A construction, therefore, becomes interesting only when it can be ranged beside other analogous constructions, forming species of the same genus. If the quadrilateral is something besides the juxtaposition of two triangles, this is because it belongs to the genus polygon. Moreover, one must be able to demonstrate the properties of the genus without being forced to establish them successively for each of the species. To attain that, we must necessarily mount from the particular to the general, ascending one or more steps. The analytic procedure 'by construction' does not oblige us to descend, but it leaves us at the same level. We can ascend only by mathematical induction, which alone can teach us something new. Without the aid of this induction, different in certain respects from physical induction, but quite as fertile, construction would be powerless to create science. Observe finally that this induction is possible only if the same operation can be repeated indefinitely. That is why the theory of chess can never become a science, for the different moves of the same game do not resemble one another. CHAPTER II MATHEMATICAL MAGNITUDE AND EXPERIENCE To learn what mathematicians understand by a continuum, one should not inquire of geometry. The geometer always seeks to represent to himself more or less the figures he studies, but his representations are for him only instruments; in making geometry he uses space just as he does chalk; so too much weight should not be attached to non-essentials, often of no more importance than the whiteness of the chalk. The pure analyst has not this rock to fear. He has disengaged the science of mathematics from all foreign elements, and can answer our question: 'What exactly is this continuum about which mathematicians reason?' Many analysts who reflect on their art have answered already; Monsieur Tannery, for example, in his _Introduction à la théorie des fonctions d'une variable_.