As the Unlimited is spatial, the Limit must be spatial too, and we should naturally expect to find that the point, the line, and the surface were regarded as all 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. According to this view, then, the point has one dimension, the line two, the surface three, and the solid four.[[790]] 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.”[[791]] It was out of such elements that it seemed possible to construct a world.

The numbers as magnitudes.

146. It is clear that 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]). The science of geometry had already made considerable advances, but the old view of quantity as a sum of units had not been revised, and so a doctrine such as we have indicated was inevitable. This is the true 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 upon 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.”[[792]] Zeller holds that this is only an inference of Aristotle’s,[[793]] 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 ὄγκοι.[[794]]

Nor is Zeller’s other argument against the view that the Pythagorean numbers were spatial any more inconsistent with the way in which we have now stated it. He himself 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.[[795]] Now it appears to me that this is just the meaning of a passage in which Aristotle criticises 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?[[796]] 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 quasi-geometrical construction of the universe. And this is, after all, really Zeller’s own view. He has shown that in the Pythagorean cosmology the numbers were regarded as spatial,[[797]] and he has also shown that the cosmology was the whole of the system.[[798]] We have only to bring these two things together to arrive at the interpretation given above.

The numbers and the elements.

147. When we come to details, we seem to see 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.” It is just this which makes it necessary for us to take up the consideration of the system once more in connexion with the pluralists. When the Pythagoreans returned to Southern Italy, they must have found views prevalent there which imperatively demanded a partial reconstruction of their own system. We do not know that Empedokles 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 Philolaos played a part in the history of medicine.[[799]] This discovery gives us the clue to the historical connexion, which formerly seemed obscure. The tradition is that the Pythagoreans explained the elements as built up of geometrical figures, a theory which we can study for ourselves in the more developed form which it attained in Plato’s Timaeus.[[800]] 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 which we find in Plato’s Timaeus. It has been mentioned already that there is good reason for believing they only knew three of the regular solids, the cube, the pyramid (tetrahedron), and the dodecahedron.[[801]] Now it is very significant that Plato starts from fire and earth,[[802]] and in the construction of the elements 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,[[803]] 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 Hippasos 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 Empedokles 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.[[804]] All this is, of course, problematical; but it will not be found easy to account otherwise for the facts.

The dodecahedron.

148. The most interesting point in the theory is, perhaps, 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.”[[805]] 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 connexion with the word “keel” applied to the central fire.[[806]] The structure of the world was compared to the building of a ship, an idea of which there are other traces.[[807]] The key to what we are told of the dodecahedron is given by Plato. In the Phaedo we read that the “true earth,” if looked at from above, is “many-coloured like the balls that are made of twelve pieces of leather.”[[808]] 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.”[[809]] 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. This points to the Pythagoreans having had at least the rudiments of the “method of exhaustion” formulated later by Eudoxos. They must have studied the properties of circles by means of inscribed polygons and those of spheres by means of inscribed solids.[[810]] That gives us a high idea of their mathematical attainments; but that it is not too high, is shown by the fact that the famous lunules of Hippokrates date from the middle of the fifth century. The inclusion of straight and curved in the “table of opposites” under the head of Limit and Unlimited points in the same direction.[[811]]

The tradition confirms in an interesting way the importance of the dodecahedron in the Pythagorean system. According to one account, Hippasos was drowned at sea for revealing its construction and claiming the discovery as his own.[[812]] What that construction was, we may partially infer from the fact that the Pythagoreans 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.[[813]]