Mathematics, From the Points of View of the Mathematician, and of the Physicist

Whilst the greatest effort has been made to ensure the quality of this text, due to the historical nature of this content, in some rare cases there may be minor issues with legibility. An interesting instance in which some very abstract departments of Mathematics were largely dependent for their origin upon concrete physical problems is the case of the problems connected with the vibra tion of strings and bars. These problems were discussed by D. Bernoulli, Euler, and Lagrange, in the eighteenth century, and led to the attempt to represent arbitrarily prescribed functions by means of trigonom etrical series. To the discussions as to the possibilities in this direction are largely owing the progressive development of the notion of a Mathe matical function, culminating in the extreme generality of the notion as it is at present accepted. In the hands of Fourier, the representa tion by trigonometrical series acquired the form which has become of so much importance in the resolution of disturbances, such as the tides, into harmonic components. But for those of enquiring mind it was also the starting point of a long series of purely Mathematical investigations, and it gave rise to all sorts of difficult questions of theoretic interest. The writings of Dirichlet, Riemann, G. Cantor, and more recently of Lebesgue, are witnesses to the very general nature of the discussions which have arisen in this connection and which, to a student of the history of the Science, appear as the natural evolution in an order of ideas which had its origin in the problems of vibrations and of the conduction of heat. The developments which arose in this way have been a dominant factor in the modern transformation of Mathematical Analysis.