Arithmetics of Boethius

A decorative initial A depicts a seated figure in academic robes. Otherness, inequality, and composition come from discord and strife, provided you interpret this as a withdrawal from the highest unity, as those who labor to vindicate Empedocles from error suggest Empedocles was a Greek philosopher who taught that the universe was governed by the opposing forces of Love/Unity and Strife/Discord. For the more fully it recedes from that unity, the greater otherness, composition, and multiplicity arise in them. It is no different than how numbers emerge more composed as they recede further from unity. Whence those things which are eternally in that immense unity, which folds and binds all things in truth, in the highest concord full of the discrete light of divinity, with the divine intellect segregating all things, finally found the cause of their plurality, change, and otherness through strife. By these things it is more established that unity is the limit of both all division and composition. However, some things are terminated by unity in themselves, in which genus is the odd number, which most closely reflects the principle of its origin, namely the triad of equality. For in this, unity is the beginning, unity the end, and unity the middle and bond. In the even number, on the contrary, there is one middle and another. Therefore, in an even number, a greater otherness becomes known. Other things, however, are defined through something else, and of these, some through many things, others on the contrary through fewer. This is surely the same as in magnitudes. For a line is defined immediately and most closely by a point. A surface is defined only by a middle line. But a body uses surface and line, using both as its middle. Wherefore, the body is the most composite of all things and also the most abject. To which indeed, among even numbers, the evenly even corresponds. To the line corresponds the evenly odd. To the surface, however, corresponds that which is established as the middle between both, the oddly even. By this it again becomes clear why one must discuss the lowest things in each genus through the evenly even numbers in arithmetic.

1 The first property of the evenly even number is this: every part of an evenly even number is even in name and quantity. It is even in name because the numerical designation of what part it is of the whole sum is named from an even number. It is even in quantity because that part itself is an even number and divisible into two equals. As, so we do not depart from the example, in the evenly even 64, the parts 32, 16, 8, 4, 2, and 1 are even in both name and quantity, and not only even, but evenly even. For 32 is even in name, insofar as it is the second part of the sum 64, and it is named from the number two, which is an even number. Then, since the same is not only an even number but also evenly even, especially since its parts receive division all the way to unity and that into two equals, it is evenly even in quantity. Wherefore it is both in name and quantity simultaneously evenly even. Thus 16 is even in name because it is the fourth part of the sum, taking its name from 4. It is also even in quantity because it is an even number. So too 8: which is both the eighth part and an even number. This is weighed no differently in the other parts. Take note, however, that unity is exempt from this property, for although it is a part, it cannot be even in quantity, because it is by no means divisible. But it is at least even in name in the whole genus of evenly even numbers. This is surely like a footprint of those things in the soul and those most imperfect entities of this world which are seen to be called homeomeria parts that are the same as the whole. Namely, those which have parts that are named the same as the whole and each other. For indeed, every part of earth is earth, of water is water, of air is air. And the more imperfect they are, the more fully they surely acknowledge this section. Furthermore, the Pythagoreans, because they ascribed the evenly even number to justice, took the occasion from this property and definition. For in the greatest as well as the smallest distribution of justice, equity must be preserved, even to the limit of the whole distribution and exchange, and even to unity. A similar and equal ratio of equity must be observed, especially so that no fuel for complaint arises at all. Add that justice is not only a medium of reason, but also of the thing itself. This does not apply to other virtues, which only walk in the medium of reason. And by this one reason alone, justice is recognized as being a medium in both name and quantity. In name, indeed, because concluding the circumstances of prudence within its own circuit along with the other virtues, it is in mediocrity. In quantity, however, because it is itself the medium of the thing and the quantity. Its limit and ratio is to be distant equally from its own ends. But to this perhaps some will object: namely, that we have made the same number both distinguished and abject. As that which we have attributed to justice, and to the lowest entities of things. But while I pass over in silence for now that no number is abject, they ought to have known that, even with the senses blinking at it, according to the diverse nature of things, the same causes work diverse and even contrary effects. In which genus, fire is recognized to dry and to moisten. What, therefore, prohibits that, according to a variegated nature, that which is an argument for imperfection in some, is on the contrary perfection in others? Who does not know that according to physicians, the same signs sometimes portend contrary outcomes in the healthy and the sick? Thus nothing forbids if the equality of division down to unity, and the same into similar parts of the same ratio, is an imperfection in natural things, but in the entities of our mind, on the contrary, it is no small perfection. For by this each of us is proven most skilled: insofar as he has not receded even a fingernail's breadth from the type and

Book I

Page 20

A decorative initial E features a small bird and curling vines. exemplar. And thus dissimilarity and departure are marks of imperfection. For he who deforms letters, if he cannot do the same a second or third time, is soon rejected by us as having an imperfect art of writing. To this he aspires: that the habit of the soul is not acquired except by similar functions. Furthermore, the same does not happen in nature. Namely, in its compositions, the discretion of parts, and their diverse ratio and denomination, is an argument for perfection. Therefore anhomcomeria things made of dissimilar parts are judged more perfect than homeomeria things made of similar parts. But these things are to be discussed more fully when we come to figured numbers, where with the caution of a particular sign of numbers, we must leave some things to ourselves.

16 By the second property, the generation of evenly even numbers becomes known. For, as it suggests, they proceed by taking continuous doubles from unity. Thus each individual continuous double from unity, and those alone, are evenly even. Furthermore, they are called continuous doubles taken from unity when the series is arranged by this law, whose beginning is unity, whereby the following term, compared to its closest neighbor, sorts into a double proportion. In which genus is this series: 1, 2, 4, 8, 16, 32, 64, 128. For 2 is double to unity. 4 to 2. 8 to 4. And so on, even as the multitude increases as much as you wish. In this series, every number that occurs is evenly even, and those alone are to be taken. And this is the legitimate, and therefore easy, invention of evenly even numbers. It should not be passed over in silence that the same are procreated if the binary the number two is led into unity, then into the product. For if you multiply unity by the binary, with unity taken once and again, two are produced. Which, multiplied again by the binary, leave a four. Which, increased by the lead of the binary, establishes 8. And by this ratio, any evenly even numbers are obtained without great expense. From which it is easy to perceive that the binary is a numerical part of evenly even numbers. Therefore, the binary is called the primordial part of the multiplied evenly even numbers, even if it is their latest division. Wherefore evenly even numbers are to be considered material. And for that reason, they are not without cause attributed to imperfect things and those covered by the mass of matter. Namely, those which are restored by the binary, the source of division. I omit that the same bear no dimension except of the even.

17 The third property suggests that an evenly even number arises by the mutual multiplication of its numerical parts. And thus they respond to each other and undergo mutual denominations. The parts responding to each other are the denominating part and the denominated part. The part is called denominating: the number indicating how many times that which takes its name from it is found in the sum. As however many times four is in eight: two is surely the denominating part. For four is indeed the second part of eight, and it is called that from the binary. Furthermore, the author expresses that this happens in two ways. For in a series both even and odd, first indeed in an even series. As in the subject: 1, 2, 4, 8, 16, 32, which is even for the reason that the numbers are placed evenly and in an even number. Therefore, there is not a single middle, but many. For 4 and 8 are the two middles of that series: which respond to each other and denominate each other. For 8 is called the fourth part of the whole sum, namely 32, from the number four. Likewise 4 is the eighth part of the same sum from the number eight. Those who are around them suffer the same, such as the binary and 16. For they denominate each other, and the binary is of the sum, for 32, the sixteenth part. And 16 is the second. But if the series is odd, there is a single middle. And that indeed denominates itself. As in this series, 1, 2, 4, 8, 16: four is indeed the middle. And of the sum 16, it is called the fourth part, and that from itself. But those who are around it respond to each other in the prior manner. It is not difficult to perceive this from the text. It is established, therefore, that each part of an evenly even number sorts its name from a part of the same number and is told what part it is. From this and the above, it is ready to be elicited that the individual parts of each evenly even number are denominated by numbers, and those same ones are evenly even. For from the first property, all are evenly even; from this one, however, they respond to each other and undergo mutual denominations. Therefore, such a nomenclature emanates from evenly even numbers. And so it happens that in both name and quantity, they should be called not only even, but evenly even. Which things respond and aim exactly at the imperfect things themselves: but that was noted above.

18 The fourth property expresses that evenly even numbers, when heaped and aggregated by the series of the center and the whole complex, restore one less than the following number. Which becomes clear through this example. Let there be a series of evenly even numbers, 1, 2, 4, 8, 16, 32. Aggregate the two first marks, namely unity and the binary: they are 3, which are surpassed by 4 by one. Now add 4 to those already heaped: soon 7 are produced, but which are distant from the sum of eight by one. Then if you proceed to aggregate 8 to the whole prior sum: 15 occur, a smaller sum indeed, and that by a unity from the following number, namely 16. And so on. In which matter a great constancy of divinity shines back, namely that each individual thing aims toward its own principle. For unity is smaller than itself by the first evenly even binary. Therefore, the binary by the growth of unity passes beyond the first unity. Which type of increase indeed every evenly even number observes toward the prior sums.