Construction of Geometric Sequence in Lowest Terms/Porism

Porism to Construction of Geometric Sequence in Lowest Terms

In the words of Euclid:

From this it is manifest that, if three numbers in continued proportion be the least of those which have the same ratio with them, the extremes of them are squares, and if four numbers, cubes.

(The Elements: Book $\text{VIII}$: Proposition $2$ : Porism)

Proof

Apparent from the construction.

From Construction of Geometric Sequence in Lowest Terms, such a geometric sequence is of the form:

$P = \tuple {q^n, p q^{n - 1}, p^2 q^{n - 2}, \ldots, p^{n - 1} q, p^n}$

Thus when $n = 2$:

$P = \tuple {q^2, p q, p^2}$

and when $n = 3$:

$P = \tuple {q^3, p q^2, p^2 q, p^3}$

$\blacksquare$

Historical Note

This proof is Proposition $2$ of Book $\text{VIII}$ of Euclid's The Elements.

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 2 (2nd ed.) ... (previous) ... (next): Book $\text{VIII}$. Propositions