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:
\[\frac\alpha\beta=\frac\gamma\delta,\]
the alternate ratio is given by
\[\frac\alpha\gamma=\frac\beta\delta.\]
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References
Bibliography
 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)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"