Commentary on Proposition X.3¶
Back to Proposition X.3.
This proposition for two commensurable magnitudes is, mutatis mutandis, exactly the same as VII. 2 for numbers. We have the process
$$ \begin{array}{l} b )\, a \,( p \\ \underline{\phantom{b )\,} pb\phantom{ )\,}} \\ \phantom{b )\,} c )\, b \,( q \\ \phantom{b )\,} \underline{\phantom{c )\,} qc\phantom{ )\,}} \\ \phantom{b )\, \phantom{c )\,}} d )\, c \,( r \\ \phantom{b )\, \phantom{c )\,}} \underline{\phantom{d )\,} rd\phantom{ )\,}} \end{array} $$
where $c$ is equal to $rd$ and therefore there is no remainder, It is then proved that $d$ is a common measure of $a$, $b$ ; and next, by a reductio ad absurdum, that it is the greatest common measure, since any common measure must measure $d$, and no magnitude greater than $d$ can measure $d$. The reductio ad absurdum is of course one of form only.
The Porism corresponds exactly to the Porism to VII. 2.
The process of finding the greatest common measure is probably given in this Book, not only for the sake of completeness, but because in X. 5 a common measure of two magnitudes A, B is assumed and used, and therefore it is important to show that such a measure can be found if not already known.