EXCURSUS I — PYTHAGORAS AND THE PYTHAGOREANS.

The problem of determining how much of the Pythagorean discoveries in mathematics can be attributed to Pythagoras himself is not only difficult; it may be said to be insoluble. Tradition on the subject is very meagre and uncertain, and further doubt is thrown upon it by the well-known tendency of the later Pythagoreans to ascribe everything to the Master himself (αὐτὸς ἔφα, Ipse dixit). Pythagoras himself left no written exposition of his doctrines, nor did any of his immediate successors, not even Hippasus, about whom the different stories ran (1) that he was expelled from the school because he published doctrines of Pythagoras, and (2) that he was drowned at sea for revealing the construction of the dodecahedron in the sphere and claiming it as his own, or (as others have it) for making known the discovery of the irrational or incommensurable. Nor is the absence of any written record of Pythagorean doctrines down to the time of Philolaus to be put down to a pledge of secrecy binding the school; at all events this did not apply to their mathematics or their physics; and it may be that the supposed secrecy was invented to account for the absence of documents. The fact seems to be that oral communication was the tradition of the school, while their doctrines would in the main be too abstruse to be understood by the generality of people outside. Even Aristotle felt the difficulty; he evidently knew nothing for certain about any ethical or physical doctrines going back to Pythagoras himself; when he speaks of the Pythagorean system, he always refers it to “the Pythagoreans,” sometimes even to “the so-called Pythagoreans.”

Since my note on Eucl. I. 47 was originally written the part of Pythagoras in the Pythagorean mathematical discoveries has been further discussed and every scrap of evidence closely, and even meticulously, examined in two long articles by Heinrich Vogt, “Die Geometrie des Pythagoras” (Bibliotheca Mathematica $\text{IX}_3$, 1908/9, pp. 15—54) and “Die Entstehungsgeschichte des Irrationalen nach Plato und anderen Quellen des 4. Jahrhunderts” (Bibliotheca Mathematica $\text{X}_3$, 1910, pp. 97—155). These papers would not indeed have enabled me to modify greatly what I have written regarding the supposed discoveries of Pythagoras and the early Pythagoreans, because I have throughout been careful to give the traditions on the subject for what they are worth and no more, and not to build too much upon them. It is right however to give, in a separate note, a few details of Vogt’s arguments.

G. Junge had, in his paper “Wann haben die Griechen das Irrationale entdeckt?” mentioned above (p. 351), tried to prove that Pythagoras himself could not have discovered the irrational; and the object of Vogt’s papers is to go further on the same lines and to show (1) that it was only the later Pythagoreans who (before 410 B.C.) recognised the incommensurability of the diagonal with the side of a square, (2) that the theory of the irrational was first discovered by Theodorus, to whom Plato refers (Theaetetus 147 D), and (3) that Pythagoras himself could not have been the discoverer of any one of the things specifically attributed to him, namely (a) the theorem of Eucl. I. 47, (b) the construction of the five regular solids in the sense in which they are respectively constructed in Eucl. XIII, (c) the application of an area in its widest sense, equivalent to the solution of a quadratic equation in its most general form.

Vogt’s main argument as regards (a) the theorem of I. 47 is based on a new translation which he gives of the well-known passage of Proclus’ note on the proposition (p. 426, 6—9), Τῶν μὲν ἱστορεῖν τὰ ἀρχαῖα βουλομένων ἀκούοντας τὸ θεώρημα τοῦτο εἰς Πυθαγόραν ἀναπεμπόντων ἐστὶν εὑρεῖν καὶ βουθύτην λεγόντων αὐτὸν ἐπὶ τῇ εὑρέσει. Vogt translates this as follows: “Unter denen, welche das Altertum erforschen wollen, kann man einige finden, welche denen Gehör geben, die dieses Theorem auf Pythagoras zurückführen und ihn als Stieropferer bei dieser Gelegenheit bezeichnen,” “Among those who have a taste for research into antiquity, we can find some who give ear to those who refer this theorem to Pythagoras and describe him as sacrificing an ox on the strength of the discovery.” According to this version the words τῶν…βουλομένων and the words ἀναπεμπόντων…καὶ…λεγόντων refer respectively to two different sets of persons, in fact two different generations; the latter are older authorities who are supposed to be cited by the former; the former are a later generation, perhaps contemporaries of Proclus, some of whom accepted the view of the older authorities while others did not. But this would have required the article τῶν before ἀναπεμπόντων, or some such expression as ἄλλων τινῶν οἳ ἀναπέμπουσι instead of ἀναπεμπόντων. Vogt’s interpretation is therefore quite inadmissible. The persons denoted by ἀναπεμπόντων are some of the persons denoted by τῶν βουλομένων; hence Tannery’s translation, to which mine (p. 350 above) is equivalent, is the only possible one, namely “Si l’on écoute ceux qui veulent raconter l’histoire des anciens temps, on peut en trouver qui attribuent ce théorème à Pythagore et lui font sacrifier un bœuf après sa découverte” (La Géométrie grecque, p. 103). ἀκούοντας agrees with the assumed subject of εὑρεῖν; ἀναπεμπόντων and λεγόντων should, strictly speaking, have been ἀναπέμποντας and λέγοντας agreeing with τινὰς (the direct object of εὑρεῖν) understood, but are simply attracted into the case of βουλομένων; the construction is quite intelligible. I agree with Vogt that Eudemus’ history contained nothing attributing the theorem to Pythagoras. The words of Proclus imply this; but I do not think that they imply (as Vogt maintains) any pronouncement by Proclus himself against such attribution. In my opinion, Proclus is simply determined not to commit himself to any view; his way of evading a decision is the sentence following, ἐγὼ δὲ θαυμάζω μὲν καὶ τοὺς πρώτους ἐπιστάντας τῇ τοῦδε τοῦ θεωρήματος ἀληθείᾳ, μειζόνως δὲ ἄγαμαι τὸν στοιχειωτήν…; the plural τοὺς πρώτους ἐπιστάντας is, I hold, used for the very purpose of making the statement as vague as possible; he will not even allow it to be inferred that he attributed the discovery to any single person. Returning to ἡ τῶν ἀλόγων πραγματεία (Proclus, p. 65, 19), we may now concede (following Diels) that we should read τῶν ἀνὰ λόγον (“proportionals”) instead of τῶν ἀλόγων (“irrationals”) and that the author intended to attribute to Pythagoras a theory of proportion (the arithmetical theory applicable to commensurable magnitudes only) rather than the theory of irrationals. But I do not agree in Vogt’s contention that the theory of the irrational was first discovered by Theodorus. It seems to me that we have evidence to the contrary in the very passage of Plato referred to. Plato (Theaetetus 147 D) mentions $\sqrt{3}$, $\sqrt{5}$, … up to $\sqrt{17}$ as dealt with by Theodorus, but omits $\sqrt{2}$. This fact, along with Plato’s allusions elsewhere to the irrationality of $\sqrt{2}$, and to approximations to it, in the expressions ἄρρητος and ῥητὴ διάμετρος τῆς πεμπάδος, as if those expressions had a well-known signification, implies that the discovery of the irrationality of $\sqrt{2}$ had been made before the time of Theodorus. The words ἡ τῶν ἀλόγων πραγματεία might well be used even if the reference is only to $\sqrt{2}$, because the first step would be the most difficult, and πραγματεία need not mean the establishment of a complete theory or anything more than “investigation” of a subject.

Junge and Vogt hold that the theory of the irrational was not discovered by the early Pythagoreans any more than Pythagoras because, if it had been so discovered, an impossibly long period would intervene between the investigation of the particular case of $\sqrt{2}$ and the extension of the theory by Theodorus to the cases of $\sqrt{3}$, $\sqrt{5}$ etc. But might not this well be due to the fact that in the meantime the minds of geometers were engrossed by other problems of importance, namely the quadrature of the circle (Hippocrates of Chios and his quadratures of lunes), the trisection of any angle (Hippias of Elis and his curve, afterwards known as the quadratrix), and the doubling of the cube (reduced by Hippocrates to the problem of finding two mean proportionals in continued proportion between two given straight lines), the last of which problems, which meant finding geometrically the equivalent of $\sqrt[3]{2}$, would naturally follow the investigation of $\sqrt{2}$? Now Hippias was probably born about 460 B.C., while Hippocrates seems to have been in Athens during a considerable portion of the second half of the fifth century, perhaps from about 450 to 430 B.C. Moreover Vogt has to get over the fact that Democritus (born 470/469 B.C.) wrote a book περὶ ἀλόγων γραμμῶν καὶ ναστῶν, On irrational lines and solids (or atoms). This difficulty he seeks to overcome by maintaining that ἀλόγων does not here mean “irrational” at all, but “without ratio” (“verhältnislos”), in the sense that any two straight lines are “without ratio” because they both contain an infinite number of the indivisible (or atomic) lines, and therefore their ratio, being of the form $\infty/\infty$, is indeterminate. But, if these were so, all lines (including commensurable lines) would be “without a ratio” to one another, whereas the title of Democritus’ work clearly implies that ἄλογοι γραμμαί are a class or classes of lines distinguished from other lines. The fact is that Democritus was too good a mathematician to have anything to do with “indivisible lines.” This is confirmed by a scholium to Aristotle’s De caelo (p. 469 b 14, Brandis) which implicitly denies to Democritus any theory of indivisible lines: “of those who have maintained the existence of indivisibles, some, as for example Leucippus and Democritus, believe in indivisible bodies, others, like Xenocrates, in indivisible lines.” Moreover Simplicius tells us that, according to Democritus himself, even the atoms were, in a mathematical sense, divisible further and in fact ad infinitum.

Coming now to (b) the construction of the cosmic figures, ἡ τῶν κοσμικῶν σχημάτων σύστασις (Proclus, p. 65, 20), I agree with Vogt to the following extent. It is unlikely that Pythagoras or even the early Pythagoreans “constructed” the five regular solids in the sense of a complete theoretical construction such as we find, say, in Eucl. XIII.; and it is possible that Theaetetus was the first to give these constructions, whether ἔγραψε in Suidas’ notice, πρῶτος δὲ τὰ πέντε καλούμενα στερεὰ ἔγραψε, means “constructed” or “wrote upon.” But σύστασις in the above phrase of Proclus may well mean something less than the theoretical constructions and proofs of Eucl. XIII.; it may mean, as Vogt says, simply the “putting together” of the figures in the same way as Plato puts them together in the Timaeus, i.e. by bringing a certain number of angles of equilateral triangles and of regular pentagons together at one point. There is no reason why the early Pythagoreans should not have “constructed” the five regular solids in this sense; in fact the supposition that they did so agrees well with what we know of their having put angles of certain regular figures together round a point (in connexion with the theorem of Eucl. I. 32) and shown that only three kinds of such angles would fill up the space in one plane round the point. But I do not agree in the apparent refusal of Vogt to credit the Pythagoreans with the knowledge of the theoretical construction of the regular pentagon as we find it in Eucl. IV. 10, 11. I do not know of any reason for rejecting the evidence of the Scholia IV. Nos. 2 and 4 which say categorically that “this Book” (Book IV.) and “the whole of the theorems” in it (including therefore Props. 10, 11) are discoveries of the Pythagoreans. And the division of a straight line in extreme and mean ratio, on which the construction of the regular pentagon depends, comes in Eucl. Book II. (Prop. 11), while we have sufficient grounds for regarding the whole of the substance of Book II. as Pythagorean.

I will permit myself one more criticism on Vogt’s first paper. I think he bases too much on the fact that it was left for Oenopides (in the period from, say, 470 to 450 B.C.) to discover two elementary constructions (with ruler and compasses only), namely that of a perpendicular to a straight line from an external point (Eucl. I. 12), and that of an angle equal to a given rectilineal angle (Eucl. I. 23). Vogt infers that geometry must have been in a very rudimentary condition at the time. I do not think this follows; the explanation would seem to be rather that, the restriction of the instruments used in constructions to the ruler and compasses not having been definitely established before the time when Oenopides wrote, it had not previously occurred to anyone to substitute new constructions based on that principle for others previously in vogue. In the case of the perpendicular, for example, the construction would no doubt, in earlier days, have been made by means of a set square.