30. axparoy, not of course absolutely dxparov (for then there would be no ratio), but axparéorepor.
30—109371. Aristotle’s statement that ‘3 to 2’, not ‘3 times 2’, is the right way of stating the formula ofa mixture is evidently right. A mixture is got not by taking the same thing over and over again but by taking two or more things in a given ratio. Multiplication means the addition of things of the same kind (1d aird det yévos ctvat év tats rokAarAacuiceow |. 32).
1093*1-1g. There seem to be two arguments used here, (1) ll. 3-9. If all things share in number (this is not a mere concession to the adversary, but is Aristotle’s own view), there is nothing surprising in the fact that some things (e.g. the periods of movement of sun and moon, the periods of the life of animals) should be designated by square and cube numbers, or by numbers related as equal or as double and half to one another. This is no warrant for treating the numbers as the cause of the phenomena. (2) Il. 9-13. It was assumed that different things could fall under the same number. Then, onthe view which is being considered, they will be the same thing, which is absurd.
This interpretation involves treating sober te l. 9 asnot depending on «i (Bekker, Bz.) but starting a fresh sentence. re often introduces a new objection (e. g. A. 991 27).
4. For the number of the motions required to account for the motions of the sun and the moon cf. A. 1073? 17, 35.
8. dvdykn év TouTois otpépeo Par, ‘all things must move within these limits’ (i.e. be designable by square or cube, equal or double, numbers); ch, KikXw otpédecOau.
12, The sun and the moon on Aristotle’s view have, each of them, five proper motions (A. 107317, 35), and this makes them a good enough illustration of his meaning. The Pythagoreans themselves assigned dzfferen/ numbers to them (2, 7).
13-» 4. Syrianus says that no important Pythagorean quoted triviali- ties such as those which Aristotle here ridicules, as instances of the power of numbers. He refers to the Pythagorean Prorus as having written about the number 7 (the treatise really belonged in all proba-
bility to Alexandrian times, Diels, Vors, 267. 22), but says he confined himself to showing how many things ‘nature does in seven years, months, or days’; while others wrote about the number 10. He retorts on Aristotle by pointing out that Aristotle himself does rever- ence to the number 3 in the De Caelo (2684 13), and forcibly reduces the flavours and the colours to seven each (De Sensu 442219). On the number 7 in Greek cultus, mythology, philosophy, and medicine cf. Roscher in Adbh. der phil.-hist. Kl. der K. Sachs. Gesellschaft d. Wissenschaften, vols. xxi, xxiv, esp. xxiv. 24-43 on the Pythagoreans. The numerical fact about 7 which interested the Pythagoreans was that within the decade it alone has neither product nor factor (Philol. fr. 20 Diels).
14. xopdat 7 dppovios (JA*T Al.) is not very natural, since seven chords are very different from seven modes or harmonies. E’s reading xopdal 4 dppovia is strongly confirmed by Prod. 9184 13 (= 922 3) Awa zi of dpxaio. éxtaxdpdovs mowidvres dppovias kth. Cf. g1g) 21 (= 922% 22). Alexander finds a difficulty in reckoning seven harmonies (832. 20).
év émra Sé d8dvtas Badder, cf. Solon fr. 27. 1 Bergk.
15. evi ye, eva 8 ov. The horse, e.g., does not, H. A. 5764 6.
1g. thy S€ dpxtov ye Sd8exa. Galen (ix. 935 K.), who evidently has this passage in view, says that seven stars were reckoned in each of the two Bears, and the same account is given in Philo de Mundi Opif. 39. 114, Anatol. p. 36. 4 Heiberg, Hermipp. Beryt. ap. Clem. S¢rom. vi. 16. 143. 1. Ptolemy, however, assigned 27 stars to the Great Bear, and Hevelius, much later, assigned 12.
oi 8€ (XaAdator Al.) mhetous. We know that the Chaldeans had a name for the Bear, viz. Narkabti, but we do not know how many stars they reckoned in it. Cf. Ginkel in Kiz0, vol. i. 5.
20-21. om... tpia. Alexander says that £ was connected with the fourth, € with the fifth, y with the octave.
21. Sri Sé pupia &v ein toaita. Tannery has published (Bull. de Corr. Hell. xx. 422) a table of the fourth century B.c. found at Delphi giving simple symbols for various groups of consonants.
22. ei 8 ot, ‘if they say that’; cf. pacity |. 20. ‘If they meet our objection (that there are many combinations of letters for each of which a single sign might be devised) by saying that € y ¢ are the only combinations each of which takes twice as long to pronounce as a single consonant, and if the cause of this is’, etc.
23. Tpidy dvtwy téTrwy, the palate, the lips, the teeth, against which the tongue is placed when é, w, ¢ respectively are formed. Syrianus says that Archinus (the introducer of the Ionian orthography in the second half of the fifth century) gave this explanation of the fact that there were in Greek only these three double letters.
24. tv éh Exdotou émpéepetar TH olypa, ‘ a single letter (guttural, labial, or dental) is applied to the sigma in each region’. Alexander seems to have read 76 ofypa, assuming the sigma to be sounded after the other element in the double sound. But ev ... 76 o¢ypa would
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be an unnatural combination, and further £ = a8, not dao (A. 993° 5). émupeperar must be taken as implying nothing with regard to the order of the two sounds combined.
26. mdelous ye al cupdwviar, i.e. new combinations (such as the eleventh) can be made out of the primary concords. But you cannot combine two double consonants to produce a new consonant.
27-28. Tots dpxalois ... mapopdow. The reference is probably to allegorizing interpreters of Homer such as Pherecydes of Syros (c. 600-525), Theagenes of Rhegium (fl. 525), Metrodorus of Lampsa- cus (d. 464), Anaxagoras (c. 500-427), and Democritus (fl. 420). Cf. Diels, Vors? i. 376. 15-17, 414. 9-24, ii. 67. 21, 22, 203. 27—204. 7, 205. 26—206. 10, Sandys, rst. of Classical Scholarship, i. 29, 30.
29. at te péoar h pev evvean Se dxTS. The peoae are the fourth and the fifth, the chief notes intermediate between the keynote and the octave. The fourth and the fifth answer to the ratios 8: 6, 9: 6.
30. 75 €mos Sexaemré. The epic verse has seventeen syllables, if we assume a spondee only in the last place. Alexander takes év pév 76 ded to mean ‘in the first half of the verse’, and this is confirmed by the fact that ‘ right’ and ‘left’ were technical terms for the first and second parts of a lyric system. The alternative is to suppose that ‘right’ is the part after the caesura xara tpitov tpoxatov, the common- est caesura in Homer; this too has nine syllables. Wilamowitz in Berl. Classthertexte,v. 2. 141, and S. E. Bassett in Class. Phil. xi. 458- 460 defend Alexander’s interpretation of 76 defidv. For the identi- fication of the right with the dépyy of movement cf. H. A. 498° 6, I, A. 70518, De Caelo 285” 16 (cf. 284» 6, where the Pythagoreans are referred to). Bassett quotes many instances to show that 70 defov was used by the metricists of the section of the verse which came first, e.g. Mar. Vict. 108. 16 Keil ‘arma virumque cano... Huius incisioni quae syllaba clauditur, si alteras duas adicias, ut tertium pedem trisyllabon compleas, erit hoc trimetrum defidv’, i.e. the three dactyls at the beginning of the line form the colon dexitrum or dpxrixdv (cf. Mar. Vict. 74. 8 K., Plotius 514. 28 K.). Further, Aristotle would naturally mention the first half of the line first. And, finally, BatveoOar refers not to caesura but to scansion in feet. The earliest Greek definition of caesura is in Aristides Quintilianus (p. 52 Meibom), in the second or third century a. p.
b 2-4. The meaning is that there are 24 notes on the flute, from the BouBvé (an onomatopoeic word for the deepest note) to the highest. guwvny is to be supplied with 6gvrarnv.
4. js. 24 is reckoned as the number of the highest note, so that Diels’s ois is unnecessary.
ioos TH odAopeneta, Alexander suggests that the number of the uni- verse is 24 because there are 12 signs of the zodiac, 8 spheres (i. e. the sphere of the fixed stars and those of Saturn, Jupiter, Mars, Venus, Mercury, Sun, Moon) and 4 elements. ovAouéAea is an Ionic word meaning the whole nature of a thing (uéAos = limb) ;_ cf. Hipp. De Artic. iv. 108 L., De Anatom. viii. 540, De Gland. viii. 556,
De Nutr. ix. 106. Here, however, there is probably a reference to the music of the spheres, cf. A. 9864 2 f. (uéAos = tune), Nicom. ap. Phot. Bzd/. 14425 says it was used as a name for the number -7, and in Theol. Arithm. p. 36 Ast we are told that the Pythagoreans, following Orpheus, applied it to the number 6 ; but the present passage shows that it was with the number 24 that it was identified.
Q. otTwot ye oKkorroupévois, ‘if we look at them in the critical way in which we have been looking at them, they seem to vanish away ’. For the dative cf. 1090? 20 ois dé ras id€as TiHeuévois TodTO péev expevyer.
10. T&v Siwpiopevav Tept Tas dpxds, Asets.2:
11. Bekker and Bonitz read airwyv éeorw. exelvo pevtTor mo odor gavepov rv. Christ reads airidy éoriv. ws pévTou rovodor, pavepov KTA. as is better attested than éxeivo, but Christ’s punctuation gives a false antithesis between as pev Aéyovod ties Kal aitia rovodet THs Pioews and @s pevto. odor. Diels’s punctuation removes this difficulty. Aris- totle makes a concession to the Pythagoreans. Their view that numbers are the causes of good and evil evaporates on examination ; but they make it clear that in some sense the good ‘ belongs’ to certain numbers. The seasons of the year and of life go ‘together’ with certain numbers. There is an analogical relation between things in different ‘categories’ (loosely used here for genera), whereby oddness
_in number may correspond to straightness in a line, evenness in a surface, and perhaps even to good in things which can® be either good or evil. But all this is mere correspondence and not causation
12. ovototxias, cf. A. 986% 23 n.
13. 15 tepitroy, cf. A. 986° 18 n., 23.
76 e000, cf. A. 9864 25 n.
TO iodks toov is probably the right reading, since it preserves a rare but genuine Pythagorean phrase (cf. JZ. M. 1182°14 od ydp éorw 7 dixarordvyn apiOpos ioaKis toos, sc. ‘as Pythagoras said it was’, and Pl. Theaet, 148 a1), and accounts best for the variants icov A>, iodp.6- pov B. iodxis icov = retpaywvov, which occurs inthe overorxia of the good, A. 9862 26.
14. at Suvdpers éviwv dpOpav. It is doubtful whether this means the ‘ powers’, in the mathematical sense (cf. A, 1019” 34, @. 104628, De Lin. Insec. 970% 2), of certain numbers (terpdywvov occurs in the overor- x‘a of the good in A. 9864 26, and cf. rerparywvous, ciBovs 1093* 7), Or whether itmeans the powers of certainnumbers, inthe non-technical sense of ‘power’. éviwy is in favour of the latter view; according to A. 986% 26 one would suppose the sgware of any number to be in the ovoroixia of goods. Alexander gives the second interpretation, and illustrates éviéwy by TEeTpPAyOVUY, TpLrywovev (numbers like 1, 3 (Ee 1+2), 6 (= I+et+ 3)), eayovov (numbers like 1, 6 (= 1+5), 15 (=1+5+9)).
dua yap Opar Kal dpi6pds tovogdi, ‘for the seasons and a certain kind of number (the square number 4) go together’, There may also be a reference to the comparison, ascribed by Aristides Quintilianus (Musica, iii, p. 145 Meib.) to Pythagoras, of the seasons to the con-
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500 Commentary
cords; spring is to autumn the fourth, spring to winter the fifth, spring to summer the octave, so that the four seasons are to one another as 6, 8, 9, 12. Plut. Ox the Birth of the Soul in the Timaeus 1028 f. ascribes the same view to the Chaldaeans,
17. cuptrepacw, ‘chance coincidences ’,
oixeta ddAnAots mavra, i.e, the normalineach class corresponds to the normal in every other class.
20. tows, i.e. for the sake of argument Aristotle is willing to allow the superiority of the odd number, on which the Pythagoreans laid such stress.
21. It is not Idea-numbers but ordinary mathematicalnumbers that are at the base of musical harmony and the like, for equal Idea-numbers, like Idea-units, differ in kind (M. 6-8), whereas the theory of harmony implies that equal numbers are identical.