A New Look at Euclid's Second Proposition

There has been considerable interest during the past 2300 years in comparing different models of geometric computation in terms of their computing power. One of the most well-known results is Mohr's proof in 1672 that all constructions that can be executed with straightedge and compass can be carried out with compass alone. The earliest such proof of the equivalence of models of computation is due to Euclid in his second proposition of Book I of the Elements, in which he establishes that the collapsing compass is equivalent in power to the mode m compass. Therefore, in the theory of equivalence of models of computation Euclid's second proposition enjoys a singular place. However, the second proposition has received a great deal of criticism over the centuries. Here it is argued that it is Euclid's early Greek commentators and more recent expositors and translators that are at fault, and that Euclid's original algorithm is beyond reproach.