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| Titre : | Thirteen Books of Euclid's Elements Vol. 3 |
| Auteurs : | Euclid |
| Type de document : | document électronique |
| Editeur : | [S.l.] : Forgotten Books, 2012 |
| Résumé : |
The discovery of the doctrine of incommensurables is attributed toP ythagoras. Thus Proclus says {C omm, onE ucL i. p. 65, 19) that Pythagoras discovered the theory of irrationals ;and, again, the scholium on the banning of Book X., also attributed toP roclus, 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. (T hey showed that) all magnitudes can be rational {prjra) and all irrational (aA oya) in a relative sense {mirpo n); hence the commensurable and the incommensurable would be for them natural (kinds) (v rci), while the rational and irrational would rest on assumption or convention (dco-ct). 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 (Typographical errors above are due to OCR software and don't occur in the book.) ** |
