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 con- tains 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. Brought to you by | provisional account Unauthenticated Download Date | 6/15/15 8:09 AM Philolaus. 177 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 Brought to you by | provisional account Unauthenticated Download Date | 6/15/15 8:09 AM 178 Wm, Romaine Newbold, 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 ex- planation 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 dis- covered 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 dimen- sion of one is the quotient reached by dividing its other dimension into the product of the two dimensions of the other l —=gd\.