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 CHAPTER II.

we want to know what the mathematicians mean by a continuum, it is useless to appeal to geometry. The geometer is always seeking, more or less, to represent to himself the figures he is studying, but his representations are only instruments to him; he uses space in his geometry just as he uses chalk; and further, too much importance must not be attached to accidents which are often nothing more than the whiteness of the chalk.

The pure analyst has not to dread this pitfall. He has disengaged mathematics from all extraneous elements, and he is in a position to answer our question:—"Tell me exactly what this continuum is, about which mathematicians reason." Many analysts who reflect on their art have already done so—M. Tannery, for instance, in his Introduction à la théorie des Fonctions d'une variable.

Let us start with the integers. Between any two consecutive sets, intercalate one or more intermediary sets, and then between these sets others