Proposition: 5.19: Proportional Magnitudes have Proportional Remainders

Euclid's Formulation

If as the whole is to the whole so the (part) taken away is to the (part) taken away then the remainder to the remainder will also be as the whole (is) to the whole.


Modern Formulation

In modern notation, this proposition reads that if \[\frac\alpha\beta=\frac\gamma\delta,\] then \[\frac\alpha\beta=\frac{\alpha-\gamma}{\beta-\delta},\] for all positive real numbers \(\alpha,\beta,\gamma,\delta\) with \(\alpha > \gamma\) and \(\beta > \delta\).

Proofs: 1 Corollaries: 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