Commentary on Proposition VII.2

Back to Proposition VII.2.

Here we have the exact method of finding the greatest common measure given in the text-books of algebra, including the reductio ad absurdum proof that the number arrived at is not only a common measure but the greatest common measure. The process of finding the greatest common measure is simply shown thus:

b) a ( p
           pb
           c) b ( q
              qc
              d) c ( r
                 rd

We shall arrive, says Euclid, at some number, say $d$, which measures the one before it, i.e. such that $c = rd$. Otherwise the process would go on until we arrived at unity. This is impossible because in that case $a$, $b$ would be prime to one another, which is contrary to the hypothesis.

Next, like the text-books of algebra, he goes on to show that $d$ will be some common measure of $a$, $b$. For $d$ measures $c$; therefore it measures $qc + d$, that is, $b$, and hence it measures $pb + c$, that is, $a$.

Lastly, he proves that $d$ is the greatest common measure of $a$, $b$ as follows.

Suppose that $e$ is a common measure greater than $d$.

Then $e$, measuring $a$, $b$, must measure $a - pb$, or $c$.

Similarly $e$ must measure $b - qc$, that is, $d$: which is impossible, since $e$ is by hypothesis greater than $d$.

Therefore etc.

Euclid’s proposition is thus identical with the algebraical proposition as generally given, e.g. in Todhunter’s algebra, except that of course Euclid’s numbers are integers.

Nicomachus gives the same rule (though without proving it) when he shows how to determine whether two given odd numbers are prime or not prime to one another, and, if they are not prime to one another, what is their common measure. We are, he says, to compare the numbers in turn by continually taking the less from the greater as many times as possible, then taking the remainder as many times as possible from the less of the original numbers, and so on; this process “will finish either at an unit or at some one and the same number,” by which it is implied that the division of a greater number by a less is done by separate subtractions of the less. Thus, with regard to 21 and 49, Nicomachus says, “I subtract the less from the greater; 28 is left; then again I subtract from this the same 21 (for this is possible); 7 is left; I subtract this from 21, 14 is left; from which I again subtract 7 (for this is possible); 7 will be left, but 7 cannot be subtracted from

7.” The last phrase is curious, but the meaning of it is obvious enough, as also the meaning of the phrase about ending “at one and the same number.”

The proof of the Porism is of course contained in that part of the proposition which proves that G, a common measure different from CF, must measure CF. The supposition, thereby proved to be false, that G is greater than CF does not affect the validity of the proof that G measures CF in any case.