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Euclid, Elements X 3©
translated by Henry Mendell (Cal. State U., L.A.)
Prop. 3: Given two commensurable magnitudes, to find the largest common measure of them.
(diagram 1) Let the given two commensurable magnitudes be AB, GD, of which AB is smaller. It is, in fact, required to find the greatest common measure of AB, GD.
For magnitude AB either measures GD or not. (diagram 2) And so, if it measures, but it also measures itself, therefore, AB is a common measure of AB, GD. And it is obvious that is it also largest. For a larger than magnitude AB will not measure AB. (diagram 3) In fact, let AB not measure GD. And the smaller being repeatedly taken away in turn from the larger, what remains left will measure sometime what is before it due to the fact that AB, GD are not incommensurable. [X 2] And let AB in measuring out ED leave a smaller than it, EG, and let EG in measuring out ZB leave a smaller than it, AZ, but let AZ measure GE. (diagram 4) And so, since AZ measures GE, but GE measures ZB, therefore AZ will also measure ZB. But it also measures itself. Therefore, AZ will measure a whole, AB. But AB measures DE. Therefore, AZ will also measure ED. But it also measures GE. Therefore, GD also measures a whole, GD. Therefore, AZ is a common measure of AB, GD. I say, in fact, that it is also largest. (diagram 5) For if not, there will be some magnitude larger than AZ which will measure AB, GD. Let it be H. And so since H measures AB, but AB measures ED, therefore, H will also measure ED. But it also measures a whole, GD. Therefore, H will also measure a remainder, GE. But GE measures ZB. Therefore, H will also measure ZB. But it also measures a whole, AB, and will measure a remainder, AZ, the larger measuring the smaller, which is impossible. Therefore, AZ is the largest common measure of AB, GD.
Provided (Corollary): From this, in fact, it is obvious that if a magnitude measures two magnitudes, it will also measure the largest common measure of them.