Customer Reviews

The History of the Calculus and Its Conceptual Development (Dover Books on Mathematics)

5 star:		(3)
4 star:		(5)
3 star:		(3)
2 star:		(2)
1 star:		(0)

 

 

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

Are you interested in the historical and philosophical development of the calculus? If so, then purchase this book immediately. In it, Boyer attempts to "fully counteract the impression of laymen, and of many mathematicians, that the great achievements of mathematics were formulated from the beginning in final form" (Boyer Back Cover). To do so, Boyer separates chapters into periods according to their conceptual importance. For example, the book begins with an analysis of Pre-Hellenic and Greek contributions, and ends with the nineteenth century's "Rigorous Formulation" of limit principles (Table of Contents). Throughout the text, the philosophical and religious ideas that impeded the progress of the calculus are thoroughly analyzed. Boyer's extensive and frequent references are of immeasurable help to readers, allowing them to consider the sources of Boyer's cornucopia of ideas. Boyer's eloquence and terseness effectively communicate the deepest of topics. Do not settle for the history of the calculus given by standard histories of mathematics because they have not paralleled, and most likely will not parallel the depth of Boyer's treatment.

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.

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.

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 <I>terminus ad quem</I> 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!)

This work has a serious defect, one that is characteristic of the 19th and first half of the 20th centuries. This defect is that of evaluating history retrospectively instead of rehearsing it progressively, as though the only significance of a thinker is what would later become of his thought. Naturally, this kind of history flatters us by telling us that we are at the pinnacle of understanding. Although the book reeks generally of this mentality, there are always specific signs of it. For instance, an author will say that the earlier thinker "anticipated" a later one. Or he will relate how one thinker "failed" to arrive at a later theory. As though the early thinker had nothing to do but sit around and try to anticipate those who came after him! As though he did not have his own distinct problems and concerns! Every thinker has a very definite relation to the past, to certain sources and traditions upon which he draws. But his relation to the future is indefinite. (In fact, his relation to the future is only a logical one formed by inverting the dependence that later thinkers have upon him.) Leaving aside the bad fruit of this wrongheaded method, we must at the very least judge it unjust to earlier thinkers that their remarkable insights are passed off as nothing more than anticipations or falling short of ourselves. In truth, it is we who are indebted to them, not them to us.

Second, the author seems to have a less than subtle grasp of the various ancient authors he cites. I winced when he repeated the modern misunderstanding that Plato was concerned that only ruler and compass be used in constructions, as if the validity of mathematics in Plato depended upon the senses. (For Plato, to have true and certain knowledge is to contemplate the pure Ideas as they are in themselves. The author is projecting early modern concepts upon Plato.) This kind of misrepresentation is all too conspicuous in the chapter on the Ancients. And he depends too much on Heath's work, which is known today to have conceptual problems.

Third, Boyer also seems to take for granted that his logicist views in mathematics are beyond dispute, despite the catastrophes of foundational mathematics. This insertion of one's own "correct" views into a pure history of other thinkers is something that really grates on me. What is surprising is that Boyer (like many other authors) seems to have been unaware of his presumption. If he wanted us to approve of his views, he should not have slipped them in as some objective criterion of historical judgment, but defended them on their own merits.

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

Puts to rest the notion that Newton and Leibniz "invented" the calculus. This book clearly shows how this mathematical tool originated and evolved from the ancient Greeks and highlights the contributions made by many scholars, including Newton and Leibniz, over the centuries. The evolution after the 17th century (Cauchy, Cantor, Dedekind, etc.) was fascinating to read as well. Recommended.

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.

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