Commentary on Proposition V.22¶
Back to Proposition V.22.
Euclid enunciates this proposition as true of any number of magnitudes whatever forming two sets connected in the manner described, but his proof is confined to the case where each set consists of three magnitudes only. The extension to any number of magnitudes is, however, easy, as shown by Simson.
“Next let there be four magnitudes A, B, C, D, and other four E, F, G, H, which two and two have the same ratio, viz.:
as A is to B, so is E to F,
and as B is to C, so is F to G,
and as C is to D, so is G to H;
A shall be to D as E to H.
Because A, B, C are three magnitudes, and E, F, G other three, which taken two and two have the same ratio, by the foregoing case,
A is to C as E to G.
But C is to D as G is to H; wherefore again, by the first case,
A is to D as E to H.
And so on, whatever be the number of magnitudes.”