Arithmetics of Boethius | Book I 16

by virtue or by name. Thus Aristotle argues the imperfection of insects: for those that live when divided, have less inequality mixed with their equality. However, the parts correspond to each other and to the whole in name to some extent, so that they can almost be compared to straight lines. But in the division of perfect beings, you recognize inequality mixed with equality: neither the parts among themselves, nor with the whole, are of the same denomination. Thus, once divided, the parts do not retain the same name even for a moment: but then they obtain different forms, which previously existed as the same when united. In this respect, they correspond to the circle, which, because of its perfection, cannot be cut without inequality being mixed with equality. For no part of a circle is itself a circle: but every part is immensely distant from the perfection of a circle. Truly, the application of numbers from what follows is more suitable for these matters.

The same may be seen similarly in beings of reason. For if you take something away from a necessary thing: immediately there remains a parity evenness or equality joined to inequality. But in contingent things, this is not so conspicuous. Thus from this statement "man is white," if "white" is taken away, nothing less contingent remains. But if you shorten this, "man is an animal," and "animal" is removed: what remains is no longer necessary, but altered and changed into a contingent thing. So if you take away accidental goodness from essential good: the essential goodness perishes. But if you again take away other things: the goodness which is accidental nonetheless persists. But perhaps these things are discussed more than is fitting.

The Fourth Definition is immediately accessible. For since all numbers exceed each other by unity alone, and follow alternately as even and odd: it is established that each is separated from the other by unity alone. It is thus unity alone which discerns distinguishes the even from the odd. For even if the number ten is discerned from seven by three: this nevertheless does not happen in even and odd numbers that follow each other most closely. Thus you would not discern 4 from 5 if you did not embrace unity as the common distinction.

From this place, an ascent is open to the divine and super-immense discernment the ability to distinguish or judge between things. For it belongs to the divine unity alone to discern among creatures: and especially in those which follow each other closely in the inequality of nature. Of this kind are rational creatures and angels. Thus surely that fire consuming all things will discern each thing and will separate the goats from the lambs. For if all things were of a fiery nature and ignited: because of the proximity of their nature, you could not discern between them. But if an animated fire were placed there, having the power of discernment: as something most inward, it would distinctly discern all those things, perceiving the perfection of each. So indeed that supreme Unity, being far more inward to creatures than any truth is to its image, or a point to a line, or unity to a number, or fire to ignited things: is recognized to discern all creatures.

In this way, if there were many torches in a hall and many lights from them, which would persist even if the torches were removed: the hall would by no means discern the plurality of the lights by its boundaries. Yet the air, being inward to them, if it had the power of discernment: would discern them most exactly. Thus our intellect almost discerns its own notions and offspring: even if they are mixed together in the same subject and in that individual. Discernment comes about by reason of inwardness. What is more inward to number than unity? And what is more inward to creatures than God: who is the truth of all creatures, and so far that in him all creatures are as they are in truth? Therefore, the logic of discerning belongs to God by his very nature. And so that universal judgment, which they call the "discussion," by primary right belongs to that divine supreme Unity.

And just as a number has no discernment except that received from unity and particularity (for you would not know by the number five that 9 is distant from 4, if you did not know that five consists of five unities, by which 4 is discerned from 9), so surely the creature acknowledges its own discernment and its particularity as received from God. You see therefore in the vestige and in the image: that the logic of judging belongs to God alone, and that by his own nature. The power of judging which is granted to creatures is particular and received. Thus the reason for their discernment must be rendered to these. Nor should we omit that numerical discernment, being made of numbers, cannot reach many things. For by number you cannot discern the diameter from the side, the hemironfus a specific musical interval or ratio from the hemitonio a half-tone or semitone, or the means of any simple harmony. Hence those who seek numerical discernment in all things wander greatly, thinking that those things which do not differ by number coincide. In this part, many errors have happened in the squaring of the circle. But those who observe that discernment happens by unity and the minimum recognize those errors very easily. They are moved by this: that they commit all things to divine judgment, which discerns all things without error. Nor should one stand by the judgments of men, except insofar as they are believed to be subject to the divine. Thus we ourselves are forbidden to judge: but revelation is to be expected, and the discernment of that super-immense Unity in numbers and in things themselves. And not only does unity discern the even from the odd, but also evens and odds from each other: so that it is considered the discernment of all numbers. In no other way is that divine and super-immense Unity the ineffable discernment of all things, both those distinct by nature and those which are of the same nature. You see therefore that all things

correspond beautifully to numbers: so that we may say not without reason, that God has made all things in measure, number, and weight. You see moreover: the contingent is distinguished from the necessary by the established unity of reason, and no less the essential and formal good from the accidental, whose goodness is external: by the unity of goods. And these things seem to suffice for the understanding of the definitions brought forward.

ON THE PRIMACY OF UNITY. CHAPTER IIII.

A decorative drop cap 'O' features interlaced geometric knot patterns in a square frame.

Every number also is the medietas mean or middle portion of those placed around it and joined to it by a natural arrangement. And those which are above those two, which are joined in the middle, if they are added together: the aforementioned number is also their Medietas Mean, the middle portion. And again, of those who are in the second place beyond those joined, when they also are added: the first number is in the place of the mean for these. And this will be until the occurring unity makes a limit. As if one sets down the number five: on either side around it are 4 above and six below. If these are joined, they make 10: of which the number 5 is the mean. But those which are around them, that is around 6 and 4, namely 3 and 7: if these are joined, the number five is their mean. Again, of those which are placed on either side, if they are joined: these also are doubles of the number five. For above 3 is 2: above 7 is 8. These therefore if joined make 10: of which the number five is again the mean. This same thing happens in all numbers: until the limit of unity can be reached. For unity alone does not have two terms around itself: and therefore it is the mean only of that which is near it. For next to one, only the number two is naturally established: of which unity is the middle part. Wherefore it is established: that unity is the first of all numbers which are in the natural arrangement, and also it is rightly recognized as the parent of plurality, however extensive.

COMMENTARY ON THE FOURTH CHAPTER.

A decorative drop cap 'E' with foliage and a mask or face motif in a square frame.

Never did the author pass this over in silence: because the energy of the art seems to defer no small amount to it. Every art rises up whole and legitimate from these three: substance, parts, and properties. For then each single thing is explored by us: when we grasp by knowledge the substance and the essence of the thing, its parts likewise, and its properties. Parts are known by division. Substance and the being of the thing are known by definition. Truly, by rules and conclusions, properties are recognized. By which reason surely the author, as he has so far both divided and defined number: now passes to the properties of numbers. And in this chapter he assigns a remarkable property of number: through which unity is shown to be the first of all numbers. In this part also the present chapter is inscribed concerning the primacy of unity. Such then is the property of numbers. Every number is found to be the mean of the two placed around it, but in a natural series: and joined together. As given the natural series of numbers 1 2 3 4 5 6 7 8 9, in which 5 is the middle: if 6 and 4 placed around it are joined, 10 will arise, of which 5 is the mean. Likewise: if 7 and 3. The same: if 8 and 2. And also if 9 and 1. Thus: if the same numbers are arranged in the form of a circle, it performs the duty of the center. Nor is it found to be the mean only of the two placed most closely around it: but also of any whatever, provided they are equidistant from it. Numbers are said to be placed around another: of which one goes before in the natural series of numbers, and the other

original: "Omni mō dicitur quod altero possit" It is said in every way that it can be done by the other.

follows. This last point we have added: lest anyone be of such a thick skull original: "corneae fibrae," literally "of horny fiber," meaning dull-witted as to say that numbers placed in any way, and serving whatever order one wishes, are therefore said to be placed around another in the same way. As if, having first written the number four, and then joining the number five as a continuation: one then resumes the number four to be assigned to the third place. On which account: he might assert those fours are placed around 5. And that finally...