Commentary on Proposition IX.18

Back to Proposition IX.18.

Given two numbers a, b, to find the condition that they may have an integral third proportional.

  1. a, b must not be prime to one another. [IX. 16]

  2. a must measure $b^2$.

For, if a, b, c be in continued proportion,

$$ac = b^2.$$

Therefore a measures $b^2$.

Condition (1) is included in condition (2) since, if $b^2 = ma$, a and b cannot be prime to one another.

The result is of course easily seen if the three terms in continued proportion be written

$$a,\quad a\frac{b}{a},\quad a\left(\frac{b}{a}\right)^2.$$