Arithmetics of Boethius | Book I

He shows that unity primarily holds our [principle].

With much presumption, a certain critic slanders Nicomachus a 2nd-century mathematician, whom Boethius follows here. He also slanders Jordanus Jordanus de Nemore, a medieval mathematician, who is easily the prince of arithmetic. Sounding like a trumpet, he boastfully proclaims that he has corrected Jordanus. Specifically, in the second property of Jordanus's first book, he wishes to add this phrase: "namely, of which one is smaller and the other larger." Truly, it would have been more proper for him to remain silent than to scoff or, what is worse, to claim for himself that which escapes his own knowledge. Thus, it would be worthwhile for this quadrivium the four mathematical arts: arithmetic, geometry, music, and astronomy not to be defiled by the blemishes of sophists. These are people who demand subtlety without weight, and for whom it is enough if they merely seem to charge a proven author with error. For this brings the greatest harm in all arts (since authors desire a healthy understanding of their words), but especially in these arts which have the greatest weight for rising toward divine things. Now, things are equidistant at an equal distance from something if the distance of one is the same as the distance of the other, just as 4 and 10 are placed around 7 and are equidistant to it. For 5 precedes it This likely refers to a specific sequence in the original logic and 10 follows it. The distance and interval by which ten exceeds seven is 3, which is the same amount by which four is exceeded by seven. From this, unity is discovered to be the principle of numbers. It is deduced as follows: if unity had something prior to itself, such as some other number or some other distinguishing thing, that thing would be placed around unity. Therefore, according to this property, unity would be the mediety the middle or mean value of this aggregate. But unity itself is the mediety of 2. Through a common conception of the mind, such an aggregate and 2 are equal. When things are equal multiples of one and the same thing, they must be equal. Such an aggregate and the number two are equal multiples of unity, namely doubles, especially since unity is the mediety of them. Therefore, they are equal, and the whole would equal the part, which is impossible. It is therefore established that unity is the principle of numbers.

Principle

From this property, an appropriate place offers itself for rising to the relationships and intervals of things, and this is of no small importance.

The triple world: supercelestial, celestial, and sublunary.

Since the world is threefold—supercelestial, celestial, and sublunary—the celestial world obtains a certain middle nature between the two extremes. For the supercelestial is a world of light. The sublunary, on the contrary, is a world of darkness. But the celestial is tempered by both light and darkness. Its power, since it reaches the lower sublunary world, is rightly placed to be ruled by the supercelestial. Thus, it is middle in power and resistance, action and passion. Here, in the lower world, there is the change of life and death. There, in the higher, is perfect life and stability of functions. But in the celestial world, there is a certain middle nature. Although there is stability of substance, there is nevertheless change of operations, places, and many things of that kind. Here, time reaches the inner parts of substance. There, eternity. In the celestial world, time is the middle measure of eternity and age. If you imagine the triangles of light and darkness, of unity and otherness, of eternity and age, of stability and instability, of rest and motion, of peace and war, of simplicity and composition, of act and matter, of immortality and mortality, of the discrete and the continuous, as interpenetrating each other, and each cutting the other at its middle portion: you will recognize in the middle celestial world a middle interval in every part. But in the superior world, you will see an abundance of perfection, while in the inferior world, you will see a defect.

A complex geometric diagram illustrates the qualities of the four elements. A central square is divided by diagonal lines and intersecting arcs, creating sectors labeled Fire (top), Air (right), Water (bottom), and Earth (left). The qualities of Fire are Sharpness, Mobility, and Thinness. Air has Bluntness, Mobility, and Thinness. Water has Bluntness, Slowness, and Thinness. Earth has Bluntness, Slowness, and Density. The diagram visually demonstrates the shared and opposing physical properties that define the relationships between the elements.

If anyone examines the world more particularly, the same conclusion occurs. In this kind of organization, an element that lies between two nearby elements placed around it is found to have obtained a middle relationship. It communicates with both, and thus the aggregate is a certain mediety in its relationship. Thus, air communicates with fire in heat, but with water in moisture. If you connect and join these together, the joining of the qualities of air immediately becomes known. Through this, it is easy to perceive the symbol shared quality or lack of symbol among elements, as well as their middle and extreme powers. In this way, air and water are recognized as having middle power and resistance. They occupy the middle position between fire, which has the greatest vigor but yields easily, and earth, which on the contrary is most resistant but has almost no activity. Thus water undergoes the moderation of the resistance of earth, and air tempers the excess of the power of fire. In the same way, imperfect mixtures are considered to have a middle nature between elements and minerals. The same is seen in shrubs, which go between herbs

and trees as middle things. No less do the brutes irrational animals stand between humans and what are called zoophytes plant-animals. In the same way, man is approved as the middle of all corporeal and celestial nature. For he has a body from the elements, a celestial spirit, the life of plants, the senses of brutes, and angelic reason and mind. If you raise your mind higher, you will perceive Christ the mediator of the highest unity and plurality. In him alone, the highest unity exists in plurality, and plurality in the highest unity. You will also consider that the divine nature did not stoop, except to unite the human nature to itself. Indeed, it was worthwhile for Christ to have both natures, of the creator and of the creatures alike, so that he would be and be held as the middle and the mediator. Thus, in the assumed nature, the superior and inferior creatures are connected. This is seen only in human nature, in so far as man alone enjoys the name of microcosm little world. But these things will be explained more fully in what follows. If you transfer yourself to the celestial world, Mercury occurs as the middle between the Moon and Venus. Since the Moon, being the first of all, is very similar to the earth itself in opacity and spots. Venus, however, is similar to the air. But Mercury, because it is a changeable and transformable star, is to be assigned to water. They said that the Sun stands above these three, namely as the life-giving fire in heaven. They also made the Sun the fourth among the planets. Likewise, Jupiter is seen as the middle between Mars and Saturn. If you greatly desire to compare the elements to the planets, you will find it in this description:

Furthermore, if you turn to the superior world, you will find there the middle hierarchy between the first and the last, having obtained a certain intervening nature. For the three tasks of the first hierarchy are to perfect, to illuminate, and to purge. On the contrary, the tasks of the lowest hierarchy are to be illuminated, to be perfected, and to be purged. The middle hierarchy performs both. For the vigor of acting which they receive from the prior hierarchy, they express in the inferior one. Not only that, but in the same hierarchy, the middle choir places itself between as a function of the interval. Specifically, to it belongs illumination, which is the middle between perfection and purgation. Almost the same thing is found in the ecclesiastical hierarchy. There are the highest, to whom the duty of promoting others is ascribed. There are the lowest, who on the contrary are to be promoted. But those who are middle between these have both duties. Finally, if you turn to morals, you will find temperance to be the middle of continence and heroic virtue; and of the active and contemplative acts. Finally, if you turn to the beings of reason, you will find that the middle of every coordination holds a certain intervening relationship between the extremes. But these things are manifest even to one moderately practiced. From these things, it is established that just as in numbers the middle is the mediety of the extremes, it is nearly so in actual things.

THE DIVISION OF EVEN NUMBER. CHAPTER V.

An ornamental drop cap P featuring foliate and scroll motifs.
There are three species of even numbers. One is called pariter par evenly even, another pariter impar evenly odd, and the third is impariter par oddly even. Those that are contrary and hold the positions of the summits seem to be the evenly even and the evenly odd. The middle, which participates in both, is the number called oddly even.

COMMENTARY ON THE FIFTH CHAPTER.

An ornamental drop cap R with stylized leaves and floral patterns.
The author recounts the species of even numbers, which are defined by the number three. They are discovered by this reasoning: if something is divided into two equal parts, this can happen either all the way down to unity, so that the whole and the universal parts—which are called denominative parts—receive an equal section, such that no part besides unity refuses such a section; or the whole admits it, but the parts refuse this section.