Arithmetica

Finally, either the whole and some parts admit it, but not all parts. This should be understood regarding the denominative parts, which, when taken as a certain number, precisely restore the whole. The first genus is called pariter par evenly even. The second is pariter impar evenly odd. The third is impariter par oddly even. The first and second are extremes. Therefore, they are most contrary in that division. For all the parts of an evenly even number receive a division into equal parts. But none of the parts of an evenly odd number are granted a section of equality. Furthermore, the third genus, the oddly even, holds a middle nature, communicating with both and therefore having a distinction from both. Because some of its parts are divided into equal parts, it approaches the nature of the evenly even, but in this, it differs from the evenly odd. Because some of its parts are deprived of such a section, and the section of equals does not reach all the way to unity, it aligns with the evenly odd, but is separated from the evenly even. This is what the author implies when he says: "The middle, however, which participates in both, is the number called oddly even." If you strive to transfer these things to reality, you might do so in this way. Since every thing composed outside of God is made of unity and otherness, departing from the simplicity of God, the parts of the universe offer themselves to be viewed in the manner of the Pythagoreans through the even number and its species. Regarding the highest things of the universe, we look through the evenly odd. Regarding the middle things, through the oddly even. Regarding the lowest things, through the evenly even. Every composite is made from one and another; therefore, it is divisible into those two as if by equal parts. Therefore, if you compare any composite to the highest unity, it obtains a certain impotence and imperfection of evenness and division. Furthermore, the fact that intelligible substances do not admit numerical distinction (since each one totally fills the perfection of its own essence) belongs to oddness. For this reason, it would not be discordant to philosophize about them through the evenly odd. Nevertheless, when created things are viewed in themselves and referred to themselves rather than to God, because those supreme beings are units of the highest simplicity in respect to others, it is more correct to discuss them through the odd number and its species, which must be spoken of more fully in its proper place. Those things which are celestial beings are subject to various affections. They do not resist composition from one and another, and I will not omit the plurality of integral parts they admit, insofar as planets and stars are called parts of the sphere, even if of greater density. Truly, they are destitute of a plurality of discrete things. Indeed, this approaches the supercelestial nature, in which they differ from each other only by a distinction of nature. To that extent, the sun, moon, and the rest of the planets exhaust the whole perfection of their nature and species in a single subject. Thus, since there is a multiple division there, but not every kind, it happens that one rightly philosophizes about them through the oddly even. But sensible things are first cut into essential parts, then into integral parts, and third into an infinite plurality of atoms and individuals of the same essence. Thus there is almost no substance in these things of which you do not find many individuals of the same species. From this it follows that division pertains to every part, and for this reason, the philosophy of the evenly even numbers is appropriately applied to them. Furthermore, the plurality of division is an argument for imperfection. Indeed, the lowest and highest of every genus are found in this part, and not without effort. Just as in the genus of elements, fire is the highest and earth the lowest. The same is true in those things that are inanimate. For the less each thing yields to plurality, the more perfect they are considered. This is revealed in metallic things by a vein of gold. Thus, that which is multiple is neglected, but everything rare is precious. In the number of those things which are fostered by vegetative power alone, trees stand above, while the multitude of herbs is placed below. Nor should those things reduced to a few be marked with an inferior stone original: "inferiore fignanda lapillo"; an idiom meaning to be judged unfavorably. Finally, among animals, man excels all, who exhausts human perfection in one species. Only the condition of the body brings about a section of integral parts for him. Brutes, which are ignoble, undergo plurality, and not even of parts. Those of an excellent nature, on the contrary, tend toward unity. By this, all things strive to be assimilated as much as possible to their principle. By this reasoning, it is not difficult to know that rational souls are superior to all sensible things, since they bring with them no parts of their integrity. This truly is an argument for imperfection in other things. But from where does the plurality of divisions hang, if not from evenness and matter, the source of all division? This is proven by the philosophers with one voice. Therefore, because our souls proceed toward a greater unity than any other beings of the lower world, it must be judged as an effect of the distance from matter. If any things were to recede still further from matter, as if from the possibility of matter, and were entirely removed from the bond of aptitude, would not their multiplicity be more fully compressed, and would they not tend more toward unity? Hence it is not difficult to perceive that angels, because they are deprived of and immune to the potentiality of matter, aptitude, or any other link and solder

do not endure numerical distinction in the unity of nature. For no reason can even be assigned why they should further flee from division, unless many numerically discrete individuals are established in each nature. Therefore, each angel uniquely exhausts its own essence. In this way, they enjoy different offices in a most excellent manner in different substances, and this without any mark of envy. If you proceed further, that highest unity presents itself, being removed from matter by an immense distance. Therefore, from the manner of rising, it is entirely devoid of all division. Thus there are not multiple gods. Nor would even equal gods measure themselves against each other. In that way, neither would be immense. And even less so if they were unequal. For that which is smaller is no longer immense, and therefore not God. But each thing is discussed more fully in its place. By these things, it is again established that philosophy through the even number is not altogether unsuitable for the parts of the universe (especially those of this sensible world), whether highest, middle, or lowest. Truly, regarding the lowest, through the evenly even. Regarding the highest, through the evenly odd. And regarding the middle, through the oddly even. The lowest things of this world are those to which only "being" belongs, such as the inanimate. The highest are those to which "sensing" belongs, such as animals. The middle are those to which "living and being" belong, such as plants. And still in each genus, there are highest, lowest, and middle. Among the inanimate, the elements subside. The highest are those that have a more ordered nature. The middle are those that are less ordered and are imperfect mixtures. Among plants, herbs are below. Trees stand above. Shrubs hold the middle position. The lowest of animals are the zoophytes. The middle are the brutes. The highest are humans. In addition, in the genus of elements, earth subsides. Fire holds the highest place. The middle are air and water.

ON THE EVENLY EVEN NUMBER AND ITS PROPERTIES. CHAPTER VI.

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14 An even number is that which can be divided into two equals, and its part into another two equals, and the part of the part into another two equals, until the division of the parts reaches the naturally indivisible unity. As the number 64 has a half of 32. This has a half of 16. This has 8. A third division partitions this into equals, which is the double of two. But two is divided by the half of unity. This unity, being naturally singular, does not receive a section. 1

15 It seems to happen to this number that whatever part of it there may be, it is found to be evenly even both in name and in quantity. But for this reason, it seems to me that this number is called "evenly even": because all its parts are found to be equally even both in name and in quantity. How it has parts even both in name and in quantity, we shall say later. 2

16 The generation of these is as follows. If you note any numbers in double proportion starting from one, evenly even numbers are always procreated. It is impossible for them to be born otherwise than by this generation. An example of this matter seems to be this following description in order. Let all the doubles from one be 1, 2, 4, 8, 16, 32, 64, 128, 256, 512. If an infinite progression were made from here, you would find all such numbers. They are made from one in double proportion, and all are evenly even. 3

17 That matter is worthy of no small consideration: that every part of it is denominated by some one part which is within the number itself. It includes such a sum of quantity as the other part of the evenly even number is, of that quantity which contains it. Thus it happens that the parts themselves correspond to each other: just as one part is a certain fraction, the other has that same quantity.