Commentary on Proposition VII.27¶
Back to Proposition VII.27.
If $a$, $b$ are prime to one another, so are $a^2$, $b^2$ and so are $a^3$, $b^3$ ; and, generally, $a^n$, $b^n$ are prime to one another.
The words in the enunciation which assert the truth of the proposition for any powers are suspected and bracketed by Heiberg because (1) in περὶ τοὺς ἄκρους the use of ἄκροι is peculiar, for it can only mean “the last products,” and (2) the words have nothing corresponding to them in the proof, much less is the generalisation proved. Campanus omits the words in the enunciation, though he adds to the proof a remark that the proposition is true of any, the same or different, powers of $a$, $b$. Heiberg concludes that the words are an interpolation of date earlier than Theon.
Euclid’s proof amounts to this.
Since $a$, $b$ are prime to one another, so are $a^2$, $b$ [VII. 25], and therefore also $a^2$, $b^2$. [VII. 25]
Similarly [VII. 25] $a$, $b^2$ are prime to one another.
Therefore $a$, $a^2$ and $b$, $b^2$ satisfy the description in the enunciation of VII. 26.
Hence $a^3$, $b^3$ are prime to one another.