Proposition: 7.09: Alternate Ratios of Equal Fractions

Euclid's Formulation

If a number is part of a number, and another (number) is the same part of another, also, alternately, which(ever) part, or parts, the first (number) is of the third, the second (number) will also be the same part, or the same parts, of the fourth.


Modern Formulation

In modern notation, this proposition states that if $A=\frac{BC}{n}$ and $D=\frac{EF}{n}$ for some integers $n \ge 1$ and $0 < BC < EF,$ then, using the division with quotient and remainder, $$\begin{array}{rclccl} D&=&m\cdot A&+&r&0\le r < A\\ &\Updownarrow&\\ EF&=&m\cdot BC&+&nr&0\le nr < BC. \end{array}$$

Proofs: 1

Proofs: 1 2
Propositions: 3

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