Introduction to Arithmetic, Book I
The ancients, who under the leadership of Pythagoras first made 1 science systematic, defined philosophy ' as the love of wisdom. Indeed the name itself means this, and before Pythagoras all who had knowledge were called ‘wise’ indiscriminately—a carpenter, for example, a cobbler, a helmsman, and in a word anyone who was versed in any art or handicraft. Pythagoras, however, restricting the title so as to apply to the knowledge and comprehension of reality, and calling the knowledge of the truth in this the only wisdom, naturally designated the desire and pursuit of this knowledge philosophy, as being desire for wisdom.
He is more worthy of credence than those who have given other 2 definitions, since he makes clear the sense of the term and the thing defined. This ‘wisdom’ he defined as the knowledge, or science, of the truth in real things, conceiving ‘science’ to be a steadfast and firm apprehension of the underlying substance, and ‘real things’ to be those which continue uniformly and the same in the universe and never depart even briefly from their existence; these real things? would be things immaterial, by sharing in the substance of which everything else that exists under the same name and is so called is said to be ‘this particular thing,’ * and exists.
1In his introductory statements Nicomachus does not run counter to widespread beliefs of ancient times. The origin of the names‘ philosophy,’ ‘ philosopher’ (g¢:Aovogla, giddcogpos) was commonly ascribed to Pythagoras; compare the citations given by Ritter and Preller, Hist. Phil. Graec., 3, and (Plut.) Epit., I. 3. 8 (= Diels, Doxographt Graec., 280–281). As to the belief that Pythagoras corrected a wrong use of the terms, compare the following parallel with Nicomachus’s statements furnished by Ammonius (/n Porphyrii Isagogen Prooem., p.g, 7): ‘‘ Pythagoras, however, says, ‘ Philosophy is the love of wisdom,’ and he was the first to assail the error found among the ancients; for whereas they would call ‘ wise’ a man who pursued any art whatsoever. . . heshifted this epithet to God, so as to call him alone wise (God, I mean) and endowed with wisdom and knowledge of those things that are eternal” (6 uévrac Tlv@arydpas gyal, Schocodia dori g@itla codlas, xpwros Tw wapd Trois wahawrépos éwiords duaprjwari. drecdy yap éxetvon copdy wrbuatoy roy HeTivaody peridvra Téxrny. . . wellorqo: Thy wpogrryoplay Taudryy éwi rou Gedw we pbvoy éxetvov xadeioGar copdy, roy Gedy Gnu, codlay Te xal Thy Tov Svrwy didlwy Exorra yruoir).
2 See Part I, p. 92.
*+745¢ 7+: in Aristotle the technical expression for the particular thing of which being is predicated. The principles which Nicomachus calls Pythagorean are here expressed in Platonic and
For bodily, material things are, to be sure, forever involved
in continuous flow and change—in imitation of the nature and peculiar quality of that eternal matter and substance which has been from the beginning, and which was all changeable and variable! throughout. The bodiless things, however, of which we conceive in connection with or together with matter, such as qualities, quantities,? configurations, largeness, smallness, equality, relations, actuali-ties, dispositions, places, times, all those things, in a word, whereby the qualities found in each body are comprehended—all these are of themselves immovable and unchangeable, but accidentally they share in and partake of the affections of the body to which they belong.
4 Now it is with such things that ‘wisdom’ is particularly concerned, but accidentally also with things that share in them, that is, bodies.
Those things, however, are immaterial, eternal, without end, and it is their nature to persist ever the same and unchanging, abiding by their own essential being, and each one of them is called real in the proper sense. But what are involved in birth and destruction, growth and diminution, all kinds of change and participation, are seen to vary continually, and while they are called real things, by the same term as the former, so far as they partake of them, they are not actually real by their own nature; for they do not abide for even the shortest moment in the same condition, but are always passing over in all sorts of changes. To quote the words of Timaeus, in Plato,? “What is that which always is, and has no birth, and what is that which is always becoming but never is? The one is apprehended by the men-
Aristotelian terminology. Late Pythagoreans thus ascribed to their founder much that he never could have said.
1 Philoponus (scholia tn Nic., ed. Hoche) on this passage says that Ammonius criticized
Nicomachus for saying that matter is τρεπτὴ καὶ ἀλλοιωτή. ‘He ought to have said τρεπτικὴ καὶ ἀλλοιωτική, for the changes and variations take place about it; it itself does not change nor vary; for if it itself changed, there would have to be still another matter wherein it would vary and change. And so it is itself unchanging and unvarying, but its forms vary; I mean quantities, qualities, ...’’ Philoponus retorts that when change and variation take place, it really is the substrate which we say changes; the forms (qualities, etc.) predicated of it do not change; they pass away and come into being. It is to be noted that, as in the case of Plato, the question of ‘ primary’ and ‘secondary’ matter can be raised in connection with Nicomachus’s doctrines; see p. 93.
2 See Part I, p. 94.
4 Timaeus, 27D. Nicomachus closely follows the original, with only minor variations. His quotations of Plato are not usually so exact; cf. I. 3. 5, 7.
tal processes, with reasoning, and is ever the same; the other can be guessed at by opinion in company with unreasoning sense, a thing which becomes and passes away, but never really is.”’
Therefore, if we crave for the goal that is worthy and fitting for man, 3 namely, happiness of life’—and this is accomplished by philosophy alone and by nothing else, and philosophy, as I said, means for us desire for wisdom, and wisdom the science of the truth in things, and of things some are properly so called, others merely share the name—it is reasonable and most necessary to distinguish and systematize the accidental qualities of things.
Things, then, both those properly so called and those that simply 4 have the name, are some of them unified and continuous, for example, an animal, the universe, a tree, and the like, which are properly and peculiarly called ‘magnitudes’ ;? others are discontinuous, in a side-by-side arrangement, and, as it were, in heaps, which are called ‘multitudes,’ a flock, for instance, a people, a heap, a chorus, and the like.
Wisdom, then, must be considered to be the knowledge of these 5 two forms. Since, however, all multitude and magnitude® are by their own nature of necessity infinite—for multitude starts from a definite root and never ceases increasing; and magnitude, when division beginning with a limited whole is carried on, cannot bring the dividing process to an end, but proceeds therefore to infinity—and since sciences‘ are always sciences of limited things, and never of infinites, it is accordingly evident that a science dealing either with
1 The word used by Nicomachus, edfwta, is once employed by Aristotle in the Ethica Nicomachea, I. 8. 1098 b 20 ff.: cuvddec 5¢ rw Ady Kal Td ed Civ Kal 7d eb wodrrew Tov eddaluora’ aoxeddv yap eitwla ris efpyrat xal edwpatia. The ‘happy life,’ in a certain definite sense, is the goal that is becoming a man, according to Anstotle. See zbid., I. 10. 1101 a 14 ff., I. 6. 10g7 b 25—1098 a 20.
2 On this and the following definitions, and their parallels in Aristotle, see p. 112 and notes.
7 That is, multitude and magnitude per se and unqualified. This point is noted by Proclus, In Prim. Eucl. Elem. Lib. Comm., p. 6, 15 Friedl.: & re yap dp:wuds dwd povddos dptdyevos dwaverov xe: rhyv avénocw, del 52 6 AndOels wewépacra, cal } Troy peyeOw dialperis ew Awerpor xwpet, Ta Se Siacpotuera wdera Gora, cal car dvepyelay wewépacra: ta pdpta rod Shov. In the Theologumena Artihmeticae, p. 3 Ast, also there is reference to this matter in the same terminology: ‘‘ And it [sc. the monad] is evidently beginning, middle, and end of all, since it bounds the infinite division of the continuous in the direction of the smaller than itself, and in the direction of the greater it cuts off a similar increase in the discrete, and this not by our decree, but by that divine nature.” This passage is probably Nicomachean. Hero of Alexandria (Definition 119, in Hultsch’s Heronis Alexandrint Geometricorum et Stereometricorum Reliquiae, Berlin, 1864, p. 33) speaks of magnitude as “‘that which is increased and divided to infinity” (yéye@és dor: rd altardueroy cal reuvdueror els dweipor).
‘The matter included in the rest of this section is touched on by Proclus, op. cif., p. 36. 3 Friedl.: ériexoweity 3° ad rd wnXlxoy xal woody ofre péyePos dwdas ofre rXHOos GXAA Td Kad magnitude, per se, or with multitude, per se, could never be formulated, for each of them is limitless in itself, multitude in the direction of the more, and magnitude in the direction of the less. A science, however, would arise to deal with something separated from each of them, with quantity, set off from multitude, and size, set off from magnitude.
1 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.’
2 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.
3 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 wis-éxdrepoy woiruévoy* rotro yap ddedovoas roy dwelpwy ras érmiortuas Karavoeiv, ws obx évdy Thr wad éxdrepow dweplay yrwoe: wepidaBety. This hestates with regard to Pythagorean mathematics, and in view of the agreement of terminology to be seen here and in the passage cited in the preceding note, it is probable that he used the work of Nicomachus as his authority.
1 Nicomachus thus subdivides the subject matter and assigns the special fields of the four
mathematical sciences: I, treating number (rd woody) (1) as such, absolutely («xa#’ davré), Arithmetic; and (2) relative number (xpds 4\Ao), Music; II, treating quantity (rd xmXxov) (1) at rest, Geometry; (2) in motion, Astronomy (σφαιρική). Proclus, op. cil., Prol., p. 35. 21 ff., Friedl., gives the same division of the field of the mathematical sciences, using the same terms, in his report of the Pythagorean mathematics, probably drawing upon this work. It is to be noted that Nicomachus does not in fact adhere strictly to his classification, for he treats in this work of relative number, which falls in the domain of Music, and in the discussion of linear, plane and solid numbers he comes close to Geometry. The classification of Theon of Smyrna (cf. Part I, p. 113, n. 4) includes Music (i.e., the mathematical consideration of harmony) under Arithmetic and avoids this inconsistency.
? To illustrate what is meant by relative things Aristotle uses the example of double and half (Met., IV. 15. 1020 b 26).
dom,” says the Pythagorean Androcydes.’ Likewise Archytas of 4 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 5
! He is mentioned as a writer On Symbols (wept cvpBddwv) in the Theol. Arith., p. 40 Ast.
2 The passage here quoted is found also, at greater length, in Porphyry, In Plolem. Harm., p. 236; a few words of the same passage appear in Iamblichus, In Nicomachi Ariithmeticam Introductionem Liber, p.g, 1 Pistelli; cf. Diels, Die Fragmente der Vorsokraitker, vol. I’, pp. 330 ff.,and Blass, in Mélanges Graux, pp. 574 & On Archytas, and the problem of the title of this work, see pp. 20 ff.
* The Epinomis, from which Nicomachus here quotes 991 D ff., is now recognized as not genuinely Platonic. Nicomachus doubtless cited the passage from memory, for he does not give it exactly; but he can hardly have distorted it as Hoche would make him. The portion of Hoche’s text that seems to demand correction is gavfoera: 5 av & KXéyouer dpGis, ef res els Evy BASrwy wdera parédver. This reading introduces a future indicative with dy, a construction not elsewhere found in Nicomachus, and one which a writer of his class would not be likely to use, although it occurs in late Greek. Certainly he would not thrust such a palpable modernism into a quotation of Plato. Hoche perhaps has not reported the manuscripts correctly on this passage; his apparatus sometimes is faulty (see Hultsch, in Neue Jahrbiicher fiir Philologie und Pddagogik, vol. XCVII, 1868, pp. 762 ff.). I have not had access to the manuscripts, but I note that C. F. A. Nobbe (Specimen Ariihmeticae Nicomacheae, Leipzig, 1828, p. 19), who used the Niirnburg and Wolfenbiittel manuscripts (N and I), omits ef here, and that Hoche does not report the fact. Now this word ef is the cause of all the difficulty with the text. It did not occur in the original, but it could easily have been inserted by a scribe who misunderstood the preceding dv. The familiarity of the phrase ef res would be another motive; ép@os res, on the other hand, is unusual. The introduction of ed, also, might bring in pav@dve: for pardeyp. I therefore read garhceras & dy, & Aéyouey, dpOds res els by BXéxwy wavra pavOdey. The change of the last word is supported by Nobbe’s manuscripts, but is not so necessary as the omission of ed.
The original passage reads: “‘ Every geometrical figure, system of numbers, composition of harmony and the regularity in the revolution of the stars must appear to one, who properly learns, to be one in principle in all cases; and it will so appear, if, as we say, one learns correctly, looking to one thing. For it will appear to them on consideration that there naturally exists one bond of all these things; and if any one will pursue these matters in another way, he must call to his aid Fortune, as we say. For without these sciences there will never arise a fortunate being in the cities; but this is the mode, this the nurture, these the lessons, whether hard or easy; this way must one proceed, and to neglect them is impious before the gods. ... And the man that has apprehended all this in this way, this man I call most truly wise, and so I also maintain both in jest and in earnest”’ (way didypappa dpifuot re ctornua nal dpyovlas cictacw dracav rijs re TaY doTpiby wepimopds Thy duodoylay ofcay ulay ardvrwy dvadaryvac def Tw KaTd Tpdror avd vorri, dvadaricera: dé, dv, & A€youer, dpOas Tis els bv BrXerwy parOdry decuds yap weducws wdvrwy roérwy els dvadarioerat dtavoovpévas: ef 6 AdAws wws raira perayepetral ris, rixny det xaheiv, Gowep xal A€youer. ob yap Avev ye robrTwy uhroré ris év wher eldaluwy yérnrac piais, some give the title Te 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 any one 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
6 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.
7 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
ANN’ obros 6 rpéwos, aury W Tpodh, Tatra rd padtpara, efre yadewd efre Addia, ravry wopevréor:
dpedijoa 8¢ ob Oeuirdy dors Gedy. . . roy 5¢ Eduwapyra Tatra ovrws efknpéra, Tovroy Aéyw Tor ddnOécrara codwraroy, éy xal ducyupltoya: walftwr cal owovidiwr dua .. .).
Theon of Smyrna quotes the first part of this passage, p. 84, 7 ff. Hiller, and the last part, p. 2, 16 ff.
1 A reference to the voir as the highest part of the soul in accordance with the ancient view that the soul is made up of parts. 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.
! The original passage (Republic, 527 D ff.) reads: ‘“‘‘ You amuse me,’ said I, ‘ because you are like one who fears the crowd lest you seem to enjoin useless studies. It is, however, not at all a trifling matter, but a difficult one to believe that in these studies some instrument of every man’s soul is cleansed and rekindled, which was being destroyed and blinded by his other pursuits, a thing more worthy to save than countless eyes: for by it alone is truth beheld.’” (780s el, 4p 6 éyw, dri Fouwas Sedidri rods woddovs, wh Soxgs Aypyora padsuara wpocrarrey. 7d 3 toriv ob wdvu davdow d\Ad yarewdr wmigretcae Sri dv rovras rots padhuaciw dxdorov bpyardy re Puyis éxxa@alperal re xal dvatwwupeirat dwohAvuevor wal truddotperoy twd Tay Addu éwirydeupdrwy, Kpetrroy ov owbijra: puplwy dupdrwr’ poyy yap airy dd\jGea dpara:). Theon of Smyrna (p. 3, 8 ff. Hiller) quotes this passage.
* This group of studies, music, arithmetic, geometry, and astronomy, make up the ‘quadrivium,’ or, as Boethius, who apparently first used the term, called it, ‘quadruvium.’ ‘ Trivium,’ to designate the study of grammar, rhetoric, and dialectic, may also go back to his time. See Gow, op. ci., p. 72, note.
+ Plato also said that arithmetic should be first learned and that it is the basis of all other arts. Rep., §22 C: olov rovro rd xowdy, Gp waoa wpooxphyrac Téyvar Te wal didvowas xal dmiorhHua, 8 cal wavrl évy wowros dedyxy pavOdvev, woiov; fn. 1d haddor rovro, Fv 5 éyw, Td Ey re xal
Ta dvo wal rd rpla diayiyywower, Adyw 6¢ avrd éy xepadaly dpwWyudr re wal Aoyirudy. Ff ovyx ourw wepl rovrwy Eye, ws wdca tréyyn te xal dmiorhun dvayadtera: avrav péroxos yiyverGac; It is interesting to observe parallels to many of the topics of this chapter in Caxton’s Mfirrour of the World (Publications of the Early English Text Society, extra series, vol. CX, pp. 36–37): ‘The fourth scyence is called arsmetrique. This science cometh after rethoryque, ande is sette in the myddle of the vii sciences. And without her may none of the vii sciences parfyghtly ne weel and entierly be knowen. Wherfor it is expedyent that it be weel knowen and conned; ffor alle the sciences take of it their substaunce in suche wise that without her they may not be. And for this reson was she sette in the myddle of the vii sciences, and there holdeth her nombre; ffor fro her procede alle maners of nombres, and in alle thynges renne, come and goo. And no thyng is without nombre. But fewe perceyue how this may be, but yf he haue be maistre of the vii artes so longe that he can truly save the troughe.”
‘Cf. below II. 22. 3. Nicomachus of course refers merely to abolishment in thought. Arithmetic, since it treats of numbers and numerical relations fundamental to the other sciences, is logically prior to them, and if it did not exist they could not exist. Nicomachus uses both wpo-
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.
3 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.
4 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 t 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’
5 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.
yevéorepos and wpérepos in this passage with the meaning ‘ prior.’ Aristotle uses the term wpérepos and his logic forms the basis for Nicomachus’s present argument. Forinstance, in Met., 1o19 a1 ff., after discussing several forms of wpérepa and worepa, he has the following: 7a pévy 3% ovrw Aévyera: wpérepa xal vorepa, ra 8¢ ward piow Kal obelar, dca évddyerac elva: Avev Adrwr, éxeiva 3 dvev éxelvwr ut = 7) Gtaupdoe: éxpijro TAdrwy. He also uses the verb curarapeto@a (‘ abolish’) as does Nicomachus: e.g., Met., 1059 b 39: 9 8¢ cvvavaipetra: trois yéveot ra efSy, Ta yéwy rais dpyais foixe waddov’ dpxh yap Td cvvarvaipoiv. Cf. also Iamblichus Jn Nicom., p. 10, 2 Pistelli.
1This form of argument also is Aristotelian, e.g., Top., VI. 6.17: éwigdpe: yap éxdory rap
Siadopay Td olxeioy yévos, xaGdrep 74 wetdy cal 7d Slwour Td (por cuvemigépe:, Logically exhausted it stands in Top., II. 4. 111 a 25 ff. thus: ob yap dvayxaioy bea rp yévec dwdpye: xal ry elder brdpyer’ Boa 8¢ rq efSe: bwdpyer dvayeaiov xal ry yéver: doa yap ry “yéves ovx drdpxe, olde ry efSe- Soa be ry elSec py Uwdpye, ovx dvayxaioy Ty yéve: uh Uwdpxeiv.
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 didiapason, 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 3 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 preéxistent * 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 har-
1 Plato in Rep., 528 A—B points out that it is a mistake to study bodies (oreped) in motion before studying them per se (abré xaé’ airé).
? The music of the spheres. Boethius, 1.1, paraphrases: quod ermonicis modulationibus molus fpse celebratur aslrorum.
1 This chapter, with I. 4. 2, gives the fullest information we have about Nicomachus’s theories of cosmogony. See Part I, p. 107.
4 This is the eternal number, to be distinguished from the‘ scientific number’ mentioned in the next section. Cf. Part I, p. 98.
3
moniously 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.
4 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.
1 Number‘ is limited multitude or a combination of units or a flow 5 of quantity made up of units; and the first division of number is even and odd.
2 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.
3 Now this is the definition after the ordinary conception; by the Pythagorean doctrine, however, the even number is that which ad-mits of division into the greatest and the smallest parts at the same operation, greatest in size and smallest in quantity, in accordance
1 Note that the definition of harmony quoted from Philolaus in II, 19. 1 implies as much, and compare Part I, pp. 100, 120, on the general subject of harmony in the numerical system.
? That is, they are elementary, for they are formed by the two elements of number, the monad and dyad respectively, and embody by reason of this origin ‘ sameness’ and ‘ otherness,’ the fundamental cosmic forces. Ci. Part I, p. go.
*Cf. Theon of Smyrna, p. 23. 3 Hiller: “And the even and the odd numbers alternate, being observed in alternate places” (¢paAAdt & eloly dAAAAcis of re Apri: cal olf wrepirrol wap fra Gewpoveror).
‘Cf. Part I, p. 114, on these definitions.
* The translation ‘ flow’ for xéua is that adopted by Heath (Euclid, IT. 280), whose note on definitions of number may be consulted, xéua elsewhere in Nicomachus is used to mean ‘ series’ (see the Glossary), but probably the metaphorical idea is present even in those cases.
® Theon of Smyrna, p. 21, 22 Hiller: wat Apriot péw eloiv ol dwidexdperot thy els foa Sialpeciv
_. wepiogol dé ol els Avia dcarpobuerot, etA. Euclid defines thus: “An even number is one that is halved” (dprios dpiOuds doriv 4 Slya Scacpobueror), Elements, VII, Def. 6. 7See Part I, p. 122 with note 1.
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 4 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 5 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.
1 That is, halves are the greatest possible parts of a term in magnitude; and there is a smaller number of them than of any other fractional part. Thus greater magnitude of factors is associated with a smaller number of them; this is the ‘ natural contrariety ’ of magnitude and quantity. Cf. Boethius, I. 4, for a discussion of this notion, and Iamblichus, p. 12, 3 ff. Pistelli. This principle may be illustrated by what was called the “lambdoid diagram” (see Part I, p. 127) from its likeness to the Greek lambda, A. This diagram sets forth in the form of the letter lambda, converging at unity, the series of natural numbers and the series of fractions, thus:
5 t
and so on. It will be noted that the corresponding integers and fractions show the ‘ natural contrariety’ referred to. The diagram occurs in Iamblichus’s commentary (p. 14, 3 ff. Pistelli) following on the discussion of these definitions, and it is referred to in Theol. Arith., p. 3 (bottom) Ast.
? When an even number is divided into two parts, whether equal or unequal, these parts are always either both odd or both even (‘ only one species of number,’ as Nicomachus says). Iamblichus, p. 12, 14 ff. Pistelli. See Heath, History, vol. I, p. 70.
3 Tamblichus (p. 13, 7 ff. Pistelli) notes that the monad is distinguished from all the odd numbers by not even admitting division into unequal parts, and the dyad from the even numbers by admitting division into equal parts only. Theon does not notice this property of the dyad, but discusses at some length the question whether the monad Is odd or even (p. 21, 24 ff. Hiller). On Nicomachus’s doctrine, that the monad and dyad are both elements of number and not themselves numbers but its ‘ beginnings,’ compare Part I, pp. 116 ff.
‘If an odd number is divided into two parts these will always be unequal and one odd, the other even (‘ the two species of number’),
1 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
2 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.
3 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.
4 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
5 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.
6 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;
1 Thus 5s is half the sum of 4 + 6, 3 +7,2+8, etc. For a typically Pythagorean application of this principle cf. Theol. Arith., p. 28 f. Ast.
? Euclid, among the definitions of Elem., VII, defines the even-times even, even-times odd, odd-times even and odd-times odd (the latter is “one which is measured by an odd number an odd number of times’’). Nicomachus confines himself to a tripartite division of the even only; Euclid’s classification applies to all numbers. The ‘odd-times odd’ of Euclid is not found in Nicomachus’s Introduction at all, and in defining the three classes given by both Nicomachus and Euclid the former uses somewhat different formulas, which are consistently praised by Iamblichus in his commentary. (See the notes on I. 8. 7, above p. 127, and Nesselmann, p. 192.) Theon (p. 25, 5 ff. Hiller) gives the same classification as Nicomachus here, and like him refers to even numbers alone. His definitions are compared to those of Nicomachus in the following notes. It may be noted that ‘odd-times odd’ occurs in Theon as another name for the prime number
(p. 23, 14 Hiller). See Heath, History, vol. I, pp. 70 ff., on the classification of numbers.
* Theon of Smyma, p. 25, 7 ff. Hiller, gives the definition of the even-times even substantially as follows: It is a number that has three characteristics: (1) It is produced by the multiplication of two even numbers; (2) all its parts are even, down to unity; (3) none of its parts has its designation in terms of an odd number. Euclid’s definition is: ‘The even-times even number is that which is measured by an even number an even number of times” (dptidacs Aprios dpcOpyds
éoriv 6 bord dprlou dpi@uod perpovuevos xara Apruv dpiOudv), Elements, VII, Def. 8. ‘*Tts genus’ is the even; cf. I. 7.2. So too Philoponus notes (ed. Hoche, p. 15).
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.*
1 The specific things meant by ‘neither of these’ (éxdrepor rodrwr) are the ‘name’ (xporryoplay, Pp. 15,17 Hoche, or better Sua implied in dpridxts dpridvuuor, ibid.) and the ‘value’ (d¢rapus implied in p. 15, 18, dpridxis dpriodtvayor) of any part of the even-times even number. These “never share in another variety”; i1.e., another variety of number, or of even numbers, than the even-times even. They are called halves, fourths, eighths, etc. (even-times even names), and their values are always 2, 4, 8, 16, etc. (even-times even numbers). On the use of S¢vayus and dpriodévapos here, cf. on I. 8. 7.
? Philoponus writes the following scholium upon this: ‘‘Here then Euclid is convicted of making a poor definition of the even-times even number in his Seventh Book; for he says that an even-times even number is one that is measured by an even number an even number of times (4 bra dpriov dpiduot per povweros xara Apriovy dpifudy). For by this definition the merely even numbers also that are not even-times even, will be found to be even-times even; e.g., 24 15 not even-times even, for it is not subdivided to the monad; but according to Euclid it will be found to be even-times even; for, look you, it is measured by 4, an even number, an even number of times, 6; for 4 X 6 = 24. So his was a bad definition.”’
2 Cf. the note on I. 8. 6 for the meaning of this statement. The word translated ‘value’ is dvvayus, which as a mathematical term usually means ‘square’ or ‘square root,’ but in non-tech-nical Greek may bear the meaning assigned (e.g., Thuc., VI, 46, 3). The word is again similarly employed in section 10 of this chapter and in I. 9. 2; 10. §; besides which the phrase dpredxis dprioduvayos seems to have the corresponding sense of “‘even-times even in value” (I. 8. 6). This interpretation has the support of Boethius, while Philoponus understands the passage differently. Boethius, 1. 9, says, Sed ideo mihi videtur hic numerus pariler par vocatus, guod cius omnes parles et nomine ef quantiiale pares pariler inveniantur. ‘Quantity,’ however, does not represent Svvauis as well as ‘value,’ since it is strictly a case of number rather than of quantity. T,L. Heath understands évvajus in the sense proposed, and commenting on this passage of Nicomachus (Euclid, IT, 282) says: “‘He says... that any part, i.e.,any submultiple, of an even-times even number is called by an even-times even designation, while it also has an even-times even value .. . when ex- pressed as so many actual units. That is, the—th part of 2” (where m is less than 7) is called after 37m the even-times even number 2”, while its actual value (6¢vayes) in units is 2”~”, which is also an even-times even number.”
On the other hand Philoponus (schol. 56 on I. 8. 6,p. 15 Hoche) says: “Since he does not employ the ordinary language of the usage of most people, I think it reasonable first to impart the meaning of the various terms and then to interpret the whole sense of what is said here. Now he calls duvduecs the numbers from which the parts of any number take their names, e.g., the half from 2, etc.... Nowhe says that all the parts of the even-times even are themselves even-times even in name (dpridats dpriweupa), and the duvdwes, from which their names are taken, are even-times even powered (dpridais dprioddvanor). For example, the even-times even number 16 has as its second part 8, 4 as its fourth, 2 as its eighth; each of these parts is even-times even. ... So reasonably the parts are called even-times even in name because they take their names from even-times even numbers. . . . And the ‘powers,’ from which the parts are names, are even-times even powered (dpriduts dprwddvayo). .. . Similarly the expression ‘and in the number of their monads dpridxis dprioddvayow’; for 4 of 16, named from 4, has the number of monads of 4, from which it is named, dpriducs dpriodtvayor, So the expression ‘in the number of monads in it’ is to be taken either as applying to the designation of the fraction (uéprov) or as applying to the part (uépos) itself” (but he immediately states that he prefers the former explanation). What Philo-
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:
9o As you proceed from unity, as from a root, by the double ratio to
Io
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.
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 ponus understands by dprideis dpriodtvayos may be seen from his statement in this same scholium: ‘And 16 would be dpridecs dprioddvayos if from it there should be named some part of an even-times even number.”
It is quite tempting to consider Sérajus the quotient of a factor when the latter is divided into the number under consideration, but the rendering adopted seems to fit the context better, while on the other hand Philoponus himself virtually confesses that his interpretation of S¢rayis does not hold in I. 9. 2 (To take a single example, the number 18, etc.), when he says, “‘If he meant ddvauis to be the number from which the part gets its name, and the half gets its name from 2, how is it that he says here that the half is odd in S¢vayus? We must therefore read with hyperbaton, i.e., ‘ the half, 9, evenly named in dévapcs (that is, from 2, for the half is the second part), is odd in its number, 9.’”” That is, in the text, rd per fuuocv dpriaxas wropacpudvor brdpyet & wepurody TH duvdpyec, he proposes to connect rp Surduer with dvopagyévor, a hyperbaton too violent to be probable. In I. 9. 2 S¢vapus seems certainly to mean ‘value,’ and it is hardly possible that it would be used in different senses in such similar contexts. A further question suggested by this passage is whether Nicomachus would class the monad as an even number, as, strictly speaking, is implied here. In the Theol. Arith., p. 3 Ast, we find the statement that the monad “embraces all things in potentiality,” and among the specific statements following it is said that “ it is even and odd and even-times odd.” The monad was called Male-Female, and this designation is perhaps connected with the notion that it is both even and odd. See Part I and on section 13 below.
1 Reading ¢&s (found in several MSS) for Hoche’s ¢§ évés.
1 Boethius, I.9: Jilud atdem non minima consideratione dignum esi, quod cius omnis pars ab una parte quacumgue, quae intra ipsum numerum est, denominalur tanlamque summam quantilalis includit quota pars est alier numerus pariter parts illius, qut eum continel, quantilatis. Itaque fil, ut sibi partes ipsae respondcant, ul quota pars una esi, lantam habeai altera quantitatem, ef quota pars tsta est, fantam in priore summa necesse sit multiludinis invenirt. ‘This correspondence of factors is thus graphically shown in one of the MSS:
1 y, 4 8 16 32 64 128
and with an odd number of terms:
1 2 4 8 16 32 64 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
1 wapdrupor .. . drt ray évrds abrov. érrds in this passage means ‘up to,’ as, e.g., also in I. 16. 3, and reference is made to the series of even-times even numbers given above. If any number of this series be selected (e.g., 64), its factors will all be numbers that have occurred in the series before 64; each factor contains as many units as one of these numbers (e.g., the fourth part contains 16 units), receives a name, as a factor, from one of them (e.g., 16 is called fourth from
4) and reciprocally gives a name to another factor (e.g., 16 gives the name sixteenth to 4), xara dvriweploraciy xal duo fhy. It will facilitate the understanding of this and the following sections to note that when Nicomachus selects for his demonstration any specific even-times even number he considers it as the end of a series; e.g., 64 must be thought of as the series 1, 2, 4, 8, 16, 32, 64.
* That is, in accordance with the definition of ‘odd.’ A unit always ‘intervenes’ to prevent the division of the odd number into halves; see I. 7. 2.
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
13 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.
14 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 rl
1 Substituting in the formula S = Be a
Fr
— the sum of n terms of this series is 2ⁿ − 1. The
—1
(n + 1)st term is 2ⁿ, and 2ⁿ − 1 is one less, as Nicomachus says.
? See chapter 16.
3 Here treated as a member of the even-times even series. Cf. on I. 8. 7.
‘Theon of Smyrna, p. 25, 19 ff. Hiller, thus defines: ‘‘‘The even-times odd numbers are measured by the dyad and some odd number; upon division into equal halves, they always have odd halves” (dpriowépuroo: Sé eloww of bwd Suddos xai wepoco’ obrivocoty perpotpern, ofrives éx warTds wepiood pépy Exove: Ta Hyuloea card Thy els [oa dtalpeoiv). Euclid (Elements, VII, Def. 9) has ‘‘ An even-times odd number is one that is measured by an even number an odd number of times” (dpridxcs 38¢ wepioods doriv db wd dpriov dp@yod perpovperos card wepiocdr dpiOpdr).
* Read el€:xGs, with Codd. Cizensis, Monacensis 482 and Hamburgensis, instead of la«cas (Hoche, following G). The word is evidently to be contrasted with rp yé»e and both are logical terms.
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 2 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 3 is, because, although it is even, its halves are at once odd.
This number is produced from the series beginning with unity, with 4 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 5
1 For 3¢»auis in the sense of “numerical value’ (i.e., here, odd or even), cf. I. 8. 7 and the note, as well as Boethius, I. 10: Accidtt autem his quod omnes partes contrarie denominatas habent, quam sunt quantitales ipsarum partium quae denominantur. Neque unquam fieri potest ut quaelibet pars huius numeri elusdem generis denominationem quantitatemque susci pial.
2 It seems necessary to take the dative (r@ . . . dvéuare) with duoryery, and in that case the reading of C (€avrod) is to be preferred to the adr of G, read by Hoche. Nicomachus is pointing out that in this class of numbers no even factor can have an even name. Hoche’s text would be translated “‘by virtue of the same name,” which is not clear.
4 -yvduoves: This term is frequently used in arithmetic with reference to the odd numbers which are added together to make the series of square numbers, because when aes Tepeo”
sented they may be successively affixed to the es pace previous figure in the characteristic form of the gnomon (see the Figure). Nicomachus, however, also employs the term with reference to any set or series of numbers that may be successively added to form a second set or series. Thus the gnomons of the hexagonal numbers are the terms from the natural series anneal siti from 1 with a common difference of 4. (See II. rz. 1 ff. and II. 13. 4 We alan find the ee used in this wider sense by Theon of Smyrna (e.g., p. 37, 11 Hiller) and by Hero of Alexandria (Def.
59, p. 21 Hultsch). For a discussion of the origin and use of the term see Cantor, o. cit., vol. I, pp. 161 ff.
TE
|
ho
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 par-ticular 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.
The odd-times even number is the one which displays the third form of the even, belonging in common to both the previously mentioned species like a single mean between two extremes, for in one respect it resembles the even-times even, and in another the even-times odd, and that property wherein it varies from the one it shares with the other, and by that property which it shares with the one it differs from the other.
The odd-times even number ® is an even number which can be
1 That is, in the Greek manner counting in both the term from which one starts and the last. So the Olympic games, which we would say came every fourth year, were to the Greeks a ‘ fifth-year festival.’
2 E.g., 10 can be divided into two parts (5, 5), but neither is divisible by 2; while in the even-times even series each number can be divided, and its parts divided, down to the ‘ least part,’
1. ‘The greater extreme,’ speaking of the even-times odd, would be the even-times odd number itself under consideration.
? Cf. I. 8. 10, where the reciprocal relation of the factors of the even-times even numbers was treated. Each factor of such a number is itself a term in the even-times even series, of which 1 and the given number are regarded as the extremes (dxpéryres). For the relation between the product of the extremes and the square of the mean (or the product of two means), cf. I. 8. 14.
4 Thus in the even-times odd series
6, 10, 14, 18, 22, the mean term (14) is 4 the sum of the extremes (6 + 22 = 28); and in the series 6, 10, 14, 18, 22, 26, the sum of the means (14 + 18 = 32) equals the sum of the extremes (6 + 26 = 32).
§ Theon of Smyrna, p. 26, 5 Hiller, defines the odd-times even as a number produced by the multiplication of an odd by an even number, which has even halves when it is divided by 2, but on further division has some parts odd, others even (wepiooduis 5¢ Apriol elow, Gy d wohdkawha-g.aguds éx duety @vrivwrotv weptogod Kal dprlov ylvera:, mal woAAawhaciacGévres els [oa pew Apria divided into two equal parts, whose parts also can so be divided, and sometimes even the parts of its parts, but it cannot carry the division of its parts as far as unity. Such numbers are 24, 28, 40; for each of these has its own half and indeed the half of its half, and sometimes one is found among them that will allow the halving to be carried even farther among its parts. There is none, however, that will have its parts divisible into halves as far as the naturally indivisible unit.
Now in admitting more than one division, the odd-times even is 3 like the even-times even and unlike the even-times odd; but in that its subdivision never ends with unity, it is like the even-times odd and unlike the even-times even.
It alone has at once the proper qualities of each of the former two,’ 4 and then again properties which belong to neither of them; for of them one had only the highest term divisible, and the other only the smallest indivisible, but this neither; for it is observed to have more divisions than one in the greater term, and more than one indivisible in the lesser.
Furthermore, there are in it certain parts whose names are not 5 opposed to their values nor of the opposite genus,” after the fashion of the even-times even; and there are also always other parts of a name opposite and contrary in kind to their values, after the fashion of the even-times odd. For example, in 24, there are parts not opposed in name to their values, the fourth part, 6, the half, 12, the sixth, 4, and the twelfth, 2; but the third part, 8, the eighth, 3, and the twenty-fourth, 1, are opposed; and so it is with the rest.
This number is produced by a somewhat complicated method, 6 and shows, after a fashion, even in its manner of production, that it is a mixture of both other kinds. For whereas the even-times even is made from even numbers, the doubles from unity to infinity, and the even-times odd from the odd numbers from 3, progressing to infinity, this must be woven together out of both classes, as being common to péon dla diaipodvra:, card Se rds wrelovs Siaipéoas A pev Apria pedpyn, & Se wepiooda Eyovory). Euclid’s definition (Elements, VII, Def. 10) is: ‘‘The odd-times even number is one that is meas-ured by an odd number an even number of times” (repioodacs dorids doriv 6 bd wepiogod dpiPuod per povperos xaTd Aprior apiludr).
‘Cf. I. g. 6. It is to be observed that Nicomachus in speaking of these numbers conceives of them serially; e.g., to him the even-times even number 16 carries with it the series 1, 2, 4, 8, 16; and so of the others; as 3, 6 (even-times odd), 3, 6, 12, 24 (odd-times even).
2Cf.1.8.7; 9.2. ‘Of the opposite genus’ refers to evenand odd. The name is ‘contrary to its value’ if, e.g., the denominator of the fraction is odd and its value, or amount, even.
7 both. Let us then set forth the odd numbers from 3 by themselves in due order in one series:
33 5) 7) Q; If, 13) 15, 17; TQ, . and the even-times even, beginning with 4, again one after another in a second series after their own order: 4, 8, 16, 32, 64, 128, 256,...
8 as far as you please. Now multiply by the first number of either series—it makes no difference which—from the beginning and in order all those in the remaining series and note down the resulting numbers; then again multiply by the second number of the same series the same numbers once more, as far as you can, and write down the results; then with the third number again multiply the same terms anew, and however far you go you will get nothing but the odd-times even numbers.
9 For the sake of illustration let us use the first term of the series of odd numbers and multiply by it all the terms in the second series in order, thus: 3 X 4, 3 X 8, 3 X 16, 3 X 32, and so on to infinity. The results will be 12, 24, 48, 96, which we must note down in one line. Then taking a new start do the same thing with the second number, 5X 4,5 X 8, 5 X 16, 5 X 32. The results will be 20, 40, 80, 160. Then do the same thing once more with 7, the third number, 7 X 4, 7X 8, 7 X 16, 7 X 32. The results are 28, 56, 112, 224; and in the same way as far as you care to go, you will get similar results.
Odd numbers
Se ee es
Even-times even
192 | 384] 768
28 | 56 | 112 | 224 | 448 | 8096 | 1792
360 | 72 | 144 | 288 | 576 | 1152 | 2304
44 | 88 | 176 | 352 | 704 | 1408 | 2816
Odd-times even
numbers
Breadth
1o Now when you arrange the products of multiplication by each term in its proper line, making the lines parallel, in marvelous fashion there will appear along the breadth of the table the peculiar property of the even-times odd, that the mean term is always half the sum of the extremes, if there should be one mean, and the sum of the means equals the sum of the extremes if two. But along the length of the table the property of the even-times even will appear; for the product of the extremes is equal to the square of the mean, should there be one mean term, or their product, should there be two. Thus this one species has the peculiar properties of them both, because it is a natural mixture of them both.
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
1 There is great disagreement among the ancient authorities upon this classification. In the first place, Nicomachus confines these species to odd numbers, thus securing a threefold classification to balance that of the even numbers (see above). He is followed in this by Iamblichus
(p. 26, 18), but Euclid (Elem., VII, Deff., 11–14) and Theon (p. 23, 6 ff. Hiller) make it a classification of both the even and the odd. Nicomachus then divides into (a) prime and incomposite;
(b) secondary and composite; (c) that which is absolutely composite but relatively prime. Nesselmann, op. cif., p- 194, points out that the latter two classes are not mutually exclusive, for b includes c. The difficulty is overcome by Iamblichus, who thus classifies: (¢) the absolutely prime, which is a priori relatively prime as well; (6) the absolutely secondary, which includes as sub-classes the relatively prime and relatively secondary; the two sub-classes are dependent upon the association of terms in specific instances. Euclid (foc. cit.) gives definitions of primes, relative primes, composite, and relatively composite numbers. This need not of course imply a strict classification along these lines. Theon, however, seems to understand it as such, and to establish his classification after this model, making his definitions agree with those of Euclid: (a) absolutely prime; (6) relatively prime; (c) absolutely composite; (d) relatively composite. The last class does not correspond to any set up by Nicomachus; it consists of numbers like 8 and g taken in connection with 6. Cf. T. L. Heath on Euclid, Joc. cit., for an extended discussion; also his History, vol. I, pp. 72 ff.
? Euclid defines a prime number as ‘one measured by unity alone’ (6 povdd: pdry uerpovpevos), Elem., VII, Def. 11. The number 2 satisfies his definition and is also called prime by Aristotle (Top., VIII. 2.157 a 39). But in Nicomachus prime numbers are a class of odd numbers, not of number in general. See Heath on the matter, Euclid, II, 284–85. Theon of Smyrna (p. 23, 9 Hiller) defines the ‘absolutely prime and incomposite’ number as one ‘measured by no number but by unity alone’ (of brd pndevds wey dp@uod, urd pbyns 6¢ povddos merpovuercc). He states that these numbers were sometimes called ‘linear’ and ‘rectilinear’ (ypappsxol, eb@vperpixol) ‘because lengths and lines are viewed in one dimension,’ and that they are also called ‘ odd-times odd’ (weptaadeis wepiocol). Theon leaves it vague whether he regards the dyad as prime; for alter stating that the even numbers are not prime because they are measured by other numbers than unity alone, he says that the dyad is an exception and is therefore called ‘odd-like’ (xa ol Norwol Aprioe kara ra adra bw Tivwy pectdvww THs powddos dpiPudy Karayerpourrac wAHY THs Suddos: ratry yap pdvy cupBéBnxev Swep xal evlas Tov wepico Gy, ro bwd wovddos perpeioPar pdbvor. dwak yap B’, 8”: 3d xal wepiccoadhs elpyrat ravrd Tois wepiogois wmemovOvia).
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.
2 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.
3 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 X 1, and 7 is 7 X 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.
1 The secondary, composite,number ® is an odd number, indeed, because it is distinguished as a member of this same class, but it has no
1 As 2 in the case of 3.
2dox%. Cf. the discussion of element (¢ro:yeior), IT. 1.
3 Nicomachus does not admit even numbers into the class of composites, doubtless because he has already exhausted their classification. Theon of Smyrna, however, makes the composite a division of number in general and gives even numbers among his examples. As noted above, he distinguishes the ‘absolute composite’ numbers that can be measured by some smaller numbers and ‘relative composites,’ those which are measured by some measure, but are prime to each other, as 8, 6, 9, with the measures 2 and 3. In this connection 1 is not considered a common measure, for, as he states, it is not itself a number but the beginning of number. Euclid, Elements, VII, Def. 14, defines a composite number as ‘one measured by some number’ (cbvGeros dpiduds éorin b dpiGucp Tum per povperos).
elementary quality, for it gets its origin by the combination of some-thing else. For this reason it is characteristic of the secondary number to have, in addition to the fractional part with the number itself as denominator, yet another part or parts with different denominators, the former always, as in all cases, unity, the latter never unity, but always either that number or those numbers by the combination of which it was produced. For example, 9, 15, 21, 25, 27, 33, 35, 393 each one of these is measured by unity, as other numbers are,! and like them has a fractional part with the same denominator as the number itself, by the nature of the class common to them all; but by exception and more peculiarly they also employ a part, or parts, with a different denominator; 9, in addition to the ninth part, has a third part besides; 15 a third and a fifth besides a fifteenth; 21 a seventh and a third besides a twenty-first, and 25, in addition to the twenty-fifth, which has as a denominator 25 itself, also a fifth, with a different denominator.
It is called secondary, then, because it can employ yet another measure along with unity, and because it is not elementary, but is produced by some other number combined with itself or with something else; in the case of 9, 3; in the case of 15, 5 or, by Zeus, 3; and those following in the same fashion. And it is called composite for this, or some such, reason: that it may be resolved into those numbers out of which it was made, since it can also be measured by them. For nothing that can be broken down is incomposite, but by all means composite.
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.
1 That is, the primes.
? Theon of Smyrna (p. 24, 8 ff.) has this class, which he calls ‘ prime to one another and not absolutely prime,’ and he points out that 8, 9, and 10 are prime to the ‘absolute primes.’ Euclid defines relatively prime numbers as ‘those that are measured only by unity as the common measure’ (xp@rot rpds ddAAHAous dpOuol elo of pordd: pbvy perpovpern corm uérpy), Elements, VII, Def. 13.
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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.
3 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.
4 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.
s 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.
6 And analogously throughout, this process will go on without in- w
terruption, 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
‘Tt is generally assumed (as by Heath, History, vol. I, p. roo) that in the ‘sieve of Eratosthenes ‘ only the odd prime numbers take on successively the measuring function, and indeed this is all that is necessary, for, e.g., 9 is a multiple of 3 and all its multiples are likewise multiples of 3. The text, however, seems to imply that all the odd numbers should be used, although perhaps Nicomachus did not intend that he should be so strictly interpreted.
* Reading ¢avrod (instead of édavray) with G,. The numbers referred to are the squares of odd prime numbers. 9, e.g., has ninths, of course, and as it is measured by 3, it will also have
thirds, the denomination of which is derived from 3.
3 Thus, if a, 6, m are odd numbers greater than unity, and m = ab, m is measured by a in the quantity of b, and vice versa, and m will have the factors ~ and: named respectively from 6 and a. ad odd numbers, though numbers like 15 (3 X 5) and 77 (7 K 11), when compared, would satisfy his requirements equally well.
4 Nicomachus evidently contemplates admitting into this division only the squares of prime
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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 sub-
' This mode of determining common factors is found in Euclid (VII. 1; X. 2) and is commonly termed the Euclidean method of finding the greatest common divisor of numbers.
2 Reference to the second example following will show that the term ‘ produced twice’ (did¢o-
povperov) by this process is the final subtrahend, which is equal to the final remainder. Boethius, I. 18, is somewhat more explicit in describing the operation: Dafis enim duobus numeris tnaequailt-bus, auferre de matore minorem oportebtl, ef qui relictus fuertt, st maior est, auferre ex eo rursus minorem, si vero minor fueril, eum ex reliquo matore detrahere atque hoc co usque faciendum, quoad untlas ullima vicem retractionis impedial, aul aliguis numerus, impar necessario, si ulrigue numeri impares proponantur; sed eum, qui relinguilur, numerum stbi ipst videbis aequalem.
tract 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,
1 Nicomachus refers here to the general relation of inequality which is opposed to equality (I. 17. 2) as one of the primary divisions of relative number. Technically equality and inequality are oyéees, ‘relations,’ the term applied here to inequality as it is to equality in I. 17. 4; sub-classes of the unequal, first the greater and the less (cf. I. 17. 6) and then the specific ratios (al Séxa dpifunrinal oxéoes, I. 23. 4) are also called ‘relations.’ For the notion oxéo:s and the kindred term Aéyos, ‘ratio,’ cf. on IL. 21. 2.
2 According to Aristotle virtue is the mean, and vice is excess or deficiency. Cf. Eth. Nic., IT. 6. 1106 b 33, xat 3:4 rar’ ofv THs wey Kaxlas 7 UwepBory Kal H EANewus, THs O dperHs 7 perdrys. These are just the varieties that Nicomachus assigns to inequality,
$Cf. Arist., Eth. Nic., as quoted in the preceding note and 1106 b 24: 4% 5 dperh wepl dn cal wpdtes dorly, év ols 7 pév UwepBody auaprdvera: xal 7 Ehrer~is [Yeyerar], Td be udoow dwarvetrar cal xaropGotrac: raira 8 dugw rs dperas. pecdtys res Apa é¢orly 7 dperi, croxacTih ye ofca TOU pécov.
4 bwepredhs dpiGuds. Theon of Smyma, pp. 45, 10 ff., 46, 4, includes this class, but calls them vrepréheco, 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
4 parts, therefore, are greater than the whole itself. And 24 has a half,
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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.’
The deficient number * is one which has qualities the opposite of those pointed out, and whose factors added together are less in comparison than the number itself. It is as if some animal should fall short of the natural number of limbs or parts, or as if a man should have but one eye, as in the poem, “And one round orb was fixed in his brow’’;* or as though one should be one-handed, or have fewer than five fingers on one hand, or lack a tongue, or some such member. Such a one would be called deficient and so to speak maimed, after the peculiar fashion of the number whose factors are less than itself, such as 8or 14. For 8 has the factors half, fourth, and eighth, which are 4, 2, and 1, and added together they make 7, and less than the original number. The parts, therefore, fall short of making up the whole. Again, 14 has a half, a seventh, a fourteenth, 7, 2, and 1, respectively; and all together they make 10, less than the original number. So this number also is deficient in its parts, with respect to making up the whole out of them.
1 The reference is to Homer’s description of Scylla, Odyssey, XII. 85 ff.
* See also L. E. Dickson, History of the Theory of Numbers, Washington, 1909, vol. I, chapter I. 3Cf. Theon, p. 46, 9 ff.
‘The Cyclops. Cf. Hesiod, Theogony, 145.
While these two varieties are opposed after the manner of extremes, the so-called perfect number ' appears as a mean, which is discovered to be in the realm of equality, and neither makes its parts greater than itself, added together, nor shows itself greater than its parts, but is always equal to its own parts. For the equal is always conceived of as in the mid-ground between greater and less, and is, as it were, moderation between excess and deficiency, and that which is in tune, between pitches too high and too low.
Now when a number, comparing with itself the sum and combination of all the factors whose presence it will admit, neither exceeds them in multitude nor is exceeded by them, then such a number is properly said to be perfect, as one which is equal to its own parts. Such numbers are 6 and 28; for 6 has the factors half, third, and sixth, 3, 2, and 1, respectively, and these added together make 6 and are equal to the original number, and neither more nor less. Twenty-eight has the factors half, fourth, seventh, fourteenth, and twenty-eighth, which are 14, 7, 4, 2 and 1; these added together make 28, and so neither are the parts greater than the whole nor the whole greater than the parts, but their comparison is in equality, which is the peculiar quality of the perfect number.
It comes about that even as fair and excellent things are few and easily enumerated, while ugly and evil ones are widespread, so also the superabundant and deficient numbers are found in great multitude and irregularly placed—for the method of their discovery is irregular—but the perfect numbers are easily enumerated and arranged with suitable order; for only one is found among the units, 6, only one other among the tens, 28, and a third in the rank of the hundreds, 496 alone, and a fourth within the limits of the thousands, that is, below ten thousand, 8,128.2. And it is their
1 Euclid’s definition, Elem., VII. 22, is: “A perfect number is one that is equal to its own parts.” Similarly Theon of Smyma defines it, p. 45, 10. See Heath, History, vol. I, p. 74.
2 Subsequent investigation has determined seven more perfect numbers. T.L. Heath, History, vol. I, pp. 74–75, and L. E. Dickson, op. c#., vol. I, chapter I.
I. 2(2?—1) = 6, VII. 238 (219—x),
II. 2? (28—1) = 28, VIII. 2% (2%—1),
Ill. 24(22—1) = 406. IX. 2 (26—1), with 37 digits.
IV. 2®(27—1) = 8,128. X. 288 (28°—1),
V. 22 (28—1) = 33,550,336. MI. 2126 (2127—7).
VI. 2’® (217—1) = 8,568,910,416.
Theon, in his notice of the perfect numbers, mentions only 6 and 28.
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