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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

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Silvestre François Lacroix (Paris, 1765 - ibid., 1843) was a most influential mathematical book author. His most famous work is the three-volume Traité du calcul différentiel et du calcul intégral (1797-1800; 2^nd ed. 1810-1819) an encyclopedic appraisal of 18th-century calculus which remained the standard reference on the subject through much of the 19^th century, in spite of Cauchy's reform of the subject in the 1820's.

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From the Calculus to Set Theory traces the development of the calculus from the early seventeenth century through its expansion into mathematical analysis to the developments in set theory and the foundations of mathematics in the early twentieth century. It chronicles the work of mathematicians from Descartes and Newton to Russell and Hilbert and many, many others while emphasizing foundational questions and underlining the continuity of developments in higher mathematics. The other contributors to this volume are H. J. M. Bos, R. Bunn, J. W. Dauben, T. W. Hawkins, and K. Møller-Pedersen. ... Read more

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"An attempt has been made in these pages to trace the evolution of intellectual thought in the progress of astronomical discovery, and, by recognising the different points of view of the different ages, to give due credit even to the ancients. No one can expect, in a history of astronomy of limited size, to find a treatise on practical or on theoretical astronomy, nor a complete descriptive astronomy, and still less a book on speculative astronomy. Something of each of these is essential, however, for tracing the progress of thought and knowledge which it is the object of this History to describe. " ... Read more

Customer Reviews (1)

Excellent book of early astronomy
This starts with the ancient Chinese, then goes through the Chaldeans, Greeks, and Arabs, then Copernicus and others of the Renaissance, and lastly the 18th and 19th centuries. There are chapters about the telescope and other instruments, the sun, moon, planets and the stars.

The author does a good job of showing how astronomers used the findings of earlier astronomers to increase their own knowledge of the subject. It's amazing to read how much was known about astronomy in the past, and how accurate their findings were. It's also funny to read things which were thought to be true at the time when the book was written. Several people reported having seen a planet inside Mercury's orbit. One man thought Mars had artificially made canals with vegetation growing on their banks. There are lots more. Maybe in 100 years astronomers will be laughing at us for thinking that dark matter and dark energy exist.

The table of contents is active, which is unusual for these free books. There are footnotes and an index at the end.

This is a great book loaded with historical information. I recommend that you have at least a basic knowledge of astronomy before reading this book, because it's not written for beginners.
... Read more

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The problem of approximating a given quantity is one of the oldest challenges faced by mathematicians. Its increasing importance in contemporary mathematics has created an entirely new area known as Approximation Theory. The modern theory was initially developed along two divergent schools of thought: the Eastern or Russian group, employing almost exclusively algebraic methods, was headed by Chebyshev together with his coterie at the Saint Petersburg Mathematical School, while the Western mathematicians, adopting a more analytical approach, included Weierstrass, Hilbert, Klein, and others.

This work traces the history of approximation theory from Leonhard Euler's cartographic investigations at the end of the 18th century to the early 20th century contributions of Sergei Bernstein in defining a new branch of function theory. One of the key strengths of this book is the narrative itself. The author combines a mathematical analysis of the subject with an engaging discussion of the differing philosophical underpinnings in approach as demonstrated by the various mathematicians. This exciting exposition integrates history, philosophy, and mathematics. While demonstrating excellent technical control of the underlying mathematics, the work is focused on essential results for the development of the theory.

The exposition begins with a history of the forerunners of modern approximation theory, i.e., Euler, Laplace, and Fourier. The treatment then shifts to Chebyshev, his overall philosophy of mathematics, and the Saint Petersburg Mathematical School, stressing in particular the roles played by Zolotarev and the Markov brothers. A philosophical dialectic then unfolds, contrasting East vs. West, detailing the work of Weierstrass as well as that of the Goettingen school led by Hilbert and Klein. The final chapter emphasizes the important work of the Russian Jewish mathematician Sergei Bernstein, whose constructive proof of the Weierstrass theorem and extension of Chebyshev's work serve to unify East and West in their approaches to approximation theory.

Appendices containing biographical data on numerous eminent mathematicians, explanations of Russian nomenclature and academic degrees, and an excellent index round out the presentation.

Customer Reviews (1)

\footnote{\dots}
My SIAM review is quoted above. I am impressed how well amazon.com managed to re-interpret my words. Congratulations. ... Read more

Product Description
From the reviews of the second edition:

"This book covers many interesting topics not usually covered in a present day undergraduate course, as well as certain basic topics such as the development of the calculus and the solution of polynomial equations. The fact that the topics are introduced in their historical contexts will enable students to better appreciate and understand the mathematical ideas involved...If one constructs a list of topics central to a history course, then they would closely resemble those chosen here."

(David Parrott, Australian Mathematical Society)

"The book...is presented in a lively style without unnecessary detail. It is very stimulating and will be appreciated not only by students. Much attention is paid to problems and to the development of mathematics before the end of the nineteenth century... This book brings to the non-specialist interested in mathematics many interesting results. It can be recommended for seminars and will be enjoyed by the broad mathematical community."

(European Mathematical Society)

"Since Stillwell treats many topics, most mathematicians will learn a lot from this book as well as they will find pleasant and rather clear expositions of custom materials. The book is accessible to students that have already experienced calculus, algebra and geometry and will give them a good account of how the different branches of mathematics interact."

(Denis Bonheure, Bulletin of the Belgian Society)

This third edition includes new chapters on simple groups and combinatorics, and new sections on several topics, including the Poincare conjecture. The book has also been enriched by added exercises. ... Read more

Customer Reviews (4)

An intellectually satisfying history of mathematics
This is a brilliant book that conveys a beautiful, unified picture of mathematics. It is not an encyclopedic history, it is history for the sake of understanding mathematics. There is an idea behind every topic, every section makes a mathematical point, showing how the mathematical theories of today has grown inevitably from the natural problems studied by the masters of the past.

Math history textbooks of today are often enslaved by the modern curriculum, which means that they spend lots of time on the question of rigor in analysis and they feel obliged to deal with boring technicalities of the history of matrix theory and so on. This is of course the wrong way to study history. Instead, one of the great virtues of a history such as Stillwell's is that it studies mathematics the way mathematics wants to be studied, which gives a very healthy perspective on the modern customs. Again and again topics which are treated unnaturally in the usual courses are seen here in their proper setting.This makes this book a very valuable companion over the years.

Another flaw of many standard history textbooks is that they spend too much time on trivial things like elementary arithmetic, because they think it is good for aspiring teachers and, I think, because it is fashionable to deal with non-western civilisations. It gives an unsound picture of mathematics if Gauss receives as much attention as abacuses, and it makes these books useless for understanding any of the really interesting mathematics, say after 1800. Here Stillwell saves us again. The chapter on calculus is done by page 170, which is about a third of the book. A comparable point in the more mainstream book of Katz, for instance, is page 596 of my edition, which is more than two thirds into that book.

Petty details aside, the main point is the following: This is the single best book I have ever seen for truly understanding mathematics as a whole.

Relationship between algebra and geometry
It is a very good book.It has presented very clearly some difficult-to-understand relationship especially the link between algebra and geometry.It is a very good balance - history, Mathmatics, biography all mixed very well together.Highly recommended.

concise and well written summary of mathematics
Stillwell covers a lot of ground in a short undergraduate text intended to unify various mathematical disciplines.Naturally, _Mathematics_and_its_History_ begins with the early Greeks and in particular geometry (which is how mathematics was typically expressed then).The development of algebra and polynomial forms is described followed by perspective geometry.The invention of calculus and the closely related discovery of infinite series provide the backdrop for short biographies of prominent mathematicians (mostly dead white males to multicultural deconstructionists).The development of elliptic integrals (used in solving functions with specified boundary conditions such as a Neumann problem found in fluid mechanics).The treatment then diverges to physical problems including the vibrating string and hydrodynamics, together with a note on the renown Bernoulli family.Then Stillwell returns to the esoteric in complex numbers, topology, group theory and logic with some comments on computation at the end.Some mathematicians may find the overview to lack comprehensiveness, but the book's brevity for each topic and biographical notes present a balanced approach to the more casual reader about this important field of study and how it developed.

see below
This is an overall good text. It offers a very in depth history of many many mathematical ideas.It gets quite technical at times, which can be a good or bad thing, depending on what you are looking for. ... Read more

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Customer Reviews (1)

Convoluted
This book is marginally useful at best. It consists almost entirely of convoluted and muddled exposition of sample theorems and proofs of one mathematician after another without much cohesion. Baron's tendency to obscure or even severely distort the point of an argument may be illustrated by the following example, where she is in addition promoting the modern propaganda myth that 17th century mathematicians committed numerous mistakes and were guided by "a happy instinct" (p. 109) rather than reason.

"When [Kepler] argued by analogy he sometimes made mistakes. Unable to determine theoretically the length of an elliptic arc he argues that, since the area of an ellipse is equal to that of a circle, the radius of which is the geometric mean of the major and minor axes (area = pi ab = pi r^2, where a/r = r/b), then the circumference of the ellipse should correspondingly be equal to the circumference of a circle the radius of which is the arithmetic mean of the semi-axes, i.e. pi (a+b)." (p. 109)

The only one making a "mistake" here is Baron. What Baron portrays as a crackpot "analogy" is in fact a perfectly sound and intelligent argument. The argument is this: since the circumference of the circle with the same area as the ellipse is 2 pi sqrt(ab), the circumference of the circle must be somewhat greater (since the circle has the least possible perimeter for a given area). Kepler then proposes to use the *approximation* pi (a+b) for the circumference, being fully aware and completely explicit that it is an approximation only ("elliptica circumferentia est proxime...", we read on the very page of the Opera Omnia referred to by Baron)---and indeed a very good and very convenient approximation at that for ellipses of small eccentricity (such as the orbit of Mars, which is Kepler's interest). ... Read more

Product Description
Prepares students for calculus courses.Thoroughcoverage of first-year college math, including algebraic, trigonometric,exponential, and logarithmic functions and their graphs.Includessolutions of linear and quadratic equations, analytic geometry,elementary statistics, differentiation and integration, determinants,matrices, and systems of equations.Problem-solving strategies areincluded at the beginning of every chapter for each topic covered. ... Read more

Customer Reviews (7)

Unclear if this book is a good resource or not
As a previous reviewer pointed out, some rather simple mistakes on the 10th problem (not corrected in the 1985ed, btw) cast a shadow on the reliability of this book.Many other problems seem to be worked out correctly, but it's beyond me to check every one.The book is helpful in spots, but the fact that they answered "105%" for "What percent of 100 is 99.5?" leaves me more than a bit unsure of their subsequent answers.As a math teacher, I would suggest a basic textbook instead of this (you'll have graphics anyway, which are critical to understanding math well).

Examples
This book provides numerous examples that aid in understanding the complex world of precalculus.

disappointment
sure it is somewhat helpful but in limited terms.I have come across solved algebra and precalculus books sold for Turkish college students. I recommend these excellent books written by Nesime Aydýn, Kerim Yeniay, Hasan Özer, Mevlut Gündoðdu, Emrullah Eraslan to math-lovers all around the world because the math language is universal and the books are written an easy fashion to follow the steps in any way.

In a word: Excellent
This is the book to go to to remember how to do all those mathematical things, before calculus, we used to know but either forgot, took for granted, or shortcut and circumvented. Perfect example: solving inequalities involving absolute values -> the method I was using was producing right answers, however it was 'shotcutting' rather than solving the problem properly and robustly. Had a look inside...found an example...instant recollection and on my way again, the right way!

As far as I can tell there are not obvious mistakes (the review further down obviously has a really old copy or is just mistaken) and the coverage is quite comprehensive (though there are no problems for you to solve...as the title says, it is entirely solved problems and LOTS of them!). This isn't a book to teach you everything (though, I think if you worked through every example in whatever section, you would be a significantly better mathematician then when you started). As Einstein said, learng by example isn't just a way to learn...it's the only way to learn! It is a fine supplemental text and reference. The solutions are very clear, explicit and step by step. There are no logic jumps that can leave you wondering how the hell did they get from here to there? It is very systematic (even with explanations of what operation they are doing along the way- like a good teacher explaning how to do something without jumping steps).

Personally I regard the $18 as money well spent. It is an enormous book for the price, rich in content and extremely helpful. And given the price, what more could you want? It's a very useful addition to your library, if only as a reference work. There are basic attack strategies at the beginning of each chapter and masses of problems! Sure, the theory it covers is contained within the problems, not explicitly...hence you may need your course book along with this (or maybe as I said earlier: just try to learn from this book, which might be kinda weird and fun).

In all: well worth 5 stars! YOu can't expect a magical panacea for all your mathematical woes to be found within...but it tries! And it does deliver a great deal...

Happy Mathematics!

If you're mathematically competent but lazy, get this book!
This book is not a tutorial or self-teaching guide, it's more of a reference book. If you are not familiar with intermediate algebra and trigonometry this book will not be useful to you. But if you know the basic concepts of alg/trig you will find this book useful for that one difficult problem that is in every workset of your textbook. So, if you don't want to spend time concentrating on one problem, you can spend time searching for a similar problem in this book that will get you started on the original problem. But if you need the entire problem worked out with explanations this is not the book for you. ... Read more

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Customer Reviews (6)

Unsubstantiated clichés
This light-weight book spends much of its time gullibly asserting unsubstantiated clichés and bombastically touting Cauchy's alleged "revolutionary transformation" (p. 15) of the calculus. So, for example, we are told that "Abel's reaction to the Cours d'analyse was almost like a religious conversion" (p. 13), for which the only evidence offered is a one-sentence quotation in which Abel calls Cauchy's book excellent. Needless to say, only a second-rate historian blinded by a predetermined agenda could extrapolate a "religious conversion" from such flimsy evidence. Similarly, Grabiner uncritically swallows the party line regarding the motivations for rigourisation; which includes, for example, grotesque exaggerations of the impact of Berkeley's trifling critique. We are told that "Lagrange took Berkeley's criticisms with the utmost seriousness" (p. 27) and "became so convinced of the validity of Berkeley's criticisms that he could not remain content with the existing foundations" (p. 37). Not even the flimsiest of evidence is offered in support of these claims; instead we read in the endnotes that "unfortunately, there is no evidence about when, if ever, [Lagrange] read [Berkeley's critique]" (p. 189). The "unfortune" referred to here is that of not finding one's predetermined thesis borne out by evidence; an unfortune that hampers the book throughout.

I was always told Newton and Leibniz invented Calculus?
At least that's what every math teacher I've ever had stated.And that is true, they did invent it.But, the really fascinating thing is: they had no idea why it worked!This was a stunning revelation to me.Often times mathematics education is presented as a laundry list of 400 years of theory and results with little motivation of how things came about or why the theorems are relevant.Maybe exceptional instructors talk about these things, I don't know.The idea that these demi-gods didn't understand why Calculus worked casts the subject (and in a way all of mathematics) into a different light.It really has been a journey of discovery.

In this book the author explores how the logical foundation of Calculus was discovered over time, mainly in the 19th century long after the subjects inventors had passed on.This book isn't especiallylight reading.I think a course in Analysis (Advanced Calculus at some schools) is necessary to understand the more mathematical parts of the book (that is, most of the book).Also, there is very little biographical discussion about the mathematicians mentioned in the book.This is a book that explores the ideas and trends that lead to Cauchy's work, not about Cauchy himself.

Also, the book excludes Bolzano from the title, although he is also a prominent figure in the discoveries.Although excluded from the title he is not excluded from the material.

Highly Recommended - But Grabiner's thoughtful, detailed work requires careful reading
The Origins of Cauchy's Rigorous Calculus by Judith V. Grabiner is more technically challenging thanmany books on the history of mathematics.A year or two of calculus is a prerequisite for full appreciation of Grabiner's work; a class in real analysis would be helpful. Grabiner's approach is scholarly, and does require careful, thoughtful attention. At points I found it useful to have an introductory analysis text nearby.Nonetheless, I fully enjoyed this fascinating book. (My university classes included several applied math courses.In recent years I have developed an interest in mathematical logic and analysis.)

Augustin-Louis Cauchy's lectures at the Ecole Polytechnique in Paris in 1820s played the key role in focusing interest on the development of a rigorous basis for calculus; the epsilon and delta notation first appeared in their now standard roles in Cauchy's lectures in 1823.Based on Cauchy's work, Abel, Riemann, Weierstrass, Dedekind, Cantor, and others subsequently made major contributions to analysis in the nineteenth century. Grabiner does not discuss this later work in any detail.

Augustin-Louis Cauchy's precise definition of the limit and his fundamental definitions and theorems on continuity, convergence, the derivative, and the integral were quickly accepted.Grabiner observes that Cauchy's work was so superior to earlier efforts that today it seems to have emerged from a void and to be a unique creation of genius.

Grabiner demonstrates, however, that not only does Cauchy's work owe much to previous efforts by Newton, Maclaurin, Euler, d'Alembert, and especially Lagrange, but that theologian Bernard Bolzano in Prague independently had many of the same ideas as Cauchy.Unfortunately, Bolzano's work had little immediate impact as it was largely published in either obscure eastern European journals or in personally funded pamphlets.

Grabiner's first chapter concisely establishes the importance and lasting influence of Cauchy's work. The subsequent five chapters examine how Cauchy himself was influenced by earlier mathematicians and by his contemporaries.Chapter 2, The Status of Foundations in Eighteenth Century Calculus, explores why earlier mathematicians had seemingly little interest in developing a rigorous foundation for calculus.

The next chapter, The Algebraic Background of Cauchy's New Analysis, argues that the tools, especially the algebra of inequalities, that Cauchy required to prove his fundamental theorems were products of the eighteenth century. Key topics include the theory of algebra and the certainty of universal arithmetic, Lagrange's contributions to approximation techniques, and other eighteenth century efforts to measure the speed of convergence and bounds on errors.

Having established a historical framework, Grabiner focuses more closely in Chapters 4, 5, and 6 on Cauchy's definitions and theorems. Chapter 4 is titled The Origins of the Basic Concepts of Cauchy's Analysis: Limit, Continuity, Convergence. The final two chapters examine his theory of the derivative and the integral.

Some sections are a bit dry (not parched, however), while other chapters flow quite smoothly.I found it awkward flipping back and forth from the text to the end notes, and I eventually began using two book marks to keep track of my locations. There is also an extensive bibliography.

Moreover, an appendix contains Grabiner's translation of several of Cauchy's key proofs found in his Cours d'analyse and his Calcul infinitesimal. These proofs seem surprisingly modern even though they date from the 1820s.

Dry Writing, Fascinating Subject
Grabiner details how Cauchy re-purposed and built on the work of his predecessors to help transform the calculus into the rigorous system we know today. While her writing is rather dry, and at times repetitive, The Origins of Cauchy's Rigorous Calculus provides a glimpse into a fascinating period of math history. Caution: This is a book about math, not a biography. A year of calculus is pretty much a prerequisite for reading this book. For those with less mathematical training, or an interest in the earlier history of the calculus, I'd recommend Eli Maor's excellent e: The Story of a Number.

Reference work for History and Philosophy of Science
This is primarily a reference work for history and philosophy of science researchers, academic or amateur, with a special interest in mathematics, and in particular, the foundations of the calculus. It helps the reader to appreciate how much slow and painstaking effort was required to build modern knowledge. My main interests here are to be able to read the work of a master -- i.e. some of Cauchy's original papers translated into English, in the appendix, and to browse through the main text from time to time for enlightening historical snippets and elucidating explanations of the evolving development of the calculus. Reading original or seminal works always seems to have a more profound effect on apprehension than any number of derivative or subsequent texts -- e.g. A. Einstein, "On the electrodynamics of moving bodies" (just the part on synchronising clocks) or the paper by Claude E. Shannon on switching algebra, "A symbolic analysis of relay and switching circuits" showing how to use boolean algebra for this purpose. It would be nice if more works of Cauchy, Lagrange, Euler and many other mathematicians were readily available in English. The 4-star rating is only a subjective measure of my own enthusiasm for the book. As a literary, scholarly work in the field of History and Philosophy of Science, Grabiner's contribution may be unparalleled. ... Read more

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Grandoise claims
The title is a bit misleading; this book represents Marx's efforts to put Calculus on a sound, rigorus footing.

As a mathematician, I have to say that Marx succeeds only in moving the handwaving from one area toanother.

If the author was not a mathematician, he should have made anattempt to familarize himself with the actual rigorization of Calculus inthe nineteenth century (in, for example, the work of Cauchy).If theauthor was a mathematician, he most certainly should have known better.

Icannot recommend this book to anyone who does not have a solidunderstanding of mathematics. ... Read more

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Discusses the results from a study conducted as a part of a larger effort by NSF to evaluate the impact of reform at the undergraduate level. Softcover. ... Read more

Customer Reviews (2)

Calculus reformers need to be more self-critical
This book is a useful synthesis of research on calculus reform.

A positive aspect of reform courses is that they do away with the authoritarian presentation of static theory. "Many students have expressed that they feel more positive about the active role that they are able to take in the learning process while in reform courses" (p. 40). Also reported are "significant positive increases in attitudes about their capacity to think mathematically and contribute valuable insights to problem-solving endeavours" (pp. 38-39).

Also positive is how reform courses do away with the aversion to visual thinking that infested the mathematical community in the 20th century. "Students ... feel that the more visual approach taken by some reform courses is very helpful" (p. 41).

Student's negative reactions are more interesting, since they illustrate how reformists like Ganter are blind to the shortcomings of their own movement.

"Student feedback on reform calculus textbooks was generally negative" (p. 41). As it should be, since the staples that make up these books are stupid "real world" applications in place of the wonderful classical applications, faked and bottled "discovery learning," glossy pictures of baseball players, etc. But Ganter refuses to listen and sweeps this under the rug with the ridiculous rationalisation that "Students ... readily admit ... that their problems with reading the book were actually the result of not knowing how to read mathematics" (p. 40).

"The biggest opponents of reform efforts are usually students who have been very successful in mathematics ... Students in this particular group often complain that reform classes are 'not mathematics' ... and are afraid that they will be at a disadvantage in future courses because they have not been doing 'real calculus.'" (pp. 37-38). These are good arguments; if Newton was alive today he too would say that reform calculus is, to a considerable extent, not real calculus. This is so for example because of the reformers' retention of the limit concept, aversion to fluency in calculations, bogus applications, etc. But again a rationalisation is forthcoming to prevent reformers from listening to reason. "The negative responses of these seemingly successful mathematics students toward reform may simple be a reflection of positive experiences that most of these students have had with traditional mathematics. It is not surprising that they would resist a course that is so different from the courses with which they have had so much success." (p. 38).

I say: what is the point of all this research if the interpretation is so blatantly self-serving? When students don't like your books you say that they are incapable of reading any book. When intelligent students don't like your course you say that they are against any change. If you are so determined to concoct opportunistic and unsubstantiated excuses for the shortcomings of your reform program then you might as well not waste your time carrying out the research in the first place.

While good students are apparently too privileged to count to the politically correct reformers, it is with great pride that the reformers report increased retention rates and improved confidence in weaker students, as if this was an end in itself. However, there is very little evidence that this confidence is warranted (or that high retention is a good idea):

"Virtually all of the research institutions [using technology in calculus reform courses] reported greater conceptual understanding with equivalent or lower computational skills. ... Other types of institutions were ... mixed in their results, with less conclusive studies that point only to the possibility of improved performance." (p. 27). "Results from studies using a common computational exam [for both reform and traditional courses] to compare achievement were less conclusive, with 50% resulting in equal performance between reform and traditional groups and another 45% concluding that the traditional students had significantly higher scores ... Interestingly, very little formalized testing has been conducted to compare areas of achievement stressed by reform courses, such as conceptual understanding, critical thinking, or technical writing." (p. 28).

Necessary reading if you are responsible for calculus
For decades, there is been a strong wind of change in the teaching of calculus. While much of it has been a result of the advance of technology, there are other factors driving the change. This short book is a summary of some of the consequences of the changes. It should come as no surprise that the results are all over the place. Nevertheless, if you are part of the changes, are about to become a part or just want to know what�s going on, then you should read these summaries. Some of them will no doubt surprise you. ... Read more

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Hardbound. ... Read more

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This book presents first-year calculus roughly in the order in which it first was discovered. The first two chapters show how the ancient calculations of practical problems led to infinite series, differential and integral calculus and to differential equations. The establishment of mathematical rigour for these subjects in the 19th century for one and several variables is treated in chapters III and IV. The text is complemented by a large number of examples, calculations and mathematical pictures and will provide stimulating and enjoyable reading for students, teachers, as well as researchers. ... Read more

Customer Reviews (9)

not the standard course book, but nevertheless a truly GREAT book!
Starting with the bottom line, THIS IS NOT a first-year undergraduate course in Mathematical Analysis and should never be used as such (you wouldn't use Hardy's A Course in Pure Mathematics, either, for all its venerable merits).

If you ARE doing first-course Analysis, you'll have to stick to whatever textbook your professor chooses (hopefully NOT Rudin's masterpiece). If you get stuck with that, you will do well to search for a more user friendly alternative or supplement in the line of BINMORE or BRANNAN's book (there are others, for the market is flooded in this area), some good text that emphasizes the fundamentals taking his time, that emphasizes the practical connection between Analysis and (high-school) Calculus, and that uses a lot of visual illustration for good. If you remain stuck still, then you will have 1) to rethink the wiseness in choosing a career that requires advanced mathematics, and/or 2) as a last resource there is the "Yet Another Introduction to Analysis", which isn't the standard textbook either, but it can get you unstuck and on safer ground.

Then, what on Earth do we need HAIRER and WANNER's Analysis by it's history for?
For starters, it's a VERY BEAUTIFUL book (in the line of Proofs by the Book, if you know what I mean, but a step lower). The explanations are right and straighrforward, at an accessible level that reminds me of that other gem of a book, DUNHAM's Euler: The Master of Us All. It has a lot of examples and graphic elements and takes you into a wonderful journey into what is rather more Calculus than Analysis, alohg the path taken by the great classics (without being a book on the History of Calculus -try EDWARDS for that, and forget Boyer's-). In addition, if you just don't read casually the book for the sheer pleasure of it, but you DO WORK through it and do the exercises, you will end with a sure grasp of the fundamentals of Calculus (and thet is what Analysis is supposed to be), a surer grasp than you would have memorizing the whole of Rudin's consecrated and masterful text, Another reviewer takes issue with the many short quotations that decorate the book, as epigraphs to every little section, that are intended to maintain a historic flavor. Some of them make good sense, and a few others are rather quizzical or downright cryptic: all in all, it's a rather idiosyncratic and funny feature of the book, but it's not a matter to take issue with.

IN SUMMARY: it's NOT supposed to be a standard textbook nor should it be used as one. But if you think that Calculus (and Analysis) is the greatest endeavor mankind has engaged in -take that "com grano salis"- and you remember still your high-school Calculus although you're not a working mathematician, but something more on the amateur side of it... if you appreciate mathematical books looking the beautiful way they should all look like (and remember -or get outright- Proofs from the Book)...
if you'd love getting a refresher on Calculus cum Analysis, THEN PLEASE DON'T MISS that book, so very clearly and beautifully conceived and written. One book like this occurs only a few times in a century, unfortunately you and I haven't written this one, at the very least let's share the joy of reading it, at whatever depth level we feel like!

contains very goodhistorical perspectives on analysis
This very interesting book contains very goodhistorical perspectives on analysis. If you want to know how things like trigonometric functions, logarithms, infinite series, differential and integral calculus and differential equations come about (but written from a modern viewpoint), then this is the book for you. It is not a book for casual reading like E T Bell's Men of Mathematics, but the reader will learn a lot of college and undergraduate mathematics along the way.

A quite magnificent book
I return to this book again and again just out of sheer pleasure. The depth of scholarship of the authors shines through on every page and the choice of historical material is fascinating.Topics like compactness and uniform convergencecan here be seen to have arisen out of genuine necessity-they are not (as would seem from other books)mere names in a standard syllabus. If you have any mathematica interest at all, take this book on holiday and sleep with it under the pillow to extract more from it by osmotic pressure overnight.

A Good Mix of Calculus and its History
This books gives a unique approach to Calculus using its historical development.The most notable feature of the book is that the order of topics is reversed from what has become standard in current textbooks.It begins with the analysis of areas and volumes.This is followed by derivatives, continuity, and the notion of function.This is the order in which analysis developed, but not the order one would follow if building understanding of the subject from a foundation upward.Historically, the foundations were laid last.

The book is not intended as a history of analysis.It is rather intended as a textbook or reference in which the topics are presented in historical order.The historical background is intended to give insight into a modern view of the subject.It accomplishes this admirably.

The book is filled with examples, quotes, vignettes, historical background, computer graphics, and copies of original documents.Special topics are interspersed throughout.The book gives us a fresh and envigorating view of Calculus.It is an invaluable resource.

A somewhat useful scrapbook with a poor second half
Chapters 1 and 2 treat classical differential and integral calculus. This is a disorganised mess of historical and mathematical tidbits. It's not a great place to learn calculus, but it's good side reading since there are many interesting topics, some of which are often neglected in today's books: continued fractions (!), complex functions already on page 56, an interesting section on differential geometry, Euler-Maclaurin summation, etc. The authors also have the very commendable habit of including charming facsimiles of figures from original works.

Chapter 3 "Foundations of Classical Analysis" and chapter 4 "Calculus in Several Variables" are almost completely ahistorical. The "by its history" part of the exposition is restricted to some scattered superficial remarks, including silly nonsense such as that if Leibniz had know of the intricate progression of theorems needed for a modern proof of the "fundamental theorem" then "he might not have had the courage to state and use this theorem" (p. 239). And in another parodic misuse of the historical perspective, the authors introduce Descartes's folium merely for the purpose of practising the determination of stationary points (p. 322)---of course, Descartes introduced the folium for a much more interesting purpose, but to learn that story we must look for an "Analysis by Its History" book worthy of its name. ... Read more

Product Description
The first large-scale study of the development of vectorial systems, awarded a special prize for excellence in 1992 from France’s prestigious Jean Scott Foundation. Traces the rise of the vector concept from the discovery of complex numbers through the systems of hypercomplex numbers created by Hamilton and Grassmann to the final acceptance around 1910 of the modern system of vector analysis. Concentrates on vector addition and subtraction, the forms of vector multiplication, vector division (in those systems where it occurs), and the specification of vector types. 1985 corrected edition of 1967 original.
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Customer Reviews (6)

great book
This is a great book.It's refreshing to read a work of history by a historian, rather than the usual assortment of stories that you hear mathematicians themselves tell.The problem with that is that history books can be difficult, but this is very light, and also (as you can already tell from the title) structured like an unfolding drama.Anyone with knowledge of vector calculus and related subjects (!) could take great pleasure in reading here about the emergence and early history of the subject.Not only Grassmann, Heaviside, Gibbs, but several other lesser known figures are woven into the tale.It's a shame it's out of print.Now I feel like I lucked out, finding my copy for sale in a local bookstore.

Far More Exciting Than I Would Have Ever Dreamed
The vector story is very smart, very passionate, and very, very, very good.

To most science and engineering graduates, nowadays, the algebra and calculus of vectors, i would imagine, strikes us all as a tradition that might have been handed down from the ancient Greeks.But such a sense of historical omnipresence stands in sharp contrast to the actual story of their historically brief existence and the extraordinarily dramatic events that have led to their widespread adoption.

This book is that story.

A reception-of-ideas history of vectorial systems
The story of vectorial systems is the story of a search for an algebra of space. In chapter 1 we see that the need for such a theory was recognised already by Leibniz. We also study the rise of the geometry of complex numbers. Since complex numbers are an extremely successful fusion of plane geometry and algebra, one is tempted to look for a three-dimensional number system to do the same for space. Hamilton did so (chapter 2), and although he had to settle for four-dimensional quaternions, their "vectorial part" may still serve the purpose of an algebra of space quite well. Grassmann achieved much the same things when working to form a sort of general algebra of multidimensional magnitudes (chapter 3). In fact, Grassmann didn't even know about the geometry of complex numbers, and had to be told about it by Gauss. As is perfectly sensible, the ideas of Hamilton and Grassmann were poorly received. Both were inclined to an annoying "metaphysical style of expression" (Hamilton's phrase; p.36), and neither of them solved a single outstanding mathematical problem. One instead needs to be "astounded" by things like "the simplicity of the calculations resulting from this method" (Grassmann; p. 56). Basically this is what happened once vectorial ideas were freed from the smothering love of their creators (chapter 4); for instance we have Maxwell claiming that vector methods are useful "especially in electrodynamics" where things "can be expressed far more simply by a few expressions of Hamilton's, than by the ordinary equations" (p. 135). By now all the main ideas of the modern theory is in place, so the rest of the story is less interesting. A new generation began to detach vector ideas from quaternions (chapter 5), which led to a heated debate with quaternionists (chapter 6), but of course the reformists succeeded and the modern formulation of the theory was well established by the turn of the century (chapter 7).

On the whole, this book is little more than a compilation of historical information. Crowe barely treats the mathematics at all, and certainly not to the extent that would be necessary to understand "the evolution of the idea of a vectorial system".

Interesting summary of the history of an important idea
Although several others made important contributions, this book is primarily a study about the ideas of four people: Hamilton, Grassmann, Gibbs, and Heaviside. Hamilton's creation of the algebra of quaternions, while an important mathematical innovation, was thought of in many minds as primarily a physical tool, to be used in many of the applications that today are done by vectorial methods (and, in fact, the terms "scalar" and "vector" were invented by Hamilton, but with slightly different meanings than their present ones). Grassmann developed a quite different system, much closer to our present vector algebra, but unrecognized because of his obscurity and his books' unreadability. The true founders of modern vector analysis were the American physical chemist Josiah Willard Gibbs and the British physicist Oliver Heaviside, working independently of each other. What is interesting is that both Gibbs and Heaviside arrived at identical systems by modifying Hamilton's quaternion algebra to make it more accurately reflect the needs of physical scientists. While both Gibbs and Heaviside started with Hamilton's methods, the system they both arrived at was closer to Grassmann's in structure. And all this is clearly put forth in Crowe's book.

One other thing that the book makes clear is that J. Willard Gibbs, far more humbly than most scientists involved in priority disputes, clearly recognized that Grassmann had anticipated his ideas, although Grassmann's books had not come to Gibbs' attention until Gibbs had completely worked out his own system. And Gibbs, though he had based his ideas on Hamilton's, also recognized that Grassmann had the superior approach. (Though this may have NOT been a sign of humility, because in this regard Gibbs ended up using Grassmann's ideas to justify his own.)

Crowe's book is very readable, makes all these points quite clearly, and is highly recommended if you are interested in the subject.

Thoughtful, Detailed History of Vector Analysis
How were the concepts of vector analysis developed?How did modern vector notation become widely accepted?Who were the key players and why did quaternions fail to gain acceptance?This book is extensively documented,scholarly in its approach, sometimes a bit slow, but overall it is afascinating look at these specific questions as well as the fundamentalissue of what factors promote or delay acceptance of revolutionary ideas inscience and mathematics.

I did not become immediately engaged withCrowe's style and even set the book aside after reading the prefaces andfirst chapter.A few months later I returned to chapter two (in part dueto a previous reviewer's high rating).And what a surprise - I suddenlyfound myself intrigued with Crowe's discussion of Sir William Hamilton'ssingle minded focus on quaternions, the perseverance and genius of HermannGrassmann, the critical roles played by Peter Tait and James Maxwell, andthe pragmatic way in which Josiah Gibbs and Oliver Heaviside independentlyextracted key vectorial concepts from Hamiliton-Tait's quaternionanalysis.

Crowe's book was originally published in 1967 by University ofNotre Dame, Dover reprinted it in 1985, Crowe recieved the Jean Scott Prizeby the Maison des Sciences de l'Homme (Paris)in 1992, and Dover reprintedit again in 1992.Dover should be commended for making such reprintsreadily available at affordable prices.

The discussion of Hamilton'squaternions does not require familiarity with quaternions, but some prioracquaintance might be helpful. I encountered quaternions in another Doverreprint: Matrices and Transformations by Pettofrezzo.Section 2-3introduces quaternion notation, simple manipulations, and shows thataddition and multiplication of quaternions is isomorphic with twoparticular sets of matrices.

Has quaternion analysis survived?SeeQuaternions and Rotation Sequences: A Primer With Applications to Orbits,Aerospace, and Virtual Reality by Jack Kuipers.The reviewsby readers are all five stars. ... Read more

Product Description
Shortly after the invention of differential and integral calculus, the calculus of variations was developed. The new calculus looks for functions that minimize or maximize some quantity, such as the brachistochrone problem, which was solved by Johann Bernoulli, Leibniz, Newton, Jacob Bernoulli and l'Hôpital and is sometimes considered as the starting point of the calculus of variations. In Woodhouse's book, first published in 1810, he has interwoven the historical progress with the scientific development of the subject.

The reader will have the opportunity to see how calculus, during its first one hundred years, developed by seemingly tiny increments to become the highly polished subject that we know today. Here, Woodhouse's interweaving of history and science gives his special point of view on the mathematics. As he states in his preface: "Indeed the authors who write near the beginnings of science are, in general, the most instructive; they take the reader more along with them, show him the real difficulties and, which is the main point, teach him the subject, the way they themselves learned it." ... Read more

Product Description
Webster's bibliographic and event-based timelines are comprehensive in scope, covering virtually all topics, geographic locations and people. They do so from a linguistic point of view, and in the case of this book, the focus is on "Differential and Integral Calculus," including when used in literature (e.g. all authors that might have Differential and Integral Calculus in their name). As such, this book represents the largest compilation of timeline events associated with Differential and Integral Calculus when it is used in proper noun form. Webster's timelines cover bibliographic citations, patented inventions, as well as non-conventional and alternative meanings which capture ambiguities in usage. These furthermore cover all parts of speech (possessive, institutional usage, geographic usage) and contexts, including pop culture, the arts, social sciences (linguistics, history, geography, economics, sociology, political science), business, computer science, literature, law, medicine, psychology, mathematics, chemistry, physics, biology and other physical sciences. This "data dump" results in a comprehensive set of entries for a bibliographic and/or event-based timeline on the proper name Differential and Integral Calculus, since editorial decisions to include or exclude events is purely a linguistic process. The resulting entries are used under license or with permission, used under "fair use" conditions, used in agreement with the original authors, or are in the public domain. ... Read more

Product Description
This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more