Commentary on Proposition IX.17

Back to Proposition IX.17.

[Folio 0413 | p. 407]

If $a, a_2, a_3, \dots a_n$ be a geometrical progression, and $a, a_n$ are prime to one another, then $a, a_2, a_n$ can have no integral fourth proportional.

For, if possible, let

$$a : a_2 = a_n : x.$$

Therefore

$$a : a_n = a_2 : x,$$

and hence [VII. 20, 21] a measures $a_2$.

Therefore $a_2$ measures $a_3$, [VII. Def. 20]

and hence a measures $a_3$, and therefore also ultimately $a_n$.

Thus $a, a_n$ are both measured by a : which is contrary to the hypothesis.