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. . . (page 94). Euclid's logical method of treating mathematics is a useless precaution, a crutch for sound legs. . . (page 95). 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 triangle:

In the case of unequal catheti also, and indeed generally in the case of every possible geometrical truth, it is quite possible to obtain such a conviction based on perception. . . (page 96). It is the analytical method in general that I wish for the exposition of mathematics, instead of the synthetical method which Euclid made use of.

In the above we have Schopenhauer's famous characterization of mathematical reasoning as "mouse-trap proofs" (Mausefallenbeweise). These quotations and other passages which space does not permit us to quote indicate that his objections are directed almost entirely against Euclid. Schopenhauer discloses no acquaintance with such modern mathematical concepts as that of a function, of a variable, of coordinate representation, and the use of graphic methods. With him Euclid and mathematics are largely synonymous. Because of this one-sided and limited vision we can hardly look upon Schopenhauer as a competent judge of the educational value of modern mathematics.

If Schopenhauer's criticism of Euclid is taken as the expression of the feelings, not of an advanced mathematician, but of a person first entering upon the study of geometry and using Euclid's "Elements," then we are willing to admit the validity of Schopenhauer's criticisms, in part. Euclid did not write his geometry for children. It is a historical puzzle, difficult to explain, how Euclid ever came to be regarded as a text suitable for the first introduction into geometry. Euclid is written for trained minds, not for immature children. Of interest is Schopenhauer's reference to the method of proof, called the reductio ad absurdum. The experience of teachers with this method has been much the same in all countries. Some French critics called it a method which "convinces but does not satisfy the mind." De Morgan says: "The most serious embarrassment in the purely reasoning part is the reductio ad absurdum, or indirect demonstration. This form of argument is generally the last to be clearly understood, though it occurs almost on the threshold of the 'Elements.' We may find the key to the difficulty in the confined ideas which prevail on the modes of speech there employed."