1 Arist. AZet. N, 5. 1092 Ὁ 8 (R. P. 76a). Aristotle does not quote the authority of Archytas here, but the source of his statement is made quite clear by Theophr. 2722. p. vi. a 19 (Usener), τοῦτο yap (sc. τὸ μὴ μέχρι Tov προελθόντα παύεσθαι) τελέου καὶ φρονοῦντος, ὅπερ ᾿Αρχύτας wor’ ἔφη ποιεῖν Εὔρυτον διατιθέντα τινὰς ψήφους" λέγειν γὰρ ὡς ὅδε μὲν ἀνθρώπου ὁ ἀριθμός, ὅδε δὲ ἵππου, ὅδε δ᾽ ἄλλου τινὸς τυγχάνει.
2 Arithmetic is older than geometry, and was much more advanced in Egypt, though still in the form which the Greeks called λογιστική rather than as ἀριθμητική proper. Even Plato puts Arithmetic before Geometry in the Republic in deference to the tradition. His own theory of number, however, suggested the inversion of this order which we find carried out in Euclid.
112 Early Greek Philosophy
It is, therefore, very significant that we do not find
any adequate account of what Aristotle can have meant
by “those who bring numbers into figures like the
triangle and the square” till we come to certain late
writers who called themselves Pythagoreans, and
revived the study of arithmetic as a science inde-
pendent of geometry. These men not only abandoned
the linear symbolism of Euclid, but also regarded the
alphabetical notation, which they did use, as something —
conventional, and inadequate to represent the true
nature of number. Nikomachos of Gerasa says ex-
pressly that the letters used to represent numbers are
only significant by human usage and convention. The
most natural way would be to represent linear or prime
numbers by a row of units, polygonal numbers by units
arranged so as to mark out the various plane figures,
and solid numbers by units disposed in pyramids and
so forth. He therefore gives us figures like this :—
a aaa a aa aaa
a aa aa aaa aa aa aaa
aa aaa
Now it ought to be obvious that this is no innovation,
but, like so many things in Neopythagoreanism, a
reversion to primitive usage. Of course the employ-
ment of the letter a/pha to represent the units is derived
from the conventional notation ;. but otherwise we are
clearly in presence of something which belongs to the
very earliest stage of the science—something, in fact,
1 Nikomachos of Gerasa, /utrod. Arithm. Ὁ. 83, 12, Hoche, Πρότερον δὲ ἐπιγνωστέον ὅτι ἕκαστον γράμμα @ σημειούμεθα ἀριθμόν, οἷον τὸ t, @ τὸ δέκα, τὸ κ, ᾧ τὰ εἴκοσι, τὸ ὦ, ᾧ τὰ ὀκτακόσια, νόμῳ καὶ συνθήματι ἀνθρωπίνῳ, ἀλλ᾽ οὐ φύσει σημαντικόν ἐστι τοῦ ἀριθμοῦ x.7.A. The same symbolism is used by Theo, Zxfosztio, pp. 31 sqq. Cf. also lambl. “ηέγοα. p. 56, 27, Pistelli, ἰστέον γὰρ ws τὸ παλαιὸν φυσικώτερον οἱ πρόσθεν ἐσημαίνοντο τὰς τοῦ ἀριθμοῦ ποσότητας, ἀλλ᾽ οὐχ ὥσπερ οἱ νῦν συμβολικῶς.
Science and Religion 113
which gives the only possible clue to the meaning of
Aristotle’s remark, and to what we are told of the
method of Eurytos.
48. This is still further confirmed by the tradition Triangular,
which represents the great revelation made by Pytha- κε στῶ Bes
goras to mankind as having been precisely a figure of a
this kind, namely the ¢etraktys, by which the Pytha-
goreans used to swear,' and we have no less an
authority than Speusippos for holding that the whole
theory which it implies was genuinely Pythagorean.”
In later days there were many kinds of ¢etrakiys, but
the original one, that by which the Pythagoreans
swore, was the “tetraktys of the dekad.” It was a
figure like this—
and represented the number ten as the triangle of four.
In other words, it showed at a glance that 1+2+3+
4=10. Speusippos tells us of several properties
which the Pythagoreans discovered in the dekad. It
is, for instance, the first number that has in it an equal
number of prime and composite numbers. How much
1 Cf. the formula Οὐ μὰ τὸν ἁμετέρᾳ γενεᾷ παραδόντα τετρακτύν, which is all the more likely to be old that it is put into the mouth of Pythagoras by the forger of the Χρυσᾷ ἔπη, thus making him swear by him- self! See Diels, Arch. iii. p. 457. The Doric dialect shows, however, that it belongs to the later generations of the school,
2 Speusippos wrote a work on the Pythagorean numbers, based chiefly on Philolaos, and a considerable fragment of it is preserved in the Theologumena Arithmetica. It will be found in Diels, Vorsokratzker, Ῥ. 235, 15, and is discussed by Tannery, Sczence hel/éne, pp. 374 564.
3 For these see Theon, Zxfosztio, pp. 93 sqq. Hiller. The τετρακτύς used by Plato in the Zimaeus is the second described by Theon (Zx/. Ῥ. 94, 10 sqq.). It is no doubt Pythagorean, but hardly as old as Pythagoras.
114 Early Greek Philosophy
of this goes back to Pythagoras himself, we cannot
tell; but we are probably justified in referring to him
the conclusion that it is “according to nature” that all
Hellenes and barbarians count up to ten and then
begin over again.
It is obvious that the ¢etraktys may be indefinitely
extended so as to exhibit the sums of the series of
successive numbers in a graphic form, and these sums
are accordingly called “triangular numbers.”
For similar reasons, the sums of the series of
successive odd numbers are called “square numbers,”
and those of successive even numbers “oblong.” If
odd numbers are added to the unit in the form of
gnomons, the result is always a similar figure, namely a
square, while, if even numbers are added, we get a
series of rectangles,! as shown by the figure :—
Square Numbers. Oblong Numbers.
πον epeehite te : ἀπ ρα τόνον j diguseesers ἐ maskuacteke : pare
i setae Ἶ ἘΝ ᾿ : ἢ diveucnieys Ἴ sdb a ones ὃ ΚΘ Ὁ 7
ie : zl : i ἜΤ ΚΎΣΤΙΝ ΤᾺ x fioles ς :
It is clear, then, that we are entitled to refer the study of sums of series to Pythagoras himself; but
1 Cf. Milhaud, Phzlosophes géométres, pp. 115 544. Aristotle puts the matter thus (Piys. T, 4. 203 ἃ 13): περιτιθεμένων yap τῶν γνωμόνων περὶ τὸ ἕν καὶ χωρὶς ὁτὲ μὲν ἄλλο del γίγνεσθαι τὸ εἶδος, ὁτὲ δὲ ἕν. This is more clearly stated by Ps.-Plut. (Stob. i. p. 22, 16), Ἔτι δὲ τῇ μονάδι τῶν ἐφεξῆς περισσῶν περιτιθεμένων ὁ γινόμενος ἀεὶ τετράγωνός ἐστι: τῶν δὲ ἀρτίων ὁμοίως περιτιθεμένων ἑτερομήκεις καὶ ἄνισοι πάντες ἀποβαίνουσιν, ἔσως δὲ ἰσάκις οὐδείς. I cannot feel satisfied with any of the explanations which have been given of the words καὶ χωρίς in the Aristotelian passage (see Zeller, p. 351, n. 2), and I would therefore suggest ταῖς χώραις, compar- ing Boutheros (Stob. i. p. 19, 9), who says, according to the MS. reading, Kal ὁ μὲν (ὁ περισσός), ὁπόταν γεννῶνται ἀνὰ λόγον καὶ πρὸς μονάδας, ταῖς αὑτοῦ χώραις καταλαμβάνει τοὺς ταῖς γραμμαῖς περιεχομένους (sc. ἀριθμούς).
Science and Religion 15
whether he went beyond the oblong, and studied
pyramidal or cubic numbers, we cannot say.’
49. It is easy to see how this way of representing Geometry and
| Σ harmonics.
numbers would suggest problems of a geometrical nature. The dots which stand for the pebbles are regularly called “boundary-stones” (ὅροι, dcermint, “terms”), and the area which they occupy, or rather mark out, is the “field” (χώρα). This is evidently a very early way of speaking, and may therefore be referred to Pythagoras himself. Now it must have struck him that “fields” could be compared as well as numbers,’ and it is even likely that he knew the rough methods of doing this which were traditional in Egypt, though certainly these would fail to satisfy him. - Once more the tradition is singularly helpful in suggest- ing the direction that his thoughts must have taken. He knew, of course, the use of the triangle 3, 4, 5 in constructing right angles. We have seen (p. 24) that it was familiar in the East from a very early date, and that Thales introduced it to the Hellenes, if they did not know it already. In later writers it is actually called the “Pythagorean triangle.” Now the Pytha- gorean proposition par excellence is just that, in a right- angled triangle, the square on the hypotenuse is equal
1 In the fragment referred to above (p. 113, n. 2), Speusippos speaks of four as the first pyramidal number ; but this is taken from Philolaos, so we cannot safely ascribe it to Pythagoras.
2 We have ὅροι of a series (ἔκθεσις), then of a proportion, and in later times of a syllogism. The signs :, ::, and .*. are a survival of the original use. The term χώρα is often used by the later Pythagoreans, though Attic usage required χωρίον for a rectangle. The spaces between the γραμμαί of the adacus and the chess-board were also called χῶραι.
3 In his commentary on Euclid i. 44, Proclus tells us on the authority of Eudemos that the παραβολή, ἔλλειψις, and ὑπερβολή of χωρία were
Pythagorean inventions. For an account of these and the subsequent application of the terms in Conic Sections, see Milhaud, Pézlosophes
géomeétres, pp. 81 sqq.
Incom- mensurability.
116 Early Greek Philosophy
to the squares on the other two sides, and the so-
called Pythagorean triangle is the application of its
converse to a particular case. The very name
“hypotenuse” affords strong confirmation of the in-
timate connexion between the two things. It means
literally “the cord stretching over against,” and this is
yi
surely just the rope of the “ harpedonapt. An early
tradition says that Pythagoras sacrificed an ox when
he discovered the proof of this proposition, and indeed
it was the real foundation of scientific mathematics.”
50. One great disappointment, however, awaited
Pythagoras. It follows at once from the Pythagorean
proposition that the square on the diagonal of a square
is double the square on its side, and this ought surely
to be capable of numerical expression. As a matter
of fact, however, there is no square number which can
be divided into two equal square numbers, and so the
problem cannot be solved. In this sense, it is doubtless
true that Pythagoras discovered the incommensurability
of the diagonal and the side of a square, and the proof
mentioned by Aristotle, namely, that, if they were
commensurable, we should have to say that an even
number was equal to an odd number, is distinctly
Pythagorean in character.» However that may be, it
1 The verb ὑποτείνειν is, of course, used intransitively. The explana- tion suggested in the text seems to me much simpler than that of Max C. P. Schmidt (Aulturhistorische Bettrage, Heft i. pp.64sqq.). He explains the hypotenuse as the longest string in a triangular harp ; but my view seems more in accordance with analogy. So ἡ κάθετος is, litérally, a plumb-line.
2 The statement comes from Eudemos; for it is found in Proclus’s commentary on Euclid i. 47. Whether historical or not, it is no Neo- pythagorean fancy. |
3 Arist. dv. Pr. A, 23. 41 a 26, ὅτι ἀσύμμετρος ἡ διάμετρος διὰ τὸ γίγνεσθαι τὰ περιττὰ toa τοῖς ἀρτίοις συμμέτρου τεθείσης. The proofs. given at the end of Euclid’s Tenth Book (vol. iii. pp. 408 sqq., Heiberg) turn on this very point. They are not Euclidean, and may be substantially Pythagorean. Cf. Milhaud, Phzlosophes géométres, p. 94.
Science and Religion 117
is certain that Pythagoras did not care to pursue the
subject any further. He had, as it were, stumbled
on the fact that the square root of two is a surd, but
we know that it was left for Plato’s friends, Theodoros
of Kyrene and Theaitetos, to give a complete theory
of the matter.. The fact is that the discovery of the
Pythagorean proposition, by giving birth to geometry,
_had really superseded the old view of quantity as a
sum of units; but it was not till Plato’s time that the
full consequences of this were seen.” For the present,
the incommensurability of the diagonal and the square
remained, as has been said, a “scandalous exception.”
Our tradition says that Hippasos of Metapontion was
drowned at sea for revealing this skeleton in ‘the
cupboard.®
51. These last considerations show that, while it is pigs ἐν τ
quite safe to attribute the substance of the First Book
of Euclid to Pythagoras, the arithmetic of Books VII.-
IX:, and the “geometrical algebra” of Book II. are
certainly not his. They operate with lines or with
areas instead of with units, and the relations which they
establish therefore hold good whether they are capable
of numerical expression or not. That is doubtless why
arithmetic is not treated in Euclid till after plane
geometry, a complete inversion of the original order.
For the same reason, the doctrine of proportion which
we find in Euclid cannot be Pythagorean, and is
1 Plato, Zheaet. 147 ἃ 3 sqq.
2 How novel these consequences were, is shown by the fact that in Zaws, 819 ἃ 5, the Athenian Stranger says that he had only realised them late in life.