Proposition: 5.02: Multiplication of Numbers is Right Distributive over Addition

(Proposition 2 from Book 5 of Euclid's “Elements”)

If a first (magnitude) and a third are equal multiples of a second and a fourth (respectively), and a fifth (magnitude) and a sixth (are) also equal multiples of the second and fourth (respectively), then the first (magnitude) and the fifth, being added together, and the third and the sixth, (being added together), will also be equal multiples of the second (magnitude) and the fourth (respectively).


Modern Formulation

If we are given two positive real numbers \(\alpha\), \(\beta\),1 and the following multiples of aliquot parts \(m > 1\), \(n > 1\): \[m\alpha=m\beta,\quad n\alpha=n\beta\quad\quad( * )\] then adding both equations gives us \[(m+n)\alpha=m\beta+n\beta.\]

General Modern Formulation

See distributivity law for real numbers.

Proofs: 1

Proofs: 1 2 3
Sections: 4

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


  1. From a geometrical point of view, \(\alpha,\beta\) could e.g. mean the lengths of some segments, the areas of some plane figures or the volumes of some solids.