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Carl B. Boyer

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

by Carl B. Boyer (Author)

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Part of: Dover Books on Mathematics (306 books)

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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.

ISBN-10

0486605094
ISBN-13

978-0486605098
Edition

First Edition
Publisher

Dover Publications
Publication date

June 1, 1959
Language

English
Dimensions

5.41 x 0.71 x 7.96 inches
Print length

368 pages
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Product details

Publisher ‏ : ‎ Dover Publications; First Edition (June 1, 1959)
Language ‏ : ‎ English
Paperback ‏ : ‎ 368 pages
ISBN-10 ‏ : ‎ 0486605094
ISBN-13 ‏ : ‎ 978-0486605098
Item Weight ‏ : ‎ 13.1 ounces
Dimensions ‏ : ‎ 5.41 x 0.71 x 7.96 inches

Best Sellers Rank: #201,579 in Books (See Top 100 in Books)
- #37 in Calculus (Books)
- #137 in Mathematics History

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4.4 out of 5

86 global ratings

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Tom

nice and fine

Reviewed in the United States 🇺🇸 on January 18, 2023

worth reading

An interesting overview with a significant amount of detail

Reviewed in the United States 🇺🇸 on February 10, 2019

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I am still reading it, but am half way through. The book has a lot of interesting information. For the first time I understand how Archimedes found the volume of a sphere by "weighing" it, and that is only one of many great insights. I am not giving it five stars because, as others have said, it is written a style that may have been common at the time, but to the modern reader can be frustrating. I think the same information, with the same nuance where necessary, could be written in a less tedious manner today.

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Alan U. Kennington

Insightful and thoroughly well researched

Reviewed in the United States 🇺🇸 on December 4, 2013

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This is a truly enlightening history of calculus from Eudoxus to Weierstrass. This 1949 book by Carl Benjamin Boyer, republished by Dover, places the developments of Newton and Leibniz within the long sequence of historical developments of calculus including Pythagoras, Zeno, Eudoxus, Aristotle, Euclid, Archimedes, Oresme, Viète, Stevin, Cavalieri, Torricelli, Kepler, Galileo, Descartes, Fermat, Wallace, Barrow, Leibniz, Newton, Maclaurin, Euler, Lagrange, Lacroix, Bolzano, d'Alembert, Cauchy, Weierstrass, Cantor, and Dedekind, roughly in that order.

Within this context, it becomes crystal clear that the old arguments about the relative precedence of Newton and Leibniz are a relatively minor matter. Both of them relied heavily on a very long sequence of earlier developments, and both of them fell very far short of a satisfactory, logically self-consistent, meaningful formalism, which required another 200 years to develop.

This book has about a thousand footnotes. It is very thoroughly researched indeed. Boyer describes how various opposing forces were at work in the development of the calculus. Very important, for example, was the weight of tradition, such as the method of exhaustion of the ancient Greeks, which was held in high esteem. There was a strong resistance to abandonment of geometrical intuition as the basis of calculus, although ultimately a satisfactory axiomatization of the real numbers permitted calculus to be liberated from its geometric origins. Unsatisfactory concepts of infinity and the infinitesimal strongly constrained or encouraged many mathematicians to reject some formulations while accepting others.

One thing which worried me a lot is how much of modern calculus is still taught in the same way as many centuries ago, using ways of thinking which are obsolete, meaningless or logically circular. In fact, one of the themes of this book is how ancient ill-founded concepts have frequently been "rediscovered" and adopted by mathematicians, long after they had been superseded by better-founded concepts.

I ended with the suspicion that our modern-day calculus (or "analysis") is not the end of the road. Even our current calculus is a mixture of intuition, metaphysics, pragmatism and sometimes empty formalism. This book seems to put some of our currently accepted calculus concepts in doubt.

7 people found this helpful

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Stan Vernooy

Fascinating material, questionable presentation

Reviewed in the United States 🇺🇸 on July 9, 2004

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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!)

40 people found this helpful

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Dinesh Yadav

Good Book

Reviewed in the United States 🇺🇸 on May 12, 2021

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I bought this book for my class course and I found that this book is written and edited in a very properway. Every next chapter has some kind of connectivity with the previous chapter. I will recommend this book to others.

One person found this helpful

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mjheath

Five Stars

Reviewed in the United Kingdom 🇬🇧 on April 26, 2017

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excellent

One person found this helpful

Daniel

Wonderful book on the history of Calculus

Reviewed in France 🇫🇷 on March 12, 2012

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Reading this book today makes one feel tremendously satisfied and sad alike. Modern books on the history of mathematics pale in contrast to this one (I would mention one exception, The History of Pi, but it was written in the 60s or 70s so I think its not really an actual book either, and you have to forgive his Woodstock Festival reamrks about the world).
The title of the book mentions the "conceptual development", and thats were the book excels at, describing how the procedural evolution of maths got to the (invention? discovery?...who knows...) of Calculus.
It has never ceased to amaze me how accurately the differential ecuations describe the real physical fenomena or how Riemanns topology gave Einstein the mathematical foundation needed for his theory 100 years later and so on (if you are intrigued by the inner nature of mathematics DONT buy "Is God a Mathematician", a book more suited for an Oprah show than for someone really interested on the real nature of mathematics). Some reviewers were more critic with certain aspects of the book, fair enough, as humans we are fable, but I seriously doubt a better book on this subject has ever been written. Regrettably there are many books on the history of mathematics but most of them fail, not this one, as someone said , Boyer is the Edward Gibbon of the history of maths.
Thanks Newton, Euler, Rieman, Pitagoras, Pointcare, Leibniz, Cantor...I saw further because I stood on your shoulders.

2 people found this helpful

sue edwards

Five Stars

Reviewed in the United Kingdom 🇬🇧 on February 18, 2016

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Excellent

One person found this helpful

Didomania

Interessant und aufschlußreich

Reviewed in Germany 🇩🇪 on September 4, 2013

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Das Werk eines kompetenten Autors. Eine spannende Lektüre. Allerdings werden für das volle Verständnis mathematikhistorische Grundkenntnisse vorausgesetzt, also für Einsteiger ist das Buch weniger geeignet.

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