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The History of the Calculus and Its Conceptual Development (Dover Books on Mathematics), by Carl B. Boyer
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This book, for the first time, provides laymen and mathematicians alike with a detailed picture of the historical development of one of the most momentous achievements of the human intellect ― the calculus. It describes with accuracy and perspective the long development of both the integral and the differential calculus from their early beginnings in antiquity to their final emancipation in the 19th century from both physical and metaphysical ideas alike and their final elaboration as mathematical abstractions, as we know them today, defined in terms of formal logic by means of the idea of a limit of an infinite sequence.
But while the importance of the calculus and mathematical analysis ― the core of modern mathematics ― cannot be overemphasized, the value of this first comprehensive critical history of the calculus goes far beyond the subject matter. This book will fully counteract the impression of laymen, and of many mathematicians, that the great achievements of mathematics were formulated from the beginning in final form. It will give readers a sense of mathematics not as a technique, but as a habit of mind, and serve to bridge the gap between the sciences and the humanities. It will also make abundantly clear the modern understanding of mathematics by showing in detail how the concepts of the calculus gradually changed from the Greek view of the reality and immanence of mathematics to the revised concept of mathematical rigor developed by the great 19th century mathematicians, which held that any premises were valid so long as they were consistent with one another. It will make clear the ideas contributed by Zeno, Plato, Pythagoras, Eudoxus, the Arabic and Scholastic mathematicians, Newton, Leibnitz, Taylor, Descartes, Euler, Lagrange, Cantor, Weierstrass, and many others in the long passage from the Greek "method of exhaustion" and Zeno's paradoxes to the modern concept of the limit independent of sense experience; and illuminate not only the methods of mathematical discovery, but the foundations of mathematical thought as well.
- Sales Rank: #238698 in Books
- Published on: 1959-06-01
- Released on: 1959-06-01
- Original language: English
- Number of items: 1
- Dimensions: 7.96" h x .71" w x 5.41" l, .82 pounds
- Binding: Paperback
- 368 pages
Most helpful customer reviews
0 of 0 people found the following review helpful.
Four Stars
By Qinglai Xiao
Bits and pieces of history of Calculus cannot be found elsewhere.
37 of 51 people found the following review helpful.
Fascinating material, questionable presentation
By Stan Vernooy
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!)
62 of 62 people found the following review helpful.
What, calculus is boring? Never!
By V. N. Dvornychenko
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
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