translated by Henry Mendell (Cal. State U., L.A.) (diagram 1) Let there be unequal magnitudes, AB, G, and let there be a larger, AB, and another, whatever happens, D. I say that AB has a larger ratio to D than does G to D, and D has a larger ratio to G than to AB. (diagram 7) But, in fact, let AE be larger than EB. (diagram 8) In fact, the smaller, EB, be being multiplied, will sometime be larger than D. Let it be multiplied, and let HQ be a multiple of EB, but larger than D. (diagram 9) And as-many-multiples-as HQ is of EB, let so-many-multiples also come-about, ZH of AE and K of G. Similarly, in fact, we will show that ZQ, K are equal-times multiples of AB, G. (diagram 10) And let N be taken similarly as a multiple of D but first to be larger than ZH. Thus, again ZH is not smaller than M. But HQ is larger than D. Therefore, a whole, ZQ exceeds D, M, that is N. But K does not exceed N, since, in fact, ZH, in being larger than HQ, that is than K, does not exceed N. And by following after the things above, in the same way, we finish the demonstration. Therefore, the larger of unequal magnitudes has a larger ratio to the same than the smaller, and the same has a larger ratio to the smaller than to the larger, which it was required to show.