Definition:Triplicate Ratio

Definition

Let $a, b, c, d$ be magnitudes such that:

$a : b = b : c = c : d$

Then $a$ has the triplicate ratio to $d$ of the ratio it has to $b$.

That is:

$a : d$ is the triplicate ratio of $a : b$.

In the words of Euclid:

When four magnitudes are $<$ continuously $>$ proportional, the first is said to have to the fourth the triplicate ratio of that which it has to the second, and so on continually, whatever be the proportion.

(The Elements: Book $\text{V}$: Definition $10$)