=265.= There is probably no other science which presents such different appearances to one who cultivates it and to one who does not, as mathematics. To this person it is ancient, venerable, and complete; a body of dry, irrefutable, unambiguous reasoning. To the mathematician, on the other hand, his science is yet in the purple bloom of vigorous youth, everywhere stretching out after the “attainable but unattained” and full of the excitement of nascent thoughts; its logic is beset with ambiguities, and its analytic processes, like Bunyan’s road, have a quagmire on one side and a deep ditch on the other and branch off into innumerable by-paths that end in a wilderness. --CHAPMAN, C. H. _Bulletin American Mathematical Society, Vol. 2 (First series), p. 61._ =266.= Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts. For with all the variety of mathematical knowledge, we are still clearly conscious of the similarity of the logical devices, the _relationship_ of the _ideas_ in mathematics as a whole and the numerous analogies in its different departments. We also notice that, the farther a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected relations are disclosed between hitherto separated branches of the science. So it happens that, with the extension of mathematics, its organic character is not lost but manifests itself the more clearly.--HILBERT, D. _Mathematical Problems, Bulletin American Mathematical Society, Vol. 8, p. 478._ =267.= The mathematics have always been the implacable enemies of scientific romances.--ARAGO. _Oeuvres (1855), t. 3, p. 498._ =268.= Those skilled in mathematical analysis know that its object is not simply to calculate numbers, but that it is also employed to find the relations between magnitudes which cannot be expressed in numbers and between functions whose law is not capable of algebraic expression.--COURNOT, AUGUSTIN. _Mathematical Theory of the Principles of Wealth [Bacon, N. T.], (New York, 1897), p. 3._