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. In other
1 Arist. Phys. Τ', 4. 204 a 20 sqq., especially a 26, ἀλλὰ μὴν ὥσπερ ἀέρος ἀὴρ μέρος, οὕτω καὶ ἄπειρον ἀπείρου, εἴ ye οὐσία ἐστὶ καὶ ἀρχή.
2 See Chap. II. § 53.
3 Cf. Speusippos in the extract preserved in the Zheologumena arith- metica, p. 61 (Diels, Vors. p. 235), τὸ μὴν yap ἃ στιγμή, τὸ δὲ B γραμμή, τὸ δὲ τρία τρίγωνον, τὸ δὲ ὃ πυραμίς. We know that Speusippos is following Philolaos here. Arist. 7722, Z, 11. 1036 Ὁ 12, καὶ ἀνάγουσι πάντα εἰς τοὺς ἀριθμούς, καὶ γραμμῆς Tov λόγον τὸν τῶν δύο εἶναί φασιν. The matter is clearly put in the Scholia on Euclid (p. 78,'19, Heiberg), οἱ δὲ Πυθαγόρειοι. τὸ μὲν σημεῖον ἀνάλογον ἐλάμβανον μονάδι, δυάδι δὲ τὴν γραμμήν, καὶ τριάδι τὸ ἐπίπεδον, τετράδι δὲ τὸ σῶμα. καίτοι ᾿Αριστοτέλης τριαδικῶς προσεληλυ- θέναι φησὶ τὸ σῶμα, ὡς διάστημα πρῶτον λαμβάνων τὴν γραμμήν.
The Pythagoreans 337
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
"1 It was out of such
as a unit “having position.
elements that it seemed possible to construct a
world.
146. It is clear that this way of regarding the point, The numbers the line, and the surface is closely bound up with the ec 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.”* 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
The identification of the point with the unit is referred to by Aristotle, Phys. B, 3. 227 a 27. 2 Arist. Met. M, 6. 1080 b 18 sqq-, 1083 Ὁ 8 sqq.; de Caelo, Τ', τ. 300
a 16(R. P. 76a). 3. Zeller, p. 381.
338 Early Greek Philosophy
magnitude. It does seem probable, however, that they called them ὄγκοι.ἦ
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.” 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?*® This means
2 Zeller, p. 382. ᾿
3. Arist. Met. A, 8. 990 a 22 (Κ. P. 81 ε). I read and interpret thus: “For, seeing that, according to them, Opinion and Opportunity are in a given part of the world, and a little above or below them Injustice and Separation and Mixture,—in proof of which they allege that each of these
The Pythagoreans 339
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 construc-
tion 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,’ and
he has also shown that the cosmology was the whole
of the system.” We have only to bring these two
things together to arrive at the interpretation given
above.
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 recon-
struction 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
is a number,—and seeing that it is also the case (reading συμβαίνῃ jwith Bonitz) that there is already in that part of the world a number of com- posite magnitudes (z.e. composed of the Limit and the Unlimited), because those affections (of number) are attached to their respective regions ;— {seeing that they hold these two things), the question arises whether the number which we are to understand each of these things (Opinion, etc.) to be is the same as the number in the world (2.6. the cosmologicaljnumber) or a different one.” I cannot doubt that these are the extended numbers which are composed (συνίσταται) of the elements of number, the limited and the unlimited, or, as Aristotle here says, the ‘‘ affections of number,” the odd and the even. Zeller’s view that ‘‘ celestial bodies” are meant comes near this, but the application is too narrow. Nor is it the number (πλῆθος) of those bodies that is in question, but their magnitude (μέγεῤο)
For other views of the passage, see Zeller, p. 391, n. I.
1 Zeller, p. 404. 2 Ibid. pp. 467 544.
The numbers
and the
elements.
340 Early Greek Philosophy
played a part in the history of medicine.’ This dis-
covery 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 7zmaeus.” 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 Z7zmaeus.
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.* Now it is very significant that Plato
starts from fire and earth,‘ and in the construction οἱ
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 50, and the dodeca-
1 All this has been put in its true light by the publication of the extract from Menon’s Ἰατρικά, on which see p. 322, ἢ. 2. . 2 In Aet. ii. 6, 5 (R. P. 80) the theory is ascribed to Pythagoras, which is an anachronism, as the mention of ‘‘ elements” shows it must be later than Empedokles. In his extract from the same source, Achilles says oi Πυθαγόρειοι, which doubtless represents Theophrastos better. There is. a. fragment of ‘‘ Philolaos” bearing on the subject (R. P. 79), where the
regular solids must be meant by ra ἐν τᾷ σφαίρᾳ σώματα.
3 See above, p. 329, ἢ. I.
4 Plato, Zim. 31 Ὁ 5.
5 Plato, Zim. 54 5 4. Itis to be observed that in 77m. 48 Ὁ 5 Plato says. of the construction of the elements οὐδείς rw γένεσιν αὐτῶν μεμήνυκεν, which implies that there is some novelty in the theory as he makes Timaios- state it. If we read the passage in the light of what has been said in § 141, we shall be inclined to believe that Plato is working out the Pythagorean doctrine on the lines of the discovery of Theaitetos. There is another
The Pythagoreans 341
hedron 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 7zmaeus
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 re-
sult 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. All this is, of course, problematical; but it
will not be found easy to account otherwise for the
facts. |
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.” Whatever we may think of the authen- _ ticity 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”
indication of the same thing in Arist. Gen. Corr. B, 3. 330 b 16, where we are told that, in the Διαιρέσεις, Plato assumed three elements, but made the middle one a mixture. This is stated in close connexion with the ascrip- tion of Fire and Earth to Parmenides.
1 See above, Chap. IV. p. 213, n. 2.
2 Aet. ii. 6, 5 (R. P. 80) ; ““ Philolaos,” fr. 12 (=20 M.; R. P. 79). On . the ὁλκάς, see Gundermann in Rhein. Mus. 1904, pp. 145 sqq. I agree with him in holding that the-reading is sound, and that the word means ‘*ship,” but I think that it is the structure, not the motion, of a ship which is the point of comparison.
The dodeca- hedron.
342 Early Greek Philosophy
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 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
"3 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.” *
made of twelve pieces of leather.
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.” That gives us
a high idea of their mathematical attainments; but
1 Aet. ii. 4, 15, ὅπερ τρόπεως δίκην προὔπεβάλετο τῇ τοῦ παντὸς «σφαίρᾳ» ὁ δημιουργὸς θεός.