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I noticed that the second paragraph of this section is incoherent. It says
"A corollary to this theorem yields the complete structure of all finite projective geometry. In their paper on "Non-Desarguesian and non-Pascalian geometries" in the 1907 Transactions of the American Mathematical Society, Wedderburn and Veblen showed that in these geometries, Pascal's theorem is a consequence of Desargues' theorem. They did so by constructing finite projective geometries which are neither "Desarguesian" nor "Pascalian" (the terminology is Hilbert's)."
You can't use the construction of a non-D non-P geometry to show that D[esargues] implies P[ascal]: it's contrary to logic. Veblen and Wedderburn say here at the outset that it's already known that D and P are equivalent for finite geometries; in fact the D-implies-P part apparently uses the commutativity of finite division rings. In this 1907 paper Wedderburn and Veblen just give (very interesting and ingenious) examples of finite geometries which don't satisfy D and hence don't satisfy P.
So this paragraph needs rewriting, but it ought to be done by someone with expert knowledge (not me).
[The paper is Trans. Am. Math. Soc. 8 (1907) 379, and the commutativity of finite division rings is in Trans. A.M.S. 6 (1905) 349 and in a nearby paper of Dickson.]
On another topic, does anyone else feel that "a significant algebraist" (in the introduction) is rather a patronising description of this mathematician? It can be read that way.