De docta ignorantia

I say therefore that if there were an infinite line, it would be straight, it would be a triangle, it would be a circle, it would be a sphere; and likewise if there were an infinite sphere, it would be triangle, circle, and line; and the same must be said of the infinite triangle and the infinite circle. But first, that the infinite line is straight is evident. The diameter of a circle is a straight line, and the circumference is a curved line greater than the diameter; if therefore the curved line admits less of its curvature the greater the circumference of the circle; therefore the circumference of the greatest circle, than which none can be greater, is least curved, and therefore maximally straight. The minimum thus coincides with the maximum, so that it is seen even by the eye to be necessary that the maximal line be maximally straight and minimally curved. Nor can any scruple of doubt remain here, when it is seen in the figure placed here at the side, in the way

FIGURA pg076: geometric diagram with three nearhorizontal lines ab, cd, ef and a large arc gh below

in which the arc cd of the larger circle recedes more from curvature than the arc ef of the smaller circle, so that the straight line ab will be the arc of the greatest circle, than which none can be greater. Thus it is seen how the maximal and infinite line is necessarily most straight, to which curvature is not opposed; indeed curvature itself in the maximal line is straightness, and this is the first point to be proved. Secondly, it has been said that the infinite line is the maximal triangle, circle, and sphere, and to show this we must see, as in beginnings, what lies in the potency of the finite line; and since whatever is in the potency of the finite line the infinite line is in act, what we inquire into will be clearer to us. And first we know that a finite line in length can be longer and straighter, and it has already been proved that the maximal is the longest and straightest. Secondly, if the line ab, with the point a remaining fixed, were rotated until b came to c, a triangle has arisen; if the rotation is completed

FIGURA pg077: circle with centre a and points b, c on circumference, d below

the rotation, until b returns to the beginning where it began, a circle is made. If again ac, with b remaining fixed, is rotated until it arrives at the opposite place where it began, which let be d, from the line ab and ad one continuous line has been effected, and a semicircle has been described; and if, with bd as fixed diameter, the semicircle is rotated, a sphere will arise, and the sphere itself is the ultimate thing existing wholly in act from the potency of the line, since the sphere is not in potency toward any

FIGURA pg077: semicircle with diameter endpoints d and b, midpoint mark a

further figure. If therefore these figures are in the potency of the finite line, and the infinite line is in act all the things toward which the finite is in potency, it follows that the infinite is triangle, circle, and sphere, which was to be proved. And since you might wish to see this more clearly, how what in potency is finite is the same in act in the infinite, I shall render you still most certain of this.