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CONTENTS OF VOLUME III
The discovery of the doctrine of incommensurables is attributed to Pythagoras. Thus Proclus says (Comm. on Eucl. 1. p. 65, 19) that Pythagoras "discovered the theory of irrationals”; and, again, the scholium on the beginning of Book X., also attributed to Proclus, states that the Pythagoreans were the first to address themselves to the investigation of commensurability, having discovered it by means of their observation of numbers. They discovered, the scholium continues, that not all magnitudes have a common measure. “They called all magnitudes measurable by the same measure commensurable, but those which are not subject to the same measure incommensurable, and again such of these as are measured by some other common measure commensurable with one another, and such as are not, incommensurable with the others. And thus by assuming their measures they referred everything to different commensurabilities, but, though they were different, even so (they proved that) not all magnitudes are commensurable with any. (They showed that) all magnitudes can be rational (pyra) and all irrational (āloya) in a relative sense (ws após tl); hence the commensurable and the incommensurable would be for them natural (kinds) (pvoel), while the rational and irrational would rest on assumption or convention (déo el).” The scholium quotes further the legend according to which "the first of the Pythagoreans who made public the investigation of these matters perished in a shipwreck,” conjecturing that the authors of this story "perhaps spoke allegorically, hinting that everything irrational and formless is properly concealed, and, if any soul should rashly invade this region of life and lay it open, it would be carried away into the sea of becoming and be overwhelmed by its unresting currents. There would be a reason also for keeping the discovery of irrationals secret for the time in the fact that it rendered unstable so much of the groundwork of geometry as the Pythagoreans had based upon the imperfect theory of proportions which applied only to numbers. We have already, after Tannery, referred to the probability that the discovery of incommensurability must have necessitated a great recasting of the whole fabric of elementary geometry, pending the discovery of the general theory of proportion applicable to incommensurable as well as to commensurable magnitudes.
It seems certain that it was with reference to the length of the diagonal of a square or the hypotenuse of an isosceles right-angled triangle that Pythagoras made his discovery. Plato (Theaetetus, 147 D) tells us that Theodorus of Cyrene wrote about square roots (duvápels), proving that the square roots of
1 I have already noted (Vol. 1. p. 351) that G. Junge (Wann haben die Griechen das Irrationale entdeckt ?) disputes this, maintaining that it was the Pythagoreans, but not Pythagoras, who made the discovery. Junge is obliged to alter the reading of the passage of Proclus, on what seems to be quite insufficient evidence; and in any case I doubt whether the point is worth so much labouring.
H. E. III.
three square feet and five square feet are not commensurable with that of one square foot, and so on, selecting each such square root up to that of 17 square feet, ởt which for some reason he stopped. No mention is here made of 12, doubtless for the reason that its incommensurability had been proved before, i.e. by Pythagoras. We know that Pythagoras invented a formula for finding right-angled triangles in rational numbers, and in connexion with this it was Hevitable that he should investigate the relations between sides and hypotenuse in other right-angled triangles. He would naturally give special attention to the isosceles right-angled triangle ; he would try to measure the diagonal, he would arrive at successive approximations, in rational fractions, to the value of 12; he would find that successive efforts to obtain an exact expression for it failed. It was however an enormous step to conclude that such exact expression was impossible, and it was this step which Pythagoras (or the Pythagoreans) made. We now know that the formation of the side- and diagonal-numbers explained by Theon of Smyrna and others was Pythagorean, and also that the theorems of Eucl. 11. 9, 10 were used by the Pythagoreans in direct connexion with this method of approximating to the value of 12. The very method by which Euclid proves these propositions is itself an indication of their connexion with the investigation of 12, since he uses a figure made up of two isosceles right-angled triangles.
The actual method by which the Pythagoreans proved the incommensurability of /2 with unity was no doubt that referred to by Aristotle (Anal. prior. 1. 23, 41 a 26–7), a reductio ad absurdum by which it is proved that, if the diagonal is commensurable with the side, it will follow that the same number is both odd and even. The proof formerly appeared in the texts of Euclid as X. 117, but it is undoubtedly an interpolation, and August and Heiberg accordingly relegate it to an Appendix. It is in substance as follows.
Suppose AC, the diagonal of a square, to be commensurable with AB, its side. Let a : ß be their ratio expressed in the smallest numbers.
Then a > B and therefore necessarily > I.
AC' : ABP = a: B, and, since
ACP= 2 AB, [Eucl. 1. 47]
a' = 2B.
a= 2y; therefore
4y = 2B,
BP = 2y, so that ß?, and therefore B, must be even.
But ß was also odd: which is impossible.
This proof only enables us to prove the incommensurability of the diagonal of a square with its side, or of 12 with unity. In order to prove the incommensurability of the sides of squares, one of which has three times the area of another, an entirely different procedure is necessary; and we find in fact that, even a century after Pythagoras' time, it was still necessary to use separate proofs (as the passage of the Theaetetus shows that Theodorus did) to establish the incommensurability with unity of V3, 15, ... up to 117.