Philolaus* by Prof

IV. Philolaus* By Prof. Will. Romaine Newbold, University of Pennsylvania, America,

Boeckh's monograph upon Philolaus ("Philolaos, des Pythagor ers Lehren, nebst den Bruchst cken seines Werkes", Berlin 1819, pp. 200) h s left little for others to do. His collection of fragments contains nearly everything of importance, and in his Interpretation both of individual passages and of the System s a whole, he clis- plays an insight the more astonishing when one remembers how imperfectly the development of early Greek philosophy was at that time understood. Yet there remain in the fragments several passages of which neither Boeckh nor any of his A successors has been able to gi\7e a satisfactory Interpretation. Of these I shall examine two. The first is the function of "embodying" and "Split- ting" ratios, which function Philolaus ascribes to number, and the meaning of the reference to the gnomon. The second is the nature of the principles assumed by Philolaus and termed by him περαίνοντá and άπειρα. If the results which I shall reach are sound, they will lead to a revision, in certain important particulars, of the accepted views s to'the details of the Philolaic cosmology. *) Stobaeus Ecl. L proem, cor 3 (p. 16, 20 Wachsmuth) FV p. 253—254.

*) I quote s "FV" Prof. Diels' epochrnaking work, "Die Fragmente^ der Vorsokratiker", Berlin. 1903: s UD", his "Doxographi Graeci*·*, Berlin, 1879. In quoting passages co tained in either of these books, I use Diels' text and orthography, unless the contrary is stated.

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θεωρεΐν δεé τá έργá êáé την "One should judge the deeds ουσίαν τù άριομώ καττάν δυναμιν and the essence of Number by the ατις εστίν εν ταé δεκάδι· μεγάλá virtue that is in the Ten, for it ãáñ êáΐ παντελής êáΐ παντοεργδς. is great, all-perfecting and all- êáΐ θείù êáΐ ουρανίù βίù êáé achieving, and is the origin and άν^ρωπίνù άρχά êáé άγεμών κοι- leader both of divine and celestial νωνουσá . .. δυναμις êáé τας δέκα- life and of that of man, sharing δος. άνεõ δε τουτας πάντ' άπειρá ... power of the Ten also. Without êáé άδηλá êáé αφανή. it all things are indefinite and vague and indistinct. γνωμικά ãáñ á φύσις á τù αριθμώ For the nature of Number is êáé ηγεμονικά êáé διδασκαλικά τù able to enlighten/guide and teach άπορουμένù παντός êáé άγνοουμένù every man s regards every thing παντί. οõ ãáñ ης δήλον ουδέν! perplexing and unknown. For of ουδέν των πραγμάτων ούτε αυτών ποθ' Things not one would have beeil αυτά ούτε άλλù προς άλλο, εé μη plain to anyone, either s they ης αριθμός êáé ά τοότù ουσία, are to themselves or s one is νυν δε ούτος κατάν ψόχάν αρμόζων to another, did not Number and αίσΟήσεί πάντá γνωστά êáé ποτάγορá its essence exist. As it is, fitting

άλλάλοις κατά γνώμονος φυσιν them to the sou!5 it makes them απεργάζεταé σωμάτων êáé σχίζων all knowable to sense and com- τους λόγους χωρίς εκάστους των parable one with another, in the πραγμάτων των τε άπειρων êáé των manner of the gnomon, embodying περαινόντων. and Splitting the several ratios apart of the Things both of the indeterminate and of the deter- mining. ίοοις δε êá οõ μόνον εν τοις δαι- You would see the nature of μονίοις êáé θείοις πράγμασé τάν τù Number and its virtue prevailing αριθμώ φυσιν êáé τάν δυναμιν not only in Things supernatural ισχύουσαν, áëëά êáé εν τοις άνΟρω- and divine, but also in the deeds πικοις εργοις êáé λόγοις πασé πάντá and words of men, in them all êáé κατά τάς δημιουργίας τας τεχνι- and through them all, and in all κάς πάσας êáé κατά τάν μουσικάν. operations of the crafts and in music. ψευδός δε ουδέν δέχεταé ά τù The nature of Number, ofwhich

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φύσις, οõ αρμονία* οõ ãáñ is fiarmoDia, adinits no untruth, oixetov αύτù έοτι. τας το> [iot?J for untruth is not akin to i t άπειρù xotl ανόητοé χαΐ άλάγù φύ- Untruth and envy belong to the αιος το ψευδός êáé ό φθόνος εστί. indeterminate and unintelligent ψευδός δε ουδαμώς ες αριθμόν and irrational nature. επιπνεΓ· πολέμιον ãáñ êáé έχθρον Untruth never breathes2) into ταé φύσεé το ψευδός, ά δ' αλήθειá Nurnber, for untruth is a foe and ρίκεΐον êáé συμφυτον ταé τù άριΟμÁ enemy to its_nature, but truth. γενεαι. is of N rnbergs house and bred \vith its kin." There is in this fragment but one passage the meaning of which is not clear. Philolaus ascribes both to number and to the gnomon the iunction of making .things "knowable to sense and comparable one with another", and at least to number, that of '"embodying" and usplitting" the ratios of things both indeterminate and deter- mining. The word "einbodying" (σωμάτων) was first suggested by Boeckh s a substitute for the meaningless σωμα'των of the MSS. He thinks that Number icembodies" things because Body is the third power of number; the meaning of "Splitting" he does not explain, but connects it.witli the notion that the void enters into and splits the nuinbers.3) It is not safe to allege that this or ny other possible parallel between numbers and things was not and would not have been drawn by a Fythagoreau. But the function liere ascribed to number

2) Probably an allusion to the well known Pythagorean doctrine that the original One was split np by inhaling the \roid from without. 3) p. 144. Im folgenden habe ich nichts mit der Lesart σωμάτων anzu- fangen gewu t und daher aus Vermutung σωμάτων gesetzt : wonach nun zweierlei von der Zahl bewirkt w rde, erstlich, da sie den Dingen Korper gibt, indem der Korper die dritte Potenz der Zahl ist: dann, da sie die Verh ltnisse des Be- grenzten und Unbegrenzten scheidet: durch beides, durch das K rperlichmachen und die Sonderung, letztere mag nun gedacht sein, wie sie wolle (vgl. St. 12 im Anfang), werden die Dinge allerdings erst bestimmt erkennbar f r die Empfindung." In the section referred to, 12, p. 108, Boeckh says, in speaking of the Pythagorean notion that the ουρανός inhales το κενόν from without, — "dies Leere sei zuerst in den Zahlen und trenne ihre Natur, und die Getrennt- heit der Naturen habe berhaupt hierin ihren Grund."

Brought to you by | provisional account Unauthenticated Download Date | 6/15/15 8:09 AM Philolaus. .179 is certainly not in keeping with the point of view of the Philolaus fragments in general or of this one in particular. Number is h re identified with what later philosophy termed the formal s against the rnaterial elernent in things. It is the principle of unity that Blocks together" into a αρμονίá the diverse things of which the world is made; it is the sole object of knowledge, and is also the instru- ment by which we compare one thing with another and determine their relation. It is the defining principle: without it all things would be confused and indistinguishable. In this context it is meaningless to add that number makes things corporeal. The functions of number of which Philolaus is speaking are of the opposite type. The "gnomon" Boeckh takes in its geometrical sense; the object known is surrounded and comprehended by the knower s each of the gnomons a b c surrounds and comprehends that part of the square that lies within it, thus syin- bolising the original concord and adaptation between knower and known.4) This Interpretation is fairly satisfactory so far s the function of "fitting things to the soul" is concerned, but it leaves the other unexplained. Yet it is not difficult to give a more probable explanation of the way in which the gnomon makes things knowable to perception and comparable one with another. At a quite early period the Pythagoreans devised a method of performing mathematical operations which has since been happily termed "geometrical algebra." 5) The substance of this geometrical algebra is founcl in the second book of Euclid and in a few propo- sitions of the first and sixtL By its aid the Pythagoreans discovered raany formulae for the multiplication and division of binomials and were enabled to perform many of the operations

4) p. 144" . . . nach dem Philolaischen Bruchstucke scheint man in den gno- monischen Verbindungen ein Bild der Befreundung und Vereinigung erblickt zu haben, welche unser Schriftsteller nicht ungeschickt auf die Erkennbarkeit der Dinge anwendet, indem das Erkannte von dem Erkennenden umfa t und er- grillen wird: wobei eine urspr ngliche Obereinstimmung und Anpassung, wie des Gnomon um sein Quadrat herum, vorausgesetzt wird." δ) Zeuthen, "Die Lehre von den Kegelschnitten im Altertum", pji. 1—38. Efeath, "Apollonius of Perga", pp. ci-cxi.

Brought to you by | provisional account Unauthenticated Download Date | 6/15/15 8:09 AM º80 Wm. Romaine Newbold, * foi wliich we now omploy arithmotic or algebra. In fact, the ablost Grook * raathematicians long preforrcd these geometrical mclhods to tho diroct Manipulation of nurabers on account of thoir groater simplicity s compared with the clumsy and laborious processes of λογιστική. Now the methods of geometrical algebra were in large part bused upon the properties of the gnomon. If through a point e on the diagonal cb of the parallelogram ab cd parallels fg and hi bc passed, it is easy to prove (Euclid I. 43) that the complemen- tary parallelograms ae and ed of the Ë \ /» I. gnomon acdief are equal. Since each is equal to the product of its base by its height, or, in a rectangle, of one side by the other, those products are equal g /ie — egXc/d), and either dimension of one is the quotient reached by dividing its other dimension into the product of the two dimensions of the other l —=gd\. The famous "application theorem" (Euclid I. 44) either uses or proves the following principles: (1) Any triangle can be conyerted into a parallelogram (ahef) containing a given angle (ahe) (Euclid I. 42). (2) Upon the Prolongation of one side (fe) of the parellelo- gram a quantum (eg) can be taten equal to any giveii line. (3) The remainder of the figure including the parallelogram ed can be constructed. By the aid of these principles it was easy to express in terms of lines or planes the results of adding, subtracting and dividing all quantities capable of being represented by a parallelogram such s ahef, and that by the use of no other Instruments than a ruler, compass and square. Numbers were in no way mvolved in the process. Suppose the quanta are given s rectilinear polygons. They are in the first place all resolved into triangles; all the triangles are reduced to parallelogram scontaining a given angle— preferably a right angle. By the aid of the application theorem the parallelograms are reduced to a series of parallelograms containing

Brought to you by | provisional account Unauthenticated Download Date | 6/15/15 8:09 AM Philolaus. 181 thc common side ey. Tlioy are then "knowable to senso uud com- parablo ono with another", They can be addecl by juxtaposition, — tiiibtracted by superposing one upon another and noting the differeoce. The height of one compared with that of another gives the quotient of that one by the other. By measuring the resulting quanta they can, if commensurable, be expressed in numbers, but if not commensurable they can still be used in further computa- tions by similar geometrical methods. Of the many uses to which this theorem was put, perhaps the inost common was that of dwision. So universal indeed was its use in that connection that the very word "apply" came to be synonymous with tcdivide'\-and the scholiast upon Euclid VI. 27 can remark, "Division is termecl among the mathematicians Appli- cation', for to 'apply' a number to a nurnber" ( s for example ahef to ecf) "is to divide the larger by the smaller, i. e. to show how many times the smaller is contained in the larger.'56) Philolaus' meaning is now, l trust, clear. It is no longer necessary, in order to compare various areas, or quanta represented by areas, to reduce them by the aid of the gnomon to parallelograms having a common side. It is now possible to measure their areas directly and, by the newly discovered properties of number, to add, subtract, multiply, and divide the numlers with f ll assurance that the results will faithfully represent the figures. Nuinber then takes the place of the gnomon in making things uknoËvable to perception and comparable one with another." Turning now to the seconcl question, — in what sense can number be said to "embody and split the ratios"? Among the many meanings of the word λόγος is that which I have rendered by "ratio". The term is not quite appropriate, — certainly not in the earlier days of Greek arithmetic, for λόγος then denoted a concept of much more limited content than does our "ratio". It was essentially an "expression" for one of two quanta in terms of the other. It was reached by measuring the larger by the smaller

°) Vol. V. Heiberg, 347.20 παραβολή παρά τοϊς μαΟ^ματικοϊς λέγεταé 6 μερισμός· παραβαλείν ãáñ αριθμόν παρά dpeftptfv εστί το μερίσαé τον μείζονá εις τον έλαττονá ήτοé δειςαι, ποσάκις b έλα'ττων περιέχεταé υπό τοõ μείζονος.

Brought to you by | provisional account Unauthenticated Download Date | 6/15/15 8:09 AM 182 Wni. Romaine Newbold, and tho smalJor by tho romain cr, repcating thc process until there was no remainder. Tliroughout tho cntire history of Greek arithmetic the λόγοé rotaiued names suggestiog the actual performance of lim process. lliiity, oquality, is the fundamental λίγος. Those that dilTor from unity by pnly one aliquot· part of the smaller quautum are named by reference to that part, e. g. επίτρπτος cca third over" = 1£, έπιτέταρτος "a fourth over?7=l^, etc. And the other terras are formed in analogous lashion, — I need not herc recount them. In a passage dealing vvith Number the mathematical use of λόγος has a prima facie claim to consideration. 4iSplitting" a λόγος is a famili r term in the later arithmetic7) although the word generally used to designate it is not σχίζειν but otoiipstv. Its correlate is ucompoundingu, συντιΟεναι. Let the lines A A1 and B B* represent the λόγος %, and let A- <%", D D1 represent the λόγος f. The first λόγος is •c an expression for A A1 in Ë- 1 rS -D terms of BB ^ the second,

_ an expression for CC' in JLt · terms of D D'. To "com- c. Á' i-E' 3 1) pound" the λόγοé ineans „to takeof^4J.' a part (Af;J') which when measured in terms of A A' is the λόγος |, and then to express that part in terms of BB'(= V)· This is what \ve term multiplying the fractions | and |. Conversely, to udivide?' the λόγος A A', 55'(^)means, 4CIn- stead of expressing A A1 in terms of EB\ express i t'in terms of a third quantum, say E E* (= f) and the latter in terms of BB'(£). This is equivalent to our "factoring" the fraction f into f X $. The process which.- we now term division of fractions \vas not distinguished'by the Greek arithmeticians from multiplication (f-hf

T) e. g. cf. Nicom..hit. Arith. II. 5, pp. 80—82, Iloche.. Porph. COIB. in Ptol. Harm., ed. Wallis, pp. 292, 318sqq..-passim.

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If σχίζων h re mcaus "lactor'V σωμάτων must be a corrupt reading behiad which lurks a tcrm s nearly equivalent to συντιοείς s σχίζων is to διαφών. I am inclined to believe that it was συναπτών, although a better .Suggestion may be forthcoming. Number then not only makes quanta directly comparable one with another, but also inultiplies and factors the ratios. No doubt before arithmetical methods of multiplying and tactoring fractions were discovered the Pythagoreans were compelled to resort in each case to direct measurement. To what arithmetical methods in particular Philolaus here refers I would not venture to say, s all the evidence that I.have met with belongs to a very late period. But it is safe to say that they must have been very crude and cumbersome. The scholia upon Euclid VL def. 5 give many illustrations of the later methods of compounding and dividing ratios. One only (No. 7, Heiberg V. 329, 18) employs our present ordinary

fJL (* fi ^*\ ( y X 4 = j-jj. The most comm^on method seems to have been that of No. 6 (i. e. 1^ X 1^ = (l X 1) +.0 X 1) + G-X 1) + CiXi) = l + i + i + i = 2), though the ratios were often (Scholium 3) first expressed in sexagesimal ternis. In both these passages, then, Philolaus celebrates the glories of Number regarded s a substitute for geometry in the actual performance of mathematical calculations. The significance of this point of view I shall consider later. Thus far I have tacitly assumed that the περαίνοντá and the άπειρá are things the λόγοé of which were capable of deterrnination by the geometrical and arithmetical methods of the Pythagoreans. If so they must both be not only quanta but iinite and, in theory at least, measurable quanta. If so, what are they? At the risk of seeming to beg the question I shall render the words by uln- detenninates" and "Determinants55. The relevant fragments are these: 1. Diog. Laert. VIII, 85, FV p. 249, 38. ά φύσις ο εν τώé κόαμωé άρμόχθη aThe Nature in the Cosmos is ες airstpcov ί^κτε περαινόντων xcci joined out of both Indeterminates

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όλο« 6 κόαμος êáé τá εν αύτώé and Determinante, olh the whole πάντα. (Usmos and all the things in it." 2. Stob. Ecl. I, 21, 7 a (p. 187, 14. W), FV p. 250, 4. ανάγκá τá έόντá εφεν πάντá "Uf necessity existing thingsare ή περαίνοντá ή άπειρá ή περαίνοντá all either Determinante or Inde- rs êáé άπειρα, άπειρá οϊ μόνον (ή terminates, or both Determinants περαίνοντá μόνον) οõ êá ειη. έπεί and Indeterminates; Indetermi- τοίνυν φαίνεταé ουτ' εê περαινόντων nates only {or Determinants only) πάντων έόντá oi>V ες απείρων they could not be. Sincethenthey πα'ντων, δήλον τάρá δτé εê περαι- plainly are not (composed) either νόντων τε êáé απείρων δ τε κόσμος entirely of Determinants or enti- χαΐ τá εν αυτώé συναρμόχΟη. δήλοé rely of Indeterminates, it is there- δέ êáé τá εν τοις εργοις. τá μεν fore clear that the Cosmos and the ãáñ αυτών εê περαινόντων περαίνοντι, things in it are joined together τá $ εê περαινόντων τε êáé απείρων out of both Determinants and περαίνοντί τε êáé οõ περαίνοντι, τá Indeterminates. This is made δ' εξ άπειρων άπειρá φανέονται. plain by those in the works also. For those of them (that consist) of Determinants, determine (or are determinate?) those (that consist) of both Determinants and Indeterminates determine and do not determine (are and are not determinate?) those (that consist) of Indeterminates are plainly indeterminate". 3.' Jambl. in Nicom. p. 7, 24 Pistelli; FV, p. 250, 16. άρχάν ãáñ ουδέ το γνωσουμενον uThe very thing to know will

έσσειταé πάντων άπειρων έόντων, bsolutely not exist5 all things κατά τον Φιλόλαον. being .indeterminate - according to JPhilolaus." 4. Stob. Ecl., L 21, 7b (p. 188, 5 W) FV, p. 250, 19. êá! πάντá ãá μάν τá γιγνωσκόμενá "Every thing one can know αριθμόν εχοντι· οõ ãáñ οίον τε ουδέν contains number, for nothing can ούτε νοηθήμεν ούτε γνωσθήμεν άνεõ be either thought or known τούτου. without it.59

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5. Ibid., L 21, 7c (p, 188, 9 W) FV, p. 251, 1. ο ãá μάν άρι&μος έχεé δυο μεν "Nu.inber possesses two proper ίδιá εί'δη, περισσον êáé άρτιον, τρίτον kinds, odd and even, and a third δε dhr' αμφοτέρων μειχθέντων άρτιο- blended of both, the Even-odd" πέριττον έκατέρù δε τù εί'δεος (i. e. product of an odd number πολλά! μορφαι', Ας εκαστον αυταυτδ by two), "but of either kind σημαίνει. there are inany forins, which each thing its very seif exhibits." Ibid. 7d. FV, p. 251, 6. περί δε φυσιος êáé αρμονίας ώδε "As regards Nature and Har- έ'χει· ά μεν έστù των πραγμα'των monia8) the case Stands thus: άίδιος έ'σσá êáé αυτά μεν á φύσις the substance of things, which is Οείαν ãε êáé ουê άνΟρωπίνην ένδέ- oternaL and their very Nature, χεταé γνώσιν πλέον ãá η δτé ουχ1 niay be known by the gods and οίον τ' ην ουδέν των Ιόντων êáé not by men, except indeed that γιγνωσκόμενον υφ' άμών ãε γενέ- none of the things that exist σθαé μη υπάρχουσας τας Ιστούς των could have become known by πραγμάτων, εξ ùν συνέστá ό κόσμο«, us did not the substance of the êáé τώ\Á περαινόντων êáé των άπει- things of which the Cosmos con- ρων. έπεé δε ταé άρχαé υπάρχον sists exist, both of the Deter- ούχ ομοΐαé ουδ? δμοφυλαé εσσαι, minants and ofthelndeterminates. ήδη αδύνατον ης êá αυταΤς κοσ- But inasmuch s the principles μηθηναι, εé μη αρμονίá έπεγένετο were not alike nor of the same ώιτινιών αδε τρόπωé έγένετο. τá kind, it would haA7e been im- μέν ùν όμοιá êáé ομόφυλá αρμονίας possible for thein to be ordered ουδέν έπεδέοντο, τá δε ανόμοιá μηδέ in a Cosmos, had not Harmonia ομόφυλá μηδέ ισολαχη ανάγκá ταé been superimposed upon thein,

*) The nearest Englisb equivalent of Philolaus' untranslatable "Harmo- ia" is perhaps "harmonious System". To the musical theorist it denoted priraarily a combination of Fourths of such a kind (hat tlie notes ia each Fourth are melodically related to those i o the others. The diatonic octave is the simplest form of such a ITarrnonia. But in applying it to the constituents of the unrverse the melodic element proper drops into the background and the word denotes (1) such a combination of these constituents s will present to the mind of the mathematician the numerical relations of the musical scalcs: (2) those very relations, conceived s a substantial soraething capable of holding the constituents together. Archiv ittr Gesciiichtc der Philosophie. XIX. 2, 13

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τοιαύταé άρμονιαé βογχεκλεΐβΟαι, in whatsocver way it was donc. οιαé μελλοντé εν κόσμαη κατέχεσΟαι. Νυνν things that were alikc and of Ihc same kind stood in no lurthor need of flarmonia, but the u alikc and not of the same kind and not cqually apportioned noeds must be locked together by some such Hannonia s they would be lield in a Cosmos by." (The remainder of this fragment identifies "Ilarmonia" with the octave and describes it s consisting of a Fifth, in which are three tones and a semitone, and a Fourth, in which there are two tones and a semitone.) That the Determinants and Indeterininates are not numbers is made highly probable by the passage already discussed, in which n um her combines and splits the λόγοé of both and is thereby distinguished from them. If not among numbers, it is most likely, in view of Pythagorean traditions, that they will be found among the objects of geometry. One is at first inclined to identify the άπειρá with the in- finitely divisible parts of extension. This notion is known to Plato and is extremely common in the later writers. But it cannot be introduced here without doing violence to the language. In Np. 2. it is clear that Philolaus supposes that the "works" to which he appeals are objects famili r to Ins readers. What famili r objects can be said to consist either of limits without extension or of extension without limits? These "works" will, I think, supply the needed clue. . The word has so many meanings that it is difficult without the aid of V clear context to fix that which it bears in a given case. But it must denote famili r objects and probably denotes objects capable of falling under the scope of geometry.. - One of its most common earlier meanings is that of "cultivated fields", and I think this notion makes the context intelligible. The άπειρá would be patches of indeterrninate, i. e. of no definable, shape, the περαίνοντá would be patches of sorne definable shape or

Brought to you by | provisional account Unauthenticated Download Date | 6/15/15 8:09 AM Philoiaus. 187 skapes. The form of tke statement suggests Euclid VI, 20, "Simi- lar polygons di\7ide into similar triangles equal in nuinber", and if one endeavors to apply the same conception here it still further narrows the ineaning of περαίνοντα. Farms wkich consist of similar component patckes are themselves similar, those tkat divide into dissimilar patckes are dissimilar, tkose tkat divide into patckes some of wkich are similar wkile otkers are dissimilar, are themselves in part similar and in part dissimilar. Tkis statement does not express all tke necessary conditions. Tke lines by vvkick the farms are divided into patckes must be drawn from and to koinologous points upon tke peripkery of eack, tke component patckes must be equal in number, and eack must kave tke same Situation witk reference to tke otker components in tke farm of wkick it is a part wkick tke corresponding patch in tke otker farm k s witk reference to its otker components. But suck defects of statement are to be expected in tke early days of geometry. We kave indeed Proclus' evidence to tke fact tkat more tkan a Century after tke time of Pkilolaus Euclid found muck to correct and to Supplement in tke material ke used in preparing tke "Elements".9) Tke language of fragment (6) points in tke direction of tkis conception of tke περαίνοντα. It would seem tkat tke άρχαί of wkick Pkilolaus kere speaks, are, s Prof. Diels indicates, identical \yii\\ tke περαι'νοντá and tke άπειρα. If tkey were δμοΤα, ομόφυλá and ίσολαχή, says Pkilolaas, tkey would stand in no need of a "Har- monia" to kold tkem togetker in a COSHLOS. Tkat tke first two of tkese tkree words are eniinently appropriate to similar figures needs no detnonstration. Tke tkird, ίσολαχή, a reading restored by Meineke from tke meaningless ισοταχή of tke MS., is still more so. Tke word does not, so far s I know, occur elsewhere. Etyrnologically i t skould mean "equally allotted" or "equally apportioned", and tkis exactly expresses tke relation wkick tke components of a figure bear to tke komologous components of anotker wkick is similar to

*) Proclus in Euch Fried!., 68, 7 ... Ευκλείδης ... τá μαλαχώτερον δεικνυ- μενá τοις έμπροσθεν είς ανελέγκτους άποδείςεις άναγαγών. 10) La Geometrie Grecque, Paris, 1887, pp. 97—99. 13*

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it. What vcr tho ratio wltirh an elctnent of one bears to its homolog us elemeut in thc otliur, thut saine ratiu in repeated botvvcon ovcry pair of homologous olements, — the second figure is "cqually apportioned" a* rcgard* tho iirst. Thiss conccption of tue Determinante s eimilar plane iigures makcs intelligible tho referonce to thc έργα, and it is perhaps pos- siblc to iutorprot the other Fragments in accordance therewith. Bat it is not altogether satisfactory. There is little evidence to show that the Pythagoreans of Philolaus3 tiine had attained a de nite conception of geometrical similarity. Tannery10) did indeed believe that it was known even to Pythagoras and that the subordinate Position assigned it by Euclid is due to the influence of Eudoxus' investigations into the laws of proportion. I ain not prepared to deny that such was the case, but I have not inet with sufficient evidence to support so broad a generalization and it is not necessary to the reconstruction of PMlolaus' theory s I understand it. It is enough to assume that by περαίνοντá Philolaus meant some iigures which at least are similar, whatever other characters they may possess. Among them most probably were the circle, the semicircle, the equilateral triaogle, probably the isosceles right triangle and the scalene right with the acute angles 60° and 30 °, the square, the pentagon, hexagon, decagon, dodecagon, pentedecagon and perhaps some others. And the word περαίνοντá properly denoted, not concrete individual iigures of these types, but the essential principles rnanifested in each. It is for this reason that Philolaus uses. the active participle. It expresses the essential principle that "limits" a large number of different iigures in the sense of bringing all under a common class. It is in this sense that Plato uses .πέρας in the Philebus, πέρας is a concept of a higher order than είδος. It is the common function performed by all είδη, that of intro- ducing into the essentially indefinite and indeterminate, definiteness, likeness and consequent classification. By way of Illustration let ine quote Nicomachus' comrnentary on Philolaus' theory. After explaining the difference between square numbers, "othersided" numbers (έτερομήκεις), i. e. prbducts of the form Ë? (Ë· + 1)? and ulong-sidedj3 numbers (προμήκεις), i. e. products

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one factor of which exceeds the other by more than unity, he pro- ceeds (Int Arith. II, 18, 3, p! 113, 19 Hoche). ουκουν ότé μεν οé τετράγωνοé υπό ccConsequently,sincethesquares τίνων άρι&μών ιδίù μήκεé μηκυν&έν- result from lengthening certain των γίνωνται, ταυτον έχοντες το numbers by their own length, μήκος τφ πλάτει, ίδιομήκεις áν their lenght being equal to their κυρίως êáé ταυτομήκεις λέγοιντο, breadth, · they might properly be ofov δις β, τρις ã, τετράκις δ êáé termed "own-sided" and "same- οί εφεξής· εé δε τούτο, επιδεκτικοί sided", s, twice t wo, thrice three, [read επιδεικτικοί with SH] πάντως four times four, etc., and if that, ταυτότητος êáé ισότητος, διόπερώρισ- they might properly be termed μένοé τε êáé περαίνοντες· το ãáñ Ισον perfectly expressive of sameness êáé το ταυτον ένΐτρόπφκαί ώρισμένø and equality, hence both definite τοιούτον δτé δε êáé ο! έτερομήκεις and deterrninant, for the Equal αριθμοί ουê ιδίù μήκει, áëë' έτεροõ and the Same is equal and same μηκυνβέντος αποτελούνται, έτερο- in one definite mode. And since μήκεις τε διá τούτο êáé ετερότητας the "other-sided" numbers are επιδεκτικοί [read επιδεικτικοί] άπει- produced by -the lengthening of ρίας τε êáé αοριστίας, τη δε άρá a nuinber not by its own length διχοστατεé êáé διανενέμηταé êáé but by that of some other, they εναντίá άλλήλοις φαίνεταé τá τε τοõ might properly for this reason be αριθμού πάντá êáé τá εν (τψ)11) termed both "other-sided5' and κοσμώ προς ταύτá άποτελεσ&έντá expressive of otherness and of êáé καλώς οé παλαιοί φυσιολογεΓν indeterminateness and indefini- άρχόμενοé την πρώτην διαίρεσιν της teness. At this point then all κοσμοποιίας ταύτγ/ ποιούνται· Πλά- the (attributes) of nuinber and all των μεν της ταυτοΰ φύσεως êáé the things in the Cosmos due to της θατέροõ όνομάζων êáé πάλιν their influence are brought into της αμέριστοõ êáé αεί κατά τá αυτά variance, disunion, and plain έχουσης ουσίας της τε áõ μεριστής Opposition one to another. And γινομένης, Φιλόλαος δε άναγκαιον rightly do the ancients in be· τá έόντá πάντá εΤμεν ήτοé ά'πειρá ginning* their theories of nature η περαίνοντá η περαίνοντá άμá êáé place at this point the iirst part- απειρα, όπεñ μάλλον συγκατατίί)εταé ing of the Oosmos-process—Plato,

n) εν κ^σμù Iloche.

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είναι, εê περαινυντων άμá êáé arrsi- in charging it to thc Nature ptov συνεατάναé τον κόσμον, κατ' of the Same and that of the εικόνá δηλονότé τοõ αριθμού· xal Other, and again to the undi- ãáñ ούτος σόμπας εê μονάδος êáé vided and always uniform Reality ουα'δος σύγκειταé αρτίοõ τε êáé ττεριτ- on the one hand and to that του, á or/ ισότητός τε êáé άνιαοτητος which suffers di Vision on theother; εμφαντικά ταυτότητος τε êáé έτερο- Philolaus, in saying, 'The existing τητος περαίνοντός τε êáé απείροõ thingsall mustbeeitherIndetermi- ώρισμένοõ τε êáé αορίστου. natesorDeterminante, orDetermi- nante and at the sametimelndeter- minates', which latter he votes a fact — that the Cosmos consists of Determinants and at the same time of Indeterminates, after the likeness of numbw of course, or it also consists entirely of Unit and Two, Even and Odd, which indeed are expressive of Equality and

Inequality3Determinant and Inde- terminate, Definite and Indefi- nite." I shall later endeavor to show that Nicomachus in\rerts the true relation of the Determinants and Indeterminates to "Number. The attributes of which he is spealdng were first discerned in them and afterwards transferred to Nuraber, not vice versa. The περαίνοντá then are, strictly speaking, the principles of similarity in those similar figures which Philolaus had chiefly studied.

But5 if we may put reliance upon the criticisms of Aristotle, or are even to^judge Philolaus by the practice of later Pythagoreans, it woi d be safe to say that the word could in his mouth deuote also the figures themselves which these principles make similar. That the Pythagoreans of this period had a real knowledge of the abstractions which are the proper objects of geometry and arithmetic is proved by the fact that they were able to reason about them correctly. But it is equally certain that they had no such analytic knowledge s the Greeks possessed after Aristotle had

Brought to you by | provisional account Unauthenticated Download Date | 6/15/15 8:09 AM Philolaus. 191 subjected these sciences to bis searching criticism (Metaph. ΧΠΙ, 3). They slill at times confused, certainly in language, probably in thougbt, the true objects of geometry and arithmetic vvith tbeir cpncrete embodiment in individual cases. I suspect tbat when Philolaus describes the Universe s consisting of both Determinants and Indeterminates, the words should be taten with some such further limitation. The Determinants are not similar figures s such, nor yet are they the pure abstractions which serve s the principles of identity in the relatively few sirailar figures which the Pythagoreans had at that time subjected to careful study. They were certain individual circles, semicircles, triangles, squares, hexagons, dodecagons, pentedecagons, etc. And these were those which the Pythagorean astronomers were in the habit of conceiving to be inscribed in the visible universe s a means of expressing geometrically and arithmetically its more important phenomena. The evidence for this inference I shall endeavor to draw from certain s yet little understood passages of the later writers. I shall not be able to put it beyond question, but I hope to give it at least some measure of probability. Proclus Comm. on Euclid, 166, 14 Friedlein. Quoted in part FV p. 246, 24. [Text of Friedlein.] * ^ οί δε Πυθαγόρειοé το μεν "The Pythagoreans hold that τρίγωνον απλώς αρχήν γενέσεως the triangle is absolutely the εΐναί φασé êáé της των γενητών principle of genesis and becoming, είδοποιίας. διό êáé τους λόγους that is to say of the processes τους φυσικούς êáé της των στοι- by which things that have come χείων δημιουργίας τριγωνικούς είναé into being receive their forrns. φησιν 6 Τίμαιος, êáé ãáñ τριχ·ή It is for this reason that (Plato's) διίστανταé êáé συναγωγοΙ των πα'ντη Timaeus makes triangul r the μεριστών εισé êáé πολύ μεταβάλù ν, rational principles found in nature της τε απειρίας άναπίμπλανταé της which create the four elements. υλικής êáé τους συνδέσμους λυτούς For these principles stand apart προίστανταé των ένύλων σωμάτων, tridimensionally and kecp their ώσπεñ δη êáé τá τρίγωνá περί- iniinitely d i visible and indeiin- έχονταé μεν υπό ευθειών, γωνίας]] itely changeful Contents together,

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Si έχεé τάς το πλήθος των γραμ- thcy are fillecl with the inaterial μων συνάγουσας êáΐ κοινωνίαν επί- Indeterminate aud reodcr disso- κτητον αύταΐς χαΐ συναφήν παρεχό- Juble tbe bonds that enable ma- μενας. efaotcu? apct êáé δ Φιλόλαος terial bodies to be, precisely s την τοõ τριγώνοõ γωνίαν τέτταρσιν triangles, though bounded by άνέοηκεν Οεοίς, Κρόνù êáé Ã'Áé8η straight lines, yet possess angles χαΐ *Αρεú êáé Διονυσω, πασαν την which bring together those many τετραμερή των στοιχείων διακόσ- lines and confer upon them an μησιν την άνωθεν άπδ τοõ οδρανοΰ adventitious coramunion and con- καοήκουσαν είτε άπδ των τεττάρων junction. Quite justly tben did τοõ ζωδιακού τμημάτων εν τούτοις Philolaus also dedicate tbe angle περιλαβών. δ μεν ãáñ Κρόνος of the triangle to four gods, παααν οφιστησé την υγράν êáé Kronos, Hades, Ares and Diony- ψυχράν ουσίαν, o os νΑρης πασαν sus, cornprising in these the entire την εμπυρον φύσιν, êáé δ μέν'Άιδης disposition of the elements, ex- τήν χΟονίην δλην συνέχεé ζωήν, δ tending with its f ur divisions δε Διόνυσος την ύγράν êáé θερμήν from heaven above downwards έπιτροπευεé γένεσιν, ης êáé δ οίνος or eise from the four divisions συμβολον υγρός ùν êáé θερμός, of the zodiac. For Kronos sustains πα'ντες δε ούτοé κατά μεν τάς εις. all the wet and cold substance, τá δευτέρá ποιήσεις διεστήκασι, and Ares all the fiery entity, ηνωνταé δε άλλήλοις. διδ êáé κατά Hades rnaintains the whole of μίαν αυτών γωνιαν συνάγεé την earth born life, and Dionysus ενωσιν δ Φιλόλαος, εé δε êáé áί presides over the wet and hot των τριγώνων διαφορά! συνεργουσé process of which wine, s wet προς την γένεσιν, εικότως. áν δμο- and hot, is the Symbol. All these, λογοΐτο το τρίγωνον άρχηγδν είναé so far s their secondary opera- της των υπδ σελήνην συστάσεως, tions are concerned, are diverse, ή μεν ãáñ ορθή γωνίá την ουσίαν but they are united one with αυτοις παρέχεταé êáé τδ μέτρον another. It is for this reason that αφορίζεé τοõ είναι, êáé δ τοõ ορθό- Philolaus effects their unification γωνίοõ τριγώνοõ λόγος ούσιοποιός in ne angle. εστί των γενητών στοιχείων, ή δε And if thedifferences also bet- άμβλεΐá την έπίπαν διάστασιν]αδτοΐς ween the various types of triangle ένδίδωσι, êáé δ τοõ άμβλυγωνίοõ co-operate in the processes of λόγος ε?ς μέγεοος αυξεé êáé παντοίαν origination and becoming, one

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εκτασιν τá είδη τá ενυλα, ή δε might justly own the triangle οξείá γωνίá διαιρετήν αυτήν απο- to be supreine over the consti- τελεί την φύσιν, êáΐ δ τοõ δξυγω- tution of sublunary things. For νίοõ λόγος έπ3 άπειρον αδτοΐς τάς right angle provides them with διαιρέσεις παρασκευάζεé γενέσθαι, substantial beiog and deter- απλώς δε 6 τριγωνικός λόγος ούσίαν mines the measure oftheirexis- διαστατήν êáé πάντη μεριστήν υφί- tence, and the rational principle στησé την των ένυλων σωμάτων. of the right angled triangle is constitutive of the substantial essence of the elements which adinit of origination and dissolution. The obtuse angle gives them their extension in all the dimensions and the rational principle of the obtuse angled triangle causes the concrete of Matter and Form to increase in size and in every sort of expansion. The acute angle renders their very substance divisible and the rational principle of the acute angled triangle enables this division to proceed to infinity. In general, the rational principle of the triangle constitutes the indefinitely divisible substance of the material bodies." 173,2. δοκέ? δε êáé τοις Ðõèá- "It is the view of the Pythag- γορείοις τούτο διαφερόντως των oreans also that among the quad- τετραπλεύρων εικόνá φέρειν της rilaterals the square in a notable θείας ουσίας· την τε ãáñ αρχαντον degree presents the likeness of τάςιν διá τούτοõ μάλιστá σημαίνίυσιν the Divine Nature. By i t in — η τ= ãáñ ίρΟότης το άκλιτον chief do they betolcen the Un- êáé ή ισότης την μόνιμον δύναμιν foroken Order, for the square's απομιμείται, κίνησις ãáñ άνισότητος Rectangularity portrays that Or- έ'κγονος, στάαις δε ισοτητος* οé dei^s attribute of being Unde-

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τοίνυν της σταθεράς ιδρύσεως αίτιοé viating, and its Equality portrays τοις δλοις êáΐ της αχράντοõ êáΐ the powor of Enduring, For άκλιτου δυνάμεως εικότως διá τοõ Motion is the oflspring of Ine- τετραγωνικού σχήματος ùς απ' quality, but Rest of Equality. εικόνος εμφαίνονται—καΐ πρδς του- Those beings therefore, that are τοις ο Φιλόλαος κατ' αλλην έπι- for the Whole the grounds of its βολήν την τοõ τετραγώνοõ γωνίαν firm establishment and itspowerto €Ρέας êáΐ Δήμητρος êáé Εστίας keep unbroken and undeviating, αποκαλεί* διότé ãáñ την γήν το quite properly .are manifested τετράγωνον υφίστησé êáΐ στοιχεΐόν through the figure of the square εστίν αυτής προσεχές, ùς παρά τοõ s though in a likeness. In Τιμαίοõ μεμα&ήκαμεν, άπδ δε πασών addition to this, Philolaus, taking τούτων των θεαινών απόρροιας ή another line of attack, declares ãη δέχεταé êáé γονίμους δυνάμεις, the angle of the square that of είκότως την τοõ τετραγώνοõ γωνίαν Rhea and Demeter and Hestia, άνήκεν ταυταις ταΐς ζωογόνοις θεαΐς. for since the square underlies êáé ãáñ Έστίαν καλοΰσé την γήν earth and is its proper consti- êáé Δήμητρα' τίνες, êáé της όλης tuent, s we have learned from Ρέας αυτήν μετέχειν φασί, êáé (Plato's) Timaeus, and since the πάντá εστίν εν αυτ-Q τá γεννητικά earth receives from all these αίτιá χθονίως. την τοίνυν μίαν goddesses effluxes and fertilizing ενωσιν των θείων τούτων γενών influences, he justly dedicated την τετραγωνικήν φησé γωνίαν περί- the angle of the square to these εχειν 14,2 . , δεé δε μη the life-grvnng goddesses. For in λανοάνειν, όπως την μεν τριγωνικήν fact some call the earth Hestia γωνίαν δ Φιλόλαος τέτταρσιν άνήκεν and Demeter and declare that it Οεοΐς, την δε τετραγωνικήν τρισίν, partakes of Rhea in her entirety, ένδεικνυμενος αυτών την δι3 άλλη- and all the causes ofgeneration ëùν χώρησιν êáé την εν πασé πάντων exist in the earth in subterranean κοινωνίαν των τε περισσών εν τοις fashion. So Philolaus affirms that αρτίοις êáé των αρτίων εν τοις the quadrangular angle embraces περισσοΐς. τριάς οδν τετραδική [êáé the one unification of these divine τετράς τριαδική] των τε γονίμων Orders. μετέχουσαé êáé ποιητικών άγαβών ... One should not fail to note την δλην συνέχουσé των γενητών how Philolaus dedicates the tri- διακόσμησιν. αφ3 ùν ή δυωδεκάς angular angle to four gods and

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είς μιαν μονάδá την τοõ Διός αρχήν the quadraogular to tliree, setting άνατείνεται. την ãáñ τοõ δώδεκα- forth their reciprocal interpene- γώνοõ γωνίαν Διός είναé φησιν δ tration and the fact that in them Φιλόλαος, ùς κατά μιαν ενωσιν τοõ all one fmds a participation in Διός όλον συνέχοντος τον της δύω- all, in the even a participation δεκάδος αριθμόν, ηγείταé ãáñ êáé in the odd, and in the odd a παρά τφ Πλα'τωνé δυωδεχάδος ο Ζευς participation in the even. Thus χαé απολύτως επιτροπευεé το παν. a Fourfold Three and a Three- fold Four, participating in the fertilizing and creating goods, hold together the entire System of things that have coine into being. And far from them the Twelve spreads itself into one Unit, the Dominion of Zeus. For the angle of the dodecagon Philolaus declares to be that of Zeus, his idea being that Zeus in one unification holds together the number Twelve. For in Plato, too, Zeus presides over the Twelve and administers absolutely the all.'5 P. 130, 9. χαé ãáñ παρά τοις "In fact among the Pythagoreans Πυθαγορείοις εύρήσομεν αλλάς ãù- we shall find the several angles νιας άλλοις ΟεοΤς άναχειμένας, ως- dedicated to the several gods, s πεñ êáé δ Φιλόλαος πεποίηκε τοις Philolaus also did when he con- μέν την τριγωνικήν γωνίαν τοις δε secrated the triangul r angle to την τετραγωνικήν άφιερώσας, êáé these gods and the quadrangular αλλάς άλλοις êáé την αυτήν πλείοαé angle to those, one to one and θεοις êáé τù αυτφ πλείους, κατά another to another, and the same τάς διαφόρους εν αυτù δυνάμεις to several and several to the άνείς. * same, making the ascription cor- respond to the diverse powers in it." Damascius Comm. in Parm. Plat. II, 127,7, Ruelle: 7 V p. 246, 39. διá τί ãáñ τώé μεν τον κυκλον u\Vhy did the Pythagoreans con- άνιέρουν οé Πυθαγόρειοι, τώé οέ secrate the circle to this god, the

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τρίγωνον, tun δε τετράγωνον, τώé triunglo to that, the square to os άλλο χαΐ άλλο των ευθυγράμμων another, to yet anothcr some [των] σχημάτων, <õς δε êáé μικτών, other rectilinear figure, so too ùς τá ημικύκλιá τοις Διόσκουροι*; with the figures both rectilinear πολλάκις δε τωé αύτώé άλλο êáΐ and curvilinear, s the semicircles άλλο απονέμων κατ1 άλλην ?διότητá to the Dioskouroi? Indeed Philo- χαί άλλην ο Φιλόλαος εν τούτοις laus, who is wise in such matters, στοφός, êáé μήποτε ùς καθόλοõ often assigns to the same god ειπείν τί> μεν περιφερές κοινδν now this, now .that figure, in σχήμá εστίν πάντων των νοερών correspondence with now this θεών ήé νοεροί, τá δε ευθύγραμμá now that property. Perhaps, gene- Γδιá έκαστων άλλá άλλων κατά τάς rally speaking, the curvilinear των αριθμών, των γωνιών êáé των figure is couiinon to all the ra- πλευρών Ιδιότητας- οίον 'Αθήνας tional gods, qua rational, while μεν το τρίγωνο ν, Έρμοõ δε το the rectilinear figures are proper τετράγωνον. ήδη δε φησιν ό Φίλο- each to each in correspondence λαός. êáé τοõ τετραγώνοõ ήδε μεν with the properties of the num- ή γωνίá της 'Ρέας, ηδε δε της bers, angles and sides, s for Ã/Ηρας, άλλη δε άλλης θεού. êáé example, the triangle belongs to ^λος εστίν δ θεολογικός περί των Athena, the square to Hermes; σχημάτων αφορισμός. % this in fact Philolaus says. And this angle of the square belongs to Rhea, this to Hera, each to soine one goddess. And all 'theological' exposition has to do with the figures." Plut. de leide et Osiride, 30, p. 363A, 7 V, 247, 2. VII p. 432 Reiske φαίνονταé δε êáé όί Πυθαγορικοé "The Pythagoreans clearly re-. τον Τυφώνá οαιμονικήν ηγούμενοé gard Typhon s a dernon-like δυναμιν. λέγουσé ãáñ εν άρτίωé Power, for they say that Typhon μέτρωé Ικτωé êáé πεντηκοστωé γεγο- comes into being in an even meas- νέναé Τυφώνα^ êáé πάλιν την μεν ure,' the fifty-sixth. And again

12) This is the reading of the MSS." The emendation έκκαιπεντηκονταγώνοõ was first suggcsted by Xylander, was approved by Reiske and has been introduced into the text by Wyttenbach, Parthey, Bernardakis and Diels.

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τοõ τριγώνοõ Á ιδού êáé Διονύσοõ they. say that the (Power?)13) χαé "Αρεος είναι· την δε τοõ τετρα- of the triangle belongs to Hades γώνοõ :Ρεας êáé 'Αφροδίτης êáé and Dionysus and Ares, that of .Δήμητρας êáé Εστίας êáé "Ηρας· the square to Rhea and Aphrodite την δε τοõ δωδεκαγώνοõ Διός· την δε and Demeter and Hestia and οκτωκαιπεντηκονταγωνίου,12)Τυφώ- Hera, — that of the dodecagon νος, ùς Εΰδοξος ίατόρηκεν. to Zeus and that of the fifty- eight sided figure to Typhon, s Eudoxus has narrated." What was the significance of Philolaus' dedication of "the angle of the triangle" to certain divinities? The accounts of the three sources are not wholly concordant, but s that of Proclus is the niost specific I shall take it s my starting point. In the lirst passage quoted, Proclus supports by three argu- ments the doctrine that the triangle is the principle of genesis and becoming: 1. The ultiinate constituents of matter are composed of triangles. This is Plato's theory. 2. Philolaus, in ascribing the "angle of the triangle" to four gods, is really connecting the triangle with the four elements whicli these gods syinbolize and which are the principles of genesis and becoming.

3%The three species of triangle symbolize the three fundamental attributes of material things, existence, extension and divi- sibility. In the second passage quoted he adduces three similar argu- ments in favor of the notion that the square symbolizes the uni- formity and pernianence of n ture. 1. (Corresponding to (3) above.) It does so by virtue of its Rectangularity and Equality. 2. (Corresponding to (1) above.) Plato teaches that earth consists of cubes, the faces of <\vhich are squares. Now the earth is the seat of the powere that sustain the System of living things. 3. (Corresponding to (2) above.) Philolaus dedicates the angle

13) With Reiste I supply ούναμιν, not γωνίαν.

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of thc gqiuu-c to thrcc godde&ses, and thetfo goddosses symbolize tho carth and ite powor suataiu tho systern of üviüg tliiogs. I'rocliu does not ascribc to Philolaus Plato's thcory, but on the coutrary expressly dbcrimiuates them. As I know of no other evidenco than such äs these passages alford for the ascription of Plato's concoption to Philolaus, I shall set that theory aside, uotwithstanding Boeckh's final declaratiou in its favor (op. cit. 161—2). The explanation of Philolaus' conception which Proclus does give, that the divinities serve to associate the geometrical figures with the elements, may or may not be true. It is in any case insufficient Why is the "angle of the triangle" dedicated to four and that "of the square" to three? Why not vice versa? What is meant by "their reciprocal interpenetration"? Why is the angle of the dodecagon dedicated to only one god? What of the arithinetically absurd statement that one finds in the even a participation in the odd and in the odd a participation in the even? The terms "angle of the triangle" and "angle of the square" u) supply a clue to the solution of these puzzles. I have found but one passage in which their meaning is explained. The author of the anonymous commentary upon Ptolemy's Tetrabiblos%published at Basel in 1559 and ascribed by the editor to Proclus, while engaged in showing that when triangles, squares and hexagons are inscribed in the zodiac, the angles, sides and subtended arcs bear to one another the harmonic ratios, takes occasion to explain that the "angle of the triangle" i. e. of the equilateral triangle, is 120°,

u) After this paper was entirely finished, and while making a final search through the' periodical literature, I discovered that Tannery, in a paper to which I have never met with a reference, (Sur un fragment de Philolaus, Arch. f. Gesch. d. Phil. II, 379), had anticipated me in.regarding the triangles and squares reierred to by Philolaus äs those inscribed in the ecliptic and in quoting Geininus and the later astrologers in Illustration of the conception. He also adduces other illustrations which had escaped me. As he does not endeavor to bring the theory into relation with other aspects of Philolaus' doctrine or with the astronomical methods of the Pythagoreans, I have thought it best to let my paper remain in its original form.

Brought to you by | provisional account Unauthenticated Download Date | 6/15/15 8:09 AM Philolaus. 199 that of the square 90°, that oi' the hexagon 60V5) The angle of tho triangle or square or hexagon is then the angle at the centre of the circle forrned by rad drawn to the vertices of two conti- guous peripheral angles of the inscribed polygon. If froin the centre of the universe rad be drawn to the ecliptic forming with one another angles of 120°, eaoh angle will include four signs of the zodiac. If the iirst, thesecond, the third and the fourth sign in each of the three divisions be connected by straight lines Avith corresponding points of the corresponding signs, four equilateral triangles will result and the twelve vertices will He severally in the twelve signs of the zodiac. In like manner three squares may be inscribed in the ecliptic, two hexagons, one dodecagon. At a very early period the Greeks had learned to inscribe in the sphere of the fixed stars circles the positions of which were determined by its more important phenoinena. Many of these, e. g. the korizon, the equator, the meridian, the two tropics, the ecliptic, are still in use. The arctic circle was described about the north pole of the heavens with a radius equal to the distance from that pole to the northerly horizon. It thus included all the stars which in a given latitude never set. The antarctic circle in like nianner included the stars which in a given latitude are always invisible. I know of no reliable evidence which would show the date of this metbod's adoption. It is described in detail by the earliest writers on Astronomy whose works have come down to us, Auto- lycus and Euclid, and by the later Pythagoreans it was ascribed t "Pythagoras and his successors."16) But it was almost certainly in use long before the time of Philolaus. In the later Greek astronomy celestial dimensions are usually

l5) P. 3l. δ ãáñ εις êáé τρίτον αριθμός 6 της τοõ τριγώνοõ γωνίας προς τον ένá τοõ τετραγώνοõ τον έπίτριτον σώζεé λόγον, 8ς r/v της διá τεσαο?ρων συμ- φωνίας· 6 6έ εΓς ό τοõ τετραγώνοõ προς την τοõ εξαγώνοõ γωνίαν, ήτις έχεé δί- μοιρον, τον ήμκίλιον έχεé λίγον, 2χεé ãáñ αυτόν êáé το ήμισõ αδτου, 8ς ην της διá πέντε συμφωνίας. IC) PluL Epit. H, 12; Stob. Ecl. I, 23; D. 340 a 10; b 10.

Brought to you by | provisional account Unauthenticated Download Date | 6/15/15 8:09 AM 200 Wm, Romaine Newbold, measured in torms of tho clogreo or 360th pari of a circle, and ite soxagcsimal fractions. But in a numbor of passages in tbese later writors one finds auother rnotbod omployed for the oxpression of eertai elementary relations, — that in \vhich the unit is the arc snbtended by ono sido of an inscribed polygon of a given number of sides, The earliest Illustration of this method that I have met with is pcrhaps to be found in Aratus, who, s is well known, generally follows Eucloxus. Aratus states that in his latitude five- eighths oi the Tropic of Cancer are always above the horizon, and thrce-eighths below it, while of the Tropic of Capricom three-eighths are above and five-eighths below.17) Although there is no mention of an inscribed octagon, it is practically certain that the circle was divided into eighths in that way. But the first definite allusion to inscribed polygons is made by Geminus (ca. 77 B. G.)· He describes at length18) the four triangles and the three squares inscribed in the ecliptic. The four triangles he employs to express certain alleged ineteorological generalizations of the astrological type. The triangles are termed Northern, Southern, Eastern and Western. If the north wind blows while the inoon is in any one of the signs touched by the vertices of-the northern triangle (Aries, Leo, Sagit- tarius), the weather will not soon change; if the moon be elsewhere it will. And so of the other triangles and the corresponding winds. The first square, with its corners touching the equinoctial and solstitial points, serves to indicate the Situation of the sun at the beginning of each of the four seasons. The seconcl indicates the sun's position at the beginning of the second third of each season, the third its position at the beginning of the last third of. each season. Geminus also states that some persons use the ^squares to indicate the relative positions of point of rising, zenith, point of

ir) Aratus Phaen. 497 Maass τοá μεν, όσον τε μάλιοτα, δι* οκτώ μετρηθέντος πέντε μεν ενδιá στρέφεταé καδ' Υπέρτερá γαίης, τá τρίá δ' εν περατηι· θέρεος δε οé εν τροπαί είσιν áëë5 8 μεν εν βορέù περί καρκίνον έστήρικται. é 18) Geminus Eiern. Astr. ed. Petavius Uranologion; 1630, -pp. 7—9.

Brought to you by | provisional account Unauthenticated Download Date | 6/15/15 8:09 AM Philolaus. 201 settiDg, nadir, but points out that owing to the obliquity of the ecliptic, only the first sq are will even roughly express these relations. Vitruvius, who wrote probably not much later than Geminus, uses inscribed triangles to express the relations of the planets to the sun. He asserts19) that the planets lying outside the sun generally cease to more away from the sun, pause and retrace their paths, at the tinies when they are in the same triangle in which the sun is. Vitruvius affords another and very curious Illustration of the use of inscribed geoinetrical figures for the expression of supposed natural phenomena. Inscribe a triangle in the celestial meridian by drawing lines from the celestial north pole to the most northerly and southerly points upon the horizon, and draw from the longer side perpendiculars to the earth. The shorter any perpendicular the higher in pitch will be the voices of the inhabitants of that part of the earth upon which the perpendicular falls.20) It is probable that a dodecagon was inscribed in the ecliptic to mark off the twelve signs of the zodiac, and another in the celestial equator to mark off the twelve double hours into which the twenty four hours of the νυχθήμερον was divided. This was

19) De Arch. IX, l, 11, Rose. Ei autem qui supra solis iter circinationes peragunt, maxime cum in trigono fuerint quod is inierit, turn non progrediuntur sed regressus facientes morantur, doneque cum idem sol de eo trigono in aliud signuin transitionem fecerit. 20) Op. cit., VJ, l, δ, igitur cum id (i. e. the horizon) habemus certum animo sustinentes, ab labro quod est in regione septentrionali, linea traiecta ad id quod est supra meridianum axem ab eoque altera obliqua in altitudinem ad sumnmm cardinem qui est post stellas septentrionuin, sine dubitatione animadvertemus ex eo esse Schema trigoni mundo, uti organi qnam σαμβύκην Graeci dicunt, itaque quod est spatium proximum imo cardini ab axis liuea in meridianis finibus, sub eo loco quae sunt nationes propter brevitatem alti- tudinis ad mundum sonitura vocis faciunt tenuem-et acutissimum, uti in organo chorda quae est proxima angulo. secundum eam autem reliquae ad mediarn Graeciam remissiores efficiunt in nationibus sonorum scansiones, item a medio in ordinem crescendo ad extremes septentriones sub altitudines caeli nationum Spiritus sont bus gravioribus a natura renim exprimuntur. ita videtur mundi conceptio tota propter inclinationein consonanlissime per solis temperaturam ad hannoniam esse composita. etc. Archiv f r Geschichte der Philosophie. XIX. 2. .14

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the δο)δεκάωρο; llio roprcdeutation of which ιιροη thc monumcuto Boll has «o brilliantly demon tratod (Sphacra, 2U5 sqq.). The iDscriptiou ο Ã orie of thesc polygoiu and the parpose for which it was uscd is dircctly attested by J'roclus. in his commentary upou Euclid, l, 8, after noting that I 7 is not essential to the demonstration of I 8, he affirms that I 7 and many other pro- positions of Euclid are introduced, not because they are essential to the development of geometry s such, but because of their use- fulness to the astronomer. He then adduces by vvay of Illustration the inscription of a polygon of fifteen sides in the great circle passing through the poles of the ecliptic and of the equator in order to express by one of its sides the distance between the poles*21) This estimate of the are s one-fifteenth of the circle is but approximately correct. One siele of a polygon of fifteen sides subtends an are of 24°. At the present time the axis of the ecliptic makes with hat of the equator an angle of 23° 27'. But in the year 400 B. Cl it was about 23° 44' 40". Hence, if the estimate was made in the late fiith or early fourth Century, the discrepancy would be only about 15', or one-half the moon's diameter, and inight well have escaped notice. And if it be true, s tradition alleges, that the Greek astronomers obtained from the Babylonians arid Egyptians observations derived from remote antiquity, the estimate inay have been made at a time when it was still nearer to the true value. About the year 2300 B.C. it would have exactly expressed the true value.3a)

21) Procl. in Eucl. 269, 8—18 êáé οõ τούτο μόνον τον στοιχειωτην ùς προς άατρονομιαν ήμίν συντελούν οδού παφεργον δεικνυναι, áëëά êáé δλλá θεωρήματá πολλά τε êáé προβλήματα, το γο&ν τελευταιον εν τø τετα'ρτω, êáè5 δ την τοõ πεν- τεκαιδεκαγώνοõ πλευράν εγγράφεé τø κύκλφ, τίνος ένεκá φησίν τις αί>τόν προβάλλειν •ή της προς άστρονομίαν τοότοõ τοõ προβλήματος αναφοράς; έγγράψαντες ãáñ εις τον διá των πόλων κύκλον το πεντεκαιδεκάγωνον εχουσé την άπόστασιν των πόλων τοõ τε ισημερινού êáé τοõ ζωδιακού, πεντεκαιδεκαγωνικην ãáñ αλλήλων πλευράν άφεστήκασιν. Cf. also Theon Smyr. 199, 6—8; 203, 11—14, Hiller; Boeckh " ber die vierj hrigen Sonnenkreise der Alten", p. 187. 22) For these figures and for several other suggestions I am indebted to my friend and colleague, Professor H. B. Evans.

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The propositiou to which Proclus refers is tlie last of the fourth book of Euclid, in which book are laid down the methods of inscribing and of circumscribing circles in and about polygons and polygons in and about circles. Of this book the scholiast declares that it is in toto the work of the Pythagoreans.*3) It seems to me not unreasonable to infer that these few illustrations which I have * been able to cull out of the later literature are survivals of a method at one tiine in general use among the Pythagorean students of geometry and astronomy, — a method which was soon displaced by the introduction of the more con- venient rnethods of calculation by degrees, minutes and seconds, and by the discovery of trigonoinetrical methods of expressing chords in terms of the radius. Yet like many another conception and method at one time honored by science, when displaced by something better it passed into the Service of superstition and has persisted to the present day in the "trine", "quartile" and "sextile" "aspects" of astrology. It is likely enough that the Pythagoreans inscribed the regul r solids in the sphere of the fixed stars for similar reasons. Accord- ing to lamblichus24) the schism in the Pythagorean school was caused by Hippasos betraying to the outer world the method of inscribing the dodecahedron in the sphere. The twelve pentagonal faces of the dodecahedron would divide the surface of the sky into twelve fields each of which is readily resolved into five triangles. Such a division inight have been useful in defining the location of the various constellations. It is perhaps to this practice thai Plato alludes in the puzzling words, Tim. 55c έτé δε ούσης ξυστάσεως μιας πέμπτη: επί το παν δ θεός αυτ-fl χατεχρήαατο έκεΓνο διαζωγραφών.

This hypothesis supplies, too? a plausible explanation of the other- Avise incredible Statement found in the Scholia on Euclid (Vol. V, Heiberg, p. 654, 1) and elsewhere, that of the five regul r solids the Pythagoreans knew only three, the cube, the pyramid and the dodecahedron, and that Theaetetus was the iirst to discover the

i3) SchoL in Euch, Heiberg, Vol. V, 273, 13. τá όλá δε θεωρήματá τοõ προκειμένοõ βιβλίοõ é; οντά πυβαγορείων ευρήματα. .«) Vit. Pyth., 88, 247: quoted FV p. 34 § 4.

Brought to you by | provisional account Unauthenticated Download Date | 6/15/15 8:09 AM 204 Wm* Romaine Newholri, or.talicdron and icosahodron. Xo Statement bclittliug thc acliiev- rnentö tho early l'ythagoreaii* glumUMio hostily rejcctcd, for thc prosumption is usually in its favor. ßut the inscriptioii of the dodeeaheclron is more difficult than that of the octahedron and icosahedron, and the statomcnt can scarcely be true äs it Stands, it may be, however, that the Pythagoreans used only these three for practical astronomical purposes. If these are the triangles and squares which Philolaus has in inind, the divinities to whom they are cledicated are^possibly those oi the zodiac.35) Many ancient peoples, the Greeks ainong them, associatecl each sign of the zodiac with some divinity. Eudoxus seems to have drawn up a species of "saints9 calendar" in which he set forth the gods to whom each sign and the month in which the sun traverses that sign were dedicated. Boll has shown that the patron gods of the months in the later Roman calendar were connected with and probably derived from that of Eudoxus.26) The divinities associated with each sign might then readily be transferred to the vertices of the polygons which are in contact with that sign. Then each triangle would be associated with three divinities, each square with four, etc. But again some one of the three divinities associated with each triangle might be selected äs the "patron saint" of that triangle. This was the rnethod of the later astrologers. The first triangle, for example, is Aries, Leo and Sagittarius. These signs are the uhouses55 of the Sun, Jupiter and Mars, but only the Sun and Jupiter are "lords of the triangle", the Sun by day, Jupiter by night. And so of the others, — the second is mied by the Moon and Venus, the third by Saturn and Mercury, the fourth by Mars, but in conjunction with the Moon and Venus.,. Thus though there will be four vertices in contact with every sign, — those of a triangle, a square, a hexagon, and a dodecagon, — the divinity of the sign might be associated prima- rily with one of the four triangles or three squares or two hexagons.

25) Tannery in the article above referred to (Note 14) suggests that these divinities may represent the planets of the Philolaic cosmology. This does not seem to ine probable. 2G) Sphaera, pp. 472—8.

Brought to you by | provisional account Unauthenticated Download Date | 6/15/15 8:09 AM Philolaxis. 205 or with the one dodecagon. . The "angle of the triangle" (120°) inight subtend that section of the zodiac in which are the four signs to whose divinities the triangles are dedicated, the "angle of the 'square" (90°) that in which are the three signs to whose divinities the squares are dedicated, and that of the dodecagon (30°) inight subtend one sign, — that to which the dodecagon is dedicated. But besides these, so to speak, "patron saints", the triangles will -have secondary relations with all the other eight gods, the squares with all the other nine, the hexagons with the other ten, the dodecagon with the other eleven. It is perhaps to these complex relations that Proclus refers in the third passage quoted. They rnay be thought äs sytnbolized by the way in which the figures pavtially coincide, and this I believe to be the meaning of the phrase "reciprocal interpenetration". The set of triangles again is a "threefold four", the set of squares a Ätfourtbld three'5, and in "them all", i. e. in the whole number of twelve signs or divinities, one always finds a participation of even in odd and odd in even, for an even number of triangles each with an odd number of vertices, or an odd number of squares each with an even number of vertices, will exhaust the twelve. Daniascius' account I cannot reconcile with that of Proclus. He makes the triangle sacred to Athena and the square to Hermes. Prochis mentions neither, and he makes all his triangles sacred to gods and his squares to goddesses.26a) Plutarch is mach inore nearly in accord with Proclus, although he oinits one of the gods of the triangle and adds two to the goddesses of the square. If one suppose that Eudoxus spoke not only of the "patron saints'' but also of the divinities associated \vith the other angles of each figure, and that the significance of the various relations was either not understood by Plutarch or disregarded by him äs unimportant, the discrepancies disappear. Proclus' words, uor eise frotti the four divisions of the zodiac", vrouM seem to intimate that Kronos, Hades, Ares and Dionysus

-Ga) Plutarch also Knows of the association betweeii the equilateml triangle and Athena, cf. de Lside p. 381, eh. 76.

Brought to you by | provisional account Unauthenticated Download Date | 6/15/15 8:09 AM 206 Wm. Romaine Newbold, • ar'o conuectcd with the wintcr solstice, spring equinox, sumracr solstico and auturan equinox, i. e. the signs Capricorn, Aries, Cancer, Libra. Each of these signs cloes in fact belong to a different triangle. But I con find no other indication of what assign- mont Philolaus had in mind, nor can I establish any connection between Eudoxus' gods of the calendar s reconstructed by Boll and Philolaus' gods of the triangles and squares. It is probable that the associations which Philolaus is endeavoring to express are of Egyptian origin, and that these Greek names represent Egyptian divinities. Whether this be the case or not, I must leave to the determination of those that are better acquainted with Egyptian mythology than I am. Damascius5 statement that the Pythagoreans dedicated the "semicircles" to the Dioskouroi aifords an incidental confirmation of the theory that all these notions are to be referred to celestial phenomena. Sextus Empiricus and Joannes Laurentius Lydus record the fact that earlier writers had associated Castor and Poly- deukes with the upper and lower hemispheres of the u diverse and had regarded the myth of their alternate sojourn in the lower world s connected with the alternation of day and night. Who these writers were is not st'ated. They may have been Pythago- reans, but they may also have been some of the rationalizing rnythologists who sought to explain the tales of the gods s nature myths. Still, the fact that the association was known makes it somewhat more probable that the "semicircles" of Damascius sh.ould be, conceived s halves of the great circles of longitude passing through the poles of the equator.37)

27 ) SextvEmp. adv. Math. IX, 37 Fabricius τá ãáñ δύο ημισφαίρια, το τε όπεñ γην êáé το υπό γην, Διοσκούρους οé σοφοί των τότε ανθρώπων ελεγον. διό êáé δ ποιητής τούτο αινιττο'μενο'ς φησιν έπ"* αυτών ά'λλοτε μεν ζώουσ5 έτερήμεροι, άλλοτε δ' αυτέ τεθνασιν ν τιμήν δε λελόγχασιν ίσá θεοΐσé πίλους τ' έπιτιθέασιν αδτοϊς êáé επί τούτοις αστέρας αίνισσομενοé την των ημι- σφαιρίων κατασκευήν. Jo. Laur. Lydus de Mensibus IV, 13 Roether; οί φιλόσοφοé φασé Διοσκόρους είναé το υπό γήν êáé υπέñ γήν ήμισφαίριον τελευτώσé δε άμοιβαδόν μυδικώς, οιονεί, ί>πό τους αντίποδας εξ αμοιβής φερόμενοι.

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That tlie association of Typhon with tlie number 56 and with the polygon of 58 sides is to be explained upon similar principles, is highly probable. Typhon symbolizes some astronomical phenoinena, the polygon of 58 sides and its 56th side have something to do with the measnrement of those phenomena. Precisely what phenomena can not be deterniined with certainty, but I may be pardoned for venturing a guess, In the Greek mythology Typhon was one of the earth-born giants vanquished by Zeus. He 'was supposed to be buried beneath the earth, and to him volcanoes and earthquakes were ascribecl. Bat Plutarch, — and no doubt Eudoxus, from whom he is dravring his Information, — means by Typhon *the Egyptian god Set, the eneniy of the good gods Osiris, Horus and Isis. Students of mythology in Plutarch's day, although they agreed in regarding Typhon s representing sorne natnral phenomenon, differed wideJy s to what phenomenon he represented. Some thought it was the sea, sorne the heat of the sun, some the north winds, some the shadow of the earth which causes the eclipse of the moon. Plutarch thinks there is some truth in all these notions; that in fact Typhon is a symbol of the EviJ in the World.28) He would have been more nearly right had he said that Typhon was a god to whose agency the Egyptians ascribed all the evii and terrifying phenomena of nature. The Egyptians seein to have identified him with the con- stellation Great Bear.29) If one suppose that the phenomenon here symbolized by Typhon is the eclipse of the moon, which the Egyptians described s a battle between Isis and Set, a very striking significance can be attached to the words "in an even measure the fifty-sixth". For if a polygon of 58 sides be inscribed in the moon's orbit, the

2S) De Iside et Osiride, c. 44, 368, [Vil 454, Reiste] Text of Parthey είσί δε τίνες οί το ακίασμá της γης, εις δ την σελήνην ολισϋαίνουσαν έκλείιτειν νομ(£ου3ΐ, Τυφώνá καλούντες, (c. 45) όθεν ουê άζέοικεν ειπείν, ùς ιδίá μεν οδê ορθώς έκαστος, όμοû οέ τ:α'ντες <ίρ9ώς λέγουσιν οõ ãáñ αύχμόν ούδ^ ανεμον ουδέ ι)άλατταν ουοέ r/οτος, áëëά παν δσον ή φύσις βλαβερον χαΐ φθαρτικόν έχεé μ(!ριον τοõ Τυφώνός έατιν. ™) Plut. de J*ide, é·. 21, ñ. 359 (Reiske VII, 418). Boll, Sphaera, p. 162£sqq.

Brought to you by | provisional account Unauthenticated Download Date | 6/15/15 8:09 AM 208 Wm. Ilornairie Newbold, eclipso limit äs calculated by Ptolemy will not only lic in the iifty—sixth siele but very near the middle of that side. And the monsure is properly even, for the Even belongs in the Pythiigorean symbolism to the category of evih The apparent path of the moon L crosses that of the sun E at two points NN', termed nodes, diametri- cally opposite one to another. Eclip- ses take place at or near the nodes. An eclipse of the moon begins \vhen the füll moon M intrenches upon the shadow of the earth S. This takes place at a certain distance NM from the node, the distance being determined by the diameter of the shadow and by the dimensions of the angle ENL, which are both variables. One side of a polygon of fifty-eight sides subtends an arc of 6.207 °-{- or 6° 12' -K If Plutarch's words "in an even measure, the fifty-sixth", be supposed to mean in the middle of the fifty- sixth side, the eclipse limit will lie two and one-half sides or 15° 30' from the node. That is to say, all eclipses of the moon must begin and end at or within that distance on the one or the other side of the node. Ptolemy fixes this value at 15Q12', which differs from that above reached by only 18', a difference so small äs to be quite negligible under the circumstances.30) But this dees not explain the choice. of a polygon of fifty- eight sides. Such a polygon cannot even to-day be inscribed in a circle by any means known to rnathematics. The Pythagoreans must have been lecl to it by endeavoring to< express in terms of a circle some pärticular arc the dimensions of which they had independently determined. Among the complex phenoinena presented by the moon are -certain variations in the velocity of its motion. The inoon's

3°) Synt. Math. VI, 5 (Basel, 1538, Vol. I, p. 144; Heiberg Vol. I, p. 484 l sqq.)·

Brought to you by | provisional account Unauthenticated Download Date | 6/15/15 8:09 AM Philolaus. . 209 average daily motion is determined by dividing 360° or some mul- tiple thereof by the number of days taken to traverse that distance. But owing to the variations in her raotion the moon's actual position at any given time may be before or behind that thus determined. The difference between the two was termed by the Greeks the moon's anomaly. The anomaly has two maxima. When the moon is nearest the earth and presents its greatest apparent diameter, the maximum anomaly is, according to Ptolemy's observations, 7° 40'; when the moon is furthest from the earth it is, according to the same authority, about5°. The mean value is 6° 20', which diifers from one side of a polygon of fifty-eight sides by less than eight seconds, a difterence much smaller than conld be measured by any Instruments possessed by the Pythagoreans. If the polygon of fifty-eight sides represents a primitive eifort to express the moon's maximum anomaly, its dedication to Set becomes intelligible and appropriate. The irregularity in the moon's motion was iucompatible with the perfect symmetry and order which Pythagorean ideals demanded of the phenomena of the celestial Cosrnos, and might well be connected with the malign influence of Set, the devil of Egyptian theology. Bevond these two co-incidences I have not been able to find any evidence in favor of this suggested interpretation of the polygon of fifty-eight sides and the "even measure, the fifty-sixth". The coincidences are very striking, — almost too striking to be accepted. It is difficult to believe that the eclipse limits and the maximum anomaly of the moon had been determined so accurately even before the time of Eudoxus. Of the eclipse-limits I have inet with no mention prior to Ptolemy. The irregularity of the moon's motion rnust have been known at a very early date. The earliest attempt to measure it that t have inet with is that of Hipparchus, reported by Ptolemy. Ptolemy states that Hipparchus, using the data supplied by eclipses

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s 5°49' upon thc first assumption and s 4° 34' upon the second, the mean being 5° 11 */,/. Ptolemy, using the sarae material, cal- culaled it s about 5°.31) Ptolemy found that this result sometimes cloes and sometimes does not agree with the observed position of the moon. ITe has himself observed an anomaly of 7° 40' and he quotes an observation of Hipparchus' own from which he (Ptolemy) iinds the same anomaly.37) Ptolemy does not say that Hipparchus had himself observed the discrepancy. But Delambre's inference33) that he had and that he was seeking material upon which to base inore accurate calculations does not seem unreasonable. The discrepancy amounts to somewhat more than three times the moon's diameter, and it is difficult to believe that it could have been overlooked by so able an astronomer s Hipparchus. On the whole, I do not think that the value of the maximum anomaly calculated by Hipparchus from eclipse-data makes it inx- possible to suppose that its mean value had been more accurately determined by direct observation long before the time of Hipparchus. Plutarch has preserved one other curious scrap of Pythagorean mimber-symbolism which I cannot leave unmentioned, although I can throw no new light upon it. The Pythagoreans, he says34) explain the fact that the Egyptian celebration of the death of

31) Ptol. Synt. Math. IV, 11 (Basel, 1538, Vol. I. p. 104; Heiberg I, 338.5—339.3); Delambre, Hist. de l'Astr. Anc. II, p. 180. 32) Synth. Math. V, 1; 3 (Basel, 1538, I, pp. 104; 110-111; Heiberg I, 350.14—351.11; 362.7—365.9; Delambre II, 185,. 206.) 33) II, 205. Hipparque avait trouve Pequation qui satisfait aux syzygies; ii aperf-ut la necessite d'une autre equation pour les quadratures. II fit des observations qui suffisaient pour trouver cette seconde equation; mais il n'eut pas le tems de" les combiner assez pour en docouvrir la loi. 34) De Iside et Osiride, c. 42, p. 367, 39 (VII, 450 Reiske) Text of Parthey. έβδομη επί δέκá την ^Οσίριδος γενέσ&αé τελευτ/jv Αιγύπτιοé μυδολογοΰσιν, εν ή μάλιστá γίνεταé πληρουμένη κατάδηλος ή πανσέληνος, διό êáé τη ν ήμέρανταυτηνάντίφραζιν οί Πυθαγόρειοé καλουαι, êáé δλως τον αριθμόν τούτον άφοσιουνται. τοõ ãáñ έξκαί- δεκá τετραγώνοõ êáé τοõ όκτωκαίδεκá έτερομήκους, οΓς μόνοις αριθμών επιπέδων συμβέ|3ηκε τάς περιμέτρους ί'σας εχειν τοις περιεχομένοις υπ' αυτών χωρι'οις, μέσος ό των έπτακαίδεκá παρεμπίπτων αντιφράττεé êáé διαζευγνυσιν απ* αλλήλων, êáé διαιρεί τον έπόγδοον λόγον, εις άνισá διαστήματá τεμνόμενος.

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Osiris begins upon the seventeenth of the month Athyr by pointing to the unlucky character of the number seventeen. It is termed άντίφραξις, "blocking", because it falls in between the square six- teen and the oblong eighteen, the only plane numbers whose peri- naeters are equal to their areas (i, e. 44-4-+-4-4-4 = 4χ4; 6 -f- 3 + 6 -f- 3= 6 X 3), blocks off and disjoins them, and thus 18 "splits" the tone-ratio 9/8, cutting it into unequal parts (i. e. /17 X /i6 = /IG)· The "Isia", s the celebration was commonly termed, was generally believed by the Greeks to symbolize and to occur at the time of the winter solstice.3S) Boeckh has made it probable that it originally celebrated the descent of the sun froni the summer to the winter side of the equator, i. e. the autuinn equinox.36) But in iact, before the adoption of the fixed calendar, it coincided with no special season of the year. The Egyptian calendar year was not connected with either the solar or the lunar year and the seventeenth of Athyr in course of time travelled through all the seasons and coincided with all the phases of the moon. The Pythagoreans seem to Im7e thought that it was connected with the eclipse, probably because they regarded the death of Osiris s representing the eclipse of the sun. The number syrabolism is plain enotigh up to a certain point. The numbers nine and eight are perhaps associated with the eclipse because the interval between earth and sun is regarded s representing a tone in the celestial Harmonia, in which case the moon would represent the semitone. But this is not satisfactory, for it is not possible in any diatonic Harmonia that two semitones slioold occur in succession. Moreover, Plutarcli's assertion that the moon becomes f ll on the 17 th of Athyr is quite unintelligible. Tt is not true either in the Egyptian or in the Greek calendar. It is just possible that at the time the Pythagoreans in question made their study of Egyptian religion the seventeenth of Athyr 'ooincided with the f ll moon, and they may have been Ignorant of the fact that the coincidence was but

3δ) Gemiaus Klein. Astr. p. 33 Petavius. S6) Ober die vierj hrigen Sonnen-kreise der Alten, p. 417—34.

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temporary. Hut this is a rjuestion which I am not competeot to discuss. Nor shall I attempt to solve the difficult problems presepted by the Pythagoreans' posscssion of such knowledge of Egyptian rcligion s these scraps of Information would imply. The fact is apparently attested by no less an authority than Eudoxus and is not lightly to be rejected. If accepted, it would tend to verify the tradition of the school, now so g nerally discredited, that the source of Pythagoreanism is to be sought in Egypt. The conception of Philolaus5 scheme which I have above outlined supplies an intelligible Interpretation of a passage in Aristotle's Metaphysic which has hitherto baffled the commentators. In the eighth chapter of the first J3opk, 989 b 29 — 990 a 32, Aristotle criticizes certain Pythogorean conceptions. He does not mention Philolaus either here or elsewhere in Ms works, b t in ny of the conceptions which he criticizes s Pythagorean are precisely those which are ascribed by later writers to Philolaus,37) and one is quite justified in anticipating that.others may be found to belong to the same System of thinking. The leading points which he makes are these. The Pythagoreans take s the objects of philosophy all natural phenomena without distinction. They describe the process by which the world came into being, they care- fully observe the phenomena "connected with its parts", and employ their principles in the explanation of these phenomena precisely s though there existed nothing real to explain save what one can perceive by the senses. Yet their principles are better fitted to explain a higher orcler of reality. Those principles are Limit and Unlimitecl (= Intederminate), Odd and Even. Howcan they by such principles s these explain motion and change? And how without explaining motion and change cah they explain genesis and dissolution, the peculiar characteristics of the sublunary world, or the phenomena presented by the heavenly bodies (ή τá των φερομένων έργá κατά τον ουρανόν)? Even if one grant their claim that the extension with which geometry deals can be derived from

37) Cf. Zeller, Aristoteles und Philolaus, Hermes X, 178 sqq.

Brought to you by | provisional account Unauthenticated Download Date | 6/15/15 8:09 AM Philolaus. 213 thcse principles, how is one to clerive from tlrem the othcr characteristics of the extended bodies ibuud in nature, such s weight and lightness? And yet if one is to judge from their assumptions and assertions, they suppose themselves to be dealing no more with geometrical solids than with those of sensef and consequently their Statements are absolutely ineaningless — they contain nothing proper to sense objects s such. (εξ ùν ãáñ υποτι'Οενταé êáé λέγουσιν, ουδέν μάλλον περί των μαθη- ματικών λέγουσé σωμάτων ή περί των αισθητών διό περί πυρός ή γης, ή των άλλων των τοιούτων σωμάτων ούδ5 δτιοΰν ειρήκασιν, ατ' ουδέν περί των αίσοητών οΐμαé λέγοντες ί'διον). All this applies directly to Philolaus5 conceptions s I understand thein. The remainder of the chapter I translate. 990a 18 (Text of Christ), έτé "Further, how is one to take δε πώς δεé λαβείν αίτιá μεν είναé τá the assertion that the properties of τοõ άρι&μοõ πά&η καίτοναριθμόν των number and number itself have κατά τον ούρανον όντων êáé γιγνομέ- ever been and now are the causes νων êáé εξ αρχής êáé νυν, αριομον δ5 of all things that exist or come into άλλον μηδένá είναé παρά τον αριθμόν existence in the (sublunary) world, τούτον ες οõ συνέστηκεν δ κόσμος; and yet that there is no other όταν ãáñ εν τψδé μεν τù μέρεé number save this of which the δόςá êáé καιρός αυτοΤς ή. μικρόν (supra-lunar) Cosmos consists? δ5 άνωθεν ή κάτωθεν αδικίá êáé For when they locate Belief and κρίσις ή μΐςις, άπόδειςιν δε λέγωσιν Opportunity in this part (of the ότé τούτων μεν εν εκαστον αριθμός Cosmos), and Wickedness or εστί, συμβαίνεé δε κατά τούτον τον Jbdgment or Blending a little τόπον ήδη πλήθος είναé των συνι- higher or lower, and allege s σταμένων μεγεθών διá το τá πάθη proof that each one of these is ταύτá άκολουΟειν τοις τόποις έκάσ- Number, and that concurrently τοις, πότερο ν ούτος ό αυτός εστίν in this (i. e. any glven) place αριθμός ό εν τù ουράνιο, ov δεé there already is a multiplicity λαβείν ότé τούτων εκαστον εστίν, of the resulting geometrical figures ή παρά τούτον άλλος; ο μεν ãáñ because these attributes (i. e. the Πλάτων έτερον είναé φησιν καίτοé figures) accompany each of the κάκεινος αριθμούς οιεταé êáé ταύτá places — is, (I ask), this (num- ειναé êáé τάς τούτων αιτίας, áëëά bcr in the supra-lunar Cosmos)

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του? μεν νοητούς αίτιου?, τούτου? that wy tuunber in the (sub- oh αισθητού?, lunary) vvorld with which we must suppose each of these to be identical, or is it anotber Number in additionthereto? Plato holds it is another, and yet he too believes that both these things and their causes are n ambers, but (holds) that Ithe intelligible (numbers) are causes, while these are objects of sense." I have here assumed that Aristotle is using κόσμος and ουρανός in the senses in which, according to the doxographers, they were used by Philolaus, the former of the celestial world of eternal order which lies above the moon, *the latter of the sublunary world of eternal change. I would paraphrase the passage somewhat s follows;· "The Pythagoreans describe s 'numbers' not only the material objects of the sublunary world which can be measured and thus in a sense can be reducecl to numbers, but also concepts such s Wickedness, Judgment, Blending. They also maintain that the only real 'Number5 is that of which the supra-lunar Cosmos 'consists'. This latter 'number' can be nothing other than the dimensiom of the celestial Cosmos and is therefore extended. That this is their ineaning is shown by their method of locating these concepts in some defmite place in the Cosmos. They allege that the concept in question, "— say cJudgment3, is such and such a number because it presents these or those resemblances to that number; that the. place in question — it may be the orbit of some planet £cf. Alex. ad. loc.) or some sign of the zodiac or some. region of an inscribed dodecahedron or ottier regul r solid — is necessarily connected with many of the geometrical figures inscribed in the Cosmos because vertices of more than one of them will be in contact with it; that these figures are in turn reducible to numbers among which is the number in question. If this arg- ument means anything, it means that this number, that of which the celestial cosmos consists, that which is got by reducing the

Brought to you by | provisional account Unauthenticated Download Date | 6/15/15 8:09 AM Philolaus. 215 cclestial iigurcs to iiumbers,. is the same numbor s that in the sublunary woiid with which those acts and processes wliich we call wickedness, judgtnent, and blending must be identified. Nothing could be more absurd, for the celestial 'numbers5 express extended objects, vvhile these words stand for non-extended concepts. This Plato saw. He too believed that wickedness, judgment, blending and the like are uurnbers, although he distinguished, s the Pytha- goreans hacl not done, between the acts and processes themselves (ταύτα) and their proxiinate causes, the several Ideas, but the only Numbers which he lield to be causes were the Intelligible Numbers; those of which the cosmos and other extended things consist he acknowledged to be nothing more than objects of sense." Pythagorean arithmetic was essentially the offspring of Pytha- gorean geoinetry. It is likely enough that both studies were pur- sued simultaneously in the school froin the earliest times, but in the form in which it has come down to us, arithmetic presupposes geometry. Not only has its terminology been almost wholly borrowed directly from geometry, but the conceptions also of the aim and methods of arithmetic are identical with those of geometry. This is seen most clearly in Euclid's formal demonstrations of the properties of nurnbers (Books VII—IX), but it is still recognisable in Nicomachus3 presentation of those properties s static truths, so to speak. Methods of cornputation had no place in this arithmetic. They belonged to a different art, that of λογιστική. Of the nature of the earlier λογιστική we known practically nothing, but even to the latest period it remained s compared with our own, incredibly clumsy and difficult On the other hand geometrical methods of computation early reached a high degree of simjllicity and convenience. It was in- evitable that many mathematical truths and methods of computation capable of expression in numbers shoulcl iirst be discovered in geometrical form, and it is easy to believe that in each case a successful reduction of the formulae of geometry to numbers would awaken renewed interest in the ever widening powers of number. It is not probable that there was ever at any one time a "diseovery" of arithmetic. But it is quite certain that there must

Brought to you by | provisional account Unauthenticated Download Date | 6/15/15 8:09 AM 216 Wm. Romaine Newbold, liavo comc a timc vvhon l'ythagorean scholars becamc convinced that many of the metlioils and rcsults of geometry could be ex- pressod by arithmetic in still more goneral form. They must then also havo seen that mauy of the phenomena of tiature which cannot be treated by geometry were amenable to arithmetic. The importance of this step can harclly be over-estimated. It was in every way analogous to that taken by Descartes when he provecl to the mathematicians of Europe that many mathematical truths hitherto expressed geometrically coald be -translated into algebraic terms. Jndeed, the step from geometry to arithmetic was more than analogous—it was essentially the first step in a process of development of which thex transition from Euclidean to ana- lytical geometry was but a later step. It must have awakened in the Pythagorean school the most intense interest, and there would naturally follow an eager desire to exploit the new knowledge to the utmost for the benefit of general philosophy. It is to this period that I would assign Philolaus. He had probably inherited from the Pythagorean school the georaetricai analysis of the universe into a limited number of similar figures. So far s it went it was quite satisfactory to the geometer. It explained to him the composition of the All. That these similar figures cohere of themselves and stand in no need of a "Harmonia" was self-evident to his mind's eye. But there remained a residuum. For that very analysis necessarily left remaining even in the Cosmos some portions of space which refused to conform to the few simple types which he termed περαίνοντα, and when he turned to the ουρανός, this present world of sense and eternal change, his theory all but broke down. Of these things very few indeed present those simplQ Jfbrms; the overwhelming majority are of an infinite diversity of shape. The new discovery, or the new consciousness, of the powers of number, offered a ready way of escape. Even these things can be reduced to number, and once so expressed they can be compared with one another and with the περαίνοντá when similarly reduced, all the famili r complex relations can be discerned between thein — all these things are once more held together in a Harmonia.

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But there still remained. those without. These are the things that inen talk of, pray for and fight for but cannot see — Justice, Ternperance, Life, and all things eise of the kind. They too must be brought vvithin the Harmonia. They cannot be directly rneas- ured äs extended things can be, but all manner of resemblances can be detected between them and the numbers, and to the Pytha- gorean mind, trained in the rnethods of mathematics, such resemblances had a species of Substantive reality not perhaps elsewhere feit until Socrates and Plato had done their work. Here then was the method, — each was identified with the. nuniber it reseinbled and was given a local habitation in that part of the Cosmos in which that number was conspicuously manifested. Such would be my tentative reconstruction of Philolaus' general scheine and of the System of ideas of which it is a modification. The evidence is scanty, ainbiguous, of disputable authority, .and to more than conjecture one cannot at present attain. But some such conception äs this makes much intelligible that was not intelligible before, and it is not in conflict with any well established facts. ·

Archiv für Geschieht«» der Philosoph i e. XIX. 2. 15

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