_The Departments of Mathematics, and their Mutual Relations; Journal of Speculative Philosophy, Vol. 5, p. 164._ =135.= Mathematics is the science of the functional laws and transformations which enable us to convert figured extension and rated motion into number.--HOWISON, G. H. _The Departments of Mathematics, and their Mutual Relations; Journal of Speculative Philosophy, Vol. 5, p. 170._ CHAPTER II THE NATURE OF MATHEMATICS =201.= Mathematics, from the earliest times to which the history of human reason can reach, has followed, among that wonderful people of the Greeks, the safe way of science. But it must not be supposed that it was as easy for mathematics as for logic, in which reason is concerned with itself alone, to find, or rather to make for itself that royal road. I believe, on the contrary, that there was a long period of tentative work (chiefly still among the Egyptians), and that the change is to be ascribed to a _revolution_, produced by the happy thought of a single man, whose experiments pointed unmistakably to the path that had to be followed, and opened and traced out for the most distant times the safe way of a science. The history of that intellectual revolution, which was far more important than the passage round the celebrated Cape of Good Hope, and the name of its fortunate author, have not been preserved to us.... A new light flashed on the first man who demonstrated the properties of the isosceles triangle (whether his name was _Thales_ or any other name), for he found that he had not to investigate what he saw in the figure, or the mere concepts of that figure, and thus to learn its properties; but that he had to produce (by construction) what he had himself, according to concepts _a priori_, placed into that figure and represented in it, so that, in order to know anything with certainty _a priori_, he must not attribute to that figure anything beyond what necessarily follows from what he has himself placed into it, in accordance with the concept. --KANT, IMMANUEL. _Critique of Pure Reason, Preface to the Second Edition [Müller], (New York, 1900), p. 690._ =202.= [When followed in the proper spirit], there is no study in the world which brings into more harmonious action all the faculties of the mind than the one [mathematics] of which I stand here as the humble representative and advocate. There is none other which prepares so many agreeable surprises for its followers, more wonderful than the transformation scene of a pantomime, or, like this, seems to raise them, by successive steps of initiation to higher and higher states of conscious intellectual being.--SYLVESTER, J. J.