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18 of 20 people found the following review helpful:
2.0 out of 5 stars
Terrific subject but a disappointing treatment, December 28, 2008
The subject of "The Fifth Postulate" is the history of Euclid's parallel postulate and the development of non-Euclidean geometry in the 19th century. During this period two mathematicians, working independently of one another, established a consistent geometry that is independent of Euclid's parallel postulate (the fifth one, hence the title). The non-Euclidean geometry developed by Janos Bolyai and others is an important subject that deserves a popular treatment, so I was excited when I spied this volume in a bookstore. I also have some familiarity with the subject, having read a couple of other books related to this topic, so I thought this would be a good opportunity for the author to clear up some popular misconceptions that recent works have corrected. But alas, this is not the case, as the author merely seems to echo the mistakes propagated by earlier predecessors. I found the book overall to be tedious, sloppily written, and poorly researched. It seems hastily put together, with some glaring grammatical errors that a thorough editor should have caught and corrected. I only give this book two stars because of the subject.
The author begins by discussing ancient Greek mathematics and mathematicians, which is a logical starting point. Many characters not related to the subject are discussed, perhaps to the detriment of telling a cohesive story. For example, the author relates the story of the death of Archimedes twice, even though Archimedes is not connected to the parallel postulate. Also, the author does a poor job of telling this particular story, which is actually an interesting one, at least as related by E.T. Bell is his book "Men of Mathematics." Another nit I have to pick is that the author seems to state opinions that are not backed up by references. For example, he implies that Pythagoras and Euclid were not first-rate mathematicians. According to Bell in Men of Mathematics, Euclid's major contribution to geometry was to organize and arrange existing knowledge, so the author may be on the mark here, but Pythagoras is credited with the discovery of irrational numbers and the concept of proof in mathematics (his eponymous theorem is not thought to have orginated with him since it was known to others before him, including the Babylonians).
Another bigger issue is that the author fully credits Gauss along with Bolyai and Lobatchevsky with the discovery and development of non-Euclidean geometry. This has little basis in fact since Gauss published nothing on the subject in his lifetime and never intended to do so. All we have are his published statements that he claimed to have developed it, despite the public acknowledgement that he was still trying to prove the fifth postulate for many years. According to his notes that were collected after his death, Gauss did make some progress in establishing a geometry independent of Euclid's fifth postulate, but it does not appear to be as mature or thorough a treatment as that by Bolyai or Lobatchevsky. Indeed one of the shocking things about Bolyai's treatment of the subject is that one short appendix he not only developed the subject but answered all the important questions related to the topic (something the author fails to mention). However, Gauss' supporters for a long time insisted giving him credit for non-Euclidean geometry, an error which has only recently been corrected.
Another set of errors repeated by the author involve the work of Janos Bolyai apart from his work on non-Euclidean geometry. He fails to mention Bolyai's novel work on complex numbers (the "Responsio" that he and his father submitted in request to a treatment on complex numbers). This was years ahead of its time, but was never published since its worth was not recognized, possibly because Janos included some work on the relationship of complex numbers to his hyperbolic geometry. Also, as noted in Elemer Kiss' "Mathematical Gems from the Bolyai Chest," Janos Bolyai also made original contributions to number theory, the theory of algebraic equations, and laid the foundations for topology. All of this was unpublished during his lifetime because he was a man of modest means and could not afford to publish his work. But the only reference to this corpus of work made by the author is to 14,000 pages of manuscripts, despite including Kiss' book in his bibliography! This is in contrast with Gauss, where any mention of a subject in private notes establishes priority, while the notes of others are completely ignored (think about how much would have to be credited to Lebiniz if we went solely on what was written in unpublished notes). The few mentions the author makes to Bolyai's life after 1832 include the assertion that he was on the "verge of insanity." This is not backed up with a reference, and is contradicted by Bolyai's ability to continue to produce original mathematical work during this period (contrast this with the zero productivity of the brilliant John Nash after the onset of full schizophrenia). While the notion of a man driven insane by lack of recognition for his work may have a certain romantic appeal, the real tragedy of Janos Bolyai's later life was that he was poor, with little opportunity to publish his additional work. The author had an opportunity to rectify this error which was propagated by earlier authors (before the wave of more recent research on the subject), but again he is lazy and prefers to merely to echo the mistakes and oversights of previous works on the subject.
The author builds up a great deal of suspense in the book about the applications of non-Euclidean geometry but spends little time on the subject near the end of the book. The book seems truncated, with a few references to relativity theory. A better treatment of the applications of non-Euclidean geometry should have been attempted.
My main disappointment with the book is the treatment of geometry. You might imagine that any book that purports to be a book about geometry might contain a few figures or illustrations to help explain the concepts. But the book contains not one figure! Instead the author relies entirely upon words to explain geometry. I suppose the intent is for this to be for the general lay public with little mathematical sophistication. However I don't think the subject can be treated with any justice without including some geometrical figures to help discuss some of the properties of Euclidean and non-Euclidean geometry.
If you want to understand non-Euclidean geometry, a good technical treatment is Jeremy Gray's "Janos Bolyai, Non-Euclidean Geometry, and the Nature of Space," and if you want a more accurate depiction of Janos Bolyai's life and works outside of geometry, Elemer Kiss' "Mathematical Gems from the Bolyai Chest" is a good place to start.
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6 of 6 people found the following review helpful:
3.0 out of 5 stars
Long on History and Biography, Extremely Short on Mathematics and Cosmology, January 25, 2009
Recent years have witnessed a veritable explosion in popularized writing about mathematics, creating a minor publishing industry phenomenon fed by writers including Simon Singh, Mario Livio, David Berlinski, John Derbyshire, and Eli Maor. In many of these books, a mathematical topic is chosen as the centerpiece: pi, e, I (the square root of -1), Poincare's Conjecture, Fibonacci numbers, the divine proportion, symmetry, Fermat's last theorem, Goldbach's Conjecture, and solving for the roots of higher-order polynomial equations, among others. However, these books often turn out less to be reader-friendly explications of the mathematics than extended regurgitations of mathematical history and meandering biographies of its most famous practitioners.
Jason Bardi has joined their ranks, first with THE CALCULUS WARS and now with THE FIFTH POSTULATE. Both these efforts continue this same pattern of favoring history over mathematical explanation, perhaps forgivable in the former book but a serious deficiency in the latter. THE FIFTH POSTULATE is built around the long and fruitless efforts to remove as unnecessary, by means of derived proof, the last of the five basic postulates from which Euclid constructed his system of geometric theorems and proofs. The so-called Fifth Postulate states (in one of its many forms) that given a straight line and a single point not on that line, one and only one line parallel to that line can be drawn through the given point. After many centuries of failure, mathematicians in the 1800s adopted a new approach, one in which they explored the implications for geometry if the fifth postulate was not true, leading to the eventual development of what is now known as non-Euclidian geometry.
While the human story of non-Euclidian geometry's development is interesting enough, Bardi approaches it as though the underlying mathematics were totally irrelevant. He offers no examples of Euclid's method of deductive proof, nor how his postulates (particularly the fifth) were incorporated. His book contains not a single mathematical expression or diagram, hardly something one would expect in a book written about Euclidean and non-Euclidean geometry. There are no representations of the multiple parallel lines of longitude intersecting at the poles of a globe, no illustrations of parallels in hyperbolic or elliptic geometries, no Klein discs, no demonstrations of the fundamental notion of geodesics - nothing that would help the slightly less informed reader to picture concepts that are otherwise just words on a page.
Bardi's book contains plenty of biographical background on Pythagoras, Carl Friedrich Gauss, Janos Bolyai, and Nikolai Lobachevsky, and it also traces extensively the passage of Euclid's Elements over time from Egypt and Greece to Persia and finally back again to Europe. Curiously, the author makes Gauss the central character in his story despite the fact that great mathematician never published a single word on what was to become non-Euclidean geometry. While it is true that Gauss may have thought about some of these implications before Bolyai and Lobachevsky, he certainly did little to advance these ideas. Perhaps Gauss is central because he's so much easier to write about from a biographical standpoint - by book's end, Bardi has certainly given us his full life's story and then some. Left relatively in Gauss's dust are others who advanced the field, including Klein, Beltrami, Poincare, Riemann, and even the great David Hilbert.
THE FIFTH POSTULATE carries prominently on its cover the eye-catching subtitle, "How Unraveling a Two-Thousand Year Old Mystery Unraveled the Universe." This is an attractive promise, but the author leaves it largely unfilled in what is perhaps the book's greatest shortcoming. Following 205 pages tracing the development of Euclidean and non-Euclidean geometry from Thales to Lobachevsky, Bardi devotes a mere and unsatisfying four pages to his subtitle's unraveling of the universe. How did non-Euclidean geometry contribute to the notions of curved space-time? What role does it play in general relativity? How have these ideas evolved in the last century's studies of cosmology and the Big Bang? Perhaps most interesting, why are the basic tenets of non-Euclidean geometry not taught in our schools?
Readers who approach THE FIFTH POSTULATE as a work in the history of mathematics will find reasonable satisfaction therein. However, those who are hoping for at least a rudimentary explanation of non-Euclidean geometry will be sorely disappointed. As for the physics of the universe and the role of non-Euclidean geometry in our understanding of space-time, don't believe everything you read on book covers.
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3 of 3 people found the following review helpful:
4.0 out of 5 stars
Biography & History, Not Math, February 9, 2009
I have taught math and physics in high school for some fifteen years. Of the math classes I have taught, my favorite is probably geometry. I like it because it invites a logical development that is often passed-over in other math classes. Of course, at this level, what we teach is Euclidean geometry, which is based on a number of postulates; most famously, the "fifth postulate" or "parallel postulate," which, essentially, assumes the existence of parallel lines.
Even in antiquity, however, this assumption was rather controversial. For millennia, mathematicians tried various, often very clever, ways of proving it. All failed. And yet, it seemed absolutely essential for the soundness of mathematics that it be proved. Ultimately, in the nineteenth century, a trio of mathematicians working independently each came to the conclusion that it was possible to develop a completely consistent geometry without the parallel postulate. These "non-Euclidean" geometries, so called, would have a tremendous impact, not only on mathematics but also on our understanding of the universe.
In his new book, The Fifth Postulate, Mr. Bardi tells the history of this idea: attempts to prove it and its ultimate rejection, leading to a new mathematics. Primarily, this is a work of biography. The focus of the book is on one of the greatest mathematicians of all time, Carl Friedrich Gauss, with additional background on Nikolai Lobachevsky and Janos Bolyai as well as a number of other mathematicians, both major and minor, who impacted the problem. But it is these three mathematicians who really gave the world non-Euclidean geometry though, as Bardi rightly points out, it would be decades after their deaths before their work would receive the recognition it was due--not so much a problem for Gauss, who was a well-known, world-class mathematician from an early age, but more of an issue for the provincial Russian, Lobachevsky, and the rattled young soldier, Bolyai. They would never know the ultimate importance of their work.
It might not matter to some but, if there is a weakness in this book for me, it is that there is simply not much math in it. Granted, I'm already familiar with the math involved; however, I would have liked to have seen some examples that illustrated the importance of the parallel postulate to Euclidean geometry and some examples illustrating the meaning of changing it. I never really got a sense from this book as to what non-Euclidean geometry "looks like." And I don't mean problems to work out. I mean, maybe some diagrams and physical examples. I'm always looking for good readings to bring into the classroom. This one will be great for history but won't help me much with the math.
But that is a small quibble. As a history and a biography, it works well. He is also very good at showing how non-Euclidean geometry influenced our understanding of the universe; in particular, we live in a non-Euclidean universe, as Einstein showed us. And, anyway, I'm sure it was not Mr. Bardi's intention to delve deeply into the math; however, I would have liked it more had he done more. As long as the reader understands what's here, there's no reason to complain.
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