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Introduction to Arithmetic, Book I

Nicomachus of Gerasa · Introduction to Arithmetic
Section 2 of 2
Underlined text was edited by AI to fix errors.
accompanying characteristic! to end alternately in 6 or 8, and always to be even.
4 There is a method of producing them,? neat and unfailing, which neither passes by any of the perfect numbers nor fails to differentiate any of those that are not such, which is carried out in the following way.
You must set forth the even-times even numbers from unity, ad- vancing in order in one line, as far as you please: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1,024, 2,048, 4,096. ... Then you must add them to-gether, one at a time, and each time you make a summation observe the result to see what it is. If you find that it is a prime, incomposite number, multiply it by the quantity of the last number added, and the result will always be a perfect number. If, however, the result is secondary and composite, do not multiply, but add the next and ob-serve again what the resulting number is; if it is secondary and composite, again pass it by and do not multiply; but add the next; but if it is prime and incomposite, multiply it by the last term added, and the result will be a perfect number; and so on to infinity. In similar fashion you will produce all the perfect numbers in succession, overlooking none.
For example, to 1 I add 2, and observe the sum, and find that it is 3, a prime and incomposite number in accordance with our previous demonstrations; for it has no factor with denominator different from the number itself,’ but only that with denominator agreeing. Therefore I multiply it by the last number to be taken into the sum, that is, 2; I get 6, and this I declare to be the first perfect number in actuality,‘ and to have those parts which are beheld in the numbers of which it is composed. For it will have unity’ as the factor with denominator the same as itself, that is, its sixth part; and 3 as the half, which is seen in 2, and conversely 2 as its third part.
5 Twenty-eight likewise is produced by the same method when another number, 4, is added to the previous ones. For the sum of the three, 1,
! Cf. Boethius, I. 20: Et semper hi numeri duobus paribus terminantur, 6 et 8, et semper allernatim in hos numeros summarum fine provenient, See p. 52.
* Euclid (LX. 36) and Theon of Smyrna (p. 45, 14 ff. Hiller) report this method of discovering perfect numbers. * That is, 4.
‘* Actual’ (évepyelg) is here specified because in sect. 8 Nicomachus designates 1 as a potential (Svvdue.) perfect number. On his use of Aristotelian terminology, as here, see p. 35.
’ Boethius, I. 20, speaking of this matter says of 6: kabel unam quidem a se denominatam par-iem, id est, sextam, 3 vero medictatem secundum dualilatem, al vero 2 secundum coacervationem, id est, secundum lernarium, quontam coacervalt 3 mulliplicati sunt,
2, and 4, is 7, and is found to be prime and incomposite, for it admits only the factor with denominator like itself, the seventh part. Therefore I multiply it by the quantity of the term last taken into the summation, and my result is 28, equal to its own parts, and having its factors derived from the numbers already adduced, a half corresponding to 2; a fourth, to 7; a seventh, to 4; a fourteenth to offset the half; and a twenty-eighth, in accordance with its own nomenclature, which is 1 In all numbers.
When these have been discovered, 6 among the units and 28 in the 6 tens, you must do the same to fashion the next. Again add the next 7 number, 8, and the sum is 15. Observing this, I find that we no longer have a prime and incomposite number, but in addition to the factor with denominator like the number itself,’ it has also a fifth and a third, with unlike denominators. Hence I do not multiply it by 8, but add the next number, 16, and 31 results. As this is a prime, incomposite number, of necessity it will be multiplied, in accordance with the general rule of the process, by the last number added, 16, and the result is 496, in the hundreds; and then comes 8,128 in the thousands, and so on,” as far as it is convenient for one to follow.
Now unity is potentially a perfect number, but not actually; for 8 taking it from the series as the very first I observe what sort it is, according to the rule, and find it prime and incomposite; for it is so in very truth,* not by participation like the rest, but it is the primary
? Thomas Taylor (Theoretic Arithmetic, p. 33) gives the following table, showing how the perfect numbers may be formed by Nicomachus’s method:
Evenly even numbers:
I, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1,024, 2,048, 4,006; Odd numbers produced by adding the above:
I, 3, 7) 15’, 31, 63, 127, 255”, 1,023’, 2,047’, 4,005’, 8,191’; Perfect numbers:
1, 6, 28, 496, 8,128.
The odd numbers which are not prime, and hence cannot be used to make perfect numbers, are marked with an accent.
+ This statement is to be understood in the light of Nicomachus’s essentially Pythagorean view of arithmetic, and with it should be compared IT. 17. 2 and incidental remarks elsewhere, e.g., II. 17. 4,5; 18. 1, 4 (end); 20. 2; 1.1 (end), etc. In II. 17. 2 it is stated that ‘sameness’ is found fundamentally in the monad, and most of the other passages cited bring out the principle that the odd numbers and the squares participate in ‘ sameness’ indirectly, through the monad, the monad being that which ‘determines the specific form’ of the odd numbers (e/doroetv), and the latter in turn acting as the bases (yrwyores) of the squares. Likewise ‘otherness’ inheres fundamentally in the dyad, and is hence conveyed secondarily, as it were, into the even numbers and the heteromecic numbers. A similar construction may be placed on the passage at hand; the only difference is that ‘primeness,’ instead of ‘sameness,’ is in question. The monad Is per se prime, g number of all, and alone incomposite. I multiply it, therefore, by the last term taken into the summation, that is, by itself, and my re- ro sult is 1; for 1 times 1 equals 1. Thus unity is perfect potentially; for it is potentially equal to its own parts, the others actually.
1 Now that we have given a preliminary systematic account of absolute quantity we come in turn to relative quantity.”
2 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.
3 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,
4 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 and other prime numbers are secondarily or by participation prime, for they are combinations of the monad (cf. I. 11. 3), though of no other numbers. The monad on the other hand can be broken up into no smaller components and is therefore elementary. It is further to be noted that the word for ‘ prime’ (#rpédros) is slightly ambiguous, and even in this specialized use of the word there must be a suggestion of the original sense, ‘ first.’ The monad is obviously ‘ first’ in a higher degree than any prime number.
5 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;
1 According to I. 3. 1, this subject belongs to music rather than to arithmetic. Cf. p, 114.
2 That is, a thing of one class can never be said to be equal to a thing of a different class. Nicomachus does not state this principle in its broadest form, namely, that it is impossible to establish any ratio between objects of entirely different classes. The latter is the form in which Theon of Smyrna, p. 73, 16 ff., puts the matter, following, as he says, Adrastus. (See above, p. 41.) Nicomachus, however, demonstrates elaborately in I. 23. 6 ff. and II. x and 2 the proposition that the relation of equality is the element of all ratio, so that, if the connecting link be supplied for him, it may be said that he implies that only homogeneous things may have a ratio. Theon’s statement is as follows: “The ratio of analogy between two homogeneous terms is their definite relation (wo.d oxéors) to one another; e.g., double or triple. For as to the relation between unlike things, Adrastus says that it cannot be known; e.g., a cubit and a mina, a choenix and a kotyle, ‘white’ and ‘sweet’ or ‘warm,’ these things cannot be brought together and compared. But homogeneous things may be, e.g., lengths with lengths, surfaces with surfaces, solids with solids, weights with weights, . . . and whatever things are of the same genus or species and therefore have some mutual relation.”
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-.
! The terms here used are adapted from Boethius’s translations and are employed in Thomas Taylor’s Theoretic Arithmetic. The present classification was no doubt the ordinary scientific one, Theon (pp. 74, 20 ff.; 76, 1 &.), however, gives two different classifications, of which the latter is like that of Nicomachus, except that Theon adds the unnecessary class olf ovdérepo:. In p. 74, 20 ff. he divides ratios first into greater, less, and equal, and then the greater into multiples, superparticulars and ‘those of neither class’ (ovdé¢repo:), the less into submultiples, sub-superparticulars and ‘those of neither class.’ It may be noted that in this context oddérepo: is properly used, and it might cover ratios of all the classes mentioned by Nicomachus other than those which Theon specifically includes. He proceeds (p. 74, 23 fi.) to enumerate the members of these classes which are also ‘concords,’ cvygwyrlac, in music, citing as obdérepo: the ratios 9:8 (i.e., the ‘ tone’) and 256: 243 (i.e., the ‘ limma’), which, he says, are the ‘ beginnings’ of concord and are therefore neither themselves concords nor yet outside of concord. Next (p. 75, 17) he goes on to say that there are, however, certain other ratios spoken of in arithmetic, with which he will deal in due time; besides the ones given, ‘also superpartients, multiple superpartients and still others.” These he enumerates, as has been said, in his final statement (p. 76, 1 ff.) which agrees with Nicomachus save for the inclusion of the otd¢repo:; he takes pains to tell us that this is ‘the arithmetical tradition’ (xara rhv dpi@unrixhy wapddoc.v) and is the classification of Adrastus. It would seem fair to conclude, then, that the former classification is the musical tradition, and was not taken from Adrastus. Perhaps he has carried over from the ‘ musical’ list the ovddrepo:; it is not suitable in the second list, as Hiller notes (see his critical note), He uses this class to cover ‘ the ratio of number to number’ (a direct reference to Timaeus, 36 B) in p. 8, 7, 1.€., the ‘limma,’ as before. Compare with Nicomachus’s list also Johannes Pediasimus,
Geomeiria, in Neue Jahrb. f. Phil. u. Paed., vol. XCII, pp. 366 ff. (f. 43 b of the Munich MS there cited).
r 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.
2 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.
3 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, sub-triple, as 1 of 3; if four times, subquadruple, as 1 of 4, and so on in succession.
4 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
5 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
1 Theon’s definition (p. 76, 8 Hiller) is: ‘It is the multiple ratio when the greater term contains the smaller more than once, 1.e., when the greater term is exactly measured by the smaller with no remainder,”’ Euclid (VII, Def. 5) has: ‘‘ A greater number is multiple of the less when it is measured by the less” (wodAawAdouos 52 6 peltwr rod é\drrovos Sray xarayerpHrac td Tov éAde-govos); the same definition as Euclid’s is given by Hero of Alexandria, Definitton 121, ed. Hultsch.
2 Nicomachus more often uses the terms half, third, etc., for these fractions.
‘’. «*® That is, from the natural series.
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 7 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 2 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, 3 third to third—3: 2,6: 4, 9:6, 12: 8—and the analogous numbers to the ones corresponding to them in position.
ro
1 rots dws dprios: that is, numbers not otherwise characterized than as even (as opposed, e.g., to the alternately occurring even numbers, or to even-times even numbers).
? The reference is now to the ‘natural series’ 1, 2, 3, 4, etc., of which the author was speaking in sections 4 and 5 before he digressed in section 6 to point out how the even multiples are placed in the series of even numbers.
+ Theon’s definition (p. 76, 21 ff.) is: “It is the superparticular ratio, when the greater term contains the less once and some one part of the less, i.e., when the difference between the greater and the less is such as to be a factor of the less.”
‘xadeira:... eidixds: It is to be noted that this is a technical expression of logic. The genus in this case is the ‘superparticular,’ and the ‘ sesquialter’ is the species. The superparticular was itself treated above (I. 17. 7) as a species of the genus ‘ greater inequality.’ Boethius here, and generally, misses the technical force of yerxds and ¢ldixés in Nicomachus, and simply omits them in his translation.
‘ The words translated ‘ antecedent ’ and ‘ consequent’ (wpédoyos, Uwédoryos) mean respectively the larger and the smaller terms in a ratio between unequal quantities. Boethius, I. 24, adopts the translations duces and comites (Voco autem matores numeros duces, minores comites). er.
4 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,
5 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.
6 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;
7 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.
8 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
1 wu@uéves: properly ‘stock’ but here translated ‘root.’ wu@u4» is the technical designation for that one in a series of equal ratios which is expressed in the lowest terms; in the words of Theon of Smyma, p. 80, 15 ff.: “Of all the ratios grouped in one species (e.g., double sesquialter, etc.) those that are expressed in the smallest numbers and numbers prime to one another are called primary among those bearing the same ratio and roots (wv@uéves) of those of the same species.” wvOuyv is so used by Plato in the famous passage on the marriage number (Kep., 546 B-c). Apollonius of Pergae used the term wv@u%» in a somewhat similar way, to designate the units which serve as the ‘stock’ in numbers consisting of those units multiplied by 10 or its powers; thus 5 is the ruv@uyy of 50, 500, 5,000, etc. See Cantor, op. cit., vol. I, p. 347. Another use of the word wv@uyv is described by Heath, History, vol. I, p. 116. In the present case, if the sesquialters are derived from the double and the triple series,
2 4 6 8rto, ete, 3 6 9 215, etc, as described above, the ‘root sesquialter’ is the ratio 3: 2, the ‘second from the root numbers’ is the ratio 6:4, etc. Boethius, I. 25, observes that the number of the ‘intervening numbers,’ of which Nicomachus is here speaking, is always one less than the number designating the order of the ratio in the series. E.g., in the third ratio of the series above, g: 6, there are two intervening numbers between 9 and 6 (8 and 7).
2 The number giving the name to each variety of superparticular (e.g., In the case of the sesquitertian, one and one third, éwl r ptt os) is to be observed always in the first instance of that ratio, that is, in the ‘root numbers’ explained in the preceding note; and more specifically this number is always the smaller of the two in each ‘root ratio,’ e.g., 2 in 3: 2 (whence sesquialier, 7 us 6X05), 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 exact-
ness. Let the diagram be as follows: 9 I 2 3 4 5 6 7 8 0) 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 4 8 12 16 20 24 28 32 36 40 5 10 I5 20 25 30 35 40 45 5° 6 I2 18 24 30 36 42 48 54 60 7 14 21 280 350 (i QtiéSGH—ts—s«éC“B 7° 8 16 24 32 40 48 56 64 72 80 9 18 27 36,45 54 «463 72s go
10 20 30 40 5° 60 7° 80 go 100
Let there be set forth in the first row the natural series from unity, 10 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
=

I

! The top row and the left-hand column of the table, which we are directed to use, meeting at 1 in a right angle, present the form of the Greek capital letter gamma, I’. With the terms in these series we compare those of row 2 and column 2, which meet in the term 4 and make the same figure, aT’; but these second rows are regarded, not as ending with 4, but as continuing to the terms 2 in the first row and the first column, as Nicomachus’s immediately following observation shows and as Boethius (I. 27) interpreted him. The first term of the first series, 1, is surpassed by the first term of the second, 2, by 1, and son; cf. section 6 above.
? Row 3 and column 3, which we now compare with row 1 and column 1, meet at the term 9 at right angles and run beyond g to 3 in both the first row and the first column. They therefore present the appearance of the Greek capital letter chi (X), the two lines of which in inscriptional forms often make right angles.
12 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.
13 The fourth row, whose common origin in both directions is 16, and whose cross-lines 7 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.
14 Incomparison 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.
15 The sesquitertians, the second species of superparticular, proceed with a regular, even advance from 4:3, 8:6, 12:9, 16:12, and so
16 0n; 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.
1 The point is again that it is nature, and not we ourselves, that is responsible for this regularity. Cf. section 8 above and the note on section 14.
1 y:acyol: That is, as before (see on section 11), lines in the form of the letter chi. 16 is their common origin, as their meeting point, but they run beyond it in all four directions.
* These successive differences, that is, are nothing but the series of triples and therefore of course appear in the line above the quadruples in the table.
‘ Nicomachus here contrasts ¢@vois, ‘nature,’ and védyos, ‘law’ or ‘convention,’ a common topic of philosophy since the sophistic period. Cf. Burnet, Greek Philosophy, Part I, Thales to Plato (London, 1914), pp. 105 ff. See p. 120.
‘That is,3—2=1; 6—4=2; 9—6 = 3; etc. The differences in order are the natural series, which was the case with the differences in the series of doubles (cf. section 11).
‘In the series of sesquitertian ratios, 4: 3, 8:6, 12:9, 16:12, etc., the differences are, as before, I, 2, 3, 4, etc., so that ‘equal,’ feos, as applied here, means ‘ regular,’ i.e., ‘ regularly increasing.’
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 roo 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,
1 Ast, Theol. Arith., pp. 254–55, has in his note on this passage: “‘ Unitates sunt tres, prima 1, secunda 10, tertia 100. . . é» dsadoptoe in distractione, h.e., oppositione vel decussatione.”” As Iamblichus explains (Jn Nicom., p. 88, 24 ff. Pistelli), ‘monads of the second and third courses’ (uovds devrepwooupérn, Tprwdovunérn) are Pythagorean terms for 10 and 100. This designation depends on their belief that the first decade epitomizes all number, and the following numbers simply repeat, in a sense, the first ro. So in the Theol. Arith. (p. 59 Ast) we are informed that they called 10 Pan “because no number js naturally greater, but if any is so conceived it somehow circles about to it again in repetition; for the hundred is 10 decads, the thousand is 10 hundreds, and each of the others will come, taking the return path either to it or to one of the numbers up to it.” This notion is of necessity linked with the doctrine that 10 is the perfect number, and as the author (probably Nicomachus himself) says just before the words quoted, 10 exists as the epitome of all numbers in itself in order to offset and control unlimited multitude, to act as ‘“‘a measure for the whole and as it were a gnomon and a straight edge” in the hands of the creating deity. The units then form the first course, the tens the second, the hundreds the third, and so on, circling around 10 and its powers as the turning points of a race-course. Cf. also Nesselmann, Geschichte der Algebra, 1. Th., p. 239 (Berlin, 1842). The ‘ product of the first two,’ then, is 1 X 100 = 100, and the ‘square of the other’ is ro or 100.
2 Hoche here (p. 54,17 of the text) reads dmoo. It is hard to see what Amoco: ¢repophxes would be. It is better to read drlco:s with G, and two other MSS referring to wAevpais (or perhaps dvod«:s has dropped out of the text; dmodas Anco would be easily explicable and would balance lodxts feor). To illustrate the meaning of the passage it may be observed that the numbers ‘ flanking’ 4 are 2 and 6, which are respectively 1 X 2 and 2 X 3; 9g is flanked by 6, and 12 (2 X 3,3 X 4). Ingeneral m?*is flanked by (m—1)mand (m+ 1)m.’ On heteromecic numbers, see II. 17.
+ Thus 6 is the heteromecic number between 4 and 9 and 4 + 9 + (2 X 6) = 25 = 5%. The general formula for this proposition would be m* + (m+ 1)? + 2 m(m + 1) = (2m + 1)? Again, 6and 12 flank 9; now 6 + 12 + (2 X 9) = 36 = 6, or, in general, (m—1)m+(m+1)m + 27) = 4m’.
1g
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 sub-superparticular, there are in the greater division of inequality the superpartient, and in the less its opposite, the subsuperpartient.
It is the superpartient’ relation when a number contains within itself the whole of the number compared and in addition more than one part of it; and ‘more than one’ starts with 2 and goes on to all the numbers in succession. ‘ Thus the root-form of the superpartient is naturally the one which has in addition to the whole two parts of the number compared, and as a species* will be called superbipartient; after this the one with three parts besides the whole will be called supertripartient as a species; then comes the superquadripartient, the superquintipartient, and so forth.
The parts have their root and origin with the third, for it is impossible in this case to begin with the half. For if we assume that any number contains two halves of the compared number, besides the whole of it, we shall inadvertently be setting up a multiple instead of a superpartient, because each whole, plus two halves of it, added together makes double the original number. Thus it is most necessary to start with two thirds, then two fifths, two sevenths, and after these two ninths, following the advance of the odd numbers; for two quarters, for example, again are a half, two sixths a third, and thus again superparticulars will be produced instead of superpartients, which is not the problem laid before us nor in accord with the systematic construction of our science.
3 After the superpartient the subsuperpartient immediately ® is produced, whenever a number is completely contained in the one compared with it, and in addition several parts of it, 2, 3, 4, or 5, and so on.
Lam!
iN
! Defined by Theon of Smyrna, p. 78, 6 ff. Hiller.
2 See on I. 19. 2 (p. 215).
4 That is to say, given a superpartient the existence of a subsuperpartient naturally follows. For if 9 is a superpartient of 7, being 17 of it, then 7 is contained in 9 1 times and is a sub-superpartient of 9.
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 sub-superpartient, 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.

Table of the Superpartients

Root-forms 5 3 Ir 6 Iz 7 IolClC«O 22 12 26 I4 5 9 33–18 39 21 20 12 44 24 52 28 25 15 55 30 65 35 30 «18 66 36 78 42 35 2! 77 42 QI 49 40 24 88 48 104 56 45 27 99 54 117 63
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,’
1 The superpartient and subsuperpartient.
2 xadapods. . . repiocovs pdvovs. Here xadapds means ‘pure’ in the sense of ‘with no admixture from another class of terms,’ as in I. 22. 3, 4; IT. 27. 4.
4 That is, when a superpartient contains, besides the lesser number, two parts of the lesser number, it is understood that those parts are thirds, etc. Cf. Boethius, I. 28: Hoc guoque tviden-dum esi, quoniam, cum duae partes ex minore plus in matoribus sunt, tertit semper vocabulum sub-auditur, ut superbipartiens. . . . dicatur superbipartiens fertias. . .
‘ These terms represent the ratios, respectively, of 19, 13, 1$, 18 to 1.
=
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th
with five, ‘sixth,’ and so on, so that the order of nomenclature is some-thing 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-.
Now the multiple superparticular is a relation * in which the greater of the compared terms contains within itself the lesser term more than once and in addition some one part of it, whatever this may be.
As a compound, such a number is doubly diversified after the peculiarities of nomenclature of its components on either side; for inasmuch as the multiple superparticular is composed of the multiple and superparticular generically, it will have in its subdivisions according to species a sort of diversification and change of names proper both to the first part of the name and to the second. For instance, in the first part, that is, the multiple, it will have double, triple, quadruple, quintuple, and so forth, and in the second part, generically from the superparticular, its specific forms in due order, the sesquialter, sesquitertian, sesquiquartan, sesquiquintan, and so on, so that the combination will proceed in somewhat this order:
Double sesquialter, double sesquitertian, double sesquiquartan, double sesquiquintan, double sesquisextan, and analogously.
Beginning once more: triple sesquialter, triple sesquitertian, triple sesquiquartan, triple sesquiquintan.
! See I. 19. 2 and the note, on the terms ‘antecedent’ and ‘ consequent.’
* That is, just as submultiple superpartient corresponds to multiple superpartient, so submultiple superbipartient (a subclass) answers to multiple superbipartient, etc. * Theon of Smyrna, p. 78, 23 ff. Hiller, defines this ratio.
Again: quadruple sesquialter, quadruple sesquitertian, quadruple sesquiquartan, quadruple sesquiquintan.
Again: quintuple sesquialter, quintuple sesquitertian, quintuple sesquiquartan, quintuple sesquiquintan, and the forms analogous to these ad infinitum. Whatever number of times the greater contains the whole of the smaller, by this quantity the first part of the ratio of the terms joined together in the multiple superparticular is named; and whatever may be the factor, in addition to the whole several times contained, that is, in the greater term, from this is named the second kind of ratio of which the multiple superparticular is compounded.
Examples of it are these: 5 is the double sesquialter! of 2; 7 the 3 double sesquitertian of 3; 9 the double sesquiquartan of 4; 11 the double sesquiquintan of 5. You will furthermore always produce them in regular order, in this fashion, by comparing with the successive even and odd numbers from 2 the odd numbers, exclusively, from 5, first with first, second with second, third with third, and the others each with the one in the same position in the series. The successive terms beginning with 5 and differing by 5 will be without exception double sesqulalters of all the successive even numbers from 2 on, when terms in the same position in the series are compared; and beginning with 3, if all those with a difference of 3 be set forth, as 3, 6, 9, 12, 15, 18, 21, and in another series there be set forth those that differ by 7, to infinity, aS 7, 14, 21, 28, 35, 42, 49, and the greater be compared with the smaller, first to first, second to second, third to third, fourth to fourth, and so on, the second species will appear, the double sesquitertian, disposed in its proper order.
Then again, to take a fresh start, if the simple series of quadruples 4 be set forth, 4, 8, 12, 16, 20, 24, 28, 32, and then there be placed beside it in another series the successive numbers beginning with 9g, and increasing by 9, as 9, 18, 27, 36, 45, 54, we shall have revealed once more the multiple superparticular in a specific form, that is, the double sesquiquartan in its proper order; and any one who desires can contrive this to an unlimited extent.
The second kind begins with the triple sesquialter, such as 7: 2,14: 4, 5 and in general the numbers that advance by steps of 7 compared with the even numbers in order from 2. Then once more, 1o: 3 is the first 6 triple sesquitertian, 20:6 the second, and, in a word, the multiples of 10 in succession, compared with the successive triples. This indeed we can observe with greater exactitude and clearness in the table studied above, for in comparison with the first row the succeeding rows in order,! compared as whole rows, display the forms of the multiple in regular order up to infinity when they are all compared in each case to the same first row; and when each row is compared to all those above it, in succession, the second row being taken as our starting point, all the forms of the superparticular are produced in their proper order; and if we start with the third row,’ all of those beginning with the fifth that are odd in the series when they are compared with this same third row, and those following it, will show all the forms of the superpartient in proper order. In the case of the multiple superparticular, the comparisons will have a natural order of their own if we start with the second row and compare the terms from the fifth, first to first, second to second, third to third, and so on, and then the terms of the seventh row to the third, those of the ninth to the fourth, and follow the corresponding order as far as we are able to go.
1 Because it contains 2 twice, plus 1; 1e., is 24 X 2.
7 It is plain that here too the smaller terms have names corresponding to the larger ones, with the prefix sub-, according to the nomenclature given them all.
1 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,
2 besides. These parts ‘ are not halves, for the reasons mentioned above, but either thirds, fourths, or fifths, and so on.
3 From what has already been said it is not hard to conceive of the
1 Referring to the table in chapter 19, the successive rows of which are multiples of the first (since this is simply the multiplication table).
* That is, the comparisons are to be 5th row with the 3d, 7th with the 4th, oth with the sth, etc. Hence we will have:
5 = 3 = 45 =—p = 14 1 7 * oa; etc “. superbipartient; a Ol wio¥: ee ere 4 8 7a etc. = ; , supertripartient; 9 _ 18 27 ¢
I F: me = etc. = 7) superquadripartient, etc.
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> Theon’s definition is found p. 79, 15 &. Hiller. 4 See 20. 2 above.
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 4 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
! The principle about to be stated is that of the ‘three rules’ (Cantor, op. cil., vol. I, p. 431;
+
Nesselmann, op. cil., p. 198), by following which, starting from three equal terms, other sets of three in different ratios may be derived, and by the reversal of which any proportion in three terms may be reduced to the orginal equality. The present purpose is to show that equality is more elementary than any form of inequality as measured by ratios (cf. II. 1. 1), and it follows for Nicomachus as a Pythagorean that what is true of numbers is also true of the universe, and that ‘equality’ and ‘sameness’ are therefore elements and principles. The proposition was undoubtedly not original with Nicomachus, for its history can be traced back several centuries. In Theon of Smyrna (p. 107, 24 Hiller) it is given on the authority of Adrastus, a Peripatetic, whose date is stated in the Pauly-Wissowa encyclopedia to be the middle of the second century A.p.
E. Hiller (Rhein. Mus., vol. XVI, pp. 582 ff.) has shown that the book of Adrastus which Theon is probably quoting is his commentary on Plato’s Timaeus. It is further probable from the context of Theon that Eratosthenes (ca. 276–194 B.C.) knew the ‘three rules.’ He is there cited in these words: “‘So we shall take three magnitudes and the proportion residing in them and change the terms, and we shall show that all mathematics is made up of the proportion of quantities and that their source and element are the principle of the proportion” (AaBérres 84 rpla peydby xal rihy év robro dvadoylay xivicouer robs Spovs Kal delEouew Ore wdvra ra év Trois pwalhuaoiy €f dvahoylas woody rivwy ovyxeira: xal Eoriw avriw dpyh xal orocyeiow % THs dvadoylas pvcis), Another citation of Eratosthenes (Theon, p. 82, 22 ff.) informs us that the ‘principle (gers) of the proportion’ is ratio, which should be taken into consideration in connection with the statements above. Theon immediately adds, after the passage first cited, ‘But Eratosthenes says that he will omit the demonstrations’ (rds é¢ dwodeiteas 6 pew “EparocOérns pool wapahelvecr), and proceeds to give the ‘three rules’ as stated by Adrastus. Eratosthenes’s reference to ‘three magnitudes’ and ‘changing the terms,’ however, seems, especially in view of the context of Theon, to apply to nothing else than the ‘three rules,’ and it must be inferred from his own statement that he would ‘omit the demonstrations,’ that the latter were familiar to him. E. Hiller (Philologus, vol. XXX, pp. 60 ff.) has shown that this quotation of Eratosthenes is probably taken from his Il\arweexés, and that this, like the book of Adrastus, was a commentary on the Timaceus. ©Cr.. Lig.
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.
5 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.
6 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.
7 Let there be given us equal numbers in three terms, first, units, then two’s in another group of three, then three’s, next four’s, five’s, 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,
1 Whatever is absolutely indeterminate can never remain the same or even retain the same name even for an instant; for then it would be determined. Objects of this sort are to be looked for among the material things mentioned in I. 1.3. When they are impressed with form, they are no longer indeterminate but determined thereby; they remain like themselves and like their pattern, the ideal form, from one moment to the next, and can be called by the same name from time to time (‘partake of likeness and similarity of name’). As Nicomachus points out above (I. 1. 2—2.1) it is only by virtue of the form with which they are impressed and not of themselves that such things have any existence and appellation; both their being and their name are those of the forms and not their own.
? Aristotle gives sobriety (swppoatrn) as the mean in matters concerning pleasure and pain with licentiousness (d«oAacia) as the excess and no extreme to match it on the side of deficiency (Eth. Nic., 1107 b 4 ff.). Courage (dvdpela) he calls the mean between fear and recklessness, pé6fos and @dppy (ibid., 1107 a 33). Gentleness (wpaérys; the extremes dpy:Aérns, dopynola): see thid., 1108 a 4 ff.; self-control and patience (¢yxpdreu, xaprepla) are discussed together, ibid., 1145 b 8 ff.
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 witha 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,
1 xal edOds rd éwererrapepés (p. 66, 14 Hoche) is omitted by Codex G.
? As stated by Theon of Smyrna, p. 107, 24 ff., Adrastus thus formulated the rule: ‘Given three terms in any proportion, if three others be taken formed from these, the first equal to the first, the second equal to the sum of the first and second, and the third the sum of the first, twice the second and the third, those thus taken will again be proportional.” Algebraically this method obtains from a, ar, ar? the series, a, a(1 +r), a(1 +r). All of the remaining results of this chapter are included in this formula. The examples given by Theon start with three equal terms, as here.
* The results thus produced will be:
I I 1 equality
I 2 4 doubles
I 3 9 triples
I 4 16 quadruples
I 5 25 quintuples, etc.
Theon gives like results. ‘ Theon includes this process in his discussion; its results are as follows:
4 2 1, doubles reversed, giving 4 6 g, sesquialters 0 3 1, triples reversed, giving 9 12 16, sesquitertians 16 4 1, quadruples reversed, giving 16 20 25, sesquiquartans
25 5 I, quintuples reversed, giving 25 30 36, sesquiquintans, etc.
Io
a
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the superbipartient from the sesquialter,’ the supertripartient from the sesquitertian, the superquadripartient from the sesquiquartan, and 11 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 12 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. 13 The following must suffice as illustrations of all that has been said hitherto, the production of these numbers and their sequence, and the 14 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 sesquiquintan; for instance, from 25, 20, 16 comes either 25, 45, 81 or 16, 36, 81. 15 In the case of all these relations that are thus differentiated? and
1 Theon reports this matter as well. The results:
Reversed superparticulars: Resulting superpartients: 9 6 4 (sesquialter) 9 15 25 (superbipartient) 16 12 9g (sesquitertian) 16 28 49 (supertripartient) 25 20 16 (sesquiquartan) 25 45 81 (superquadripartient), etc. 2 This gives the following results: Superparticulars: Multiple superparticulars: 4 6 _—g (sesquialter) 4 10 25 (double sesquialter) 9 12 16 (sesquitertian) g 21 49 (double sesquitertian) 16 20 25 (sesquiquartan) 16 36 81 (double sesquiquartan). + What Nicomachus meant by racay. . . raw diafevy dew dy, ‘the contrasted ratios,’ is shown by Iamblichus’s commentary, which here has ray rhaccouévwy cxécewr, They are the pairs of 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 16 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 g, 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 oF 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.’
“|
ratios that may be derived from any given ratio by the application of the rules under discussion to the given ratio taken in direct and reversed order in turn, and it is because of the latter circumstance that they are called ‘contrasted’ (so Ast, Theol. Arith., p. 268, distunclis et inter se oppost-fis, una mimirum recta, allera conversa). In further illustration of the meaning the ratios mentioned by the author may be examined:
Original ratios: Direct order, 4 6 9g 9 12 16 16 20 25;
Reverse order, 9 6 4 16 12 9 25 20 16; Derived forms, 9 15 25 16 28 49 25 45 81; 4 10 25 9 21 49 16 36 81
Now whenever these derivative ratios are produced, (1) the last term, a square, is the same in each (25, 49, 81 in the scheme above); (2) the first term in the first derivative is the larger square of the original ratio, but in the second it is the smaller (‘it changes from the larger to the smaller’); (3) all the extreme terms are squares.
1 Certain of the MSS (See the critical note, p. 70, 15 Hoche) here add: “Moreover in all the given series the extremes are always squares; and the mean terms are derived from their sides multiplied together; and the first term of the generating ratio becomes the smaller term of the ratio generated. And in both the ratios generated the last and greater square is the same.” This material was used by Ast to reconstruct the text of section 15, which would then read much like the addition to the text just translated. In comparing the ratios given in the preceding note it may be observed that in 4, 6, 9, for example, the mean, 6, is 2 X 3 (the product of the sides of the squares 4 and g) and the same is true of the rest. Then again the first term of 9, 6, 4 is the smallest of the series 9, 15, 25 derived from it, while the first term of 4, 6, 9 is the smallest of the derivative series 4, 10, 25, and so with the rest.