Proposition: 7.15: Alternate Ratios of Multiples

Euclid's Formulation

If a unit measures some number, and another number measures some other number as many times, then, also, alternately, the unit will measure the third number as many times as the second (number measures) the fourth.


Modern Formulation

This proposition is a special case of [Prop. 7.9]: if \[1=\frac bn\quad\text{ and }\quad c=\frac dn,\] then if \[1=\frac cl,\] then \[b = \frac dl,\] where all symbols denote numbers.

Proofs: 1

Proofs: 1 2 3 4 5

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Adapted from (subject to copyright, with kind permission)

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

Adapted from CC BY-SA 3.0 Sources:

  1. Prime.mover and others: "Pr∞fWiki",, 2016