# Definition: 5.17: Ratio ex Aequali

There being several magnitudes, and other (magnitudes) of equal number to them, (which are) also in the same ratio taken two by two, a ratio via equality (or ex aequali) occurs when as the first is to the last in the first (set of) magnitudes, so the first (is) to the last in the second (set of) magnitudes. Or alternately, (it is) a taking of the (ratio of the) outer (magnitudes) by the removal of the inner (magnitudes).

### Notes

• This is not a definition, but rather a proposition and will be proven in Proposition 5.22.
• Therefore, the definition is merely verbal; it gives a convenient name to a certain concept.

### Modern Formulation

In other words, if $$\alpha_1,\ldots,\alpha_n$$ and $$\beta_1,\ldots,\beta_n$$ are positive real numbers such that the following corresponding pairs of numbers have the same ratio: $\frac{\alpha_1}{\alpha_{2}}=\frac{\beta_1}{\beta_{2}},\quad\frac{\alpha_2}{\alpha_{3}}=\frac{\beta_2}{\beta_{3}},\quad\ldots,\quad\frac{\alpha_{n-1}}{\alpha_{n}}=\frac{\beta_{n-1}}{\beta_{n}}$ then the ratio ex aequali or (the ratio via equality) is the equation $\frac{\alpha_1}{\alpha_{n}}=\frac{\beta_1}{\beta_{n}}.$

Proofs: 1 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 29 30 31
Propositions: 32 33 34 35 36 37

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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

#### Adapted from (subject to copyright, with kind permission)

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