Definition: 5.18: Perturbed Proportion

(Definition 10 from Book 5 of Euclid's “Elements”)

There being three magnitudes, and other (magnitudes) of equal number to them, a perturbed proportion occurs when as the leading is to the following in the first (set of) magnitudes, so the leading (is) to the following in the second (set of) magnitudes, and as the following (is) to some other (i.e., the remaining magnitude) in the first (set of) magnitudes, so some other (is) to the leading in the second (set of) magnitudes.

Modern Formulation

In other words, if $$\alpha_1,\alpha_2,\alpha_3$$ and $$\beta_1,\beta_2,\beta_3$$ are positive real numbers, then they are said to be in perturbed proportion, if the following have the same ratios:

$\frac{\alpha_1}{\alpha_{2}}=\frac{\beta_2}{\beta_{3}},\quad\frac{\alpha_2}{\alpha_{3}}=\frac{\beta_1}{\beta_{2}}.$

Proofs: 1 2
Propositions: 3 4

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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