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Corollary: 8.02: Construction of Geometric Progression in Lowest Terms

(related to Proposition: 8.02: Construction of Geometric Progression in Lowest Terms)

(Corollary to Proposition 2 from Book 8 of Euclid's “Elements”)

So it is clear, from this, that if three numbers in continued proportion are the least of those (numbers) having the same ratio as them then the outermost of them are square, and, if four (numbers), cube.

Modern Formulation

Obviously, in the geometric progression $A^nq^i$, $i=0,\ldots,n$ with $q=\frac AB$ ($A$ and $B$ being co-prime), the first and the last numbers are the powers $A^n$ and $B^n.$

Table of Contents

Proofs: 1

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Proofs: 1 2 3 4

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References

Adapted from (subject to copyright, with kind permission)

1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"