Euclid, Elements V 8©
translated by Henry Mendell (Cal. State U., L.A.)

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Prop. 8: 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.

(general diagram)

Diagrams for book 5 proposition 8, following the display, construction, and demonstration(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 2) For since AB is larger than G, let BE be posited equal to G. In fact, the smaller of AE, EB, in being multiplied, will sometime be larger than D. (diagram 3) Let AE first be smaller than EB, and let AE be multiplied, and let ZH be a multiple of it that’s larger than D, (diagram 4) and as-many-multiples-as ZH is of AE, let so-many-multiples also come-about, HQ of EB and K of G. (diagram 5) And let a double, L, of D be taken, and a triple, M, and successively more by one, until the taken is a multiple of D but first to be larger than K. Let it be taken, and let N be as four-times D, first to be larger than K. (diagram 6) And so, since K is first to be smaller than N, therefore K is not smaller than M. And since ZH is equally-times a multiple of AE as HQ of EB, therefore, ZH is equally-times a multiple of AE as ZQ of AB. (v 1) But ZH is equally-times a multiple of AE as K of G. Therefore, ZQ is equally-times a multiple of AB as K of G. Therefore, ZQ, K are equal-times multiples of AB, G. Again, since HQ is equally-times a multiple of EB as K of G, but EB is equal to G, therefore, HQ is also equal to K. But K is not smaller than M. Therefore, HQ is not smaller than M. But ZH is larger than D. Therefore, a whole, ZQ is larger than both together, D, M. But both together, D, M, are equal to N, since, in fact, M is three-times D, but both together, M, D, are four-times D. Therefore, both together, M, D, are equal to N. But ZQ is larger than M, D. Therefore, ZQ exceeds N. But K does not exceed N. And ZQ, K are equal-times multiples of AB, G, while N is another multiple, whatever happens, of D. Therefore, AB has a larger ratio to D than G to D.

I say, in fact, that D has a larger ratio to G than D to AB. For when the same are constructed we will show similarly that N exceeds K, while N does not exceed ZQ. And N is a multiple of D and ZQ, K, other magnitude, whatever happens, are equal-times multiples of AB, G. Therefore, D has a ratio to G larger than D to AB.


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(general diagram)

(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.

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