OSE, Chapter 6, Note 9

The Geometric Program

9 Cp. Gorgias, 488e, ff.; the passage is more fully quoted and discussed in section VIII below (see note 48 to this chapter, and text). For Aristotle’s theory of slavery, see note 3 to chapter 11 and text. The quotations from Aristotle in this paragraph are: (1) and (2) Nicom. Ethics, V, 4, 7, and 8; (3) Politics, III, 12, 1 (1282b; see also notes 20 and 30 to this chapter. The passage contains a reference to the Nicom. Eth.); (4) Nicom. Ethics, V, 4, 9; (5) Politics, IV (VI), 2, 1 (1317b).—In the Nicom. Ethics, V, 3, 7 (cp. also Pol., III, 9, 1; 1280a), Aristotle also mentions that the meaning of ‘justice’ varies in democratic, oligarchic, and aristocratic states, according to their different ideas of ‘merit’. *(What follows here was first added in the American edition of 1950.)

For Plato’s views, in the Laws, on political justice and equality, see especially the passage on the two kinds of equality (Laws, 757b-d) quoted below under (1). For the fact, mentioned here in the text, that not only virtue and breeding but also wealth should count in the distribution of honours and of spoils (and even size and good looks), see Laws, 744c, quoted in note 20 (1) to the present chapter, where other relevant passages are also discussed.

(1) In the Laws, 757b—d, Plato discusses ‘two kinds of equality’. ‘The one of these .. is equality of measure, weight, or number [i.e. numerical or arithmetical equality]; but the truest and best equality .. distributes more to the greater and less to the smaller, giving each his due measure, in accordance with nature ... By granting the greater honour to those who are superior in virtue, and the lesser honour to those who are inferior in virtue and breeding, it distributes to each what is proper, according to this principle of [rational] proportions. And this is precisely what we shall call “political justice”. And whoever may found a state must make this the sole aim of his legislation ..: this justice alone which, as stated, is natural equality, and which is distributed, as the situation requires, to unequals.’ This second of the two equalities which constitutes what Plato here calls ‘political justice’ (and what Aristotle calls ‘distributive justice’), and which is described by Plato (and Aristotle) as ‘proportionate equality’—the truest, best, and most natural equality—was later called ‘geometrical’ (Gorgias 508a; see also 465b/c, and Plutarch, Moralia 719b, f.), as opposed to the lower and democratic ‘arithmetical’ equality. On this identification, the remarks under (2) may throw some light.

(2) According to tradition (see Comm. in Arist. Graeca, pars XV, Berlin, 1897, p. 117, 29, and pars XVIII, Berlin, 1900, p. 118, 18), an inscription over the door of Plato’s academy said: ‘Nobody untrained in geometry may enter my house.’ I suspect that the meaning of this is not merely an emphasis upon the importance of mathematical studies, but that it means: ‘Arithmetic (i.e. more precisely, Pythagorean number theory) is not enough; you must know geometry!’ And I shall attempt to sketch the reasons which make me believe that the latter phrase adequately sums up one of Plato’s most important contributions to science. See also Addendum, p. 319.

As is now generally believed, the earlier Pythagorean treatment of geometry adopted a method somewhat similar to the one nowadays called ‘arithmetization’. Geometry was treated as part of the theory of integers (or ‘natural’ numbers, i.e. of numbers composed of monads or ‘indivisible units’; cp. Republic, 525e) and of their ‘logoi’, i.e. their ‘rational’ proportions. For example, the Pythagorean rectangular triangles were those with sides in such rational proportions. (Examples are 3:4 15; or 5: 12: 13.) A general formula ascribed to Pythagoras is this: 2n + 1: 2n(n + 1): 2n(n + 1) + 1. But this formula, derived from the ‘gnomon’, is not general enough, as the example 8: 15: 17 shows. A general formula, from which the Pythagorean can be obtained by putting m = n + 1, is this: m2—n2: 2mn: m2 + n2 (where m > n). Since this formula is a close consequence of the so-called ‘Theorem of Pythagoras’ (if taken together with that kind of Algebra which seems to have been known to the early Pythagoreans), and since this formula was, apparently, not only unknown to Pythagoras but even to Plato (who proposed, according to Proclus, another non-general formula), it seems that the ‘Theorem of Pythagoras’ was not known, in its general form, to either Pythagoras or even to Plato. (See for a less radical view on this matter T. Heath, A History of Greek Mathematics, 1921, vol. I, pp. 80-2. The formula described by me as ‘general’ is essentially that of Euclid; it can be obtained from Heath’s unnecessarily complicated formula on p. 82 by first obtaining the three sides of the triangle and by multiplying them by 2/mn, and then by substituting in the result m and n and p and q.)

The discovery of the irrationality of the square root of two (alluded to by Plato in the Greater Hippias and in the Meno; cp. note 10 to chapter 8; see also Aristotle, Anal. Priora, 413a6 f.) destroyed the Pythagorean programme of ‘arithmetizing’ geometry, and with it, it appears, the vitality of the Pythagorean Order itself. The tradition that this discovery was at first kept secret is, it seems, supported by the fact that Plato still calls the irrational at first ‘arrhētos’, i.e. the secret, the unmentionable mystery; cp. the Greater Hippias, 303b/c; Republic, 546c. (A later term is ‘the non-commensurable’; cp. Theaetetus, 147c, and Laws, 820c. The term ‘alogos’ seems to occur first in Democritus, who wrote two books On Illogical Lines and Atoms (or and Full Bodies) which are lost; Plato knew the term, as proved by his somewhat disrespectful allusion to Democritus’ title in the Republic, 534d, but never used it himself as a synonym for ‘arrhetos’. The first extant and indubitable use in this sense is in Aristotle’s Anal. Post., 76b9. See also T. Heath, op. cit., vol. I, pp. 84 f., 156 f. and my first Addendum on p. 319, below.)

It appears that the breakdown of the Pythagorean programme, i.e. of the arithmetical method of geometry, led to the development of the axiomatic method of Euclid, that is to say, of a new method which was on the one side designed to rescue, from the breakdown, what could be rescued (including the method of rational proof), and on the other side to accept the irreducibility of geometry to arithmetic. Assuming all this, it would seem highly probable that Plato’s role in the transition from the older Pythagorean method to that of Euclid was an exceedingly important one—in fact, that Plato was one of the first to develop a specifically geometrical method aiming at rescuing what could be rescued from, and at cutting the losses of, the breakdown of Pythagoreanism. Much of this must be considered as a highly uncertain historical hypothesis, but some confirmation may be found in Aristotle, Anal. Post., 76b9 (mentioned above), especially if this passage is compared with the Laws, 818c, 895e (even and odd), and 819e/820a, 820c (incommensurable). The passage reads: ‘Arithmetic assumes the meaning of “odd” and “even”, geometry that of ‘irrational”..’ (Or ‘incommensurable’; cp. Anal. Priora, 41a26 f., 503a7. See also Metaphysics, 983a20, 1061b1-3, where the problem of irrationality is treated as if it were the proprium of geometry, and 1089a, where, as in Anal. Post., 76b40, there is an allusion to the ‘square foot’ method of the Theaetetus, 147d.) Plato’s great interest in the problem of irrationality is shown especially in two of the passages mentioned above, the Theaetetus, 147c-148a, and Laws, 819d-822d, where Plato declares that he is ashamed of the Greeks for not being alive to the great problem of incommensurable Magnitudes.

Now I suggest that the ‘Theory of the Primary Bodies’ (in the Timaeus, 53c to 62c, and perhaps even down to 64a; see also Republic, 528b-d) was part of Plato’s answer to the challenge. It preserves, on the one hand, the atomistic character of Pythagoreanism—the indivisible units (‘monads’) which also play a role in the school of the Atomists—and it introduces, on the other hand, the irrationalities (of the square roots of two and three) whose admission into the world had become unavoidable. It does so by taking two of the offending rectangular triangles—the one which is half of a square and incorporates the square root of two, and the one which is half of an equilateral triangle and incorporates the square root of three—as the units of which everything else is composed. Indeed, the doctrine that these two irrational triangles are the limits (peras; cp. Meno, 75d-76a) or Forms of all elementary physical bodies may be said to be one of the central physical doctrines of the Timaeus.

All this would suggest that the warning against those untrained in geometry (an allusion to it may perhaps be found in the Timaeus, 54a) might have had the more pointed significance mentioned above, and that it may have been connected with the belief that geometry is something of higher importance than is arithmetic. (Cp. Timaeus, 31c.) And this, in turn, would explain why Plato’s ‘proportionate equality’, said by him to be something more aristocratic than the democratic arithmetical or numerical equality, was later identified with the ‘geometrical equality’, mentioned by Plato in the Gorgias, 508a (cp. note 48 to this chapter), and why (for example by Plutarch, loc. cit.) arithmetic and geometry were associated with democracy and Spartan aristocracy respectively—in spite of the fact, then apparently forgotten, that the Pythagoreans had been as aristocratically minded as Plato himself; that their programme had stressed arithmetic; and that ‘geometrical’, in their language, is the name of a certain kind of numerical (i.e. arithmetical) proportion.

(3) In the Timaeus, Plato needs for the construction of the Primary Bodies an Elementary Square and an Elementary Equilateral Triangle. These two, in turn, are composed of two different kinds of sub-elementary triangles—the half-square which incorporates √2, and the half-equilateral which incorporates √3 respectively. The question why he chooses these two sub-elementary triangles, instead of the Square and the Equilateral itself, has been much discussed; and similarly a second question—see below under (4)—why he constructs his Elementary Squares out of four sub-elementary half-squares instead of two, and the Elementary Equilateral out of six sub-elementary half-equilaterals instead of two. (See the first two of the three figures below.)

Concerning the first of these two questions, it seems to have been generally overlooked that Plato, with his burning interest in the problem of irrationality, would not have introduced the two irrationalities √2 and √3 (which he explicitly mentions in 54b) had he not been anxious to introduce precisely these irrationalities as irreducible elements into his world. (Cornford, Plato’s Cosmology, pp. 214 and 231 ff., gives a long discussion of both questions, but the common solution which he offers for both—his ‘hypothesis’ as he calls it on p. 234—appears to me quite unacceptable; had Plato wanted to achieve some ‘grading’ like the one discussed by Cornford—note that there is no hint in Plato that anything smaller than what Cornford calls ‘Grade B’ exists—it would have been sufficient to divide into two the sides of the Elementary Squares and Equilaterals of what Cornford calls ‘Grade B’, building each of them up from four elementary figures which do not contain any irrationalities.) But if Plato was anxious to introduce these irrationalities into the world, as the sides of sub-elementary triangles of which everything else is composed, then he must have believed that he could, in this way, solve a problem; and this problem, I suggest, was that of ‘the nature of (the commensurable and) the uncommensurable’ (Laws, 820c). This problem, clearly, was particularly hard to solve on the basis of a cosmology which made use of anything like atomistic ideas, since irrationals are not multiples of any unit able to measure rationals; but if the unit measures themselves contain sides in ‘irrational ratios’, then the great paradox might be solved; for then they can measure both, and the existence of irrationals was no longer incomprehensible or ‘irrational’.

But Plato knew that there were more irrationalities than √2 and √3, for he mentions in the Theaetetus the discovery of an infinite sequence of irrational square roots (he also speaks, 148b, of ‘similar considerations concerning solids’, but this need not refer to cubic roots but could refer to the cubic diagonal, i.e. to √3); and he also mentions in the Greater Hippias (303b-c; cp. Heath, op. cit., 304) the fact that by adding (or otherwise composing) irrationals, other irrational numbers may be obtained (but also rational numbers—probably an allusion to the fact that, for example, 2 minus √2 is irrational; for this number, plus √2, gives of course a rational number). In view of these circumstances it appears that, if Plato wanted to solve the problem of irrationality by way of introducing his elementary triangles, he must have thought that all irrationals (or at least their multiples) can be composed by adding up (a) units; (b) √2; (c) √3; and multiples of these. This, of course, would have been a mistake, but we have every reason to believe that no disproof existed at the time; and the proposition that there are only two kinds of atomic irrationalities—the diagonals of the squares and of cubes—and that all other irrationalities are commensurable relative to (a) the unit; (b) √2; and (c) √3, has a certain amount of plausibility in it if we consider the relative character of irrationalities. (I mean the fact that we may say with equal justification that the diagonal of a square with unit side is irrational or that the side of a square with a unit diagonal is irrational. We should also remember that Euclid, in Book X, def. 2, still calls all incommensurable square roots ‘commensurable by their squares’.) Thus Plato may well have believed in this proposition, even though he could not possibly have been in the possession of a valid proof of his conjecture. (A disproof was apparently first given by Euclid.) Now there is undoubtedly a reference to some unproved conjecture in the very passage in the Timaeus in which Plato refers to the reason for choosing his sub-elementary triangles, for he writes (Timaeus, 53c/d): ‘all triangles are derived from two, each having one right angle ..; of these triangles, one [the half-square] has on either side half of a right angle,.. and equal sides; the other [the scalene].. has unequal sides. These two we assume as the first principles .. according to an account which combines likelihood [or likely conjecture] with necessity [proof]. Principles which are still further removed than these are known to heaven, and to such men as heaven favours.’ And later, after explaining that there is an endless number of scalene triangles, of which ‘the best’ must be selected, and after explaining that he takes the half-equilateral as the best, Plato says (Timaeus, 543/b; Cornford had to emend the passage in order to fit it into his interpretation; cp. his note 3 to p. 2 14): ‘The reason is too long a story; but if anybody puts this matter to the test, and proves that it has this property, then the prize is his, with all our good will.’ Plato does not say clearly what ‘this property’ means; it must be a (provable or refutable) mathematical property which justifies that, having chosen the triangle incorporating √2, the choice of that incorporating √3 is ‘the best’; and I think that, in view of the foregoing considerations, the properly which he had in mind was the conjectured relative rationality of the other irrationals, i.e. relative to the unit, and the square roots of two and three.

(4) An additional reason for our interpretation, although one for which I do not find any further evidence in Plato’s text, may perhaps emerge from the following consideration. It is a curious fact that √2 + √3 very nearly approximates π (Cp. E. Borel, Space and Time, 1926, 1960, p. 21b; my attention was drawn to this fact, in a different context, by W. Marinelli.) The excess is less than 0.0047, i.e. less than 1 1/2 pro mille of π, and a better approximation to π than 0.0047 was hardly known at the time. A kind of explanation of this curious fact is that the arithmetical mean of the areas of the circumscribed hexagon and the inscribed octagon is a good approximation of the area of the circle Now it appears, on the one hand, that Bryson operated with the means of circumscribed and inscribed polygons (cp. Heath, op. cit., 224), and we know, on the other hand (from the Greater Hippias) that Plato was interested in the adding of irrationals, so that he must have added √2 + √3. There are thus two ways by which Plato may have found out the approximate equation √2 + √3 = π, and the second of these ways seems almost inescapable. It seems a plausible hypothesis that Plato knew of this equation, but was unable to prove whether or not it was a strict equality or only an approximation.

Plato’s Elementary Square, composed of four sub-elementary isosceles rectangular triangles.Plato’s Elementary Equilateral, composed of six sub-elementary scalene rectangular triangles.

The rectangle ABCD has an area exceeding that of the circle by less than 1 1/2 pro mille.

But if this is so, then we can perhaps answer the ‘second question’ mentioned above under (3), i.e. the question why Plato composed his elementary square of four sub-elementary triangles (half-squares) instead of two, and his elementary equilateral of six sub-elementary triangles (half-equilaterals) instead of two. If we look at the first two of the figures above, then we see that this construction emphasizes the centre of the circumscribed and inscribed circles, and, in both cases, the radii of the circumscribed circle. (In the case of the equilateral; the radius of the inscribed circle appears also; but it seems that Plato had that of the circumscribed circle in mind, since he mentions it, in his description of the method of composing the equilateral, as the ‘diagonal’; cp. the Timaeus, 54d/e; cp. also 54b.)

If we now draw these two circumscribed circles, or more precisely, if we inscribe the elementary square and equilateral into a circle with the radius r, then we find that the sum of the sides of these two figures approximates rπ; in other words, Plato’s construction suggests one of the simplest approximate solutions of the squaring of the circle, as our three figures show. In view of all this, it may easily be the case that Plato’s conjecture and his offer of ‘a prize with all our good will’, quoted above under (3), involved not only the general problem of the commensurability of the irrationalities, but also the special problem whether √2 + √3 squares the unit circle.

I must again emphasize that no direct evidence is known to me to show that this was in Plato’s mind; but if we consider the indirect evidence here marshalled, then the hypothesis does perhaps not seem too far-fetched. I do not think that it is more so than Cornford’s hypothesis; and if true, it would give a better explanation of the relevant passages.

(5) If there is anything in our contention, developed in section (2) of this note, that Plato’s inscription meant ‘Arithmetic is not enough; you must know geometry!’ and in our contention that this emphasis was connected with the discovery of the irrationality of the square roots of 2 and 3, then this might throw some light on the Theory of Ideas, and on Aristotle’s much debated reports. It would explain why, in view of this discovery, the Pythagorean view that things (forms, shapes) are numbers, and moral ideas ratios of numbers, had to disappear—perhaps to be replaced, as in the Timaeus, by the doctrine that the elementary forms, or limits (‘peras’; cp. the passage from the Meno, 75d-76a, referred to above), or shapes, or ideas of things, are triangles. But it would also explain why, one generation later, the Academy could return to the Pythagorean doctrine. Once the shock caused by the discovery of irrationality had worn off, mathematicians began to get used to the idea that the irrationals must be numbers, in spite of everything, since they stand in the elementary relations of greater or less to other (rational) numbers. This stage readied, the reasons against Pythagoreanism disappeared, although the theory that shapes are numbers or ratios of numbers meant, after the admission of irrationals, something different from what it had meant before (a point which possibly was not fully appreciated by the adherents of the new theory). See also Addendum I, p. 319, below.*