Unit III — Coincidence of Infinite Circle, Straight Line, Triangle & Sphere¶
De Docta Ignorantia · Liber I, Capitula XIII–XV (1440)
Si esset linea infinita, illa esset recta, illa esset triangulus, illa esset circulus et esset sphaera; immo curvitas in ipsa maxima linea est rectitudo.
If there were an infinite line, it would be a straight line, a triangle, a circle, and a sphere; nay rather, within that maximal line itself curvature is rectitude. — De Docta Ignorantia I.13 §§1–2
Book I, Chapter 13 — On the Properties of the Maximal and Infinite Line¶
Original GEOMETOR Translation from the Latin · h I, nn. 35–36 · Paris 1514 fol. 5r–5v · Rotta 1913 pp. 27–29 (restoring Klibansky’s collation on arc gh)
I declare, therefore, that if there were an infinite line, it would be a straight line, it would be a triangle, it would be a circle, and it would be a sphere; and in like manner, if there were an infinite sphere, it would be a circle, a triangle, and a line; and the very same must be said concerning an infinite triangle and an infinite circle.
First, that an infinite line is straight is plain: the diameter of a circle is a straight line, whereas the circumference is a curved line greater than the diameter. If, then, the curved line admits less curvature the larger the circle whose circumference it is, the circumference of the maximal circle—which cannot be greater—is minimally curved, and therefore maximally straight. Thus the Minimum coincides with the Maximum, so that before our very eyes it appears necessary that the maximal line be maximally straight and minimally curved. Nor can the slightest scruple of doubt linger here when in the figure alongside one sees how the arc cd of the larger circle recedes further from curvature than the arc ef of the smaller circle, and that arc recedes further from curvature than the arc gh of a still smaller circle; wherefore the straight line ab will be the arc of the maximal circle, which cannot be greater. Thus we see how a maximal and infinite line is necessarily most straight, to which curvature is not opposed—nay rather, within that maximal line itself curvature is rectitude; and this was the first point to be proved.
Second, it was stated that the infinite line is the maximal triangle, circle, and sphere. To demonstrate this, we must observe in finite lines what lies in the potency of a finite line; and because whatever lies in the potency of a finite line is what the infinite line is in act, what we seek will become clearer to us.
First we know that a finite line can be longer and straighter in length, and it has already been proved that the maximal line is longest and straightest. Second, if the line ab, while the point a remains fixed, is swung around until b arrives at c, a triangle arises; if the revolution is completed until b returns to the starting-point where it began, a circle is produced. Again, if while a remains fixed, b is swung around until it reaches the place opposite where it began—let it be d—from the lines ab and ad one continuous line is formed and a semicircle is described. And if, while the diameter bd remains fixed, the semicircle is revolved around it, a sphere arises; and this sphere itself is the ultimate realization of the potency of the line, existing wholly in act, because the sphere does not stand in potency toward any further figure. If, therefore, these figures reside in the potency of a finite line, and the infinite line is in act all things toward which a finite line is in potency, it follows that the infinite line is triangle, circle, and sphere. Which was to be proved.
And since perhaps you would wish to see still more clearly how the infinite is in act those things that lie in the potency of the finite, I shall render you most certain of this as well.
Book I, Chapter 14 — That the Infinite Line Is a Triangle¶
Original GEOMETOR Translation from the Latin · h I, nn. 37–39 · Paris 1514 fol. 5v–6r · Rotta 1913 pp. 30–31 (restoring Klibansky’s collation on bd and dc)
The imaginative power, which does not transcend the genus of sensible things, does not grasp how a line can be a triangle, since among magnitudes these differ without proportion. For the intellect, however, this will prove easy. For it has already been established that only one Maximum and Infinite is possible. Next it is established—because any two sides of whatever triangle taken together cannot be lesser than the third—that in a triangle of which one side is infinite, the remaining sides are not lesser. And since every part of the infinite is infinite, for every triangle having one infinite side the other sides must likewise be infinite. Moreover, since there cannot be several infinites, you understand transcendently that an infinite triangle cannot be compounded out of several lines, even though it is the maximal and truest triangle, uncompounded and most simple; and because it is the truest triangle, which cannot exist without three lines, that single infinite line itself will necessarily be three, and the three will be one most simple line. And so too regarding angles: for there will be only one infinite angle, and that angle is three angles, and the three angles are one. Nor will this maximal triangle be compounded of sides and angles, but infinite line and angle are one and the same, so that line is also angle because triangle is line.
Furthermore, you can aid yourself toward understanding this by ascending from a quantified triangle to a non-quantified one. Clearly every quantified triangle has three angles equal to two right angles; hence the larger one angle is, the smaller are the others. And although any one angle can be enlarged up to two right angles exclusively—and never maximally, according to our first principle—let us nevertheless admit by supposition that it is enlarged maximally right up to two right angles inclusively while the triangle remains. Then it is manifest that the triangle has one angle that is three, and that the three are one.
In like manner you can see that a triangle is a line: because any two sides of a quantified triangle taken together exceed the third side all the more according as the angle they form is much less than two right angles—for instance, because the angle bac is much smaller than two right angles, the lines ba and ac joined together are much longer than bc—therefore, the larger that apex angle becomes, such as bdc, the less do the lines bd and dc surpass the line bc, and the smaller is the enclosed area. Wherefore, if by supposition the angle were equal to two right angles, the entire triangle would resolve into a simple line. Hence by means of this supposition, which among quantified things is impossible, you can aid yourself in ascending to non-quantified realities, among which what is impossible in quantities you behold to be altogether necessary. And herein it is plain that the infinite line is the maximal triangle. Which was to be shown.
Book I, Chapter 15 — That That Triangle Is a Circle and a Sphere¶
Original GEOMETOR Translation from the Latin · h I, nn. 40–41 · Paris 1514 fol. 6r · Rotta 1913 pp. 31–32
Next it will be seen still more clearly that this triangle is a circle. Let the triangle abc be generated by supposition through swinging the line ab around while a remains fixed until b arrives at c. Beyond doubt, if the line ab were infinite and b were swung completely around until it returned to its starting-point, a maximal circle would be generated, of which bc is a portion. Now because bc is a portion of an infinite arc, bc is a straight line. And since every part of the infinite is infinite, bc is therefore not smaller than the whole arc of the infinite circumference. Hence bc will be not merely a portion, but the most complete circumference itself. Consequently the triangle abc must be the maximal circle.
Moreover, because the circumference bc is a straight line, it is not greater than the infinite line ab, since nothing is greater than the infinite; nor are there two lines, because there cannot be two infinites. Therefore the infinite line, which is a triangle, is also a circle. Which was proposed.
Furthermore, that the infinite line is a sphere becomes utterly manifest as follows: the line ab is the circumference of the maximal circle—nay rather, it is the circle itself, as has already been proved—and in the triangle it was carried from b to c, as stated above. Yet bc is the infinite line, as has likewise just been proved. Therefore ab has returned upon itself in c with a complete return upon itself. And when this happens, it follows of necessity that a sphere has arisen from such a revolution of a circle upon itself. Because it was proved above that abc is circle, triangle, and line, we have now proved that it is also a sphere. And these are the things we set out to investigate.