Commentary on Proposition VII.21

Back to Proposition VII.21.

[Folio 0329 | p. 323]

In other words, if $a$, $b$ are prime to one another, the ratio $a : b$ is “in its lowest terms.”

The proof is equivalent to the following.

If not, suppose that $c$, $d$ are the least numbers for which

$$a : b = c : d.$$

[Euclid only supposes some numbers $c$, $d$ in the ratio of $a$ to $b$ such that $c < a$, and (consequently) $d < b$. It is however necessary to suppose that $c$, $d$ are the least numbers in that ratio in order to enable VII. 20 to be used in the proof.]

Then [VII. 20] $a = mc$, and $b = md$, where $m$ is some integer.

Therefore

$$a = cm, \quad b = dm, \qquad \text{[VII. 16]}$$

and $m$ is a common measure of $a$, $b$, though these are prime to one another : which is impossible. [VII. Def. 12]

Thus the least numbers in the ratio of $a$ to $b$ cannot be less than $a$, $b$ themselves.

Where I have quoted VII. 16 Heiberg regards the reference as being to VII. 15. I think the phraseology of the text combined with that of Def. 15 suggests the former rather than the latter.