De docta ignorantia

The imagination, which does not transcend the genus of sensible things, does not grasp that a line can be a triangle, since these differ disproportionately in quantity; yet with the intellect this will be easy. For it is already established that only one maximum and infinite is possible. Then it is established, since any two sides of any triangle joined together cannot be less than the third, that of a triangle whose one side is infinite the others are not less. And since every part of the infinite is infinite, it is necessary that every side of a triangle whose one side is infinite be likewise infinite. And since there cannot be several infinites, you understand transcendently that the infinite triangle cannot be composed of several lines, although it is the maximal and truest triangle, incomposite and most simple. And since it is the truest triangle, which cannot be without three lines, it will be necessary that the one infinite line itself be three, and the three be one most simple. So too of the angles, since there will be but one infinite angle, and it is three angles and three angles are one. Nor will this maximal triangle be composed of sides and angles, but the infinite line and the angle are one and the same, so that the line too is an angle, since the triangle is a line. You will still be able to aid yourself toward understanding this by ascending from the quantitative triangle to the non-quantitative. For it is manifest that every quantitative triangle has three angles equal to two right angles; and thus, the greater one angle is, the less the others; and although each angle can be increased up to two right angles exclusively and not maximally, according to our first principle, let us yet admit that it is increased maximally up to two right angles inclusively, the triangle remaining; then it is manifest that the triangle has one angle which is three, and the three are one. Likewise you will be able to see that the triangle is a line, since when any two sides of a quantitative triangle joined together are longer than the third by so much

FIGURA pg079: tall thin triangle with apex a, base bc, and inner lines forming a smaller triangle

as the angle which they make is less than two right angles, as the angle bac, since it is much less than two right angles, hence the lines ba and ac joined together are much longer than bc. Therefore the greater that angle shall have been, as bdc, the less the surface; and if by supposition the angle were to amount to two right angles, the whole triangle would be resolved into a simple line. Whence with this supposition, which in quantities is impossible, you can aid yourself, ascending to non-quantitative things, in which what in quantities is impossible you see to be in all respects necessary, and in this it is evident that the infinite line is the maximal triangle, which was to be shown.