The World as Will and Idea, Vol. I — Arthur Schopenhauer, trans. Haldane & Kemp

was not distinguished from empirical perception. Certain passages from the works of Proclus, the commentator of Euclid, which Kepler translated into Latin in his book, “ De Harmonia Mundi,” seem to show that he fully recognised this distinction. But Proclus did not attach enough importance to the matter ; he merely mentioned it by the way, so that he remained unnoticed and accomplished nothing.

Tlmrefore, not till two thousand years later will the doctrine of Kant, which is destined to make such great changes in all the knowledge, thought, and action of European nations, produce this change in mathematics also.

For it is only after we have learned from this great man that the intuitions or perceptions of space and time are quite different from empirical perceptions, entirely independent of any impression of the senses, conditioning it, not conditioned by it, i.e., are a priori, and therefore are not exposed to the illusions of sense ; only after we have learned this, I say, can we comprehend that Euclid’s logical method of treating mathematics is a useless precaution, a crutch for sound legs, that it is like a wanderer who during the night mistakes a bright, firm road for water, and carefully avoiding it, toils over the broken ground beside it, content to keep from point to point along the edge of the supposed water.

Only now can we affirm with certainty that what presents itself to us as necessary in the perception of a figure, does not come from the figure on the paper, which is perhaps very defectively drawn, nor from the abstract concept under which we think it, but immediately from the form of all knowledge of which we are conscious a priori.

This is always the principle of sufficient reason ; here as the form of perception, i.e., space, it is the principle of the ground of being, the evidence and validity of which is, however, just as great and as immediate as that of the principle of the ground of knowing, i.e., logical certainty. Thus we need not and ought not to leave the peculiar province of mathematics 2in 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 : “Axpi^earepa S’ €7naT7)firj eiriarypLyt Kai Trporepa, rjTe tov oti Kai rov Biotc 17 avTf}, aWa firj 'X^copi'i tov ort, T17? tov Start ” (Svjbtilior autem et praestantior ea est sdentia, qud quod aliquid sit, et CUR sit una simulqtie intelligimus non separatim QUOD, et cub siC).

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 35 th 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