The Origins of Cauchy's Rigorous Calculus (Dover Books on Mathematics) [Paperback]

Judith V. Grabiner (Author), Mathematics (Author)

Book Description

Publication Date: October 20, 2011 | Series: Dover Books on Mathematics

This text for upper-level undergraduates and graduate students examines the events that led to a 19th-century intellectual revolution: the reinterpretation of the calculus undertaken by Augustin-Louis Cauchy and his peers. These intellectuals transformed the uses of calculus from problem-solving methods into a collection of well-defined theorems about limits, continuity, series, derivatives, and integrals. 1981 edition.

Product Details

• Paperback: 272 pages
• Publisher: Dover Publications (October 20, 2011)
• Language: English
• ISBN-10: 0486438155
• ISBN-13: 978-0486438153
• Product Dimensions: 9.2 x 6.1 x 0.6 inches
• Shipping Weight: 13.4 ounces (View shipping rates and policies)
• Average Customer Review: 4.2 out of 5 stars  See all reviews (6 customer reviews)
• Amazon Best Sellers Rank: #327,042 in Books (See Top 100 in Books)

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9 of 10 people found the following review helpful:

5.0 out of 5 stars Highly Recommended - But Grabiner's thoughtful, detailed work requires careful reading, November 4, 2007

By 

Michael Wischmeyer (Houston, Texas) - See all my reviews
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This review is from: The Origins of Cauchy's Rigorous Calculus (Dover Books on Mathematics) (Paperback)

The Origins of Cauchy's Rigorous Calculus by Judith V. Grabiner is more technically challenging than many 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.

3 of 3 people found the following review helpful:

3.0 out of 5 stars Unsubstantiated clichés, January 10, 2010

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Viktor Blasjo - See all my reviews
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This review is from: The Origins of Cauchy's Rigorous Calculus (Dover Books on Mathematics) (Paperback)

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.

12 of 16 people found the following review helpful:

4.0 out of 5 stars suitable historical perspective, March 15, 2006

By 

W Boudville (Terra, Sol 3) - See all my reviews
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This review is from: The Origins of Cauchy's Rigorous Calculus (Dover Books on Mathematics) (Paperback)

Calculus was discovered by Newton and Liebniz in the 1600s. They used a method of infinitesimals to derive their results. To this day, infinitesimals are used as an intuitive explanation of calculus, to scientists and engineers. But to many mathematicians, there is an unsettling lack of rigour.

Grabiner explains how this state of affairs from the 1600s persisted until Cauchy [and others] came along. He found an epsilon -delta approach that permits a strict derivation of theorems. This book should give an appreciation of that approach, in a suitable historical perspective.