Proposition: 7.19: Relation of Ratios to Products

Euclid's Formulation

If four number are proportional then the number created from (multiplying) the first and fourth will be equal to the number created from (multiplying) the second and third. And if the number created from (multiplying) the first and fourth is equal to the (number created) from (multiplying) the second and third then the four numbers will be proportional. * Let $A$, $B$, $C$, and $D$ be four proportional numbers, (such that) as $A$ (is) to $B$, so $C$ (is) to $D$. * And let $A$ make $E$ (by) multiplying $D$, and let $B$ make $F$ (by) multiplying $C$. * I say that $E$ is equal to $F$.


Modern Formulation

In modern notation, this proposition reads that if \[\frac ab=\frac cd,\] then \[a\,d=b\,c,\] and vice versa, where all symbols denote numbers.

Proofs: 1

Proofs: 1 2 3 4 5 6 7 8 9

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