Definition:Proportion/Continued

Definition

Four magnitudes $a, b, c, d$ are in continued proportion if and only if $a : b = b : c = c : d$.

Also known as

A sequence of terms in continued proportion is more frequently referred to as a geometric sequence or geometric progression.

The term continuously proportional can also be found in Euclid's The Elements to mean in continued proportion:

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 $10$)

Historical Note

The term continued proportion is rarely seen outside Euclid's The Elements.

In fact, while Euclid used the term continued proportion throughout Book $\text{VIII}$ of The Elements, he never formally defined it.

In the words of Euclid:

If there be as many numbers as we please in continued proportion, and the extremes of them be prime to one another, the numbers are the least of those which have the same ratio with them.

(The Elements: Book $\text{VIII}$: Proposition $1$)