Proof: By Euclid

(related to Proposition: Prop. 9.19: Condition for Existence of Fourth Number Proportional to Three Numbers)



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Adapted from (subject to copyright, with kind permission)

  1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"


  1. The proof of this proposition is incorrect. There are, in fact, only two cases. Either $A$, $B$, $C$ are in continued proportion, with $A$ and $C$ prime to one another, or not. In the first case, it is impossible to find a fourth proportional number. In the second case, it is possible to find a fourth proportional number provided that $A$ measures $B$ times $C$. Of the four cases considered by Euclid, the proof given in the second case is incorrect, since it only demonstrates that if $A/B=C/D$ then a number $E$ cannot be found such that $B/C=D/E$. The proofs given in the other three cases are correct (translator's note).