Proposition: 7.17: Multiples of Ratios of Numbers

(Proposition 17 from Book 7 of Euclid's “Elements”)

If a number multiplying two numbers makes some (numbers) then the (numbers) generated from them will have the same ratio as the multiplied (numbers). * For let the number $A$ make (the numbers) $D$ and $E$ (by) multiplying the two numbers $B$ and $C$ (respectively). * I say that as $B$ is to $C$, so $D$ (is) to $E$.


Modern Formulation

In modern notation, this proposition states that if \[d = a\,b\quad\text{ and }\quad e=a\,c,\] then \[\frac de=\frac bc,\] where all symbols denote numbers.

Proofs: 1

Proofs: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

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