Book Description

Customer Reviews (9)

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!)

What, calculus is boring? Never!
Most of us got our first glimpse of the fascinating history behind the calculus in first-year calculus. That is, we did if we were lucky -- for the fast pace in acquiring basic calculus skills leaves little extra time. Perhaps we managed to learn that Newton and Leibnitz are regarded co-discoverers of the calculus, but that their splendid contributions were marred by a bitter - at times positively ugly - rivalry.We may also have learned something about their precursors, for example Descartes, Fermat and Cavalieri.

If these glimpses left a taste for more, Boyer's "The History of the Calculus and Its Conceptual Development" is just the book. Boyer begins by tracing the calculus roots back to Ancient Greece. During this period two figures emerge preeminent: Eudoxus and Archimedes. Archimedes was a pioneer whom many consider the "grandfather" of calculus. But lacking modern notation he was limited in how far he could go.
The role played by Eudoxus is more ambiguous. He represents that vein of mathematics which treats "infinity" with the greatest caution - if not abhorrence. Although magnitudes are allowed to become arbitrarily large, they can never actually become infinite. This has given rise to two schools of thought: 1) those that consider a circle to be a polygon of infinite number of sides (completed infinity), and 2) those that allow that a circle can be approximated arbitrarily closely by means of polygons, but disallow this process ever being completed (incomplete infinity or "exhaustion" method). Both schools remain with us to the present.
Their relevance to calculus is this: the first gave rise to "infinitesimals" (infinitely small quantities); the second to the "limit" or "epsilon-delta" approach.

In chapters II and IV Boyer discusses the contributions of the precursors of Newton and Leibnitz. These include Occam, Oresme, Stevin, Kepler, Galileo, Cavalieri, Torricelli, Roberval, Pascal, Fermat, Descartes, Wallis, and Barrow. The tremendous contributions of Descartes are well known. Fermat came very close to anticipating Newton and Leibnitz. Barrow is important in that he was the mentor of Newton.

Chapter V deals with the works of Newton and Leibnitz, as well as their monumental feud. During this feud Newton often exhibited a cruel and vindictive streak. (There are those who think this aspect of his personality was a source of his power. Others, following Freud, attribute his powers to sexual sublimation. He never married.)

Chapter VI deals with the period of rapid development which followed after the methods of Newton and Leibnitz became widely known. As Newton was the more secretive, the methods and notation of Leibnitz gained the upper hand. The great luminaries of this period were the Bernoullis, Euler, Lagrange and Laplace. Benjamin Robins carried on the work of Newton in his home country, using Newton's notation and methods. However, this increasingly became a rearguard action.During this phase technique progressed at a tremendous rate, but the logical foundations of the calculus remained shaky. Many of these pioneers thought in term of infinitesimals (a type of completed infinity).

Chapter VII deals with the revolution that took place from approximately 1820 to 1870. During this time the foundations of the calculus were completely recast and put on a rigorous basis. The principal names associated with this phase are Cauchy, Riemann and Weierstrass. The results of this revolution were that "infinitesimals" were discarded.These were replaced by the now-familiar epsilon-delta methodology (limits) - a complete triumph for the followers of Eudoxus!

In chapter VIII Boyer seems to express the opinion that with the triumph of the epsilon-delta method the evolution of calculus has been completed. One cannot help but harbor a suspicion that this triumph is ephemeral.There are several reasons for this. Most beginning calculus student instinctively dislike the epsilon-delta formulation as something artificial. Maybe they are right. Just as the method of Eudoxus in geometry was largely made irrelevant by the discovery of irrational numbers, so one feels there may be something "lurking out there" which will "blow away" the deltas and epsilons. In fact, recent research in "non-standard analysis" seems to have rehabilitated infinitesimals so some degree. Finally, it is of great interest that the maximum rate of progress was during the period when infinitesimals (completed infinity) were allowed. Using apparently fallacious methods these pioneers obtained profound results - and rarely made mistakes!

In a lighter vein, an apparently serious problem with infinitesimals is that there appears to be a need for an unending chain of these: first-order infinitesimals, second-order infinitesimals, etc. Between every two "ordinary" numbers (finite magnitudes) lie infinitely many first-order infinitesimals. But, between any two of these lies an infinity of second-order infinitesimals, and so on. This endless chain brings to mind the following jingle: Big fleas have little fleas/ Upon their back to bite 'em /And little fleas have lesser fleas / And so ad infinitum. / Ogden Nash

The history of an amazing and extremely useful idea
Since Boyer writes from the perspective of a math professor in the thirties and forties, some of his style is dated. Nevertheless, his content is not and remains just as accurate as it was when first written. There are few mathematical tools that are more useful than the calculus and yet it is based on several abstractions that are never achieved. However, we act as if it they are, manipulating limits as if they were whole numbers and manipulating infinities as if they are real objects.
The original ideas that began the development of the calculus are very old, the first known exposition of the problems of limits is the well known paradox proposed by Zeno, which dates back to ancient Greece. Zeno's arguments involving the Tortoise and Achilles still serve as intellectual fodder for many a philosophical debate. Therefore, the second chapter deals with the mathematics of antiquity that began the long intellectual journey towards the dual creation of calculus by Newton and Liebniz.

While there were some advancements in the medieval years, they were relatively unsubstantial and therefore Boyer spends only a brief time with them. Unfortunately, he concentrates on the activity in Europe, ignoring some of the work in other parts of the world. The fourth chapter deals with the century before Newton, where the last of the foundation ideas were set down and Newton's giants did their work and puffed out their shoulders.
The fifth chapter is devoted largely to the parallel work of Newton and Liebniz, where they independently invented what we now call differential and integral calculus. While the utility of the new mathematics could not be denied, there were many people who found great fault in it. It is easy for us to think of these critics as short sighted, but in fact many of their arguments were valid. Despite all the genius of Newton and Liebniz, there were still many gaps in the calculus that had to be corrected, which is the subject of the remaining chapters.
Written at a level so that mathematicians and laypeople alike can understand the ideas and how they expanded over the centuries, this is a book that is still of use in histories of mathematics and the centuries long development of ideas.

Vivid history of the calculus, fascinating!
This book clearly shows how the underlying concept of the caculus had been developed from geometric intuiton to formal logical elaboration. It feels good to know that even Newton himself was having trouble defining the infinitesimals. Now I can understand why modern calculus books are filled with so many strict definitions of continuity and limits. Mathematicians had to establish rigorous formulation of the calculus to free themselves from the vague definition of physical realities which modern physicist cannot understand even now. ... Read more

Product Description
This Elibron Classics edition is a facsimile reprint of a 1861 edition by Macmillan and Co., Cambridge. ... Read more

Book Description
This book describes the development of what we would now regard as a class of statistical fitting procedures between 1750 and 1900. The book contains detailed algebraic descriptions of the fitting of linear relationships by the method of least squares and the closely related least absolute deviations and minimax absolute deviations procedures. The prerequisite is a basic course in mathematical statistics. The primary audience for this book will be statisticians concerned with the fitting of linear models. However, it will also be of interest to engineers and scientists concerned with the empirical determination of linear relationships. ... Read more

Book 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

Book Description

More than three centuries after its creation, calculus remains a dazzling intellectual achievement and the gateway into higher mathematics. This book charts its growth and development by sampling from the work of some of its foremost practitioners, beginning with Isaac Newton and Gottfried Wilhelm Leibniz in the late seventeenth century and continuing to Henri Lebesgue at the dawn of the twentieth--mathematicians whose achievements are comparable to those of Bach in music or Shakespeare in literature. William Dunham lucidly presents the definitions, theorems, and proofs. "Students of literature read Shakespeare; students of music listen to Bach," he writes. But this tradition of studying the major works of the "masters" is, if not wholly absent, certainly uncommon in mathematics. This book seeks to redress that situation.

Like a great museum, The Calculus Gallery is filled with masterpieces, among which are Bernoulli's early attack upon the harmonic series (1689), Euler's brilliant approximation of pi (1779), Cauchy's classic proof of the fundamental theorem of calculus (1823), Weierstrass's mind-boggling counterexample (1872), and Baire's original "category theorem" (1899). Collectively, these selections document the evolution of calculus from a powerful but logically chaotic subject into one whose foundations are thorough, rigorous, and unflinching--a story of genius triumphing over some of the toughest, most subtle problems imaginable.

Anyone who has studied and enjoyed calculus will discover in these pages the sheer excitement each mathematician must have felt when pushing into the unknown. In touring The Calculus Gallery, we can see how it all came to be.

Customer Reviews (10)

Another masterpeice by William Dunham
If you enjoyed "Journey through Genius" by the same author, you will also enjoy the present volume. It requires more math knowledge (at least a working knowledge of calculus), but the level is aimed at a bright high school AP student, or a college undergraduate I would recommend it for even serious mathematicians who would like to know more about how the present state of knowledge of analysis came about. I would especially recommend it for teachers and students of calculus. Too often, ideas which took literally centuries to mature are presented in finished form, as if some mathematician sat down one day and wrote out finished, rigorous theorems. Seeing how even venerable mathematicians like Newton and Cauchy got results without the rigour which we see as necessary today is an eye-opener, and should be an encouragement to experiment and "learn by doing", and not to be afraid to go boldly forth, even if you haven't dotted all the "i" and crossed all the "t".

stresses the important aspects.
wonderful book, adds mathematical context to the ideas developed. good to read along a textbook on analysis.

Great Read
If you are up on your math it almost reads like a novel.I can't say anything about it that hasn't already been said, but just affirm all the positive comments.If you like math you will love this book.

Very Good IF you have a solid background
And by "solid background", I mean a good understanding of honors-level algebra (high school senior variety) AND a reasonably thorough understanding of basic calculus notions such as limits, integration, and differentiation.Even if your background is not quite this strong, I feel you will understand parts of the Newton Leibnitz Euler chapters, but you may start to struggle a touch after that.And, as was pointed out in an earlier review, this book is NOT for the General Reader.Very far from it, in spite of professor Dunham's substantial skill as a science/math writer.(A pleasure to read the gentleman!)

In fine: I studied applied math at UCLA some 55 years ago, and I have not done any serious math since about 1970, so I may be overstating the requirements necessary to enjoy this work, given my own very dim memory of some key elements of mathematics and of the mathematical proof. But for those who feel qualified, this is yet another wonderful piece of work by a truly gifted author.

NafFebruary 2007; Los AltosCA

Great side reader for a calculus course
I wish this book had been around when I was taking calculus a few decades ago.It is extremenly well written and explains all the reasons why mathematicians had to introduce all the concepts and definitions you encounter in a calculus course.Reading this book on the side will tell you exactly why you're doing what you're doing, and where you are going.All students of calculus will benefit from this book. ... Read more

Product Description
After completing his famous Foundations of Analysis (See AMS Chelsea Publishing, Volume 79.H for the English Edition and AMS Chelsea Publishing, Volume 141 for the German Edition, Grundlagen der Analysis), Landau turned his attention to this book on calculus. The approach is that of an unrepentant analyst, with an emphasis on functions rather than on geometric or physical applications. The book is another example of Landau's formidable skill as an expositor. It is a masterpiece of rigor and clarity. ... Read more

Customer Reviews (1)

The meaning of analysis
This book would more correctly be described as a book on analysis, because of the completeness of its proofs. It assumes you know just arithmetic of natural numbers, integers, rational, real and complex numbers-that is to say, just how to sum, multiply, subtract and divide. Edmund Landau wrote a masterpiece, because nothing is left without proof. You will not find a single step missing. From arithmetic to all the concepts of calculus, like differentiation, integration, infinite series and sequences, this book contains the mathematics that most mathematicians should know presented in a perfect way. It is not easy reading, though. Landau strives to reach the perfect axiomatic presentation, so like Euclid's "Elements" the book is the clear and beautiful presentation of a doctrine. I dare say that no book in analysis approaches Euclid's ideal of presentation better than Landau's, never in the past and never in the future. What you will find in this book is, in its best appearance, truth, that will not change even in a thousand years. ... Read more

Book Description

Customer Reviews (11)

Tangentially integrated
Isaac Newton invented calculus in 1665 and 1666, but chose not to publish due to criticism (by Hooke) of his published work on light. Leibniz invented calculus independently ten years later and published his findings. Things seemed fine between the two men until, primarily through the actions of good-hearted meddlers, controversy was stirred up, words like "plagiarism" were used and bad feelings were had all around. Of the situation, one might wonder, "Who cares?" Well, smart guy, skilled researcher and appropriately named Jason Socrates Bardi did and does and so chose to compose an entire book on the subject. Unfortunately what he produced is unlikely to enlighten the reader much beyond the basic facts, which are set forth early on and detailed later, nor is it likely to entertain due to its repetitive nature (in facts and words), and the awkward tangents taken, whereby he switches from objective writer to, in the very next sentence, commentator, generally without even the use of parentheses. Some form of the word "society" comes up 11 times on page 189 and ten on page 191, "many" occurs five times in one paragraph on page 146, and "escape" is used thrice in one sentence (page 135), "He escaped again, his second escape apparently helped by the fact that he was allowed to escape." Transitions from storyteller mode to commentator mode occur regularly (Pp 120, 156, 167), for example, after explaining that Newton described in a notebook his use of a needle to perform experiments on his own eyes, the author writes, "I saw a copy of this notebook...[p 29]," and gets further off track in telling of an encounter with a mother and son viewing same. Upon describing a process for manufacturing phosphorus, he comments, "I get this picture when I think about it:...[p 105]" Additionally, he chooses to bestow a seemingly official name, used as a chapter title, on an eyebrow-raising occurrence (The Affair of the Eyebrow). And inexplicably writes, "As it was, court intrigues in Hanover at the time were enough to make a soap-opera-loving housewife blush [p 163-164]." Neither man achieves a clear victory in the wars, and although the topic is book worthy, the repetition and clumsy transitions take away from what would otherwise be interesting reading. Better books on math and science: American Prometheus by Kai Bird and Martin J. Sherwin, Obsessive Genius by Barbara Goldsmith, A Beautiful Mind by Sylvia Nasar, and The Double Helix by James D. Watson.

Great insights
I really enjoyed this book and found it offered a great many wonderful tidbits to fill in my understanding of the issues. My most recent previous reading on Leibniz was the wonderful book "The Courtier and the Heretic," which covers Leibniz' interrelationship with Spinoza and the two books fit nicely together. It is clear from Bardi's book that there are many more wonderful possibilities out there many of which have been available for years - a book relating Huygens and Leibniz for example. Perhaps this is one of the most wonderful aspects of a book like this. It points to many sources to explore if one is interested in following up. This book clearly details how the situation got so mixed up and why it will forever remain an embarrassment to those who value the advance of reason and wish human frailties would not create so many bumps on the path.

I did not check these reviews before picking up the book and (not plagiarizing them but independently noting them myself!) found the sheer number of editorial mistakes annoying. One wants to send it back to have it corrected out of habit. These are the sorts of mistakes Word doesn't let happen. I bet I could not even reproduce many of them here without Word automatically correcting them. But I agree this seems to be the editors fault not Bardi's since even if they were Bardi's the editor should have easily caught them. But I myself have seen multiple errors magically appear in a published text that were not there in the original. Perhaps the paperback is corrected? I did not see any mention of this on Bardi's web page either.

But I have a major point to question concerning Bardi's view that Leibniz's vortex argument has been disposed of by Newton's gravity. Would not Einstein's view of the curvature of space achieve essentially the same explanation of Leibniz'? in short, though the short history following the controversy seemed to make Newton's position on gravity the winner (not as an explanation of movements) hasn't more recent history at least shown both theories useful for different purposes and therefore both correct in context?

Perhaps my understanding of this issue is wrong? After all, in the short introduction to the Principia in "On the Shoulders of Giants" edited with commentary by Stephen Hawking he seems to suggest the same thing. What gives? What happened to Relativity?

This is a good read!
I thorougly enjoyed this book.I was not aware of the history of Newton and Leibnitz, and so this was a new subject for me.I really feel that in history classes we should read books like this, because it really opens up mathematics.I am going for phD later this year, and so I am starting to review my mathematics textbooks, such as discrete mathematics and calculus.Reading about the extraordinary men that created calculus and battled over it, made calculus seem to me like a living thing, and actually I am looking forward to reviewing my calculus
textbook!On the other hand, if you aren't a science geek, this book is still a good read, because it also gives us psychological insights into two brilliant men and the time period in which they lived.

Bardi Reestablishes What "Genius" Means
Jason Bardi wisely decided not to write a book about mathematics.Instead, "The Calculus Wars" is an informative story about a great era of mathematical discovery.We learn much not only about the primary figures, Leibniz and Newton, and their peers (such as John Wallis and Jacob Bernoulli), but also about the contemporary dilettantes and sycophants that buzzed around them, the most damaging of which were the nobility.Too often scientists and science have been used to prop the worthless ambitions of fops, even to today's Al Gore.

Bardi is quite right in noting the superiority of Leibniz's notation.He doesn't quite see that Leibniz's universal language was the beginnings of symbolic logic as developed only in the 1800's.Bardi misses one important (and debatable) point, and it is that Isaac Newton created calculus as an indispensable tool for the mathematical development of his new physics.It is the mark of his towering genius that a revolutionary new mathematics was for him simply a means to an end.For Leibniz, calculus was a form of verification that reason could triumph over everything, certainly over the vast landscape of his endeavors.Leibniz is much more the completed Renaissance man, whereas Newton is the scientist of a future that would be molded by his thoughts.

There are grammatical and typographical errors scattered throughout the book, which jar one's reading.These are clearly not Bardi's fault, rather, some numskull editor at the Avalon Publishing Group cut corners and rushed the book to print.Typical error:Bardi correctly states on page 237 that Newton died in March of 1727, but the incompetent editor didn't catch this line one paragraph later, "He was interred in the nave of Westminster Abbey on March 28, 1726...".

Despite such potholes, Bardi's book is good reading.In these days, when every Hollywood celebrity is called a "genius", it is good to reestablish the word by proper examples, such as Sir Isaac Newton and Baron Gottfried Leibniz.

Fascinating
I just got my copy of The Calculus Wars and have been reading it every night. I enjoy the story as well as the information I am learning about history. I am no math buff, but find the insight into both Newton and Leibniz lives interesting and filled with great drama.
... Read more

Book Description

Book Description
This is a lucid account of the highlights in the historical development of the calculus from ancient to modern times from the beginnings of geometry in antiquity to the nonstandard analysis of the twentieth century. It emphasizes the genesis and evolution of both fundamental concepts and computational techniques. The intended audience includes not only students of the history of mathematics, but also the wider mathematical community, specifically those who study, teach and use calculus. Among the distinctive features of this exposition are historically motivated exercises and carefully chosen illustrative examples. Numerous sections of the book are suitable for use in courses in introductory and advanced calculus as well as the general history of mathematics. ... Read more

Customer Reviews (1)

Useful resource, but dry and incomplete
The historical path is often more sensible than modern textbooks, as we see here in numerous cases: the logarithm should be understood as the area under the hyperbola y=1/x, Taylor series should be understood in terms of the Gregory-Newton interpolation formula, etc. But if there is one lesson history should teach calculus textbook authors it is this: power series. Power series were always indispensable and inseparable from the calculus at every stage of the development. Modern authors shoot themselves (and their students) in the foot by postponing power series as far as possible. Euler, in his Introductio, beautifully derives the derivatives of the elementary functions by power series methods, which is neat and systematic and makes use of concepts of great power and scope. By contrast, modern authors, suffering from rigour hiccups, insists that these derivatives must be deduced from "the definition" of the derivative, using horrendously ad hoc limit-manipulation tricks. This book is useful and certainly much better than Boyer's awful book, but it is still very far from being a satisfactory history of the calculus. In particular there is no physics, which is of course utterly absurd if it is to be a true history of the calculus. Also, it treats only the very basics of the calculus, essentially ignoring differential equations, several variables, the calculus of variations, etc. ... Read more

Customer Reviews (1)

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 whatýs going on, then you should read these summaries. Some of them will no doubt surprise you. ... Read more

Product Description
This volume is produced from digital images created through the University of Michigan University Library's preservation reformatting program. ... Read more

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

Customer Reviews (7)

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.

This book adheres to the highest editorial standards
With all due respect to the reviewer who found two errors in this book, these mistakes were corrected over 16 years ago. Our assumption is that he has a very outdated edition. The reviews of this book have been excellent and we are confident that it will help any student in pre-calculus. ... Read more

Book Description

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.

Book Description
This is a serious - but not solemn - textbook that attempts to make a clear, conceptual understanding of calculus accessible to all liberal arts students. It presents mathematics as growing out of the classical liberal arts to form a natural bridge between the humanities and the sciences, integrating the history and pedagogy of mathematics in a way that may be of interest to prospective teachers as well. Instead of a pre-calculus review, this book offers an historical development of much of the geometry and algebra needed, emphasizing the fundamental need for students to develop a clear style of writing. Calculus is here largely restricted to the study of algebraic functions, but all the usual aspects of the interplay between functions and derivatives are covered: optimization, instantaneous rates, Newton's method, freely falling bodies, antiderivatives, integrals, areas, volumes, etc. The fundamental theorem is prominently featured and carefully treated. A brief final chapter about the intellectual climate surrounding the development of calculus offers students further insight into the place of mathematics as an element in the history of thought. ... Read more

Customer Reviews (1)

Great Survey of the Calculus
Priestley's work is a gem in that it incorporates the full-orbed liberal arts approach to the calculus. Professor Priestley places the differential and integral calculus in the context of history (the best methodology for teaching any aspect of mathematics). You will also find a good number of brain-expanding exercises and real world applications strategically placed in the text. This book is a "must-have" text for any teacher of mathematics and for a student who desires to understand the calculus in the context of history, science, philosophy, and literature. This is education at its best! ... Read more

Book Description

Customer Reviews (5)

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).

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.

Thorough, intelligent and impeccably unprejudiced.
This book is a model of science history. Crowe manages to give one a clear view of the trends and moods of the time (1840-1900), the personalities of the various figures involved (Esp. Hamilton, Grassman and Gibbs), withoutsacrificing his most significant asset: the facts. ... Read more