Introduction to Arithmetic, Book II

Nicomachus of Gerasa · Introduction to Arithmetic

Section 2 of 3

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I

serving of the honor, let the arithmetic proportion have its discus-sion at our hands before the others.

It is an arithmetic proportion,’ then, whenever three or more terms are set forth in succession, or are so conceived, and the same quantita-tive difference is found to exist between the successive numbers, but not the same ratio among the terms, one to another. For example, 1, 2,3, 4, 5,6, 7, 8, 9, 10, 11, 12, 13; for in this natural series of numbers, examined consecutively and without any omissions, every term what-soever is discovered to be placed between two and to preserve the arith-metic proportion to them. For its differences as compared with those ranged on either side of it are equal; the same ratio, however, is not preserved among them.

And we understand that in such a series there comes about both a continued and a disjunct proportion; for if the same middle term an-swers to those on either side as both antecedent and consequent, it would be a continued proportion, but if there is another mean along with it, a disjunct proportion comes about.

Now if we separate out of this series any three consecutive terms whatsoever, after the form of the continued proportion, or four or more terms after the disjunct form, and consider them, the difference of them all would be 1, but their ratios would be different throughout. If, however, again we select three or more terms, not adjacent, but sepa-rated, separated nevertheless by a constant interval, if one term was omitted in setting down each term, the difference in every case will be 2; and once more with three terms it will be a continued proportion; with more, disjunct. If two terms are omitted, the difference will always be 3 in all of them, continued or disjunct; if three, 4; if four, 5; and so on.

Such a proportion,® therefore, partakes in equal quantity in its

| That is, the ‘fourth, fifth and sixth’ forms. Cf. section 1. The first three are the ‘leaders’

of the subcontraries because the latter are based on them. 2 Cf. the definition in Theon, p. 113, 18 ff. 7Cf. on IL. 20.3; 21.5; 22. 2.

differences, but of unequal quality; for this reason it is arithmetic. If on the contrary it partook of similar quality, but not quantity, it would be geometric instead of arithmetic.

A thing is peculiar’ to this proportion that does not belong to any other, namely, the mean is either half of, or equal to, the sum of the extremes, whether the proportion be viewed as continuous or disjunct or by alternation; for either the mean term with itself, or the mean terms with one another, are equal to the sum of the extremes.

It has still another peculiarity; what ratio each term has to itself,’ this the differences have to the differences; that is, they are equal.

Again, the thing which is most exact, and which has escaped the notice of the majority,’ the product of the extremes when compared to the square of the mean is found to be smaller than it by the product of the differences, whether they be 1, 2, 3, 4, or any number whatever.

In the fourth place,* a thing which all previous writers also have noted, the ratios between the smaller terms are larger, as compared to those between the greater terms. It will be shown that in the har-monic proportion, on the contrary, the ratios between the greater terms are greater than those between the smaller; for this reason the harmonic proportion is subcontrary to the arithmetic, and the

1 The translation here follows the reading of Ast (7d card cUvOeciy TOv Axpwy Sierddovow Tob pécov A loov Trois péoos elvar). Stated in algebraic form, and letting a >) >c >d, the proposi-tions in this section are:

I. (1)¢—6 =c—d, or (2)a—b = 5b—« (typical forms of the progression), then Il. (1)a +d =b+c,or (2) a+ c = 256. But if I is given, it is evident that the following also are true: Il, a—c =b—d (by alternation), then IV. a+d=6+c. The proposition is noticed by Theon, Joc. cit., and by Nicomachus himself in his Handbook of Music (c. 8, p. 251, 13 Von Jan).

? Boethius, II. 43: Namgue omnis lerminus sibi aequalts est et differentiae differentits sunt aequales.

* Boethius, IL. 43, says that this was discovered by Nicomachus. In general this proposition may be stated: If a—b = 6b—c, ®—ac = (2—b) (6b—cc) = (a2—BY = (6—c)*. Nicoma-chus in his Handbook of Music (c. 8, p. 251, 15, Von Jan) again mentions this property of the proportion.

‘ Thus in the series 1, 2, 3, comparing the ratio of the lesser terms (1, 2) and that of the greater (2, 3),

S51. 33 2:

But in a similar comparison of the terms of the harmonic progression 3, 4, 6, ' 4:3<.6:4.

In a geometric progression, as 1, 2, 4,

2:1 = 4:2, as Nicomachus proceeds to say. Nicomachus states that this fact had been noted by all previous writers. The statement is borne out by the fact that it occurs as early as Archytas (fragment 2, Diels, Die Fragmente der Vorsokrattker, I’, p. 334, in Porphyry, Jn Ptol. Harm., p. 267). For its enunciation by Archytas, see p. 21, where he is quoted.

“a

‘. geometric is midway between them, as it were, between extremes,

for this proportion has the ratios between the greater terms and those between the smaller equal, and we have seen that the equal is in the middle ground between the greater and the less. So much, then, about the arithmetic proportion.

The next proportion’ after this one, the geometric, is the only one in the strict sense of the word to be called a proportion, because its terms are seen to be in the same ratio. It exists whenever, of three or more terms, as the greatest is to the next greatest, so the latter is to the one following, and if there are more terms, as this again is to the one following it, but they do not, however, differ from one another by the same quantity, but rather by the same quality of ratio, the opposite of what was seen to be the case with the arithmetic proportion.

For an example, set forth the numbers beginning with 1 that advance by the double ratio, 1, 2, 4, 8, 16, 32, 64, and so on, or by the triple ratio, 1, 3, 9, 27, 81, 243, and so on, or by the quadruple, or in some | similar way. In each one of these series three adjacent terms, or four, or any number whatever that may be taken, will give the geometric proportion to one another; as the first is to the next smaller, so is that to the next smaller, and again that to the next smaller, and so on as far as you care to go, and also by alternation. For instance, 2, 4, 8; the ratio which 8 bears to 4, that 4 bears to 2, and conversely; they do not, however, have the same quantitative difference. Again, 2, 4, 8, 16; for not only does 16 have the same ratio to 8 as before, though not the same difference, but also by alternation it preserves a similar relation—as 16 is to 4, so 8 is to 2, and conversely, as 2 is to 8, so 4 is to 16; and disjunctly, as 2 is to 4, so 8 is to 16; and conversely and in dis-junct form, as 16 is to 8 so 4 is to 2; for it has the double ratio.

The geometric proportion has a peculiar property shared by none of the rest, that the differences of the terms? are in the same ratio to

1 dvadoyla: Cf. Theon, pp. 107, § and 114,1 ff.,0n this proportion. Euclid defines numbers in proportion as follows: “‘ Numbers are in proportion when the first is the same multiple of the second as the third of the fourth, or the same part of it, or the same parts”’ (dp:@uol dvddoydy ela.v ray 6 wpwros Tov Sevrépov xal 6 tplros rot rerdprov lodecs 7 wokAawAdows 4 7d alrd pépos ra atra pépyn dow), Elements, VII, Def. 21.

? Thus in the series 1, 2, 4, 8, 16, the ratio is double and the ratio between the successive dif-ferences (1, 2, 4, 8) is also double.

each other as the terms to those adjacent to them, the greater to the less, and vice versa. Still another property is that the greater terms have as a difference, with respect to the lesser, the lesser terms them-selves, and similarly difference differs from difference, by the smaller difference itself, if the terms are set forth in the double ratio; ! in the triple ratio both terms and differences will have as a difference twice the next smaller, in the quadruple ratio thrice, in the quintuple four times, and so on.

Geometric proportions come about not only among the multiples, 4 but also among all the superparticular, superpartient, and mixed forms, and the peculiar property of this proportion in all cases is preserved, that in the continued proportions the product of the extremes is equal to the square of the mean, but in disjunct proportions, or those with a greater number of terms,” even if they are not continued, but with an even number of terms, that the product of the extremes equals that of the means.

As an illustration of the fact that in all the relations, all kinds of mul- 5 tiples, superparticulars, superpartients, and mixed ratios the peculiar property of this proportion is preserved, let that suffice * and be suffi-cient for us wherein we fashioned, beginning with equality, by the three rules all the kinds of inequality out of one another, when they were in both direct and reverse order; for each act of fashioning and each series set forth is a geometric proportion with all the aforesaid properties as well as a fourth, namely, that they keep the same ratio‘ in both the greater and the smaller terms. Moreover, if we set forth the series shared by both heteromecic and square numbers, one by one,

1 Doubles I 2 4 8 16 32 64 128 Differences I 2 4 8 16 32 64 Differences of differences 1 2 4 8 16 32

In general, 2”*!—2” = 2” Triples I 3 9 27 81 243 729 Differences 2 6 18 54 162 486

Differences of differences 4 12 36 108 324 In general, 3**!—3” = 3°(3—1) = 2 X 3” Quadruples 1 4 16 64 256 1,024 4,006 Differences 3 #12 48 192 768 3,072 Differences of differences 9 36 144 576 2,304 In general, 4”*!—4” = 4”(q—1) = 3 X 4” ? Thus in the series 2, 4, 8, 16, 32, 64 2 64 = 4% 32 = 8 & 16 (= 128). 7Cf. 1. 23. 7 ff. ‘ That is, in the series of doubles (1, 2, 4, 8, 16, 32, etc.) the ratio between 32 and 16 is the Same as that between 2 and 1.

containing the terms in both series, and then selecting the terms by groups of three beginning with 1, examine them, in each case setting down the last of the former group as the starting point of the next, we shall find that from the multiple relation—that is, the double—all the kinds of superparticulars * appear one after the other, the sesqui-alter, sesquitertian, sesquiquartan, and so on.

6 It would be most seasonable, now that we have reached this point, to mention a corollary that is of use to us for a certain Platonic theo-rem :* for plane numbers are bound together always by a single mean, solids by two, in the form of a proportion. For with two consecutive t The series of squares and heteromecic numbers is 1, 2, 4, 6, 9, 12, 16, 20, 25, etc. Taking them in groups of three as directed the following ratios appear:

1,2,4 (double)

4,6, 9 (sesquialter)

9, 12, 16 (sesquitertian) 16, 20, 25 (sesquiquartan)

? The reference is to Timaeus, 32 A-B, and Nicomachus endeavors to elucidate a real difficulty in the Platonic text. In stating the case as he does at first briefly and summarily (“‘planes are always joined by one mean, solid numbers by two”), he doubtless quotes from memory, for he does not report Plato precisely. Plato does not say that planes can have but one mean, but that one suffices (el of» érlwedov. . . Eder ylyver@ar rd Tov warrds cOpa, ula perdrys Av ef jpxer rd re pel” daurns Evrdety xal daurhe viv 3é—orepecesdy yap avréy wpocixer elvat, ra 5¢ oreped ula pév odddwore, Sto 52 del perdryres Evvappérroverw...). But Nicomachus, going on to restrict the application of these two principles to consecutive squares and consecutive cubes would seem to be trying to impose upon the Platonic passage an interpretation which would stand mathematical scrutiny.

The words used by Plato, éwlredov and orepedyv, are capable of a very broad interpretation and difficulties would then arise. For example, a plane number could be any number of the form ab, and supposing a, 5, c, and d to be prime integers, it would be impossible to find one rational mean between the plane numbers ab and cd, for V abcd would be irrational. However, it would always be possible to find a single mean between two successive squares, for if the squares are a? and (a + 1)?, a(a + 1) will be a geometrical mean between them. Furthermore, Nicomachus’s statement about the cubes helps to dismiss a real difficulty in the second part of the Platonic theorem, for there are certain solid numbers that can be put into a geometrical proportion with but one mean (e.g., Archer-Hind, ad /oc., cites 8, which is 2%, and 512, which is 8’, and the proportion 8:64 = 64:512). But if by solid numbers Plato meant consecutive cubes, as Nicomachus says, then it will be found that no single rational geometrical mean can be inserted between two such. For if the cubes are a? and (a + 1)’, the geometrical mean would be a(a + 1)Wa(a +1) and would be irrational.

At the hands of modern commentators the Platonic passage has been subjected to somewhat similar restriction. Archer-Hind in his note follows Martin for the most part, and declares his belief that Plato meant érlwredov and crepedr in the strictest possible sense, the former a number of two factors only, the latter of three, all the factors being prime integers, and that in the case of the solid numbers he restricts himself to cubes. Then it would be possible always to find one geometrical mean between two squares (as a?: ab = ab: %), though in other plane numbers two means might be possible; and the possibility of two cubes with but one rational geometrical mean will be excluded, for if x is the mean between a’ and 6, it will have the irrational value abV, ab, a and 6 being prime integers.

squares’ only one mean term is discovered which preserves the geo-metric proportion, as antecedent to the smaller and consequent to the greater term, and never more than one. Hence we conceive of two intervals? between the mean term and each extreme, in the relation of similar ratios. Again, with two consecutive cubes* only two middle 7 terms in proper ratio are found, in accordance with the geometric proportion, never more; hence there are three intervals, one, that be-tween the mean terms compared to one another, and two between the extremes and the means on either side. Thus the solid forms are called 8 three-dimensional and the plane ones two-dimensional; for example, 1 and 4 are planes, and 2 a middle term in proportion, or again 4 and 9, two squares, and their middle term 6, held by the greater and holding the lesser term in the same ratio* as that in which one difference holds the other. The reason® for this is that the sides of the two squares, one 9 belonging peculiarly to each, both together produced this very number

6. In cubes, however, for example 8 and 27, no longer one but two mean terms are found, 12 and 18, which put themselves and the terms

1 Cf. Euclid, Elements, VIII, 11: ‘There is one mean term in proportion between two square numbers, and the square has to the square double the ratio of side to side” (300 rerpaywdewy dp:Oudy els péoos dvddoydy deriv dpiGuds, xal 6 rerpdywvos wpds Tov Terpdywrow Simdaclova héyor tye. Hrep i whevpa wpds Thy whevpdy). Theon of Smyrna does not include this proposition, nor the following one, concerning cubes; this is strange, since he is professedly offering helps to the study of Plato.

? Siagrjyara: The Greek word may also be translated ‘intervals.’ On the meaning of the word in this connection, cf. on II. 6.3. These differences will bear to each other the ratio of the terms (cf. sect. 3 above).

3Cf. Euclid, Elements, VIII, 12: “There are two mean terms in proportion between two cubes, and cube has to cube thrice the ratio of side to side” (S00 cUBwy dpiOuay d00 péoo dvddoyér claw dpcBpuol, kal 6 xuBos wpds rd xiBor rprwhaclova Advyor Eye Hrep H rhevpa wpds Thy whevpdy).

‘In general a proportion between successive squares would be a?:a(a +1) = a(e +1): (a-+1)*. The ratio of the sides would be a:a-+ 1. The differences on either side would be a-+a—a’=a,anda’+2a+1-—a?—a=a+1. So the differences have the same ratio as the terms themselves. In this kind of proportion the only ‘intervals’ are those between the first and middle terms and between the middle and last terms, whereas in any proportion with cubes as extremes, as a?: m = n: 5, there are three, between a and m, m and n, and a and PB. It is to be remarked that the same word, διάστημα, can be translated ‘interval’ and ‘dimension’ in speaking of geometric squares or cubes. To Pythagoreans such a coincidence would mean much.

* If for example the cubes are a? and 6’, the proportion may be of the form a*: a4) = a*b: ab? = gb?: &; the constant ratio is a:b, and the differences will be a*—a%b, att—a, and abt—&. But (a?—a%b)(ab?—b*) = (a*b—ab*)*. The differences therefore may be put in continued proportion

#0) 5 oh oe

ab—ah® ah—B That is, the ratio between the differences is the same as that between the terms. In the case of cubes of prime numbers there would be eight further possible forms of the proportion, all of which obey this law, as may be readily tested. If, however, the original numbers were not prime, the number of forms increases with the number of factors.

, which reduces to 7