Proposition: 7.16: Natural Number Multiplication is Commutative

(Proposition 16 from Book 7 of Euclid's “Elements”)

If two numbers multiplying one another make some (numbers) then the (numbers) generated from them will be equal to one another. * Let $A$ and $B$ be two numbers. * And let $A$ make $C$ (by) multiplying $B$, and let $B$ make $D$ (by) multiplying $A$. * I say that $C$ is equal to $D$.


Modern Formulation

see multiplication of natural numbers is commutative.

Proofs: 1

Proofs: 1 2 3 4 5

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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