Bulletin of the American Mathematical SocietySociety, 1902 - Mathematics |
Contents
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Common terms and phrases
A₁ abelian group algebraic American Mathematical Society analytic functions Analytical geometry angle application assemblage Auflage axioms Berlin BULLETIN C₁ calculus of variations chapter Cloth coefficients College congruence conic continued fraction convergent coördinates corresponding cubic cubic surface curvature curve defined definition determined Deutsche Mathematiker-Vereinigung E. H. MOORE elementary elements fessor finite number four hours func G. A. Miller given Göttingen group of order hours.-By Dr hours.-By Professor infinite infinitesimal invariant L. E. DICKSON lectures Leipzig linear differential equations lineoids London Mathematical Society matics mechanics meeting Member method obtained operator of order paper Paris Ph.D physics plane problem Professor of Mathe Professor of Mathematics proof Read ruled surface Seminar solution subgroup surface theorem Theory of Numbers three hours three hours.-By tion Transactions transformation Ueber University values variables vector volume X₁ X₂ y₁ York
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Page 416 - History teaches the continuity of the development of science. We know that every age has its own problems, which the following age either solves or casts aside as profitless and replaces by new ones. If we would obtain an idea of the probable development of mathematical knowledge in the immediate future, we must let the unsettled questions pass before our minds and look over the problems which the science of today sets and whose solution we expect from the future. To such a review of problems the...
Page 426 - When we are engaged in investigating the foundations of a science. we must set up a system of axioms which contains an exact and complete description of the relations subsisting between the elementary ideas of that science. The axioms so set up are at the same time the definitions of those elementary ideas; and no statement within the realm of the science whose foundation we are testing is held to be correct unless it can be derived from those axioms by means of a finite number of logical steps.
Page 420 - It remains to discuss briefly what general requirements may be justly laid down for the solution of a mathematical problem. I should say first of all, this : that it shall be possible to establish the correctness...
Page 256 - If pa is the highest power of a prime, p, which divides the order of a group, G, the number of sub-groups, H, of order pa is congruent to unity, modulo p.
Page 417 - As long as a branch of science offers an abundance of problems, so long is it alive: a lack of problems foreshadows extinction or the cessation of independent development.
Page 424 - After seeking in vain for the construction of a perpetual motion machine, the relations were investigated which must subsist between the forces of nature if such a machine is to be impossible; and this inverted question led to the discovery of the law of the conservation of energy, which, again, explained the impossibility of perpetual motion in the sense originally intended. This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us...
Page 458 - But, we ask, with the extension of mathematical knowledge will it not finally become impossible for the single investigator to embrace all departments of this knowledge ? In answer let me point out how thoroughly it is ingrained in mathematical science that every real advance goes...
Page 423 - ... possession of a method which is applicable also to related problems. The introduction of complex paths of integration by Cauchy and of the notion of the ideals in number theory by Kummer may serve as examples. This way...