This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1894 Excerpt: ...the paper by Minkowski, Ueber Eigenschaften von ganzen Zahlen, die durch rdumliche Anschauung erschlossen sind, which I had the pleasure of presenting to you in abstract at the Congress of Mathematics. Here geometry is used directly for the development of new arithmetical ideas. 58 To-day I wish to speak on the ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1894 Excerpt: ...the paper by Minkowski, Ueber Eigenschaften von ganzen Zahlen, die durch rdumliche Anschauung erschlossen sind, which I had the pleasure of presenting to you in abstract at the Congress of Mathematics. Here geometry is used directly for the development of new arithmetical ideas. 58 To-day I wish to speak on the composition of binary algebraic forms, a subject first discussed by Gauss in his Disquisitiones arithmetics and of Rummer's corresponding theory of ideal numbers. Both these subjects have always been considered as very abstruse, although Dirichlet has somewhat simplified the treatment of Gauss. I trust you will find that the geometrical considerations by means of which I shall treat these questions introduce so high a degree of simplicity and clearness that for those not familiar with the older treatment it must be difficult to realize why the subject should ever have been regarded as so very intricate. These considerations were indicated by myself in the Gottinger Nachrichten for January, 1893; and at the beginning of the summer semester of the present year I treated them in more extended form in a course of lectures. I have since learned that similar ideas were proposed by Poincare in 1881; but I have not yet had sufficient leisure to make a comparison of his work with my own. I write a binary quadratic form as follows: / = ax? + bxy + cy2, i.e. without the factor 2 in the second term; some advantages of this notation were recently pointed out by H. Weber, in the Gottinger Nachrichten, 1892-93. The quantities a, b, c, x, y are here of course all assumed to be integers. It is to be noticed that in the theory of numbers a common factor of the coefficients a, b, c cannot be introduced or omitted arbitrarily, as in projective geometry; in other words, ...
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