Proposition: 7.10: Multiples of Alternate Ratios of Equal Fractions
(Proposition 10 from Book 7 of Euclid's “Elements”)
If a number is parts of a number, and another (number) is the same parts of another, also, alternately, which(ever) parts, or part, the first (number) is of the third, the second will also be the same parts, or the same part, of the fourth.
 For let a number $AB$ be parts of a number $C$, and another (number) $DE$ (be) the same parts of another $F$.
 I say that, also, alternately, which(ever) parts, or part, $AB$ is of $DE$, $C$ is also the same parts, or the same part, of $F$.
Modern Formulation
In modern notation, this proposition states that if $C=AB\cdot n+r_0$ with $0 < r_0 < AB$ and $F=DE\cdot n+r_1$ with $0 < r_1 < DE,$ then, using the division with quotient and remainder,
$$\begin{array}{rclccl}
D&=&AB\cdot m&+&r&0\le r < AB\\
&\Updownarrow&\\
F&=&C\cdot m&+&s&0\le s < C,
\end{array}$$
for some integers $m,s,r.$
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