# Proposition: 8.02: Construction of Geometric Progression in Lowest Terms

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

To find the least numbers, as many as may be prescribed, (which are) in continued proportion in a given ratio.

• Let the given ratio, (expressed) in the least numbers, be that of $A$ to $B$.
• So it is required to find the least numbers, as many as may be prescribed, (which are) in the ratio of $A$ to $B$.

### Modern Formulation

If $q:=\frac AB$ is a reduced ratio, i.e. $A,B$ are co-prime, then

• $C=A^2,$
• $D=AB=A^2q,$
• $E=B^2=A^2q^2.$

It follows $\frac CD=\frac DE=\frac AB.$

• $F=A^3,$
• $G=A^2B=A^3q,$
• $H=AB^2=A^3q^2,$
• $K=B^3=A^3q^3.$

It follows $\frac FG=\frac GH=\frac HK=\frac AB.$

And so forth with the numbers $A^n,A^nq,\ldots,A^nq^n.$

Proofs: 1 Corollaries: 1

Proofs: 1 2 3 4 5 6 7

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### References

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

#### Adapted from CC BY-SA 3.0 Sources:

1. Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016