Fig. 3.
Which Euclid demonstrates as follows:—
"Let a b c be a triangle having the angle a b c equal to the angle a c b, then the side a c must be equal to the side a b also.
"For, if side a b be not equal to side a c, one of them is greater than the other. Let a b be greater than a c; and from b a cut off b d equal to c a, and draw d c. Then, in the triangles d b c, a b c, because d b is equal to a c, and b c is common to both triangles, the two sides d b and b c are equal to the two sides a c, a b, each to each; and the angle d b c is equal to the angle a c b, therefore the base d c is equal to the base a b, and the triangle d b c is equal to the triangle a b c, the less triangle equal to the greater,—which is absurd. Therefore a b is not unequal to a c, that is, a b is equal to a c."
Now, in this demonstration we have a reason of knowing for the truth of the proposition. But who bases his conviction of that geometrical truth upon this proof? Do we not rather base our conviction upon the reason of being, which we know intuitively, and according to which (by a necessity which admits of no further demonstration, but only of evidence through intuition) two lines drawn from both extreme ends of another line, and inclining equally towards each other, can only meet at a point which is equally distant from both extremities; since the two arising angles are properly but one, to which the oppositeness of position gives the appearance of being two; wherefore there is no reason why the lines should meet at any point nearer to the one end than to the other.
It is the knowledge of the reason of being which shows us the necessary consequence of the conditioned from its condition—in this instance, the lateral equality from the angular equality—that is, it shows their connection; whereas the reason of knowing only shows their coexistence. Nay, we might even maintain that the usual method of proving merely convinces us of their coexistence in the actual figure given us as an example, but by no means that they are always coexistent; for, as the necessary connection is not shown, the conviction we acquire of this truth rests simply upon induction, and is based upon the fact, that we find it is so in every figure we make. The reason of being is certainly not as evident in all cases as it is in simple theorems like this 6th one of Euclid; still I am persuaded that it might be brought to evidence in every theorem, however complicated, and that the proposition can always be reduced to some such simple intuition. Besides, we are all just as conscious à priori of the necessity of such a reason of being for each relation of Space, as we are of the necessity of a cause for each change. In complicated theorems it will, of course, be very difficult to show that reason of being; and this is not the place for difficult geometrical researches. Therefore, to make my meaning somewhat clearer, I will now try to bring back to its reason of being a moderately complicated proposition, in which nevertheless that reason is not immediately evident. Passing over the intermediate theorems, I take the 16th:
"In every triangle in which one side has been produced, the exterior angle is greater than either of the interior opposite angles."
This Euclid demonstrates in the following manner (see fig. 4):—
Fig. 4.