Euclid, Elements X 4©
translated by Henry Mendell (Cal. State U., L.A.)

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Prop. 4: Given three commensurable magnitudes, to find the largest common measure of them.
Corollary

(general diagram)

(diagram 1) Let there be the given three commensurable magnitudes, A, B, G. It is required, in fact, to find the largest common measure of A, B, G.

(diagram 2) For let the largest common measure of two, A, B, be taken, and let it be D. [X 3] D, in fact, either measures or it does not measure G. (diagram 3) First let it measure. And so, since D measures G, but it also measures A, B, therefore D measures A, B, G. Therefore, D is a common measure of A, B, G. And it is obvious that it is also largest. For a larger than magnitude D does not measure A, B.

(diagram 2) Let D, in fact, not measure G. I say first that G, D are commensurable. For since A, B, G are commensurable, some magnitude will measure them, which clearly will also measure A, B. Thus, the largest common measure of A, B will also measure D. [X 3 cor.] But it also measures G. Thus, the mentioned magnitude will measure G, D. Therefore G, D are commensurable. (diagram 4) And so let there be taken the largest common measure of them, and let it be E.[X 3] And so, since E measures D, but D measures A, B, therefore, E also will measure A, B. But it also measures G. Therefore, E measures A, B, G. Therefore, E is a common measure of A, B, G. I say, in fact, that is also largest. (general diagram) For if it is possible, let there be some magnitude larger than E, namely Z, and let it measure A, B, G. And since Z measures A, B, G, therefore, it will also measure A, B and will also measure the largest common measure of A, B. [X 3 cor.] But the largest common measure of A, B is D. Therefore, Z measures D. But it also measures G. Therefore, Z measures G, D. Therefore, Z will also measure the largest common measure of G, D. But it is E. Therefore, Z will measure E, the larger the smaller, which is impossible. Therefore, some magnitude larger than E does not measure A, B, G. Therefore E is the largest common measure of A, B, G, if D does not measure G, but if it does measure G, it is D itself.

Therefore, given three commensurable magnitudes, the largest common measure of them has been found, just what it was required to show.

Provided (Corollary): From this, in fact, it is obvious that if a magnitude measures three magnitudes, it will also measure the largest common measure of them.

In fact, the largest common measure will be similarly taken also in the case of more commensurable magnitudes, and the corollary will proceed, just what it was required to show

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