Page:Dictionary of Greek and Roman Biography and Mythology (1870) - Volume 3.djvu/633

PYTHAGORAS. ponded on seems to liave been derived from the writings of Philolaus and Archytas, especially the former (Ritter, I. c. p. 62, &c.). On the philosophy of Archytas Aristotle had composed a treatise in three books, which has unfortunately perished, and had instituted a comparison between his doctrines and those of the Timaeus of Plato (Athen. xii. 12 ; Diog. Laert. v. "25). Pythagoras resembled greatly the philosophers of what is termed the Ionic school, who undertook to solve by means of a single primordial principle the vague problem of the origin and constitution of the universe as a whole. But, like Anaximander, he abandoned the physical hypotheses of Thales and Anaximenes, and passed from the province of physics to that of metaphysics, and his predilection for mathematical studies led him to trace the origin of all things to number, this theory being suggested, or at all events confirmed, by the ob- servation of various numerical relations, or analo- gies to them, in the phenomena of the imiverse. " Since of all things numbers are by nature the first, in numbers they (the Pythagoreans) thought they perceived many analogies to things that exist and are produced, more than in tire, and earth, and Avater ; as that a certain affection of numbers was justice ; a certain other affection, soul and intel- lect ; another, opportunity ; and of the rest, so to say, each in like manner ; and moreover, seeing the affections and ratios of what pertains to harmony to consist in numbers, since other things seemed in their entire nature to be formed in the likeness of immbers, and in all nature numbers are the first, they supposed the elements of numbers to be the elements of all things" (Arist. Met. i. 5, comp. especially Met. xiii. 3). Brandis, who traces in the notices that remain more than one system, developed by different Pythagoreans, according as they recognised in numbers the inherent basis of things, or only the patterns of them, considers that all started from the common conviction that it was in numbers and their relations that they were to find the absolutely certain principles of know- ledge (comp. Philolaus, ap. Stob. Eel. Phys. i. p. 458; Bockh, Philolaos, p. 62 ; Stob. I.e. i. p. 10 ; Bockh, I.e. p. 145, |/euSos ovZajxdos is dpiOixdv eTriTTj/e? a d* dxddaa oiKelou Koi (TVjiKpvTOv Ta Tc5 dpiOfjiw yeve^), and of the objects of it, and ac- cordingly regarded the principles of numbers as the absolute principles of things ; keeping true to the common maxim of the ancient philosophy, that like takes cognisance of like (/caflaTrep eA.676 teal 6 4»/A.o- Kaos, dewp-qTiKoP re oura {tou Koyov rov diro twi> ixadrixdrwv irepiyevS/xevov) ttjs tvv oKwv (f)vaews ex^tJ' Ttj/ct (Tvyy^veiav 7rp6? ravr-qu^ iirciijep iJTrd Tou ofiulov TO ojjLoiou KUTaafx§duea6ai. Sext. Emp. adv. Math. vii. 92 ; Brandis, I. c. p. 442). Aristotle states the fundamental maxim of the Py- thagoreans in various forms, as, (palvovrai 5rj koI ouTui TOU dpidfxov vo/xi^ovTis dpX')v (Jvai Kal ds vKt]V TOis odai koi cis Trct^Tj re /cat e^eis (Met. i. 5) ; or, Toi' dpiQpLov eluai T-qu ovaiau dtravTuv (ibid. p. 987. 19, ed. Bekker) ; or, toi)? dpiO/xovs alrlous ehai rois aWois ttjs ovaias (Met, i. 6. p. 987. 24) ; nay, even that numbers are things themselves (Ibid. p. 987. 28). According to Phi- lolaus (Syrian. 2M Arist. Met. xii. 6. p, 1080, b. 16), number is the " dominant and self-produced bond of the eternal continuance of things." But number has two forms (as Philolaus terms them, ap. Stob. l. c. p. 456 ; Bockh, /. c, p. 58), or elements (Arist. PYTHAGORAS. 621 Met. i. 5), the even and the odd, and a third, re- sulting from the mixture of the two, the even-odd {dprioTTcpKTffov, Philol. /. c). This third species is one itself, for it is both even and odd (Arist. I.e. Another explanation of the dpTioircpicraov^ which accords better with other notices, is that it was an even number composed of two uneven numbers. Brandis, /. c. p. 465, &c.). One, or unity, is the essence of number, or absolute num- ber, and so comprises these two opposite species. As absolute number it is the origin of all numbers, and so of all tilings. (Arist. Alet. xiii. 4. ev dpxd irdvrwv Philol. ap. Bockh, § 19. According to another passage of Aristotle, Met. xii. 6. p. 1080, b. 7. number is produced e/c rov-^ov — rov ivos — Koi aKKov rivos.) This original unity they also termed God (Ritter, Gesch. der FML vol. i. p. 389). These propositions, however, would, taken alone, give but a very partial idea of the Pythagorean system. A most important part is played in it by the ideas of limit, and tlie unlimited. They are, in fact, the fundamental ideas of the whole. One of the first declarations in the work of Philolaus [Philolaus] was, that all things in the universe result from a combination of the unlimited and the limiting ((puffis Se ev rw KOffjucf) dp/jLoxdr} e| direipwp re Kal irepaivourwv, koi oos icoa/JLos Koi rd ev avrtp irdvra. Diog. Laert. viii, 85 ; Bockh, p. 45) ; for if all things had been unlimited, nothing could have been the object of cognizance (Phil. /. c. ; Bockh, p. 49). From tlie unlimited were deduced immediately time, space, and motion (Stob. Eel. Phys. p. 380 ; Simplic. iii Arist. Phys. f. 98, b. ; Brandis, I.e. p. 451). Then again, in some extra- ordinary manner they connected the ideas of odd and even with the contrasted notions of the li- mited and the unlimited, the odd being limited, the even unlimited (Arist. Met. i. 5, p. 986, a. 18, Bekker, comp. Phys. A use. iii. 4, p. 203. 10, Bek- ker). They called the even unlimited, because in itself it is divisible into equal halves ad infinitum, and is only limited by the odd, which, when added to the even, prevents the division (Simpl. ad Arist. Phys. Ausc. iii. 4, f. 1 05 ; Brandis, p. 450, note). Limit, or the limiting elements, they con- sidered as more akin to the primary unity (Syrian. in Arist. Met. xiii. 1). In place of the plural ex- pression of Philolaus (to irepaiuoura) Aristotle sometimes uses the singular irepas, which, in like manner, he connects with the unlimited (t^ aireipov. Met. i. 8, p. 990, 1. 8, xiii. 3. p. 1091, 1. 18, ed. Bekk.). But musical principles played almost as im- portant a part in the Pythagorean system as mathematical or numerical ideas. The opposite principia of the unlimited and the limiting are, as Philolaus expresses it (Stob. /. c. p. 458 ; Bcickh, i. c. p. 62), *■' neither alike, nor of the same race, and so it would have been impossible for them to unite, had not harmony stepped in." This harmony, again, was, in the conception of Philolaus, neither more nor less than the octave (Brandis, /. c. p. 456). On the investigation of the various harmo- nical relations of the octave, and their connection with weight, as the measure of tension, Philolaus bestowed considerable attention, and some impor- tant fragments of his on this subject have baen pre- served, which Biickh has carefully examined (/. c. p. 60 — 89, comp. Brandis, /. c. p. 457, &c.). We find running through the entire Pythagorean systpm the idea tiiat order, or harmony of relation, is the