Arithmetica

A circular diagram illustrates musical and arithmetical proportions with concentric arcs labeled with musical terms: Diapason Octave, Diapente Fifth, Diatessaron Fourth, Diapente diapason Twelfth, and Disdiapason Double Octave. These arcs are linked to the numbers 1, 2, 3, and 4 at the top, showing the mathematical ratios of harmony.

If the human mind, by a threefold distinction, enfolds every distinction and subsequently every harmony, should we not believe that the immense and Divine Mind—the super-immense truth of the human mind—likewise enfolds every super-eminent harmony and distinction by its own threefold distinction? For what the human mind is in the human craft of numbers, the Divine Mind is in the divine craft of numbers and things. This later point will become manifest. Now let us examine the number of our mind: whether it consists of itself and nothing less than the even and the odd. Rejecting the opinions of the ancients and following Aristotle, we say that number consists of itself, and likewise of the even and the odd. In this way, we knit opposites into one composition and melody. We strive to prove this as follows: The units of any number, if viewed separately and without the mind's reason, union, and connection binding and welding them together, by no means form a number. For example: suppose there are two units in the mind. There is not yet the number two binarius unless there is the mind's union, connection, and formal reason, from which the number two has its existence and is considered one. For those units are like the parts of a house. Their connection and bond are like the form and figure of the house. Therefore, just as the parts of a house are distinct and divided in themselves, but are welded together by that figure which is one, making them one thing—from which the house both is and is judged to be one—so the units of the number two, considered in themselves and separately, are distinct, divided, and different. To that extent, not without reason, they bear the name of evenness, since it is proper to the even to be divided, and division is ascribed to it, thus corresponding to matter. But the union and formal reason, by which those units are one and a single number, is one without division, and a single bond of the divided parts, so that it rightly takes the name of oddness, insofar as it corresponds to form. Thus you recognize that although all numbers agree in the essence of their units, they are distinguished only by their union and bond, just as things share matter but not form.

It is established that the number of the mind's measure consists of the even and the odd.

It is established, therefore, that the number of our mind consists of the even and the odd, as if from opposites, but which does not shun every proportion of reason. Furthermore, what is the bond and union of two units in subject and in reality, other than the number two? And what is the number two other than two units? You see, therefore, that the number two consists of itself, and nothing less than the even and the odd. In this way, you can show that any number is restored to these elements. And since the units of the number, as well as their bond, are not at all absent from its substance—though these are not entirely similar, they are also not separate in substance and nature, but rather point toward each other by a certain ratio and proportion, like the even and the odd, or form and matter—it is established that number is composed neither from those things that are the same in every part, nor from those that have no proportion. This is what Boethius implies. However, the claim of some ancients that units do not correspond to each other and have no ratio of proportion is shown here to be untrue. For they understand this either only in different numbers—such as that the units of the number two are not of the same ratio as the units of the number three—or that they do not agree even within the same number. But the second view cannot stand. Otherwise, something one would be made from things that hold no proportion of reason. In that case, the opinion of Democritus would be true, who asserted that all things, even if they are diverse, are one.

No proportion could be assigned because in all cases they have no proportion to each other... Unity and composition are opposites.

Indeed, no reason can be assigned why one thing should not be made from all things, except that not all things have a ratio of proportion and correspondence to each other. Nor is it consistent with truth that these units should not correspond at least in different numbers, since if they were so, the proportions, distinctions, and harmonies of numbers would not stand. Indeed, we say the number three is sesquialter one and a half to the number two for no other reason than that it contains it and its half. But how could the unit of the number three be called the half of two if the units were not of the same nature? As for those things said to consist of themselves, it should not seem a wonder. Since that which first flowed and was first "principled" brought into being from a principle, so to speak, it is worthwhile that it consist of itself. For that which is principled, departing from unity and simplicity, must necessarily be a composite. Indeed, unity and composition are opposites. To move away from one of the opposites is to approach the other. Yet it cannot be composed of other things, for those things of which it consisted would have to be prior to it, perhaps even by nature. Especially since parts are prior to the whole, being simpler than it. In this way, it could not even be the first principled thing. Therefore, the first principled thing, which first

Number is the principle of means in these institutions.

departs from the highest simplicity of truth, is both composite and necessarily consists of itself. Truly, it must seem so: if the number about which the human mind first philosophizes as a principled thing is asserted by the Pythagoreans to consist of itself. This did not escape Plato, who established the Infinite and the Finite as the principles of numbers. He called the multitude deserted by unity "infinite," and that union from which every number has its name as "one" he called "finite." He suggested nothing different from those Pythagoreans who believe number consists of the one and the other, or the even and the odd. By this, it is further explored that number consists of units and their union. Furthermore, it should be noted that this composition of number carries much weight for divine theories original Greek: θεορίας. In this regard, it becomes known that the craft of our mind responds to the divine craft. To these things, it seems necessary to add that Nicomachus, in one or two places in his second book, composes every number of even and odd. There, he takes number collectively, for the entire body and series of numbers. There is no doubt that the series of numbers has alternating even and odd, and thus number and the series of numbers consist of even and odd. But the discussion of that matter should be deferred to that place.

ON THE DEFINITION AND DIVISION OF NUMBER AND

the various definitions of even and odd. CHAPTER III.

A decorative initial letter 'T' depicts a seated scholar in a classical setting, likely representing Boethius or a teacher of the liberal arts.
First, we must define what number is. Number is a collection of units, or a heap of quantity poured out from units. The first division of this is into the odd and the even. The even is that which can be divided into two equal parts without a middle unit intervening. The odd is that which no one can divide into equal parts without the aforementioned one unit intervening in the middle. This type of definition is common and well-known.

But according to the Pythagorean discipline, the definition is as follows: An even number is that which, It can be resolved into the largest parts and the smallest in indivisible parts... called by the name of these quantities as magnitude in measures. under the same division, can be divided into the largest and the smallest parts. It is largest in space but smallest in quantity, according to the contrary properties of those two types. An odd number is that to which this cannot happen, but whose natural section is into two unequal sums. This is the exemplar: if any given even number is divided, no middle part will be found larger than it in terms of the spaces of division; but in terms of quantity, none will be smaller than the partition made into two. For example, if the even number 8 is divided into 4 and 4, there will be no other division that produces larger parts. Furthermore, there will be no other division that divides the whole number into a smaller quantity. Multitude. For in a division into two parts, nothing is smaller. For when someone divides a whole into three parts, the space of the sum is diminished, but the number of the division is increased. What was said about the "contrary properties of two types" is of this sort: It is said by the property of... a certain... We previously taught that quantity grows into infinite pluralities, while spaces (that is, magnitudes) are diminished into the most infinite smallness. And therefore the opposite happens here. For this division of the even is largest in space but smallest in quantity. According to an even older method, there is another definition of an even number: An even number is that which into two equal parts, etc.