we need not and ought not to leave the peculiar province of mathematics in order to put our trust only in logical proof, and seek to authenticate mathematics in a sphere which is quite foreign to it, that of concepts. If we confine ourselves to the ground peculiar to mathematics, we gain the great advantage that in it the rational knowledge _that_ something is, is one with the knowledge _why_ it is so, whereas the method of Euclid entirely separates these two, and lets us know only the first, not the second. Aristotle says admirably in the Analyt., post. i. 27: “Ακριβεστερα δ᾽ επιστημη επιστημης και προτερα, ἡτε του ὁτι και του διοτι ἡ αυτη, αλλα μη χωρις του ὁτι, της του διοτι” (_Subtilior autem et praestantior ea est scientia, quâ_ QUOD _aliquid sit, et_ CUR _sit una simulque intelligimus non separatim_ QUOD, _et_ CUR _sit_). In physics we are only satisfied when the knowledge that a thing is as it is is combined with the knowledge why it is so. To know that the mercury in the Torricellian tube stands thirty inches high is not really rational knowledge if we do not know that it is sustained at this height by the counterbalancing weight of the atmosphere. Shall we then be satisfied in mathematics with the _qualitas occulta_ of the circle that the segments of any two intersecting chords always contain equal rectangles? That it is so Euclid certainly demonstrates in the 35th Prop. of the Third Book; _why_ it is so remains doubtful. In the same way the proposition of Pythagoras teaches us a _qualitas occulta_ of the right-angled triangle; the stilted and indeed fallacious demonstration of Euclid forsakes us at the _why_, and a simple figure, which we already know, and which is present to us, gives at a glance far more insight into the matter, and firm inner conviction of that necessity, and of the dependence of that quality upon the right angle:— [Illustration]