Mathematical Thought from Ancient to Modern Times, Vol. 1

Item met all expectations.

Great purchase.

The only reader I think Kline's book would be right for is one who wants a single source for the history of mathematics and who is not willing to use more specialized books. I have had Kline's history for years and I sometimes look something up in it, am disappointed by his presentation, and then look for the topic in another book.For a reader who wants an accessible and reliable general history of mathematics I recommend Victor Katz's "A History of Mathematics". Kline covers European mathematics in more detail than Katz does, but Katz is a better one volume work, and I suggest that anyone who wants more detail than what Katz gives should use one of the following references instead of turning to Kline.The two volume "Abrégé d'histoire des mathématiques" edited and partially written by Dieudonne, Moritz Cantor's "Vorlesungen über Geschichte der Mathematik", and the two volume "Companion Encyclopedia" edited by Ivor Grattan-Guinness are all reliable and cover in detail much material. Dieudonne's Histoire is not comprehensive, but it is excellent for the material it does cover, mostly in function theory and the theory of numbers.For a mathematically knowledgeable reader who wants a structural history of certain parts of mathematics, I recommend Bourbaki's "Elements of the History of Mathematics". That book however is not meant to be a comprehensive history of mathematics, and really should be thought of as a history of the parts of mathematics that interested Bourbaki, written from their point of view. It is however reliable and specific in its details.For the history of Greek mathematics one cannot do better than to read Heath's books and translations. A one volume summary of Greek mathematics is given in Heath's "A Manual of Greek Mathematics", available from Dover.The best book to use for the number theory of Fermat and Euler is Weil's "Number Theory: An Approach Through History". It is exhaustive, completely reliable, and has excellent analysis of the material.A reader who wants to know anything about Newton's mathematics needs to consult Whiteside's "The Mathematical Papers of Isaac Newton". This will probably never be improved on and Whiteside gives extremely good comments on the papers. However, the commentary is given as footnotes to papers, and the book is hard to jump into. It would probably be hard casually to use Whiteside's edition. Joseph E. Hofmann's "Leibniz in Paris" is a good first place to turn for questions about the mathematics of Leibniz. Finally, for any question about the history of rational mechanics, you should see what Clifford Truesdell has to say, especially in his "Rational Mechanics of Flexible or Elastic Bodies", published as part of Euler's Opera omnia.

As one might expect from a 3-volume history, _Mathematical Thought_ is comprehensive; Kline covers basically all the important mathematical developments from ancient times (e.g. the Babylonians) until about 1930. Note that (as Klein himself mentions) the coverage of ancient mathematics, while taking up a good half of the first volume, is necessarily modest, and if that is the reader's primary interest, s/he would do best to seek out specific histories on the Greeks, Chinese, etc. [Kline gives several useful references, as always].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!

I've been learning more about the history of analytic geometry and the mathematization of physics, so I consulted the last chapters of Vol. 1. Kline explains very clearly the innovations Descartes made but also his misunderstanding of Galileo's mathematization of physics. For example, Galileo (and Newton) described motion and gravity mathematically, but they did not attempt to explain WHY; Descartes mistakenly criticized Galileo's work because of that.There was an incredible change in thinking about science during the seventeenth century. Kline explains clearly how that unfolded mathematically.I only gave the book four stars because of others' critique of the early part of the book, which I didn't read.

if flawed. Not only do you have to wade through the gentleman amateur flavour of the first couple of hundred pages or so, but Kline manages to describe William Hamilton as 'the greatest English theoretical physicist after Newton'; even an Irishman would concede that the greatest English theoretical physicist after Newton was Maxwell - Hamilton was third. However with the first impact tremors announcing the approach of Leonard Euler, when the technical issues start to thicken, things improve enormously. Kline is clearly in awe of Euler, and does a good job of communicating why awe is appropriate.It is nevertheless fortunate that the history of mathematics, unlike that of science, is a discipline essentially invulnerable to whiggish prejudice.

Unique achievement and, as usual, in Kline's masterly, explanatory style !A must read if you want to understand the evolution of mathematics, on the shoulders of giants, throught this beautifully written 3-volume set.

OK a good read.

Great