_Introduction to Mathematics (New York, 1911), p. 113._ =1404.= The union of philosophical and mathematical productivity, which besides in Plato we find only in Pythagoras, Descartes and Leibnitz, has always yielded the choicest fruits to mathematics: To the first we owe scientific mathematics in general, Plato discovered the analytic method, by means of which mathematics was elevated above the view-point of the elements, Descartes created the analytical geometry, our own illustrious countryman discovered the infinitesimal calculus--and just these are the four greatest steps in the development of mathematics. --HANKEL, HERMANN. _Geschichte der Mathematik im Altertum und im Mittelalter (Leipzig, 1874), pp. 149-150._ =1405.= Without mathematics one cannot fathom the depths of philosophy; without philosophy one cannot fathom the depths of mathematics; without the two one cannot fathom anything. --BORDAS-DEMOULINS. _Quoted in A. Rebière: Mathématiques et Mathématiciens (Paris, 1898), p. 147._ =1406.= In the end mathematics is but simple philosophy, and philosophy, higher mathematics in general.--NOVALIS. _Schriften (Berlin, 1901), Teil 2, p. 443._ =1407.= It is a safe rule to apply that, when a mathematical or philosophical author writes with a misty profundity, he is talking nonsense.--WHITEHEAD, A. N. _Introduction to Mathematics (New York, 1911), p. 227._ =1408.= The real finisher of our education is philosophy, but it is the office of mathematics to ward off the dangers of philosophy.--HERBART, J. F. _Pestalozzi’s Idee eines ABC der Anschauung; Werke [Kehrbach], (Langensalza, 1890), Bd. 1, p. 168._ =1409.= Since antiquity mathematics has been regarded as the most indispensable school for philosophic thought and in its highest spheres the research of the mathematician is indeed most closely related to pure speculation. Mathematics is the most perfect union between exact knowledge and theoretical thought.--CURTIUS, E. _Berliner Monatsberichte (1873), p. 517._ =1410.= Geometry has been, throughout, of supreme importance in the history of knowledge.--RUSSELL, BERTRAND. _Foundations of Geometry (Cambridge, 1897), p. 54._ =1411.= He is unworthy of the name of man who is ignorant of the fact that the diagonal of a square is incommensurable with its side.--PLATO. _Quoted by Sophie Germain: Mémoire sur les surfaces élastiques._