Commentary on Proposition VI.5

Back to Proposition VI.5.

This proposition is the complete converse, VI. 6 a partial converse, of VI. 4.

Todhunter, after Walker, remarks that the enunciation should make it clear that the sides of the triangles taken in order are proportional. It is quite possible that there should be two triangles ABC, DEF such that

AB is to BC as DE to EF, and BC is to CA as DF to ED (instead of EF to FD), so that AB is to AC as DF to EF (ex aequali in perturbed proportion);

in this case the sides of the triangles are proportional, but not in the same order, and the triangles are not necessarily equiangular to one another. For a numerical illustration we may suppose the sides of one triangle to be 3, 4 and 5 feet respectively, and those of another to be 12, 15 and 20 feet respectively.

In VI. 5 there is the same apparent avoidance of indirect demonstration which has been noticed on I. 48.