Commentary on Proposition XI.20¶
Back to Proposition XI.20.
After excluding the obvious case in which all three angles are equal, Euclid goes on to say “If not, let the angle BAC be greater,” without adding greater than what. Heiberg is clearly right in saying that he means greater than BAD, i.e. greater than one of the adjacent angles. This is proved by the words at the end “Similarly we can prove,” etc. Euclid thus excludes as obvious the case where one of the three angles is not greater than either of the other two, but proves the remaining cases. This is scientific, but he might further have excluded as obvious the case in which one angle is greater than one of the others but equal to or less than the remaining one.
Simson remarks that the angle BAC may happen to be equal to one of the other two and writes accordingly “If they [all three angles] are not [equal], let BAC be that angle which is not less than either of the other two, and is greater than one of them DAB.” He then proves, in the same way as Euclid does, that the angles DAB, DAC are greater than the angle BAC, adding finally : “But BAC is not less than either of the angles DAB, DAC; therefore BAC, with either of them, is greater than the other.”
It would be better, as indicated by Legendre and Rausenberger, to begin by saying that, “if one of the three angles is either equal to or less than either of the other two, it is evident that the sum of those two is greater than the first. It is therefore only necessary to prove, for the case in which one angle is greater than each of the others, that the sum of the two latter is greater than the former. Accordingly let BAC be greater than each of the other angles.” We then proceed as in Euclid.