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The reader interested in the 18th and 19th centuries will find plenty of food for thought. For example, the story of non-Euclidean geometry is covered well, and Kline does a good job of putting the discoveries in the light of the times. One notable thing I learned is that Lobachevsky and Bolyai were not the discoverers of non-Euclidean geometry, nor were they the first to publish material on that subject. Others before had expressed the opinion that non-Euclidean gometry was consistent and as viable a geometry as Euclidean (e.g. Kluegel, Lambert...even Gauss!) It remained for Beltrami to later show that if Euclidean geometry were consistent, so is non-Euclidean. Of course, important events like the invention of Galois theory are also mentioned. Really, if it's a major mathematical development before 1930, Kline will have it somewhere in these 3-volumes.
Incidentally, Kline advances the interesting theory that Lobachevsky and Bolyai somehow learned of Gauss' work on non-Euclidean geometry (which he kept secret and was not learned of until after his death) through close friends of Gauss: Bartel (mentor to Lobachevsky) and Bolyai's father, Farkas. [I understand that this theory has been shown false by recent research into Gauss' correspondence] Kline is careful to indicate it is only speculation by phrasing words carefully, e.g. "might have..." and "perhaps he..." I can appreciate Kline's various speculations and opinions, usually they are very interesting, and (at least in these volumes) he always does a good job of highlighting where the account of history ends and his ideas begins. Even so, luckily for those who like unbiased historical accounts, he inserts himself into the text rarely. This may surprise readers who have read his other books, like _Mathematics: the Loss of Certainty_. This history is a scholarly work, although one can't really say that about his other works.
Kline also writes quite a bit about the development of the calculus, as one should expect, given its major role in forming modern mathematics. I got a much deeper appreciation of calculus from reading various sections, which explained how this or that area was influenced or invented because of certain calculus problems.
I debated about giving this book 4 stars since there are a few minor flaws. One I've mentioned above; I think Kline should have kept his voice objective, instead of occasionally going into a little diatribe on his pet peeves. This is minor, since he doesn't do it too often, and I suppose he can be excused for being human. Another is that the index is rather weak. For a work of this magnitude, one expects that one ought to be able to find the phrase "hyperbolic geometry" in the index. Surprisingly one doesn't. "Non-Euclidean geometry" is there, but not the other phrase, which is synonymous and more common nowadays. There are other examples, but this is the one that comes to mind now.
Finally, I should add that I have not read every page of this history nor am I even close to doing that. I have read parts of all three volumes, and the quality seems consistent. That said, this is not a history one should read straight through. It is meticulous and well-documented, which can make for rather dry reading, so I suggest you do plenty of skipping around. I found (and will probably still find) Kline useful for helping me understand the context of the various mathematical concepts I was studying. Not only that, but I found his explanations of some topics to be even better than those in standard textbooks. Because of the insights I've gained, I've decided to overlook the little flaws, so...five stars!
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