The (almost really) Complete Works of Lewis Carroll

[1495]—In your “Chats abont Geometrical Measurement,” p. 337, line 5 from end, you say that Euclid’s 12th Axiom is “no axiom,” as “the converse is demonstrated in the 17th Proposition.” If this were the logical “converse” of the axiom, so as to follow immediately from it, there would be some absurdity in making the first statement an axiom and the second a theorem. But this is not so. The two statements are of the form “all X is Y,” and “all Y is X;” and it is so far from being the case that, if one of these be axiomatic, the other is axiomatic, that it may easily happen that one is axiomatic, while the other is not even true.

Again, at p. 334, col. 1, last line, you say “it is evident that $KL=CH$.” If this were once granted, you would not need the diagram in col. 2, you need only say “join $HK$. These triangles have all their sides equal. Therefore angle $CKH$ is equal to angle $KHL$. Therefore the three angles of the triangle $HCK$ are equal to the three angles $KCH$, $HCK$, $KHL$, i. e., to two right-angles. But any given right-angled triangle may be treated like this, and any triangle may be divided into two right-angled triangles. Hence the angles of any triangle are equal to two right-angles.” This proves Euc. I. 32, after which all is easy.

Again, in col. 2, you claim to have proved Simson’s axiom that two lines through a point cannot both be parallel to a third line. But you have only proved this for “parallels” as you define them, viz., “lines which have a common perpendicular,” and so have not proved Simson’s axiom at all. This is, I fear, a logical flaw in your argument.