Content of type 'introductory_note'

Book V Introductory Note

The anonymous author of a scholium to Book V. (Euclid, ed. Heiberg, Vol. V. p. 280), who is perhaps Proclus, tells us that “some say” this Book, containing the general theory of proportion which is equally applicable to geometry, arithmetic, music, and all mathematical science, “is the discovery of Eudoxus, the teacher of Plato.” Not that there had been no theory of proportion developed before his time; on the contrary, it is certain that the Pythagoreans had worked out such a theory with regard to numbers, by which must be understood commensurable and even whole numbers (a number being a “multitude made up of units,” as defined in Eucl. VII). Thus we are told that the Pythagoreans distinguished three sorts of means, the arithmetic, the geometric and the harmonic mean, the geometric mean being called proportion (ἀναλογία) par excellence; and further Iamblichus speaks of the “most perfect proportion consisting of four terms and specially called harmonic,” in other words, the proportion

Book VI Introductory Note

The theory of proportions has been established in Book v. in a perfectly general form applicable to all kinds of magnitudes (although the representation of magnitudes by straight lines gives it a geometrical appearance); it is now necessary to apply the theory to the particular case of geometrical investigation. The only thing still required in order that this may be done is a proof of the existence of such a magnitude as bears to any given finite magnitude any given finite ratio; and this proof is supplied, so far as regards the subject matter of geometry, by VI. 12 which shows how to construct a fourth proportional to three given straight lines.

Book X Introductory Note

We have seen (Vol. i., p. 351 etc.) that the discovery of the irrational is due to the Pythagoreans. The first scholium on Book x. of the Elements 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 measureable 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 (ῥητά) and all irrational (ἄλογα) in a relative sense (ὡς πρός τι); hence the commensurable and the incommensurable would be for them natural (kinds) (φύσει), while the rational and irrational would rest on assumption or convention (θέσει).” 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.