"Let a b c be a triangle; and let the side b c be produced to d; then the exterior angle a c d shall be greater than either of the interior opposite angles b a c or c b a. Bisect the side a c at e, and join b e; produce b e to f, making e f equal to e b, and join f c. Produce a c to g. Because a e is equal to e c, and b e to e f; the two sides a e, e b, are equal to the two sides c e, e f, each to each; and the angle a e b is equal to the angle c e f, because they are opposite vertical angles; therefore the base a b is equal to the base c f, and the triangle a e b is equal to the triangle c e f, and the remaining angles of one triangle to the remaining angles of the other, each to each, to which the equal sides are opposite; therefore the angle b a e is equal to the angle e c f. But the angle e c d is greater than the angle e c f. Therefore the angle a c d is greater than the angle a b c."
"In the same manner, if the side b c be bisected, and the side a c be produced to g, it may be demonstrated that the angle b c g, that is, the opposite vertical angle a c d is greater than the angle a b c."
My demonstration of the same proposition would be as follows (see fig. 5):—
Fig. 5.
For the angle b a c to be even equal to, let alone greater than, the angle a c d, the line b a toward c a would have to lie in the same direction as b d (for this is precisely what is meant by equality of the angles), i.e., it must be parallel with b d; that is to say, b a and b d must never meet; but in order to form a triangle they must meet (reason of being), and must thus do the contrary of that which would be required for the angle b a c to be of the same size as the angle a c d.
For the angle a b c to be even equal to, let alone greater than, the angle a c d, line b a must lie in the same direction towards b d as a c (for this is what is meant by equality of the angles), i.e., it must be parallel with a c, that is to say, b a and a c must never meet; but in order to form a triangle b a and a c must meet and must thus do the contrary of that which would be required for the angle a b c to be of the same size as a c d.
By all this I do not mean to suggest the introduction of a new method of mathematical demonstration, nor the substitution of my own proof for that of Euclid, for which its whole nature unfits it, as well as the fact that it presupposes the conception of parallel lines, which in Euclid comes much later. I merely wished to show what the reason of being is, and wherein lies the difference between it and the reason of knowing, which latter only effects convictio, a thing that differs entirely from insight into the reason of being. The fact that Geometry only aims at effecting convictio, and that this, as I have said, leaves behind it a disagreeable impression, but gives no insight into the reason of being—which insight, like all knowledge, is satisfactory and pleasing—may perhaps be one of the reasons for the great dislike which many otherwise eminent heads have for mathematics.
I cannot resist again giving fig. 6, although it has already been presented elsewhere; because the mere sight of it without words conveys ten times more persuasion of the truth of the Pythagorean theorem than Euclid's mouse-trap demonstration.
