Nicomachus, Arithmetic

Again, to start afresh, since of quantity one kind is viewed by itself, having no relation to anything else, as ‘even,’ ‘odd,’ ‘perfect,’ and the like, and the other is relative to something else and is conceived of together with its relationship to another thing, like ‘double,’ ‘greater,’ ‘smaller,’ ‘half,’ ‘one and one-half times,’ ‘one and one-third times,’ and so forth, it is clear that two scientific methods will lay hold of and deal with the whole investigation of quantity; arithmetic, absolute quantity, and music, relative quantity.

And once more, inasmuch as part of ‘size’ is in a state of rest and stability, and another part in motion and revolution, two other sciences in the same way will accurately treat of ‘size,’ geometry the part that abides and is at rest, astronomy that which moves and revolves.

Without the aid of these, then, it is not possible to deal accurately with the forms of being nor to discover the truth in things, knowledge of which is wisdom, and evidently not even to philosophize properly, for “just as painting contributes to the menial arts toward correctness of theory, so in truth lines, numbers, harmonic intervals, and the revolutions of circles bear aid to the learning of the doctrines of wisdom,” says the Pythagorean Androcydes, “is the knowledge of these things.” Likewise, Archytas of Tarentum, at the beginning of his treatise On Harmony, says the same thing, in about these words: “It seems to me that they do well to study mathematics, and it is not at all strange that they have correct knowledge about each thing, what it is. For if they knew rightly the nature of the whole, they were also likely to see well what is the nature of the parts. About geometry, indeed, and arithmetic and astronomy, they have handed down to us a clear understanding, and not least also about music. For these seem to be sister sciences; for they deal with sister subjects, the first two forms of being.”

Plato, too, at the end of the thirteenth book of the Laws, to which some give the title The Philosopher, because he investigates and defines in it what sort of man the real philosopher should be, in the course of his summary of what had previously been fully set forth and established, adds: “Every diagram, system of numbers, every scheme of harmony, and every law of the movement of the stars, ought to appear one to him who studies rightly; and what we say will properly appear if one studies all things looking to one principle, for there will be seen to be one bond for all these things, and if anyone attempts philosophy in any other way he must call on Fortune to assist him. For there is never a path without these; this is the way, these the studies, be they hard or easy; by this course must one go, and not neglect it. The one who has attained all these things in the way I describe, him I for my part call wisest, and this I maintain through thick and thin.” For it is clear that these studies are like ladders and bridges that carry our minds from things apprehended by sense and opinion to those comprehended by the mind and understanding, and from those material, physical things, our foster-brethren known to us from childhood, to the things with which we are unacquainted, foreign to our senses, but in their immateriality and eternity more akin to our souls, and above all to the reason which is in our souls.

And likewise in Plato’s Republic, when the interlocutor of Socrates appears to bring certain plausible reasons to bear upon the mathematical sciences, to show that they are useful to human life—arithmetic for reckoning, distributions, contributions, exchanges, and partnerships, geometry for sieges, the founding of cities and sanctuaries, and the partition of land, music for festivals, entertainment, and the worship of the gods, and the doctrine of the spheres, or astronomy, for farming, navigation, and other undertakings, revealing beforehand the proper procedure and suitable season—Socrates, reproaching him, says: “You amuse me, because you seem to fear that these are useless studies that I recommend; but that is very difficult, nay, impossible. For the eye of the soul, blinded and buried by other pursuits, is rekindled and aroused again by these and these alone, and it is better that this be saved than thousands of bodily eyes, for by it alone is the truth of the universe beheld.”

Which then of these four methods must we first learn? Evidently, the one which naturally exists before them all, is superior and takes the place of origin and root and, as it were, of mother to the others. And this is arithmetic, not solely because we said that it existed before all the others in the mind of the creating God like some universal and exemplary plan, relying upon which as a design and archetypal example the creator of the universe sets in order his material creations and makes them attain to their proper ends; but also because it is naturally prior in birth, inasmuch as it abolishes other sciences with itself, but is not abolished together with them.

For example, ‘animal’ is naturally antecedent to ‘man,’ for abolish ‘animal’ and ‘man’ is abolished; but if ‘man’ be abolished, it no longer follows that ‘animal’ is abolished at the same time. And again, ‘man’ is antecedent to ‘schoolteacher’; for if ‘man’ does not exist, neither does ‘schoolteacher,’ but if ‘schoolteacher’ is nonexistent, it is still possible for ‘man’ to be. Thus since it has the property of abolishing the other ideas with itself, it is likewise the older.

Conversely, that is called younger and posterior which implies the other thing with itself, but is not implied by it, like ‘musician,’ for this always implies ‘man.’ Again, take ‘horse’; ‘animal’ is always implied along with ‘horse,’ but not the reverse; for if ‘animal’ exists, it is not necessary that ‘horse’ should exist, nor if ‘man’ exists, must ‘musician’ also be implied.

So it is with the foregoing sciences; if geometry exists, arithmetic must also needs be implied, for it is with the help of this latter that we can speak of triangle, quadrilateral, octahedron, icosahedron, double, eightfold, or one and one-half times, or anything else of the sort which is used as a term by geometry, and such things cannot be conceived of without the numbers that are implied with each one. For how can ‘triple’ exist, or be spoken of, unless the number 3 exists beforehand, or ‘eightfold’ without 8? But on the contrary 3, 4, and the rest might be without the figures existing to which they give names. Hence arithmetic abolishes geometry along with itself, but is not abolished by it, and while it is implied by geometry, it does not itself imply geometry.

And once more is this true in the case of music; not only because the absolute is prior to the relative, as ‘great’ to ‘greater’ and ‘rich’ to ‘richer’ and ‘man’ to ‘father,’ but also because the musical harmonies, diatessaron, diapente, and diapason, are named for numbers; similarly all of their harmonic ratios are arithmetical ones, for the diatessaron is the ratio of 4:3, the diapente that of 3:2, and the diapason the double ratio; and the most perfect, the di-diapason, is the quadruple ratio.

More evidently still astronomy attains through arithmetic the investigations that pertain to it, not alone because it is later than geometry in origin—for motion naturally comes after rest—nor because the motions of the stars have a perfectly melodious harmony, but also because risings, settings, progressions, retrogressions, increases, and all sorts of phases are governed by numerical cycles and quantities.

So then we have rightly undertaken first the systematic treatment of this, as the science naturally prior, more honorable, and more venerable, and, as it were, mother and nurse of the rest; and here we will take our start for the sake of clearness.

All that has by nature with systematic method been arranged in the universe seems both in part and as a whole to have been determined and ordered in accordance with number, by the forethought and the mind of him that created all things; for the pattern was fixed, like a preliminary sketch, by the domination of number preexistent in the mind of the world-creating God, number conceptual only and immaterial in every way, but at the same time the true and the eternal essence, so that with reference to it, as to an artistic plan, should be created all these things, time, motion, the heavens, the stars, all sorts of revolutions.

It must needs be, then, that scientific number, being set over such things as these, should be harmoniously constituted, in accordance with itself; not by any other but by itself. Everything that is harmoniously constituted is knit together out of opposites and, of course, out of real things; for neither can non-existent things be set in harmony, nor can things that exist, but are like one another, nor yet things that are different, but have no relation one to another. It remains, accordingly, that those things out of which a harmony is made are both real, different, and things with some relation to one another.

Of such things, therefore, scientific number consists; for the most fundamental species in it are two, embracing the essence of quantity, different from one another and not of a wholly different genus, odd and even, and they are reciprocally woven into harmony with each other, inseparably and uniformly, by a wonderful and divine Nature, as straightway we shall see.

Number is limited multitude or a combination of units or a flow of quantity made up of units; and the first division of number is even and odd.

The even is that which can be divided into two equal parts without a unit intervening in the middle; and the odd is that which cannot be divided into two equal parts because of the aforesaid intervention of a unit.

Now this is the definition after the ordinary conception; by the Pythagorean doctrine, however, the even number is that which admits of division into the greatest and the smallest parts at the same operation, greatest in size and smallest in quantity, in accordance with the natural contrariety of these two genera; and the odd is that which does not allow this to be done to it, but is divided into two unequal parts.

In still another way, by the ancient definition, the even is that which can be divided alike into two equal and two unequal parts, except that the dyad, which is its elementary form, admits but one division, that into equal parts; and in any division whatsoever it brings to light only one species of number, however it may be divided, independent of the other. The odd is a number which in any division whatsoever, which necessarily is a division into unequal parts, shows both the two species of number together, never without intermixture one with another, but always in one another’s company.

By the definition in terms of each other, the odd is that which differs by a unit from the even in either direction, that is, toward the greater or the less, and the even is that which differs by a unit in either direction from the odd, that is, is greater by a unit or less by a unit.

Every number is at once half the sum of the two on either side of itself, and similarly half the sum of those next but one in either direction, and of those next beyond them, and so on as far as it is possible to go. Unity alone, because it does not have two numbers on either side of it, is half merely of the adjoining number; hence unity is the natural starting point of all number.

By subdivision of the even, there are the even-times even, the odd-times even, and the even-times odd. The even-times even and the even-times odd are opposite to one another, like extremes, and the odd-times even is common to them both like a mean term.

Now the even-times even is a number which is itself capable of being divided into two equal parts, in accordance with the properties of its genus, and with each of its parts similarly capable of division, and again in the same way each of their parts divisible into two equals until the division of the successive subdivisions reaches the naturally indivisible unit. Take for example 64; one half of this is 32, and of this 16, and of this the half is 8, and of this 4, and of this 2, and then finally unity is half of the latter, and this is naturally indivisible and will not admit of a half.

It is a property of the even-times even that, whatever part of it be taken, it is always even-times even in designation, and at the same time, by the quantity of the units in it, even-times even in value; and that neither of these two things will ever share in the other class. Doubtless it is because of this that it is called even-times even, because it is itself even and always has its parts, and the parts of its parts down to unity, even both in name and in value; in other words, every part that it has is even-times even in name and even-times even in value.

There is a method of producing the even-times even, so that none will escape, but all successively fall under it, if you do as follows: As you proceed from unity, as from a root, by the double ratio to infinity, as many terms as there are will all be even-times even, and it is impossible to find others besides these; for instance, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512 . . .

10 Now each of the numbers set forth was produced by the double ratio, beginning with unity, and is in every respect even-times even, and every part that it may be found to have is always named from some one of the numbers before it in the series, and the sum of units in this part is the same as one of the numbers before it, by a system of mutual correspondence, indeed, and interchange. If there is an even number of terms of the double ratio from unity, not one mean term can be found, but always two, from which the correspondence and interchange of factors and values, values and factors, will proceed in order, going first to the two on either side of the means, then to the next on either side, until it comes to the extreme terms, so that the whole will correspond in value to unity and unity to the whole. For example, if we set down 128 as the largest term, the number of terms will be even, for there are eight in all up to this number; and they will not have one mean term, for this is impossible with an even number, but of necessity two, 8 and 16. These will correspond to each other as factors; for of the whole, 128, 16 is one eighth and conversely 8 is one sixteenth. Thence proceeding in either direction, we find that 32 is one fourth, and 4 one thirty-second, and again 64 is one half, and 2 one sixty-fourth, and finally at the extremes unity is one one-hundred-twenty-eighth, and conversely 128 is the whole, to correspond with unity.

If, however, the series consists of an odd number of terms, seven for example, and we deal with 64, there will be of necessity one mean term in accordance with the nature of the odd; the mean term will correspond to itself because it has no partner; and those on either side of it in turn will correspond to one another until this correspondence ends in the extremes. Unity, for example, will be one sixty-fourth, and 64 the whole, corresponding to unity; 32 is one half, and 2 one thirty-second; 16 is one fourth, and 4 one sixteenth; and 8 the eighth part, with nothing else to correspond to it.

It is the property of all these terms when they are added together successively to be equal to the next in the series, lacking one unit, so that of necessity their summation in any way whatsoever will be an odd number, for that which fails by a unit of being equal to an even number is odd. This observation will be of use to us very shortly in the construction of perfect numbers. But to take an example, the terms from unity preceding 256 in the series, when added together, are within 1 of equaling 256, and all the terms before 128, the term immediately preceding, are similarly equal to 128 save for one unit; and to the next terms the sums of those below them are similarly related. Thus unity itself is within one unit of equaling the next term, which is 2, and these two together fail by 1 of equaling the next, and the three together are within 1 of the next in order, and you will find that this goes on without interruption to infinity.

This too it is very needful to recall: If the number of terms of the even-times even series dealt with is even, the product of the extremes will always be equal to the product of the means; if there is an odd number of terms, the product of the extremes will be equal to the square of the mean. For, in the case of an even number of terms, 1 times 128 is equal to 8 times 16 and further to 2 times 64 and again to 4 times 32, and this is so in every case; and with an odd number of terms, 1 times 64 equals 2 times 32, and this equals 4 times 16, and this again equals 8 times 8, the mean term alone multiplied by itself.

1 The even-times odd number is one which is by its genus itself even, but is specifically opposed to the aforesaid even-times even. It is a number of which, though it admits of the division into two equal halves, after the fashion of the genus common to it and the even-times even, the halves are not immediately divisible into two equals, for example, 6, 10, 14, 18, 22, 26, and the like; for after these have been divided their halves are found to be indivisible.

It is the property of the even-times odd that whatever factor it may be discovered to have is opposite in name to its value, and that the quantity of every part is opposite in value to its name, and that the numerical value of its part never by any means is of the same genus as its name. To take a single example, the number 18, its half, with an even name, is 9, odd in value; its third part, again, with an odd designation, is 6, even in value; conversely, the sixth part is 3 and the ninth part 2; and in other numbers the same peculiarity will be found.

It is possibly for this reason that it received such a name, that is, because, although it is even, its halves are at once odd.

This number is produced from the series beginning with unity, with a difference of 2, namely, the odd numbers, set forth in proper order as far as you like and then multiplied by 2. The numbers produced would be, in order, these: 6, 10, 14, 18, 22, 26, 30, and so on, as far as you care to proceed. The greater terms always differ by 4 from the next smaller ones, the reason for which is that their original basic forms, the odd numbers, exceed one another by 2 and were multiplied by 2 to make this series, and 2 times 2 makes 4.

Accordingly, in the natural series of numbers the even-times odd numbers will be found fifth from one another, exceeding one another by a difference of 4, passing over three terms, and produced by the multiplication of the odd numbers by 2.

They are said to be opposite in properties to the even-times even, because of these the greatest extreme term alone is divisible, while of these former the smallest only proved to be indivisible; and in particular because in the former case the reciprocal arrangement of parts from extremes to mean term or terms makes the product of the former equal to the square or product of the latter; but in this case by the same correspondence and comparison the mean term is one half the sum of the extremes, or if there should be two means, their sum equals that of the two extremes.

Again, while the odd is distinguished over against the even in classification and has nothing in common with it, since the latter is divisible into equal halves and the former is not thus divisible, nevertheless there are found three species of the odd, differing from one another, of which the first is called the prime and incomposite, that which is opposed to it the secondary and composite, and that which is midway between both of these and is viewed as a mean among extremes, namely, the variety which, in itself, is secondary and composite, but relatively is prime and incomposite.

Now the first species, the prime and incomposite, is found whenever an odd number admits of no other factor save the one with the number itself as denominator, which is always unity; for example, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31. None of these numbers will by any chance be found to have a fractional part with a denominator different from the number itself, but only the one with this as denominator, and this part will be unity in each case; for 3 has only a third part, which has the same denominator as the number and is of course unity, 5 a fifth, 7 a seventh, and 11 only an eleventh part, and in all of them these parts are unity.

It has received this name because it can be measured only by the number which is first and common to all, unity, and by no other; moreover, because it is produced by no other number combined with itself save unity alone; for 5 is 5 × 1, and 7 is 7 × 1, and the others in accordance with their own quantity. To be sure, when they are combined with themselves, other numbers might be produced, originating from them as from a fountain and a root, wherefore they are called ‘prime,’ because they exist beforehand as the beginnings of the others. For every origin is elementary and incomposite, into which everything is resolved and out of which everything is made, but the origin itself cannot be resolved into anything or constituted out of anything.

Now while these two species of the odd are opposed to each other a third one is conceived of between them, deriving, as it were, its specific form from them both, namely the number which is in itself secondary and composite, but relatively to another number is prime and incomposite. This exists when a number, in addition to the common measure, unity, is measured by some other number and is therefore able to admit of a fractional part, or parts, with denominator other than the number itself, as well as the one with itself as denominator. When this is compared with another number of similar properties, it is found that it cannot be measured by a measure common to the other, nor does it have a fractional part with the same denominator as those in the other. As an illustration, let 9 be compared with 25. Each in itself is secondary and composite, but relatively to each other they have only unity as a common measure, and no factors in them have the same denominator, for the third part in the former does not exist in the latter nor is the fifth part in the latter found in the former.

The production of these numbers is called by Eratosthenes the ‘sieve,’ because we take the odd numbers mingled together and indiscriminate and out of them by this method of production separate, as by a kind of instrument or sieve, the prime and incomposite by themselves, and the secondary and composite by themselves, and find the mixed class by themselves.

The method of the ‘sieve’ is as follows. I set forth all the odd numbers in order, beginning with 3, in as long a series as possible, and then starting with the first I observe what ones it can measure, and I find that it can measure the terms two places apart, as far as we care to proceed. And I find that it measures not as it chances and at random, but that it will measure the first one, that is, the one two places removed, by the quantity of the one that stands first in the series, that is, by its own quantity, for it measures it 3 times; and the one two places from this by the quantity of the second in order, for this it will measure 5 times; and again the one two places further on by the quantity of the third in order, or 7 times, and the one two places still farther on by the quantity of the fourth in order, or 9 times, and so ad infinitum in the same way.

Then taking a fresh start I come to the second number and observe what it can measure, and find that it measures all the terms four places apart, the first by the quantity of the first in order, or 3 times; the second by that of the second, or 5 times; the third by that of the third, or 7 times; and in this order ad infinitum.

Again, as before, the third term 7, taking over the measuring function, will measure terms six places apart, and the first by the quantity of 3, the first of the series, the second by that of 5, for this is the second number, and the third by that of 7, for this has the third position in the series.

And analogously throughout, this process will go on without interruption, so that the numbers will succeed to the measuring function in accordance with their fixed position in the series; the interval separating terms measured is determined by the orderly progress of the even numbers from 2 to infinity, or by the doubling of the position in the series occupied by the measuring term, and the number of times a term is measured is fixed by the orderly advance of the odd numbers in series from 3.

Now if you mark the numbers with certain signs, you will find that the terms which succeed one another in the measuring function neither measure all the same number—and sometimes not even two will measure the same one—nor do absolutely all of the numbers set forth submit themselves to a measure, but some entirely avoid being measured by any number whatsoever, some are measured by one only, and some by two or even more. Now these that are not measured at all, but avoid it, are primes and incomposites, sifted out as it were by a sieve; those measured by only one measure in accordance with its own quantity will have but one fractional part with denominator different from the number itself, in addition to the part with the same denominator; and those which are measured by one measure only, but in accordance with the quantity of some other number than the measure and not its own, or are measured by two measures at the same time, will have several fractional parts with other denominators besides the one with the same as the number itself; these will be secondary and composite.

The third division, the one common to both the former, which is in itself secondary and composite but primary and incomposite in relation to another, will consist of the numbers produced when some prime and incomposite number measures them in accordance with its own quantity, if one thus produced be compared to another of similar origin. For example, if 9, which was produced by 3 measuring by its own quantity, for it is 3 times 3, be compared with 25, which was produced from 5 measuring by its own quantity, for it is 5 times 5, these numbers have no common measure except unity.

We shall now investigate how we may have a method of discerning whether numbers are prime and incomposite, or secondary and composite, relatively to each other, since of the former unity is the common measure, but of the latter some other number also besides unity; and what this number is.

Suppose there be given us two odd numbers and some one sets the problem and directs us to determine whether they are prime and incomposite relatively to each other or secondary and composite, and if they are secondary and composite what number is their common measure. We must compare the given numbers and subtract the smaller from the larger as many times as possible; then after this subtraction subtract in turn from the other, as many times as possible; for this changing about and subtraction from one and the other in turn will necessarily end either in unity or in some one and the same number, which will necessarily be odd. Now when the subtractions terminate in unity they show that the numbers are prime and incomposite relatively to each other; and when they end in some other number, odd in quantity and twice produced, then say that they are secondary and composite relatively to each other, and that their common measure is that very number which twice appears.

For example, if the given numbers were 23 and 45, subtract 23 from 45, and 22 will be the remainder; subtracting this from 23, the remainder is 1, subtracting this from 22 as many times as possible you will end with unity. Hence they are prime and incomposite to one another, and unity, which is the remainder, is their common measure.

But if one should propose other numbers, 21 and 49, I subtract the smaller from the larger and 28 is the remainder. Then again I subtract the same 21 from this, for it can be done, and the remainder is 7. This I subtract in turn from 21 and 14 remains; from which I subtract 7 again, for it is possible, and 7 will remain. But it is not possible to subtract 7 from 7; hence the termination of the process with a repeated 7 has been brought about, and you may declare the original numbers 21 and 49 secondary and composite relatively to each other, and 7 their common measure in addition to the universal unit.

To make again a fresh start, of the simple even numbers, some are superabundant, some deficient, like extremes set over against each other, and some are intermediary between them and are called perfect. Those which are said to be opposites to one another, the superabundant and deficient, are distinguished from one another in the relation of inequality in the directions of the greater and the less; for apart from these no other form of inequality could be conceived, nor could evil, disease, disproportion, unseemliness, nor any such thing, save in terms of excess or deficiency. For in the realm of the greater there arise excesses, overreaching, and superabundance, and in the less need, deficiency, privation, and lack; but in that which lies between the greater and the less, namely, the equal, are virtues, wealth, moderation, propriety, beauty, and the like, to which the aforesaid form of number, the perfect, is most akin.

Now the superabundant number is one which has, over and above the factors which belong to it and fall to its share, others in addition, just as if an animal should be created with too many parts or limbs, with ten tongues, as the poet says, and ten mouths, or with nine lips, or three rows of teeth, or a hundred hands, or too many fingers on one hand. Similarly if, when all the factors in a number are examined and added together in one sum, it proves upon investigation that the number’s own factors exceed the number itself, this is called a superabundant number, for it oversteps the symmetry which exists between the perfect and its own parts. Such are 12, 24, and certain others, for 12 has a half, 6, a third, 4, a fourth, 3, a sixth, 2, and a twelfth, 1, which added together make 16, which is more than the original 12; its parts, therefore, are greater than the whole itself. And 24 has a half, a third, fourth, sixth, eighth, twelfth, and twenty-fourth, which are 12, 8, 6, 4, 3, 2, 1. Added together they make 36, which, compared to the original number, 24, is found to be greater than it, although made up solely of its factors. Hence in this case also the parts are in excess of the whole.

Nicomachus, Arithmetic. Book I, Chapter 15

Now that we have given a preliminary systematic account of absolute quantity we come in turn to relative quantity.

Of relative quantity, then, the highest generic divisions are two, equality and inequality; for everything viewed in comparison with another thing is either equal or unequal, and there is no third thing besides these.

Now the equal is seen, when of the things compared one neither exceeds nor falls short in comparison with the other, for example, 100 compared with 100, 10 with 10, 2 with 2, a mina with a mina, a talent with a talent, a cubit with a cubit, and the like, either in bulk, length, weight, or any kind of quantity. And as a peculiar characteristic, also, this relation is of itself not to be divided or separated, as being most elementary, for it admits of no difference. For there is no such thing as this kind of equality and that kind, but the equal exists in one and the same manner. And that which corresponds to an equal thing, to be sure, does not have a different name from it, but the same; like ‘friend,’ ‘neighbor,’ ‘comrade,’ so also ‘equal’; for it is equal to an equal.

The unequal, on the other hand, is split up by subdivisions, and one part of it is the greater, the other the less, which have opposite names and are antithetical to one another in their quantity and relation. For the greater is greater than some other thing, and the less again is less than another thing in comparison, and their names are not the same, but they each have different ones, for example, ‘father’ and ‘son,’ ‘striker’ and ‘struck,’ ‘teacher’ and ‘pupil,’ and the like.

Moreover, of the greater, separated by a second subdivision into five species, one kind is the multiple, another the superparticular, another the superpartient, another the multiple superparticular, and another the multiple superpartient. And of its opposite, the less, there arise similarly by subdivision five species, opposed to the foregoing five varieties of the greater, the submultiple, subsuperparticular, subsuperpartient, submultiple-superparticular, and submultiple-superpartient; for as whole answers to whole, smaller to greater, so also the varieties correspond, each to each, in the aforesaid order, with the prefix sub-.

Once more, then; the multiple is the species of the greater first and most original by nature, as straightway we shall see, and it is a number which, when it is observed in comparison with another, contains the whole of that number more than once. For example, compared with unity, all the successive numbers beginning with 2 generate in their proper order the regular forms of the multiple; for 2, in the first place, is and is called the double, 3 triple, 4 quadruple, and so on; for ‘more than once’ means twice, or three times, and so on in succession as far as you like.

Answering to this is the submultiple, which is itself primary in the smaller division of inequality. It is the number which, when it is compared with a larger, is able to measure it completely more than once, and ‘more than once’ starts with twice and goes on to infinity. If then it measures the larger number that is being compared twice only, it is properly called the subdouble, as 1 is of 2; if thrice, subtriple, as 1 of 3; if four times, subquadruple, as 1 of 4, and so on in succession.

While each of these, the multiple and the submultiple, is generically infinite, the varieties by subdivision and the species also are observed naturally to make an infinite series. For the double, beginning with 2, goes on through all the even numbers, as we select alternate numbers out of the natural series; and these will be called doubles in comparison with the even and odd numbers successively placed beginning with unity.  All the numbers from the beginning two places apart, and third in order, are triples, for example, 3, 6, 9, 12, 15, 18, 21, 24. It is their property to be alternately odd and even, and they themselves in the regular series from unity are triples of all the numbers in succession as far as one wishes to go on with the process.

6 The quadruples are those in the fourth places, three apart, for instance, 4, 8, 12, 16, 20, 24, 28, 32, and so on. These are the quadruples of the regular series of numbers from unity going on as far as one finds it convenient to follow. It belongs to them all to be even; for one needs only to take the alternate terms out of the even numbers already selected. Thus necessarily it is true that the even numbers, with no further designation, are all doubles, the alternate ones quadruples, those two places apart sextuples, and those three places apart octuples, and this series will go on, on this same analogy, indefinitely.

The quintuples will be seen to be those four places apart, placed fifth from one another, and themselves the quintuples of the successive numbers beginning with unity. Alternately they are odd and even, like the triples.

The superparticular, the second species of the greater both naturally and in order, is a number that contains within itself the whole of the number compared with it, and some one factor of it besides.

If this factor is a half, the greater of the terms compared is called specifically sesquialter, and the smaller subsesquialter; if it is a third, sesquitertian and subsesquitertian; and as you go on throughout it will always thus agree, so that these species also will progress to infinity, even though they are species of an unlimited genus.

For it comes about that the first species, the sesquialter ratio, has as its consequents the even numbers in succession from 2, and no other at all, and as antecedents the triples in succession from 3, and no other. These must be joined together regularly, first to first, second to second, third to third—3:2, 6:4, 9:6, 12:8—and the analogous numbers to the ones corresponding to them in position.

If we care to investigate the second species of the superparticular, the sesquitertian (for the fraction naturally following after the half is the third), we shall have this definition of it—a number which contains the whole of the number compared, and a third of it in addition to the whole. We may have examples of it, in the proper order, in the successive quadruples beginning with 4 joined to the triples from 3, each term with the one in the corresponding position in the series, for example, 4:3, 8:6, 12:9, and so on to infinity. It is plain that that which corresponds to the sesquitertian but is called, with the prefix sub-, subsesquitertian, is the number, the whole of which is contained and a third part in addition, for example, 3:4, 6:8, 9:12, and the similar pairs of numbers in the same position in the series.

And we must observe the never-failing corollary of all this, that the first forms in each series, the so-called root numbers, are next to one another in the natural series; the next after the root-forms show an interval of only one number; the third two; the fourth three; the fifth four; and so on, as far as you like. Furthermore, that the fraction after which each of the superparticulars is named is seen in the lesser of the root numbers, never in the greater.

That by nature and by no disposition of ours the multiple is a more elementary and an older form than the superparticular we shall shortly learn, through a somewhat intricate process. And here, for a simple demonstration, we must prepare in regular and parallel lines the multiples specified above, according to their varieties, first the double in one line, then in a second the triple, then the quadruple in a third, and so on as far as the tenfold multiples, so that we may detect their order and variety, their regulated progress, and which of them is naturally prior, and indeed other corollaries delightful in their exactness. Let the diagram be as follows:

Let there be set forth in the first row the natural series from unity, and then in order those species of the multiple which we were bidden to insert.

Now then in comparison with the first rows beginning with unity, if we read both across and up and down in the form of the letter gamma, the next rows both ways, themselves in the form of a gamma, beginning with 4, are multiples according to the first form of the multiple, for they are doubles. The first differs by unity from the first, the second from the second by 2, the third from the third by 3, the next by 4, those following by 5, and you will find that this follows throughout.

The third rows in both directions from 9, their common origin, will be the triples of the terms in that same first row according to the second form of the multiple; the cross-lines like the letter chi, ending in the term 3 in either direction, are to be taken into consideration. The difference, for these numbers, will progress after the series of the even numbers, being 2 for the first, 4 for the next, 6 for the third; and this difference nature has of her own accord interpolated for us between these rows that are being examined, as is evident in the diagram.

The fourth row, whose common origin in both directions is 16, and whose cross-lines end with the terms 4, exhibits the third species of multiple, the quadruple, when it is compared with that same first row according to corresponding positions, first term with first, second with second, third with third, and so on. Again, the differences of these numbers are 3, 6, then 9, then 12, and the quantities that progress by steps of 3. These numbers are detected in the structure of the diagram in places just above the quadruples, and in the subsequent forms of the multiple the analogy will hold throughout.

In comparison with the second line reading either way, which begins with the common origin 4 and runs over in cross-lines to the term 2 in each row, the lines which are next in order beneath display the first species of the superparticular, that is, the sesquialter, between terms occupying corresponding places. Thus by divine nature, not by our convention or agreement, the superparticulars are of later origin than the multiples. For illustration, 3 is the sesquialter of 2, 6 of 4, 9 of 6, 12 of 8, 15 of 10, and throughout thus. They have as a difference the successive numbers from unity, like those before them.

The sesquitertians, the second species of superparticular, proceed with a regular, even advance from 4:3, 8:6, 12:9, 16:12, and so on; having also a regular increase of their differences. And in the other multiple and superparticular relations you will see that the results are in harmony and not by any means inconsistent as you go on to infinity.

The following feature of the diagram, moreover, is of no less exactness. The terms at the corners are units; the one at the beginning a simple unit, that at the end the unit of the third course, and the other two units of the second course appearing twice; so that the product (of the first two) is equal to the square (of the last).  Furthermore, in reading either way there is an even progress from unity to the tens, and again on the opposite sides two other progressions from 10 to 100.

The terms on the diagonal from 1 to 100 are all square numbers, the products of equals by equals, and those flanking them on either side are all heteromecic, unequal, and the products of sides of which one is greater than the other by unity; and so the sum of two successive squares and twice the heteromecic numbers between them is always a square, and conversely a square is always produced from the two heteromecic numbers on the sides and twice the square between them.

An ambitious person might find many other pleasing things displayed in this diagram, upon which it is not now the time to dwell, for we have not yet gained recognition of them from our Introduction, and so we must turn to the next subject. For after these two generic relations of the multiple and the superparticular and the other two, opposite to them, with the prefix sub-, the submultiple and the subsuperparticular, there are in the greater division of inequality the superpartient, and in the less its opposite, the subsuperpartient.

The regular arrangement and orderly production of both species are discovered when we set forth the successive even and odd numbers, beginning with 3, and compare with them simple series of odd numbers only, from 5 in succession, first to first—that is, 5 to 3,—second to second—that is, 7 to 4,—third to third—that is, 9 to 5,—fourth to fourth—that is, 11 to 6,—and so on in the same order as far as you like. In this way the forms of the superpartient and the subsuperpartient, in due order, will be disclosed through the root-forms of each species, the superbipartient first, then the supertripartient, superquadripartient, and superquintipartient, and further in succession in similar manner; for after the root-forms of each species the ones which follow them will be produced by doubling, or tripling, both the terms, and in general by multiplying after the regular forms of the multiple.

It must be observed that from the two parts in addition to the whole which are contained in the greater term, we are to understand ‘third,’ in the case of three parts, ‘fourth,’ with four parts, ‘fifth,’ with five, ‘sixth,’ and so on, so that the order of nomenclature is something like this: superbipartient, supertripartient, superquadripartient, then superquintipartient, and similarly with the rest.

Now the simple, uncompounded relations of relative quantity are these which have been enumerated. Those which are compounded of them and as it were woven out of two into one are the following, of which the antecedents are the multiple superparticular and multiple superpartient, and the consequents the ones that immediately arise in connection with each of the former, named with the prefix sub-; together with the multiple superparticular the submultiple superparticular, and with the multiple superpartient the submultiple superpartient. In the subdivision of the genera the species of the one will correspond to those of the other, these also having names with the prefix sub-.

The multiple superpartient is the remaining relation of number. This, and the relation called by a corresponding name with the prefix sub-, exist when a number contains the whole of the number compared more than once (that is, twice, thrice, or any number of times) and certain parts of it, more than one, either two, three, or four, and so on, besides. These parts are not halves, for the reasons mentioned above, but either thirds, fourths, or fifths, and so on.

From what has already been said it is not hard to conceive of the varieties of this relation, for they are differentiated in the same way as, and consistently with, those that precede: double superbipartient, double supertripartient, double superquadripartient, and so on. For example, 8 is the double superbipartient of 3, 16 of 6, and in general the numbers beginning with 8 and differing by 8 are double superbipartients of those beginning with 3 and differing by 3, when those in corresponding places in the series are compared, and in the case of the other varieties one could ascertain their proper sequence by following out what has already been said. In this case, too, we must conceive that the nomenclature of the number compared goes along and suffers corresponding changes, with the addition of the prefix sub-.

Thus we come to the end of our speculation upon the ten arithmetical relations for a first Introduction. There is, however, a method very exact and necessary for all discussion of the nature of the universe which very clearly and indisputably presents to us the fact that that which is fair and limited, and which subjects itself to knowledge, is naturally prior to the unlimited, incomprehensible, and ugly, and furthermore that the parts and varieties of the infinite and unlimited are given shape and boundaries by the former, and through it attain to their fitting order and sequence, and like objects brought beneath some seal or measure all gain a share of likeness to it and similarity of name when they fall under its influence. For thus it is reasonable that the rational part of the soul will be the agent which puts in order the irrational part, and passion and appetite, which find their places in the two forms of inequality, will be regulated by the reasoning faculty as though by a kind of equality and sameness. And from this equalizing process there will properly result for us the so-called ethical virtues, sobriety, courage, gentleness, self-control, fortitude, and the like.

Let us then consider the nature of the principle that pertains to these universal matters. It is capable of proving that all the complex species of inequality and the varieties of these species are produced out of equality, first and alone, as from a mother and root.

Let there be given us equal numbers in three terms, first, units, then twos in another group of three, then threes, next fours, fives, and so on as far as you like. For them, as the setting forth of these terms has come about by a divine, and not human, contrivance, nay, by Nature herself, multiples will first be produced, and among these the double will lead the way, the triple after the double, the quadruple next, and then the quintuple, and, following the order we have previously recognized, ad infinitum; second, the superparticular, and here again the first form, the sesquialter, will lead, and the next after it, the sesquitertian, will follow, and after them the next in order, the sesquiquartan, the sesquiquintan, the sesquisextan, and so on ad infinitum; third, the superpartient, which once more the superbipartient will lead, the supertripartient will follow immediately upon it, and then will come the superquadripartient, the superquintipartient, and according to the foregoing as far as one may proceed.

Now you must have certain rules, like invariable and inviolable natural laws, following which the whole aforesaid advance and progress from equality may go on without failure. These are the directions: Make the first equal to the first, the second equal to the sum of the first and second, and the third to the sum of the first, twice the second, and the third. For if you fashion according to these rules you would get first all the forms of the multiple in order out of the three given terms of the equality, as it were, sprouting and growing without your paying any heed or offering any aid. From equality you will first get the double; from the double the triple, from the triple successively the quadruple, and from this the quintuple in due order, and so on. From these same multiples in their regular order, reversed, there are immediately produced by a sort of natural necessity through the agency of the same three rules the superparticulars, and these not as it chances and irregularly but in their proper sequence; for from the first, the double, reversed, comes the first, the sesquialter, and from the second, the triple, the second in this class, the sesquitertian; then the sesquiquartan from the quadruple, and in general each one from the one of similar name. And with a fresh start, if the superparticulars are set forth in the order of their production, but with terms reversed, the superpartients, which naturally follow them, are brought to light, the superbipartient from the sesquialter, the supertripartient from the sesquitertian, the superquadripartient from the sesquiquartan, and so on ad infinitum.  If, however, the superparticulars are set forth with terms not in reverse but in direct order, there are produced through the three rules the multiple superparticulars, the double sesquialter out of the first, the sesquialter; the double sesquitertian from the second, the sesquitertian; the double sesquiquartan from the third, the sesquiquartan, and so on. From those produced by the reversal of the superparticular, that is, the superpartients, and from those produced without such reversal, the multiple superparticulars, there are once more produced, in the same way and by the same rules, both when the terms are in direct or reverse order, the numbers that show the remaining numerical relations.

The following must suffice as illustrations of all that has been said hitherto, the production of these numbers and their sequence, and the use of direct and of reverse order. From the relation and proportion in terms of the sesquialter, reversed so as to begin with the largest term, there arises a relation in superpartient ratios, the superbipartient; and from it in direct order, beginning with the smallest term, a multiple superparticular relation, the double sesquialter. For example, from 9:6:4, we get either 9:15:25 or 4:10:25. From the relation in terms of sesquitertians, beginning with the greatest term, is derived a superpartient, the supertripartient; beginning with the smallest term, a double sesquitertian. For example, from 16:12:9 comes either 16:28:49 or 9:21:49. And from the relation in terms of sesquiquartans, when it is arranged to begin with the largest term, is derived a superpartient, the superquadripartient; when it starts with the smallest term, a multiple superparticular, the double sesquiquartan; for instance, from 25:20:16 comes either 25:45:81 or 16:36:81.

In the case of all these relations that are thus differentiated, and of the one from which both of the differentiated ones are derived, the last term is always the same and a square; the first term becomes the smallest, and invariably the extremes are squares.

Moreover the multiple superpartients and superpartients of other kinds are made to appear in yet another way out of the superpartients; for example, from the superbipartient relation arranged so as to begin with the smallest term comes the double superbipartient, but, arranged so as to start with the greatest, the superpartient ratio of 8:5. Thus from 9:15:25 comes either 9:24:64 or 25:40:64. From the supertripartient, beginning with the smallest term, we have the double supertripartient, and, beginning with the largest, the ratio of 11:7. Thus, from 16:28:49 comes either 16:44:121 or 49:77:121.  Again, from the superquintipartient, as, for example, 25:45:81, beginning with the lesser term we derive the double superquintipartient in the terms 25:70:196, but beginning with the greater a superpartient again, the ratio of 14:9, in the terms 81:126:196. And you will find the results analogous and in agreement with the foregoing in all successive cases to infinity.

An element is said to be, and is, the smallest thing which enters into the composition of an object and the least thing into which it can be analyzed. Letters, for example, are called the elements of literate speech, for out of them all articulate speech is composed and into them finally it is resolved. Sounds are the elements of all melody; for they are the beginning of its composition and into them it is resolved. The so-called four elements of the universe in general are simple bodies, fire, water, air, and earth; for out of them in the first instance we account for the constitution of the universe, and into them finally we conceive of it as being resolved.

We wish also to prove that equality is the elementary principle of relative number; for of absolute number, number per se, unity and the dyad are the most primitive elements, the least things out of which it is constructed, even to infinity, by which it has its growth, and with which its analysis into smaller terms comes to an end. We have, however, demonstrated that in the realm of inequality advance and increase have their origin in equality and go on to absolutely all the relations with a certain regularity through the operation of the three rules. It remains, then, in order to make it an element in very truth, to prove that analyses also finally come to an end in equality. Let this then be considered our procedure.

Suppose then you are given three terms, in any relation whatsoever and in any ratio, whether multiple, superparticular, superpartient, or a compound of these, multiple superparticular or multiple superpartient, provided only that the mean term is seen to be in the same ratio to the lesser as the greater to the mean, and vice versa. Subtract always from the mean the lesser term, whether it be first or last in order, and set down the lesser term itself as the first term of your new series; then put as your second term what remains from the second after the subtraction; then after having subtracted the sum of the new first term and twice the new second term from the remaining number—that is, the greater of the numbers originally given you—make the remainder your third term, and the resulting numbers will be in some other ratio, naturally more primitive. And if again in the same way you subtract the remainder from these same terms, it will be found that your three terms have passed back into three others more primitive, and you will find that this always takes place as a consequence, until they are reduced to equality, whence by every necessity it appears evident that equality is the elementary principle of relative quantity.

There follows upon this speculation a most elegant principle, extremely useful in its application to the Platonic psychogony and the problem of all harmonic intervals; for in the Platonic passage we are frequently bidden, for the sake of the argument, to set up series of intervals of two, three, four, five, or an infinite number of sesquialter ratios, or two sesquitertians, sesquiquartans, sesquioctaves, or superparticulars of any kind whatsoever, and in each case three, four, or five of them, or as many as may be directed. It is reasonable that we should do this not in an unscientific, unintelligent fashion, it may be even blunderingly, but artistically, surely, and quickly, by the following procedure.

1 We must make a similar table in illustration of the triple:

In the foregoing table we shall observe that in the same way the first triple, 3, stands at the head of but one sesquitertian ratio, 4, its own sesquitertian, which immediately shuts off the development of another like it; for 4 is not divisible by 3, and hence will not have a sesquitertian. The second triple is 9, and hence will begin a series of only two sesquitertian ratios, 12, its own, and 16, that of 12; but 16 cuts off further progress, for it is not divisible by 3 and hence will not have a sesquitertian. Next in order is the triple 27, three times removed from 1, for the triples progress thus: 1, 3, 9, 27. Therefore this number will stand at the head of three sesquitertian ratios and no more. The first is its own, 36; the second the sesquitertian of 36, 48; the third that of the last, 64, and this no longer has a third part and therefore will not admit of a sesquitertian. The fourth leads a series of four sesquitertians and the fifth, of course, five.

Such, then, is the illustration; and for the other multiples let the manner of your tables be the same. Observe that likewise here, as we found to be true in our previous discussion, Nature shows us that the doubles are more nearly original than the triples, the triples than the quadruples, these latter than the quintuples, and so on throughout. For the highest rows of figures, across the breadth of the tables, if they are doubles, will have doubles lying parallel to them, and the numbers lying diagonally, on the hypotenuse, will be of the next succeeding variety, greater by 1, that is, triples, seen also in a series of parallel lines. If, however, there are triples across the breadth, the diagonals will by all means be quadruples; if the former are quadruples, then the latter are quintuples, and so forth.

It remains, after we have explained what other ratios are produced by combination of ratios, to pass on to the succeeding topics of the Introduction.

Now the first two ratios of the superparticular, combined, produce the first ratio of the multiple, namely, the double; for every double is a combination of sesquialter and sesquitertian, and every sesquialter and sesquitertian combined will invariably produce a double.

For example, since 3 is the sesquialter of 2, and 4 the sesquitertian of 3, 4 will be the double of 2, and is a combination of sesquialter and sesquitertian. Again, as 6 is the double of 3, we shall find between them some number that will of necessity preserve the sesquitertian ratio to the one and the sesquialter to the other; and indeed 4, lying between 6 and 3, gives the sesquitertian ratio to 3 and the sesquialter to 6.

It was rightly said, then, that the double, when resolved, is resolved into the sesquialter and the sesquitertian, and that when sesquialter and sesquitertian are combined there arises the double, and that the first two forms of the superparticular combined make the first form of the multiple.

But again, to take another start, this first form of the multiple which has thus been produced, together with the first form of the superparticular, will produce the next form of the same class, that is, the second multiple, the triple; for from every multiple and sesquialter combined a triple of necessity arises. For example, as the double of 6 is 12, and the sesquialter of this is 18, then immediately 18 is the triple of 6; and to take another method, if I do not care to make 12 the mean term, but rather 9, the sesquialter of 6, the same result will come about, without deviation and harmoniously; for while 18 is the double of 9 it will preserve the triple ratio to 6. Hence from the sesquialter and the double, the first forms of the superparticular and the multiple, there arises by combination the second form of the multiple, the triple, and into them it is always resolved. For look you; 6, which is the triple of 2, will have a mean term, 3, which will exhibit two ratios, the sesquialter with regard to 2, and the double ratio of 6 to itself.

But if this triple ratio, likewise, the second form of the multiple, is combined with the sesquitertian, which is the second form of the superparticular, there would be produced from them the next form of the multiple, namely, the quadruple, and this also will of necessity be resolved into them after the same fashion as the cases previously set forth; and the quadruple, taking into combination the sesquiquartan, will make the quintuple, and, once more, the latter with the sesquiquintan will make the sextuple, and so on to the end. Thus the multiples in regular order from the beginning with the superparticulars in regular order from the beginning will be found to produce the next larger multiples. For the double with the sesquialter makes the triple, the triple with the sesquitertian the quadruple, the quadruple with the sesquiquartan the quintuple, and as far as you wish to proceed no contrary result will appear.

Up to this point then we have sufficiently discussed relative number, by a process of selection measuring out what is easily comprehended and appropriate to the nature of the matters thus far introduced. Whatever remains to be said on this topic will be filled in after we have put it aside and have first discussed certain subjects which involve a more serviceable inquiry, having to do with the properties of absolute number, not relative. For mathematical speculations are always to be interlocked and to be explained one by means of another. The subjects which we must first survey and observe are concerned with linear, plane, and solid numbers, cubical and spherical, equilateral and scalene, ‘bricks,’ ‘beams,’ ‘wedges,’ and the like, the tradition concerning which, to be sure, since they are more closely related to magnitude, is properly given in the Geometrical Introduction. Yet the germs of these ideas are taken over into arithmetic, as the science which is the mother of geometry and more elementary than it. For we recall that a short time ago we saw that arithmetic abolishes the other sciences with itself, but is not abolished by them, and conversely is of necessity implied by them but does not itself imply them.

First, however, we must recognize that each letter by which we indicate a number, such as iota, the sign for 10, kappa for 20, and omega for 800, designates that number by man’s convention and agreement, not by nature. On the other hand, the natural, unartificial, and therefore simplest indication of numbers would be the setting forth one beside the other of the units contained in each. For example, the writing of one unit by means of one alpha will be the sign for 1; two units side by side, that is, a series of two alphas, will be the sign for 2; when three are put in a line it will be the character for 3, four in a line for 4, five for 5, and so on. For by means of such a notation and indication alone could the schematic arrangement of the plane and solid numbers mentioned be made clear and evident, thus:

The number 1: α
The number 2: α α
The number 3: α α α
The number 4: α α α α
The number 5: α α α α α
and further in similar fashion.

Unity, then, occupying the place and character of a point, will be the beginning of intervals and of numbers, but not itself an interval or a number, just as the point is the beginning of a line, or an interval, but is not itself line or interval. Indeed, when a point is added to a point, it makes no increase, for when a non-dimensional thing is added to another non-dimensional thing, it will not thereby have dimension; just as if one should examine the sum of nothing added to nothing, which makes nothing. We saw a similar thing also in the case of equality among the relatives; for a proportion is preserved—as the first is to the second, so the second is to the third—but no interval is generated in the relation of the extremes to each other, as there is in all the other relations with the exception of equality. In exactly the same way unity alone out of all number, when it multiplies itself, produces nothing greater than itself.

Unity, therefore, is non-dimensional and elementary, and dimension first is found and seen in 2, then in 3, then in 4, and in succession in the following numbers; for ‘dimension’ is that which is conceived of as between two limits.

The first dimension is called ‘line,’ for ‘line’ is that which is extended in one direction. Two dimensions are called ‘surface,’ for a ‘surface’ is that which is extended in two directions. Three dimensions are called ‘solid,’ for a ‘solid’ is that which is extended in three directions, and it is by no means possible to conceive of a solid which has more than three dimensions, depth, breadth, and length. By these are defined the six directions which are said to exist in connection with every body and by which motions in space are distinguished, forward, backward, up, down, right, and left; for of necessity two directions opposite to each other follow upon each dimension, up and down upon one, forward and backward upon the second, and right and left upon the third.

The statement, also, as it happens, can be made conversely thus: If a thing is solid, it has by all means three dimensions, length, depth, and breadth; and conversely, if it has the three dimensions, it is always a solid, and nothing else.

That which has but two dimensions, therefore, will not be a solid, but a surface, for the latter admits of but two dimensions. Here too it is possible similarly to reverse the statement; directly stated, a surface is that which has two dimensions, and conversely, that which has two dimensions is always a surface.

The surface, then, is exceeded by the solid by one dimension, and the line is exceeded by the surface by one, for the line is that which is extended in but one direction and has only one dimension, and it falls short of the solid by two dimensions. The point falls short of the latter by one dimension, and hence it has already been stated that it is non-dimensional, since it falls short of the solid by three dimensions, of the surface by two, and of the line by one.

The point, then, is the beginning of dimension, but not itself a dimension, and likewise the beginning of a line, but not itself a line; the line is the beginning of surface, but not surface; and the beginning of the two-dimensional, but not itself extended in two directions. Naturally, too, surface is the beginning of body, but not itself body, and likewise the beginning of the three-dimensional, but not itself extended in three directions.

Exactly the same in numbers, unity is the beginning of all number that advances unit by unit in one direction; linear number is the beginning of plane number, which spreads out like a plane in one more dimension; and plane number is the beginning of solid number, which possesses a depth in the third dimension, besides the original ones. To illustrate and classify, linear numbers are all those which begin with 2 and advance by the addition of 1 in one and the same dimension; and plane numbers are those that begin with 3 as their most elementary root and proceed through the next succeeding numbers. They receive their names also in the same order; for there are first the triangles, then the squares, the pentagons after these, then the hexagons, the heptagons, and so on indefinitely, and, as we said, they are named after the successive numbers beginning with 3.

The triangle, therefore, is found to be the most original and elementary form of the plane number. This we can see from the fact that, among plane figures, graphically represented, if lines are drawn from the angles to the centers each rectilinear figure will by all means be resolved into as many triangles as it has sides; but the triangle itself, if treated like the rest, will not change into anything else but itself.

Hence the triangle is elementary among these figures; for everything else is resolved into it, but it into nothing else. From it the others likewise would be constituted, but it from no other. It is therefore the element of the others, and has itself no element. Likewise, as the argument proceeds in the realm of numerical forms, it will confirm this statement.

Now a triangular number is one which, when it is analyzed into units, shapes into triangular form the equilateral placement of its parts in a plane. 3, 6, 10, 15, 21, 28, and so on, are examples of it; for their regular formations, expressed graphically, will be at once triangular and equilateral. As you advance you will find that such a numerical series as far as you like takes the triangular form, if you put as the most elementary form the one that arises from unity, so that unity may appear to be potentially a triangle, and 3 the first actually.

Their sides will increase by the successive numbers, for the side of the one potentially first is unity; that of the one actually first, that is, 3, is 2; that of 6, which is actually second, 3; that of the third, 4; the fourth, 5; the fifth, 6; and so on.

The triangular number is produced from the natural series of number set forth in a line, and by the continued addition of successive terms, one by one, from the beginning; for by the successive combinations and additions of another term to the sum, the triangular numbers in regular order are completed. For example, from this natural series, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, I take the first term and have the triangular number which is potentially first, 1; then adding the next term I get the triangle actually first, for 2 plus 1 equals 3. In its graphic representation it is thus made up: Two units, side by side, are set beneath one unit, and the number three is made a triangle:

α
α α

Then when next after these the following number, 3, is added, simplified into units, and joined to the former, it gives 6, the second triangle in actuality, and furthermore, it graphically represents this number:

α
α α
α α α

Again, the number that naturally follows, 4, added in and set down below the former, reduced to units, gives the one in order next after the aforesaid, 10, and takes a triangular form:

α
α α
α α α
α α α α

5, after this, then 6, then 7, and all the numbers in order, are added, so that regularly the sides of each triangle will consist of as many numbers as have been added from the natural series to produce it:

Side 5:

α
α α
α α α
α α α α
α α α α α

Side 6:

α
α α
α α α
α α α α
α α α α α
α α α α α α

Side 7:

α
α α
α α α
α α α α
α α α α α
α α α α α α
α α α α α α α

Nicomachus, Arithmetic. Book II, Chapter 9

The square is the next number after this, which shows us no longer 3, like the former, but 4, angles in its graphic representation, but is nonetheless equilateral. Take, for example, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100; for the representations of these numbers are equilateral, square figures, as here shown; and it will be similar as far as you wish to go:

It is true of these numbers, as it was also of the preceding, that the advance in their sides progresses with the natural series. The side of the square potentially first, 1, is 1; that of 4, the first in actuality, 2; that of 9, actually the second, 3; that of 16, the next, actually the third, 4; that of the fourth, 5; of the fifth, 6, and so on in general with all that follow.

This number also is produced if the natural series is extended in a line, increasing by 1, and no longer the successive numbers are added to the numbers in order, as was shown before, but rather all those in alternate places, that is, the odd numbers. For the first, 1, is potentially the first square; the second, 1 plus 3, is the first in actuality; the third, 1 plus 3 plus 5, is the second in actuality; the fourth, 1 plus 3 plus 5 plus 7, is the third in actuality; the next is produced by adding 9 to the former numbers, the next by the addition of 11, and so on.

So much, then, concerning the three proportions celebrated by the ancients, which we have discussed the more clearly and at length for just this reason, that they are to be met with frequently and in various forms in the writings of those authors. The succeeding forms, however, we must only epitomize, since they do not occur frequently in the ancient writings, but are included merely for the sake of our own acquaintance with them and, so to speak, for the completeness of our reckoning. They are set forth by us in an order based on their opposition to the three archetypes already described, since they are fashioned out of them and have the same order.

The fourth, and the one called subcontrary, because it is opposite to, and has opposite properties to, the harmonic proportion, exists when, in three terms, as the greatest is to the smallest, so the difference of the smaller terms is to that of the greater, for example 3, 5, 6. For the terms compared are seen to be in the double ratio, and it is plain wherein it is opposite to the harmonic proportion; for whereas they both have the same extreme terms, and in double ratio, in the former the difference of the greater terms as compared to that of the lesser preserved the same ratio as that of the extremes, but in this proportion just the reverse, the difference of the smaller compared with that of the greater. You must know that its peculiar property is this: The product of the greater and the mean terms is twice the product of the mean and the smaller; for 6 times 5 is twice 5 times 3.

The two proportions, fifth and sixth, were both fashioned after the geometrical, and they differ from each other thus.

The fifth form exists, whenever, among three terms, as the middle term is to the lesser, so their difference is to the difference between the greater and the mean, as in 2, 4, 5, for 4 is the middle term, the double of 2, the lesser, and 2 is the double of 1—the difference of the smallest terms as compared with that of the largest. That which makes it contrary to the geometric proportion is that in the former, as the middle term is to the lesser, so the excess of the greater over the mean is to the excess of the mean over the lesser term, whereas in this proportion, on the contrary, it is the difference of the lesser compared to that of the greater. Nevertheless it is peculiar to this proportion that the product of the greatest by the middle term is double that of the greatest by the smallest, for 5 times 4 is twice 5 times 2.

The sixth form comes about when, in a group of three terms, as the greatest is to the mean, so the excess of the mean over the lesser is to the excess of the greater over the mean, for example 1, 4, 6, for both are in the sesquialter ratio. There is in this case also a reasonable cause for its opposition to the geometrical; for here, too, the likeness of the ratios reverses, as in the fifth form.

These are the six proportions commonly spoken of among previous writers, the three prototypes having lasted from the times of Pythagoras down to Aristotle and Plato, and the three others, opposites of the former, coming into use among the commentators and sectarians who succeeded these men. But certain men have devised in addition, by shifting the terms and differences of the former, four more which do not much appear in the writings of the ancients, but have been sparingly touched upon as an over-nice detail. These, however, we must run over in the following fashion, lest we seem ignorant.

The first of them, and the seventh in the list of them all, exists when, as the greatest term is to the least, so their difference is to the difference of the lesser terms, as 6, 8, 9, for on comparison the ratio of each is seen to be the sesquialter.

The eighth proportion, which is the second of this group, comes about when, as the greatest is to the least term, so the difference of the extremes is to the difference of the greater terms, as 6, 7, 9; for this also has sesquialters for the two ratios.

The ninth in the complete list, and third in the number of those subsequently invented, exists when there are three terms and whatever ratio the mean bears to the least, that also the difference of the extremes has in comparison with that of the smallest terms, as 4, 6, 7.

The tenth, in the full list, which concludes them all, and the fourth in the series presented by the moderns, is seen when, among three terms, as the mean is to the lesser, so the difference of the extremes is to the difference of the greater terms, as 3, 5, 8, for it is the superbipartient ratio in each pair.

To sum up, then, let the terms of the ten proportions be set forth in one illustration, for the sake of easy comprehension:

First: 1, 2, 3
Second: 1, 2, 4
Third: 3, 4, 6
Fourth: 3, 5, 6
Fifth: 2, 4, 5
Sixth: 1, 4, 6
Seventh: 6, 8, 9
Eighth: 6, 7, 9
Ninth: 4, 6, 7
Tenth: 3, 5, 8

It remains for me to discuss briefly the most perfect proportion, that which is three-dimensional and embraces them all, and which is most useful for all progress in music and in the theory of the nature of the universe. This alone would properly and truly be called harmony rather than the others, since it is not a plane, nor bound together by only one mean term, but with two, so as thus to be extended in three dimensions, just as a while ago it was explained that the cube is harmony.

When, therefore, there are two extreme terms, both of three dimensions, either numbers multiplied thrice by themselves so as to be a cube, or numbers multiplied twice by themselves and once by another number so as to be either ‘beams’ or ‘bricks,’ or the products of three unequal numbers, so as to be scalene, and between them there are found two other terms which preserve the same ratios to the extremes alternately and together, in such a manner that, while one of them preserves the harmonic proportion, the other completes the arithmetic, it is necessary that in such a disposition of the four the geometric proportion appear, on examination, commingled with both mean terms—as the greatest is to the third removed from it, so is the second from it to the fourth; for such a situation makes the product of the means equal to the product of the extremes. And again, if the greatest term be shown to differ from the one next beneath it by the amount whereby this latter differs from the least term, such an array becomes an arithmetic proportion and the sum of the extremes is twice the mean. But if the third term from the greatest exceeds and is exceeded by the same fraction of the extremes, it is harmonic and the product of the mean by the sum of the extremes is double the product of the extremes.

Let this be an example of this proportion: 6, 8, 9, 12. 6 is a scalene number, derived from 1 times 2 times 3, and 12 comes from the successive multiplication of 2 times 2 times 3; of the mean terms the lesser is from 1 times 2 times 4, and the greater from 1 times 3 times 3. The extremes are both solid and three-dimensional, and the means are of the same class. According to the geometric proportion, as 12 is to 8, so 9 is to 6; according to the arithmetic, as 12 exceeds 9, by so much does 9 exceed 6; and by the harmonic, by the fraction by which 8 exceeds 6, viewed as a fraction of 6, 8 is also exceeded by 12, viewed as a fraction of 12.

Moreover 8:6 or 12:9 is the diatessaron, in sesquitertian ratio; 9:6 or 12:8 is the diapente in the sesquialter; 12:6 is the diapason in the double. Finally, 9:8 is the interval of a tone, in the superoctave ratio, which is the common measure of all the ratios in music, since it is also the more familiar, because it is likewise the difference between the first and most elementary intervals.

And let this be sufficient concerning the phenomena and properties of number, for a first Introduction.

THE END