Euclid and his modern rivals
The book might also serve as an illustration of the resistance to innovation in acacdemic circles. It's difficult to judge whether Carroll's criticisms were justified, since we now live in a different era, but I would suspect that they were the sort of minor faults one can find with any book. It seems that Carroll thought that the way to make a subject more interesting was to add jokes, rather than to structure it in a more understandable way.
One of the most surprising things about the book is that versions of the parallel axiom are discussed at great length, but Carroll seems totally unaware of the invention of non-Euclidean geometries 50 years before. Indeed, the book might be of interest to historians of mathematics, showing the sort of ideas which were circulating concerning the treatment of mathematics via axioms.