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Commentary on Proposition X.4¶

This proposition again corresponds exactly to VII. 3 for numbers. As there Euclid thinks it necessary to prove that, $a, b, c$ not being prime to one another, $d$ and $c$ are also not prime to one another, so here he thinks it necessary to prove that $d, c$ are commensurable, as they must be since any common measure of $a, b$ must be a measure of their greatest common measure $d$ (X. 3, Por.).

The argument in the proof that $e$, the greatest common measure of $d, c$, is the greatest common measure of $a, b, c$, is the same as that in VII. 3 and X. 3.

The Porism contains the extension of the process to the case of four or more magnitudes, corresponding to Heron’s remark with regard to the similar extension of VII. 3 to the case of four or more numbers.