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.


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 $m,s,r.$

Proofs: 1

Proofs: 1

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



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