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A History of Analysis (History of Mathematics, V. 24), by Hans Niels Jahnke
Download A History of Analysis (History of Mathematics, V. 24), by Hans Niels Jahnke
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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.
- Sales Rank: #2828223 in Books
- Brand: Brand: American Mathematical Society
- Published on: 2003-08-01
- Original language: English
- Dimensions: 9.75" h x 7.00" w x 1.00" l, 2.05 pounds
- Binding: Hardcover
- 432 pages
- Used Book in Good Condition
Review
"This book is very good at tying biographies to the development of the mathematics and to the related developments in science ... for the majority of the mathematicians listed here, this [book] is an outstanding starting point ... recommended for academic and research mathematics, physics, and history of science collections." ---- E-Streams
"It is a highly readable and informative account of the history of analysis." ---- Zentralblatt MATH
Most helpful customer reviews
7 of 8 people found the following review helpful.
Acceptable
By Viktor Blasjo
This is an acceptable survey of the history of analysis, which, however, is not without its flaws. I would like to object in particular to the following passage:
"When Bernoulli wrote this in 1692, he could not carry out the final integration [of the differential equation for the catenary] since he did not yet know that the integral of 1/x is the logarithm. ... [Instead] he reduced the construction of the catenary to squaring a hyperbola. This example shows clearly the role of geometry in infinitesimal calculus at the beginning of the 18th century." (p. 111)
The claim that Bernoulli "did not yet know that the integral of 1/x is the logarithm" is a drastic oversimplification at best. Indeed, he explicitly proves that the area under 1/x is "logarithmic" earlier in the very same work containing this construction of the catenary (his 1692 lectures on the calculus; the proof appears at the end of the tenth lecture). And the numerical method for finding areas under 1/x using logarithm tables was also well known at the time (having been explicitly outlined decades earlier, e.g. by Huygens, Horologium Oscillatorium, 1673, Part III, right before Proposition X).
Furthermore, immediately after giving his construction, Bernoulli goes on to discuss Leibniz's construction of the same curve. This is a geometrical construction corresponding precisely to the modern analytic formula (1/2)(e^x+e^-x). In the course of this discussion Bernoulli writes "per naturam Logarithmicae, zdy=adz" (lecture 37), thus demonstrating his complete understanding of this differential equation, contrary to Jahnke's claim. But this did not suggest to neither Bernoulli nor Leibniz that an analytic expression involving log(x) or e^x should be considered a "solution" of the differential equation for the catenary.
On the contrary, Leibniz explicitly states that the catenary is second to no transcendental curve in terms of simplicity. Thus expressing the catenary in terms of log(x) or e^x does not reduce it to a simpler form. True to his word, Leibniz does indeed proceed to discuss how logarithms can be found using the catenary.
One may also ask: if it was not known in 1692 that the antiderivative of 1/x is log(x) then when was this discovered? The answer is anticlimactical to say the least. So far as I have been able to ascertain, the first explicit use of log as a solution to a differential equation occurs in an article by Leibniz in 1694, where he quite casually writes "dv : v =dy : a. Ergo log. v = y : a" (MS.V.313), as if this was the most trivial thing in the world (as indeed it probably was to Bernoulli and the other intended readers). The fact that the analytic logarithm function as an antiderivative enters the world in this inconspicuous way speaks strongly against the notion that it filled a lacuna in contemporary knowledge. Instead it reinforces what our reading of Bernoulli already had suggested, namely that the analytic expression is a rather trivial codification of well-established knowledge and that no one had written it down in this form earlier for the simple reason that they had not found it useful to do so.
Clearly, then, deeper reasons than mere ignorance lay behind Bernoulli's preferred mode of construction. Indeed, by the standards of his day, writing "log(x)" would amount to little more than replacing concrete geometrical imagery by a pretentious use of analytical jargon in order to express a complicated curve in terms of another curve no less complicated than the first.
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