Corollary: 5.19: Proportional Magnitudes have Proportional Remainders
(related to Proposition: 5.19: Proportional Magnitudes have Proportional Remainders)
(Corollary to Proposition 19 from Book 5 of Euclid's “Elements”)
And since it was shown (that) as $AB$ (is) to $CD$, so $EB$ (is) to $FD$, (it is) also (the case), alternately, (that) as $AB$ (is) to $BE$, so $CD$ (is) to $FD$.
 Thus, composed magnitudes are proportional.
 And it was shown (that) as $BA$ (is) to $AE$, so $DC$ (is) to $CF$.
 And (the latter) is converted (from the former).
Modern Formulation
In modern notation, this corollary reads that if \[\frac\alpha\beta=\frac\gamma\delta,\] then \[\frac\alpha{\alpha\beta}=\frac\gamma{\gamma\delta,}\]
for all positive real numbers \(\alpha,\beta,\gamma,\delta\) with \(\alpha > \beta\) and \(\gamma > \delta\).
Table of Contents
Proofs: 1
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References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"