Introduction to Arithmetic, Book II

Nicomachus of Gerasa · Introduction to Arithmetic

Section 3 of 3

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I

in the same ratio as that which the differences bear to one another; and the reason of this is that the two mean terms are the products of the sides of the cubes commingled, 2 times 2 times 3 and 3 times 3 times 2.

In general, then, if a square takes a square, that is, multiplies it, it always makes a square; but ifa square multipliesa heteromecic number, or vice versa, it never makes a square; and if cube multiplies cube, a cube will always result, but if a heteromecic number multiplies a cube, or vice versa, never is the result a cube. In precisely the same way if an even number multiplies an even number, the product is always even and if odd multiplies odd always odd; but if odd multiplies even or even odd, the result will always be even and never odd.!’ These mat-ters will receive their proper elucidation in the commentary on Plato,’ with reference to the passage on the so-called marriage number in the Republic introduced in the person of the Muses. So then let us pass over to the third proportion, the so-called harmonic, and analyze it.

The proportion that is placed in the third order is one called the harmonic,’ which exists whenever among three terms the mean on examination is observed to be neither in the same ratio to the extremes, antecedent of one and consequent of the other, as in the geometric proportion, nor with equal intervals, but an inequality of ratios, as in the

1 The propositions stated here are: (1) m’n? is always a square; (z) m’n(n + 1) is never a square; (3) mn? is always a cube; (4) n(n + 1) is never a cube; (5) 2m X 2n is always even; (6) (2 m+ 1)(2” + 1) is always odd; (7) 2 m(2 + 1) is always even.

The formula for the ‘marriage number’ occurs in the Republic, 546 a ff. The meaning of the passage is still disputed. Nicomachus may perhaps refer to some work of his in which he commented on the Republic.

* Iamblichus (p. 100, 19 ff.) names this among the three kinds of proportion known to Pythag-oras and his school, by whom it was called twrevarria because it was considered to be subcon-trary to the arithmetic; the name, however, was changed to dpyomx?, harmonic, by the schools of Archytas and Hippasus, because they found in it the harmonic ratios. Iamblichus adds

(p. 108, 3 ff.) that the fundamental forms (rv@uéves) of this proportion are 2, 3, 6 and 3, 4, 6, the multiples or superparticulars of which terms give other examples of it. On account of this limi-tation the name ‘fixed,’ ‘established’ (¢écrn«via) was given to the harmonic proportion by some. It will be noted that Nicomachus uses the examples mentioned, although he does not speak of such a limitation.

arithmetic, but on the contrary, as the greatest term is to the small-est, so the difference between greatest and mean terms is to the differ-ence between mean and smallest term.’ For example, take 3, 4, 6, or 2, 3,6. For 6 exceeds 4 by one third of itself, since 2 is one third of 6, and 3 falls short of 4 by one third of itself, for 1 is one third of 3. In the first example, the extremes are in double ratio and their differences with the mean term are again in the same double ratio to one another; but in the second they are each in the triple ratio.

It has a peculiar property, opposite, as we have said,” to that of the arithmetic proportion; for in the latter* the ratios were greater among the smaller terms, and smaller among the greater terms. Here, however, on the contrary, those among the greater terms are greater and those among the smaller terms smaller, so that in the geometric pro-portion, like a mean between them, there may be observed the equality of ratios on either side, a midground between greater and smaller.

Furthermore, in the arithmetic proportion’ the mean term is seen to be greater and smaller than those on either side by the same fraction of itself, but by different fractions of the terms that flank it; in the har-monic, however, it is the opposite, for the middle term is greater and less than the terms on either side by different fractions of itself, but always the same fraction of those terms at its sides, a half of them or a third; but the geometric,® as if in the midground between them,

' The general formula then is a:¢c = a—6:b—cc. For Theon’s discussion see p. 114, 14 ff., Hiller.

2 Cf. II. 23.6. TIamblichus (p. 110, 18 ff.) says that this was the opinion of the Pythagoreans but that some considered the harmonic proportion contrary to both the arithmetic and geometric. He then argues elaborately for the view expressed in the text, that it is subcontrary to the arith-metic only.

* Compare the ratios of the terms in the harmonic series 3, 4,6 and the arithmetic series 3, 4, 5:

4 = 1} (less) 4 = 11 (greater) S$ = 1} (greater) 5 = 11 (less)

‘ The following examples will illustrate Nicomachus’s meaning.

Series a“. 2 Ss £ fae £ 2 fp oh h F 31 4) 5 3r 4 6 4, 6, 9 Differences ie se ee I, I 1,2 2,3 Which are the following fractions of the mean 4 4 4 i i}

And the following fractions of the extremes

xt 54 44 In each case there are two differences. These are (a) in the arithmetic, the same fraction of the middle term but different fractions of the extremes; (6) in the harmonic, different fractions of the mean, but the same fraction of the extremes; (c) in the geometric, different fractions of both mean and extremes.

’ What is meant by the elliptic statement of the text, ‘the geometric. . . neither in the mean alone, nor in the extremes alone, but in both mean and extreme,’ may be seen from the example in the preceding note. The middle term does not differ from both extremes by the same

shows this property neither in the mean term exclusively nor in the extremes, but in both mean and extreme.

Once more, the harmonic proportion has as a peculiar property the fact that when the extremes are added together’ and multiplied by the mean, it makes twice the product of themselves multiplied by one another.

The harmonic proportion was so called because the arithmetic proportion was distinguished by quantity, showing an equality in this respect with the intervals from one term to another, and the geometric by quality, giving similar qualitative relations between one term and another, but this form, with reference to relativity, appears now in one form, now in another, neither in its terms exclusively nor in its differences exclusively, but partly in the terms and partly in the differ-ences; for as the greatest term is to the smallest, so also is the differ-ence between the greatest and the next greatest, or middle, term to the difference between the least term and the middle term, and vice versa.

In the classification of Being previously set forth we recognized the relative? as a thing peculiar to harmonic theory; but the musical ratios of the harmonic intervals are also rather to be found in this proportion. The most elementary is the diatessaron, in the sesquitertian ratio, 4:3, which is the ratio of term to term ® in the example in the double ratio, or of difference to difference in that which follows, the triple, for these differences are of 6 to 2 or again of 6to3. Immediately following is the diapente, which is the sesquialter, 3: 2 or again, 6: 4, the ratio of part of itself nor by the same part of the extremes; but from the first term by the same frac-tion of itself as the fraction of the last term by which it differs from the last term; and con-versely, sameness and difference of the fraction involved in comparing the terms do not lie, in this type of proportion, exclusively in the relations to the middle term, or in those of the middle term to the extremes.

! Thus in 2, 3,6, (2 +6) X 3 = 24and2 X 2 & 6.

In general qo 8 whence 6 = 2% and b(a +0) = 2 a€. c b—-e a+c

? See I. 3. 1.

* The examples referred to are the harmonic proportions cited in II. 25.1. The proportion in double ratio is 3, 4, 6, and that in triple ratio is 2, 3,6. The first two terms are in sesquiter-tian ratio in the former (4: 3) and the differences of 6 to 2 and of 3 respectively (4, 3) give the same ratio in the latter.

term to term.’ Then the combination? of both of these, sesquialter and sesquitertian, the diapason, which comes next, is in the double ratio, 6: 3 in both of the examples, the ratio of term to term.* The fol-lowing interval, that of the diapason and diapente together,* which preserves the triple ratio of the two of them together, since it is the com-bination of double and sesquialter, is as 6: 2, the ratio of term to term in the example in the triple ratio, and likewise of difference to differ-ence in the same, and in the proportion with double ratio it is the ratio of the greatest term to the difference between that term and the mean term, or of the difference between the extremes to the difference be-tween the smaller terms. The last and greatest interval, the so-called di-diapason, as it were twice the double, which is in the quadruple ratio,® is as the middle term in the proportion in the double ratio to the difference between the lesser terms, or as the difference between the extremes, in the example in the triple ratio, to the difference between the lesser terms.

Some, however, agreeing with Philolaus, believe that the proportion is called harmonic because it attends upon all geometric harmony, and they say that ‘geometric harmony’ is the cube because it is harmo-nized in all three dimensions, being the product of a number thrice multiplied together. For in every cube® this proportion is mirrored; there are in every cube 12 sides, 8 angles and 6 faces; hence 8, the mean between 6 and 12, is according to harmonic proportion, for as the extremes are to each other, so is the difference between greatest and

' That is, the sesquialter ratio may be derived from the terms of the two harmonic proportions cited without calling in differences. 6:4 comes from the double ratio; 3: 2, from the triple.

3 erie eee (2, 3, 6 and 3, 4, 6) the diapason occurs among the terms (3: 6).

‘ This, a triple ratio, is seen in the triple harmonic series 2, 3, 6 (a) in the terms 6 and 2; (6) in the differences (6—3 = 3 and 3—2 = 1); and in the double series 3, 4, 6 (a) in the ratio of 6 to (6—4), i.e., 2, and (5) in the ratio between the difference of 6 and 3 and that between 4 and 3. On the triple ratio as the combination of the double and the sesquialter, cf. II. 5. 4.

The harmonic series 2, 3, 6 is of the type that Nicomachus used in illustration in IT. 5, as may be seen from the following arrangement:

2 | : 3 6 Sesquialter Double, (Diapente), _ (Diapason)

Triple (Diapason and Diapente together)

' In the double series 3, 4, 6, it is the ratio of 4 to (¢—3 =)1; in the triple, 2, 3, 6, it is the ratio of (6—2 =) 4 to (3—2 =) 1.

* Nicomachus shows first that this series, 6, 8, 12, conforms to all the tests of the harmonic proportion just stated, and then that all the ratios of the harmonic intervals are to be found among its terms and differences.

middle term to that between the middle and smallest terms, and, again, the middle term is greater than the smallest by one fraction of itself and by another is less than the greater term, but is greater and smaller by one and the same fraction of the extremes. And again, the sum of the extremes multiplied by the mean makes double the product of the extremes multiplied together. The diatessaron is found in the ratio 8:6, which is sesquitertian, the diapente in 12: 8, which is sesquialter; the diapason, the combination of these two, in 12:6, the double ratio; the diapason and diapente combined, which is triple, in the ratio of the difference of the extremes to that of the smaller terms, and the di-diapason is the ratio of the middle term to the difference between itself and the lesser term. Most properly, then, has it been called harmonic.

Just as in the division of the musical canon,’ when a single string is stretched or one length of a pipe is used, with immovable ends, and the mid-point shifts in the pipe by means of the finger-holes, in the string by means of the bridge, and as in one way after another the aforesaid proportions, arithmetic, geometric, and harmonic, can be produced, so that the fact becomes apparent that they are logically? and very properly named, since they are brought about through changing and shifting the middle term in different ways, so too it is both reasonable and possible to insert the mean term that fits each of the three pro-portions between two arithmetic terms, which stay fixed and do not change, whether they are both even or odd. In the arithmetic pro-portion this mean term is one that exceeds and is exceeded by an equal amount; in the geometric proportion it is differentiated from the

1 The woveixds kava» was a measuring rod corresponding to, and placed by the side of, the monochord, on which by means of a movable bridge experiments were made to determine the musical intervals precisely, by the use of mathematics instead of by ear. The procedure in such experiments may be gathered from Boethius, Inst. Mus., IV. 5: Sit chorda intensa AB. Huic aequa sit regula, quae propostits pariitionibus dividatur, ut ca regula chordae apposita eaedem divi-stones in nervi longiludine signentur quas antea signaveramus in regula. Nos vero nunc tla dividi-mus quasi ipsam chordam et non regulam partiamur. He then describes the actual division. Apparently similar experiments could be made on one pipe of the flute. The word used by Nicomachus for ‘bridge,’ traywyevs, is properly referred to a movable, as opposed to a fixed, bridge (uayds); Ast in his note on the present passage cites a scholium on Ptolemy, Harm., 1. 8: payas 7 wh ayoudrn, bwayuryels 52 db dyopevos” Karaxpyorimds 82 xal wayds A€yera trayoryevs. The expression xdvwvros xararouy is the title of a work of Euclid.

2 That is, they are properly called yeorédryres (‘means’) because the mean term, uéoos Spos, determines their character.

extremes by the same ratio, and in the harmonic it is greater and smaller than the extremes by the same fraction of those same extremes.

Let there be given then, first, two even terms, between which we must find how the three means would be inserted, and what they are. Let them be 10 and 40.

First, then, I fit to them the arithmetic mean. It is 25, and the at-tendant properties of the arithmetic proportion are all preserved; for as each term is to itself, so also is difference to difference; they are in equality, therefore. And as much as the greater exceeds the means by so much the latter exceeds the lesser term; the sum of the extremes? is twice the mean; the ratio of the lesser terms? is greater than that of the greater; the product of the extremes’? is less than the square of the mean by the amount of the square of the differences; and the middle term‘ is greater and less than the extremes by the same fraction of it-self, but by different fractions regarded as parts of the extremes.

If, however, I insert 20 as a mean between the given even terms, the properties of the geometrical proportion come into view and those of the arithmetic are done away with. For as the greater term is to the middle term, so is the middle term to the lesser; the product of the extremes is equal to the square of the mean; the differences are ob-served to be in the same ratio to one another as that of the terms; °® neither in the extremes ° alone nor in the middle term alone does there reside the sameness of the fraction concerned in the relative excess and deficiency of the terms, but in the middle term and one of the extremes by turns; and both between greater and smaller terms there is the same ratio.

But if I select 16 as the mean, again the properties of the two former proportions disappear and those of the harmonic are seen to remain l4o +10 = 2% 25.

225 > 4g.

325?—(10 X 40) = 225 = 15°.

4 As the numerical difference is constant, it follows that the mean is both exceeded and exceeds by the same fraction of itself (i-e., 4 or }). But 15 compared with the extremes is } of 10 and 3% of 40.

* The differences in the series 10, 20, 40 are 10 and 20. Both terms and differences are in the double ratio.

® The reference is to the peculiarity of the geometrical proportion noted in IT. 25. 3. In this case the difference between 10 and 20 is the whole of 10 and half of 20; that between 20 and go is the whole of 20 and half of 40. If both the differences are viewed in relation either to the mean or to the respective extremes, the fraction is not constant; but if one difference be regarded as a fraction of the mean, while the other is regarded as a fraction of the extreme, there will be “‘iden-tity of the fraction of excess and defect.”

fixed, with respect to the two even terms. For as the greatest term? is to the least, so is the difference of the greater terms to that of the lesser; by what fractions, seen as fractions of the greater term, the mean is smaller than the greater term, by these the same mean term is greater than the smallest term when they are looked upon as fractions of the smallest term; the ratio between the greater terms is greater, and that of the smaller terms, smaller, a thing which is not true of any other proportion; and the sum of the extremes multiplied by the mean is double the product of the extremes.

6 If, however, the two terms that are given are not even but odd, like 5, 45, the same number, 25, will make the arithmetic proportion; and the reason for this is that the terms on either side overpass it and fail to come up to it by an equal number, keeping the same quantitative difference with respect to it. 15 substituted makes the geometric proportion, as it is the triple and subtriple of each respectively; and if g takes over the function of mean term it gives the harmonic; for by those parts of the smaller term by which it exceeds, namely, four fifths of the smaller, it is also less than the greater, if they be regarded as parts of the greater term, for this too is four fifths, and if you try all the previously mentioned properties of the harmonic ratio you will find that they will fit.

7 And let this be your method whereby you might scientifically fashion the mean terms that are illustrated in the three proportions. For the two terms given you, whether odd or even, you will find the arithmetic? mean by adding the extremes and putting down half of them as the mean, or if you divide by 2 the excess of the greater over the smaller, and add this to the smaller, you will have the mean. As for the geo-metric mean, if you find the square root of the product? of the ex-tremes, you will produce it, or, observing the ratio‘ of the terms to one

1Jn the series 10, 16, 40 (2) @ = 216 4, 10 16—10 6 (b) The difference between 40 and 16, 24, is 3 of 40, and the difference between 16 and ro,

6, is 2 of ro.

*In general terms, the arithmetic mean between a and b is a7 b, or, ifa <b, itisa + b = g,

In general terms the geometric mean is ab. ‘ Nicomachus is stating the following proposition put into general terms: given a and ar, @ 2 will form a geometric proportion with them and a: = = = :ar. This will hold good only for another, divide this by 2 and make the mean, for example, the double, in the case of a quadruple ratio. For the harmonic mean,’ you must multiply the difference of the extremes by the lesser term and divide the product by the sum of the extremes, then add the quotient to the lesser term, and the result will be the harmonic mean.

So much, then, concerning the three proportions celebrated by the ancients, which we have discussed the more clearly and at length for just this reason, that they are to be met with frequently and in various forms in the writings of those authors.2/ The succeeding forms,* how-ever, we must only epitomize, since they do not occur frequently in the ancient writings, but are included merely for the sake of our own proportions of the type cited in Nicomachus’s example, those where the ratio is the quadruple.

For if a: “2 =@:ar, then ar = ar and 4r—r?=o0. Only two values of r satisfy this con-2 2 4 dition, 4 and o, of which the latter need not be taken into consideration when dealing with the number system of Nicomachus. Nicomachus should have directed that the square root of r be taken, not its half; then, provided that r were a perfect square, he could set up the proportion, a: avr =aVyr:ar. The number 4 is the only one of which the half and the square root are identical (among the positive integers).

1 Given a and ¢ (if a > c), the harmonic mean is (a—ce +c. This is equal to 2 a¢ | hwwhich a+ec ate was given above as the general formula for the harmonic mean (see on II. 25. 4) and would be a simpler one to follow.

? That is, in the sections of the works of ancient authors read and interpreted in the schools (dvayviouara).

*Theon of Smyrna also enumerated other proportions in addition to the first and principal three. In addition he says (p. 106, 13) there are the ‘subcontrary’ (sc. to the harmonic; cf. p. 115, 5 ff.), fifth and sixth, and then six more, subcontraries to the first six. This second set of six, however, he leaves without explanation or illustration. For the sake of the conspectus, the general formulae for the proportions as the two authors give them may be set down, letting a >6> cineach:

NICOMACHUS THEON t. Arithmetic a—b=b—c (Il. 23.5) a-b=b—c (p. 113, 18). : b a 2. Geor f=? IT. = . 2 f 2 metric hae (IT. 24) ale (p. 114, 1) 3. Harmonic ¢ 6–2 (II. 25) 2–8 (p. 114, 14). c b—ce c b—e 4. Subcontrary to harmonic ey aa (II. 28. 3) pay aaa, (p. 115, 5). ¢ a-—b a b-—e¢ , b b-—€e c_a-—b . Fifth f = . 28, = = . IIS, 12). 5. Fifth form a (II. 28. 4) ta ar) (p. 115, 12) . b—c¢ b a-—hb 6. Sixth f = . 28. = . 118, 20). ixth form =—(II. 28. 5). (p. 115, 20)

In the last three forms Theon consistently reverses the formulae of Nicomachus.

acquaintance with them and, so to speak, for the completeness of our

2 reckoning. They are set forth by us in an order based on their opposi-tion to the three archetypes already described, since they are fashioned out of them and have the same order.

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

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

The fifth form exists,* whenever, among three terms, as the middle term is to the lesser, so their difference is to the difference between the greater and the mean, as in 2, 4, 5, for 4 is the middle term, the double of 2, the lesser, and 2 is the double of 1—the difference of the smallest terms as compared with that of the largest. That which makes it con-trary to the geometric proportion is that in the former,‘ as the middle term is to the lesser, so the excess of the greater over the mean is to the excess of the mean over the lesser term, whereas in this proportion, on the contrary, it is the difference of the lesser compared to that of the a_a—5b

1 The harmonic proportion is - = j The one now under discussion is of the form, ¢—€

¢ = ° In the example given the ‘elements compared’ are the extremes and differences G = respectively.

? The statement is not correct; it should be that the product of the greater by the middle equals the product of the middle by the less multiplied by the ratio. For if = 2; where a >b>¢c, let r be the ratio. Then dcr will equal bc X Z or ab. Or the mean multiplied by the sum of the extremes equals the sum of the squares of the extremes, b(a¢ +c) = a? +c’,

* That is, if > b > ¢, the proportion is 2 = *—<. iI—

‘ That is, 6 4–9 f derived from 2 = 2) cf. IT. 24. c b—e b oc greater. Nevertheless it is peculiar to this proportion’ that the product of the greatest by the middle term is double that of the greatest by the smallest, for 5 times 4 is twice 5 times 2.

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

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

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

1 This statement again should be corrected in the same way as the former one. It can be stated that if I; where a > 6 > cc, andr is the constant ratio,ab = acr. Forr = and ¢ an ¢ ac X ®—gb. Nicomachus’s proposition applies only to cases where r = 2. c

0—¢

* General form, if a > 5 > cc,

This gives from the series, 1, 4, 6, D iw :—. the 4—4

= 7 (See note on section 4). Here the ratio of differences is inverted. (Cf. Euclid, Elem., V., Def. 16: dvacrpoph Adyou éorl ARyus Tov Hyouuérou wpds Thy Uwepoy hp, U bwepéxye: rd Fyobperor rou érouévov. That is, as Heath puts it, conversion is taking instead of the ratio of ato 6 the ratio of a toa—b. This is evidently not the sort of conversion here intended.)

? Boethius, II. 52, says: ef Ace quidem sunt sex mediectates, quarum tres usque a Pythagora ad Platonem Aristotelemque manserunt. Post vero qui inseculi sunt has tres alias, de quibus supra disseruimus, suis commentariis addiderunt. See note on II. 22. 1.

‘ The general form is = ;

= Z iia >b>c. In the given series 6, 8, 9, we have

aS 8—6 2

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

9 Theninth? in the complete list, and third in the number of those sub-sequently invented, exists when there are three terms and whatever ratio the mean bears to the least, that also the difference of the ex-tremes has in comparison with that of the smallest terms, as 4, 6, 7.

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

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

First: 1, 2, 3 Sixth: 1, 4, 6 second: 1, 2,4 Seventh: 6, 8, 9 Third: 3, 4, 6 Eighth: 6, 7,9 Fourth: 3, 5, 6 Ninth: 4, 6, 7 Fifth: 2, 4, 5 Tenth: 3, 5, 8 t It remains® for me to discuss briefly the most perfect proportion, that which is three-dimensional and embraces them all, and which is most useful for all progress in music and in the theory of the nature of the universe. This alone would properly and truly be called harmony °

1 The general form, ifa > 5 > cis pe—;» and in the given series 6, 7, 9, we have 9 = 9—6 = 3, 6° 90–7 2 b_a-c

The general form, if ¢>b>¢, is, and in the given series 4, 6, 7, we

c b-ce fave OO 48, =) 3

4 6–4 2 ? That is, ifa>b>c, : ap? and in the given series 3, 5, 8, $53 = 5

4It is to be noticed that all the proportions can be formed from numbers in the first decade.

§ Iamblichus calls this proportion woverx#, and says of it (p. 118, 23 Pistelli): evpyua 3 adryr gacw elvac BaBvAwvlwy, xal 8d T[lu@aydpov xpdrov els “EAAnvas é\Geir (“they say it was a dis-covery of the Babylonians, and that it was by Pythagoras first introduced among the Greeks’’).

§ Cf. IL. 26. 2.

b_a-—e

rather than the others, since it is not a plane, nor bound together by only one mean term, but with two, so as thus to be extended in three dimen-sions,’ just as a while ago it was explained that the cube is harmony.

When, therefore, there are two extreme terms, both of three dimen-sions, either numbers multiplied thrice by themselves so as to be a cube, or numbers multiplied twice by themselves and once by another num-ber so as to be either ‘beams’ or ‘bricks,’ or the products of three un-equal numbers, so as to be scalene, and between them there are found two other terms which preserve the same ratios to the extremes alter-nately and together,’ in such a manner that, while one of them pre-serves the harmonic proportion, the other completes the arithmetic, it is necessary that in such a disposition of the four the geometric proportion appear,* on examination, commingled with both mean terms—as the greatest is‘ to the third removed from it, so is the second from

1 Or, ‘three intervals’; for the sense, cf. IT. 24. 6.

2 Ast thus comments on the words translated (¢vaAAdé, dvaylt): “ évadddé est permutato s. inverso ordine, et dvaulé promiscue s. inter se; dvaulE medii termini 8 et 9 [referring to the ex-ample 6, 8, 9, 12] ad extremos 6 et 12 ita se habent, ut aequalem inter se servent proportionem; 8 enim ad 6 sesquitertiam habet rationem, ut 12 ad 9g; inverso autem ordine 12 ad 8 ita se habet, ut 9 ad 6; utriusque enim ratio est sesquialtera.”” Ast apparently means ‘alternately’ or ‘by alternation’ by ‘permutato sive inverso ordine’ (= ¢raAAdE), as his illustration shows. évaA\d£ is so used in II. 21. 6, but in the same section dvaplf is used in precisely the same way, both mean-ing ‘by alternation.’ So we must assume either that the terms are here used as synonyms, as in II. 21. 6, or that ¢évadAdé means ‘alternately’ and drawl something else, ‘directly,’ perhaps, as Ast would imply. It is quite certain that évad\\d=—means alternately. Cf. T. L. Heath on Eu-clid, V. Def. 12: ‘‘The word évaddd£ is of course a common term which has no exclusive reference to mathematics. But this same use of it with reference to proportions already occurs in Aristotle, Anal. Post., I. 5. 74a 18, wal 7d dvddovyor Ere évaddd£, ‘and that a proportion (is true) alternately, or alternando.’ Used with Adyor as here, the adverb é»adddé has the sense of an adjective, ‘alternate’; we have already had it similarly used of ‘alternate angles’ (al éva\Adf -ywrlac) in the theory of parallels.” It is also clear that rods abrods \éyous refers to geometric propor-tion, not to the harmonic and arithmetic referred to after Gere, for Aéyos is not used of the rela-tion existing between the terms in arithmetic proportion. I have translated dvaulf ‘together’ but with some diffidence, taking the sense to be approximately that given by Ast.

* Such a proportion is of the form a, pat tion, for the product of the extremes equals that of the means. Nicomachus further specifies that both @ and 6 are to be of the general forms m’, m*n or Imn. In giving his example he considers unity a factor. Boethius, IJ. 54, thus describes this proportion: Haec aulem huiusmodt tnveniclur, si duobus terminis constilutis, qui ipsi tribus creverint intervallis, longitudine latitudine et profundi-tale, duo huiusmodi termini medti fuerini constiluti et ipst tribus intervallis notati, qui vel ab acgua-libus per aequales aequaliler sint producti vel ab tnaequalibus ad tnaequalia aequaliter, vel ab in-acqualibus ad aequalia aequaliter vel quolibet alio modo, alque ita, cum armonicam proporlionem custodiant, alio lamen modo comparati faciant arithmeticam medtetatem, hisque geometrica medietas, quae tnler ulrasque versalur, dcesse non possi.

‘The Greek fashion of counting, that is, reckoning in both ends, is used in the specification of these terms. ‘By alternation’ here is duwAéydnr ‘in interlocking or interwoven fashion’; practically equivalent to éraAAde.

a+b, 6. These will form a geometric propor- it to the fourth; for such a situation makes the product of the means equal to the product of the extremes. And again, if the greatest term be shown to differ from the one next beneath it by the amount whereby this latter differs from the least term, such an array becomes an arith-metic proportion and the sum of the extremes is twice the mean. But if the third term from the greatest exceeds and is exceeded by the same fraction of the extremes, it is harmonic and the product of the mean by the sum of the extremes is double the product of the ex-tremes.

3 Let this be an example of this proportion, 6, 8,9, 12. 61s ascalene number, derived from 1 times 2 times 3, and 12 comes from the succes-sive multiplication of 2 times 2 times 3; of the mean terms the lesser is from 1 times 2 times 4, and the greater from 1 times 3 times 3. The extremes are both solid and three-dimensional, and the means are of the same class. According to the geometric proportion, as 12 is to 8, so g is to 6; according to the anthmetic, as 12 exceeds 9, by so much does 9 exceed 6; and by the harmonic, by the fraction by which 8 exceeds 6,’ viewed as a fraction of 6, 8 is also exceeded by 12, viewed as a fraction of 12.

4 Moreover 8:6 or 12:9 is the diatessaron, in sesquitertian ratio; 9:6 or 12:8 is the diapente in the sesquialter; 12:6 1s the diapason in the double. Finally, 9:8 is the interval of a tone, in the superoctave ratio, which is the common measure of all the ratios in music, since it is also the more familiar, because it is likewise the difference? between the first and most elementary intervals.

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

18 exceeds 6 by 2, or by $ of 6; 12 exceeds 8 by 4, or by 4 of 12; so 6,8, 12 is a harmonic series.

2 Boethius has the following explanation: unde notum est, quod inter diatessaron et dtapente consonantiarum tonus differentia est, sicul inter sesquitertiam et sesquialleram proportionem sola est epogdous differentia (II. 54).