Euclid, Elements X 1 (with scholia)©
translated by Henry Mendell (Cal. State U., L.A.)

Go to prop. 2

Note on the translation: in Euclid, ‘plêthos’ is a general word in Euclid for a countable collection, often translated ‘multitude’. The word 'arithmos' is restricted to collections of units. So two lists are equal-in-plêthos if there is a matching of the members of the two. In these translations, I avoid translating either word.

Prop. 1: Upon two unequal magnitudes being displayed, if from the larger a larger than half is taken away, and a larger than half of what’s left over, and this repeatedly comes about, there will be left a certain magnitude which will be smaller than the smaller displayed magnitude.

Different proof

Scholia (diagram 1) Let there be two unequal magnitudes, AB, G, of which AB is larger. I say that if a larger than half is taken from AB, and a larger than half of what’s left over, and this repeatedly comes about, there will be left a certain magnitude which will be smaller than G. (diagram 2) For G being multiplied will sometime be larger than AB. [V def. 4] Let it be multiplied, and let DE be a multiple of G, but larger than AB, (diagram 3) and let DE be divided into equals to G, i.e., DZ, ZH, HE, and let more than half, BQ, be taken away from AB, and more than half, QK, from AQ, and let this repeatedly come about, until the divisions in AB become equal-in-plêthos to the divisions in DE.

(diagram 4) And so, let AK, KQ, QB be divisions, being equal-in-plêthos to DZ, ZH, HE. And since DE is larger than AB, and a smaller than half, EH, is taken away from DE,sch 31 and a larger than half, BQ, from AB, therefore, a remainder, HD, is larger than a remainder, QA. And since HD is larger than QA, and a half of HD, i.e. HZ, is taken away, and a larger than half of QA,sch 32 i.e., QK, therefore, a remainder, DZ is larger than a remainder, AK. But DZ is equal to G. Therefore, G is also larger than AK. Therefore, AK is smaller than G. Therefore, magnitude AK is left over from magnitude AB, being smaller than the displayed smaller magnitude, G, which it was required to show. But it will be shown similarly even if the magnitudes taken away are half.

Differently

(general diagram) (diagram 1) Let two unequal magnitudes be displayed, AB, G. And since G is smaller, by being multiplied it will sometime be larger than magnitude AB. [V def. 4] (diagram 2) Let it have come-about as ZM and let it be divided into equals to G, and let them be MQ, QH, HZ, (diagram 3) and let a larger than half, BE, be taken away from AB, and from EA a larger than half, ED, and let this repeatedly come-about, until the divisions in ZM become equal to the divisions in AB. Let them have come-about as BE, ED, DA, and (general diagram) let each of KL, LN, NX be equal to DA, and let this come-about until the divisions in KX become equal to those in ZM.

And since BE is larger than half of BA, BE is larger than EA. Therefore it is larger by much than DA. But DA is equal to XN. Therefore BE is larger than NX. Again, since ED is larger than half of EZ, it is larger than DA. But DA is equal to NA. Therefore ED is larger than NL. Therefore, a whole, DB, is larger than XL. But DA is equal to LK. Therefore, a whole, BA, is larger than XK. But MZ is larger than BA. Therefore, MZ is larger by much than XK And since XN, NL, LK are equal to one anohter, but MQ, QH, HZ are also equal to one another, and the plethos of those in MZ is equal to the plethos of those in XK, therefore, it is: as KL to ZH so KX to ZM. [V 15] But ZM is larger than KX. Therefore, HZ is also larger than LK. [V 14] And ZH is equal to G, and KL to AD. Therefore, G is larger than AD, just what it was required to show.

Scholia: Heiberg reports readings from Vat. 190 (P), Laurentius 28.3 (F), Ox. D'Orville X 1 (B), Bon. 18-19 (b), Vind 103 (V), Vat. 204 fol. 198-205 (V), Vat. 1038 (v), Laur. 28.6 (f), Laur. 28.2 (l), Laur. 28.8 (λ), Magliabechianae Flor. 11.53 (Maglb), Paris 2344 (q), Paris 2345 (r), Par. 2346 (s), Par. 2373 (t), Par. 2762 (u), Par. 2366 (x), Par. 2343 (y), Par. 2466 (p), Coil. 174 (Coisl.), Ambros. C311 (A), Marc. 300 (n), Marc. 302 (μ), Marc. 317 (ν)

26. He also shows through this theorem that there isn't a smallest magnitude, as the Democriteans say, if it is, at least, possible to take a smaller than any given magnitude. (manuscripts P V q; see v A l)

27. larger than half: One must conceive here of the larger segment of the initial given larger magnitude as larger in comparison with the half of it, and not in comparison with the smaller initially displayed magnitude. Similarly, one must also think of the half in this way. (manuscripts V q; see v P A)

28. It becomes clear through this theorem, the 1st, that there is incommensurability in magnitudes. For if it is possible to take some magnitude smaller than the displayed magnitude and smaller than this and ever smaller, magnitudes will be cut ad infinitum, and not into a determinate smallest measure, just as the unit in the case of arithmoi. And so, if there is no determinate smallest magnitude, some magnitudes are incommensurable, which are not measured by some common magnitude due to the unboundedness. (manuscripts P B F Vat. V q r)

29. Due to the definition of the 5th book which says: but the larger is a multple of the smaller when it is measured by the smaller. For the larger and the smaller in a word is a ratio, that is, only a relation of finite magnitudes . (manuscripts V q, see P)

30. It is the same to say that the magnitude is divided ad infinitum. (manuscript V)

31. And a smaller than half is taken away from DE: For DE was divided into a 3rd, and the 3rd of it is smaller than the half of it. (manuscripts V q)

32. For since a whole, magnitude DE, was constructed larger than magnitude AB, but a smaller than the half of it , EH, is taken away from magnitude DE, but BQ, a larger than the half of it, is taken away from AB, thus what is made clear of AQ is the case. (manuscript q; see P)

33. He denies that it is required to take away from AB a larger than half of G, but the larger than its own half, AB. For example, if AB is 100, take away from 100 60. Remainder is 40. Again from 40 take away the larger than a half, e.g., 24, and so in the case of the rest. (manuscripts V q; see P)