← euclid / The Thirteen Books of Euclid’s Elements

Commentary on Proposition V.6¶

This proposition corresponds to V. 2, with subtraction taking the place of addition. It asserts namely that, if $n$ is less than $m$, $ma - na$ is the same multiple of $a$ that $mb - nb$ is of $b$. The enunciation distinguishes the cases in which $m - n$ is equal to $1$ and greater than $1$ respectively.

Simson observes that, while only the first case (the simpler one) is proved in the Greek, both are given in the Latin translation from the Arabic ; and he supplies accordingly the proof of the second case, which Euclid leaves to the reader. The fact is that it is exactly the same as the other except that, in the construction, CK is made the same multiple of F that GB is of E, and at the end, when it has been proved that KC is equal to HD, instead of concluding that HD is equal to F, we have to say “Because GB is the same multiple of E that KC is of F, and KC is equal to HD, therefore HD is the same multiple of F that GB is of E.”