This is on the whole a standard semi-historical text on non-Euclidan geometry. What sets this text apart from the others in this genre is Trudeau's candidness regarding conclusions and interpretation of this story. Trudeau's conclusions and interpretations are precisely those of the blue-eyed research mathematician of today, not realising that he is distorting history for the purpose of propaganda. His propaganda goal is simple: intuition is bad, formalism is good. The history of non-Euclidean geometry does not support this conclusion, so Trudeau has to fake it. He does this in the form of fictitious dialogues with a student, whose stupid "intuitions" are always set straight. For example, this alleged student has the alleged "intuition" that Playfair's axiom is necessarily true. His feeble struggle with Trudeau's iron intellect even needs to be condensed by this statement:

"Time goes by. More proofs are suggested. Trudeau, with obvious pleasure, demonstrates how each proposal turns on some assertion logically equivalent to Postulate 5. Finally the other lapses into exasperated silence." (p. 161).

The conclusion sanctioned by Trudeau is that intuition should be given up in favour of formalism. But this is complete bogus-history. The whole point of the debate concerning Postulate 5 was precisely that it was *not* intuitively obvious. Which is why Trudeau is reduced to fighting, "with obvious pleasure," a fictitious opponent. While the result was not intuitive, the nature of the formal system suggested that it could be proved. Thus *it was the formal system that fooled us,* not our intuitions about geometry. Therefore, if any lessons are to be learned from the history of non-Euclidean geometry it is the exact opposite of that sanctioned by Trudeau.

Among the many other things which Trudeau gets backwards because of his doctrinal blindness are the relation of logic to mathematics (p. 14) and why Kant was wrong (p. 250). But I do not have the patience to detail these things.

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