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Commentary on Proposition VII.10¶

If $a = \frac{m}{n} b$ and $c = \frac{m}{n} d$, then, whatever fraction $a$ is of $c$, the same fraction is $b$ of $d$.

To prove this, $a$ is divided into its $m$ parts equal to $b/n$, and $c$ into its $m$ parts equal to $d/n$.

Then, by VII. 9, whatever fraction one of the $m$ parts of $a$ is of one of the $m$ parts of $c$, the same fraction is $b$ of $d$.

And, by VII. 5, 6, whatever fraction one of the $m$ parts of $a$ is of one of the $m$ parts of $c$, the same fraction is the sum of the parts of $a$ (that is, $a$) of the sum of the parts of $c$ (that is, $c$).

Whence the result follows.

In the Greek text, after the words “so that, in addition” in the last line but one, is an additional explanation making the reference to VII. 5, 6 clearer, as follows: “whatever part or parts AG is of DH, the same part or the same parts is GB of HE also;

therefore also, whatever part or parts AG is of DH, the same part or the same parts is AB of DE also. [VII. 5, 6]

But it was proved that, whatever part or parts AG is of DH, the same part or the same parts is C of F also;

therefore also” etc. as in the last two lines of the text.

Heiberg concludes, on the authority of P, which only has the words in the margin in a later hand, that they may be attributed to Theon.