iii - ... former presentation or record recent advances. This plan of publication has been arranged in order to meet the demand of teachers and the convenience of classes, but it is also thought that it may prove advantageous to readers in special lines of mathematical literature. It is the intention of the publishers and editors to add other monographs to the series from time to time, if the call for the same seems to warrant it. Among the topics which are under consideration are those of elliptic functions,...

Page 51 - Analyst" (1808), first deduced the law of facility of error, (j>(x) = tY"*"*1, c and h being constants depending on precision of observation. He gave two proofs, the second being essentially the same as Herschcl's (1850). Gauss gave the first proof which seems to have been known in Europe (the third after Adrain's) in 1809. To him is due much of the honor of placing the subject before the mathematical world, both as to the theory and its applications. Further proofs were given by Laplace...

Page 26 - ... Of value, too, have been the labors of Killing on the structure of groups, Study's application of the group theory to complex numbers, and the work of Schur and Maurer. ART. 8. DETERMINANTS. The Theory of Determinants* may be said to take its origin with Leibniz (1693), following whom Cramer (1750) added slightly to the theory, treating the subject, as did his predecessor, wholly in relation to sets of equations. The recurrent law was first announced by Bezout (1764). But it was Vandermonde (1771)...