Definition: 5.12: Alternate Ratio

(Definition 12 from Book 5 of Euclid's “Elements”)

An alternate ratio is a taking of the (ratio of the) leading (magnitude) to the leading (of two equal ratios), and (setting it equal to) the (ratio of the) following (magnitude) to the following.

Modern Formulation

Given two equal ratios of positive real numbers:


the alternate ratio is given by


Thank you to the contributors under CC BY-SA 4.0!




  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"