Essays on Ancient Greek Philosophy

John Burnet's Early Greek Philosophy:
Chapter 7: The Pythagoreans

138. The Pythagorean School

After losing their supremacy in the Achaean cities, the Pythagoreans concentrated themselves at Rhegion; but the school founded there did not maintain itself for long, and only Archytas stayed behind in Italy. Philolaus and Lysis, the latter of whom had escaped as a young man from the massacre of Croton, had already found their way to Thebes. We know from Plato that Philolaus was there towards the close of the fifth century, and Lysis was afterwards the teacher of Epaminondas. Some of the Pythagoreans, however, were able to return to Italy later. Philolaus certainly did so, and Plato implies that he had left Thebes some time before 399 B.C., the year Socrates was put to death. In the fourth century, the chief seat of the school is the Dorian city of Taras, and we find the Pythagoreans heading the opposition to Dionysius of Syracuse. It is to this period that the activity of Archytas belongs. He was the friend of Plato, and almost realized the ideal of the philosopher king. He ruled Taras for years, and Aristoxenus tells us that he was never defeated in the field of battle. He was also the inventor of mathematical mechanics. At the same time, Pythagoreanism had taken root in the East. Lysis remained at Thebes, where Simmias and Cebes had heard Philolaus, while the remnant of the Pythagorean school of Rhegion settled at Phlius. Aristoxenus was personally acquainted with the last generation of this school, and mentioned by name Xenophilus the Chalcidian from Thrace, with Phanton, Echecrates, Diocles, and Polymnastus of Phlius. They were all, he said, disciples of Philolaus and Eurytus, and we learn from Plato that Simmias and Cebes of Thebes and Echecrates of Phlius were also associates of Socrates. Xenophilus was the teacher of Aristoxenus, and lived in perfect health at Athens to the age of a hundred and five.

139. Philolaus

This generation of the school really belongs, however, to a later period; it is with Philolaus we have now to deal. The facts we know about his teaching from external sources are few in number. The doxographers, indeed, ascribe to him an elaborate theory of the planetary system, but Aristotle never mentions his name in connection with that. He gives it as the theory of "the Pythagoreans" or of "some Pythagoreans." It seems natural to suppose, however, that the Pythagorean elements of Plato's Phaedo and Gorgias come mainly from Philolaus. Plato makes Socrates express surprise that Simmias and Cebes had not learnt from him why it is unlawful for a man to take his life, and it seems to be implied that the Pythagoreans at Thebes used the word "philosopher" in the special sense of a man who is seeking to find a way of release from the burden of this life. It is probable that Philolaus spoke of the body (sôma) as the tomb (sêma) of the soul. We seem to be justified, then, in holding that he taught the old Pythagorean religious doctrine in some form, and that he laid special stress on knowledge as a means of release. That is the impression we get from Plato, who is far the best authority we have.

We know further that Philolaus wrote on "numbers"; for Speusippus followed him in the account he gave of the Pythagorean theories on that subject. It is probable that he busied himself mainly with arithmetic, and we can hardly doubt that his geometry was of the primitive type described in an earlier chapter. Eurytus was his disciple, and we have seen (§ 47) that his views were still very crude.

We also know now that Philolaus wrote on medicine, and that, while apparently influenced by the theories of the Sicilian school, he opposed them from the Pythagorean standpoint. In particular, he said that our bodies were composed only of the warm, and did not participate in the cold. It was only after birth that the cold was introduced by respiration. The connection of this with the old Pythagorean theory is clear. Just as the Fire in the macrocosm draws in and limits the cold dark breath which surrounds the world (§ 53), so do our bodies inhale cold breath from outside. Philolaus made bile, blood, and phlegm the causes of disease; and, in accordance with this theory, he had to deny that the phlegm was cold, as the Sicilian school held. Its etymology proved it to be warm. We shall see that it was probably this preoccupation with the medicine of the Sicilian school that gave rise to some of the most striking developments of later Pythagoreanism.

140. Plato and the Pythagoreans

Such, so far as I can judge, was the historical Philolaus, though he is usually represented in a very different light and has even been called a predecessor of Copernicus. To understand this, we must turn our attention to the story of a literary conspiracy.

We have seen that there are one or two references to Philolaus in Plato, but these hardly suggest that he played an important part in the development of Pythagorean science. The most elaborate account we have of this is put by Plato into the mouth of Timaeus the Locrian, of whom we know no more than he has chosen to tell us. It is clear at least that he is supposed to have visited Athens when Socrates was still in the prime of life, and that he must have been practically a contemporary of Philolaus. It hardly seems likely that Plato should have given him the credit of discoveries which were really due to his better known contemporary. However, Plato had many enemies and detractors, and Aristoxenus was one of them. We know he made the extraordinary statement that most of the Republic was to be found in a work by Protagoras, and he seems also to be the original source of the story that Plato bought "three Pythagorean books" from Philolaus and copied the Timaeus out of them. According to this, the "three books" had come into the possession of Philolaus; and, as he had fallen into great poverty, Dion was able to buy them from him, or from his relatives, at Plato's request, for a hundred minae. It is certain, at any rate, that this story was already current in the third century; for the sillographer Timon of Phlius addresses Plato thus: "And of thee too, Plato, did the desire of discipleship lay hold. For many pieces of silver thou didst get in exchange a small book, and starting from it didst learn to write Timaeus." Hermippus, the pupil of Callimachus, said that "some writer" said Plato himself bought the books from the relatives of Philolaus for forty Alexandrian minae and copied the Timaeus out of it; while Satyrus, the Aristarchean, says he got it through Dion for a hundred minae. There is no suggestion in any of these accounts that the book was by Philolaus himself; they imply rather that what Plato bought was either a book by Pythagoras, or at any rate authentic notes of his teaching, which had come into the hands of Philolaus. In later times, it was generally supposed that the forgery entitled The Soul of the World, which goes by the name of Timaeus the Locrian, was meant; but it has now been proved that this cannot have existed earlier than the first century A.D. Moreover, it is plain that it is based on Plato's Timaeus itself, and that it was written in order to bolster up the story of Plato's plagiarism. It does not, however, fulfil the most important requirement, that of being in three books, which is always an essential feature of that story.

Not one of the writers just mentioned professes to have seen these famous "three books"; but at a later date there were at least two works which claimed to represent them. Diels has shown how a treatise in three sections, entitled Paideutikon, Politikon, Phusikon, was composed in the Ionic dialect and attributed to Pythagoras. It was largely based on the Puthagorikai apophaseis of Aristoxenus, but its date is uncertain. In the first century B.C., Demetrius Magnes professes to quote the opening words of the work published by Philolaus. These, however, are in Doric. Demetrius does not actually say this work was written by Philolaus himself, though it is no doubt the same from which a number of extracts are preserved under his name in Stobaeus and later writers. If it professed to be by Philolaus, that was not quite in accordance with the original story; but it is easy to see how his name may have become attached to it. We are told that the other book which passed under the name of Pythagoras was really by Lysis. Boeckh has shown that the work ascribed to Philolaus probably consisted of three books also, and Proclus referred to it as the Bacchae, a fanciful Alexandrian title which recalls the "Muses" of Herodotus. Two of the extracts in Stobaeus bear it. It must surely be confessed that the whole story is very suspicious.

141. The "Fragments of Philolaus"

Boeckh argued that all the fragments preserved under the name of Philolaus were genuine; but no one will now go so far as that. The lengthy extract on the soul is given up even by those who maintain the genuineness of the rest. It cannot be said that this position is plausible. Boeckh saw there was no ground for supposing that there ever was more than a single work, and he drew the conclusion that we must accept all the remains as genuine or reject all as spurious. As, however, many scholars still maintain the genuineness of most of the fragments, we cannot ignore them altogether. Arguments based on their doctrine would, it is true, present the appearance of a vicious circle at this stage, but there are two serious objections to the fragments, which may be mentioned at once.

In the first place, we must ask whether it is likely that Philolaus should have written in Doric? Ionic was the dialect of science and philosophy till the time of the Peloponnesian War, and there is no reason to suppose the early Pythagoreans used any other. Pythagoras was himself an Ionian, and it is not likely that in his time the Achaean states in which he founded his Order had adopted the Dorian dialect. Alcmaeon of Croton seems to have written in Ionic. Diels says that Philolaus and then Archytas were the first Pythagoreans to use the dialect of their homes; but Philolaus can hardly be said to have had a home, and it is hard to see why an Achaean refugee at Thebes should write in Doric. Nor did Archytas write in the Laconian dialect of Taras, but in what may be called "common Doric," and he is a generation later than Philolaus, which makes a great difference. In the time of Philolaus and later, Ionic was still used even by the citizens of Dorian states for scientific purposes. The Syracusan historian Antiochus wrote in Ionic, and so did the medical writers of Dorian Kos and Cnidos. The forged work of Pythagoras, which some ascribed to Lysis, was in Ionic; and so was the book on the Akousmata attributed to Androcydes, which shows that, even in Alexandrian times, it was believed that Ionic was the proper dialect for Pythagorean writings.

In the second place, there can be no doubt that one of the fragments refers to the five regular solids, four of which are identified with the elements of Empedocles. Now Plato tells us in the Republic that stereometry had not been adequately investigated at the time that dialogue is supposed to take place, and we have express testimony that the five " Platonic figures," as they were called, were discovered in the Academy. In the Schoha to Euclid we read that the Pythagoreans only knew the cube, the pyramid (tetrahedron), and the dodecahedron, while the octahedron and the icosahedron were discovered by Theaetetus. This sufficiently justifies us in regarding the "fragments of Philolaus" with suspicion, and all the more so as Aristotle does not appear to have seen the work from which these fragments come.

142. The Problem

We must look, then, for other evidence. From what has been said, it will be clear that it is above all from Plato we can learn to regard Pythagoreanism sympathetically. Aristotle was out of sympathy with Pythagorean ways of thinking, but he took great pains to understand them. This was because they played so great a part in the philosophy of Plato and his successors, and he had to make the relation of the two doctrines as clear as he could to himself and his disciples. What we have to do, then, is to interpret what Aristotle tells us in the spirit of Plato, and then to consider how the doctrine we thus arrive at is related to the systems which preceded it. It is a delicate operation, no doubt, but it has been made much safer by recent discoveries in the early history of mathematics and medicine.

Zeller has cleared the ground by eliminating the Platonic elements which have crept into later accounts of the system. These are of two kinds. First of all, we have genuine Academic formulae, such as the identification of the Limit and the Unlimited with the One and the Indeterminate Dyad; and secondly, there is the Neoplatonic doctrine which represents the opposition between them as one between God and Matter. It is not necessary to repeat Zeller's arguments here, as no one will now attribute the doctrine in that form to the Pythagoreans.

This simplifies the problem, but it is still very difficult. According to Aristotle, the Pythagoreans said Things are numbers, though that is not the doctrine of the fragments of "Philolaus." According to them, things have number, which makes them knowable, while their real essence is something unknowable. We have seen reason for believing that Pythagoras himself said Things are numbers (§ 52), and there is no doubt as to what his followers meant by the formula; for Aristotle says they used it in a cosmological sense. The world, according to them, was made of numbers in the same sense as others had said it was made of "four roots" or "innumerable seeds." It will not do to dismiss this as mysticism. The Pythagoreans of the fifth century were scientific men, and must have meant something quite definite. We shall, no doubt, have to say that they used the words Things are numbers in a somewhat non-natural sense, but there is no difficulty in that. The Pythagoreans had a great veneration for the actual words of the Master (autos epha); but such veneration is often accompanied by a singular licence of interpretation. We shall start, then, from what Aristotle tells us about the numbers.

143. Aristotle on the Numbers

In the first place, Aristotle is quite clear that Pythagoreanism was intended to be a cosmological system like the others. "Though the Pythagoreans," he tells us, "made use of less obvious first principles and elements than the rest, seeing that they did not derive them from sensible objects, yet all their discussions and studies had reference to nature alone. They describe the origin of the heavens, and they observe the phenomena of its parts, all that happens to it and all it does." They apply their first principles entirely to these things, "agreeing apparently with the other natural philosophers in holding that reality was just what could be perceived by the senses, and is contained within the compass of the heavens," though "the first principles and causes they made use of were really adequate to explain realities of a higher order than the sensible."

The doctrine is more precisely stated by Aristotle to be that the elements of numbers are the elements of things, and that therefore things are numbers. He is equally positive that these "things" are sensible things, and indeed that they are bodies, the bodies of which the world is constructed. This construction of the world out of numbers was a real process in time, which the Pythagoreans described in detail.

Further, the numbers were intended to be mathematical numbers, though they were not separated from the things of sense. On the other hand, they were not mere predicates of something else, but had an independent reality of their own. "They did not hold that the limited and the unlimited and the one were certain other substances, such as fire, water, or anything else of that sort;. but that the unlimited itself and the one itself were the reality of the things of which they are predicated, and that is why they said that number was the reality of everything." Accordingly the numbers are, in Aristotle's own language, not only the formal, but also the material, cause of things.

Lastly, Aristotle notes that the point in which the Pythagoreans agreed with Plato was in giving numbers an independent reality of their own; while Plato differed from the Pythagoreans in holding that this reality was distinguishable from that of sensible things. Let us consider these statements in detail.

144. The Elements of Numbers

Aristotle speaks of certain "elements" (stoicheia) of numbers, which were also the elements of things. That is clearly the key to the problem, if we can discover what it means. Primarily, the "elements of number" are the Odd and the Even, but that does not seem to help us much. We find, however, that the Odd and Even were identified with the Limit and the Unlimited, which we have seen reason to regard as the original principles of the Pythagorean cosmology (§ 53). Aristotle tells us that it is the Even which gives things their unlimited character when it is contained in them and limited by the Odd, and the commentators are at one in understanding this to mean that the Even is in some way the cause of infinite divisibility. They get into difficulties, however, when they try to show how this can be. Simplicius has preserved an explanation, in all probability Alexander's, to the effect that they called the even number unlimited "because every even is divided into equal parts, and what is divided into equal parts is unlimited in respect of bipartition; for division into equals and halves goes on ad infinitum. But, when the odd is added, it limits it; for it prevents its division into equal parts." Now it is plain that we must not impute to the Pythagoreans the view that even numbers can be halved indefinitely. They must have known that the even numbers 6 and 10 can only be halved once. The explanation is rather to be found in a fragment of Aristoxenus, where we read that "even numbers are those which are divided into equal parts, while odd numbers are divided into unequal parts and have a middle term." This is still further elucidated by a passage which is quoted in Stobaeus and ultimately goes back to Posidonius. It runs: "When the odd is divided into two equal parts, a unit is left over in the middle; but when the even is so divided, an empty field is left, without a master and without a number, showing that it is defective and incomplete." Again, Plutarch says: "In the division of numbers, the even, when parted in any direction, leaves as it were within itself ... a field; but, when the same thing is done to the odd, there is always a middle left over from the division." It is clear that all these passages refer to the same thing, and that can hardly be anything else than the "terms" or dots with which we are already familiar (§ 47). The division must fall between these; for, if it meets with an indivisible unit, it is at once arrested.

145. The Numbers Spatial

Now there can be no doubt that by his Unlimited Pythagoras meant something spatially extended; for he identified it with air, night, or the void. We are prepared, then, to find that his followers also thought of the Unlimited as extended. Aristotle certainly regarded it so. He argues that, if the Unlimited is itself a reality, and not merely the predicate of some other reality, then every part of it must be unlimited too, just as every part of air is air. The same thing is implied in his statement that the Pythagorean Unlimited was outside the heavens. Further than this, it is not safe to go. Philolaus and his followers cannot have regarded the Unlimited as Air; for, as we shall see, they adopted the theory of Empedocles as to that "element," and accounted for it otherwise. One of them, Xouthos, argued that rarefaction and condensation implied the void; without it the universe would overflow. We do not know, however, whether he was earlier than the Atomists or not. It is enough to say that the Pythagoreans meant by the Unlimited the res extensa.

As the Unlimited is spatial, the Limit must be spatial too, and we should expect to find that the point, the line, and the surface were regarded as forms of the Limit. That was the later doctrine; but the characteristic feature of Pythagoreanism is just that the point was not regarded as a limit, but as the first product of the Limit and the Unlimited, and was identified with the arithmetical unit instead of with zero. According to this view, then, the point has one dimension, the line two, the surface three, and the solid four. In other words, the Pythagorean points have magnitude, their lines breadth, and their surfaces thickness. The whole theory, in short, turns on the definition of the point as a unit "having position" (monas thesin echousa). It was out of such elements that it seemed possible to construct a world.

146. The Numbers as Magnitudes

This way of regarding the point, the line, and the surface is closely bound up with the practice of representing numbers by dots arranged in symmetrical patterns, which we have seen reason for attributing to the Pythagoreans (§ 47). Geometry had already made considerable advances, but the old view of quantity as a sum of units had not been revised, and so the point was identified with 1 instead of with 0. That is the answer to Zeller's contention that to regard the Pythagorean numbers as spatial is to ignore the fact that the doctrine was originally arithmetical rather than geometrical. Our interpretation takes full account of that fact, and indeed makes the peculiarities of the whole system depend on it. Aristotle is very decided as to the Pythagorean points having magnitude. "They construct the whole world out of numbers," he tells us, "but they suppose the units have magnitude. As to how the first unit with magnitude arose, they appear to be at a loss." Zeller holds that this is only an inference of Aristotle's, and he is probably right in this sense, that the Pythagoreans never felt the need of saying in so many words that points had magnitude. It does seem probable, however, that they called them onkoi.

Zeller, moreover, allows, and indeed insists, that in the Pythagorean cosmology the numbers were spatial, but he raises difficulties about the other parts of the system. There are other things, such as the Soul and Justice and Opportunity, which are said to be numbers, and which cannot be regarded as constructed of points, lines, and surfaces. Now it appears to me that this is just the meaning of a passage in which Aristotle criticizes the Pythagoreans. They held, he says, that in one part of the world Opinion prevailed, while a little above it or below it were to be found Injustice or Separation or Mixture, each of which was, according to them, a number. But in the very same regions of the heavens were to be found things having magnitude which were also numbers. How can this be, since Justice has no magnitude? This means surely that the Pythagoreans had failed to give any clear account of the relation between these more or less fanciful analogies and their geometrical construction of the universe.

147. The Numbers and the Elements

We seem to see further that what distinguished the Pythagoreanism of this period from its earlier form was that it sought to adapt itself to the new theory of "elements." This is what makes it necessary to take up the consideration of the system once more in connexion with the pluralists. When the Pythagoreans returned to Southern Italy, they would find views prevalent there which demanded a partial reconstruction of their own system. We do not know that Empedocles founded a philosophical society, but there can be no doubt of his influence on the medical school of these regions; and we also know now that Philolaus played a part in the history of medicine. This gives us the clue to what formerly seemed obscure. The tradition is that the Pythagoreans explained the elements as built up of geometrical figures, a theory we can study for ourselves in the more developed form it attained in Plato's Timaeus. If they were to retain their position as the leaders of medical study in Italy, they were bound to account for the elements.

We must not take it for granted, however, that the Pythagorean construction of the elements was exactly the same as that we find in Plato's Timaeus. As we have seen, there is good reason for believing they only knew three of the regular solids, the cube, the pyramid (tetrahedron), and the dodecahedron. Now Plato makes Timaeus start from fire and earth, and in the construction of the elements he proceeds in such a way that the octahedron and the icosahedron can easily be transformed into pyramids, while the cube and the dodecahedron cannot. From this it follows that, while air and water pass readily into fire, earth cannot do so, and the dodecahedron is reserved for another purpose, which we shall consider presently. This would exactly suit the Pythagorean system; for it would leave room for a dualism of the kind outlined in the Second Part of the poem of Parmenides. We know that Hippasus made Fire the first principle, and we see from the Timaeus how it would be possible to represent air and water as forms of fire. The other element is, however, earth, not air, as we have seen reason to believe that it was in early Pythagoreanism. That would be a natural result of the discovery of atmospheric air by Empedocles and of his general theory of the elements. It would also explain the puzzling fact, which we had to leave unexplained above, that Aristotle identifies the two "forms" spoken of by Parmenides with Fire and Earth.

148. The Dodecahedron

The most interesting point in the theory is, however, the use made of the dodecahedron. It was identified, we are told, with the "sphere of the universe," or, as it is put in the Philolaic fragment, with the "hull of the sphere." Whatever we may think of the authenticity of the fragments, there is no reason to doubt that this is a genuine Pythagorean expression, and it must be taken in close connection with the word "keel" applied to the central fire. The structure of the world was compared to the building of a ship, an idea of which there are other traces. The key to what we are told of the dodecahedron is also given by Plato. In the Phaedo, which must have been written before the doctrine of the regular solids was fully established, we read that the "true earth," if looked at from above, is "many-colored like the balls that are made of twelve pieces of leather." In the Timaeus the same thing is referred to in these words: "Further, as there is still one construction left, the fifth, God made use of it for the universe when he painted it." The point is that the dodecahedron approaches more nearly to the sphere than any other of the regular solids. The twelve pieces of leather used to make a ball would all be regular pentagons; and, if the material were not flexible like leather, we should have a dodecahedron instead of a sphere. That proves that the dodecahedron was well known before Theaetetus, and we may infer that it was regarded as forming the "timbers" on which the spherical hulk of the heavens was built.

The tradition confirms in an interesting way the importance of the dodecahedron in the Pythagorean system. According to one account, Hippasus was drowned at sea for revealing "the sphere formed out of the twelve pentagons." The Pythagorean construction of the dodecahedron we may partially infer from the fact that they adopted the pentagram or pentalpha as their symbol. The use of this figure in later magic is well known; and Paracelsus still employed it as a symbol of health, which, is exactly what the Pythagoreans called it.

149. The Soul a "Harmony"

The view that the soul is a "harmony," or rather an attunement, is intimately connected with the theory of the four elements. It cannot have belonged to the earliest form of Pythagoreanism; for, as shown in Plato's Phaedo, it is quite inconsistent with the idea that the soul can exist independently of the body. It is the very opposite of the belief that "any soul can enter any body." On the other hand, we are told in the Phaedo that it was accepted by Simmias and Cebes, who had heard Philolaus at Thebes, and by Echecrates of Phlius, who was the disciple of Philolaus and Eurytus. The account of the doctrine given by Plato is quite in accordance with the view that it was of medical origin. Simmias says: "Our body being, as it were, strung and held together by the warm and the cold, the dry and the moist, and things of that sort, our soul is a sort of temperament and attunement of these, when they are mingled with one another well and in due proportion. If, then, our soul is an attunement, it is clear that, when the body has been relaxed or strung up out of measure by diseases and other ills, the soul must necessarily perish at once." This is clearly an-application of the theory of Alcmaeon (§ 96), and is in accordance with the views of the Sicilian school. It completes the evidence that the Pythagoreanism of the end of the fifth century was an adaptation of the old doctrine to the new principles introduced by Empedocles.

It is further to be observed that, if the soul is regarded as an attunement in the Pythagorean sense, we should expect it to contain the three intervals then recognized, the fourth, the fifth and the octave, and this makes it extremely probable that Posidonius was right in saying that the doctrine of the tripartite soul, as we know it from the Republic of Plato, was really Pythagorean. It is quite inconsistent with Plato's own view of the soul, but agrees admirably with that just explained.

150. The Central Fire

The planetary system which Aristotle attributes to "the Pythagoreans" and Aetius to Philolaus is sufficiently remarkable. The earth is no longer in the middle of the world; its place is taken by a central fire, which is not to be identified with the sun. Round this fire revolve ten bodies. First comes the Antichthon or Counter-earth, and next the earth, which thus becomes one of the planets. After the earth comes the moon, then the sun, the planets, and the heaven of the fixed stars. We do not see the central fire and the antickthon because the side of the earth on which we live is always turned away from them. This is to be explained by the analogy of the moon, which always presents the same face to us, so that men living on the other side of it would never see the earth. This implies, of course, from our point of view, that these bodies rotate on their axes in the same time as they revolve round the central fire, and that the antichthon revolves round the central fire in the same time as the earth, so that it is always in opposition to it.

It is not easy to accept the statement of Aetius that this system was taught by Philolaus. Aristotle nowhere mentions him in connection with it, and in the Phaedo Socrates gives a description of the earth and its position in the world which is entirely opposed to it, but is accepted without demur by Simmias the disciple of Philolaus. It is undoubtedly a Pythagorean theory, however, and marks a noticeable advance on the Ionian views current at Athens. It is clear too that Socrates states it as something of a novelty that the earth does not require the support of air or anything of the sort to keep it in its place. Even Anaxagoras had not been able to shake himself free of that idea, and Democritus still held it along with the theory of a flat earth. The natural inference from the Phaedo would certainly be that the theory of a spherical earth, kept in the middle of the world by its equilibrium, was that of Philolaus himself. If so, the doctrine of the central fire would belong to a later generation.

It seems probable that the theory of the earth's revolution round the central fire really originated in the account of the sun's light given by Empedocles. The two things are brought into close connection by Aetius, who says that Empedocles believed in two suns, while "Philolaus" believed in two or even in three. His words are obscure, but they seem to justify us in holding that Theophrastus regarded the theories as akin. We saw that Empedocles gave two inconsistent explanations of the alternation of day and night, (§ 113) and it may well have seemed that the solution of the difficulty was to make the sun shine by reflected light from a central fire. Such a theory would, in fact, be the natural issue of recent discoveries as to the moon's light and the cause of its eclipses, if these were extended to the sun, as they would almost inevitably be.

The central fire received a number of mythological names, such as the "hearth of the world," the "house," or "watch-tower" of Zeus, and "the mother of the gods." That was in the manner of the school, but it must not blind us to the fact that we are dealing with a scientific hypothesis. It was a great thing to see that the phenomena could best be "saved" by a central luminary, and that the earth must therefore be a revolving sphere like the other planets. Indeed, we are tempted to say that the identification of the central fire with the sun was a detail in comparison. It is probable, at any rate, that this theory started the train of thought which made it possible for Aristarchus of Samos to reach the heliocentric hypothesis, and it was certainly Aristotle's successful reassertion of the geocentric theory which made it necessary for Copernicus to discover the truth afresh. We have his own word for it that he started from what he had read about the Pythagoreans.

In the form in which it was now stated, however, the theory raised almost as many difficulties as it solved, and it did not maintain itself for long. It is clear from Aristotle that its critics raised the objection that it failed to "save the phenomena" inasmuch as the assumed revolution of the earth would produce parallaxes too great to be negligible, and that the Pythagoreans gave some reason for the belief that they were negligible. Aristotle has no clear account of the arguments on either side, but it may be pointed out that the earth was probably supposed to be far smaller than it is, and there is no reason why its orbit should have been thought to have an appreciably greater diameter than we now know the earth itself to have.

A truer view of the earth's dimensions would naturally suggest that the alternation of night and day was due to the earth's rotation on its own axis, and in that case the earth could once more be regarded as in the center. It does not appear that Aristotle knew of anyone who had held this view, but Theophrastus seems to have attributed it to Hicetas and Ecphantus of Syracuse, of whom we know very little otherwise. Apparently they regarded the heaven of the fixed stars as stationary, a thing Aristotle would almost have been bound to mention if he had ever heard of it, since his own system turns entirely on the diurnal revolution.

Both theories, that of the earth's revolution round a central fire and that of its rotation on its own axis, had the effect of making the revolution of the fixed stars, to which the Pythagoreans certainly adhered, very difficult to account for. They must either be stationary or their motion must be something quite different from the diurnal revolution. It was probably this that led to the abandonment of the theory.

In discussing the views of those who hold the earth to be in motion, Aristotle only mentions one theory as alternative to that of its revolution round the central fire, and he says that it is that of the Timaeus. According to this the earth is not one of the planets but "at the center," while at the same time it has some kind of motion relatively to the axis of the universe. Now this motion can hardly be an axial rotation, as was held by Grote; for the whole cosmology of the Timaeus implies that the alternation of day and night is due to the diurnal revolution of the heavens. The fact that the earth is referred to a little later as "the guardian and artificer of night and day" proves nothing to the contrary, since night is in any case the conical shadow of the earth, which is thus the cause of the alternation of day and night. So far, Boeckh and his followers appear to be in the right.

When, however, Boeckh goes on to argue that the word illomenên in the Timaeus does not refer to motion at all, but that it means "globed" or "packed" round, it is quite impossible for me to follow him. Apart from all philological considerations, this interpretation makes nonsense of Aristotle's line of argument. He says that, if the earth is in motion, whether "outside the center" or "at the center," that cannot be a "natural motion"; for, if it were, it would be shared by every particle of earth, and we see that the natural motion of every clod of earth is "down," i.e. towards the center. He also says that, if the earth is in motion, whether "outside the center" or "at the center," it must have two motions like everything else but the "first sphere," and therefore there would be excursions in latitude (pairodoi) and "turnings back" (tropai) of the fixed stars, which there are not. It is clear, then, that Aristotle regarded the second theory of the earth's movement as involving a motion of translation equally with the first, and that he supposed it to be the theory of Plato's Timaeus. It is impossible to believe that he can have been mistaken on such a point.

When we turn to the passage in the Timaeus itself, we find that, when the text is correctly established, it completely corroborates Aristotle's statement that a motion of translation is involved, and that Boeckh's rendering is inadmissible on grammatical and lexicological grounds. We have therefore to ask what motion of translation is compatible with the statement that the earth is "at the center," and there seems to be nothing left but a motion up and down (to speak loosely) on the axis of the universe itself. Now the only clearly attested meaning of the rare word illomai is just that of motion to and fro, backwards and forwards. It may be added that a motion of this kind was familiar to the Pythagoreans, if we may judge from the description of the waters in the earth given by Socrates in the Phaedo on the authority of some unnamed cosmologist.

What was this motion intended to explain? It is impossible to be certain, but it is clear that the motions of the circles of the Same and the Other, i.e. the equator and the ecliptic, are inadequate to "save the appearances." So far as they go, all the planets should either move in the ecliptic or remain at an invariable distance from it, and this is far from being the case. Some explanation is required of their excursions in latitude, i.e. their alternate approaches to the ecliptic and departures from it. We have seen (§ 19) that Anaximander already busied himself with the "turnings back" of the moon. Moreover, the direct and retrograde movements of the planets are clearly referred to in the Timaeus a few lines below. We are not bound to show in detail that a motion of the kind suggested would account for these apparent irregularities; it is enough if it can be made probable that the fifth-century Pythagoreans thought it could. It may have seemed worth while to them to explain the phenomena by a regular motion of the earth rather than by any waywardness in the planets; and, if so, they were at least on the right track.

To avoid misunderstanding, I would add that I do not suppose Plato himself was satisfied with the theory which he thought it appropriate for a Pythagorean of an earlier generation to propound. The idea that Plato expounded his own personal views in a dialogue obviously supposed to take place before he was born, is one which, to me at least, is quite incredible. We know, moreover, from the unimpeachable authority of Theophrastus, who was a member of the Academy in Plato's later years, that he had then abandoned the geocentric hypothesis, though we have no information as to what he supposed to be in the center of our system. It seems clear too from the Laws that he must have attributed an axial rotation to the earth.

151. The Antichthon

The existence of the antichthon was also a hypothesis intended to account for the phenomena of eclipses. In one place, indeed, Aristotle says the Pythagoreans invented it in order to bring the number of revolving bodies up to ten; but that is a mere sally, and Aristotle really knew better. In his work on the Pythagoreans, he said that eclipses of the moon were caused sometimes by the intervention of the earth and sometimes by that of the antichthon; and the same statement was made by Philip of Opus, a very competent authority on the matter. Indeed, Aristotle shows in another passage how the theory originated. He tells us that some thought there might be a considerable number of bodies revolving round the center, though invisible to us because of the intervention of the earth, and that they accounted in this way for there being more eclipses of the moon than of the sun. This is mentioned in close connection with the antickthon, so Aristotle clearly regarded the two hypotheses as of the same nature. The history of the theory seems to be this. Anaximenes had assumed the existence of dark planets to account for lunar eclipses (§ 29), and Anaxagoras had revived that view (§ 135). Certain Pythagoreans had placed these dark planets between the earth and the central fire in order to account for their invisibility, and the next stage was to reduce them to a single body. Here again we see how the Pythagoreans tried to simplify the hypotheses of their predecessors.

152. The Harmony of the Spheres

We have seen (§ 54) that the doctrine commonly, but incorrectly, known as the "harmony of the spheres" may have originated with Pythagoras, but its elaboration must belong to a later generation: and the extraordinary variations in our accounts of it must be due to the conflicting theories of the planetary motions which were rife at the end of the fifth and the beginning of the fourth centuries B.C. We have the express testimony of Aristotle that the Pythagoreans whose doctrine he knew believed that the heavenly bodies produced musical notes in their courses. Further, the pitch of the notes was determined by the velocities of these bodies, and these in turn by their distances, which were in the same ratios as the consonant intervals of the octave. Aristotle distinctly implies that the heaven of the fixed stars takes part in the celestial symphony; for he mentions "the sun, the moon, and the stars, so great in magnitude and in number as they are," a phrase which cannot refer solely or chiefly to the five planets. We are also told that the slower bodies give out a deep note and the swifter a high note, and the prevailing tradition gives the high note of the octave to the heaven of the fixed stars, which revolves in twenty-four hours. Saturn, of course, comes next; for, though it has a slow motion of its own in a contrary direction, that is "mastered" (krateitai) by the diurnal revolution. The other view, which gives the highest note to the Moon and the lowest to the fixed stars, is probably due to the theory which substituted an axial rotation of the earth for the diurnal revolution of the heavens.

153. Things Likenesses of Numbers

We have still to consider a view, which Aristotle sometimes attributes to the Pythagoreans, that things were "like numbers." He does not appear to regard this as inconsistent with the doctrine that things are numbers, though it is hard to see how he could reconcile the two. There is no doubt, however, that Aristoxenus represented the Pythagoreans as teaching that things were like numbers and there are other traces of an attempt to make out that this was the original doctrine. A letter was produced, purporting to be by Theano, the wife of Pythagoras, in which she says that she hears many of the Hellenes think Pythagoras said things were made of number, whereas he really said they were made according to number.

When this view is uppermost in his mind, Aristotle seems to find only a verbal difference between Plato and the Pythagoreans. The metaphor of "participation" was merely substituted for that of "imitation." This is not the place to discuss the meaning of the so-called "theory of ideas"; but it must be pointed out that Aristotle's ascription of the doctrine of "imitation" to the Pythagoreans is abundantly justified by the Phaedo. When Simmias is asked whether he accepts the doctrine, he asks for no explanation of it, but replies at once and emphatically that he does. The view that the equal itself is alone real, and that what we call equal things are imperfect imitations of it, is quite familiar to him, and he is finally convinced of the immortality of the soul just because Socrates makes him see that the theory of forms implies it.

It is also to be observed that Socrates does not introduce the theory as a novelty. The reality of the "ideas" is the sort of reality "we are always talking about," and they are explained in a peculiar vocabulary which is represented as that of a school. The technical terms are introduced by such formulas as "we say." Whose theory is it? It is usually supposed to be Plato's own, though some call it his "early theory of ideas," and say that he modified it profoundly in later life. But there are serious difficulties in this view. Plato is very careful to tell us that he was not present at the conversation recorded in the Phaedo. Did any philosopher ever propound a new theory of his own by representing it as already familiar to a number of distinguished living contemporaries? It is not easy to believe that. It would be rash, on the other hand, to ascribe the origin of the theory to Socrates, and there seems nothing for it but to suppose that the doctrine of "forms" (eidê, ideai) originally took shape in Pythagorean circles, though it was further developed by Socrates. There is nothing startling in this. It is a historical fact that Simmias and Cebes were not only Pythagoreans but disciples of Socrates, and there were, no doubt, more "friends of the ideas" than we generally recognize. It is certain, in any case, that the use of the words eidê and ideai to express ultimate realities is pre-Platonic, and it seems most natural to regard it as of Pythagorean origin.

We have really exceeded the limits of this work by tracing the history of Pythagoreanism down to a point where it becomes practically indistinguishable from the theories which Plato puts into the mouth of Socrates; but it was necessary to do so in order to put the statements of our authorities in their true light. Aristoxenus is not likely to have been mistaken with regard to the opinions of the men he had known personally, and Aristotle's statements must have had some foundation.