Commentary on Proposition V.14¶
Back to Proposition V.14.
Algebraically, if $a : b = c : d$, then, according as $a > = < c$, $b > = < d$.
Simson adds the specific proof of the second and third parts of this proposition, which Euclid dismisses with “Similarly we can prove….”
“Secondly, if A be equal to C, B is equal to D ; for A is to B as C, that is A, is to D ;
Thirdly, if A be less than C, B shall be less than D. For C is greater than A ; and, because C is to D as A is to B,
D is greater than B, by the first case.
Wherefore B is less than D.”
Aristotle, Meteorol. III. 5, 376 a 11—14, quotes the equivalent proposition that, if $a > b$, $c > d$.