Commentary on Proposition XI.22¶
Back to Proposition XI.22.
The Greek text gives an alternative proof, which is relegated by Heiberg to the Appendix. Simson selected the alternative proof in preference to that given above; he objected however to words near the beginning, “If not, let the angles at the points B, E, H be unequal and that at B greater than either of the angles at E, H,” and altered the words so as to take account of the possibility that the angle at B might be equal to one of the other two.
As will be seen, Euclid takes no account of the relative magnitude of the angles except as regards the case when all three are equal. Having proved that one base is less than the sum of the two others, he says that “similarly we can prove” the same thing for the other two bases.
If a distinction is to be made according to the relative magnitude of the three angles, we may say, as in the corresponding place in XI. 21, that, if one of the three angles is either equal to or less than either of the other two, the bases subtending those two angles must obviously be together greater than the base subtending the first. Thus it is only necessary to prove, for the case in which one angle is greater than either of the others, that the sum of the bases subtending those others is greater than that subtending the first. This is practically the course taken in the interpolated alternative proof.