The reputed motto of the Platonic lecture-room, “Αγεωμετρητος μηδεις εισιτω,” of which mathematicians are so proud, was no doubt inspired by the fact that Plato regarded the geometrical figures as intermediate existences between the eternal Ideas and particular things, as Aristotle frequently mentions in his “Metaphysics” (especially i. c. 6, p. 887, 998, _et Scholia_, p. 827, ed. Berol.) Moreover, the opposition between those self-existent eternal forms, or Ideas, and the transitory individual things, was most easily made comprehensible in geometrical figures, and thereby laid the foundation of the doctrine of Ideas, which is the central point of the philosophy of Plato, and indeed his only serious and decided theoretical dogma. In expounding it, therefore, he started from geometry. In the same sense we are told that he regarded geometry as a preliminary exercise through which the mind of the pupil accustomed itself to deal with incorporeal objects, having hitherto in practical life had only to do with corporeal things (_Schol. in Aristot._, p. 12, 15). This, then, is the sense in which Plato recommended geometry to the philosopher; and therefore one is not justified in extending it further. I rather recommend, as an investigation of the influence of mathematics upon our mental powers, and their value for scientific culture in general, a very thorough and learned discussion, in the form of a review of a book by Whewell in the _Edinburgh Review_ of January 1836. Its author, who afterwards published it with some other discussions, with his name, is Sir W. Hamilton, Professor of Logic and Metaphysics in Scotland. This work has also found a German translator, and has appeared by itself under the title, “_Ueber den Werth und Unwerth der Mathematik_” _aus dem Englishen_, 1836. The conclusion the author arrives at is that the value of mathematics is only indirect, and lies in the application to ends which are only attainable through them; but in themselves mathematics leave the mind where they find it, and are by no means conducive to its general culture and development, nay, even a decided hindrance. This conclusion is not only proved by thorough dianoiological investigation of the mathematical activity of the mind, but is also confirmed by a very learned accumulation of examples and authorities. The only direct use which is left to mathematics is that it can accustom restless and unsteady minds to fix their attention. Even Descartes, who was yet himself famous as a mathematician, held the same opinion with regard to mathematics. In the “_Vie de Descartes par Baillet_,” 1693, it is said, Liv. ii. c. 6, p. 54: “_Sa propre expérience l’avait convaincu du peu d’utilité des mathématiques, surtout lorsqu’on ne les cultive que pour elles mêmes.... Il ne voyait rien de moins solide, que de s’occuper de nombres tout simples et de figures imaginaires_,” &c. Chapter XIV. On The Association Of Ideas.