Product Description
"Here is the story of algebra."With this deceptively simple introduction, we begin our journey.Flanked by formulae, shadowed by roots and radicals, but escorted by an expert who navigates unerringly on our behalf, we are guaranteed safe passage through even the most treacherous mathematical terrain.

Our first encounter with algebraic arithmetic takes us back thirty-eight centuries to the time of Abraham and Isaac, Jacob and Joseph, Ur and Haran, Sodom and Gomorrah.Moving deftly from Abel’s proof to the higher levels of abstraction developed by Galois, we are eventually introduced to what algebraists have been focusing on during the last century.

As we travel the ages, it becomes apparent that the invention of algebra was more than the start of a specific discipline of mathematics – it was also the birth of a new way of thinking that clarified both basic numeric concepts as well as our perception of the world around us.Algebraists broke new ground when they discarded the simple search for solutions to equations and concentrated instead on abstract groups.This dramatic shift in thinking revolutionized mathematics.

Written for those among us who are unencumbered by a fear of formulae, Unknown Quantity delivers on its promise to present a history of algebra.Astonishing in its bold presentation of the math and graced with narrative authority, our journey through the world of algebra is at once intellectually satisfying and pleasantly challenging. ... Read more

Customer Reviews (38)

excellent work in need of a layman's touch
This is allegedly written for the educated layman but I know few who could grasp (or care about) field vs group theory, rings, tensors or manifolds much less "movements" like "Intuitionism" and "Constructivism". The author thankfully does not attempt a deep inquiry into every crevice of modern math but instead presents a breezy if sometimes obtuse overview.

Even mathematicians can be informed here, particularly on the personalities.The author attempts to stay within the bounds of algebra while explaining that various disciplines necessarily cross lines with geometry, topology other algebras.The author thankfully includes illustrations although many more are required.Many times, it was next to impossible trying to picture the prose.Then again, much of the latter part of the book is conceptual without basis in reality - parallel lines that meet, many dimensions (presented in three, of course), complex numbers, vectors, etc. Many times, the creation of the mathematical concept (matrix, field permutation, etc) is as astounding as the principles by which they are solved. My Grade - B+

A nice book on the history of algebra
This book is a relatively easy to read history of algebra, togehter with short introductions to some topics in algebra that a layman might never have encountered (or needs to review).

The author gives good concrete examples of concepts such as groups, rings and fields, so that one can get an idea of what these ideas mean and why they are important.So I think it's a good introduction to some of abstract algebra; however, to understand that subject well would require more in-depth study than one could get from this book.

The narratives about prominent mathematicians are interesting and well-written too. (They also provide a sometimes-needed break from reading the more technical material.) The author also does a good job of showing the historical context for the development of certain ideas in mathematics.

Toward the end of the book, I found myself sometimes losing track of what was going on - the concepts get more and more abstract, and the explanations seem to get shorter.

In summary: this book has enough technical material so that one not familiar with the subject can learn something.I think much of it would be understandable to a motivated reader who didn't have any math background beyond what is taught in high-school algebra.

Abstraction not quite brought down to earth
Don't get me wrong, Derbyshire makes a lot of VERY abstract ideas a lot more accessible to the non-specialist reader than he/she should reasonably expect.Even so, he takes a lot more for granted this time round than he did in his last book (on prime numbers and the Riemann Hypothesis).By the end of that book, I felt that I really understood (albeit in a not especially technical way) a very difficult area of mathematical thought.Granted, UQ has as it's focus the entire mathematical branch of algebra over its 3,000 year history.I should not have expected a thorough explication of mathematical details along the way.As a survey, UQ certainly whetted my appetite to further read and explore in the field of algebra.And it was packed with fascinating information.Less one star only because JD had set such a high bar with his last book.This one doesn't quite reach it.

Group theory for beginners
This is simply brilliant.However, it is also different from John Derbyshire's other book "Prime Obsession".It is more abstract and I myself had to read in a second time to understand the concepts fully.But what a reward!Once the reader comprehends his deliberation on groups and invariance, he or she can enbark on other journeys in understanding the nature of physics (and by corollary the universe itself).So in summary, it is as clear as it can be, and as engaging as Derbyshire has been.

Good explanations + history
The historical attributions balance & complement the algebraic examples beautifully.Fun to read. ... Read more

Product Description
Mathematics Across Cultures: A History of Non-WesternMathematics consists of essays dealing with the mathematicalknowledge and beliefs of cultures outside the United States andEurope. In addition to articles surveying Islamic, Chinese, NativeAmerican, Aboriginal Australian, Inca, Egyptian, and Africanmathematics, among others, the book includes essays on Rationality,Logic and Mathematics, and the transfer of knowledge from East toWest. The essays address the connections between science and cultureand relate the mathematical practices to the cultures which producedthem. Each essay is well illustrated and contains an extensivebibliography. Because the geographic range is global, the book fills agap in both the history of science and in cultural studies. It shouldfind a place on the bookshelves of advanced undergraduate students,graduate students, and scholars, as well as in libraries serving thosegroups. ... Read more

Product Description
In 1859, Bernhard Riemann, a little-known thirty-two year old mathematician, made a hypothesis while presenting a paper to the Berlin Academy titled"On the Number of Prime Numbers Less Than a Given Quantity."Today, after 150 years of careful research and exhaustive study, the Riemann Hyphothesis remains unsolved, with a one-million-dollar prize earmarked for the first person to conquer it.

Alternating passages of extraordinarily lucid mathematical expositionwith chapters of elegantly composed biography and history, PrimeObsession is a fascinating and fluent account of an epicmathematical mystery that continues to challenge and excite the world.Amazon.com Review
Bernhard Riemann was an underdog of sorts, a malnourished son of a parson who grew up to be the author of one of mathematics' greatest problems. In Prime Obsession, John Derbyshire deals brilliantly with both Riemann's life and that problem:proof of the conjecture, "All non-trivial zeros of the zeta function have real part one-half." Though the statement itself passes as nonsense to anyone but a mathematician, Derbyshire walks readers through the decades of reasoning that led to the Riemann Hypothesis in such a way as to clear it up perfectly. Riemann himself never proved the statement, and it remains unsolved to this day. Prime Obsession offers alternating chapters of step-by-step math and a history of 19th-century European intellectual life, letting readers take a breather between chunks of well-written information. Derbyshire's style is accessible but not dumbed-down, thorough but not heavy-handed. This is among the best popular treatments of an obscure mathematical idea, inviting readers to explore the theory without insisting on page after page of formulae.

Customer Reviews (90)

entertaining and informative
This is a very well-written account of the most famous unsolved problem in mathematics, the Riemann Hypothesis. The author has organized it into odd-numbered chapters which have most of the math (targeted at the reader with only a high-school level background in math and no calculus background), and even-numbered chapters which have most of the history. The history is interesting enough that the math chapters could be skipped if the reader is so inclined. The math chapters are as gentle as possible given the formidable nature of the topic so many readers will find them readable. Somewhat more ambitious readers might find this book a good stepping-off point for further investigations, but be forewarned that truly comprehending this material will require a solid grounding in calculus and complex analysis, so if your background is only high-school math you probably have a minimum of two years equivalent of college math to get through first.

The Riemann Hypothesis relates to number theory, specifically the distribution of prime numbers, which is fairly understandable at the elementary level to the non-mathematically trained, so even without understanding the math behind the Hypothesis itself, most readers can still come away with an understanding of what the Hypothesis is trying to say and why it's so important.

What I found most interesting is the way in which seemingly completely unrelated areas of math and science have been found to have connections with the Hypothesis. This is true of a lot of mathematical propositions, and is perhaps the most mysterious aspect of math in general. As Shakespeare once wrote, "there's more in heaven and earth than is dreamt of in your philosophy, Horatio!" Lots more!

One minor correction: on page 54 the author expresses his awe at the ability of the great mathematician Gauss to extract the primes from blocks of 1000 integers in 15 minutes, using only a pencil, paper, and a list of primes up to 829. In fact, according to Havil in his excellent book "Gamma, Exploring Euler's Constant", on p. 174, Gauss began this task at age 15 after receiving a table of logarithms which had appended a list of primes up to one million. So what he was actually doing in those 15 minutes was just adding up the number of primes listed in the table for blocks of 1000 integers, a task even ordinary mortals could have easily acomplished.

History of a mystery
Prime numbers hold an unmistakable appeal. The basic concept seems so simple - once you have the idea of multiplying some numbers together to make others, the question seems natural: which numbers can not be formed by multiplying others? Those are the primes.

So, starting from such a simple statement, it's quite amazing that so much subtlety inheres in the primes, and that they occur with such a maddening mix of predictability and randomness, and that they materialize in places so seemingly unrelated to their definition. This book traces one thread of that inquiry into the primes, the Riemann Hypothesis. With a little mathematical magic, the primes transform into function with wild behavior - but with a startling regularity of its own. At least, it seems regular, but "seeming" isn't good enough for mathematicians. It must be proven. Despite a hundred and fifty years of attack, the Hypothesis remains firmly defended against any effort at explanation. Derbyshire traces the history of that attack, and gives us glimpses into the baffling personalities of those attackers.

Although I enjoyed this book, it might not hold great appeal unless the math itself holds your interest. There's a balance to maintain when tracing the history of a phenomenon like this, between the personalities of those seeking out this mathematical beast and the beast itself. Derbyshire includes plenty of math, despite his efforts to keep this popular. I could have wished for a bit more of the human side of the quest, and I'm sure other readers will wish the same. Even so, it makes a fascinating story and a clear statement of the problem that has become such a consuming passion for so many of mathematics' finest minds.

- wiredweird

Math's Greatest Mystery for Everyone
I just started reading this book and I literally can't put it down. The book presents in a relatively simple language the hypothesis that has plagued mathematicians for such a long time. The author explains crucial math concepts like infinity, limit and function for people whose math background is not strong. The alternation of chapters is very appropriate -- you read the math first, and then the historical context of that math. For example, I never really thought of the fact that Gauss counted primes using... a paper and a pencil (well, there were no computers in his time); this made me appreciate the math even more.

Personally, I would recommend this book to anyone who is interested in math, no matter how strong his/her background is.

Becoming obsessed
I am becoming obsessed with the Riemann Hypothesis. Better, I'm obsessed to, at least, trying to understand what it is about. After peeking up some books, some more mathematically inclined then others I can surely say that this was, by far, the one that gave me a better insight. It is very delightful to read a book such as this one.

Kindled an Obsession in Me
I browsed a bookstore in 2007 and bought it after a skimming. Since then, I have read and re-read it many times. I vowed to some day understand everything in it. I bought a copy of Maple and recently of Mathematica, so I could reproduce every mathematical expression in the book and then some. My first victory was when I reproduced "The Derbyshire Spiral", as I began to call the logo on the cover. I have now reproduced Derbyshire's elucidation of the random Hermitian matrix and its strange behavior, as well as most of the other math in the book.

I could go on describing other adventures in math that were inspired by this book. Having flunked as many math courses as I ever passed,long ago,the clear presentations by Derbyshire were a pleasant surprise! The historical context makes for highly interesting reading as well whenever the math gets to be too much.

If the idea of prime numbers has any attraction to a casual reader at all, this book is a must buy.

Mauri Pelto ... Read more

Product Description

The Language of Mathematics is Devlin's second iteration of theapproach he used in Mathematics: The Science ofPatterns. It covers all the same ground (and uses many of thesame words) as the latter, but with fewer glossy pictures, sidebars,and references. Devlin has also added chapters on statistics and onmathematical patterns in nature. --Mary Ellen Curtin ... Read more

Customer Reviews (24)

To see the world in a grain of sand.
Devlin's language of mathematics is an outstanding book "about" mathematics. It's best suited to lay audiences without any background in mathematics, I particularly disagree with other reviews which claimed that this work require any higher understanding of mathematics. In fact, if you are one of the people (including myself) who have not done any math or took a class or read a book relating to math since secondary school, this is a good place for you to start. I'm a lawyer who have spent much of a lifetime as a math-loathing individual. My mathematics education ceased 13 years ago when I was 17. But thanks to this book, with its blend of engaging narative history as well as concise-but-sharp explanations on the concepts of mathematics, I am now in love with mathematics for the first time in my life.

This book needs not be read from cover to cover. Every discussion can stand on their own, and you are free to explore the content as your curiosity may lead. The first two chapters about what mathematical endeavor has been to manknid is brief but effective. I especially like the chapter on the mathematical induction; the subject was presented very clear yet powerful. The power of that simple exposition awakened something in me. I can only regret that I wasn't exposed to something like this when I was in school. It would have changed my life.

However, if you want a proper mathematics education. This book is not up to that task. It is written only to educate lay people or just to whet an appetite for more serious treatment of math. In short, this is a book "about" mathematics, not a mathematics book. Therefore if you have a serious goal in reeducating yourself about math, I recommend "What is mathematics?" by Courant (it has got to be one of the best math books written in the past 60 years). Also books written by Prof John Stillwell definitely worth checking out. Stillwell wrote many books on fundamental concepts of mathematics without requiring deeper background than highschool mathematics.

An excellent way to begin the study of the universal language of the universe
It has long been argued that to understand the universe, it is first necessary to understand the language used to describe it, namely mathematics. Devlin opens with that position and then proceeds to give a historical account of many areas of mathematics and how they are used to describe the actions of the physical world. Devlin does an excellent job in explaining the mathematics at a level of basic understanding, although to his credit he does not hesitate to include formulas and equations when they are necessary.
Some of the topics covered are:

*) Formal grammars
*) Number theory
*) Conic sections
*) Logical reasoning
*) The fundamental geometries
*) Basic topology
*) Fundamental probability

In each section, Devlin's goal is to explain the material well enough so that the reader can understand and appreciate the value that it brings to the mathematical world. He is very successful in this endeavor, developing the topic in sufficient yet with no unnecessary detail so that the bright and dedicated reader will be able to appreciate the overall beauty and utility of mathematics.

A Beautiful Read
A beautiful read. And I mean beautiful. As I read I could envision see the beauty in the patterns being described. It is both history and description. I would definitely recommend this book.

Refreshing my love of math
I'm in my early 30's, got an undergraduate math degree, and have been experiencing a renewal of my interest in math.I found this book at the perfect time.

The negative reviews on this book are true.I can't imagine more than small parts of chapters being interesting to someone who hasn't already been exposed to university level maths.The symmetry and topology chapters were very entertaining, but I can imagine them being a little opaque for someone without prior study of these topics (or who hadn't at least tried to tile their bathroom).

If you're in my position, having studied math, forgot about it for a couple of years, and are now renewing your interest in it, I highly recommend this book.If you've never studied, but are interested in learning about math, find an easier introduction and save this for your 3rd or fourth book, when you're ready to see how math helps us understand abstract ideas even before they have real world application.

What is Math?
If you ever really wondered whats behind all of the numbers this is thebook that you want. Very easy to read and explains everything you wanted to know about Math. ... Read more

Product Description
In 1977, the Mathematics Department at the University of California, Berkeley, instituted a written examination as one of the first major requirements toward the Ph.D. degree in Mathematics. Its purpose was to determine whether first-year students in the Ph.D. program had successfully mastered basic mathematics in order to continue in the program with the likelihood of success. Since its inception, the exam has become a major hurdle to overcome in the pursuit of the degree.

The purpose of this book is to publicize the material and aid in the preparation for the examination during the undergraduate years since a) students are already deeply involved with the material and b) they will be prepared to take the exam within the first month of the graduate program rather than in the middle or end of the first year. The book is a compilation of approximately nine hundred problems which have appeared on the preliminary exams in Berkeley over the last twenty years. It is an invaluable source of problems and solutions for every mathematics student who plans to enter a Ph.D. program. Students who work through this book will develop problem solving skills in areas such as real analysis, multivariable calculus, differential equations, metric spaces, complex analysis, algebra, and linear algebra. The problems are organized by subject and ordered in an increasing level of difficulty. Tags with the exact exam year provide the opportunity to rehearse complete examinations. The appendix includes instructions on accessing electronic versions of the exams as well as a syllabus, statistics of passing scores, and a Bibliography used throughout the solutions. This new edition contains approximately 120 new problems and 200 new solutions. It is an ideal means for students to strengthen their foundation in basic mathematics and to prepare for graduate studies. ... Read more

Customer Reviews (5)

Excellent Problem Book
The problems in this book are excellent, they are both entertaining and instructive.I thought I knew calculus, linear algebra, and all of the other typical undergraduate subjects very well, until I purchased this book.After working several problems, mostly without success, I realized that there is a big difference between knowing theorems and knowing how to use them.Since then I have worked these problems daily to improve my "working knowledge," and it has made me a much better mathematician.Learning the definitions and theorems is just the first stage of mathematical knowledge.In this form your knowledge is simply something stored in memory.In the second stage, you must turn it into something more like "software," something that is an active part of your thinking.The only way to do this is by solving problems, and for undergraduate mathematics, this is probably the best book of problems you will find.Highly recommended.

Excellent Problems!
These are great problems for those who would like to review undergraduate mathematics or those who would like to try some challenging problems. They are not as difficult as the problems on the Putnam competitions or those in the Math Monthly , but many require a bit of thought and some ingenuity. Some of the problems are routine, and if you don't want to review the basics, you can skip those and just try the more difficult ones. Even experienced problem solvers will have fun with some of these! Anyone who teaches undergraduate mathematics should have this collection. Highly recommended.

A real pearl!
This book is a rare peak inside one of the best Ph.D. programs in Mathematics in the world. It allows you to try out and test yourself on the same problems that the best young and aspiring mathematicians are testing themselves.

The problems are neatly arranged by subject and in increasing level of difficulty, and the solutions, are not only beautifully
written, but somewhat surprising and unexpected for a seasoned student. I pull mine out of the shelf on the rainy days and try a few more, and when I get one, I really savour it!

Challenging problems, or just standard textbook stuff.
The problem with the contents of this book is that they are mostly standard, or require a simple trick. Many problems also require quite a bit of knowledge in higher mathematics, which may be good if you want to test that.

However, this book is not for problem solving enthusiasts.

TESTYOURPREPAREDNESS!
A GOOD COLLECTION OF PROBLEMS THAT HAVE BEEN THE BENCH MARK OF SELECTION TO THE PhD PROGRAMME AT UNIVERSITY OF CALIFORNIA AT BERKLEY IN MATHEMATICS. THERE IS A WIDE RANGE OF PROBLEMS THAT COMPRISE SOME SIMPLE AND SOME HARDERNUTS TO CRACK.THE PROBLEMS HAVE BEEN PUT IN THE ORDER IN WHICH THEY HAVEAPPEARED IN THE EXAMS. NOT ALL QUESTIONS FIND SOLUTIONS IN THE BOOK HENCETHIS KEEPS UP THE SPIRIT OF THE EXAMINATION BUT THE SOLVED ONES SHALL HELPTO VENT FRUSTARTION OR OVER COME FALIURE AT TIMES. THIS BOOK IS A MUST READFOR STUDENTS WHO ARE PLANNING TO TAKE UP A PhD IN MATHEMATICS TO GUAGETHEIR AREAS OF STRENGTH AND WEAKNESS.A VERY WELL COMPILED SET THAT COMESYET AGAIN WITH THE GUIDENCE OF P.R.HALMOS. ... Read more

Product Description
From creating the pyramids to exploring infinity, this pocket size book traces humankind's greatest achievements through the towering mathematical intellects of the past 4,000 years to where we stand today. ... Read more

Customer Reviews (2)

Highly illustrated general survey
This is a highly illustrated, 202 page, survey of the story of mathematics.It is divided into 9 chapters covering the development of mathematics according to specific topics, such as: numbers, geometry, algebra, the development of analytic geometry and calculus, the geometry of complex shapes (topology) and set theory.These topics are roughly chronological, tracing the development of mathematics from ancient times to the present.Given that the book contains only 202 pages of text and much of this is taken up will illustrations, the treatment of the material is, of necessity, quite cursory.

What I liked - The book gives a reasonable treatment of many subjects - not detailed, but reasonable.The illustrations generally support the text and there are many inserts that provide biographical information or additional specific information on a given topic.There was sufficient information to allow one to search out more on the Internet or from another, more detailed, book.I particularly liked the chapter that covered the development of calculus.It clearly showed that Newton did not develop it from scratch, but built on the firm foundation of the work of others, some of who came very close to finishing the job before him.

What I did not like - There is relatively little mathematical development (as opposed to the historical aspects) and I have a feeling that if someone knew little or nothing about a subject that they would be lost.This certainly was my feeling about the section about set theory.This section did not really tell me much about the subject or why I should care about it.I also found several errors, which made me feel that I had only scratched the surface of the potential flaws in the book.For instance, Julius was not the proper name of Caesar, it was his clan name (he was from the clan Julii; sub clan Caesar).His proper name (or praenomen) was Gaius.This is a trivial point to be sure, one that has nothing to do with mathematics, but one that points to the possibility that there are other lapses in the author's research.(The author is not a mathematician, but has a background in medieval English literature.Were she a mathematician, I would have been much more likely to overlook an error of this sort, which is non-mathematical.)I also found that there was a lack of completeness in some discussions that could mislead one with little mathematical background.For instance, the author includes the old deductive "proof" that 2=1, without pointing out the error (multiplying both sides of an equation by zero) that makes this possible.If this book is given to a mathematical neophyte this discussion is apt to lead to confusion and frustration.

I would recommend this book to a high school student, providing that there was some supervision to help the student over the rough spots.It is also suitable, with the same caveat, for a more advanced student interesting in the history and development of mathematics.I would not, however, recommend this to someone interested in a more detailed presentation.For them, I would recommend Derbyshire's "Unknown Quantity" or Kline's Mathematics for the "Non-mathematician".

Fantastic!
It is my son's book of the month right now. He is homeschooled 7th grader and this is his absolute favorite book right now, he reads it between classes and cannot get enough. ... Read more

Product Description
Beautifully Illustrated Activities Highlight Mathematical HistoryThese blackline activity masters highlight important achievements in the history of mathematics, from our earliest counting systems to modern developments in chaos theory. The book is beautifully illustrated with historical art. Its engaging vignettes introduce concepts, events, and influential mathematicians and show the contributions of the world's many cultures to the development of mathematics. ... Read more

Customer Reviews (4)

Enjoyable Math History
This book is a valuable resource for people, inclucing students, who want to know more about the people who created the math we know.Interesting reading.

Wonderful book of Math History!!
This is a great book to use in the classroom and provides great information on the history of math concepts.I use it in my middle school classes and the kids love it!

Multicultural Mathematics
I bought this book for my godson to help him with his study of mathematics. I can't say enough good things about it! I am a former math major and have read extensively in the history of math but this book had information that even I didn't know. It tells of mathematics as it has been done among all the major world civilizations: Europe, India, the Islamic countries, China, ancient America. As my godson's mother is Chinese, the sections on China will also help him appreciate his inheritance through her. (She's also teaching him Chinese.)

Fantastic resource for all math teachers.
This book offers practical ways to enrich any curriculum with historical perspectives.Each "unit" is a one page introduction to a person or concept from the history of mathematics.Each unit includes questions/projects for students to try.This book makes it easy to introduce the history and humanistic side of mathematics to high school students.It would also make a great supplement to a college course geared towards pre-service teachers. ... Read more

Product Description
Sweeping, monumental study traces the history of mechanical principles chronologically from their earliest roots in antiquity through the Middle Ages to the revolutions in relativistic mechanics and wave and quantum mechanics of the early 20th century. Contributions of ancient Greeks, Leonardo, Galileo, Kepler, Lagrange, many other important figures. 116 black-and-white illustrations.
... Read more

Customer Reviews (4)

When in doubt, give credit to the Frenchman
Dugas has created a beautiful history of mechanics. The book has 450 pages of "classical" mechanics and 200 pages of "modern" mechanics (up to 1950ish), with the dividing line being about the year 1900. Building on the works of Mach, Duhem, and Jouget, we follow from Aristotle on up through the modern schools of thought. We see how the scientists switch from qualitatively describing physics into the quantitative and mathematical treatments we still use today. We see how concepts evolve, namely from impetus to momentum, and the distinctions between force, work, and energy are made. The focus is on mechanics in general; any emphasis on fluids or solids comes out of context alone. The tone is a little formal and boring, but any student of mechanics- physicist or engineer- would do well to read this book and learn about the nonlinear and interwoven history of the most elegant of subjects.

Now, there is no doubt that mechanics is the product of western Europe, but there is an incredible French bias in this book. This shouldn't be surprising, knowing that the French STILL cannot get past the fact that Doppler unquestionably deserves priority over Fizeau for the effect that bearshis name. I don't mean to underemphasize the contributions of l'academie, but French scientists are given several pages of explanations...and my what geniuses they were, with their clever and unquestionably perfect experiments (slight sarcasm). The majority of non-French scientists are merely discussed in passing. The chapter on Newton does not exactly hold the man in the most reverent of lights. Not that the Principia was perfect, but it WAS a sea change in the history of mechanics. On page 325, Dugas tries to attribute the general "F=d/dt(mv)" form of Newton's second law to Lazare Carnot, even though not 100 pages and 50 years earlier he makes the case that Euler has that priority. Hooke is mentioned once, and not for his most famous law related to mechanics. Leibniz's contributions (namely, the form of calculus we use today) are underappreciated. Coulomb is given priority over Stokes for the no-slip boundary condition. Lagrange is held as the pinnacle of classical mechanics, even though most of what he did in _Mechanique Analytique_ was reproducing and/or trivially extending what Euler did. Laplace is somehow given priority for special relativity. Dugas felt the need to underscore Einstein's own words in developing relativitly, namely that he build from induction rather than axioms. Nevermind the fact that Einstein was almost scooped by Hilbert, Dugas goes on and on about how Poincare deserves all the credit for general relativity. Even though most of quantum mechanics did not come out of France, de Broglie is held as the pinnacle of that theory. The list goes on and on and on.

For solids-specific history, try Timoshenko, and for fluids-specific history, try Tokaty. For something a little less biased (and, more correct), consider some of the works by the eminent C.A. Truesdell (books Essays on the History of Mechanics and Tragicomical History of Thermodynamics, and various articles in the scientific literature).

Necessary Reading For All Believers In God
This book is important today because it deals with the poorly developed concept of `impetus.'' Impetus is defined as the motion that bodies have acquired from a motive agency.Today, the nature of the motive agency divides nonbelievers from the believers.Believers say that God is the motive agency that produces the impetus in all bodies of the universe. On the other hand, nonbelievers say that the motive agency of all bodies in the universe is an infinitely dense physical thing called the Big Bang. So, the believers say that the motive agency is spiritual whereas the nonbelievers say that it is physical.

Dugas opens his book with Aristotle's thoughts on motion. Aristotle divides all motions into natural or violent.For instance, the motion of a stone in asling is natural.The motion of the stone becomes violent when the stone is released from the sling and becomes a projectile. Aristotle also says that the motive agency is always in contact with its moving bodies but is not a part of the moving bodies. Contact means that the motive agency and the moving bodies coexist.

Since Aristotle's universe has no vacuums, when a stone becomes violent and moves through air, for instance, the stone forces air out of the space it consumes and new air fills the empty space as the stone moves forward.Aristotle thought that the motion of the new air moves the stone forward until the projectile falls to the ground due to the attraction of gravity. As seen, Aristotle's motive agency is physical.

William of Ockham (1300-1350) was the first to reject Aristotle's theory of motion.But, he offers no alternative theory. Jean Buridan (1300-1358), a student of Ockham at the University of Paris, argues that the motive agency is spiritual.However, Buridan expected theologians to show how God acts as the motive agency of all bodies in the universe.

Theologians eventually showed that God is the motive agency of all moving bodies. First, Nicole Oresme (1329-1382), a mathematician and theologian, became the forerunner of the analytical geometry of Descartes when he argued that every measurable thing has a continuous quantity. Oresme's argument says that a motive agency, spiritual or physical, must coexist continuously with the motions it produces. This argument cannot be met with the Big Bang theory because the infinitely dense physical particle becomes nonexistent upon its explosion. On the other hand, God is permanent and is thus in continuous contact with all bodies in the universe.

Nicholas of Cusa (or Cues) was the first to show how God is in continuous contact with all things found in the universe.He shows this contact by developing a positive and negative science out of precise symbols... In this science, all opposites coincide in God.Thus, `what is not moving' and `what is moving' coexist continuously.Cusa's writings inspired the cosmologies of Copernicus and Kepler, Bruno's monads, Da Vinci's `forza,' the deism of Isaac Newton and the work of Leibniz.In Part II (Chapter Two), Leibniz distinguishes dead forces and living forces.The dead forces are expressed with the calculus and the equation d (m v2/2) = F ds.When this equation is integrated,living forces areborn from the dead forces. The integrating process forms thefundamental law,m dv/dt = F.

In a new book entitled 'The First Scientific Proof of God, I conclude that God creates bodies with spiritual atoms.The spiritual atoms are one in God, are immortal and are endless in number. At creation, the spiritual atoms become distinct, different and related.Their relations form one universe with many organized bodies (stones, water, bees, horses, plants, humans, etc).The spiritual atoms enter and leave bodies at different rates continuously. This is why all organized bodies age and die. The permanency (immortality) and the motions of the spiritual atoms are guided by a complex differential equation that only God can know. We can only gain better understandings of God's Intelligent Design.In my book, I show that it is natural for all bodies to change continuous (aging, etc).Thus I view a continuum of motion as a continuum of rests.Thus, one's death is only a rest in a continuous processes that includes reincarnation. God also rests, as Moses concluded in his creation theory.But, God's rests are wise changes..

Interestingly, The Notebooks of Leonardo Da Vinci by Dover Publications (1970) shows that Da Vinci protected his thoughts about Cusawith complex coding schemes. Thus, the truths in Dan Brown's book on The Da Vinci Code can questioned.Galileo also studied Cusa.But, Galileo did not secure his research results as Leonardo did.. Eventually, the Roman Church charged Galileo with heresy and was imprisoned, after he agreed with Cusa that the universe has no center body and that planet earth is not the center of the universe, as taught by the Church.

Excellent book
This book is a fascinating overview of the history of mechanics. One learns just how much was known about mechanics before the time of Galileo, and the facts are surprising. The work of Galileo and Newton was not a sudden leap in knowledge of mechanics, but grew out of work done in the middle ages and in ancient times. Here are just a few of the highly interesting historical facts that are expounded on in the book: 1. The work of Plato, who held that the weight of an object was the result of a force that was pushing it downwards. 2. The approach of Aristotle to mechanics: he was much more elaborate than Plato, and he made an intensive effort to understand the origin of motion fundamentally. The Aristotelian conception of motion and dynamics can be summarized as follows. There are two kinds of motion, natural and violent. Aristotle gives the falling body as an example of natural motion, whereas projectile motion is an example of violent motion. Violent motion has the property of being temporary, and objects undergoing violent motion will eventually undergo natural motion. Natural motion is in turn divided up into two classes: celestial motion and terrestrial motion. Objects that undergo celestial motion are characterized by what is now called uniform circular motion. On the other hand, terrestrial motion is rectilinear, and for Aristotle rectilinear motion is either straight up or straight down.An object needs a force and a resistance acting on it if it is to be in motion. Heavy bodies fall faster than light ones. Aristotle refused to admit the existence of a vacuum since such a void would not sustain the motion of an object. In addition, a total vacuum would offer no resistance to the motion of an object, and thus the object would, according to Aristotle, move off with increasing speed. In a vacuum he concluded that heavy bodies would move with the same speed as light ones, again unacceptable he argued. 3.In the sixth century, the philosopher John Philoponus gave what is probably the first systematic attack on the Aristotelian ideas of motion. He argued that the planetary motions, as confirmed by observation, are much more complex than the simple circular motion that Aristotle imputed to them. Philoponus also proposed the so-called "impetus theory" to explain projectile motion. As the name implies, according to this theory the thrower imparts an impressed force or impetus to the projectile which keeps it moving until the "natural" motion and the resistance of the medium takes over. The projectile then falls straight down. The power given to the projectile by the impetus will gradually damp out because of the resistance of the medium and these "natural tendencies" of the projectile. 4. A somewhat more axiomatic and mathematical approach to mechanics in the thirteenth century was proposed by Jordanus of Nemore. His work is interesting in that it contains a notion of virtual work, a concept that was really not developed extensively (and proven to be practically useful) until the nineteenth century. Jordanus also attempted to quantify the idea of acceleration, and in his writings he deduced that an object accelerates when, in a fixed interval of time,a greater amount of space is covered by the object, and so when the speed of the object increases. Jordanus discussed this in the context of falling objects, but he did not in his writings relate the distance the object falls in terms of the time it took the object to fall. As for the cause of the acceleration of the object, Jordanus held to the view that as the object descended, the object caused the air surrounding it to be less resistive, thereby causing the body to accelerate. Lastly, and most interestingly from the standpoint that it occured during the thirteenth century, an anomynous author of a work entitled Liber Jordani de ratione ponderis solved correctly the problem of the equilibrium of a heavy body on an inclined plane. 5. John I. Buridan of Bethune (1300 - 1358) developed a theory of the impetus which rejected the idea that air is the motive power for projectiles, Buridan took the impetus to reside permanently in the projectile, until the object is acted upon by some other forces. This belief of Buridan regarding the impetus is very important from a modern standpoint, because it is somewhat similar to the idea of inertia that was developed in the sixteenth and seventeenth centuries. In Buridan's view a heavy body would receive more of the impetus than a lighter one. Buridan gave as an example the difference in impetus gained from a heavy iron versus that of light wood. If two objects, one of wood and one of iron have the same volume and the same shape, then the iron object will be moved farther because it will have a greater impetus imparted to it.6. The ideas of Buridan were discussed by Albert of Saxony (1316 - 1390). He held basically the same beliefs as Buridan about impetus and projectile motion, free fall, etc. Albert concluded that the speed of a falling object increases as the distance of the fall.Most interesting is that he used the idea of impetus to treat the problem of a stone dropped through a hole in the Earth, and concluded that the motion of the stone would oscillate about the center of the Earth until the impetus in the stone was exhausted. 7. Noted work on mechanics in the fourteenth century was performed by logicians at Merton College. One individual of this school was William Heytesbury, who distinguished between the velocity of an object and the measure of how much the velocity is changing: the acceleration of the object. Heytesbury gave definitions of uniform velocity, uniform acceleration, and instantaneous velocity, but these were not correct from the standpoint of modern mechanics. He rejected the Aristotelian ideas on free fall, noting that a falling body travels three times as far in the second second of its fall as in the first.

Very good, but ....
This book is refered by the publishers as the best book on history of mechanics. Indeed, it is a very complete work, showing in cronological order all the developments and contributions made by a great number of scientists and mathematicians, many of them famous but including also many not-so-famous and even not known.The book is very well written. However, when I bought this book I was expecting something more informal, more enjoyable to read. The feeling I have is that the book is somewhat too formal, arid, like the common textbooks. Although, it is still a very fine, very good book, and I suggest that every physics student to read it. ... Read more

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A History of Mathematics: From Mesopotamia to Modernity covers the evolution of mathematics through time and across the major Eastern and Western civilizations.It begins in Babylon, then describes the trials and tribulations of the Greek mathematicians.The important, and often neglected, influence of both Chinese and Islamic mathematics is covered in detail, placing the description of early Western mathematics in a global context.The book concludes with modern mathematics, covering recent developments such as the advent of the computer, chaos theory, topology, mathematical physics, and the solution of Fermat's Last Theorem. Containing more than 100 illustrations and figures, this text, aimed at advanced undergraduates and postgraduates, addresses the methods and challenges associated with studying the history of mathematics.The reader is introduced to the leading figures in the history of mathematics (including Archimedes, Ptolemy, Qin Jiushao, al-Kashi, al-Khwasizmi, Galileo, Newton, Leibniz, Helmholtz, Hilbert, Alan Turing, and Andrew Wiles) and their fields.An extensive bibliography with cross-references to key texts will provide invaluable resource to students and exercises (with solutions) will stretch the more advanced reader. ... Read more

Customer Reviews (4)

Shockingly remarkable
Although the chapter topics follow the current model of history of mathematics text books (compare the table of contents Victor J. Katz's history of mathematics; notably similar), the text has a strength, depth, and honesty found all too seldom in a text book mathematical history.This is not the typical text-book on technical history that can be dismissed (as Victor J. Katz's should be) as "a pack of lies" with only "slight exageration" (to quote William Berkson's Fields of Force).

Also, the text is bold enough to quote and translate the actual and typical style of presentation used in Bourbaki meetings: "tu es demembere foutu Bourbaki" ("you are dismmembered [..]) [a telegram sent by Bourbaki group to Cartan, informing him that his book was accepted and would be published].Luke Hodgkin's text dispenses with the asterisk (see p.241).

A History of Mathematics
A slightly more descriptive title for this book would be On the History of Mathematics, because the book is not a chronology and detailed narrative of the development of mathematics over the course of human history, but rather a careful, questioning look at selected past moments in mathematics. It does not attempt to tell a comprehensive story of its subject, and in fact ponders at times how such a story should be told. The writing style is polished and reflective. The author often compares the methods, notation, meanings, and possible intentions of earlier mathematicians to those of our own, and contemplates what the differences might imply for our understanding of the texts. The book is a scholarly, thoughtful overview, and would work well as an introductory supplement to more comprehensive general histories of mathematics.

Hodgkin refers often throughout the text to Fauvel and Gray's The History of Mathematics: A Reader.

Brief Contents
Introduction
1.Babylonian mathematics
2.Greeks and 'origins'
3.Greeks, practical and theoretical
4.Chinese mathematics
5.Islam, neglect and discovery
6.Understanding the scientific revolution
7.The calculus
8.Geometry and space
9.Modernity and its anxieties
10.A chaotic end?
Conclusion
Bibliography
Index

"We have not, unfortunately, resisted the temptation to cover too wide a sweep, from Babylon in 2000 BCE to Princeton 10 years ago. We have, however, selected, leaving out (for example) Egypt, the Indian contribution aside from Kerala, and most of the European eighteenth and nineteenth centuries. Sometimes a chapter focuses on a culture, sometimes on a historical period, sometimes (the calculus) on a specific event or turning-point. At each stage our concern will be to raise questions, to consider how the various authorities address them, perhaps to give an opinion of our own, and certainly to prompt you for one.

"Accordingly, the emphasis falls sometimes on history itself, and sometimes on historiography: the study of what historians are doing." (4)

A historiography-geek history
Hodgkin is a historiography geek with no interest in writing a history of mathematics other than to nitpick about details. Basically, each chapter summarises the conventional story---usually rather scornfully, and too briefly for anyone to gain from it---and then dwells on a myriad of minuscule objections to this version raised by highly specialised historians and published (for a reason, I would say) in highly specialised journals. This piling up of obscure historiographical hypotheses rarely makes a coherent point, let alone does it contribute to any substantial understanding of the history of mathematics.

Refreshing math history
Mr. Hodgkin gives a great overview of the history of mathematics, the current state of historical arguments, and all the references (including websites) for further study.At 262 pages it is very readable - I was not looking for a ponderous work with every possible fact catalogued.His approach is refreshingly irreverent and even funny:
"10th century Damascus must surely have been unique as a place where copying the text of Euclid could earn you a living." and
"Perhaps rather than decrying the 'low level' of geometry present in Vitruvius's architecture, we should think about the fact that it was a Roman, rather than a Greek, who bothered to write such a treatise....We have different cultures (cohabiting in the same empire) with different ideas of what a book is for."
I have slogged my way through many math histories without learning half as much, and to be entertained as well is more than one hopes for in such a book. ... Read more

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This sequel to the popular title furthers a reader's appreciation of just how mathematics is connected to the everyday world--and how a grasp of essential concepts can enrich one's life in innumerable ways. This fresh and lively approach to mathematics--appealing even to those who are intimidated by the world of numbers--unlocks the pleasures, mysteries, and practical applications of hundreds of mathematical concepts. ... Read more

Customer Reviews (2)

Countless, clear examples of the math world around you
Many have sadly been led to believe that math is a cold, lifeless subject limited only to homework assignments and balancing your checkbook.Nothing could be further from the truth, and Pappas books show this.Her "More Joy of Mathematics" shows a vast amount of instances of where math shows up, some math history, and a few visual brain teasers.How are exponents involved in the forging that creates a powerful Samuri sword?How do the properties of an elipse make your car's headlights switch to high-beam?What math can be found in an ocean wave, the strength of a honeycomb pattern, or a nautilus shell?How is math vital to the contruction of musical instruments?Is zero really a "number", and where does the concept come from?What are some currently unsolved problems in mathematics?A total layman could understand most of the book, but to understand all the mini essays you might at least want to have knowledge of math at the high school level.

The book is a fast read, and fun to flip back and forth through, because each example is summarized in its own 1 or 2 page section, with illustrations.The same goes for "Joy of Mathematics" so you don't necessarily have to read that one first; they just contain different sets of examples.And don't think that all the good ideas were already taken for the first book -- "More Joy of Mathematics" is just as exciting to read.Plus it has a single index listing the topics from both this book and the previous one, so if you buy both it's easy to find the article you want by only looking it up once.Perfect gift for a math enthusiast at any level, and it may even covert a few "mathphobes".

A Book of Equivalent Quality to Its Predecessor
If you enjoyed Pappas' "The Joy of Mathematics," then you should love this addition to the set.This book, like its predecessor, contains adiverse collection of concise, insightful discussions about mathematicaltopics and how they relate to the observed world.It develops ideas withan elegant simplicity by providing the reader with copious amounts ofillustrations and diagrams.Pappas communicates mathematical ideas clearlyand, unlike some mathematicians, stresses their relation to the lives andexperiences of humans. She reveals the appealing aspects of the subject byexcluding the technical, logical deductions that most frequently discouragepeople from studying it.The variety of topics presented in the bookdisplays the versatility of mathematics and its relevance to humanknowledge. For students interested in exploring the meaning andsignificance of mathematics or for teachers lacking the necessary materialsto enlighten their students about these topics, this book is ideal. ... Read more

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The manuscript gives a coherent and detailed account of the theory of series in the eighteenth and early nineteenth centuries. It provides in one place an account of many results that are generally to be found - if at all - scattered throughout the historical and textbook literature. It presents the subject from the viewpoint of the mathematicians of the period, and is careful to distinguish earlier conceptions from ones that prevail today.

Product Description

Mathematics Elsewhere is a fascinating and important contribution to a global view of mathematics. Presenting mathematical ideas of peoples from a variety of small-scale and traditional cultures, it humanizes our view of mathematics and expands our conception of what is mathematical.

Through engaging examples of how particular societies structure time, reach decisions about the future, make models and maps, systematize relationships, and create intriguing figures, Marcia Ascher demonstrates that traditional cultures have mathematical ideas that are far more substantial and sophisticated than is generally acknowledged. Malagasy divination rituals, for example, rely on complex algebraic algorithms. And some cultures use calendars far more abstract and elegant than our own. Ascher also shows that certain concepts assumed to be universal--that time is a single progression, for instance, or that equality is a static relationship--are not. The Basque notion of equivalence, for example, is a dynamic and temporal one not adequately captured by the familiar equal sign. Other ideas taken to be the exclusive province of professionally trained Western mathematicians are, in fact, shared by people in many societies.

The ideas discussed come from geographically varied cultures, including the Borana and Malagasy of Africa, the Tongans and Marshall Islanders of Oceania, the Tamil of South India, the Basques of Western Europe, and the Balinese and Kodi of Indonesia.

This book belongs on the shelves of mathematicians, math students, and math educators, and in the hands of anyone interested in traditional societies or how people think. Illustrating how mathematical ideas play a vital role in diverse human endeavors from navigation to social interaction to religion, it offers--through the vehicle of mathematics--unique cultural encounters to any reader. ... Read more

Customer Reviews (4)

So-so
This book didn't contain the information on Native American Mathematics that I was looking for, but I thought I could use it anyway.The index is very limited.The book is dry and uninteresting.Try her other book, Ethnomathematics, instead.It is much better.

Mathematics is everywhere if you know how to look, Ascher does
Mathematics is found in many places and in many forms, you only have to look for it in the right way. Ascher does that in this book. In chapter 1, "The Logic of Divination" randomization processes used in Madagascar, the Caroline Islands of the Pacific and the west coast of Africa are described. The processes are described using Boolean algebra and modular arithmetic.
Accurate calendars are an important component of most cultures and the computations used to maintain calendars in many cultures are described in chapters 2 and 3. The cyclic nature of the passage of the days is described using modular arithmetic. Some of the cultures whose calendars are explained are the Jewish; a tribe living on the island of Sumba in Indonesia called the Kodi, the Mayans of Central America and the Trobriand Islanders of the coast of New Guinea.
Chapter four deals with the tactics used by the Polynesian people as they navigated thousands of miles across the sea from one Pacific island to another. Their use of stick charts describing the paths based on wave patterns is an interesting form of graph. Relationships are the topic of chapter five, in particular the cyclic, sequential and circular structure used by the Basque people. Each person has a nearest neighbor on the "left" and on the "right" and they interact most strongly with those people when it comes to giving and receiving aid during critical times such as the harvest. Other relationship structures covered are the complex relationships in the Tonga island chain and among the Borana people of Ethiopia.
The sixth and final chapter describes figures that Tamil people in India draw on their doorsteps using white powders. The designs are so complex that they are fractal in nature and computer scientists have used them as models to develop descriptive picture languages.
Ascher describes many uses of mathematics, from the Pacific Islands of Polynesia to the tribal cultures of Africa and many places in between. This is a fascinating book and one that teachers of comparative cultures should examine. The mathematics is not difficult; it is well within the level of understanding of anyone with knowledge of basic algebra. I found it so interesting that I am now considering talking to the sociology department about the possibility of team teaching an honors level course on the use of mathematics in so-called "primitive" cultures.

refreshing!
there are very few books on ethnomathematics
out there (another good one is Mathematics
Across Cultures, Selin (ed.))

This book has the plus of smooth and enjoyable
reading, WITHOUT wattering down in content

Advisable for teachers, historians, and, in
addition, persons interested in the epistemological
problems in science.

Plese keep on writing, Marcia!

Excellent!
This is an excellent book.It sounds like an odd premise for a book - look at ideas in 'other' cultures and see how these are in essence mathematical ideas (in the western sense).However, what the author has done has turn what could be 'worthy but dull' material into a fascinating read.If you teach math (school, college or university) you will find lots of great topics to illustrate your lectures.If you just like math then this is a good read.The author has a nice style too - very easy to read.I loved this book.If you have any interest in math ideas then you will too. ... Read more

Product Description
This is an OCR edition without illustrations or index. It may have numerous typos or missing text. However, purchasers can download a free scanned copy of the original rare book from GeneralBooksClub.com. You can also preview excerpts from the book there. Purchasers are also entitled to a free trial membership in the General Books Club where they can select from more than a million books without charge. Original Published by: University press in 1889 in 309 pages; Subjects: Mathematics; Education / Higher; Education / Teaching Methods & Materials / Mathematics; Mathematics / General; Mathematics / History & Philosophy; Mathematics / Study & Teaching; ... Read more

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Covers 6 aspects of graduate school mathematics: Algebra, Topology, Differential Geometry, Real Analysis, Complex Analysis & Partial Differential Equations.Contains a selection of more than 500 problems & solutions based on the Ph.D. qualifying test papers of a decade of influential universities in North America.Paper.DLC: Mathematics - Problems, exercises, etc. ... Read more

Customer Reviews (2)

An Solution Encyclopedia of Pure Mathematics
When I hold this book, I feel worry free on mathematical questions because this cover-all book could tell me where to find or how to find the answers with its more than 500 problems and solutions on six aspects of graduateschool mathematics: Algebra, Topology, Differential Geometry, RealAnalysis, Complex Analysis and Partial Differential Equations.it coversmore than I expected with its brief, straightforward, and clear analysisand solutions which are easy to be understood.

An Solution Encyclopedia of Pure Mathematics
When I hold this book, I feel worry free about mathematical questions because this imperial book could tell me where to find the solution or the direction to find the answer with its more than 500 problems and solutionson six aspects of graduate school mathematics: Algebra, Topology,Differential Geometry, Real Analysis, Complex Analysis and PartialDifferential Equations. It covers the most aspects in one book I have seen.It gives me brief, straightforward and clear solutions which are easy to beunderstood. ... Read more

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Most Comprehensive
I have dozens, perhaps hundreds but certainly dozens, of math history books and this is one of the most if not THE most comprehensive math history books I own.It seems the real history lies within the history of math. ... Read more

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An Episodic History of Mathematics delivers a series of snapshots of the history of mathematics from ancient times to the twentieth century. The intent is not to be an encyclopedic history of mathematics, but to give the reader a sense of mathematical culture and history. The book abounds with stories, and personalities play a strong role. The book will introduce readers to some of the genesis of mathematical ideas. Mathematical history is exciting and rewarding, and is a significant slice of the intellectual pie. A good education consists of learning different methods of discourse, and certainly mathematics is one of the most well-developed and important modes of discourse that we have.The focus in this text is on getting involved with mathematics and solving problems. Every chapter ends with a detailed problem set that will provide the student with many avenues for exploration and many new entrees into the subject. ... Read more

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This lucid presentation examines how mathematics shaped and was shaped by the course of human events. Authors Resnikoff and Wells explore the growth, development and far-reaching applications of trigonometry, navigation, cartography, logarithms, algebra, and calculus through ancient, medieval, post-Renaissance and modern times. Preface. Index. Bibliography. 203 black-and-white figures. 7 tables. 14 photos.
... Read more

Customer Reviews (2)

A few interesting applications
The first half of the book is a fairly interesting discussion of astronomy and its applications to cartography and navigation. Aristarchus calculated the distance to the moon and the sun from three easy measurements: the shadow of the earth is about two moon radii wide at the distance of the moon (observable at lunar eclipses, of course), both the moon and the sun subtend and angle of about half a degree when viewed from the earth, and we can calculate the ratio of the distance to the sun and the distance to the moon by measuring the angle between the sun and the moon at half moon. But to carry out these calculations we need to know values of trigonometric functions, so we go on to tackle this problem from first principles by deducing Aristarchus' approximation 1/60tan(y)/y where x>y applied to a 30-60-90 triangle. The section ends with an interesting exercise: "Could the 'size of the universe' have been reasonably estimated in early times if the earth had no moon?" Astronomical observations are also crucial for navigation, of course, and then we are led to map making. We study the usual Mercator projection (x=longitude, y=log(tan(latitude/2 + 45))), but unfortunately we cannot prove its key property of being conformal (instead the authors present a "slightly involved" proof that stereographic projection is conformal, which they say is a step in the right direction because then the rest can be done by complex function theory). It follows that a constant compass course will be a straight line in the Mercator projection; on the surface of the earth itself it will be a complicated "loxodromic" curve, although later we shall see that it becomes an equiangular spiral when projected onto the equatorial plane. After having seen that both astronomy and navigation calls for quite extensive calculations we look at how the invention of logarithms reduced this burden dramatically. Again there are some interesting exercises along the way, such as "Can a table of values of the function x^2/4 be used to convert division [and multiplication] to addition and/or subtraction?" (There are Babylonian clay tablets relevant in this context.) The second half of the book is a very dumb presentation of the calculus and a few applications. For instance, one supposed application is a better way of calculating logarithms through infinite series expansion. But the series expansion is derived from Taylor's theorem (which is unnecessary and historically backwards) and thus relies on appeal to the "formula" for the derivative of the logarithm function which is stated in a table without the least indication of proof. Even if the authors had told us that we can understand the logarithm as the integral of 1/x this would not have done us any good since the one insight on which all understanding of this and a million other things rest---the fundamental theorem of calculus---is simply stated without any indication of why it is true. Thus the useful series for the logarithm, which would have been very interesting to derive from first principles, is instead derived by three layers of appeal to authority.

Entertaining and puts math in context
This book is easy and entertaining to read. It puts mathematics in context and motivates to study mathematics. The book also teaches to respect the work of all those mathematicians that have made it possible for us to reach the current level of civilization. ... Read more

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Customer Reviews (6)

A book to treasure
I sense that David Ruelle wrote this book as a labor of love, and I feel priveleged to have been able to read it (as with his wonderful book Chance and Chaos).He provides a fairly penetrating and sophisticated treatment of the nature of mathematics and what it's like to be a research mathematician.His writing style is informal and friendly without sacrificing clarity, precision, and elegance.He doesn't shy away from including some real and nontrivial mathematics (for demonstration purposes), but the book isn't overly technical and he puts the harder stuff in the endnotes.If you've at least dabbled in higher mathematics and have some rudimentary familiarity with set theory, abstract algebra, topology, number theory, Turing machines, etc., you should be able to handle the book (and love it); without that background, it may be tough going.

Perhaps the best way to describe the content of the book is to summarize some of the key points:

(1) A goal of mathematical deduction is to derive nontrivial and interesting results (particularly mathematical theories), not just any or all results which follow from the axioms.Mathematics makes progress because new theorems are built on prior theorems.As it has developed, mathematics has generally become more difficult, though breakthroughs sometimes allow the solution of many problems to be greatly simplified.

(2) Solution of mathematical problems is aided by proper (or clever) classification of problems, imagination, allowing problems to incubate in the unconscious, use of analogy as a heuristic (though not highly reliable), and brute-force use of computers (which is controversial, since such methods have little appeal to our intuition and our desire for insight).

(3) Finding proofs can sometimes be very difficult because the process is like "walking in an infinite-dimensional labyrinth," trying to connect ideas in a sequence which meets the requirements of logic.Even seemingly simple theorems may require very long proofs (eg, Fermat's last theorem).

(4) When errors and gaps in proofs are found, it's often not overly difficult to correct them, so the resulting theorems tend to be fairly stable.In other words, the same destination can often be reached by many paths.

(5) Mathematical papers generally consist of figures, sentences, and formulas.Figures make use of our visual skills, but they're rarely mandatory.Sentences in natural language are indispensible.Formulas are compact and efficient ways of expressing sentences.Putting all of this together well is an art.Formal language could be used in principle but is unworkable in practice.

(6) The conceptual or intuitive aspect of mathematics is related to its natural structures, which are not the same as the formal aspects of mathematics.These structures may reflect human and historically contingent elements, rather than being purely "natural."

(7) The different branches of mathematics are deeply related, sometimes in surprising ways.Set theory (eg, ZFC) is perhaps the most fundamental branch of mathematics.The natural structures of mathematics often guide the development of new branches of mathematics.

(8) "Active research in mathematics gives intellectual rewards different from those enjoyed by a spectator."This research is primarily an individual rather than group activity, but the overall body of mathematics is a collective achievement.

(9) Many (but not all) mathematicians are prone to a "somewhat rigid way of thinking and behaving," mathematicians are twice as likely as physicists to be religious, and, on average, mathematicians don't possess greater artistic ability than the general population.Their special aesthetic sense is therefore of a different kind from that of artists.

(10) Nature is remarkably amenable to mathematical modeling ("unreasonable effectiveness"), especially in physics, and tends to give hints regarding which models to use.

(11) There's a striking contrast between the fallibility of the human mind and the infallibility of mathematical deduction.Unlike science and other intellectual endeavors, mathematics transcends uncertainties and offers a (Platonic) "perfection, purity, and simplicity" which we naturally yearn for, even if we can't be sure how mathematics ultimately relates to us and physical reality.Moreover, "the beauty of mathematics lies in uncovering the hidden simplicity and complexity that coexist in the rigid logical framework that the subject imposes."

(12) Gödel showed that, for a consistent and nontrivial axiomatic system, the system will contain true statements which can't be proven from within the system, including its own consistency.This discovery of incompleteness doesn't overly trouble most mathematicians in their daily work, though I personally find it to be profound and somewhat disturbing, or at least very perplexing ...

If these key points interest you, I urge you not to miss this book.If you find them obvious, I recommend reading the book anyway, since a list of key points doesn't do justice to the richness and charm of Ruelle's discussion.Personally, this book ranks among my favorite mathematics books and I'm a bit saddened to have reached the end of it.Now I just hope that Ruelle will write more books for nonspecialists!

From the man who coined the term 'strange attractor'
The man who coined the term 'strange attractor' provides a contemporary and a personal look at mathematics in an easy to read fashion.This book is a little bit eclectic which may be considered as a positive point for people outside the world of mathematics, and Ruelle does not adhere to a linear organization, preferring to jump from one subject to another but manages to provide good connections and insights.

Among the mathematicians he writes about, I found the case of Alexander Grothendieck very remarkable, inspiring, sad and hilarious [1]. This is a very interesting part of the history of mathematics which includes important lessons about organizations, politics and power relations.

Ruelle's discussion on some messy parts of math and proof-checking is very good and he poses important questions about proofs getting longer and longer and formalisms required to handle things as rigorously as possible.

The closing chapters are devoted to Ruelle's area of expertise and he writes very strongly on mathematical physics and give very good examples how diverse scientific fields help each other.

1- See my blog entry 'Corporatism in Science and Math: Mathematician Missing - Part 2':

http://ileriseviye.org/blog/?p=2223

The Mathematician's Brain
The book is philosophically shallow and mathematically unfocused. The author rambles and his writing lacks intellectual vigor. The chapters read like first thoughts dashed off before drifting off to sleep. The endnotes function as next-daysupplements to give the book an illusion of depth. At most, these pages may find a respectable place online where they can be skimmed and forgotten. Anyone interested in Alexander Grothendieck, for example, who appears in chapters six and seven, can find through a simple online search, narrative portraits of more substance and value than what Ruelle offers.

Limited but intriguing
In this small book the author (a distinguished professor of mathematical physics) touches on what mathematicians do, how they do it, how they think and feel about it, and how they relate to the world at large. On such a quick tour there are bound to be some mysterious turns and bumps on the road. More than necessary occur in this book: advanced topics are frequently introduced with unhelpful advice for the novice such as "Just go through it rapidly." Nevertheless I enjoyed learning a new bits of math (now I can define algebraic geometry) and stories of mathematicians. What kept me going was the author's skeptical attitude toward the mathematical establishment of which he is a part, and his genuine compassion for colleagues whose genius can so easily turn to madness.

good insights from a real mathematician;needs more editing
The author, who is a very distinguished mathematician, gives his personal view on how mathematicians think.It is welcome to have books like this written by real mathematicians, as opposed to philosophers who doesn't know that much math.While professional mathematicians might not learn much, students of mathematics can get some very nice insights into how mathematics and mathematicians work.

Unfortunately, some parts of the book that discuss specific mathematics (as opposed to what mathematics is like in general) are not clearly written and should have been edited better.For example, it shakes the confidence of the reader when early on, the pythagorean theorem is stated incorrectly, and then on the next page a statement is asserted to follow from the pythagorean theorem, when it actually follows from the converse of the pythagorean theorem.Most readers of the book will probably know this anyway so it doesn't matter, but later, descriptions of more advanced mathematical concepts are sometimes so brief that they would probably be incomprehensible to someone who does not already know them, and puzzling to someone who does.

Disclosure: I only skimmed this in the bookstore because I didn't feel like paying 20 cents per page for it.I hope that an inexpensive paperback edition will appear, with corrections. ... Read more

Product Description

This book presents the Riemann Hypothesis, connected problems, and a taste of the body of theory developed towards its solution. It is targeted at the educated non-expert. Almost all the material is accessible to any senior mathematics student, and much is accessible to anyone with some university mathematics. The appendices include a selection of original papers that encompass the most important milestones in the evolution of theory connected to the Riemann Hypothesis. The appendices also include some authoritative expository papers. These are the “expert witnesses” whose insight into this field is both invaluable and irreplaceable.

Customer Reviews (1)

Not all papers are in English
I enjoyed reading the book which gave me more appreciation of how some of the important results relating the to Riemann Hypothesis are derived.However, I was disappointed that not all of the included reference papers were in English. I think these could have all been translated. ... Read more

Product Description
This is a gripping account, full of surprises, of one of the greatest adventures in human thought, and at the same time a user-friendly introduction to many vital mathematical concepts. It is a global survey of the history of mathematics, suitable for people with no background in mathematics, as well as for the more informed reader or teacher. "From Five Fingers to Infinity" tells the story of the history of mathematics in the form of 114 articles, organised in a chronological and thematic manner. Together the articles cover all the most important areas, and can be read as a consistent narrative history. The articles are best by writers such as Carl Boyer, Howard Eves, Morris Kline, and Dirk Struik.Among the distinctive features of this volume are a multicultural treatment, with consideration of the mathematical accomplishments of traditional peoples, native Americans and others, not usually discussed in histories of mathematics; actual translations from early groundbreaking mathematical texts; a comprehensive review of Babylonian mathematical achievements; a sensitivity to the social and cultural context of mathematical endeavours; over 300 illustrations, and 18 "historical exhibits"; and informative introductions and pointers and extensive bibliographical information. Each article is short and self-contained. "From Five Fingers to Infinity" has been conceived and designed for three uses: enjoyable personal reading; as a general reference on the history of mathematics; and as a classroom text. ... Read more

Customer Reviews (1)

Excellent resource for any high-school math teacher
I read this book during the summer, and I know it will help my teaching.The amount of historical information this book contains is extraordinary, and I truly believe that this would be a valuable resource to any highschool math teacher looking to make a class more interesting, more alive. Get the book! ... Read more