# Definition: 5.09: Squared Ratio

And when three magnitudes are proportional, the first is said to have to the third the squared1 ratio of that (it has) to the second.

### Modern Formulation

The ratio $\frac \alpha\gamma$ of positive real numbers $\alpha,\gamma$ is called a squared ratio if there is a positive real number $\beta$ sucht that $\alpha,\beta,\gamma$ are in a proportion in three terms. $\frac\alpha\beta=\frac\beta\gamma.$

Note: Note that in this case we have

$\frac\alpha\gamma=\frac{\alpha^2}{\beta^2},$

explaining the name "squared ratio".

Corollaries: 1
Proofs: 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
Propositions: 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46
Sections: 47

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### References

#### Bibliography

1. Health, T.L.: "The Thirteen Books of Euclid's Elements - With Introduction and Commentary by T. L. Health", Cambridge at the University Press, 1968, Vol 1, 2, 3