Arithmetica

triple, or sesquialter having a ratio of 3 to 2. This certainly becomes clear if we take the number three. For the number three, which cannot be divided into two equal parts, and which only unity measures—that is, insofar as it is odd and prime—is considered according to itself. But when expressed by three units and reflecting a triangular form: it is figural. But when related to the number two as its sesquialter: it is relative.

It should be noted that "even" and "odd" are homonyma words with multiple meanings. For every number does not consist of even and odd in the same way, nor does every number undergo this sectioning in the same way, whereby here one number is judged even and another odd. For (as the following proper arguments will show) from the addition of evens comes the odd, and from the following of evens, the odd. Therefore, if these are taken simply and uniquely: it would be worth the effort to consider each number restored by collecting and taking as one these same evens and odds, which the arithmeticians will never allow. Following the teachings of Plato, who after the One made two principles, the finite and the infinite, from which he wished number to be composed: we detect the homonymy of terms by this art. For if (says Plato) you separate unity from number: there is infinity. This is just as if one were to remove the point from a line; thus the odd would be removed. Therefore, by these things, it is indeterminate. Where there is infinity: there is no discernment. Therefore: no number. But unity arriving at such infinity and terminating it: restores number. Therefore, every number is from the One and the infinite.

Furthermore, the Pythagoreans expressed the infinite by the name of "evenness," on account of division; calling unity and that which is one, by contrast, "odd." For in the coordination of good things, they placed the finite, the one, and the odd. They used these as if they were the same. And in the coordination of evil things, by contrast: the infinite, the even, and the many. Moreover, lest anything seem to be lacking for drawing out theories from numbers: we have set down both coordinations below.

The Coordination of Good Things

The Coordination of Evil Things

The words Finite and Infinite on row 7 are circled in the original manuscript.

Therefore, it is said with the best right that every number consists of even and odd: and this is so when the terms are accepted in that significance. Hence, through number, they discussed any proposed subject: insofar as each thing

The Pythagorean? dogma? seems? to have been? disputed? recently?

is composed of formal and material principles, just as it is of even and odd. By the material principle, which is in itself indeterminate and indifferent: it corresponds to evenness. By the formal principle, which terminates the matter: it corresponds to the odd. But in the composite: it is an entire number. This is for the understanding: that "being" and "essence" restored to their parts are for them numbers and likewise "prime" numbers.

Nor are there lacking those who assert that the Pythagoreans, when saying that the soul is a "number moving itself," spoke symbolically. Likewise the Heavens, because they consist of act and potency, are not exempt from the ratio of number. And even less so is the composite of nature, consisting of matter and form. And this is the one cause that moved the Pythagoreans to philosophize about all things through numbers. Not without reason is the odd number attributed to form and act: because forms are in themselves individual. And the more perfect it is: the more so. Inasmuch as all union proceeds from it. But the even is attributed to the material principle: as that which exists as the cause of every division. Thus those things which possess more matter: are more subject to division. As in the elements and those same simple bodies: air is more so than fire, and earth more so than water. And in composites: inanimate things more fully than those which participate in a soul. And among living things: plants more so than animals. And this clearly depends on matter.

From this place, it is easy to rise to the supreme Unity: as that which, because it recedes by an immense degree from matter, is seen to be entirely partless and individual, and also an infinite and immense act. For since division depends on matter: it follows that what recedes from matter also departs from the ratio of division. But to recede from division: is to approach indivisibility and the ratio of indivision, for division and indivisibility are opposites. To recede from one of the opposites: is to approach the other. Therefore, those things which recede more from matter, and thus from division: approach more fully to the ratio of act, and then of indivisibility. And if they recede most: by a like and similar reason, they approach most. And if by an immense degree: likewise by an immense degree. Furthermore, the supreme Unity recedes by an immense degree from the fellowship of matter. And it is also distant from division by an immense degree. Therefore, the same will approach by an immense degree to the ratio of act as well as of indivision. That which approaches these by an immense degree: must necessarily be an immense act, and likewise entirely individual. Thus it is established that the supreme Unity: is an immense act, entirely partless and individual, and these things are not difficult to gather.

Moreover, sometimes the name "binary" the number two is used to express matter: and the name "ternary" the number three to express form. And this is because the binary exists as the source of division and multitude, so that it seems to philosophers not so much a multitude as the beginning, source, and origin of multitude. It is what first flowed from unity: and is the first birth of unity, insofar as it also responds to the first product of divine creation. For no small philosophers placed God and Nothingness as two infinites; but Nothingness is the subject of divine infinity and omnipotence, as that which came forth into being by His word. Primarily, however, matter came forth: which some placed as the necessity of the complexion, since forms would be given according to its disposition. Hence Plato said: "forms are given according to the merits of matter." In the second place, however, form came forth, which must be understood by the priority of nature: not by time. And not without reason did they ascribe the ternary to form, as that which is in the second place from unity.

But they said the first composite was the quinary the number five: which, being restored by the binary and ternary, enfolds within itself the diapente musical interval of a fifth consonance. Which indeed Plato assigned rather to it. The septenary the number seven is the second composite: as being restored from a coarser matter and a doubled binary and ternary; it touches within itself the diatessaron musical interval of a fourth consonance. For the diatessaron consonance is of the quaternary to the ternary; this Plato attributed to the worse and more imperfect things. From these is restored the duodenary the number twelve, the number of the city in Plato, which is restored by the consonance diapason musical interval of an octave: namely by the quinary and septenary, and the diatessaron and diapente harmonies. Thus he wished the city to consist of the better and the worse, by which this world is painted for us, as in a symbol: composed and conflated of perfections and imperfections. To that extent they philosophize through the quinary concerning the composite perfections of this world, and through the septenary concerning the more imperfect things. But let us return to the matter.

There is another acceptance of even and odd: according to which number is severed into even and odd, so that here a number is even, there odd. As the binary is even: the ternary is odd, as it is taken in this place. And in this manner of acceptance, I would not believe that Pythagoras made the principles of numbers. For I do not approve of the quaternary being conflated from the binary and ternary: or any of the evens being restored by joining an even and an odd. For that the quinary is said to be restored by the binary and ternary: that ought rather to be referred to its quantity than to its essence, so that it is no more from those: than from four and unity, or two binaries and unity. Since twice two and one are five, just as three and two are. Accordingly, many pairs would be the principles of numbers, which the Pythagoreans would never allow.