Proposition: 5.05: Multiplication of Real Numbers is Left Distributive over Subtraction

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

If a magnitude is the same multiple of a magnitude that a (part) taken away (is) of a (part) taken away (respectively) then the remainder will also be the same multiple of the remainder as that which the whole (is) of the whole (respectively).


Modern Formulation

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

General Modern Formulation

See distributivity law for real numbers.

Proofs: 1

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.