PYTHAGORAS, undoubtedly one of the greatest names in the history of science, was an Ionian, born at Samos about 572 B.C., the son of Mnesarchus. He spent his early manhood in Samos, removed in about 532 B.C. to Croton, where he founded his school, and died at Metapontium at a great age (75 years according to one authority, 80 or more according to others). His interests were as various as those of Thales, but with the difference that, whereas Thales’ knowledge was mostly of practical application, with Pythagoras the subjects of which he treats become sciences for the first time. Mathematicians know him of course, mostly or exclusively, as the reputed discoverer of the theorem of Euclid I. 47; but, while his share in the discovery of this proposition is much disputed, there is no doubt that he was the first to make theoretical geometry a subject forming part of a liberal education, and to investigate its first principles.1 With him, too, began the Theory of Numbers. A mathematician then of brilliant achievements, he was also the inventor of the science of acoustics, an astronomer of great originality, a theologian and moral reformer, founder of a brotherhood ‘which admits comparison with the orders of mediaeval chivalry.’2
The epoch-making discovery that musical tones depend on numerical proportions, the octave representing the proportion of 2 : 1, the fifth 3 : 2, and the fourth 4 : 3, may with sufficient certainty be attributed to Pythagoras himself,3 as may the first exposition of the theory of means, and of proportion in general applied to commensurable quantities, i.e. quantities the ratio between which can be expressed as a ratio between whole numbers. The all-pervading character of number being thus shown, what wonder that the Pythagoreans came to declare that number is the essence of all things ? The connexion so discovered between number and music would also lead not unnaturally to the idea of the ‘harmony of the heavenly bodies’.
Pythagoras left no written exposition of his doctrines, nor did any of his immediate successors in the school; this statement is true even of Hippasus, about whom the different stories arose (1) that he was expelled from the school because he published doctrines of Pythagoras,4 (2) that he was drowned at sea for revealing the construction of the dodecahedron in a sphere and claiming it as his own,5 or (as others have it) for making known the discovery of the irrational or incommensurable.6 Nor is the absence of any written record of early Pythagorean doctrine to be put down to any pledge of secrecy binding the school; there does not seem to have been any secrecy observed at all unless perhaps in matters of religion or ritual; the supposed secrecy seems to have been invented to explain the absence of any trace of documents before Philolaus. The fact appears to be merely that oral communication was the tradition of the school, and the closeness of their association enabled it to be followed without inconvenience, while of course their doctrine would be mainly too abstruse to be understood by the generality of people outside.
Philolaus was the first Pythagorean to write an exposition of the Pythagorean system. He was a contemporary of Socrates and Democritus, probably older than either, and we know that he lived in Thebes in the last decades of the fifth century.7
It is difficult in these circumstances to disentangle the portions of the Pythagorean philosophy which may safely be attributed to the founder of the school. Aristotle evidently felt this difficulty; he clearly knew nothing for certain of any ethical or physical doctrines going back to Pythagoras himself; and, when he speaks of the Pythagorean system, he always refers it to ‘the Pythagoreans’, sometimes even to ‘the so-called Pythagoreans’.8 The account which he gives of the Pythagorean planetary system corresponds to the system of Philolaus as we know it from the Doxographi
For Pythagoras’s own system, therefore, that of Philolaus affords no guide; we have to seek for traces, in the other writers of the end of the sixth and the beginning of the fifth centuries, of opinions borrowed from him or of polemics directed against him.9 On these principles we have seen reason to believe that he was the first to maintain that the earth is spherical and, on the basis of this assumption, to distinguish the five zones.
How Pythagoras came to conclude that the earth is spherical in shape is uncertain. There is at all events no evidence that he borrowed the theory from any non-Greek source.10 On the assumption, then, that it was his own discovery, different suggestions11 have been put forward as to the considerations by which Pythagoras convinced himself of its truth. One suggestion is that he may have based his opinion upon the correct interpretation of phenomena, and above all, on the round shadow cast by the earth in the eclipses of the moon. But it is certain that Anaxagoras was the first to suggest this, the true explanation of eclipses. The second possibility is that Pythagoras may have extended his assumption of a spherical sky to the separate luminaries of heaven; the third is that his ground was purely mathematical, or mathematico-aesthetical, and that he attributed spherical shape to the earth for the simple reason that ‘the sphere is the most beautiful of solid figures’.2 I prefer the third of these hypotheses, though the second and third have the point of contact that the beauty of the spherical shape may have dictated its application both to the universe and to the earth. But, whatever may have been the ground, the declaration that the earth is spherical was a great step towards the true, the Copernican view of the universe.13 It may well be (though we are not told) that Pythagoras, for the same reason, gave the same spherical shape to the sun and moon and even to the stars, in which case the way lay open for the discovery of the true cause of eclipses and of the phases of the moon.
There is no doubt that Pythagoras’s own system was geocentric. The very fact that he is credited with distinguishing the zones is an indication of this; the theory of the zones is incompatible with the notion of the earth moving in space as it does about the central fire of Philolaus. But we are also directly told that he regarded the universe as living, intelligent, spherical, enclosing the earth in the middle the earth, too, being spherical in shape.14 Further, it seems clear that he held that the universe rotated about an axis passing through the centre of the earth. Thus we are told by Aristotle that
‘Some (of the Pythagoreans) say that time is the motion of the whole (universe), others that it is the sphere itself’;15
and by Aëtius that
‘Pythagoras held time to be the sphere of the enveloping (heaven).’16
Alcmaeon, a doctor of Croton, although expressly distinguished from the Pythagoreans by Aristotle,17 is said to have been a pupil of Pythagoras;18 even Aristotle says that, in the matter of the Pythagorean pairs of opposites, Alcmaeon, who was a young man when Pythagoras was old, expressed views similar to those of the Pythagoreans, ‘whether he got them from the Pythagoreans or they from him’.19 Hence he was clearly influenced by Pythagorean doctrines. Now the doxographers’ account of his astronomy includes one important statement, namely that
‘Alcmaeon and the mathematicians hold that the planets have a motion from west to east, in a direction opposite to that of the fixed stars.’20
Incidentally, the assumption of the motion of the fixed stars suggests the immobility of the earth. But this passage is also the first we hear of the important distinction between the diurnal revolution of the fixed stars from east to west and the independent movement of the planets in the opposite direction; the Ionians say nothing of it (though perhaps Anaximenes distinguished the planets as having a different movement from that of the fixed stars); Anaxagoras and Democritus did not admit it; the discovery, therefore, appears to belong to the Pythagorean school and, in view of its character, it is much more likely to have been made by the Master himself than by the physician of Croton.21 For the rest of Alcmaeon’s astronomy is on a much lower level; he thought the sun was flat,22 and, like Heraclitus, he explained eclipses and the phases of the moon as being due to the turning of the moon’s bowl-shaped envelope.23 It is right to add that Burnet24 thinks that the fact of the discovery in question being attributed to Alcmaeon implies that it was not due to Pythagoras. Presumably this is inferred from the words of Aristotle distinguishing Alcmaeon from the Pythagoreans; but either inference is possible, and I prefer Tannery’s. It is difficult to account for Alcmaeon being credited with the discovery if, as Burnet thinks, it was really Plato’s.
But we have also the evidence of Theon of Smyrna, who states categorically that Pythagoras was the first to notice that the planets move in independent circles :
’The impression of variation in the movement of the planets is produced by the fact that they appear to us to be carried through the signs of the zodiac in certain circles of their own, being fastened in spheres of their own and moved by their motion, as Pythagoras was the first to observe, a certain varied and irregular motion being thus grafted, as a qualification, upon their simply and uniformly ordered motion in one and the same sense’ [i. e. that of the daily rotation from east to west].25
It appears probable, therefore, that the theory of Pythagoras himself was that the universe, the earth, and the other heavenly bodies are spherical in shape, that the earth is at rest in the centre, that the sphere of the fixed stars has a daily rotation from east to west about an axis passing through the centre of the earth, and that the planets have an independent movement of their own in a sense opposite to that of the daily rotation, i.e. from west to east.
1 Proclus, Comm. on Eucl. I, p. 65. 15–19.
2 Gomperz, Griechische Denker, i3, pp. 80, 81.
3 Burnet, Early Greek Philosophy, p. 118.
4 Clem. Stromat. v. 58 (Vors. i2, p. 30. 18); Iamblichus, Vit. Pyth. 246, 247 (Vors. i2, p. 30. 10, 14).
5 Iamblichus, Vit. Pyth. 88 (Vors. i2, p. 30. 2).
6 Ibid. 247 (Vors. i2, p. 30. 17).
7 Zeiler, i5, pp. 337, 338.
8 Burnet, Early Greek Philosophy, p. 100.
9 Tannery, op. cit., p. 203.
10 The question is discussed by Berger (Geschichte der wissenschaftlichen Erdkunde der Griechen, pp. 171–7) who is inclined to think that, along with the facts about the planets and their periods discovered, as the result of observations continued through long ages, by the Egyptians and Babylonians, the doctrine of a suspended spherical earth also reached the Greeks from Lydia, Egypt, or Cyprus. Berger admits, however, that Diodorus (ii. 31) denies to the Babylonians any knowledge of the earth’s sphericity. Martin, it is true, in a paper quoted by Berger (p. 177, note), assumed that the Egyptians had grasped the idea of a spherical earth, but, as Gomperz observes (Griechische Denker, i3, p. 430), this assumption is inconsistent with the Egyptian representation of the earth’s shape as explained by one of the highest authorities on the subject, Maspero, in his Hist. ancienne des peuples de l’Orient classique, Les origines, pp. 16, 17.
11 Gomperz, Griechische Denker, i3, p. 90.
12 Diog. L. viii. 35 (Vors. i2, p. 280. 1) attributes this statement to the Pythagoreans.
13 Gomperz, Griechische Denker, i3, p. 90.
14 Alexander Polyhistor in Diog. L. viii.
15 Aristotle, Phys. iv. 10, 218 a 33.
16 Aet. i. 21. 1 (D. G. p. 318; Vors. i2, p. 277. 19).
17 Aristotle, Metaph. A. 5, 986 a 27–31.
18 Diog. L. viii. 83 (Vors. i2, p. 100. 19); Iamblichus, Vit. Pyth. 104.
19 Aristotle, Metaph. i. 5, 986 a 28.
20 Aët. ii. 16. 2–3 (D. G. p. 345; Vors. i2, p. 101. 8).
21 Tannery, op. cit., p. 208.
22 Aët. ii. 22. 4 (D. G. p. 352; Vors. i2, p. loi. 10).
23 Aët. ii. 29. 3 (D. G. p. 359; Vors. i2, p. 101. 10–12).
24 Burnet, Early Greek Philosophy, p. 123, note.
25 Theon of Smyrna, p. 150. 12–18.