The Arithmetic of Boethius

measure, which is posited as twofold: this one measuring, that one measured, to which the measuring thing itself is applied. This, a wooden yardstick and cloth clearly show, which certainly helps not a little in the understanding of the following. For how would you not divide by a unit if you make number a sensible thing? And how will you recognize triangles, squares, or other figurative numbers described, and their mutual relationship? I omit that it is not primarily, but only symbolically, possible to find proportions of numbers in sensible things, unless you speak foolishness, saying two horses to one make a double interval, and three oxen to two flies make a sesquialter original: "sesquialterum"; a ratio of 3:2, which is certainly ridiculous in arithmetic. Therefore, number, defined by Boethius, is a collection of units, or a heap of quantity derived from units. Furthermore, since every definition ought to express a cause, the first definition surely expresses the formal cause, yet without neglecting what is ascribed to the material cause. For the name "units" expresses the matter itself. Union and nexus, however, represent the form, or at least that which is analogous and corresponding to form. That collection and nexus of units is made by the artifice of our mind; to that extent, number is recognized as the first composite of our mind. For as the Divine mind is to creatures, so truly is the human mind to its numbers. And as creatures proceed from God by Divine artifice, so do numbers proceed by the artifice of the human mind. Just as each creature, because it is one and is held as such, has this from the Divine mind, so also the number of our mind, because it is one and is held as such, has this from our mind. Take away the mind, and there is no number, let alone one.

The second definition of number seems to express the efficient cause beyond those, as it asserts that that heap, and that discrete quantity and multitude, is from units. And it hints that unity is the beginning of numbers. However, the mind itself is the principal cause of numbers; unity, however, is in the second place and is the instrument of our mind in forming number. Nor is anything else hinted at by such a definition, except that number proceeds from unity by a certain flow, just as a line is formed by the flow of a point, and surfaces from a line, and a body from a surface. And nonetheless, it is from units. In which it receives a distinction from a point, because although it is internal to a line, it is not a part; and this is because it has position in the continuous. And no continuous thing is made composed of indivisibles, so that by the addition of individual to individual, nothing is held to be made or increased therein. On the contrary, this happens in discrete quantity. For numbers are increased according to discretion by the additions of units, even though they are individual. Wherefore, the second definition expresses the beginning of numbers, as that from which they proceed and out of which they are constituted. Thus, indeed, insofar as number is from unity, unity bears before it a certain character of an efficient cause; but insofar as it is constituted from it, it seems to be attached to a material cause. Hence it becomes known in some part: as all things are from the Divine mind, so in a way are all things from our mind. For what God is in the creation of things, our mind is in the production of numbers. The Divine mind discerns all things; our mind also discerns all things. But the discretion of God is the production of things in their own subsistence. Our discretion, however, is only of numbers, which are similarities of Divine discretion. But furthermore, through the unity of our mind, it is given to rise to that Divine and incomprehensible unity. God For since unity is the beginning of all numbers, inasmuch as all number flows from it, and it is the end of all things as that in which all number is resolved, nor does it draw its origin from anything, nor is it cut into any number (so that it can be without numbers; but numbers, far from being able to exist without it, are so because it is most intimate to them), it is seen to be a trace of Divine unity. For God is the beginning and end of all things, so that not without reason is He called Alpha and Omega, opening all things and closing all things. Before Whom and after Whom there is nothing. And it is so far from having taken its origin from creatures, that He preceded them by an eternal interval, existing without them. Creatures, on the contrary, are not without Him. For their being, living, sensing, reasoning, understanding, and whatever is finally found in creatures, is of that supreme unity. And that supreme unity gives being to things far more, and to each other thing, than a creature does to its own image. Thus the crater of Mercury referring to the crater of Hermes/Thoth, often associated with divine knowledge warns us from the A hand pointing to the text with a decorative sleeve. Monad monad of our mind to think of the true monad. The monad, he says, is the beginning of all things, the root and the origin. Truly, without a beginning, there is nothing. A beginning, however, is not a principle, but of something else. Therefore, the monad is the principle, and it contains all number, contained by none. And it begets all number, begotten by no number. Whatever is begotten is imperfect, divisible, growing, and decreasing; to that which is perfect, none of these things happens. That which increases is increased by the virtue of the monad. It vanishes, however, by its own weakness when it can no longer grasp the monad. This says the Crater. Nor does it lack the shadowing of Divine light, because unity is the terminus and measure of all numbers, measuring all things, communicating its name to all numbers, and nameable by none.