Arithmetica of Boethius

The first acceptance was sufficiently declared in the second chapter. Now let us pursue the second. The author assigns four divisions of even and odd, taken from various books of the ancients. For the first of these, such an example is taken. The number 4 is even: because it is divided into 2 and 2, and these parts, which are restored from equal units, namely two: are equal. Indeed, of equality and inequality, there is a single measure: the most simple: which is unity. Nor indeed by any other means, though not without a vestige of divinity (as was shown before), will you bring forth equal and unequal numbers. Add: those two restore the whole quaternarius the number four itself. Thus 8 is even: because it is separated into 4 and 4 equal parts. These joined together: restore the octonarius the number eight. But 5 is odd. For it cannot in this way be cut into two equal parts that restore the whole sum, neither by exceeding nor by falling short. For if you divide it into 3 and 3: you exceed the sum, since 6 is produced. If into 2 and 2: you diminish the sum, since only 4 is held. But if into 2 and 3: the sum is indeed held, but the parts into which the partition is made are not equal. And this: I deem to be quite clear.

The second definition is that of Pythagoras, who first introduced the study of philosophy into Italy, defining an even number through the maximum and minimum, so that a number is called even: which under the same division is divided into the maximum and the minimum. Here, "maximum" refers to the greatest parts: which are called "spaces." For in this place, space, interval, and part: are the same. "Minimum" truly: refers to the division. A division is not called large or small by reason of the parts or intervals into which a number is cut: for it matters nothing whether they be small or large: but the number according to which the division is made must be considered. Since from such a number: the division is named small or large. For example, a division according to 10, namely into 10 parts: is greater than a division made according to 6, namely into six parts. and that according to 6: is greater than that according to 3. But the minimum of all is that which is made according to 2: since the binarius the number two is the smallest of all numbers. Therefore, the division of a number into two parts: is the minimum in quantity. And that indeed is discrete: that is, expressed by the number of parts into which the division is made. But if it is made into parts than which none are greater: it will be said to be made in the maximum space, that is, in the greatest intervals. And since there are two types of parts, the "constituting" and the "numbering" which others call "aliquot": here we treat only of the numbering part, which when taken several times restores the whole. This they sufficiently intimate: when they say no part is greater than the discrete half. For example, the greatest part of 4 is 2. This is by no means true of the "constituting" part: since 3, a part of the quaternarius, is a constituting part, and it is indeed greater than two. Furthermore, that the half is the greatest numbering part of any number: is deduced from this. The more a part is named from a larger number: the smaller it is. And the more from a smaller number: the larger. For one-tenth: is smaller than one-sixth, because it is named from ten, which is larger than 6. What if it is named from the smallest number: will it not be judged the greatest? But the discrete half: is named second, from two, the smallest of all numbers: therefore it is the greatest. Since therefore the discrete half is assigned to an even number: its division into two halves is said to be made in the maximum space, that is, into the greatest parts of that number. And since it is only into two: it is said to be the minimum in number of division. Specifically, that which is named from the smallest number, namely the binarius.

Wherefore an even number: is separated in the maximum space and the minimum quantity. And that indeed: under the same division. For example, 4: into 2 and 2. 6: into 3 and 3. This partition: is only into two, and therefore the minimum quantity. And those parts: are the greatest of the numbering parts. For only 2 and 1 number the quaternarius. The two: taken twice, and unity: taken four times. But for the senarius the number six: 3, 2, and 1. The 3: if taken twice. The 2: if taken thrice. And the 1: if taken six times. But 2: is greater than 1, and 3: is greater than 2 and 1. Wherefore the aforementioned divisions: were into the maximum spaces.

Otherwise, the odd number, since it lacks a discrete half (for it is divided into unequal sums according to its first definition), cannot be divided in this way. For if you divide 9 into three threes: you divide it into the greatest numbering parts (for no numbering part of 9 is found greater than 3) but that division: is not the minimum, since it is into three parts, and is named from the ternarius the number three, which is not the smallest number. For by as much as more parts of the whole are assigned: by so much smaller they are, and by so much the number of the division is increased. Hence, as much as the space and magnitude decrease: by so much the number of the division is increased further.

Furthermore, what he subjoins, "according to the contrary passions of the two": as he intimates in declaring himself, is to be understood thus. The property of number from the prologue is: to increase toward infinity; the property of magnitude, by contrast: is to decrease. And these properties indeed: are opposite, but in this division of the even: the opposite happens. For increase is attributed to the space. And decrease to the number of division. For the spaces: are held to be the maximum, and thus their increase is at its peak. But the number of division, by contrast, is the mi-

Book I

-nimum: which does not happen without the highest decrease. And it should be noted that Boethius here uses the name of the genus: for the species. For example, he uses the name "quantity": for discrete quantity.

From this definition, it is easy to philosophize about the lowest of beings through the even number: because things that are lowest are subject to alterities and to maximums and minimums. And that indeed: by reason of matter, to which they are especially subject. And thus in these: is the highest corruption of things. Hence we see them take on contrary affections at different times and even at the same times. They seem to be as far as possible alienated from the supreme unity. Thus we see the virtuous power of the sun, which is nearly simple and uniform: altered in sensible things, and bringing in contrary affections. And although a single point is uniform and most simple: yet in bodies it is in the greatest alterity. So much so that the same point: is called the beginning, middle, and end of the curved and the straight. Of the highest things, by contrast: we philosophize through the odd number, for these things do not happen to them: but they persevere in their stability, and they do not even receive contrary affections at the same time. For we see the motion of the sun persevere in its own course: its power is not diminished, it does not deviate from the duty granted to it. In that place, the same beauty: is neither worn away by age, nor overwhelmed by contraries. From which one may conjecture: that in the highest heaven it is far more perfect, as it is from there that all stability and regularity which is in the heavens proceeds. Hence one is given leave to rise to the inhabitants of that region: and to assert that which even the pagans confessed.

Psellus on the Heaven original: "Psello de coelo" chapter 9 toward the end. having life and through self-sufficiency for all eternity original: "ζωὴν ἔχοντα καὶ δι’ αὐτάρκειαν ἅπαντος αἰῶνος"

"Things there are not fit to be in a place, nor does time make them grow old. Nor is there any single change of any of those things arranged above the outermost rotation: but unchanging and impassive, they possess the best and most sufficient life throughout all eternity." original Greek: τόπῳ τἄκει πέφυκεν, οὔτε χρόνος αὐτὰ ποιεῖ γηράσκειν. οὐδ' ἔστιν οὐδενὸς οὐδὲ μία μεταβολὴ τῶν ὑπὲρ τὴν ἐξωτάτω τεταγμένων φοράν ἀλλ' ἀναλλοίωτα καὶ ἀπαθῆ τὴν ἀρίστην ἔχοντα ζωήν καὶ τὴν αὐταρκεστάτην διατελεῖ τὸν ἅπαντα αἰῶνα.

You see, therefore, that we can philosophize about stability through the odd number: because it does not admit contrary and mutually fighting affections. But about the instability and continuous flux of things: by contrast, through the even. For the supreme unity: is eternity, and like a point everywhere on the circumference. The odd number is the same: the first: like the age aevum the duration of eternal things, and the first countable line. That which is composite: is like time and surface. Furthermore, the even number: is like the age that penetrates interior things, and like a body. But if you wish to transfer these to beings of reason: necessary things receive the name of "oddness," such is their immutability; contingent things: receive the name of "evenness." Those things which obtain essential goodness: rejoice in oddness. 10 Those which have indifference, yet are not entirely destitute of the unity of goodness: rejoice in evenness. But concerning these things, on the occasion of the second definition, enough has been said.

Now let us declare the third: which belongs to several of the ancients, defining an even number through the admixture of the even and the odd. Namely, that it is called even: which divided in any way into two (understand this as two parts which precisely restore the whole sum), whether they be equal or unequal: never has oddness mixed with evenness. Such that one of those parts is even, and the other odd. Instead, in any division: either both are even, or both are odd. For example, the division of 8 into 4 and 4; into 6 and 2; into 5 and 3; into 7 and 1: in none of such sections is even mixed with odd. Rather, in the first two divisions: both parts are even. In the rest: both are odd. But as for his assertion that every even number is divided into two equals and two unequals: that must be taken with a certain determination, namely by excluding the binarius. The Pythagoreans do not call it "even" so much as: the beginning of evenness and of multitude, and it alone admits a section of equals.

An odd number, firstly, cannot be divided into two equal parts that restore the whole itself, as is clear from the first definition. Moreover, when it is divided into those which are unequal: oddness is mixed with evenness, so that one part of this section is even, the other odd. And that indeed is evident in the division into 5 and 4, 6 and 3, 7 and 2, 8 and 1. Thus in the parts of such divisions: oddness is mixed with evenness, so that always one part is even and the other odd. As in 5, the odd: 4 is even. 6 even: 3 odd: and the same in the rest.

From this it is easy to rise: to a double necessity, the "supposed" and the "absolute." Nor is this far from the school of the Peripatetics Followers of Aristotle. Necessity from hypothesis: is referred to evenness and material number. Absolute necessity: is referred to the odd and formal number. Thus form: does not subsist without matter. Matter, however: can happen to subsist without form (though this be at the time of transmutation). Just as evenness: can exist without oddness, but oddness never subsists without evenness. From this again, the even number is known to be suited to the sensible world. For the more abject things are: the more their parts obtain the same nature. Thus earth is seen to be divided into parts of the same name: so that there is no mixed oddness, either between the parts themselves, or with the whole.