92 THE WORLD AS IDEA.
machine are shown, but from whom its inner connection and construction are withheld. We are compelled by the principle of contradiction to admit that what Euclid demonstrates is true, but we do not comprehend why it is so. We have therefore almost the same uncomfortable feeling that we experience after a juggling trick, and, in fact, most of Euclid's demonstrations are remarkably like such feats. The truth almost always enters by the back door, for it manifests itself per accidens through some contingent circumstance. Often a reductio ad absurdum shuts all the doors one after another, until only one is left through which we are therefore compelled to enter. Often, as in the proposition of Pythagoras, lines are drawn, we don't know why, and it afterwards appears that they were traps which close unexpectedly and take prisoner the assent of the astonished learner, who must now admit what remains wholly inconceivable in its inner connection, so much so, that he may study the whole of Euclid through and through without gaining a real insight into the laws of space-relations, but instead of them he only learns by heart certain results which follow from them. This specially empirical and unscientific knowledge is like that of the doctor who knows both the disease and the cure for it, but does not know the connection between them. But all this is the necessary consequence if we capriciously reject the special kind of proof and evidence of one species of knowledge, and forcibly introduce in its stead a kind which is quite foreign to its nature. However, in other respects the manner in which this has been accomplished by Euclid deserves all the praise which has been bestowed on him through so many centuries, and which has been carried so far that his method of treating mathematics has been set up as the pattern of all scientific exposition. Men tried indeed to model all the sciences after it, but later they gave up the attempt without quite knowing why. Yet in our eyes this method of Euclid in mathematics
