Imprimatur

Introduction to Arithmetic, Book II

Nicomachus of Gerasa · Introduction to Arithmetic
Section 1 of 3
Underlined text was edited by AI to fix errors.
An element is said to be, and is, the smallest thing which enters into the composition of an object and the least thing into which it can be analyzed. Letters, for example, are called the elements of literate speech, for out of them all articulate speech is composed and into them finally it is resolved. Sounds are the elements of all melody; for they are the beginning of its composition and into them it is resolved. The so-called four elements of the universe in general are simple bodies, fire, water, air, and earth; ’ for out of them in the first instance we account for the constitution of the universe, and into them finally we conceive of it as being resolved.
We wish also to prove that equality is the elementary principle’ of relative number; for of absolute number, number per se, unity and the dyad * are the most primitive elements, the least things out of which it is constructed, even to infinity, by which it has its growth, and with which its analysis into smaller terms comes to an end. We have, however, demonstrated that in the realm of inequality advance and increase have their origin in equality and go on to absolutely all the relations with a certain regularity through the operation of the three rules. It remains, then, in order to make it an element in very truth, to prove that analyses also finally come to an end in equality. Let this then be considered our procedure.
Suppose then you are given three terms, in any relation whatsoever and in any ratio, whether multiple, superparticular, superpartient, or a compound of these, multiple superparticular or multiple superpar-
1 The ordinary list of elements for practically all Greek philosophy. These four were distin-guished as primitive bodies in immemorial antiquity, but the more scientific idea of them as ele-ments seems to have originated with Empedocles. On the matter see Burnet’s summary, Greek Philosophy, Part I, Thales to Plato, p. 26.
2 See on I. 23. 4. * Cf. Part I, pp. 99 ff. ‘ That is, those given in I. 23. 8.
tient, provided only that the mean term is seen to be in the same ratio to the lesser as the greater to the mean, and vice versa. Subtract always from the mean the lesser term, whether it be first or last in order, and set down the lesser term itself as the first term of your new series; then put as your second term what remains from the second after the subtraction; then after having subtracted the sum of the new first term and twice the new second term from the remaining num-ber—that is, the greater of the numbers originally given you—make the remainder your third term, and the resulting numbers will be in some other ratio, naturally more primitive.’ And if again in the same 2 way you subtract the remainder from these same terms,” it will be found that your three terms have passed back into three others more primitive, and you will find that this always takes place as a conse-quence, until they are reduced to equality, whence by every necessity it appears evident that equality is the elementary principle of relative quantity.
There follows upon this speculation a most elegant principle, ex-tremely useful in its application to the Platonic psychogony * and the problem of all harmonic intervals; for in the Platonic passage we are frequently bidden, for the sake of the argument, to set up series of intervals of two, three, four, five, or an infinite number of sesquialter ratios, or two sesquitertians, sesquiquartans, sesquioctaves, or super-particulars of any kind whatsoever, and in each case three, four, or five of them, or as many as may be directed. It is reasonable that 4 we should do this not in an unscientific, unintelligent fashion, it may be even blunderingly, but artistically, surely, and quickly, by the following procedure.
Every multiple will stand at the head * of as many superparticular 1 ratios corresponding in name with itself as it itself chances to be re-moved from unity,° and no more nor less under any circumstances.
1 This is because the process is the reverse of the former. Theon of Smyrna, p. 110, 19 ff., gives this rule, taking it from Adrastus.
2 For example, take 8, 32, 128 (quadruple series). The first term of the new series will be 8; the second will be 32—8 = 24; the third will be 128—[(2 X 24) + 8], or 72. This gives a triple series. Then similarly from 8, 24, 72 will be derived 8, 16, 32, the double series, and from the latter 8, 8, 8, a series of equal terms. 3See Plato, Timaeus, 35 A ff.
‘ irytoerac: That is, with reference to the table in section 4; ‘will head a column.’
6 That is, in the list of doubles (see the table).
2 The doubles, then, will produce!’ sesquialters, the first one, the second two, the third three, the fourth four, the fifth five, the sixth six, and neither more nor less, but by every necessity when the super-particulars that are generated attain the proper number, that is, when their number agrees with the multiples that have generated them, at that point by a divine device, as it were, there is found the number which terminates them all because it naturally is not divisible by that factor whereby the progression of the superparticular ratios went on.
From the triples all the sesquitertians will proceed, likewise equal in number to the number of the generating terms, and coming to an end, after the independence of their advance is lost, in numbers not divisible by 3. Similarly the sesquiquartans come from the quad-ruples, reaching a culmination after their independent progression in a number that is not divisible by 4.
3 As an example, since doubles generate sesquialters corresponding to them in number,’ the first row of multiples * will be 1, 2, 4, 8, 16, 32, 64. Now since 2 is the first after unity, this will be the origin of one sesquialter only, 3, which number is not divisible by 2, so that another sesquialter might arise out of it. The first double, therefore, is productive of but one sesquialter, and the second, 4, of two. For it produces its own sesquialter, 6, and that of 6, 9, but there is none for 9 because it has no half. Eight, which is the third double, is father to three sesquialters; one its own, 12; the second, 18, the sesquialter of 12; and third, 27, that of 18; there is no fourth one, however, be-cause of the general rule, for 27 is not divisible by 2. Sixteen, the fourth double, will stand at the head of four sesquialters, 24, 36, 54, and finally 81, so that they may of necessity be equal in number to what generated them; for 81 by its nature is not divisible by 2. And this, as you go on, you will find holds true in similar fashion to infinity.
4 For the sake of illustration let there be set down the table of the doubles, thus: ,
1 gUeove:: In the same sense that the even numbers ‘produced’ sesquialters by the process of I. 19.2; but each double is here regarded as the source or producer not only of its own sesqui-alter, but also that of this sesquialter itself, and so on, as far as the ratio can be carried on in integers.
2 The number of the multiple is of course that of its order in the series of doubles, or triples, etc. 3 That is, doubles, the simplest subclass.

Translation: Book [i 233

The double ratio in the breadth of the table 124 8 16 32 64 3 6 12 24 48 96 The triple ratio along 9 18 36 72 144 The sesquialter ra-
the hypotenuse 27 54 108 216 tio in the depth 81 162 324 of the table 243 480 729 We must make a similar table in illustration of the triple: I
The triple ratio in the breadth
I 3 9 27 81 243 729 4 12 36 108 324 972
The quadruple 16 48 144 432 1296 The sesquiter- ratio on the 64 192 576 1728 tian ratio in hypotenuse 256 768 2304 the depth
1024 3072 4096
In the foregoing table we shall observe that in the same way the first triple, 3, stands at the head of but one sesquitertian ratio, 4, its own sesquitertian, which immediately shuts off the development of another like it; for 4 is not divisible by 3, and hence will not have a sesquiter-tian. The second triple is 9, and hence will begin a series of only two sesquitertian ratios, 12, its own, and 16, that of 12; but 16 cuts off further progress, for it is not divisible by 3 and hence will not have a sesquitertian. Next in order is the triple 27, three times removed 2 from 1, for the triples progress thus: 1, 3,9, 27. Therefore this number will stand at the head of three sesquitertian ratios and no more. The first is its own, 36; the second the sesquitertian of 36, 48; the third that of the last, 64, and this no longer has a third part and therefore will not admit of a sesquitertian. The fourth leads a series of four ses-quitertians and the fifth, of course, five.
Such, then, is the illustration; and for the other multiples let the 3 manner of your tables be the same. Observe that likewise here, as we found to be true in our previous discussion, Nature shows us that the doubles are more nearly original than the triples, the triples than the quadruples, these latter than the quintuples, and so on throughout. For the highest rows of figures, across the breadth of the tables, if they are doubles, will have doubles lying parallel to them, and the numbers lying diagonally, on the hypotenuse, will be of the next succeeding variety, greater by 1, that is, triples, seen also in a series of parallel lines. If, however, there are triples across the breadth, the diagonals will by all means be quadruples; if the former are quadruples, then the latter are quintuples, and so forth.
t It remains, after we have explained what other ratios are produced by combination of ratios, to pass on to the succeeding topics of the Introduction.
2 Now the first two ratios of the superparticular, combined, produce the first ratio of the multiple, namely, the double; for every double is a combination of sesquialter and sesquitertian, and every sesquialter and sesquitertian ' combined will invariably produce a double.
For example,’ since 3 is the sesquialter of 2, and 4 the sesquitertian of 3, 4 will be the double of 2, and is a combination of sesquialter and sesquitertian. Again, as 6 is the double of 3, we shall find between them some number ® that will of necessity preserve the sesquitertian ratio to the one and the sesquialter to the other; and indeed 4, lying between 6 and 3, gives the sesquitertian ratio to 3 and the sesquialter to 6.
3 It was rightly said, then, that the double, when resolved, is resolved into the sesquialter and the sesquitertian, and that when sesquialter and sesquitertian are combined there arises the double, and that the first two forms of the superparticular combined make the first form of the multiple.
1 That is, when the last term of the first is the same as the first term of the second ratio; for given the general formula for the sesquialter, a + “; then the sesquitertian of the second term, | ee ee: ; ee 3.4 a+=+-+4+ ao 2 a, is the double of the first term; or, more simply, 2 = 2, 2 2 3
2 Some of the MSS diagrammatically illustrate thus: 2 3 4
sesquialter sesquitertian | | double |
1 That is, given a and 2 a, in double ratio, } K a = 3 X 24.
But again, to take another start, this first form of the multiple which 4 has thus been produced, together with the first form of the superpar-ticular,’ will produce the next form of the same class, that is, the second multiple, the triple; for from every multiple and sesquialter combined a triple of necessity arises. For example,’ as the double of 6 is 12, and the sesquialter of this is 18, then immediately 18 is the triple of 6; and to take another method, if I do not care to make 12 the mean term, but rather 9, the sesquialter of 6, the same result will come about, without deviation and harmoniously; for while 18 is the double of 9 it will preserve the triple ratio to 6. Hence from the sesquialter and the double, the first forms of the superparticular and the multiple, there arises by combination the second form of the multiple, the triple, and into them it is always resolved. For look you; 6, which is the 5 triple of 2, will have a mean term 3, which will exhibit two ratios, the sesquialter with regard to 2, and the double ratio of 6 to itself.
But if this triple ratio,’ likewise, the second form of the multiple, is
‘ The sesquialter. ? Diagrams given in the MSS:
6 12 18
double sesquialter These principles may also be | | demonstrated in general terms: arenes a)
4 16 F 20 (or, 4, 5s 20) 7 quadruple ‘ sesquiquartan 5 | 29 30 (or, 5, 6, 30)
quintuple sesquiquintan |
Algebraic statements of the matter above:
(a) m; 3m; (sm +5) = 4m,
combined with the sesquitertian, which is the second form of the super-particular, there would be produced from them the next form of the multiple, namely, the quadruple, and this also will of necessity be re-solved into them after the same fashion as the cases previously set forth; and the quadruple, taking into combination the sesquiquartan, will make the quintuple, and, once more, the latter with the sesqui-quintan will make the sextuple, and so on to the end. Thus the multi-ples in regular order from the beginning with the superparticulars in regular order from the beginning will be found to produce the next larger multiples. For the double with the sesquialter makes the triple, the triple with the sesquitertian the quadruple, the quadruple with the sesquiquartan the quintuple, and as far as you wish to proceed no contrary result will appear.
1 Up to this point then we have sufficiently discussed relative number, by a process of selection measuring out what is easily comprehended and appropriate to the nature of the matters thus far introduced. Whatever remains to be said on this topic will be filled in after we have put it aside and have first discussed certain subjects which involve a more serviceable inquiry, having to do with the properties of
‘- absolute number, not relative. For mathematical speculations’ are always to be interlocked and to be explained one by means of another.. The subjects which we must first survey and observe are concerned with linear, plane, and solid numbers, cubical and spherical, equilateral and scalene, ‘bricks,’ ‘beams,’ ‘wedges,’ and the like, the tradition concerning which, to be sure, since they are more closely related to: magnitude, is properly given in the Geometrical Introduction.? Yet
OT, m#; m+™. (sm+3™ m) = 4m, 3
(5) m; 4m; (4m+4™) = sm, Or, ™; wos (am+% tm) = 5m. (c) m; 5m; (sm+5%) «6m
OF, ; mei (sm+5™) = 6m, 5
1 Boethius, II. 4: Amat enim quodammodo matheseos speculatio alterna probationem rations constitut. ? Cf. p. 70.
the germs of these ideas are taken over into arithmetic, as the science | which is the mother of geometry and more elementary than it. For we recall that a short time ago we saw that arithmetic abolishes the other sciences with itself,’ but is not abolished by them, and conversely is of necessity implied by them but does not itself imply them.
First, however, we must recognize that each letter by which we in-dicate a number, such as lota, the sign for 10, kappa for 20, and omega for 800, designates that number by man’s convention and agreement, not by nature. On the other hand, the natural, unartificial, and there-fore simplest indication of numbers would be the setting forth one beside the other of the units contained in each. For example, the writing of one unit by means of one alpha will be the sign for 1; two units side by side, that is, a series of two alphas, will be the sign for 2; when three are put in a line it will be the character for 3, four in a line for 4, five for 5, and so on. For by means of such a notation and indication alone could the schematic arrangement of the plane and solid numbers mentioned be made clear and evident, thus:
The number 1, a
The number 2, aa
The number 3, aaa The number 4, aaaa The number 5, aaaaa and further in similar fashion.
Unity, then, occupying the place and character of a point, will be the beginning of intervals and of numbers, but not itself an interval or a number,’ just as the point is the beginning of a line, or an interval, but is not itself line or interval.* Indeed, when a point is added to a point, it makes no increase, for when an non-dimensional thing is added to another non-dimensional thing, it will not thereby have dimension; just as if one should examine the sum of nothing added to nothing,
1Cf. I. 4. 2–5. ? See p. 116.
?With this passage should be compared Theon of Smyrna, p. 81, 6 ff., where ‘interval’ (did-ornua) is defined: ‘“ ‘Interval’ and ‘ratio’ (Aéyos) are different; for ‘interval’ is that which is be-tween homogeneous unequal terms, ‘ratio’ merely the relation of homogeneous terms to one an-other. Wherefore there is in the case of equal terms no interval between, but there is one and the same ratio, that of equality; whereas in the case of unequals, there is one and the same interval from each to each, but a different and opposite ratio of each to each. For example, there is one and the same interval from 2 to 1 and from to 2, but a different ratio; 2:1 isa double ratio and 1:2 is one half.”” He then quotes Eratosthenes on the subject. This will explain what is said below as to intervals in connection with the relation of equality.
which makes nothing. We saw’ a similar thing also in the case of equality among the relatives; for a proportion is preserved—as the first is to the second, so the second is to the third—but no interval is generated in the relation of the extremes to each other, as there is in all the other relations with the exception of equality. In exactly the same way ” unity alone out of all number, when it multiplies itself, produces nothing greater than itself.
Unity, therefore, is non-dimensional and elementary, and dimension first is found and seen in 2, then in 3, then in 4, and in succession in the following numbers; for ‘dimension’ is that which is conceived of as between two limits.
4 The first dimension is called ‘line,’ for ‘line’ is that which is extended in one direction. Two dimensions are called ‘surface,’ for a ‘surface’ is that which is extended in two directions. Three dimensions are called ‘solid,’ for a ‘solid’ is that which is extended in three directions, and it is by no means possible to conceive * of a solid which has more than three dimensions, depth, breadth, and length. By these are de-fined the six directions which are said to exist in connection with every body and by which motions in space are distinguished, forward, back-ward,‘ up, down, right and left; for of necessity two directions opposite to each other follow upon each dimension, up and down upon one, for-ward and backward upon the second, and right and left upon the third.
1 The reference is the series of equal numbers employed in I. 23. 7 ff. In the series 1, 1, 1; 2, 2, 2, etc., the ratio is the same between any pair of terms; the extremes have the same ratio as the means; that is, they are all equal, so there is no interval between the extremes.
? The Neo-Pythagoreans commonly used this fact to substantiate their identification of the monad with God. Like God the monad is immutable and eternal (e.g., Chalcidius, Comm. in Tim., c. 39: sola inconcusso inure est algue in slatu suo perseverat; semper eadem. . . immutabilis, et singularitas semper). The name monad they derived from ‘remain’ (mords, uévecv) because the monad ‘remains’ the same under these conditions (cf. Theol. Arith., p. 3 Ast; Iamblichus J Nic., p. 11, 24 f.; Theon, p. 19, 7). See also II. 17. 4 below.
* Philo Judaeus, De Decalogo, 7, also states that there can be only three dimensions (w)elous yap rp.ay duacrdces obx &yévynaer).
‘ The six categories of relative position (and motion) also were frequently cited in Neo-Py-thagorean arguments; the topic was, moreover, invested with greater significance from the fact that Plato employed it, in close connection with the varieties of motion, in Timaeus, 43 8. Add-ing rotation, Plato mentions seven varieties of motion, 1bid., 34 a (cf. 40 A-B), and 10 (not all spatial however) in Laws, 894 c. The Neo-Pythagoreans regarded it significant of the peculiar virtues of 6, therefore, that there should be six ‘so-called spatial positions’ (Theol. Arith., p. 36 Ast, al Aeyéuerat cwparikal wepurrdoeis; cf. also Philo, Leg. Alleg., I. 2; (Plut.) Epét., ITI. 15, 10 = Doxog. Graec., 380, 24; M. Capella, VII, 736, who adds that the seventh, circular mo-tion, is eternal). Many of them similarly used the group of seven motions in praise of the num-ber 7 (e.g., Anatolius, ap. Theol. Arith., p.42 Ast; Lydus, De Mens., II, 11; Philo, De Mund. Op., 41, and Leg. Alleg., I. 4; Macrobius, Comm. in Somn. Scip., I. 6. 81). Nicomachus, then, is using a topic very frequently employed.
The statement, also, as it happens, can be made conversely thus: 5; If a thing is solid, it has by all means three dimensions, length, depth and breadth; and conversely, if it has the three dimensions, it is al-ways a solid, and nothing else.
That which has but two dimensions, therefore, will not be a solid, 6 but a surface, for the latter admits of but two dimensions. Here too it is possible similarly to reverse the statement; directly stated, a surface is that which has two dimensions, and conversely, that which has two dimensions is always a surface.
The surface, then, is exceeded by the solid by one dimension, and the 7 line is exceeded by the surface by one, for the line is that which is extended’ in but one direction and has only one dimension, and it falls short of the solid by two dimensions. The point falls short of the latter by one dimension, and hence it has already been stated that it is non-dimensional, since it falls short of the solid by three dimen-sions, of the surface by two, and of the line by one.
The point, then, is the beginning of dimension, but not itself a dimension, and likewise the beginning of a line, but not itself a line; the line is the beginning of surface, but not surface; and the begin-ning of the two-dimensional, but not itself extended in two directions. Naturally, too, surface is the beginning” of body, but not itself body, 2 and likewise the beginning of the three-dimensional, but not itself extended in three directions.
Exactly the same in numbers, unity is the beginning of all number 3 that advances unit by unit in one direction; linear number is the be-ginning of plane number, which spreads out like a plane in one more dimension; and plane number is the beginning of solid number, which possesses a depth ® in the third dimension, besides the original ones. To illustrate and classify, linear numbers are all those which begin with 2 and advance by the addition of 1 in one and the same
’73 dcacraréy is translated ‘that which is extended’; 8iudornyua is here translated by ‘di-mension,’ though in a general sense it might be rendered ‘extension.’ d:deracis is used as a synonym for didornpa,
* These statements are paralleled in Photius’s report of a Life of Pythagoras (Codex 240,
p. 249 a, 19 Bekk.). 3 Cf. Plato, Timaeus, 53 Cc: 1d 8é rod odparos eldos wav xal Adbos Lye.
dimension; and plane numbers are those! that begin with 3 as their most elementary root and proceed through the next succeeding num-bers. They receive their names also in the same order; for there are first the triangles, then the squares, the pentagons after these, then the hexagons, the heptagons, and so on indefinitely, and, as we said, they are named after the successive numbers beginning with 3.
4 The triangle, therefore, is found to be the most original and elemen-tary form of the plane number. This we can see from the fact that, among plane figures,” graphically represented, if lines are drawn from the angles to the centers each rectilinear figure will by all means be resolved into as many triangles as it has sides; but the triangle itself, if treated like the rest, will not change into anything else but itself.
1 Nicomachus, here and in the following chapters, adopts the broadest view of what consti-tutes the class of plane numbers. Not all the ancients agreed with him; Euclid, in Elements, VII, Def. 17, defines the plane number as we should, as that which is produced when two numbers multiply each other, the multiplier and multiplicand being its sides (Sray &é d¥o dpsfuol wodda-whacidoavres ddAHAOUs woidel Tiva, d yerduevos éwlwedos xadefrai, wAevpal dé atvrod ol wod\AawAaci-deavres d\dHAovs dpifuol), and Theon of Smyrna twice defines them similarly (p. 31, 9 Hiller, ἐστὶν ὁ ὑπὸ δύο ἀριθμῶν πολλαπλασιασθείς; p. 36, 5, of diyws wodAawhacratduern éxlredor). Th. Martin clearly explains the difference between this application of the term and its more com-prehensive use by Nicomachus: ‘“‘En effet, les nombres rectangles et carrés expriment la mesure des surfaces, et les nombres parallélipipédes rectangles et cubiques expriment la mesure des solides. Au contraire, les nombres triangles, pentagones, hexagones, etc., de méme que les nombres tétraédres, pentaédres, hexaédres, etc., n’expriment rien qu’une disposition imaginaire des unités dans l’espace” (Chapitres IX* ef XX* du Livre Second de 0’Iniroduction Arithmétique de Nicomaque de Gérase, Rome, 1858, p. 7).
In spite of his definition, Theon of Smyrna lists triangular and other polygonal numbers, like Nicomachus, and consequently must have known and shared to a certain extent Nicomachus’s conception of them, whether or not he was aware of any inconsistency; and that this conception was somewhat generally current is shown by its appearance in the works of Philo Judaeus (see p. 32). Further, it may be noted that this notion of the polygonals is found in Diophantus when (De Polygonis Numeris, vol. I, p. 450, 3 Tannery) he remarks that “each of the numbers beginning with the triad and increasing by unity is a polygonal number in the first degree from the monad, and has as many angles as the number of units in it, and its side is the next number after the monad, 2” (fxacros rar dwd ris rpiddos dpw@pay avfoudrww powdds wohktywrds dors wpiror dwd ris porddos, cal fye:ywwlas rocavras boov éorly rd rhidos Tay év abr@ porddwy whevpd Te atroi doriv 6 défjs Tis povddos dpOuds, 6 8’). From Nicomachus’s point of view, evidently, the same number could be called linear, plane, or solid, according to the assumed arrangement of its component monads.
2 Nicomachus here agrees with Plato, Timaeus, 53 c ff., in declaring the triangle to be the fundamental form of the plane surface. Plato in the passage cited uses the principle further to explain the forms of the minutest particles of the four elements. He agrees with Nicomachus in stating that all plane surfaces may be reduced to triangles (Timaeus, 53 c, 7 d¢ 690} rijs éxc-wédou Bdoews éx Tpvydrwr cuvéorynxe), but with reference to the subdivision of the triangle itself, he points out that each may be reduced by dropping a perpendicular from the apex (in-stead of drawing lines to the center, according to Nicomachus) to two elementary forms, the right-angled scalene or the right-angled isosceles. Cf. also Theologumena Arithmeticae, p. 18, Ast, and II. 12. 8 below.
Hence the triangle is elementary among these figures; for everything else is resolved into it, but it into nothing else. From it the others likewise would be constituted, but it from no other. It is therefore the element of the others, and has itself no element. Likewise, as 5 the argument proceeds in the realm of numerical forms, it will confirm this statement.
Now a triangular number is one which, when it is analyzed into 1 units, shapes into triangular form the equilateral placement of its parts in a plane. 3, 6, 10, 15, 21, 28, and so on, are examples of it; for their regular formations, expressed graphically, will be at once triangular and equilateral. As you advance you will find that such a numerical series as far as you like takes the triangular form, if you put as the most elementary form the one that arises from unity, so that unity may appear to be potentially a triangle,” and 3 the first actually.
Their sides will increase by the successive numbers, for the side of 2 the one potentially first is unity; that of the one actually first, that is, 3, is 2; that of 6, which is actually second, 3; that of the third, 4; the fourth, 5; the fifth, 6; and so on.
The triangular number is produced * from the natural series of num- 3 ber set forth in a line, and by the continued addition of successive terms, one by one, from the beginning; for by the successive combi-nations and additions of another term to the sum, the triangular num-bers in regular order are completed. For example, from this natural series, I, 2, 3, 4, 5, 6, 7, 8, 9, 10, II, 12, 13, 14, 15, I take the first term and have the triangular number which is potentially first,1, A; then adding the next term I get the triangle actually first, for 2 plus 1 equals 3. In its graphic representation it is thus made up: Two units, side by side, are set beneath one unit, and the number three is made
1 This is again the distinction between potential and actual, and according also to Theon, p. 33, 5, the monad is the first potentially triangular number. On what potentiality might be conceived to mean in this case, cf. Boethius, II. 8: Nam si cunctorum mater est numerorum (sc. unilas), quicquid in his quae ab ea nascuntur numeris inventlur necesse esl ul ipsa naturali quadam polestate contineal.
2 Theon of Smyrna, p. 32, 22 ff., notes this method of generating triangular numbers. Cf. also Johannes Pediasimus, Geometria, in Newe Jahrb. f. Phil. u. Paed., vol. XCII, 1865, pp. 366 ff. (f. 40 a of the Munich MS there cited).
a triangle: Then when next after these the following number,
3, is added, simplified into units, and joined to the former, it gives 6, the second triangle in actuality, and furthermore, it graphically repre- sents this number: Again, the number that naturally follows,
4, added in and set down below the former, reduced to units, gives the one in order next after the aforesaid, 10, and takes a triangular form: pe e\ 5) after this, then 6, then 7, and all the numbers in aadaede order, are added, so that regularly the sides of each triangle will consist of as many numbers! as have been added from the natural series to produce it:
1 The square is the next number’ after this, which shows us no longer 3, like the former, but 4, angles in its graphic representation, but is none the less equilateral. Take, for example, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100; for the representations of these numbers are equilateral, square figures, as here shown; and it will be similar as far as you wish
to go: I 4 9 16 25 2 It is true of these numbers, as it was also of the preceding, that the advance in their sides progresses with the natural series. The side
1 Theon of Smyrna, p. 37, 13 ff., states that the units in the sides will equal the last number added.
? This number is treated by Theon of Smyrna (pp. 26, 14; 28,3; 34,13 39, 10), who repeats himself several times.
of the square potentially first, 1, is 1; that of 4, the first in actuality, 2; that of 9, actually the second, 3; that of 16, the next, actually the third, 4; that of the fourth, 5; of the fifth, 6, and so on in general 3 with all that follow.
This number also is produced’ if the natural series is extended in a line, increasing by 1, and: no longer the successive numbers are added to the numbers in order, as was shown before, but rather all those in alternate places, that is, the odd numbers. For the first, I, is potentially the first square; the second, 1 plus 3, is the first in actuality; the third, 1 plus 3 plus 5, is the second in actuality; the fourth, 1 plus 3 plus 5 plus 7, is the third in actuality; the next is produced by adding 9 to the former numbers, the next by the addi-tion of 11, and so on.
In these cases, also, it is a fact that the side of each consists of as 4 many units as there are numbers taken into the sum to produce it.
The pentagonal number is one which likewise upon its resolution 1 into units and depiction as a plane figure assumes the form of an equi-lateral pentagon. 1, 5, 12, 22, 35, 51, 70, and analogous numbers are examples. Each side of the first actual pentagon, 5, is 2, for 1 is the 2 side of the pentagon potentially first, 1; 3 is the side of 12, the second of those listed; 4, that of the next, 22; 5, that of the next in order, 35, and 6 of the succeeding one, 51, and so on. In general the side con-tains as many units as are the numbers that have been added to- gether to produce the pentagon, chosen out of the natural arithmetical series set forth in a row. For in a like and similar manner, there are added together to produce the pentagonal numbers * the terms begin-ning with 1 to any extent whatever that are two places apart, that is, those that have a difference of 3.
1 Cf. Theon of Smyrna, l. c. He adds the obvious generation of squares by multiplying num-bers by themselves (implied by Nicomachus, IT. 18. 3), and adds that the squares are alternately odd and even (p. 34, 3). The method of Nicomachus was known to the old Pythagoreans; cf. Aristotle, Phys., IIT. 4, and Cantor, op. cit., vol. I, p. 160.
2 So in the first square, 1, the side is 1 and only one term is taken to produce it. In the sec-ond, 4, the side is 2 and two terms are taken to produce it (1 + 3). Generally, the algebraic sum of 1,3,5 .. . to terms is n’.
7 Cf. Theon of Smyrna, pp. 34, 11 and 39, 14, on the derivation of pentagonals.
Unity is the first pentagon,’ potentially, and is thus depicted:
5, made up of 1 plus 4, is the second, similarly represented:
12, the third, is made up out of the two former numbers with 7 added to them, so that it may have 3 as a side, as three numbers have been added to make it. Similarly the preceding pentagon, 5, was the com-bination of two numbers and had 2 as itsside. The graphic represen-tation of 12 is this:
a a a a a a a a a a a a
The other pentagonal numbers will be produced by adding together one after another in due order the terms after 7 that have the difference 3, as, for example, 10, 13, 16, 19, 22, 25, and so on. The pentagons will be 22, 35, 51, 70, 92, 117, and so forth.
1 The figures given are those found in MS G. The regular pentagonal arrangement is given by M. Martin (op. cit.) in a way to show the numbers added in each instance. He takes these from editions of Theon and Iamblichus, but cf. Hoche, p. 87, critical notes. On the other hand the statements of II. 12. 2 seem to favor the schemes given by G.
Side: q (4)
) (2) (3)
Number added: . 4 Cy 7 AY 10 5 12 22
The hexagonal, heptagonal, and succeeding numbers will be set forth in their series by following the same process, if from the natural series of number there be set forth series with their differences increas-ing by 1. For as the triangular number was produced by admitting into the summation the terms that differ by 1 and do not pass over any in the series; as the square was made by adding the terms that differ by 2 and are one place apart, and the pentagon similarly by add-ing terms with a difference of 3 and two places apart (and we have demonstrated these, by setting forth examples both of them and of the polygonal numbers made from them), so likewise the hexagons will have as their root-numbers ' those which differ by 4 and are three places apart in the series, which added together in succession will produce * the hexagons. For example, 1, 5, 9, 13, 17, 21, and so on; so that the hexagonal numbers produced will be 1, 6, 15, 28, 45, 66, and so on, as far as one wishes to go.
The heptagonals, which follow these, have as their root-numbers terms differing by 5 and four places apart in the series, like 1, 6, 11, 16, 21, 26, 31, 36, and so on. The heptagons that thus arise are 1, 7, 18, 34, 55, 81, 112, 148, and so forth.
1 That is, gnomons; the term being used in the broader sense. See on I. 9. 4, and cf. IT. 9. 3. 2? MS G gives the following diagram of the hexagonal number 15:
a
3 The octagonals? increase after the same fashion, with a difference of 6 in their root-numbers and corresponding variation in their total constitution.
4 In order that, as you survey all cases, you may have a rule generally applicable,” note that the root-numbers of any polygonal differ by 2 less than the number of the angles shown by the name of the polyg-onal—that is, by 1 in the triangle, 2 in the square, 3 in the pentagon, 4 in the hexagon, 5 in the heptagon, and so on, with similar increase.
x Concerning the nature of plane polygonals this is sufficient for a first Introduction. That, however, the doctrine of these numbers is to the highest degree in accord with their geometrical representation, and not out of harmony with it, would be evident, not only from the graphic representation in each case, but also from the following:
1 The following illustrations are from the same MS:
Derivation of heptagonals: 7 18 34 55 I, 2, 3, 4, 5, 6, 7, 8, 9, 10, II, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 Heptagonal Odagonal a a a 4 a a a a
a
* Cf. also Theon, pp. 34, 6 and p. 40, 11 ff. The principle here stated by Nicomachus had already been given by Hypsicles (ca. 180 B.c.), whose theorem is cited by Diophantus (De Polygonis Nu-meris, Prop. IV) as follows: ‘If as many numbers as you please be set out at equal interval from 1, and the interval is 1, their sum is a triangular number; if the interval is 2, a square; if 3, a pentagonal; and generally the number of angles is greater by 2 than the interval.” Dio-phantus gives this as a theorem of ‘Hypsicles ¢» 8pw,’ which may mean either that it occurred ‘in a definition’ which he made somewhere in his writings, or that it was in a book called “Opos. Cf. Nesselmann, of. cit., p. 466; Gow, op. cil., p. 87.
Every square figure ' diagonally divided is resolved into two triangles and every square number is resolved into two consecutive triangular numbers, and hence is made up of two successive triangular numbers. For example, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, and so on, are triangular numbers and 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, squares. If you add 2 any two consecutive triangles that you please, you will always make a square, and hence, whatever square you resolve, you will be able to make two triangles of it.
Again, any triangle * joined to any square figure makes a pentagon, for example, the triangle 1 joined with the square 4 makes the penta-gon 5; the next triangle, 3 of course, with 9, the next square, makes the pentagon 12; the next, 6, with the next square, 16, gives the next pentagon, 22; 10 and 25 give 35; and so on.
Similarly,’ if the triangles are added to the pentagons, following 3
1MS G gives the following figure as an illustration. The principle may be proved from the formulas of arithmetic progression,
S=”(@+),b=a+ (n—1)d.
Two successive triangular numbers, formed according to definition by the summation of s and v-+on n +1 terms respectively, will therefore be:
w+ 3n+2
and
The Neo-Pythagoreans employed an interesting development of this principle to display the relative characters of the monad and the dyad (cf. Theol. Arith., p. 9 Ast, and Iamblichus Jn Nic., p. 75, 20 ff.). The matter is stated in the Theol. Arith., I. c., as follows: The monad is the cause of squares not only because the odd numbers successively arranged about it give squares, but also “because each side, as the turning point (sc. of a double race course) from the monad as starting point to the monad as finish line has as the sum of its going forth and of its return its own square” (AX Srexal éxdory wrevpd, Gowep xaunrhp dwd vowAryyos wovddos els 5¢ vicoay pordda, wdduv elxe THs mpoddou kal dwavddou Thr cuvdecivy dp’ dauris avrdy roy rerpdywror), That is, to take the side 5, when the successive numbers up to 5 are set out as one side of the race-track, 5 is made the turning point and the other side is made up of the descending numbers to 1, e.g.,
, and their sum is n? + 22+ 1, which is (mn + 1)’, a perfect square.
to? 3 4 5 zt? $3 4 the sum of the whole series is 25, or 5. The series 1. . . 5, of course, is one triangular number, and the descending series 4. . . 1 the immediately preceding one. From its resemblance to the double race course of the Greek games this proposition was apparently recognized under the name ‘diaulos’ (cf. Iamblichus, p. 75, 25). Its further application to the heteromecic num-bers is not pertinent to the present subject.
? This may be seen by comparing the figure of the pentagon as shown in the diagrams ac-companying Chapter X; and it is an argument in favor of representing them as does MS G.
+ This proposition and the preceding are special cases of the theorem that the polygonal number of r sides with side m, plus the triangular number with side »—1, makes the polyg- onal number with ¢ +1 sides and side mn. Algebraically “ot (2 + nd) + nin to) = BE hy wd 42)), the same order, they will produce the hexagonals in due order, and again the same triangles with the latter will make the heptagonals in order, the octagonals after the heptagonals, and so on to infinity.
4 To remind us, let us set forth rows of the polygonals, written in parallel lines, as follows: The first row, triangles, the next squares, after them pentagonals, then hexagonals, then heptagonals, then if one wishes the succeeding polygonals.
Triangles I 3 6 10 I§ 21 28 8360 = «45 55 Squares Ir 4 9g 16 25 36 49 64 81 100 Pentagonals 1 5 I2 22 35 5I 7O =O2 &I17 = 145 Hexagonals 1 6 15 28 45 66 1 120 153 190 Heptagonals 1 7 18 34 55 8r «12 148 189 235
You can also set forth the succeeding polygonals in similar parallel lines.
5 In general, you will find that the squares are the sum of the triangles above those that occupy the same place in the series, plus the numbers of that same class in the next place back;! for example, 4 equals 3 plus 1, 9 equals 6 plus 3, 16 equals 10 plus 6, 25 equals 15 plus 10, 36 equals 21 plus 15, and so on.
The pentagons are the sum of the squares above them in the same place in the series, plus the elementary triangles that are one place further back in the series; for example, 5 equals 4 plus 1, 12 equals 9 plus 3, 22 equals 16 plus 6, 35 equals 25 plus 10, and so on. |
6 Again, the hexagonals are similarly the sums of the pentagons above them in the same place in the series plus the triangles one place back; for instance, 6 equals 5 plus 1, 15 equals 12 plus 3, 28 equals 22 plus 6, 45 equals 35 plus 10, and as far as you like.
7 The same applies to the heptagonals, for 7 is the sum of 6 and 1, 18 equals 15 plus 3, 34 equals 28 plus 6, and so on. Thus each polyg-onal number is the sum of the polygonal in the same place in the series with one less angle, plus the triangle, in the highest row, one place back in the series.
8 Naturally, then, the triangle is the element of the polygon? both in figures and in numbers, and we say this because in the table, reading
1 That is, in the column next to the left. 7Cf.Il. 7.4. Theol. Arith., p. 8 Ast, states that the triangle is the element of both magnitudes and numbers and is made by the congress of the monad and the dyad.
either up and down’ or across, the successive numbers in the rows are discovered to have as differences the triangles in regular order.
From this it is easy to see what the solid number is and how its 1 series advances with equal sides; for the number which, in addition to the two dimensions contemplated in graphic representation in a plane, length, and breadth, has a third dimension, which some call depth, others thickness, and some height, that number would be a solid number, extended in three directions and having length, depth, and breadth.
This first makes its appearance in the so-called pyramids. These 2 are produced from rather wide bases narrowing to a sharp apex, first after the triangular form’ from a triangular base, second after the form of the square from a square base, and succeeding these after the pentagonal form from a pentagonal base, then similarly from the hexagon, heptagon, octagon, and so on indefinitely.
Exactly so among the geometrical solid figures; if one imagines three 3 lines from the three angles of an equilateral triangle, equal in length to the sides of the triangle, converging in the dimension height to one and the same point, a pyramid would be produced, bounded by four triangles, equilateral and equal one to the other, one the original tri-angle, and the other three bounded by the aforesaid three lines. And 4 again, if one conceives of four lines starting from a square, equal in length to the sides of the square, each to each, and again converging in the dimension height to one and the same point, a pyramid would be completed with a square base and diminishing in square form, bounded by four equilateral triangles and one square, the original
1 Ast, Theol. Arith., p. 288, declares that xal wAdros (the reading of the Paris MS for Hoche’s cal kara whdros, p. 99, 5) is an interpolation, but Hoche retains the words on the authority of Philoponus. The triangular numbers are the differences in the table taken ‘in depth’ («ara Bdéos); for in reading down the second column the common difference is 1, that of the third column is 3, of the fourth 6, and so on, the differences agreeing in turn with each of the triangular numbers. This observation is omitted by Boethius, who devotes II. 19 to showing that the triangular numbers furnish the differences taken across the breadth. When the numbers of the table are compared with those of the same column but in the row next above, and the comparisons are carried across the whole table, the differences are found to be the triangular numbers. Algebraically the corresponding equation is the same as that givenin thenote above to IT. 12. 3.
? That is, successive sections parallel to the base are triangular. On pyramids, cf. Theon, pp. 42, 3 ff.
5 one. And starting from a pentagon,’ hexagon, heptagon, and how-ever far you care to go, lines equal in number to the angles, erected in the same fashion from the angles and converging to one and the same point, will complete a pyramid named from its pentagonal, hexagonal, or heptagonal base, or similarly.
6 So likewise among numbers, each linear number increases from unity, as from a point, as for example, 1, 2, 3, 4, 5, and successive numbers to infinity; and from these same numbers, which are linear and ex-tended in one direction, combined in no random manner, the polyg-onal and plane numbers are fashioned—the triangles by the combi-nation of root-numbers * immediately adjacent, the square by adding
7 every other term, the pentagons every third term, and so on. In exactly the same way, if the plane polygonal numbers are piled one upon the other and as it were built up, the pyramids that are akin to each of them are produced, the triangular pyramid from the triangles, the square pyramid from the squares, the pentagonal from the penta-gons, the hexagonal from the hexagons, and so on throughout.
8 The pyramids with a triangular base, then, in their proper order, are these: 1, 4, 10, 20, 35, 56, 84, and so on; and their origin is the piling up of the triangular numbers one upon the other, first 1, then 1, 3, then 1, 3, 6, then 10 in addition to these, and next 15 together with the foregoing, then 21 besides these, next 28, and so on to infinity.
9 It is clear that the greatest number is conceived of as being lowest,
1 The following diagrams are from Codex G: Pyramids on square, triangular and pentagonal bases:
Pyramids numerically represented:
a a om
a a a a a: a a aa a aa a a «a a a «@ a a@eéE ce aga aa a dad@aese@ a First Pyramid Second Pyramid Third Pyramid
‘They are built up in layers as it were (cf. sections 7, 9 infra), like piles of shot or spheres of any kind, and the layers are the triangular numbers in order. If all were put in triangular form, it would be clearer.
: That is, gnomons; see on I. 9. 4. In this case the gnomons are the natural series.
for it is discovered to be the base; the next succeeding one is on top of it, and the next on top of that; until unity appears at the apex and, so to speak, tapers off the completed pyramid into a point.
The next pyramids in order are those with a square base which rise 1 in this shape to one and the same point. These are formed in the same way as the triangular pyramids of which we have just spoken. For if I extend in series the square numbers in order beginning with unity, thus, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, and again set the successive terms, as in a pile, one upon the other in the dimension height, when I put 1 on top of 4, the first actual pyramid with square base, 5 is pro- 2 duced, for here again unity is potentially the first. Once more, I put this same pyramid entire, composed of 5 units, just as it is, upon the square 9, and there is made up for me the pyramid 14, with square base and side 3—for the former pyramid had the side 2, and the one potentially first 1 as a side. For here too each side of any pyramid whatsoever must consist of as many units as there are polygonal numbers piled together to create it.
Again, I place’ the whole pyramid 14, with the square g as its base, 3 upon the square 16 and I have 30, the third actual pyramid of those that have a square base, and by the same order and procedure from a pentagonal, hexagonal, or heptagonal base, and even going on farther, we shall produce pyramids by piling upon one another the correspond-ing polygonal numbers, starting with unity as the smallest and going on to infinity in each case.
1 The square pyramids might be represented thus:
a 2 a aa a @ aa aa a @ aa ada aad aaa a@aa
aaa aad
agaacda
a@adada
qaeadaada
gaadgda
These layers are to be piled one above the other in space, and thus the edges will contain as many units as there are layers, or, in other words, as many as the numbers of square numbers taken in addition.
=
ho
From this too it becomes evident that triangles are the most elemen-tary; for absolutely all of the pyramids that are exhibited and shown, with the various polygonal bases, are bounded by triangles up to the apex.
But lest we be heedless of truncated, bi-truncated, and tri-truncated pyramids, the names of which we are sure to encounter in scientific writings, you may know that if a pyramid with any sort of polygon as its base, triangle, square, pentagon, or any of the succeeding polygons of the kind, when it increases by this process of piling up does not taper off into unity, it is called simply truncated when it is left without the natural apex that belongs to all pyramids; for it does not terminate in the potential polygon, unity, as in some one point, but in another polygon, and an actual one, and unity is not its apex, but its upper boundary becomes a plane figure with the same number of angles as the base. If, however, in addition to the failure to terminate in unity it does not even terminate in the polygon next to unity and the first in actuality, such a pyramid is called bi-truncated, and if, still further, it does not have the second actual polygon at its upper limit, but only the one next beneath, it will be called tri-truncated, yes, even four times truncated, if it does not have the next one as its limit, or five times truncated at the next step, and so on as far as you care to carry the nomenclature.
While the origin, advance, increase, and nature of the equilateral solid numbers of pyramidal appearance is the foregoing, with its seed and root in the polygonal numbers and the piling up of them in their regular order, there is another series of solid numbers of a different kind, consisting of the so-called cubes, ‘beams,’ ‘bricks,’ ‘wedges,’ spheres and parallelepipedons, which has the order of its progress somewhat as follows:
The foregoing squares 1, 4, 9, 16, 25, 36, 49, 64, and so on, which are extended in two directions and in their graphic representation in a plane have only length and breadth, will take on yet a third dimension and be solids and extended in three directions if each is multiplied by its own side; 4, which is 2 times 2, is again multiplied by 2, to make 8;
9, which is 3 times 3, is again increased by 3 in another dimension and gives 27; 16, which is 4 times 4, is multiplied by its own side, 4, and 64 results; and so on with the succeeding squares throughout.
Here, too, the sides will be composed of as many units as were in the sides of the squares from which they arose, in each case; the sides of 8 will be 2, like those of 4; those of 27, 3, like those of 9; those of 64, 4, like those of 16; and so on, so that likewise the side of unity, the potential cube, will be 1, which is the side of the potential square, r.
In general, each square is a single plane, and has four angles and four sides, while each several cube, having increased out of some one square multiplied by its own side, will have always six plane surfaces, each equal to the original square, and twelve edges, each equal to and containing exactly the same number of units as each side of the original square, and eight solid angles, each of which is bounded by three edges like in each case to the sides of the original square.
Now since the cube is a solid figure with equal sides in all dimensions, in length, depth, and breadth, and is equally extended in all the six so-called directions,’ it follows that there is opposed to it that which has its dimensions in no case equal to one another, but its depth unequal to its breadth and its length unequal to either of these, for example 2 times 3 times 4, or 2 times 4 times 8, or 3 times 5 times 12, or a figure which follows some other scheme of inequality.
Such solid figures,’ in which the dimensions are everywhere unequal one to another, are called scalene in general. Some, however, using other names, call them ‘wedges,’ for carpenters’, house-builders’ and blacksmiths’ wedges and those used in other crafts, having unequal sides in every direction, are fashioned so as to penetrate; they begin with a sharp end and continually broaden out unequally in all the di-mensions. Some also call them sphekiskoi, ‘wasps,’ because wasps’ bodies also are very like them, compressed in the middle and showing the resemblance mentioned. From this also the sphekoma,’ ‘point of the helmet,’ must derive its name, for where it is compressed it imitates the waist of the wasp. Others call the same numbers ‘altars,’ using
1 Cf. IT. 6. 4 and the note.
? Cf. Theon’s brief account of the solid numbers, p. 41, 8 ff. He has only the name ‘little
altars’ (cf. below) for scalene numbers. * The point of the helmet where the plume was affixed.
|
me
their own metaphor, for the altars of ancient style, particularly the Ionic, do not have the breadth equal to the depth, nor either of these equal to the length, nor the base equal to the top, but are of varied dimensions everywhere.
Now whereas the two kinds of numbers, cube and scalene, are ex-tremes, the one equally extended in every dimension, the other un-equally, the so-called parallelepipedons are solid numbers like means betweenthem. The plane surfaces of these are heteromecic numbers,’ just as in the case of the cubes the faces were squares, as has been shown.
Again, then, to take a fresh start, a number is called heteromecic’ if its representation, when graphically described in a plane, is quadrilat-eral and quadrangular, to be sure, but the sides are not equal one to an-other, nor is the length equal to the breadth, but they differ by 1. Examples are 2, 6, 12, 20, 30, 42, and so on, for if one represents them graphically he will always construct them thus: 1 times 2 equals 2, 2 times 3 equals 6, 3 times 4 equals 12, and the succeeding ones simi-larly, 4 times 5, 5 times 6, 6 times 7, 7 times 8, and thus indefinitely, provided only that one side is greater than the other by 1 and by no other number. If, however, the sides differ otherwise than by 1, for instance, by 2, 3, 4 or succeeding numbers, as in 2 times 4, 3 times 6, 4 times 8, or however else they may differ, then no longer will such a number be properly called a heteromecic, but an oblong number. For the ancients of the school of Pythagoras and his successors saw ‘the other’ * and ‘otherness’ primarily in 2, and ‘the same’ and ‘sameness’
1 See the following chapter.
2 There is no good English equivalent for ἑτερομήκης. Boethius calls this number allera parie longior. To this class belong numbers of the type n(m + 1). The definition is repeated in II. 18. 2; cf. Theon, p. 26, 21 ff.
**The other,’ ‘difference,’ ‘the same,’ and ‘sameness’ are Platonic terms, rather than early Pythagorean. They could have been included as opposites in the lists of such (the everorx la), such as that preserved by Aristotle in Me?., I. 5; but they do not occur there. On the other hand we are informed by Simplicius (Phys., 181, 7 p), quoting Eudorus, that the Pythagoreans made the dpxq# primarily ‘the one’ (rd év), secondarily ‘the one’ and its opposite, under which were Classified respectively ‘elegant things’ (dereia) and ‘trivial things’ (@adAa). This second déox%, Eudorus further says, was called the ‘indefinite dyad’ (dépioros Suds). This latter again is a Platonic term. ‘The same’ and ‘the other’ (ratrév, @drepor) may be seen in a Platonic context in the famous account of the making of the world-soul, Timaeus, 35 A ff. (See on IT. 18. 4), and are generally considered to be Pythagorean at least in ultimate origin. Plato, however, was in 1, as the two beginnings of all things, and these two? are found to differ from each other only by 1. Thus ‘the other’ is fundamentally ‘other’ by 1, and by no other number, and for this reason customarily ‘other’ * is used, among those who speak correctly, of two things and not of more than two.
Moreover, it was shown that all odd number is given its specific 2 form * by unity, and all even number by 2. Hence we shall naturally say that the odd partakes of the nature of ‘the same,’ and the even of that of ‘the other’; for indeed there are produced by the successive additions of each of these—naturally, and not by our decree—by the addition of the odd numbers from 1 to infinity the class of the squares, and by the addition of the evens from 2 to infinity, that of the hetero-mecic numbers.‘
There is, accordingly, every reason to think that the square once 3 more shares in the nature of the same; for its sides display the same ratio, alike, unchanging and firmly fixed in equality, to themselves; while the heteromecic number partakes of the nature of the other; for just as 1 is differentiated from 2, differing by 1 alone, thus also the undoubtedly the one who contributed most to the vogue of these particular terms. Nicoma-chus’s present statements, then, may reasonably be regarded as in accord with later Pythago-reanism which was strongly influenced by Plato. Cf. also Theophrastus, Met., 33, p. 322, 14 Br. Theon of Smyrna describes the heteromecic numbers in a manner that agrees in the main with Nicomachus. He briefly defines them (p. 26, 21) as ‘those with one side greater than the other by a unit,” and notes two methods of producing them in series, (a) by adding together in succession the terms in the series of even numbers, and (b) by multiplying together successive pairs of terms in the natural series. Both methods are mentioned by Nicomachus (sections 1, 2).
1 Cf. the picturesque personification of Theon (p. 27, 1): ‘‘For the beginning of numbers, the monad, which is odd, seeking ‘otherness,’ made the dyad heteromecic by its own doubling” () yap apy Tar dpiOydy, rourdariv 7 words, wepirrh ofca rihv érepdrnra fyrovca rhy dudda érepounxy Ty auras derrkaciacuyp érolnce).
* A somewhat similar distinction in terms was adopted by the arithmologists (see p. 117, n. 4) as a topic in praise of the number 3 (See Theol. Arith., p. 14 Ast; Lydus, De Mensibus, IV. 64 Wiinsch; Anatolius, p. 31, 8 ff. Heiberg; Chalcidius, In Timaeum, c. XXXVIII; Theon of Smyrna, p. 100, 13 ff. Hiller). The purport of these passages is that of 3 we can first use the term ‘all,’ for of one thing or two things we say ‘one’ or ‘both.’ The Theologumena Arithme-ticae adds that, in expressions like ‘thrice ten thousand,’ 3 is used as a symbol of plurality. The notion that 3 was Called ‘all’ as the first possessor of beginning, middle, and end is coupled with the statement above in some of the sources cited. These passages have a bearing on the present utterance of Nicomachus so far as they illustrate the Pythagorean idea that ‘otherness,’ repre-sented by 2, and ‘plurality’ are not identical. Duality and ‘otherness,’ first seen in and typified by 2, are elementary; plurality is derived.
3 Boethius, 2. 27, gives the following explanation why the odd is founded (perfici is his expression) on unity and the even on the dyad: Nam cutuscungue medtetas unus est, ille impar est; cuius vero duo, hic paritate recepla in gemina aequa distungitur. Cf. I. 7. 2.
‘ This method of deriving the heteromecic series is given below in II. 18. 2 and 20. 3, and by Theon (p. 27, 8 ff., 31, 14 ff.).
sides of every heteromecic number differ from one another, one differ-ing from the other by 1 alone.
To illustrate, if I have set out before me the successive numbers in series beginning with 1, and select and arrange by themselves the odd numbers in the line and the even by themselves in another, there are obtained these two series:
IT, 3, 5) 7) 9, IT, 13, 15, 17, 19, 21, 23, 25, 27 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28
4 Now, then, the beginning of the odd series is unity, which is of the same class as the series and possesses the nature of ‘the same,’ and so whether it multiplies itself in two dimensions or in three! it is not made different, nor yet does it make any other number depart from what it was origi-nally,’ but keeps it just as it was. Such a property it is impossible to
5 find in any other number. Of the other series the beginning is 2, which is similar in kind to this series and imitates ‘otherness’; for whether it multiplies itself or another number, it causes a change,® for ex-ample, 2 times 2, 2 times 3.
6 But in cases like 8 times 8 times 2, or 8 times 8 times 3, such solid forms are called ‘bricks,’ * the product of a number by itself and then by a smaller number; if, however, a greater height is joined to the square, as in 3 times 3 times 7, 3 times 3 times 8, or 3 times 3 times 9, or how-ever many times the square be taken, provided only it be a greater number of times than the square itself, then the number is a ‘beam,’ the product of a number by itself and then by a larger number. The
1 éxiréSws 4 crepes: ‘as a surface or as a solid.’
? That is, when it multiplies any other number. Boethius, IJ. 28, says of unity: ... in tantum ciusdem nec mutabilis substantiae est, ut, cum vel se ipsa mulliplicaveril vel in planiiudine vel in profunditate, vel st alium quemlibel numerum per se ipsa multiplicel, a prioris quantilatis forma non discrepel. Cf. IT. 6. 3.
*%xoeracts: literally ‘a standing out of’ (sc. its former state or, as here, number), hence, ‘change.’ Cf. Aristotle, De Anima, 406 b 13, waaa xlynois Uxoracls dari Tov xtvovpévou F xuetrac. When 2 is the multiplier, the result is always different from the multiplicand; for Nicomachus’s number system, consisting of positive integers only, 2x is always different from x.
4 Such a definition as this suits well certain kinds of Roman bricks which were square in their broadest aspect and relatively thin. The Romans introduced baked brick into Greek lands, and Nicomachus would doubtless be acquainted with this variety. Theon, p. 41, 8 ff., gives the same name and definition. Theon also similarly names and defines ‘beams’ and cubes, but for the ‘wedges’ he has only the name ‘little altars’ (cf. II, 16. 2) of the several that Nicomachus uses. Hero of Alexandria (Definition 113, in Hultsch, Heronts Alexandrint Geometricorum et Stereometricorum Reliqutae, p. 31) defines ‘bricks’ as solids with the length less than the breadth and depth, the two latter being sometimes equal (on the ‘bricks,’ cf. also Theon, p. 113, 5); and the ‘beam’ (ibid., Definition 112) he defines as a solid having a length greater than the breadth or thickness, the two latter being sometimes equal.
‘wedges,’ to be sure, were the products of three unequal numbers, and cubes of three equal ones.
Among the cubes, some of them, in addition to being the product of 7 three equal numbers, have the further property of ending at every multiplication in the same number as that from which they began; these are called spherical, and also recurrent.” Such indeed are those with sides 5 or 6; for however many times I increase each one of these, it will by all means end each time in the same figure, the derivative of 6 in 6 and that of 5 in 5. For example, the product of 5 times 5 will end in 5, and so will 5 times this product and if necessary, 5 times this again, and to infinity no other concluding term will be found except 5. From 6, too, in the same fashion 6 and no other will be the concluding term; and so 1 likewise is potentially spherical and recurrent, for as is reasonable it has the same property as the spheres and circles. For each one of them, circling and turning around, ends where it begins. And so these numbers aforesaid are the only ones of the products of equal factors to return to the same starting point from which they began, in the course of all their increases. If they increase in the manner of planes, in two dimensions, they are called circular, like 1, 25, and 36, derived from 1 times 1, 5 times 5, and 6 times 6; but if they have three dimensions, or are multiplied still further than this, they are called spherical solid numbers, for example 1, 125, 216, or, again, 1, 625, 1,296.
Regarding the solid numbers this is for the present sufficient. The 1 physical philosophers, however, and those that take their start with mathematics, call ‘the same’ and ‘the other’ the principles of the uni-verse, and it has been shown that ‘the same’ inheres in unity and the odd numbers, to which unity gives specific form, and to an even greater degree in the squares, made by the continued addition of odd numbers, because in their sides they share in equality; while ‘the other’ inheres in 2 and the whole even series, which is given specific form by 2, and partic-
1 dwoxaracrarixol: So Theon of Smyrna, p. 38, 16 ff. Hiller, citing 5 and 6 asexamples. Lydus also (De Mensibus, IV. 76 Wiinsch) calls 5 a egaipa for the same reason. This property of s is mentioned also by Anatolius, p. 33, 2 ff. Heiberg; and by Capella, De Nuptiis Phil. et Merc.,
VII, 735 (who calls it apocatastaticus). Anatolius remarks on the similar property of 6; cf. also Theol. Arith., p. 35 Ast. In fact these propositions were regular topics of arithmology.
ularly in the heteromecic numbers, which are made by the continued addi- tion of the even numbers, because of the share of the original inequality’
and ‘otherness’ which they have in the difference between their sides.
Therefore it is most necessary further to demonstrate how in these two, as in origins and seeds, there are potentially existent all the peculiar properties of number, of its forms and subdivisions, of all its relations, of polygonals, and the like.
First, however, we must make the distinction whereby the oblong (promecic) number? differs from the heteromecic. The heteromecic is, as was stated above,* the product of a number multiplied by another larger than the first by 1, for example, 6, which is 2 times 3, or 12, which is 3 times 4. But the oblong is similarly the product of two differing numbers, differing, however, not by 1 but by some larger number, as 2 times 4, 3 times 6, 4 times 8, and similar numbers, which in a way exceed in length and overstep the difference of 1.
3 Therefore, since squares are produced from the multiplication of numbers by their own length, and have their length the same as their breadth, properly speaking they would be called ‘idiomecic’ or ‘tau-tomecic’;‘ for example, 2 times 2, 3 times 3, 4 times 4, and the rest. And if this is true, they will admit in every way of sameness and equal-ity, and for this reason are limited and come to an end; for ‘the equal’ and ‘the same’ are so in one definite way. But since the heteromecic numbers are produced by the multiplication of a number by not its own, but another number’s length, they are therefore called ‘hetero-mecic,’ and admit of infinity and boundlessness.
4 In this way, then, all numbers and the objects in the universe which have been created with reference to them are divided and classified and are seen to be opposite one to another, and well do the ancients at the very beginning of their account of Nature make the first subdivision bo
1That is, a unit, for it was shown in II. 17. 2 that ‘the other is fundamentally other by the unit,’ and the difference between the sides of a heteromecic number is by definition a unit.
2 epoutxns: Theon of Smyrna, p. 27, 23 ff., gives a similar definition of this class of numbers, though he calls them wapadAnAdéypaypor apiOuol, but in p. 30, 8 ff. he defines rpoujnes as the products of two unequal terms, which differ by 1, 2, or any other number, thus including the heteromecic among the oblong numbers. Theon gives the following classification of oblongs in connection with the definition just cited: (a) the heteromecic numbers are oblongs in a sense; (b) numbers that by one factoring are heteromecic, by another oblong, as 12, which is either 3 X 4 or 2 X 6; (c) numbers that are oblong by all possible factorings, e.g., 40, which is 2 X 20, 4 X10, or 5 X 8.
*Cf. I. 17. 3.
4 [3uophxeas .. . xal ravroutxes, as opposed to érepoutxes or wpouhnes.
in their cosmogony on this principle. Thus Plato! mentions the dis-tinction between the natures of ‘the same’ and ‘the other,’ and again, that between the essence which is indivisible and always the same and the one which is divided; and Philolaus® says that existent things must all be either limitless or limited, or limited and limitless at the same time, by which it is generally agreed that he means that the uni-verse is made up out of limited and limitless things at the same time, obviously after the image of number, for all number is composed of unity and the dyad, even and odd, and these in truth display equality and inequality, sameness and otherness, the bounded and the bound-less, the defined and the undefined.
That we may be clearly persuaded of what is being said, namely, that things are made up of warring and opposite elements * and have
1 Cf. Plato, Timaeus, 35 a (Archer-Hind’s translation): ‘From the undivided and ever change-less substance and that which becomes divided in material bodies, of both these he mingled in the third place the form of Essence, in the midst between the Same and the Other; and this he composed on such wise between the undivided and that which is in material bodies divided; and taking them, three in number, he blended them into one form, forcing the nature of the Other, hard as it was to mingle, into union with the Same,” etc.
? Philolaus, the Pythagorean, was a native of Croton or of Tarentum. Ritter and Preller (Hist. Phil. Gr.) give 440 B.c. as his floruit. This fragment (1 b Chaignet, 3 Mullach) is found in much fuller form in Stobaeus, Ecl. Phys., I. 21. 7 (vol. I, p. 187, Wachsmuth-Hense).
?Tt isa question whether Nicomachus here has in mind strictly Pythagorean ideas of the origin and constitution of the universe, or the Platonic account in the Timaeus, which is in fact strongly Pythagorean in tone. Elsewhere he refers to the Timaeus (I. 2.1; II. 2. 3; 18. 4; 24. 6) and emphasizes the fact that he hopes to make his work useful for the interpretation of Plato (IL. 24. 11) and of the ancient texts read in the schools, among which the Timaeus was certainly in-cluded (II. 21. 1; 28.1). There is so much in common between Plato and the Pythagoreans that probably Nicomachus would think of both in making this statement. For the Pythagorean doctrine that chaotic matter was ordered on harmonic principles cf. Philolaus, in Stobaeus, op. cit., p. 189 (fr. 4 Chaignet, 3 Mullach): éwel d¢ ral dpyal trdpxor oby duotar odd” dudpudce Yooa, fin dddvaror hs xa adrais cogunOjuer, al uh dpporla éweyévero Wrimwr tpdwy eyédrero.
Plato gives a clearer picture of the ‘warring and opposite’ constituents of the universe (Ni-comachus does not call them ¢ro:yxeia, ‘elements’) in Timaeus, 30 A: Boudrndels ydp 6 Geds dyaea pév wdvra, ddaipow de under elvar card Sivayuy, ovrw 5) war Scov hy dpardy rapadhafSuy oby hov-xlav Ayor dA Kivodperoy wAnupedas wal drdxrws, els rdiiv abro Hyayey éx rijs dratias... .
That this rdéis is a harmony, and furthermore that it is a sort of mathematical harmony, Plato makes clear by showing that it is secured by the interweaving of the world-soul into the whole extent of the universe (36 £) and that the world-soul is constituted on harmonic principles (34c ff.). We may further compare 53 B: dred éwexetpeiro xoopeieGai 7d wav, wip wp@rov kal vdwp cal yiv cal dépa, Cyvn pew Exovra abray drra, warrdwaadl ye why diaxeluera Gowep elxds Exerv day, bray dwg Tims Gebs, ovrw 3) rére wepuxéra Taira wpdror diecxqulaaro elder! re wal dprOpors,
The idea of a chaos of warring elements (frigida pugnabant calidis, umentia siccts, Ovid, Met., I. 19) is a commonplace in ancient literature after Hesiod (Cf. Classical Philology, VIII, p. 405 with note 4). Cf. with this passage I. 6. 3.
in all likelihood taken on harmony—and harmony! always arises from opposites; for harmony is the unification of the diverse and the reconciliation of the contrary-minded—let us set forth in two paral-lel lines no longer, as just previously, the even numbers from 2 by themselves and the odd numbers from 1, but the numbers that are pro-duced from these by adding them successively together, the squares from the odd numbers, and the heteromecic from the even. For if we give careful attention to their setting forth, we shall admire their mu-tual friendship and their coéperation to produce and perfect the re-maining forms, to the end that we may with probability conceive that also in the nature of the universe from some such source as this a similar thing was brought about by universal providence.’
Let the two series then be as follows: That of the squares, from unity, I, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, and that of the heteromecic numbers, beginning with 2 and proceeding thus, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240.
3 In the first place, then, the first square is the fundamental multiple* of the first heteromecic number; the second, compared to the second, is its sesquialter; the third, sesquitertian of the third; the fourth, ses-quiquartan of the fourth; then sesquiquintan, sesquisextan, and so on similarly ad infinitum. Their differences,* too, will increase ac-cording to the successive numbers from 1; the difference of the first terms is 1, of the second 2, of the third 3, and so on. Next, if first the second term of the squares be compared with the first heteromecic num-ber, the third with the second, the fourth with the third, and the rest similarly, they will keep unchanged the same ratios as before, but their differences will begin to progress no longer from 1, but from 2, remain-ing the same as before, and according to the advance observed in the former comparison, the first to the first will be the first, or root-form, multiple, the second to the second the second sesquialter from the root- bo
1 Cf. the remainder of the fragment of Philolaus quoted in the preceding note: rd pér dp duota Kal dudpura dpporlas obdévy éreddovro, rad dé dvduora unde dudpvda pydé looday i dvdyxa TG Toure dpporlg cuyKexdeioGat, al wédAdovri dy xéopyp xaréxerGar. The words are also quoted by Theon of Smyrna, p. 12, 10, and Ast on the passage cites further Iamblichus, Zn Nicom., p- 73, 1 Pistelli, and Asclepius, In Nicom.
* rpdvoa, ‘forethought,’ may be also translated ‘providence’ and has reference to teleology. Even before Plato’s time the teleological idea was in the air, but it was Plato who first made it an essential part of the theory of the constitution of the universe. This is another indication that Nicomachus is decidedly a Platonizing Pythagorean.
2 That is, the double (2 = 2 X 1). For the use of the term wv@yu%», cf. on J. 19. 6.
4 That is, comparing homologous terms of the two series. The difference between 1 and 2 is 1; between 4 and 6, 2; between 9 and 12, 3; and so on.
form, the third to the third the third sesquitertian from the root-form, and the succeeding terms will go on in similar fashion.
Furthermore, the squares! among themselves will have only the odd 4 numbers as differences, the heteromecic, even numbers. And if we put? the first heteromecic number as a mean term between the first two squares, the second between the next two, the third between the two following, and the fourth between the two next succeeding, therein will be seen still more regularly the numerical relations? in groups of three terms. For as 4 is to 2, so is 2 to 1; and as 9 is sesquialter to 6, so is 6 to 4; and as 16 to 12, so is 12 to 9, and so on, with both numbers and ratios regularly advancing. As the greater is to the mean, so will the mean be to the lesser, and not in the same ratio, but always a different one, by an increase. In all the groupings, too, the product of the extremes is equal to the square of the mean; and the extremes, plus twice the mean, by exchange will always give a square.* What is neatest of all, from the addition of both® there comes about the production of the triangles in due order, showing that the nature of these is more ancient*®
1 Squares: 1 4 9 16 25 36 49, etc.
3 5 7 9 II 13, etc.; odd differences. Heteromecic: 2 6 12 20 #30 42 56, etc. 4 6 8 10 12 14, etc.; even differences.
1Cf. Theon, p. 28, 16 ff. Hiller.
* That is, m*, m(m—+ 1), and (m+ 1)? always constitute the terms of a geometrical pro-portion. Theon notes this (c. 16), adding that the successive heteromecic numbers do not make
& proportion with the included square; i.e., m(m—1), m?, and m(m-+ 1) are not proportional. ‘ The ratios formed as directed and the additional properties of the series may be seen in this table:
Ratios Sum of extremes Plus 2 X mean term Sum I:2=2:4 5 = 4 sa 9 4:6=6:9 13 + 12 = 25 g:12 = 12:16 25 + 24 = 49 16:20 = 20:25 4! + 40 = 81
Generally, in the ratio m?:m(m-+ 1) = m(m + 1): (m+ 1)?, the sum of the extremes plus twice the mean will be 4 m* + 4 m+ 1, which is a perfect square.
‘This may best be seen by setting the squares and heteromecic numbers alternately and combining them in pairs thus:
I 2 4 6 9 12 16 20 25 30 3 6 10 15 21 28 86360045 55 = triangular numbers.
® This is significant to Nicomachus because he believed that the ‘same’ and the ‘other’ were the ultimate elements, and that they resided par excellence in the squares and the heteromecic numbers respectively (Cf. II. 17.3; 18.1). Now that into which all plane figures are ultimately analyzed is the triangle (II. 7. 4; cf. 12. 8); as Nicomachus himself shows; and it is to be re-membered that in the Timaeus the triangle is made the ultimate basis of the corpuscles of the elements, a theory which he doubtless has in mind. Therefore the present proposition confirms his position in holding ‘sameness’ and ‘otherness’ to be the most elementary things, prior even to the elementary triangle. An interesting confirmation of the interpretation above is found in
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than the origin of all things, thus: 1 plus 2, 2 plus 4, 4 plus 6, 6 plus 9, 9 plus 12, 12 plus 16, 16 plus 20, and by this process the triangles, which give rise to the polygons, come forth in order.
Still further, every square plus its own side becomes heteromecic, or by Zeus, if its side is subtracted from it. Thus, ‘the other’ is conceived of as being both greater and smaller than “the same,’ since it is produced, both by addition and by subtraction, in the same way that the two kinds of inequality’ also, the greater and the less, have their origin from the application of addition or subtraction to equality. This also is sufficient evidence that the two forms partake of sameness and other-ness, of otherness in an indefinite fashion, but of sameness definitely, 1 and 2 generically,” but the odd of sameness after the manner of a subordinate species because it belongs to the same class as 1, and the even of otherness because it is homogeneous with 2.
There is also a still clearer reason why the square, since it is the prod-uct of the addition of odd numbers, is akin to sameness, and the heteromecic numbers to otherness because it is made up by adding even numbers; for as though they were friends of one another, these two forms share in their two rows the same differences when they do not have the same ratios, and conversely the same ratios when they do not have the same differences. For the difference * between 4 and 2 in the
Theol. Arith., 8 Ast (on the dyad) in a context certainly Nicomachean: ‘‘ Wherefore the first congress of these (sc. the monad and the dyad) first perfected defined multitude, the element of things, which would be a triangle both of magnitudes and of numbers somatic and bodiless; for as the rennet coagulates the running milk by its active and effective property, so the unifying force of the monad approaching the dyad, which is the source of facile movement and of disso-lution, gave a bound and a form that is a number to the triad; for this is the beginning of ac-tuality in number as that is defined by the combination of monads.”’ In the present text ‘their nature’ refers to the squares and heteromecic numbers and ‘the origin of all things’ to the tri-angle, as the elementary figure.
1Cf. I. 17.6. The results obtained by adding to or subtracting their sides from the square numbers are as follows:
4t2=—= 6=2X%3 4—-2= 2=1%X%2
QtZ=I2=3X4 9–3= 6=2X3
16+4=20=4X5 iG—4=12=3%X%4
2a5+5=30=5X6 as—c=20=4X5 or or
m +m = (m+ 1m m—m= (m—1)m
2 See p. 101 for comment in this passage.
3 The examples in the text show sameness of difference coupled with difference of ratio. For the converse, compare 4, 6, and 6,9. The ratio is sesquialter in both cases, but the difference varies.
double ratio is found between 6 and 4 as a superparticular; and again the difference between 9 and 6, as a sesquialter, is found between 12 and 9 as a sesquitertian, and so on. What is the same in quality‘ is different in quantity, and just the opposite, what is the same in quan-tity is different in quality. Again, it is clear that in all their relations? the same difference between two terms will necessarily be called frac-tions with names that differ by 1, and be the half of one and the third of the other, or the third of one and the quarter of the other, or the fourth of one and the fifth of the other, and so on.
But what will most of all confirm the fact that the odd, and never the even, is preéminently the cause of sameness, is to be demonstrated in every series beginning with 1 following some ratio, for example, the double ratio, 1, 2, 4, 8, 16, 32, 64, 128, 256, or the triple, I, 3,9, 27, 81, 243, 729, 2,187, and as far as you like. You will find? that of necessity all the terms in the odd places in the series are squares, and no others by any device whatsoever, and that no square is to be found in an even place.
But all the products of a number multiplied twice into itself, that is, the cubes, which are extended in three dimensions and seen to share in sameness to an even greater extent, are the product of the odd num-bers, not the even,‘ 1, 8, 27, 64, 125, and 216, and those that go on analogously, in a simple, unvaried progression as well. For when the successive odd numbers are set forth indefinitely beginning with 1, observe this: The first one makes the potential cube; the next two,
1 This is substantially a repetition of the previous statement. In the terminology of Nicoma-chus two pairs of numbers have a relation (cxéots) qualitatively (wocdrqr:) alike if they exhibit the same ratio; numerically (woodryrc), if they have the same arithmetical difference. This terminology appears again in the discussion of the proportions; cf. 23, 4 below.
*gxéees: That is, the comparisons of the pairs of terms from the two series; see on 21. 2 below for a further discussion of the meaning of oxéors ‘relation.’ For illustration of the mean-ing, we may take pairs of homologous terms from the two series, as 1, 2; 4,6; 9,12; 16, 20; etc., and their differences 1, 2, 3, 4, etc. 1 is the whole of 1 and the half of 2; 2 is the half of 4 and the third of 6; 3 is the third part of 9 and the fourth of 12, etc. Or, compare 4, 2 with 4,6; 9, 6 with 9, 12; etc.
*Philo, De Mundi Opificio, 36, points out further properties of the table of doubles: “If one doubles he will find that the third from unity (i.e., counting both ends in the Greek fashion) is a square, the fourth a cube, and the seventh, arising from both the third and the fourth, a cube and square together.” Similarly Theon, p. 34, 16 ff., points out that the terms in every other place (he does not specifically say ‘odd’) of the series of multiples are squares, those in every third place cubes, and those in every fifth place both cubes and squares. He adds furthermore that squares are always divisible by 3 and 4, either as they stand or when 1 is subtracted, that those divisible by 3 when 1 is subtracted are always divisible by 4 if they are even, and vice versa, etc.
‘See Part I, p. 133.
added together, the second; the next three, the third; the four next following, the fourth; the succeeding five, the fifth; the next six, the sixth; and so on.
After this it would be the proper time to incorporate the nature of proportions, a thing most essential for speculation about the nature of the universe and for the propositions of music, astronomy, and geome-try, and not least for the study of the works of the ancients,’ and thus to bring the Introduction to Arithmetic to the end that is at once suitable and fitting.
A proportion,” then, is in the proper sense, the combination of two or more ratios, but by the more general definition the combination of
1 els rds TGv wadatdy suvavayrwces: Boethius, II. 40: ad velerum lectionum inielligentiam.
? Two Greek words, dradoyla and peodrys, the former of which is used here, may be translated ‘proportion,’ and Nicomachus points out that an dvadoyla is, strictly speaking, a combina-tion of ratios (like 1, 2, 4; i.e., in his classification only the ‘geometrical proportions,’ yewpe-Tpixal dyadoyla: or weodryres). Properly then an arithmetical progression of three or four terms (e.g., 1, 2, 3, OF I, 2, 3, 4) should not be called an dvadoyla, but in practice it is so called. It was formerly thought until Nesselmann cleared up the matter (Geschichte der Algebra, p. 210, note) that d»adoyla meant, properly, a proportion of four terms (i.e., a disjunct proportion; see section 6), and yeoérys a continued proportion (of three terms; cf. section 5). Arguing from the statements of Iamblichus (pp. 100, 15 ff. and 104, 19 ff. Pistelli; the latter passage runs) 8¢ devrépa perdrns 7 yewperpixh xuplws dvadoyla xéxAnrat, diubre Abyor roy adbrdy ol Spor wepiéxovory, dvd rév atrép héyor dueorGres) and Nicomachus himself (see II. 24. 1), Nessel-mann states that originally dvyadoyla was applied only to geometrical proportions and pécorns to the other two, the harmonic and the arithmetic, but that in later usage the distinction of terms vanished. It is certain that Nicomachus uses them indiscriminately of all three types. But the present passage makes the matter perfectly clear if the proper stress is laid on the words Aédyos and oxéo.s. Nicomachus here definitely states that a proportion (dvadoyla) is properly or strictly (xvplws) the combination of two or more ratios (Aéywr), but in a more general sense (xot-yérepor) a combination of relations (cxérewr). Now dédbyos is the term which properly means ratio, the measurement of one number in terms of another, and it is not used by Nicomachus with reference to the relations between numbers in any other sense (he does, to be sure, use Aéyos with other meanings, but not with reference to the relations between numbers; see the Glossary).
Nicomachus is undoubtedly woefully lacking in precision when he defines Aéyor (section 3), but in his usage he is consistent. oxéo. on the other hand means simply relation and can refer to any kind of relation, including Aéyo proper and mere numerical excess or deficiency as well; it is therefore a more general term and sometimes, but not always, synonymous with Aéyos. Here, however, Nicomachus uses it in the general sense, so that it includes the relation of exceed-ing and being exceeded. He admits, then, that dvadoyla is used of arithmetic and harmonic progressions as well as of the true proportions, the geometric. It may be noticed that in dis-cussing arithmetic proportions Nicomachus does not use the term Adéyos to describe their mutual relation; in fact he says that they are not in the same Adyos (II. 23.1: Gray... wh pérror Abyos 6 abrds év rots Spois wpds dAAHAOUs ylenrar). In I]. 22. 3 (dv ydp rH rod dwhod dpocOuod guouwy dwd porddos exPéoe: . . . cwherar 6 ravrns udyns Adyos, where raérys = Tis docOunrixys)
two or more relations, even if they are not brought under the same ratio, but rather a difference, or something else.
Now a ratio’ is the relation of two terms to one another, and the combination of such is a proportion, so that three is the smallest num-ber of terms of which the latter is composed, although it can be a series of more, subject to the same ratio or the same difference. For ex-ample, 1: 2 is one ratio, where there are two terms; but 2: 4 is another similar ratio; hence 1, 2, 4 is a proportion, for it is a combination of ratios, or of three terms which are observed to be in the same ratio to one another. The same thing may be observed also in greater numbers and longer series of terms; for let a fourth term, 8, be joined to the former after 4, again in a similar relation, the double, and then 16 after 8 and so on.
Now if the same term, one and unchanging, is compared to those on
éyos does not mean ‘ratio,’ but ‘scheme,’ ‘plan,’ or, practically, ‘definition.’ In I. 5. 1 he speaks of dpyomxol Aéyo:, meaning the intervals of harmony, but only to point out that these are really ratios in the strict sense.
A good illustration of the proper use of the words Aéyos and oxéois is found in Theon, p. 22, 19 Hiller: av&éuevor 52 rods Adyous rijs wpds dAAHAOUs cxdcews avray pete (‘as they increase they diminish the ratios of their relation to one another’). He is speaking of the fact that in the natural series of numbers the ratios between the terms decrease as you ascend in the series, e.g., 2:1 > 3:2> 4:3, etc. Adyos is the ratio; oxéois means the general relation existing between pairs of terms, that is, the simple pairing and comparing of the terms. It is further to be noted with regard to the terminology of proportions that, according to Iamblichus (p. 99, 25 ff.), dyvadoyla properly refers to the continued proportion (that in three terms), as opposed to To dvddo-yor, which means rather a geometrical progression and is the more proper term to apply to the disjunct proportion (in four terms). Needless to say, the distinction is not observed by Nicomachus.
1 In view of what has been said in the preceding note this definition is a poor one, for it merely asserts that a ratio (Adyos) is a relation (¢xéors). Nicomachus is either guilty of carelessness, or, as is very probable, the word rod has fallen out before ox éo:s, leaving no trace in the MSS. The addition of this one word would make the definition fairly satisfactory, although it would still lack the precision of Euclid’s, or Theon’s. In mathematical language roid oxéors ‘a rela-tion of some kind,’ ‘a qualitative relation,’ means one that can be described as ‘of some kind,’ that is, double, triple, sesquialter, or the like, in other words, a ratio proper, and it would be con-trasted with roo} oyéois, ‘a relation of a certain amount,’ which would mean a relation where a mere arithmetical difference between the terms is in question. Euclid uses this terminology in his definitions of ratio and proportion in Book V, init.: ‘‘ Ratio is the qualitative relation with reference to size between two homologous magnitudes. Proportion is the likeness of ratios” (Adyos dort Sto peyeOSy opoyeray f ard wydkixérntd wow cyéots. dvadroyla 8 éorly 4 Tray Adywy duoidrns); cf. also Hero of Alexandria, Definition 127, ed. Hultsch, p. 36. As Nesselmann (Gesch. d. Alg., 212) showed, the inclusion of dvoyerdv, rydkcxérqra and duoidrys brings out points overlooked by Nicomachus, but even more important is xoid. On the necessity of terms in a ratio being homogeneous, see on I. 17. 4; Nicomachus neglects this matter. Theon’s definitions of ratio and proportion are more like Euclid’s, but that of proportion is either poorly stated or wrongly transmitted: ‘Ratio is the qualitative relation in analogy (xar dvddoyor) existing between two terms of the same genus” (p. 73, 16); “‘Proportion is the qualitative relation of ratios to one another” (p. 74, 12).
either side of it, to the greater as consequent’ and to the lesser as ante-cedent, such a proportion is called continued; for example, 1, 2, 4 isa continued proportion as regards quality,’ for 4: 2 equals 2:1, and con-versely 1: 2 equals 2:4. In quantity, 1, 2, 3, for example, is a contin-ued proportion,’ for as 3 exceeds 2, so 2 exceeds 1, and conversely, as 1 is less than 2, by so much 2 Is less than 3.
6 If, however, one term answers to the lesser term, and becomes its antecedent and a greater term, and another, not the same, takes the place of consequent and lesser term with reference to the greater, such a mean and such a proportion is called no longer continued, but dis-junct; for example, as regards quality, 1, 2, 4, 8, for 2:1 equals 8:4, and conversely 1: 2 equals 4: 8, and again 1: 4 equals 2: 8or 4: 1 equals 8:2; and in quantity, I, 2, 3, 4, for as 1 is exceeded by 2, by so much 3 is exceeded by 4, or as 4 exceeds 3, so 2 exceeds 1, and by interchange, as 3 exceeds 1, so 4 exceeds 2, or as 1 is exceeded by 3, by so much 2 is exceeded by 4.
1 The first three proportions,* then, which are acknowledged by all the ancients, Pythagoras, Plato, and Aristotle, are the arithmetic, geometric, and harmonic; and there are three others subcontrary to
On the meaning of ‘antecedent’ and ‘consequent,’ see the note on I. 19. 2.
?On the meaning of ‘quality’ referring to ratios, compare on II. 20.3. With reference to proportions, as here, the meaning is consistent with the former usage. A proportion as regards quality (card woidrynra, xara 7d wordy, wordryTt) isa series of terms exhibiting similar ratios, and a proportion in quantity (xara rooéryra, xara Td woody, wooéryr:) is an arithmetical pro-gression, with a common difference. Cf. also II. 22.2; 23. 4 below.
3 cuvnupévn: Theon, p. 82, 10, uses the term guvexs and diypyuéry for ‘disjunct’ (in Nicoma-chus, dietevypérn, sect. 6).
‘ Iamblichus adds concerning the history of the proportions (p. 100, 19): ‘‘ But of old there were but three means in the days of Pythagoras and the mathematicians of his time, the arith-metic, the geometric, and the third in order, which was once called the subcontrary, but had its name forthwith changed to harmonic by the schools of Archytas and Hippasus, because it seemed to embrace the ratios that govern the harmonized and tuneful. And it was formerly called sub-contrary because its character was somehow subcontrary to the arithmetic. ... After this name had been changed, those who came later, Eudoxus and his school, invented three more means, and called the fourth properly subcontrary because its properties were subcontrary to the har-monic. . . and the other two they named simply from their order, fifthand sixth. The ancients and their successors thought that this number, i.e., six, of means could be set up; but the moderns have found four more in addition, devising their formation from the terms and the intervals.” Cf. also p. 113, 16 ff. He adds (p. 116, 1 ff.) that the first six were in use from Plato’s time to Eratosthenes, and that the other four were devised by Myonides and Euphranor, both Pythago-reans, who lived later. Apparently Moderatus of Gades, as well as Nicomachus, employed all ten forms (see Proclus, In Tim., II. 18. 29 ff. Diehl).
them, which do not have names of their own, but are called in more general terms the fourth, fifth, and sixth forms of mean; after which the moderns discover four others as well, making up the number ten,’ which, according to the Pythagorean view, is the most perfect possible. It was in accordance with this number indeed that not long ago the ten relations? were observed to take their proper number, the so-called ten categories,* the divisions and forms of the extremities of our hands and feet, and countless other things which we shall notice in the proper place.’
Now, however, we must treat from the beginning, first, that form of 2 proportion which by quantity ° reconciles and binds together the com-parison of the terms, which is a quantitative equality as regards the difference of the several terms to one another. This would be the arithmetic proportion, for it was previously reported that quantity is its peculiar belonging. What, then, is the reason that we shall treat 3 of this first, and not another? Is it not clear that Nature shows it forth before the rest? For in the natural series of simple numbers, beginning with 1, with no term passed over or omitted, the definition of this proportion® alone is preserved; moreover, in our previous statements,’ we demonstrated that the Arithmetical Introduction itself is antecedent to all the others, because it abolishes them together with itself, but is not abolished together with them, and because it is im-plied by them, but does not imply them. Thus it is natural that the mean which shares the name of arithmetic will not unreasonably take
1 The sacredness of the number 10 was a favorite theme of the Pythagoreans. 10 symbolized for them the universe, and by the felraktys (1 + 2 + 3 +4 = 10) their most sacred oath was taken. It is the all-inclusive nature which they discovered in the decad that gained it its peculiar reverence from them, and Nicomachus here cites evidence of the type accepted by the Pythag-orean school to substantiate that property of the decad. It is well to note that in two other Nicomachean sources similar statements are found. Photius’s report of the Theologumena Arithmeticae represents the decad as the universe because there are 10 fingers, 10 toes, 10 cate-gories and 10 parts of speech, and because it comprehends all solid and plane figures, all kinds of number and of numerical relations; and there is a close parallel passage in Ast’s Theol. Arith., p- 59. Another instance of the reverence paid to the decad and its supposed universal character among numbers was seen in I. 19.17 (Cf. the note). On the Pythagorean decad in general, cf. Philolaus, in Stobaeus, Ecl., I (Wachsmuth-Hense, vol. I, p. 16); Aristotle, Med., I. 5. 986 a, 8.
? The reference is to I. 17. 7–8 and the following discussion.
*The Aristotelian categories; cf. Part I, p. 95, notes 1 and 2. Boethius, II. 42, says that Archytas the Pythagorean first distinguished the categories, licef guibusdam sil ambiguum, and that Plato followed his distinction. There seems to have been a book on the categories (falsely) attributed to Archytas. See Part I, ibid.
‘ This seems to be a reference to the work called Theologumena Arithmeticae.
' xara 7d xogdy: Cf. the notes on IT. 20. 3; 21. 5.
84 ravrns udens Néyos; cf. on II. 21. 2. 7Cf. L. q. 2 ff. and the note.
precedence of the means which are named for the other sciences, the geometric and harmonic; for it is plain that all the more will it take precedence over the subcontraries,’ over which the first three hold the
4 leadership. As the first and original, therefore, since it is most de-