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.
 For let a unit $A$ measure some number $BC$, and let another number $D$ measure some other number $EF$ as many times.
 I say that, also, alternately, the unit $A$ also measures the number $D$ as many times as $BC$ (measures) $EF$.
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.
Table of Contents
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References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"
Adapted from CC BYSA 3.0 Sources:
 Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016