4 INTRODUCTION
of great importance, and finally, in 1882, Cantor's "transfinite numbers" were defined independently of the aggregates in connexion with which they first appeared in mathematics.
III
The investigations[1] of the eighteenth century on the problem of vibrating cords led to a controversy for the following reasons. D'Alembert maintained that the arbitrary functions in his general integral of the partial differential equation to which this problem led were restricted to have certain properties which assimilate them to the analytically representable functions then known, and which would prevent their course being completely arbitrary at every point. Euler, on the other hand, argued for the admission of certain of these "arbitrary" functions into analysis. Then Daniel Bernoulli produced a solution in the form of an infinite trigonometrical series, and claimed, on certain physical grounds, that this solution was as general as d'Alembert's. As Euler pointed out, this was so only if any arbitrary[2] function ø(x) were developable in a series of the form
- ↑ Cf. the references given in my papers in the Archiv der Mathematik und Physik, 3rd series, vol. x, 1906, pp. 255-256, and Isis, vol. i, 1914, pp. 670-677. Much of this Introduction is taken from my account of "The Development of the Theory of Transfinite Numbers" in the above-mentioned Archiv, 3rd series, vol. x, pp. 254-281; vol. xiv, 1909, pp. 289-311; vol. xvi, 1910, pp. 21-43; vol. xxii, 1913, pp. 1-21.
- ↑ The arbitrary functions chiefly considered in this connexion by Euler were what he called "discontinuous" functions. This word does not mean what we now mean (after Cauchy) by it. Cf. my paper in Isis, vol. i, 19 14, pp. 661-703.
