Product Description
This is an expanded edition of a popular book on the history of mathematics.

Where did math come from?Who thought up all those algebra symbols, and why?What's the story behind...negative numbers?...the metric system? ...quadratic equations? ...sine and cosine?The 25 independent sketches in "Math through the Ages" answer these questions and many others in an informal, easygoing style that's accessible to teachers, students, and anyone who is curious about the history of mathematical ideas.Each sketch contains Questions and Projects to help you learn more about its topic and to see how its main ideas fit into the bigger picture of history.

The 25 short stories are preceded by a 56-page bird's-eye overview of the entire panorama of mathematical history, a whirlwind tour of the most important people, events, and trends that shaped the mathematics we know today.``What to Read Next'' and reading suggestions after each sketch provide starting points for readers who want to pursue a topic further.

It is ideal for a broad spectrum of audiences, including students in math history courses in the late high school or early college level, pre-service or in-service teachers and casual readers who just want to know a little more about the origins of mathematics. ... Read more

Customer Reviews (9)

I've Read Better... AND Clearer!!!
As a college senior majoring in Mathematics Education, I needed to take a Math History class. I read two books that focus on the history of mathematics; one of those books was Math Through the Ages.I found this book, especially in comparison with the other book, Journey Through Genius, to be disjointed, redundant and vague.The first part of the book reads like a typical math history book and the second part repeats the information given in the first part but reads more like a textbook, including questions and projects that pertain, loosely, to the information offered in each section.I found that the questions were often irrelevant for anyone not specifically majoring in Math history, which is fine for a history book... unless that book claims to be great for students of math education.Also, I felt that the questions and projects asked more from the student than the book gave to the student. It is one thing for the projects to expect extra research, but the point of a book is to give you the knowledge you need, especially to answer its end-of-section questions, not just pose more questions than it answers.Really, if you want a better understanding of Mathematics and its history, check out Journey Through Genius.It reads better and offers more detail in explaining concepts that pertain to today's mathematicians.

Math through the Ages
If you like math and want to read a short history of it, you'll like this book. It is written at a "popular mathematics" level, so it is accessible to nearly anyone who would take an interest in it. The writing is vibrant and to the point. The content is exemplary for a first look at the subject. The structure of the book is practical, intelligent, and effective. It begins with a 59 page summary of the history of mathematics, and this summary hits the high points. Then there are 25 chapter sketches over the next 179 pages, the chapter lengths being either 6 or 8 pages (meaning numbered pages, where a book leaf has 2 pages). These sketches discuss a single topic from the history, going into detail not given in the summary. The sketches and the summary each conclude with 2 pages of questions and projects that are as interesting and stimulating to read as the rest of the book.

Throughout the book the authors refer the reader to books and articles listed in their bibliography, which has 141 entries. After the 25 sketches there is a 7 page section called "what to read next" which directs the reader to specific math books and also to web sites they believe will be especially helpful. They include in this discussion 15 historical books they think you ought to read. This section could be thought of as a partial annotation of the bibliography.

Here are the topics covered in the sketches:

1. writing whole numbers
2. where the symbols of arithmetic came from
3. the story of zero
4. writing fractions
5. negative numbers
6. metric measurement
7. the story of pi
8. writing algebra with symbols
9. solving first degree equations
10. quadratic equations
11. solving cubic equations
12. the pythagorean theorem
13. Fermat's last theorem
14. Euclid's plane geometry
15. the platonic solids
16. coordinate geometry
17. complex numbers
18. sine and cosine
19. the non-euclidean geometries
20. projective geometry
21. the start of probability theory
22. statistics becomes a science
23. electronic computers
24. logic and boolean algebra
25. infinity and the theory of sets

Here are the 15 historical books they think you ought to read:

Tobias Dantzig - Number the Language of Science
William Dunham - Journey Through Genius: The Great Theorems of Mathematics
E. T. Bell - Men of Mathematics
Lynn Osen - Women in Mathematics
Howard Eves - Great Moments in Mathematics (before 1650)
Howard Eves - Great Moments in Mathematics (after 1650)
Dava Sobel - Longitude
Robert Osserman - Poetry of the Universe
David Bodanis - E=mc^2: A Biography of the World's Most Famous Equation
Asger Aaboe - Episodes from the Early History of Mathematics
S. Cuomo - Ancient Mathematics
David Salsburg - The Lady Tasting Tea
Benjamin Yandell - The Honor's Class
Donald J. Albers & Gerald L. Alexanderson (eds.) - Mathematical People
Donald J. Albers & Gerald L. Alexanderson (eds.) - More Mathematical People

Just in time for class!
I saved a ton of money on this book, as compared to if I had bought it in the school bookstore. Although the book was listed as used, I really couldn't even tell that anyone had ever opened it. Thank you.

Brilliant textbook for future math teachers
I came across this book because a friend of mine uses it in a college class for math ed.It's really well written and makes the material accessible for people whose math background isn't necessarily very strong.I bet it could even be used for high school students.The exercises and projects are really good, too.

An excellent place to start
For math teachers at the high school or college level, or anyone else interested in math, this is an ideal introduction to the history of math. Start with the 55-page overview. Then read any or all of the articles that follow, on a variety of topics such as negative numbers, pi, quadratic equations, the Pythagorean theorem, the history of probability theory, and infinity, all around five pages each. Once you're finished with that, there's an extensive bibliography with plenty of suggestions for further reading on the topics that have piqued your interest.

Throughout, the authors have striven for (and succeeded at attaining) readability, accessibility, and historical accuracy. The result is a book that scores high marks for being both interesting and informative. ... Read more

Product Description
Mathematics is a wonderfully austere science, but it also has a very human side. It is embedded in a colorful history filled with extraordinary personalities, deep philosophical debates, and breath-taking advances in knowledge. This book offers a brief but penetrating synopsis of that history.

This book includes many detailed explanations of important mathematical procedures actually used by famous mathematicians. This gives the reader an opportunity to learn the history and philosophy of mathematics by way of problem solving. For example, there is a careful treatment of topics such as unit fractions, perfect numbers, linear Diophantine equations, Euclidean construction, Euclidean proofs, the circle area formula, the Pell equation, cubic equations, log table construction, the four-square theorem, quaternions, and Cantor's set theory. Several important philosophical topics such as infinity and Platonism, are pursued throughout the text.

This book is written as an undergraduate textbook, but it is intended for anyone who wants to understand how mathematics grows out of, and nourishes, the total human experience. ... Read more

Customer Reviews (6)

Worthless
This book is utterly worthless in every conceivable way. It is a mystery to me why Springer have brought disgrace upon themselves by publishing this inept drivel. A complete account of Anglin's incompetence would require a review as thick as the book itself, but hopefully a few deterring examples will suffice.

First, there are many blatant factual errors, e.g.:

"There were five planets (or so Kepler thought) and five regular polyhedra. This could not be an accident!" (p. 158)

Since Kepler's work on Mars, Jupiter and Saturn are mentioned on the same page, one wonders whether it is Mercury, Venus or the Earth that Anglin imagines Kepler to have been ignorant of.

Other statements cannot even be called false since they are such ludicrous nonsense, e.g.:

"Just as many people before Lobachevsky thought that Euclid's parallel postulate was a kind of sacred truth, so many people before Hamilton thought that the law of commutativity for multiplication was ineluctable. For us it is a commonplace that this law need not hold, since we have a ready example of noncommutativity in matrix multiplication." (p. 195)

The notion that there is some sort of "law" out there about commutativity of multiplication that may or may not hold is a very childish misconception. Whether we call certain operations with matrices and quaternions "multiplication" or not is purely a matter of convention. Thus the alleged "law" that "many people" allegedly held for "ineluctable" has no meaning whatsoever other than as a thoroughly inconsequential claim about naming conventions. Anglin's stupidity is particularly disturbing in light of his immodest description of his own book as offering "a deep penetration into the key mathematical and philosophical aspects of the history of mathematics", "giving the student an opportunity to come to a full and consistent knowledge" (p. viii).

Let us give another example of what Anglin considers to be "a deep penetration" into philosophical issues. We read that "there are various objections to formalism," e.g. that "formalism offers no guarantee that the games of mathematics are consistent." Now, presumably in order to "give the student an opportunity to come to a full and consistent knowledge," Anglin professes to offer the other side of the coin: "the formalist can reply [that] although some of the games of mathematics are indeed inconsistent, and hence trivial, others are not" (pp. 218-219). It is not clear in what sense Anglin fancies the assertion that mathematics is consistent to be a "reply" the the challenge to prove as much.

Finally, the book is full of unsubstantiated revisionist history motivated by unabashed Christian propaganda ("God" is the entry in the index with the most references by far; more, in fact, than Euclid, Archimedes, Newton and Riemann combined), e.g.:

"Most of the mathematicians at the Academy and the Museum rejected the new truths [sic] of Christ's revelation. This is unfortunate because ... if the mathematicians had joined the Christians, the Dark Ages would have been brightened by a dialogue between reason and faith. As it was, this dialogue was postponed to the later Middle Ages, when thinkers like Thomas Aquinas (1225-1275) advanced philosophies that were influenced as much by the Elements as by the Bible." (p. 111)

What a loss for mathematics that we had to wait so many centuries for the great geometer Aquinas!

Trite; filled with unnecessary religious hot air
This is a simplistic and shallow book. Proof if any is needed that relgious dogma (and the authors desire to spread it ... even in a maths book!) poisons the mind. The authors frequent attempts to bring god into the picture are unsubtle, (mostly) irrelevant and unbelievably crass. I am disappointed in Springer; how on earth did they allow this to be published?

Nice Read overall
Not a perfect book by any stretch, but I am not the type of reader who has to agree with a book to enjoy it.Many histories of mathematics books are rife with anti-god, anti-religion references, this is a balance to that.Admittedly a little pushy the other way, but not a bad read.

Nice antidote to E. T. Bell
On the math side of things, this book provides a concise overview of the history of mathematics.Actually, I found it to be a bit "too concise" - I think that a college professor would be hard-pressed to stretch the book out over a one-semester "History of Mathematics" course.The content of the book is clearly designed for liberal arts students interested in the "History of Mathematics", rather than for mathematics students interested in the "Mathematics of History".In an appendix at the end of the book the author includes a number of sample assignments, tests and exams which I personally found rather useful.

Yes, I agree with previous reviewers that the author pushes his Christian views on the reader, but I must say that I found it a refreshing change to the tiresome and offensive anti-Christian propaganda found in E. T. Bell's book "Men of Mathematics", in which Blaise Pascal is made out to be a mentally ill religious lunatic, while Augustin-Louis Cauchy is made out to be a harsh and bigoted religious fanatic!

Terrible
The whole book is infested with very annoying and irrelevant
personal thoughts on religion. Mathematicians seem to fall in two
categories: Christian and atheist. The latter are generally evil
and of little relevance, while the former are moral persons that
have produced excellent mathematics. The following quotes from
the book illustrate its general attitude:

Exercise:
Even if the solar system is gravitatinally stable, it still needs
God to keep it in existence. Comment.

Some historians feel that to be 'scientific' they must do their
work on the assumption that there is no God.

Laplace's greatest contribution to mathematics was his phrase
'it is easy to see'.
... Read more

Product Description
The writings of Newton, Liebniz, Pascal, Riemann, Bernoulli, and others in a comprehensive selection of 125 treatises, articles from the Renaissance to end of the 19th century—most unavailable elsewhere. Grouped in five sections: Number; Algebra; Geometry; Probability; and Calculus, Functions, and Quaternions. Index. 83 illustrations.
... Read more

Customer Reviews (3)

Great book
The footnotes contained in the book are very helpful. I bought this book some few years ago and stiff look upon the pages for referencing and clarification.

I have an Older 1959 version in two parts
Mathematics was never easy and looking at these classic
reproductions you really get a feel of the ignorance
of even the greatest men in mathematics.
Here you can see them struggle to invent new ideas
that made possible our scientific and technical culture.
I think this kind of book is invaluable to the student
who wishes to actually understand.
Some of the papers are almost impossibly difficult.
It is a very good book!

many original papers
For students of the history of maths, Smith provides you with a very convenient reference. He has gone back to many of the original papers by Newton, Pascal and others, and gathered 125 of these into this book. You can search for insight into how those luminaries made their important discoveries. As an added utility, the papers have been translated into English.

An amazing time saver. For he lets you access the papers without any intermediary. The alternative would be to spend months searching in some large research library. And also probably having to order copies made from other libraries. At non-trivial cost in time and money. ... Read more

Product Description

This is a novel, short, and eminently readable history of mathematics. Many histories provide a chronological history of the entire subject, which can sometimes make it difficult to follow the development of a particular branch over time. Dahan-Dalmedico and Pfeiffer succeed splendidly in tracing each branch from its beginnings forward. They also give an outstanding account of how the Arabs not only preserved Greek mathematics, but extended it in the 800 year period from 400-1200. The large number of informative illustrations support the text and contribute to what is a great read.

Customer Reviews (1)

Unexceptional
This is an unexceptional and awkwardly translated history of mathematics.

Among the many common myths unthinkingly propagated by the authors is the nonsense that the development of "the concept of function" is somehow enlightening and important. An entire chapter is devoted to this meaningless study, in which we read such bogus as:

"Descartes' idea of restricting the notion of function only to algebraic expressions was an iron collar" (p. 233) (Incidentally, this sentence illustrates the poor quality of the translation. Of course what is meant here is that Descartes restricted the notion of function to algebraic expressions alone, i.e. that he restricted too much, not that he "only" restricted the notion to some extent but not enough, as the translation clumsily suggests. Awkward expressions like this occur on virtually every page.)

No evidence is ever provided for claims such as these, of course, since they are mere prejudices without any basis in historical fact. One might as well claim that the 17th century conception of "vehicle" as necessarily animal-powered was an "iron collar" impeding the development of cars. People held this conception not because they were dense or unimaginative but because a more general conception would have served no purpose whatsoever at the time. So also with functions, which makes the study of history through this anachronistic lens a complete waste of time.

Another flaw of the book is that is that it is full of casually scattered statements that are neither explained nor supported and very often too brief and vague for anyone to gain anything from them or even to understand what they mean. For example we read that "the incompatibility of theory and observations pushed the Danish astronomer Tycho Brahe (1546-1601) to reject Copernicanism and look for a compromise" (p. 26). Since Tycho's theory is observationally equivalent to Copernicus', this statement appears to be false or at least highly misleading. But then again maybe not, for this is just another throwaway remark among many, and since no more information is given about neither Tycho's theory nor what the alleged "incompatibility" consisted in, it is quite impossible to understand what the authors are trying to say. ... Read more

Product Description
Praise for Hal Hellman

Great Feuds in Mathematics

"Those who think that mathematicians are cold, mechanical proving machines will do well to read Hellman's book on conflicts in mathematics. The main characters are as excitable and touchy as the next man. But Hellman's stories also show how scientific fights bring out sharper formulations and better arguments."
-Professor Dirk van Dalen, Philosophy Department, Utrecht University

Great Feuds in Technology

"There's nothing like a good feud to grab your attention. And when it comes to describing the battle, Hal Hellman is a master."
-New Scientist

Great Feuds in Science

"Unusual insight into the development of science . . . I was excited by this book and enthusiastically recommend it to general as well as scientific audiences."
-American Scientist

"Hellman has assembled a series of entertaining tales . . . many fine examples of heady invective without parallel in our time."
-Nature

Great Feuds in Medicine

"This engaging book documents [the] reactions in ten of the most heated controversies and rivalries in medical history. . . . The disputes detailed are . . . fascinating. . . . It is delicious stuff here."
-The New York Times

"Stimulating."
-Journal of the American Medical Association ... Read more

Customer Reviews (4)

David Foster Wallace's Righteous Twin
Wallace's seriously flawed Everything and More: A Compact History of Infinity (Great Discoveries) made me wary of buying another pop math book by a non-mathematician, so I put off buying _Great Feuds_ for some time, but eventually I gave in, and I'm glad I did.Hellman's cautious approach contrasts nicely with Wallace's bombast, and, unlike Wallace, Hellman gets almost all of the details right, with a notable exception being his claim that having a smallest element (rather than each of its nonempty subsets having smallest elements) is what makes a set well-ordered.

There's a lot of quoting of the opinions of professional historians, which is probably appropriate for a book written by an outsider, but I found it a bit tiresome after a while (just as I did when Peter Ackroyd took a similar approach in Albion: The Origins of the English Imagination).Also, I felt that Hellman didn't make it as clear as he could have who ended up winning the war of which these feuds were battles.21st century mathematics is overwhelmingly Cantorian, Zermeloian, and Hilbertian, in the sense that the existence of actual infinities and the appropriateness of using the Law of the Excluded Middle and the Axiom of Choice are all taken for granted by mainstream practitioners.There are respected researchers probing the effects of rejecting these principles, but they are few in number and those who do reject them are definitely working in the margins.

Don't let these quibbles or my 4-star rating keep you from buying this book.Within its genre, it's about as good as they come.

Readable Math
After seeing Mr. Hellman on CBS Sunday Morning News recently, I picked up his newest book, Great Feuds in Mathematics. A most enjoyable book. Although I am not a mathematician I was able to read "around" the few equations and enjoy the insight he brought to the math and its place in history. A good read.

Math is math
This book would be of most interest to advanced mathematicians and some philosophy initiates. I am a lay person with a large curiosity, however not being a math college major, limited my enjoyment of the "feuds."
Antonio Gonzalez

Fascinating Disputes But Beware The Challenging Arguments
This is an excellent book. The prose is clear and engaging and, despite the title, there are very few equations such that those who are equation-phobic have little to fear. However, many of the disputes center on nineteenth to twentieth century front-line research in pure mathematics - areas such as set theory, concepts of infinity, etc. These early ideas were prone to heated discussion and, in many cases, led to feuds. In order to allow the reader to understand the basis for these feuds, the author has included the essence of some of the key contentious mathematical arguments, often directly quoting members of each camp. I found that carefully following these arguments in detail could be difficult at times, but I certainly agree that pondering them is important if one is to clearly understand the position of each side. The final chapter poses the fascinating question: Are mathematical advances discoveries or inventions?" And here again, there are avid supporters of each side. I gave the book five stars because of the interesting subject matter and because I feel that the author has done a truly excellent job in presenting such a potentially difficult subject to as broad an audience as possible. Nevertheless, I still believe that I would benefit from reading some chapters a second time. Although anyone reading this book could learn much from it, I believe that it would be most enjoyed by serious math buffs. ... Read more

Product Description
Analysis as an independent subject was created as part of the scientific revolution in the seventeenth century. Kepler, Galileo, Descartes, Fermat, Huygens, Newton, and Leibniz, to name but a few, contributed to its genesis. Since the end of the seventeenth century, the historical progress of mathematical analysis has displayed unique vitality and momentum. No other mathematical field has so profoundly influenced the development of modern scientific thinking.

Describing this multidimensional historical development requires an in-depth discussion which includes a reconstruction of general trends and an examination of the specific problems. This volume is designed as a collective work of authors who are proven experts in the history of mathematics. It clarifies the conceptual change that analysis underwent during its development while elucidating the influence of specific applications and describing the relevance of biographical and philosophical backgrounds.

The first ten chapters of the book outline chronological development and the last three chapters survey the history of differential equations, the calculus of variations, and functional analysis.

Special features are a separate chapter on the development of the theory of complex functions in the nineteenth century and two chapters on the influence of physics on analysis. One is about the origins of analytical mechanics, and one treats the development of boundary-value problems of mathematical physics (especially potential theory) in the nineteenth century. ... Read more

Product Description
Fluent description of the development of both the integral and differential calculus. Early beginnings in antiquity, Medieval contributions and a century of anticipation lead up to a consideration of Newton and Leibniz, the period of indecison that followed them, and the final rigorous formulation that we know today.
... Read more

Customer Reviews (12)

Propaganda history
This is not a history of the calculus, but rather a profoundly biased and small-minded quasi-history of its foundations. Boyer kneels before the modern theory of the foundations of the calculus with religious awe, and crusades on its behalf with exceptional arrogance and obliviousness to reason. The purpose of a history of the calculus, according to Boyer, is apparently to condemn the infidels for their "misapprehension ... as to the logical basis of the calculus" (p. 8). His so-called "history" is in reality a thinly veiled sermon on the sins of the "logically unsatisfactory" (p. 4) and the "logically irrelevant" (p. 8).

Newton and Leibniz are chastised because they "did not fully recognize the need" for "the rigorous formulation of the concepts involved" (p. 47) and "were insensible to the delicate subtleties required in the logical development of the subject" (p. 5). To any sane person these historical facts show that there was in fact no such "need" and that Boyer's favourite "subtleties" were in fact not "required" at all. But not so to Boyer, who apparently considers his own intellect so superior that he has nothing to learn from Newton and Leibniz.

For a more specific illustration of Boyer's simplistic and dogmatic mindset, we may consider his appraisal of Berkeley's "objection to Newton's infinitesimal conceptions as self-contradictory" as "well taken" and "pertinent" (p. 226). Here it is useful to make a comparison with complex numbers (as Leibniz himself suggested, p. 215). The rules for discarding infinitesimals do not prove that infinitesimal calculus is self-contradictory any more than the fallacious reasoning that -2 = Sqrt(-2)Sqrt(-2) = Sqrt((-2)(-2)) = Sqrt(4) = 2 proves that complex numbers are self-contradictory. In both these cases, all that is shown is that these new entities do no obey all the laws of ordinary numbers. But no one ever claimed that they should, so there is nothing "self-contradictory" or "logically unsatisfactory" about either of these situations, as is obvious to anyone who, unlike Boyer and Berkeley, is not blinded by dogma.

A little technical on the historical side...
Boyer is a historian of mathematics, and I have his larger history text, which I like much better. I honestly expected a history of the calculus to be more of a fascinating read. The author does an excellent job of taking you through some of the finer points of this history and reasons why, for example, Archimedes should not be given credit for discovering the calculus, but why there is some justification for such a claim. The thing is, these finer points of the history are mentioned quite frequently even with regard to mathematicians whom I have never heard of. It seems that someone is always saying that so-and-so really discovered the calculus, and Boyer always points out why in fact they did not. The writing also can be rather verbose at times (this is sometimes entertaining). I do not see this text as appealing to a lay reader with an interest in the history of one of the greatest intellectual acheivements of all time: the calculus. I see this as appealing more to historians of mathematics or other such related fields. I started this book twice, and the second time, I made it about three fifths of the way through. It's hard to read a lot at once. It's a history book, not a book about the history. There are a fair amount of diagrams, and the math is interesting, if at times confusing, to follow. I can't say that my understanding of calculus is much deeper after reading the majority of the book, though it certainly does have a larger and more technical context.

More history, less real mathematics?
Only two things made me give the bookbetter than two stars:
the idea of an error term to:
d(x^n)/dx=n*x^(n-1)+error(f(x,n)).
And the mention of harmonic triangle:
t(n,m)=1/(n*Binomial[n,m])
The question of what would mathematics be like without Leibniz and Newton
and calculus really takes us back to what mathematics was like in 1600:
geometry, algebra and number theory. That mathematics had such a great part in the industrial revolution by making physics, a science based mainly on derivative calculus, makes me think that we would have sailing ships and horses still.
The fact that he pretty much leaves out fractional calculus is another
strike against him presenting a true history of calculus.

Mildly instructive
but atrociously written: this book is an epitome of the shift/reduce conflict -- some paragraphs defy parsing altogether. Overall OK if you're into calculus to the point of worrying about its history or if you want to get to understand how, and even more why it came about. Although the hows definitely prevail over the whys here, unfortunately. The book is far from flawless, but still, if you can get through the stultifying writing, it will enlarge somewhat your overall conceptual view of calculus. Recommended? Perhaps. If you have time.

Fascinating material, questionable presentation
The first thing I noticed about this book is that it is written with an intellectually arrogant, indecipherable style which (I hope) would today prevent its being published at all.Here is a paragraph, verbatim, from the introduction:

"At this point it may not be undesirable to discuss these ideas, with reference both to the intuitions and speculations from which they were derived and to their final rigorous formulation.This may serve to bring vividly to mind the precise character of the contemporary conceptions of the derivative and the integral, and thus to make unambiguously clear the terminus ad quem of the whole development."

I admit that back in 1939, when this book was originally written, it was common for academics to express themselves in that sort of haughty, impenetrable prose.But that doesn't make it any easier to read today, and it doesn't really provide those people with an excuse for having written that way.Didn't it occur to them that their writing might be read by real human beings?There are plenty of mathematical writers today who can write in real English without sacrificing rigor or depth.

Secondly, I recommend that everyone read the review by the reader from Phoenix (February 7, 2001).In particular, I agree with the criticism that this book takes a backwards approach to the history of Calculus, interpreting each historical idea and contribution in terms of the way we think of those ideas today.As Boyer certainly should have known, the proper way to relate the history of ideas is to place each idea in the context of its own time.Instead, he writes this book as if each ancient mathematician had tried and failed to reach the level of understanding which we superior moderns are now gifted with.I think it is important for a reader to read this book with this defect clearly in mind.

Having got those two criticisms off my chest, however, I have to admit that there is a wealth of interesting material in this book, and I don't know of any other place where it is all gathered together in one volume.If you want a detailed, in-depth account of how mathematicians and philosophers (they used to be the same people!) eventually evolved the ideas and methods of calculus, then this book is probably the best place to find it.

(I just wish the publisher would hire someone to translate it into real English!) ... Read more

Product Description
Exciting, hands-on approach to understanding fundamental underpinnings of modern arithmetic, algebra, geometry and number systems, by examining their origins in early Egyptian, Babylonian and Greek sources. Students can do division like the ancient Egyptians, solve quadratic equations like the Babylonians and more.
... Read more

Customer Reviews (2)

Excellent Historical Review of the Roots of 'Earth-Measurements'
This book can only be appreciated by a scholar of both history and mathematics.If you want the real story of the basis of modern geometry and where it had evolved from,then this book will amaze and educate.Three cultures are responsible for producing mathematical greatness;the Greeks,the Babylonians and the Egyptians.The book is recommended for high school students for enriching geometry skills,yet can be read by eager seventh and eighth graders as well.There are three chapters dedeicated to 'Euclid',the greatest of the Greek geometricians.Yet,only two chapters on 'Pythagoras',the more famous of the two philosophical mathematicians.Each chapter has exercises and answer keys for checking.For the price,it's well worth reading to broaden your geometry knowledge and eliminate your hidden numerophobia.

Using math history for teachers and students
I have used this book in the past when I taught a math history course to high school students.In its paperbook form now it is a good resource book for teachers who are looking for projects or extra math history work for their students to do and study.It is easy to read and the math is easy to follow. ... Read more

Product Description
"A fascinating compendium of information about writing systems–both for words and numbers."
–Publishers Weekly

"A truly enlightening and fascinating study for the mathematically oriented reader."
–Booklist

"Well researched. . . . This book is a rich resource for those involved in researching the history of computers."
–The Mathematics Teacher

In this brilliant follow-up to his landmark international bestseller, The Universal History of Numbers, Georges Ifrah traces the development of computing from the invention of the abacus to the creation of the binary system three centuries ago to the incredible conceptual, scientific, and technical achievements that made the first modern computers possible. Ifrah takes us along as he visits mathematicians, visionaries, philosophers, and scholars from every corner of the world and every period of history. We learn about the births of the pocket calculator, the adding machine, the cash register, and even automata. We find out how the origins of the computer can be found in the European Renaissance, along with how World War II influenced the development of analytical calculation. And we explore such hot topics as numerical codes and the recent discovery of new kinds of number systems, such as "surreal" numbers.

Adventurous and enthralling, The Universal History of Computing is an astonishing achievement that not only unravels the epic tale of computing, but also tells the compelling story of human intelligence–and how much further we still have to go.Amazon.com Review
From the I Ching to AI, tremendous human brainpower has been devoted to devising easier means of counting and thinking. Former math teacher Georges Ifrah has devoted his life to tracking down traces of our early calculating tools and reporting on them with charm and verve. The Universal History of Computing: From the Abacus to Quantum Computing gives a grand title to a grand subject, and Ifrah makes good on his promise of universality by leaping far back in time and spanning all of the inhabited continents. If his scope is vast, his stories and details are still engrossing. Readers will hang on to the stories of 19th-century inventors who converged on multiplication machines and other, more general "engines," and better understand the roots of biological and quantum computation. Ifrah has great respect for our ancestors and their work, and he transmits this feeling to his readers with humor and humility. His timelines, diagrams, and concordance help the reader who might be unfamiliar with foreign concepts of numbers and computation keep up with his narrative. By the end, his slight bias against strong artificial intelligence comes through, but he is careful to acknowledge the future's unforeseeable nature and suggest that we keep our minds open. How can we resist? --Rob Lightner ... Read more

Customer Reviews (5)

Simply A Must Read!
This is simply a must read for anyone who is interested in numerical literacy!

Indispensable and should be required curriculum for anyone who teaches who wishes to think of themselves as versed in the arts and letters.

APhilosophical Approach
If you have been looking for a more academic approach to the history of computing then this is the book for you.

The book is divided into three parts. Part One contains a very comprehensive taxonomy/chronology showing the evolution of human number systems.

Part Two is where you will find the core "History of Computing" bit: tables, logarithms, analogue/digital, mechanical calculators, automatic calculation, electronic machines etc. It also includes an interleaved, and detailed, explanation of how computing has evolved from basic number crunching into abstract information processing.

Part Three reads like a long philosophical conclusion and contains some excellent material on ethics and artificial intelligence.

It starts with the development of efficient notation
Until recently, the history of computing has tended to be tied to the goals of mathematicians, as they struggled to keep up with the increasing demands of a society growing more technical. As nations began to trade with other nations, the necessity of performing computations on larger numbers very quickly forced changes in the notation. When first introduced into Europe, the modern decimal system of notation was greeted with skepticism and some hostility. However, as is nearly always the case in human endeavors, it was accepted rather quickly, as it was so much more efficient than other systems such as Roman numerals. Therefore, the history of computing devices is bound very tightly with improvements in representation, and the historical changes in notation are the topic of the first section of the book.
Ifrah does an excellent job in recapitulating the history of the notation of computation, covering the entire world, ending up with the modern notation and the efficiency of binary numbers. Nearly forty pages are devoted to explanations of many ancient numerical notations, and many figures are included. It is this approach that differentiates this book from other histories of computing. Other authors concentrate on the history of the evolving architectures of the computing devices, ignoring the necessary precondition of a compact and efficient notation. It is very difficult to imagine computing devices that could easily perform arithmetic on Roman numerals.
The second section is a two track treatment of the development of computing devices. One track covers the mathematical preliminaries and the second the mechanical advances that led to the construction of accurate computers. Most of the early improvements were done by mathematicians, and it was not until the late nineteenth century that governments started to be interested in computers. The primary event was the work of Charles Babbage, who showed that computers were possible and how valuable they could be in performing routine computations that were highly prone to error.
In many ways, this history of computing is more a history of the requisite mathematics rather than a history of hardware. This is a second way in which this book differs from other histories. One of the reasons why computers have improved so quickly is that much of the theoretical background for their actions were developed before the machines were. Ifrah explains that in great detail, describing how some of the principles of abstract mathematics have been applied to the building of computers.
The final section is very small and deals with the future of computing. This is a wise move, as this book is a history and one thing we have learned from the recent history of computers is that predicting the future is largely impossible. We know that they will get faster, have more memory and the usage will increase, but the consequences of this are difficult to predict.
If your interest is in the preconditions necessary for computers to be widely used, then this is the book for you. Ifrah covers all of the notational and mathematical background necessary for computers to be useful, for without that, they would probably have been little more than intellectual toys.

Published in the recreational mathematics e-mail newsletter, preprinted with permission.

Methodical history but a little dry
I would have expected from the title that this book might have started in the 1940s (or at the earliest with Babbage and the Difference Engine) and told the story of the development of computers from there. No, as the subtitle indicates, this book goes way back. In fact, the first section is a summary of number systems going back to the age of the Egyptians and before. It's a very methodical and somewhat dry tale, not helped by being translated from the French by translators who feel compelled to insert their own comments at intervals.

When it does get going, it provides a history of the relevant mathematics as well as automata from the Islamic era forward. The actual computer era is touched on mostly in its early stages, with the first computers of the forties and fifties. And it concludes with about sixty pages that have nothing to do with history but rather attempt to define key words such as "information" and "computer."

All in all, it is a methodical and thorough book, perhaps a little dry but not as much as some books I have read. The author muses on the implications of various stages of discovery rather than simply relating the facts (and the translators chime in as well), which enlivens the story. Still, this book is probably for the more interested rather than the casual reader.

The Universal History of Computing : From the Abacus to
This book is really fascinating, especially if you are interested in scientific and technical achievements. Read this book and you'll find out how the computer can be traced to the Renaissance, and how Word War II influenced the development of analytical calculation. The epic tale of computing comes to life in these pages. ... Read more

Product Description
This inexpensive paperback uses lively language to put mathematics in an interesting, historical context and points out the many links to art, philosophy, music, computers, navigation, science, and technology. The arithmetic, algebra, and geometry are presented in a way that makes them relevant to daily life as well as larger issues.Topics include: Oh So Mysterious Egyptian Mathematics; Mesopotamia Here We Come; Those Incredible Greeks; Greeks Bearing Gifts; Must All Good Things Come to an End?; Europe Smells the Coffee; Mathematics Marches On; A Few Good Men; A Most Amazing Century of Mathematical Marvels!; The Age of Euler; A Century of Surprises; Ones and Zeros; Some More Math Before You Go. ... Read more

Customer Reviews (5)

I loved Dr. LeWinter's class, and I love his book
I graduated from SUNY Purchase in 2001. Dr. LeWinter's History of Mathematics was one of the last classes I took before graduating. Because I have dyscalculia, I wasn't able to pass an actual math course, so my advisor suggested this speciality class. It was amazing. Marty LeWinter brought his guitar in every day, recited poetry, and had us create math-related poems for homework. His class was geared toward students who were doing badly in math but needed a math class to pass. Thanks to Dr. LeWinter's teaching -- and his textbook, now sold on Amazon -- I had so much fun learing math, without the headaches.

easy read, but be wary of the "history" of civilizations
I enjoyed reading this book until I got to Chapter 5 which I found to be inaccurate about Islamic history, not to mention extremely offensive.It would be different if the statements were true, but they're untrue and unneccesarily harsh.Other than that, this book is easy to understand and is quite humorous.I would recommend it for people who would like to learn the history of math, but I feel there are better resources out there to learn about the history of civilizations.

I finally understand math
I never appreciated my math courses in school. As a returning adult, I really found this book easily explains the concepts it is trying to get across. I especially liked the Chinese numbers. It is nice to learn about math fromother cultures. I now want to learn more math. Thank You, Lewinter and Widulski, I hope you guys write more math books.

An Excellent Book!!!!
This book covers Egyptian mathematics through to todays computers. It takes you on a journey through "mathematics" time. But the best part is even I understood it which says quite alot. I especially enjoyed the humor (Yes, a math book with humor) and the historical tidbits. I love this book!!

math affects everything!
fun to read; full of history; math was easy to follow; showed
impact of math on science, music, art, navigation, computers and philosophy; every parent should read this for when their kid says, "i hate math"...i loved the picture proof that the first n
odd numbers add up to n squared (like 1 + 3 + 5 + 7 = 16 which is 4 squared)...i envy the authors' students! ... Read more

Product Description
Blending relevant mathematics and history, this book immerses readers in the full, rich detail of mathematics. It provides a description of mathematics and shows how mathematics was actually practiced throughout the millennia by past civilizations and great mathematicians alike. As a result, readers gain a better understanding of why mathematics developed the way it did.Chapter topics include Egyptian Mathematics, Babylonian Mathematics, Greek Arithmetic, Pre-Euclidean Geometry, Euclid, Archimedes and Apollonius, Roman Era, China and India, The Arab World, Medieval Europe, Renaissance, The Era of Descartes and Fermat, The Era of Newton and Leibniz, Probability and Statistics, Analysis, Algebra, Number Theory, the Revolutionary Era, The Age of Gauss, Analysis to Mid-Century, Geometry, Analysis After Mid-Century, Algebras, and the Twentieth Century.For teachers of mathematics. ... Read more

Product Description
In thiscarefully researched study, the author examines Egyptian mathematics, demonstrating that although operations were limited in number, they were remarkably adaptable to a great many applications—solution of problems in direct and inverse proportion, linear equations of the first degree, and arithmetical and geometrical progressions.
... Read more

Customer Reviews (2)

Gillings'errors and omissions
Gillings attempted to bring together all the known hieratic mathematical texts. Even the Akhmim Wooden Tablet (AWT) was mentioned as a footnote. Though not analyzed in any section, the AWT and the hieratic math textsoutline an exciting book. Taken together the Middle Kingdom math texts were read as one document, one text checking the another text for errors and omissions. In great part Gillings followed that rule.

Exceptions to the rule lie in the reporting of five texts, and aspects of other texts. The five under valued texts are the AWT, the Egyptian Mathematical Leather Roll, the Reisner Papyri, the Kahun Papyrus, and the RMP. All five texts were reported by Gillings with minor oversights becoming major oversights when oversights were placed in the larger context of scribal meta mathematics.

Concerning the EMLR, Gillings' oversight consisted of four of the 26 lines of texts, only reporting them as additive in scope, as were the other 22 lines. Actually a higher form of abstract arithmetic should have been discussed as potentially present.

The Reisner Papyri was discussed as containing quotients, which it does. Gillings' oversight was not mentioning the remainders that filled the scribal overseer notes from a construction site where daily worker digging rates were measured in units of 10. Hence all of the digging rates were divided by 10, and were reported by the scribe as quotient and remainder totals, a remainder arithmetic fact that escaped Gillings analysis. One scribal error was corrected by Gillings, properly listing a quotient and remainder; however, the proper modern name for the ancient arithmetic was not potentiallly commented upon by Gillings.

Finally, throughout the RMP quotients and remainders fill the document for almost every division and subtraction that Ahmes reported in his 84 problems. Yet, again, only quotients are mentioned, from time to time, with the remainder aspect of Egyptian fractions often being the major component, were not commented upon by Gillings. A clear example of Gillings' oversight is cited on page 250 "Horus-Eye fractions in terms of hin", where 29 divisions of a hekat, a volume unit, were divided by rational numbers in the range 1/64 to 64, with each answer written down as quotients and remainders. All of the two-part statements were created from the hekat unity, (64/64), being divided by a divisor n, or: (64/64)/n = Q/64 + (R5/n)*1/320, with Q the quotient and R the remainder. As a passing comment, Gillings also missed Ahmes' hin rule, 1/10 of the hekat, creating a one-part number by using 10/n hin, as listed 29 times in the table, the additive context in which Gillings incorrectly reported the totality of the table.

Returning the the Akhmim Wooden Tablet, the text reported in vivid terms a hekat unity (64/64) divided by 3, 7, 10, 11 and 13. The answers used binary quotients and scaled remainders, an abstract form of arithmetic used in the RMP 40 times. Georges Daressy first reported aspects of the AWT in 1906. Gillings cited none of Daressy' ground breaking work. Daressy's incomplete analysis OF THE AWT was finally corrected on the proof side, in 2002 by Hana Vymazalova, a Charles U., Prague, graduate student.

In summary, Gillings' main 1972 point: that Egyptian mathematics must be revisited and updated is true. 21st century math historians have taken up the 1972 challenge and completing the decoding of the hieratic texts as one body of knowledge. A Charles University grad student, Hana Vymazolva, and other young students are finding pieces of a large jig-saw puzzle. Humpty dumpty is being put back together again, thanks to the urgings of Gillings.

All I ever wanted to know about the mathematical papyri.
The Rhind, Moscow, and other important mathematical papyri decoded in every detail. A sweeping tour through the ancient Egyptian methods of calculation, parts of which are still used today in computer code! In hiswell-written account, Mr. Gillings makes it very clear that the common viewon ancient Egyptian mathematics as 'rather primitive' is definitely to berevised. Provided with a few basic tools, the scribes of the epoch wereable to carry out very complicated computations indeed, at times involvingseveral different units. Their rough-and-ready estimate of pi was off byonly 0.6 percent as compared to the correct value. The author presents arich variety of calculated examples and explains the logic behind them.Earlier researchers in the field are commented. ... Read more

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"A lucid representation of the fundamental concepts and methods of the whole field of mathematics. It is an easily understandable introduction for the layman and helps to give the mathematical student a general view of the basic principles and methods."--Albert Einstein (on the first edition)

For more than two thousand years a familiarity with mathematics has been regarded as an indispensable part of the intellectual equipment of every cultured person. Today, unfortunately, the traditional place of mathematics in education is in grave danger. The teaching and learning of mathematics has degenerated into the realm of rote memorization, the outcome of which leads to satisfactory formal ability but not to real understanding or greater intellectual independence. This new edition of Richard Courant's and Herbert Robbins's classic work seeks to address this problem. Its goal is to put the meaning back into mathematics.

Written for beginners and scholars, for students and teachers, for philosophers and engineers, What is Mathematics?, Second Edition is a sparkling collection of mathematical gems that offers an entertaining and accessible portrait of the mathematical world. Covering everything from natural numbers and the number system to geometrical constructions and projective geometry, from topology and calculus to matters of principle and the Continuum Hypothesis, this fascinating survey allows readers to delve into mathematics as an organic whole rather than an empty drill in problem solving. With chapters largely independent of one another and sections that lead upward from basic to more advanced discussions, readers can easily pick and choose areas of particular interest without impairing their understanding of subsequent parts. Brought up to date with a new chapter by Ian Stewart, What is Mathematics, Second Edition offers new insights into recent mathematical developments and describes proofs of the Four-Color Theorem and Fermat's Last Theorem, problems that were still open when Courant and Robbins wrote this masterpiece, but ones that have since been solved.

Formal mathematics is like spelling and grammar: a matter of the correct application of local rules. Meaningful mathematics is like journalism: it tells an interesting story. But unlike some journalism, the story has to be true. The best mathematics is like literature: it brings a story to life before your eyes and involves you in it, intellectually and emotionally. What is Mathematics is a marvelously literate story: it opens a window onto the world of mathematics for anyone interested to view.Amazon.com Review
A 1996 revision ofa timeless classic originally published in 1941. Highly recommendedfor any serious student, teacher or scholar of mathematics. ... Read more

Customer Reviews (35)

excellent buy
Everything is in good condition as said.I am happy of doing business with the seller.
TKS.

typos galore, terrible layout
The book is littered with mangled formulas, mostly due to the fact the minus sign is missing from most formulas. This is completely unacceptable in any math book, but particularly so in a book aimed at beginners, who will probably feel bewildered by the huge amount of nonsensical formulas.

Adding insult to injury, the poor layout of Kindle edition makes the book hard to enjoy. Section headings often appear as orphans at the bottom of pages and many formulas are displayed as small, low-quality images. The low image quality is especially visible when reading the book on a Kindle 2 as opposed to for example the Kindle Mac app. Inlining formula images with running text is particularly annoying, since it deforms the text layout.

I was so annoyed by these problems that I tried to return this book, only to discover that returns are not allowed for digital books.

Do NOT buy the Kindle edition of this book.

A fascinating book.
For those who love mathematics, I can't praise this work highly enough. It's not designed so much to help the reader solve math problems as it is to impart a solid understanding of the fundamentals of the subject. Anyone who finds mathematics endlessly intriguing will be glad to own this book.

Math education at its best and most beautiful.
This book is so beautifully written that it epitomises an ideal of mathematics education. Everything is presented and explained clearly, which makes it suitable for the layman. The contents covered are also ambitious and wide-range. Topics are presented in a logical and satisfying manner. Natural numbers and the concept of induction are introduced early. Then a more thorough discussion of number system follows in Chapter 2 which concludes the background in arithmatic. After that the book shifts its focus to the opposite pole of mathematics where geometry (both euclidian and beyond) are presented in a way that algebra is employed to give greater understanding and appreciation of the topic and to allow the readers to understand the key role of algebra which helps bringing arithmetic and geometry together in a coherent manner.

Although I believe that the book is suited for lay people, we must not expect that the authors will hold you hand-in-hand while introducing you to the beauty of mathematics. The authors make clear that they want you to think. And think hard you will, if you want to get the best out of this book. By this, we have come to the greatest merit of this tome: sure, everything is clearly explained, however the most important messages are written not conspicuously but between the lines (and much of them in the exercises). We can only come to appreciate what the authors want to say when we commit ourselves in the required thinking part and discover for ourselves what the authors want us to know but didn't say. I think the book succeeds triumphantly in its writing style and is therefore difficult to be out done in this respect. Mathematics will continue to progress but this book will never become obsolete.

I do strongly encourage everyone who reads this book to do all the exercises in every chapter. You have no idea what you miss if you skip them; the are the crown-gem of this book. Don't be deterred if you cannot solve over a third of the problems. Just by thinking about them (albeit unable to solve) will move you closer to real understanding of the topics being discussed. Don't be appalled to know that answers are not provided in the back of the book. You know it when you solve it; this only heigthens the satisfaction of doing math.

I know many math-degree holders who hesitate to refer to themselves as mathematicians (they would just say "well, I had/did a degree in math."). I think that is because while they know a lot of mathematical facts and techniques, they never really understand or appreciate it in a fundamental way. This shows what is missing in our math education. This book is the ideal antidote.

Also check out books by John Stillwell. Most of his books are as good as this volume.

Incredibly thorough
Basic Math Quick Reference Handbook
This thorough coverage of so many aspects of mathematics forms a threshold that separates the interested enthusiast from the professional mathematician.
Couldn't be better. ... Read more

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Customer Reviews (5)

From Ore with love: old but instructive introduction to number theory.
Pros:
1. The book could teach basic number theory to a wide range of readers, from mathematically inclined high-school students to much more advanced lovers of mathematics. 2. It is enlivened by nice historical allusions.
3.The author shows and shares his fascination with the subject in the writing.
4. On a less lofty side, the font is large enough to avoid eye strain.

Cons:
1. First published 59 years ago, the book has to be dated. For example, many beautiful applications of number theory had been unknown at the time of writing.
2. Not all exercises require creativity, many of them are routine drills.

Bottom line:
If number theory is not your fortress, the book could strike a balance between enjoyable reading and learning.

A book for practically anyone
Ore's book is an excellent introduction to the fascinating topic of number theory.He takes his time explaining the history of numbers and goes into Euclid's algorithm so smoothly you hardly realize what you've learned.He discusses prime numbers and I was particularly delighted to see diophantine equations explained with lots of examples and an easy to follow method.The book is filled with interesting concepts, lots of examples, and good problems to do on your own.

At the end, for example, Ore talks of how number theory relates to geometry and I wish there were more of that in it.

I took this book on a very long trip, worked through many of the problems and simply found it a wonderful companion.If you get it, enjoy.One caution: if you already know some number theory you may find this book too simplistic.Still, it's worth having.

A book for practically anyone
Ore's book is an excellent introduction to the fascinating topic of number theory.He takes his time explaining the history of numbers and goes into Euclid's algorithm so smoothly you hardly realize what you've learned.He discusses prime numbers and I was particularly delighted to see diophantine equations explained with lots of examples and an easy to follow method.The book is filled with interesting concepts, lots of examples, and good problems to do on your own.

At the end, for example, Ore talks of how number theory relates to geometry and I wish there were more of that in it.

I took this book on a very long trip, worked through many of the problems and simply found it a wonderful companion.If you get it, enjoy.One caution: if you already know some number theory you may find this book too simplistic.Still, it's worth having.

Hamony?
A noted conjecture of the author's on the harmonic mean of the divisors is tucked unobtrusively in this pleasant reader: "Every harmonic number is even." See problem B2 in Richard K. Guy's Unsolved Problem's in Number Theory.

A good book (but not a great book). Very basic. For the more advanced historical approach, Andre Weil's Number Theory: An approach through history" is to be recommended. Or even Guy's book mentioned above.

Excellent theory interspersed with history
This book goes into detail on number theory, but it is often hard to follow with the history mingled with the theory.More advanced material is referenced without proofs.Two readers will especially like this book: those who want an introduction to number theory and those who want a good introduction to the history of number theory. ... Read more

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Customer Reviews (1)

Ideological
Goldfarb, "Poincaré against the Logicists." Poincaré complained that attempts to define arithmetic formally actually presupposed it, for example in using the concept "in no case" when defining zero. Goldfarb claims to "defeat" this objection as follows. "Poincaré is ... construing the project of the foundations of mathematics as being concerned with matters of the psychology of mathemtics and faulting logicism for getting it wrong." (p. 67). But "it is a central tenet on antipshychologism that such conditions are irrelevant to the rational grounds for a proposition. Thus the objection is defeated." (p. 70). But what about the question, central to Poincaré and many others, of whether it is possible to reduce arithmetic to logic? Goldfarb is apparently happy to dismiss this as an "irrelevant" matter of "pshychologism."

Dauben, "Abraham Robinson and Nonstandard Analysis." I have only read the incompetent section on Lakatos (section 2) of this chapter. Here Dauben offers a groundless and ideologically motivated attack on Lakatos' paper on Cauchy. First there is the nonsense about Robinson's non-standard analysis. Dauben writes correctly that: "There is nothing in the language or thought of Leibniz, Euler, or Cauchy (to whom Lakatos devotes most of his attention) that would make them early Robinsonians" (p. 180). This is all true, but it is also true that Lakatos never claimed otherwise, which is why Dauben must resort to underhand insinuations like this. Leaving this straw man aside, Lakatos wrote correctly that: "The downfall of Leibnizian theory was not due to the fact that it was inconsistent, but that it was capable only of limited growth. It was the heuristic potential of growth---and explanatory power---of Weierstrass's theory that brought about the downfall of infinitesimals" (p. 181). Dauben foolishly claims that "Lakatos apparently had not made up his mind" and "even contradicts himself" (p. 182) in acknowledging that Leibnizian calculus is inconsistent. This makes no sense. There is no contradiction. The inconsistency of Leibnizian calculus is even referred to as a fact in the first quotation. Dauben also claims that Lakatos is wrong because "the real stumbling block to infinitesimals was their acknowledged inconsistency" (p. 181). Why, then, did the calculus "stumble" only after two hundred years? If Dauben thinks that classical infinitesimal calculus "stumbled" before it had dried up, I suggest that he shows us what theorems it could have reached were it not for this obstacle.

Askey, "How can mathematicians and mathematical historians help each other?" Most of this article deals with haphazard and obscure notes regarding Askey's own historical research and does nothing to answer the title question. Askey's basic perspective is that mathematicians are well-meaning saints who do nothing wrong but that mathematical historians are incompetent and prejudiced in various ways. For example, Askey amuses himself with finding errors in Kline's history, and concludes that "it is clear that mathematical historians need all the help they can get" (p. 212). But it makes no sense to blame historians, for Kline was a mathematician. He obtained his Ph.D. in mathematics and was a professor of mathematics at a mathematics department all his career. Elsewhere Askey writes: "One cannot form an adequate picture of what is really important on the basis of current undergraduate curriculum and first-year graduate courses. In particular, I think there is far too much emphasis on the emergence of rigor and the foundations of the mathematics in much of what is published on the history of mathematics." (p. 203). The obvious lesson is for mathematicians to stop teaching lousy courses that trick students into thinking that rigour is a huge deal, etc. But no. That would entail admitting a flaw among the glorified mathematicians that Askey loves so much. So instead he nonsensically blames historians without further discussion. ... Read more

Product Description
Research shows that students learn best when, as opposed to simply listening or reading, they actively participate in their learning. In particular, hands-on activities provide the greatest opportunities for gaining understanding and promoting retention. Apart from simple manipulatives, the mathematics classroom offers few options for hands-on activities. However, the history of mathematics offers many ways to incorporate hands-on learning into the mathematics classroom. Prior to computer modeling, many aspects of mathematics and its applications were explored and realized through mechanical models and devices. By bringing this material culture of mathematics into the classroom, students can experience historical applications and uses of mathematics in a setting rich in discovery and intellectual interest. Whether replicas of historical devices or models used to represent a topic from the history of mathematics, using models of a historical nature allows students to combine three important areas of their education: mathematics and mathematical reasoning; mechanical and spatial reasoning and manipulation; and evaluation of historical versus contemporary mathematical techniques.This volume is a compilation of articles from researchers and educators who use the history of mathematics to facilitate active learning in the classroom. The contributions range from simple devices such as the rectangular protractor that can be made in a geometry classroom, to elaborate models of descriptive geometry that can be used as a major project in a college mathematics course. Other chapters contain detailed descriptions on how to build and use historical models in the high school or collegiate mathematics classroom. Some of the items included in this volume are: sundials, planimeters, Napier s Bones, linkages, cycloid clock, a labyrinth and an apparatus that demonstrates the brachistocrone in the classroom. ... Read more

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Pulled together for the first time, and paired with commentary from the world's most respected scholars, God Created the Integers presents history's extraordinary moments in math, culled from 2,500 years of history and 21 distinguished mathematicians, four more than the hardcover edition. Each chapter begins with a profile of one of these mathematical masters, followed by original printings of their relevant works. This new paperback edition includes the work of Euler, Galois, Bolyai, and Lobachevsky.

Readers get a window into the minds of these geniuses and can see the unfolding thought process as it leads, inevitably, to the high-water marks in mathematical thinking. This new edition comes with an index to make it a valuable and easy-to-use research and reference tool.Amazon.com Review
"God created the integers," wrote mathematician Leopold Kronecker, "All the rest is the work of Man." In this collection of landmark mathematical works, editor Stephen Hawking has assembled the greatest feats humans have ever accomplished using just numbers and their brains. Each of the 17 sections opens with a historical introduction of the featured author, and proceeds to a faithful translation of their most famous work. While most mathematicians will already have complete editions of Isaac Newton's Principia or Georg Cantor's Contributions to the Founding of the Theory of Transfinite Numbers, this book is unique in presenting just the best bits of these and other theoretical works. The collection spans 2,500 years and covers a vast range of theories: the parallel postulate, Boolean logic, differential calculus, and the philosophy of the unknowable among them. Dense with numbers, formulae, and ideas, God Created the Integers is quite challenging, but Hawking rewards curious readers with a look at how mathematics has been built. In contrast to the towering physical edifices of great civilizations of the past, Hawking writes, "The greatest wonder of the modern world is our understanding." --Therese Littleton ... Read more

Customer Reviews (33)

integers
most of the book is reprints of what other mathematicians have published
stephen hawking's comments on their stuff is pretty good, and can be used as kind of a guide to the evolution of math up to present day, ie into the age of computers and algorhythms

Exciting to read the great proofs but needs more editing
This book gives the ancient theories and proofs of math. It is thrilling to read, for example, proofs by Descartes and Archimedes showing the area of a circle and how to use geometry to depict the product of two numbers or their square root long before the modern tools of calculus were developed.In so doing you can learn how the area of a circle is r^2.(Archimedes said that a circle with radius r and circumference 2r has the same area as a right triangle with height r and baseline 2r.The area of this triangle is 1/2 base * height = 1/2 * 2r * r = r^2).What I just clearly explained here is not so clearly explained by Stephen Hawking.Hawking famously explained to the laymen in his book on the universe the theories of Einstein.But his editors in this book fell short perhaps unable to comprehend what they read.This book would be of better use to the laymen if Hawking proceeded as does David Foster Wallace in his own book on math to walk the reader through these proofs one small step at a time.Instead Hawkins tosses out these ideas in tersely worded passages that do not assist those who are less clever in understanding what is means.This book could take one a lifetime (or even longer) to distill.Its ideas so important, clever, and yes God-like in its beauty that each time I am able to understand one proof completely I plan to post my own explanation on the Internet for other students to read.BTW this book is in it's nth edition.Each subsequent edition 1,2,3, n-1 must have been a revision made to correct logical and technical errors in the mathematics.Might I suggest that version n+1 include some improvement in the prose as well.

math
Math, math, everywhere there is math. I have not finished this book. I will be a long time finishig this book, but it is great reading for an 11 hour flight to Europe. This is a book that can be read several times and more can be learned each time read. Not a late night book, it stirs the brain into overdrive!

Awkward layout
It's not a straight through linear read, but more of a pick the stuff that interests you and read it. The paperback version of the book is very difficult to handle due to the size, also the page formatting tries to use every ounce of space with the smallest of fonts. Not as good as I thought it would be.

interesting history
If you enjoy math and the history of science, you will like this compilation of works from ancient to modern times.Hawking's brief comments are interesting for prespective and some superficial explanation of the mathematics, but the main gist of this book is the reproduction of the seminal works themselves, from Euclid to Godel, living from 325 BCE to 1978 CE.
One gets a sense of the paradigm shift from geopetry to calculus to number theory and so on.For instance, it is fascinating that geometers calculated pi to between 3 1/7 and 3 10/71, but apparently did not come up with the limit concept that made calculus possible.Looking at their papers, it is amazing what they did with triangles and proofs, compared to what many of us probably struggled with in middle to high school geometry/ trigonometry.They were just one small insight away from the calculus, it seems, from looking at their diagrams!
By the middle of the book, the math surpassed my experience, but it was still interesting to try and follow along by skimming through it.
Is this book for a layperson?Only one with a strong interest in math, and even then, maybe not for everyone.It is not a textbook.I did not understand a lot of it, but I still enjoyed it.For a mathematician, this would be a fantastic gift, I should think. ... Read more

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The year 1897 was marked by two important mathematical events: the publication of the first paper on representations of finite groups by Ferdinand Georg Frobenius (1849-1917) and the appearance of the first treatise in English on the theory of finite groups by William Burnside (1852-1927). Burnside soon developed his own approach to representations of finite groups. In the next few years, working independently, Frobenius and Burnside explored the new subject and its applications to finite group theory.They were soon joined in this enterprise by Issai Schur (1875-1941) and some years later, by Richard Brauer (1901-1977). These mathematicians' pioneering research is the subject of this book. It presents an account of the early history of representation theory through an analysis of the published work of the principals and others with whom the principals' work was interwoven. Also included are biographical sketches and enough mathematics to enable readers to follow the development of the subject. An introductory chapter contains some of the results involving characters of finite abelian groups by Lagrange, Gauss, and Dirichlet, which were part of the mathematical tradition from which Frobenius drew his inspiration.This book presents the early history of an active branch of mathematics. It includes enough detail to enable readers to learn the mathematics along with the history. The volume would be a suitable text for a course on representations of finite groups, particularly one emphasizing an historical point of view.Co-published with the London Mathematical Society beginning with Volume 4. Members of the LMS may order directly from the AMS at the AMS member price. The LMS is registered with the Charity Commissioners. ... Read more

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