5 comments about Touch and Feel: Shapes.

1. My son is only 5 months old, and he seems like picking up all the book-like objects, so I don't know if he is especially interested in this book or not.

   As a parent's opinion, I like the different texture it offers to my kid to touch in the book, that's why I have bought quite some "touch and feel" style books for my baby. But I find this less perfect.

   I find this book not too creative and imaginative. They always use "buttons" as one of the examples of many shapes. For example, when they introduce circle, there are buttons; when they introduce square, there are buttons; and even heart shape, there are buttons. They could have chosen a greater variety of objects.

   "Cushion" is another object that appears a lot. In both the square and in the star shape sections, there are cushions. I think they could have at least use star fish instead of cushion in the star shape section. That will help the kids to learn more about different things besides different shapes.




2. This Dk touch and feel book is not their best. It feels like they just slapped a few ideas together and didn't really try to come up with real-life, meaningful texture examples. I agree with the other reviewer who said it was repetitive; it is! Enough with the buttons already! Plus, the textures seem forced, like the best idea they had for "fluffy" is a pencil case? and "shiny" is a star that feels bumpy? The examples don't seem relevant to a baby's world.

   (...)




3. This is our second touch & feel book and my son really enjoys the new textures introduced in this book. Not neccesarily the best touch & feel book out there, but still decent.



4. Toddlers really seem to be drawn to the Touch and Feel books. It's a unique way for them to learn and they enjoy it!



5. A fun way to teach shapes to your little one. I thought the images in this book were not as good as they could have been, but all in all another good book by DK.

5 comments about Algebraic Geometry (Graduate Texts in Mathematics).

1. This book hardly needs a review on Amazon, because if you have as much math background as it needs, then you must already know it is indispensible for learning about schemes in algebraic geometry. The book is clear, concise, very well organized, and very long. If you do not already know the Noether normalization theorem, and the Hilbert Nullstellensatz, then you do not want this book yet--you want an introduction to commutative algebra.



2. This book is one of the most used in graduate courses in algebraic geometry and one that causes most beginning students the most trouble. But it is a subject that is now a "must-learn" for those interested in its many applications, such as cryptography, coding theory, physics, computer graphics, and engineering. That algebraic geometry has so many applications is quite amazing, since it was not too long ago that it was thought of as a highly abstract, esoteric topic. That being said, most of the books on the subject, including this one, are written from a very formal point of view. Those interested in applications will have to face up to this when attempting to learn the subject. To read this book productively one should gain a thorough knowledge of commutative algebra, a good start being Eisenbud's book on this subject. Also, it is important to dig into the original literature on algebraic geometry, with the goal of gaining insight into the constructions and problems involved. The author of this book does not make an attempt to motivate the subject with historical examples, and so such a perusal of the literature is mandatory for a deeper appreciation of algebraic geometry. The study of algebraic geometry is well worth the time however, since it is one that is marked by brilliant developments, and one that will no doubt find even more applications in this century.

   Varieties, both affine and projective, are introduced in chapter 1. The discussion is purely formal, with the examples given unfortunately in the exercises. The Zariski topology is introduced by first defining algebraic sets, which are zero sets of collections of polynomials. The algebraic sets are closed under intersection and under finite unions. Therefore their complements form a topology which is the Zariski topology. The properties of varieties are discussed, along with morphisms between them. "Functionals" on varieties, called regular functions in algebraic geometry, are introduced to define these morphisms. Rational and birational maps, so important in "classical" algebraic geometry are introduced here also. Blowing up is discussed as an example of a birational map. A very interesting way, due to Zariski, of defining a nonsingular variety intrinsically in terms of local rings is given. The more specialized case of nonsingular curves is treated, and the reader gets a small taste of elliptic curves in the exercises. A very condensed treatment of intersection theory in projective space is given. The discussion is primarily from an algebraic point of view. It would have been nice if the author would have given more motivation of why graded modules are necessary in the definition of intersection multiplicity.

   The theory of schemes follows in chapter 2, and to that end sheaf theory is developed very quickly and with no motivation (such as could be obtained from a discussion of analytic continuation in complex analysis). Needless to say scheme theory is very abstract and requires much dedication on the reader's part to gain an in-depth understanding. I have found the best way to learn this material is via many examples: try to experiment and invent some of your own. The author's discussion on divisors in this chapter is fairly concrete however.

   The reader is introduced to the cohomology of sheaves in chapter 3, and the reader should review a book on homological algebra before taking on this chapter. Derived functors are used to construct sheaf cohomology which is then applied to a Noetherian affine scheme, and shown to be the same as the Cech cohomology for Noetherian separated schemes. A very detailed discussion is given of the Serre duality theorem.

   Things get much more concrete in the next chapter on curves. After a short proof o the Riemann-Roch theorem, the author studies morphisms of curves via Hurwitz's theorem. The author then treats embeddings in projective space, and shows that any curve can be embedded in P(3), and that any curve can be mapped birationally into P(2) if one allows nodes as singularities in the image. And then the author treats the most fascinating objects in all of mathematics: elliptic curves. Although short, the author does a fine job of introducing most important results.

   This is followed in the next chapter by a discussion of algebraic surfaces in the last chapter of the book. The treatment is again much more concrete than the earlier chapters of the book, and the author details modern formulations of classical constructions in algebraic geometry. Ruled surfaces, and nonsingular cubic surfaces in P(3) are discussed, as well as intersection theory. A short overview of the classification of surfaces is given. The reader interested in more of the details of algebraic surfaces should consult some of the early works on the subject, particularly ones dealing with Riemann surfaces. It was the study of algebraic functions of one variable that led to the introduction of Riemann surfaces, and the later to a consideration of algebraic functions of two variables. A perusal of the works of some of the Italian geometers could also be of benefit as it will give a greater appreciation of the methods of modern algebraic geometry to put their results on a rigorous foundation.




3. This is THE book to use if you're interested in learning algebraic geometry via the language of schemes. Certainly, this is a difficult book; even more so because many important results are left as exercises. But reading through this book and completing all the exercises will give you most of the background you need to get into the cutting edge of AG. This is exactly how my advisor prepares his students, and how his advisor prepared him, and it seems to work.

   Some helpful suggestions from my experience with this book:
   1) if you want more concrete examples of schemes, take a look at Eisenbud and Harris, The Geometry of Schemes;
   2) if you prefer a more analytic approach (via Riemann surfaces), Griffiths and Harris is worth checking out, though it lacks exercises.




4. Here's my impression after doing the first 30 pages: What makes this a really good book is the exercises. Not too hard, always interesting. If you are new to the subject you need to look up results from commutative algebra somewhere else. It can be a little strange getting used to working with the Zariski topology. All open sets are dense, so you don't have the notion of a small neighborhood of a point. For instance any bijection between two curves is a homeomorphism.



5. Why does everyone love this book so much? Read Eisenbud and Harris first. It seems algebraic geometers love this book in a sort of cult fan club because they are proud of having mastered such a dense work. That's good, but there are other books that attempt to explain the subject rather than encapsulate it.

5 comments about Euclid's Window : The Story of Geometry from Parallel Lines to Hyperspace.

1. "That the mast and sails vanish first, Aristole saw in a flash of genius, is a sign that the earth is curved."
   
   Surprising that this got by the author and proof readers and wonder what other subtle "mix ups" are between the covers.



2. This luminous book offers the rare combination of serious scientific contemplation and reader-friendly accessibility.
   
   Starting with the mathematicians and geometers of antiquity, Mlodinow traces the progress of rational thought -- and irrational numbers -- from before Euclid's elucidation of the Elements of geometry to the possibilities which still wait for us to reveal them -- from "A point is that which has no part" straight up to the equally puzzling notion that space and time may only be shadowy hints of some more fully flowering, if abstract, function of mathematics on another plane of reality. Sound like science fiction? Rest assured that Mlodinow has both feet planted square on terra firma. The paradoxes and upsets of his discipline are not lost on the author -- nor, indeed, are the ironies and jokes of history (say what you like about death, but it was the decidedly un-mystical necessity of taxation which launched geometry as a scholarly pursuit in ancient Egypt) -- but the author reminds his reader at various points of the dangers of assuming too readily that any given idea is worthless, too far-out, or obviously and intuitively wrong. Intuition, as it turns out, resists and rebels against much of what has become higher learning in the fields of mathematics and physics.
   
   Mlodinow's dedication to the subject matter at hand matches in beautiful, if heartbreaking, counterpoint to the obscurity in which many of the scholars he discusses labored. Drawing not only on the work of famous theoreticians like Einstein and Hawking, but also on essays and ideas buried in forgotten papers and musty appendices, the author gives full credit wherever it may be due. In the process, whether by design or accident, Mlodinow imparts an even more valuable lesson: the ease with which scientific knowledge can be lost, sometimes for millennia. If Artistotle knew, nearly 2,500 years ago, that the planet must be round, why do we still hear that Columbus' sailors were terrified of sailing off the edge of a flat Earth? (This story in itself is almost certainly apocryphal.) If primitive versions of the Theory of Evolution were kicking around in ancient Greece, how is it we still face voids of serious scientific credibility in modern-day Kansas? Regrettably, superstition, fear, politics, and the manipulation of knowledge -- who gets it and who pays the price for seeking too much of it -- is also part of the history of geometry, as it is part of the history of science in general.
   
   Your reviewer himself studied a fair amount of the history of mathematics and physics in the Western World (starting, in fact, with Euclid, and progressing then through Ptolemy, Apollonius, Descartes, Newton, et al, right up through Einstein and Minkowski) and found certain parts of the curriculum cheerless, if not downright appalling. What a relief and a joy, then, to find Euclid's Window not only concise and readily understandable, but effervescent as well. Author Mlodinow clearly enjoys the subject matter and -- more importantly -- enjoys imparting it to others. As a writer and a teacher, Mlodinow demonstrates that he is gifted and enthusiastic.



3. This book is well-written, easy to follow for the most part. I really enjoyed the history of the math greats and the tidbits surrounding their lives. The Alexandria information was the most interesting to me. Miodinow was aiming for the middle-of-the-road math meddler and hit the target. It inspired me (and challenged me) to search deeper into math literature - and my journey continues...



4. I like math and I am a computer's science teacher in Brazil. But when I read that Dominicans and Franciscans sent teachers to Charlemagne's church schools I became disapointed with this book. The author is very weak in History of the Church. I recomended to him, Kenneth Scott Latourette's book: "History of Christianity".



5. Having a lifelong interest in 2 and 3D "geometry", this walk down memory lane into the future of mathematical theory and application was most informative, enlightening and a learning experience. Being introduced to many personalities old and new such as Edward Witten was a real treat! Mlodinow's approach caused me to think and ponder and his humorous style and personal experiences kept me very interested! I cannot wait to finish "The Drunkard's Walk".

5 comments about Optimization by Vector Space Methods (Series in Decision and Control).

1. Optimization by Vector Space Methods, by David Luenberger, is one of the finest math texts I have ever read, and I've read hundreds. Many years ago this book sparked my interest in optimization and convinced me that the abstract mathematics I had been immersed in actually would be applicable to real problems. Since then, Luenberger's book has inspired several of my graduate students. I merely lent them my copy, and Luenberger did the rest; he drew them in by carefully laying the foundation for an elegant theory, with just the right mix of formalism and intuition, and opened their eyes to the beauty and practicality of abstract mathematics. Anyone with an interest in higher-level mathematics (beyond multi-variable calculus, say) would benefit from exposure to this finely-crafted book. I daresay, the rampant math anxiety that is so prevalent in the West would be substantially reduced if more authors would take such meticulous care in presenting their material.

   The format of Luenberger's book is also extremely appealing in a way that I cannot quite put my finger on. The typography and illustrations are inherently crisp and inviting; they draw you in. There is nothing at all superfluous or gratuitous in this book. It is utterly to-the-point, methodical, and above all, clear. The techniques are developed starting from an elementary treatment of vector spaces, then proceeding on to Banach spaces and Hilbert spaces. Along the way, Luenberger introduces convexity, cones, basic topology, random variables, minimum-variance estimators, and least squares, among many other things. There is a recurring theme of duality, which can be used in a way analogous to the inner product of a Hilbert space. In particular, the familiar projection theorems of Hilbert spaces can be echoed in simpler normed linear spaces using duality, which Luenberger motivates and covers beautifully.

   The book also covers some of the standard fare of functional analysis, such as the Han-Banach theorem, strong and weak convergence, and the Banach inverse theorem. However, Luenberger never wanders too far off into abstract nonsense; around every corner lay tantalizing application of these ideas to optimization. Luenberger first explores optimization of functionals then covers constrained optimization, which builds upon concepts such as positive cones and Lagrange multipliers. The optimization methods themselves have endless applications in fields such as computer vision, computer graphics, economics, and physics. Indeed, the list is effectively endless as optimization techniques pervade math and science.

   I'm certain that the appeal of this book is helped immeasurably by the inherent beauty of the subject matter. Hilbert-space methods are lovely in themselves--they possess a structure that engages one's geometric intuition while at the same time admitting convenient algebraic properties. Once you are in the habit of phrasing problems in abstract settings such as Hilbert spaces, it forever changes how you look at things; you cannot help but look past the clutter to the essence of a problem (or, at least try very hard to do so). While this material is not nearly as abstract as, say, category theory, it nevertheless hits a high point in mathematics--a point more people ought to experience.

   If you've had some exposure to optimization methods, or need to apply them in the context of computer vision, graphics, or finance, to mention just a few areas, then I urge you to take a look at Luenberger's fine book. It too hits a high point in clarity of mathematical writing. Combine beautiful theory with endless applications and lucid writing, and you have a winner of a book.




2. This book is a timeless classic, filled with extraordinarily powerful mathematics and applicable to just about every serious subject area. Luenberger did a masterful job of writing a book that will "unravel the spaghetti" seen in most other books. The visual perspectives he provides to seemingly abstract ideas are the key.



3. The exposition is pretty clear and the book has a good number of worked non-trivial examples. At $40 this would be a great book, but $100 for a PAPERBACK book written 30 years ago is a bit ridiculous. The first 1/4 of the book is also a (very) basic introduction to functional analysis which, if you have had any contact with this subject before, you will probably skip making the book quite short.



4. Professor Luenberger unites many areas of optimization using a few principles from functional analysis. The explanations are clear and the proofs are compact and elegant. This book is your tool for understanding the deep connection between linear programming, convex optimization, game theory, optimal control and series approximation (e.g. Fourier series).
   
   Luenberger's book has over 1300 citations as of March 2006. In my opinion, the material in this book is essential for any graduate student or professional who intends to contribute to the literature in optimization or optimal control.



5. Although Luenberger's book is probably the only treatment out there that combines optimization with applied functional analysis, I have to say that this book does not do a very thorough job of explaining the abstract connection between these two very different fields. The book is well written in a grammatical sense, however, one of the major shortcomings I have found is that 99% of the proofs Luenberger gives throughout the text are either incomplete, left to the reader, or just plain confusing. (for example, the proof given for the Hahn-Banach Theorem is pretty much useless). Furthermore, the examples the author gives are mostly one-liners and do not offer any kind of clarification to the important theorems given in the text. Also, Luenberger does use concepts from Lebesgue and Riemann-Stieltjes integration and mildly says at the beginning of the text that it is not necessary for the reader to have a background in these fields to be able to understand this textbook. This is a very dangerous assumption, since most of functional analysis is based on Lebesgue measure theory. As a result, Luenberger's treatment of Lebesgue integration is merely "washed over" and, in some parts, is flat out wrong.
   
   Overall, don't buy Luenberger's textbook expecting to be able to learn optimization and/or functional analysis just from this book. I would recommend consulting Rudin's Real and Complex Analysis and Wheeden's Measure and Integral before pursuing this book. Luenberger's book is very much a serious applied mathematics book written by a non-mathematician which merely brushes over very deep results in optimization theory.

3 comments about Schaum's Outline of Differential Geometry (Schaum's).

1. As with all of the Schaum's Outline Series, this book is particularly useful if the readers intent is to gain a working knowledge of the subject. The subject of Differential Geometry is no exception. Dr. Lipschultz has done an excellent job of communicating the essential aspects of differential geometry to the reader. The book assumes a fairly low level of mathematical ability having calculus as the primary prerequisite. From this humble beginning, Dr. Lipschultz takes the reader through the necessary discussions of vector functions, curvature, fundamental forms, and tensor analysis. Given the theoretical nature of the subject, Dr. Lipschultz has included most of the theorems and associated proofs necessary for a general understanding of the subject. However, this book is not a substitute for a serious study of differential geometry. In addition most of the problems are limited to two dimensional surfaces and this reader would have enjoyed a more adventurous investigation of higher dimensional spaces. Like all Schaum's series, the text is chock full of problems and their solution. I recommend this book for anyone interested in quickly gaining a working knowledge of the subject.



2. This book is intended to assist upper level undergraduate and graduate students in their understanding of differential geometry, which is the study of geometry using calculus. Usually students study differential geometry in reference to its use in relativity. I personally have a rather oddball application for the subject - modeling of curved geometry for computer graphics applications. The fundamental concepts are presented for curves and surfaces in three-dimensional Euclidean space to add to the intuitive nature of the material.
   The book presumes very little in the way of background and thus starts out with the basic theory of vectors and vector calculus of a single variable in the first two chapters. The following three chapters discuss the concept and theory of curves in three dimensions including selected topics in the theory of contact.
   Great care is given to the definition of a surface so that the reader has a firm foundation in preparation for further study in modern differential geometry. Thus, there is some background material in analysis and in point set topology in Euclidean spaces presented in chapters 6 and 7. The definition of a surface is detailed in chapter eight. Chapters 9 and 10 are devoted to the theory of the non-intrinsic geometry of a surface. This includes an introduction to tensor methods and selected topics in the global geometry of surfaces. The last chapter of the outline presents the basic theory of the intrinsic geometry of surfaces in three-dimensional Euclidean space.
   Exercises are primarily in the form of proofs, and there are plenty of worked examples. Since the examples are kept to no more than three dimensions, the outline contains plenty of good instructive diagrams that illustrate key concepts. This Schaum's outline has quite a bit of instruction in it past the bare required minimum, but you might still want a good primary textbook. My personal favorite is Pressley's "Elementary Differential Geometry". Overall I find this to be a very good outline and source of solved problems on the subject and I highly recommend it.



3. I have found this to be an excellent addition to my library.

1 comments about Algebra and Trigonometry Enhanced with Graphing Utilities (5th Edition).

1. When I got my book I was shocked. I expected the book was going to be ragged but I couldn't even tell it was used! I thought it was new! I am definitely ordering my books from ya'll for now on!!!

2 comments about Geometry : Explorations and Applications.

1. Now this is type of Geometry book it should be. More examples, answer key, and a glossary. Unlike that complicated Michael Serra Geometry book from hell.



2. This book is lacking in examples, diagrams and as well as facts and information. The diagrams that are present do not provide the necessary information to actually LEARN form the book. As a result, we have to deviate from the textbook making it completely comfusing. DO NOT BUY THIS BOOK. It is simply not worth it.
   
   I suggest Prentice Hall if you need a book, after using Prentice Hall for several years; I find it completely helpful and easy to learn from.
   
   The book has few examples and it doesn't show HOW to do the problem, rather, it just explains what the problem is and quickly gives the solution. The chapters are organized in a fashionable manner but it is lacking in material, the answers are at the back of the book but what really ticks me off is the 'real life problems'. They give 4-5 of those in a lesson and they take at least half an hour of think and writing [that doesn't involve math' and then on the test, they never show up becuase the problem had nothing to do with the chapter. The book is just lacking in materials. Look at the publiciation date. 1998--this book is just wayyy out dated.

4 comments about Finite Mathematics (9th Edition).

1. As a reviewer and educator, I encounter many textbooks that can be used in the lower level classes such as basic statistics, algebra, precalculus and finite mathematics. Since the material to be covered in these classes has been established by consensus, it is a rare occasion when there is a significant difference in coverage. In addition, there is a natural order to many of the topics, so the differences in sequence also tend to be minimal. All of this leads to a routine sameness of the books, especially when examined by someone experienced in the topic.
   Therefore, when examining a book for adoption consideration, my primary concern is how easy it will be for the students to read it. This requires that you intentionally dumb yourself down, reading the explanations in detail, looking for simplistic clarity. If a book has that feature, then it is most likely a good choice. With the exception of including solutions to a large percentage of the problems, all other aspects of the book are secondary. (I consider the lack of solutions to problems in a math book to be a fatal flaw.) In general, I consider the inclusion of more problems of the same type to be trivial padding; after all, the value added by including ten more routine matrix addition problems is minimal. In finite mathematics, it is possible to include problems based on circumstances that actually occur in the real world. The cost of manufacturing some items is in fact locally linear and the allocation of resources can be described by a matrix. Therefore, the realism of the applied problems must also be a consideration when examining a finite math text.
   With these conditions of acceptance established, I can say that this book passes the test. The explanations have the simplistic clarity that students need and solutions to many of the problems are included. Many problems demonstrate reasonably practical conditions where finite mathematics is used in the world, and the appropriate background for the problems is given. While no decision has yet been made concerning what book to use next year, this one has been placed on my list of the top three finalists for further consideration.



2. I took an online class over the summer and used this book. Eventhough the teacher was not helpful, the book was. All the exercises were extremely helpful and the text was simple and easy to read. I would recommend this book for any who has to take this class.



3. This book was only used for one semester. It is in great condition looks brand new. I bought it brand new and kept it in the same condition.



4. I purchased this book brand new and full priced because it is listed as havinga MathXL Tutorial on CD. I felt the aid of interactive instruction would be beneficial. Since having recieved this book and inquiring of Amazon, I find the book does NOT come with a CD but with an online address for a website with instruction which costs and ADDITIONAL $35.00 to access. The title is incorrect and misleading, and making this transaction with Amazon has been disappointing. I will think twice before I make another.Finite Mathematics (9th Edition) (MathXL Tutorials on CD Series)

4 comments about The Pythagorean Theorem: A 4,000-Year History.

1. The Pythagorean Theorem could rightfully be called the 'Crown Jewel of Mathematics'. For from its truths and intellectual spawn come all the wonders of our modern word--high rises, automobiles, cell phones, interplanetary probes, you name it! Unfortunately, the last serious book on this subject was written over 80 years ago by an Ohio school teacher, Elisha Loomis. Enter Dr. Eli Maor! He has written an absolutely marvelous book about 'The Crown Jewel' that will captivate anyone with a good high school mathematics background. Read it and behold a wonder!



2. Eli Maor is a fine mathematician who has produced some wonderful books on math topics for a general--well, let me say, educated--readership. His book, Trigonometric Delights, is my favorite. It is very interesting and engaging. I want to say "for an educated reader" again, though it seems rather redundant. Why would anyone who didn't know anything about trig and have an interest in the subject even bother to pick up the book? Still, as someone who spent more than ten years in high school math classrooms, I also found his work useful to interest and inspire my students (and myself).
   
   Since the class I taught most often was geometry, I was very happy to see this book on the Pythagorean theorem. I have to admit, as an avid reader on the subject, I was familiar with much of what's here; particularly, the historical development and the more "Euclidean" applications of the theorem. On the other hand, he developed some proofs and problems I hadn't seen before which I found quite interesting.
   
   Overall, however, I didn't find this book quite as engaging as some of his other work. I got the feeling he started off wanted to write a book that would have more universal appeal than some of his other titles. I mean, after all, nearly everyone knows what the Pythagorean theorem is, or has at least heard of it. But there wasn't nearly enough of the "simple" stuff and the last half of the book really goes quite far afield into mathematics without which someone without a pretty decent background in the subject will have a difficult time; particularly since the development is rather sparse in what feels like an aborted effort to keep things simple. Even some of the earlier demonstrations and proofs are a bit difficult if you don't have the background in Greek mathematics which, unfortunately, is often lacking these days.
   
   Still, as someone who loves geometry and has a pretty good background in it, I found much here to like. Any reader who feels confident in their mathematical ability will probably find much here to like too.



3. I loved e: the story of a number, both the story and the mathematics in it. But for some reason this book does not catch the same spirit. It doesn't have the exciting thread of a story that makes you want to turn to the next page, and the many different proofs make it feel like it's a patchwork of items forcing itself to support the topic rather than a natural inspiring thread that helps you see the growth in the mathematics. I found it disappointing.



4. XXXXX
   
   "To this day, the theorem of [Greek mathematician] Pythagoras [which states that the square of a right-angled triangle's longest side or hypotenuse is equal to the sum of the squares of the other two sides, written in the language of mathematics as (c^2 = a^2 + b^2) or, more commonly, (a^2 + b^2 = c^2)] remains the most important single theorem in the whole of mathematics. That seems like a bold and extraordinary thing to say, yet it is not extravagant; because what Pythagoras established is a fundamental characterization of the space in which we move, and it is the first time that it is translated to numbers...In fact, the numbers that compose right-angled triangles [called Pythagorean Triples such as (3,4,5), (28, 45, 53) and (65, 72, 97)] have been proposed as messages which we might send out to planets in other star systems a test for the existence of rational life there."
   
   The above quotation is found in this fascinating book authored by history of mathematics professor and author Eli Maor. (Note that the above quotation was not said by Maor.) It catches the importance of this deceptively simple theorem, a theorem children's writer Lewis Carroll (who was also a mathematician) called "dazzlingly beautiful."
   
   What did I learn from this book? Answer: there's a lot more to the Pythagorean theorem than (a^2 + b^2 = c^2)!! Maor may be the first author who has examined all the mathematics, history of mathematics, and physics books and collected just the material directly and indirectly related to the Pythagorean theorem.
   
   The result is that Maor has brought the long history of the Pythagorean theorem back to life. Sometime around 570 BCE Pythagoras proved (notice I said "proved" and not "discovered") a theorem about right triangles that made his name immortal. He also pondered the workings of the universe and tried to relate its workings to the laws of musical harmony. In the subsequent centuries, this theorem was used and developed by others such that it has become central to almost every branch of science, pure or applied. After twenty-five centuries, this theorem was expanded and thrust into four-dimensional space-time by Albert Einstein to formulate his own picture of the universe.
   
   Yes, there is simple mathematics in this book. To understand it, all you will need is some high school algebra and geometry and a bit of elementary calculus.
   
   Do you have to follow the mathematics found in this book? NO. Personally, I found that you could skim, even skip the mathematical parts and still not lose the essential flow of the main narrative. (Actually, the more difficult mathematics is relegated to the book's appendices.)
   
   Throughout the book are diagrams and even some pictures to enhance its main narrative. As well, there are eight pages of colour photographs found near the book's center.
   
   A feature of this book is that it contains "sidebars." These are brief sections (there are ten) found at the end of some chapters that usually focus on some aspect of the Pythagorean theorem. My two favourites have the following titles: "The Pythagorean Theorem in Art, Poetry, and Prose" and "Four Pythagorean Brainteasers." You don't have to read each sidebar.
   
   Another feature of this book is its chronology. It more or less summarizes the main events in this book in chronological order. This chronology begins in the year 1800 BCE and ends in the year 1996.
   
   Finally, a note on the book's cover picture (displayed above by Amazon). It shows the detail or "zooming in" of a beautiful larger 1649 picture called "Allegory of Geometry" by artist Laurent de la Hyre (displayed on this book's inside back flap). The book's cover picture zooms in on several geometric figures, the one on the top left showing Euclid's proof of the Pythagorean theorem.
   
   In conclusion, this book is essential for anyone that wants to be familiar with the four thousand year history of the Pythagorean theorem. I leave you with some actual lines from Gilbert and Sullivan's "Pirates of Penzance:"
   
   "I'm very well acquainted, too, with matters mathematical,
   I understand equations, both simple and quadratic,
   About Binomial Theorem I'm teeming with a lot o'news,
   With many cheerful facts about the square of the hypotenuse."
   
   (first published 2007; list of colour plates; preface; prologue; 16 chapters; epilogue; main narrative 215 pages; 8 appendixes; chronology; bibliography; illustrations credits; index)
   
   <>
   
   XXXXX

3 comments about An Introduction to Manifolds (Universitext).

1. 
   i think there is a jump from ugrad analysis/alg/top etc to early grad school concepts. i didnt know category theory, i only had the flimsiest notion of a manifold, etc etc. and this book fills in that jump wonderfully. it does the right mix of analysis-differential topology-topology so that you can go read a book like bott and tu later (that's what it was designed for).
   so im having a good time with it.



2. This is an excellent book. I wish that more books on advanced mathematics were written in this style. In contrast to most books on manifolds that tend to be very difficult for beginners to follow, Prof. Tu has made every effort to make this subject understandable to the nonexpert.
   
   Greg Chirikjian
   Professor, Mechanical Engineering
   Johns Hopkins University



3. This is my favorite book on Differentiable Manifolds. After reading this book the reader will obtain a solid background on the following essential notions: Charts and atlas of a manifold; tangent vectors (as derivations); differential of a smooth function between manifolds; submanifolds and embeddings; quotient spaces; partitions of unity; vector fields; vector bundles; differential forms and de Rham cohomology. And on the road, the reader gets a gentle exposure to Lie groups, Lie algebras; and some basic notion of Category and Functors.
   
   I found the following aspects of the book especially attractive:
   > Clear style of writing: The author is the coauthor of the acclaimed "Differential Forms in Algebraic Topology". See the comments for that book. The clarity has not decreased at all.
   > Bite-sized sections: The materials contained in each section is approximately equal to that of a 50-minute lecture. This helps readers who plan self-study.
   > Right amount of topics: This is not an encyclopedia on manifolds. However, it does contain the ``absolute must'' one should know about manifolds. And it does such a good job in presenting it, the reader will be left with a solid understanding on those essential topics.
   
   I first read this book as a Physics student and had no trouble reading it. I later switched discipline to Mathematics, and I know that this book has helped me appreciate the beauty of Mathematics. I thank the author for writing such an wonderful book.