Commentary on Proposition VIII.1¶
Back to Proposition VIII.1.
What we call a geometrical progression is with Euclid a series of terms “in continued proportion” (ἑξῆς ἀνάλογον).
This proposition proves that, if $a, b, c, \dots k$ are a series of numbers in geometrical progression, and if $a, k$ are prime to one another, the series is in the lowest terms possible with the same common ratio.
The proof is in form by reductio ad absurdum. We should no doubt desert this form while retaining the substance. If $a', b', c', \dots k'$ be any other series of numbers in G.P. with the same common ratio as before, we have, ex aequali,
$$a : k = a' : k', \qquad \text{[VII. 14]}$$
whence, since $a, k$ are prime to one another, $a, k$ measure $a', k'$ respectively, so that $a', k'$ are greater than $a, k$ respectively.