Commentary on Proposition VII.3¶
Back to Proposition VII.3.
Euclid’s proof is here longer than we should make it because he distinguishes two cases, the simpler of which is really included in the other.
Having taken the greatest common measure, say $d$, of $a$, $b$, two of the three given numbers $a$, $b$, $c$, he distinguishes the cases
in which $d$ measures $c$,
in which $d$ does not measure $c$.
In the first case the greatest common measure of $d$, $c$ is $d$ itself; in the second case it has to be found by a repetition of the process of VII. 2. In either case the greatest common measure of $a$, $b$, $c$ is the greatest common measure of $d$, $c$.
But, after disposing of the simpler case, Euclid thinks it necessary to prove that, if $d$ does not measure $c$, $d$ and $c$ must necessarily have a greatest common measure. This he does by means of the original hypothesis that $a$, $b$, $c$ are not prime to one another. Since they are not prime to one another, they must have a common measure; any common measure of $a$, $b$ is a measure of $d$, and therefore any common measure of $a$, $b$, $c$ is a common measure of $d$, $c$; hence $d$, $c$ must have a common measure, and are therefore not prime to one another.
The proofs of cases (1) and (2) repeat exactly the same argument as we saw in VII. 2, and it is proved separately for $d$ in case (1) and $e$ in case (2), where $e$ is the greatest common measure of $d$, $c$, ($\alpha$) that it is a common measure of $a$, $b$, $c$, ($\beta$) that it is the greatest common measure.
Heron remarks (an-Nairīzī, ed. Curtze, p. 191) that the method does not only enable us to find the greatest common measure of three numbers; it can be used to find the greatest common measure of as many numbers as we please. This is because any number measuring two numbers also measures their greatest common measure ; and hence we can find the G.C.M. of pairs, then the G.C.M. of pairs of these, and so on, until only two numbers are left and we find the G.C.M. of these. Euclid tacitly assumes this extension in VII. 33, where he takes the greatest common measure of as many numbers as we please.