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Mathematics Emerging: A Sourcebook 1540 - 1900, by Jacqueline Stedall
Download PDF Mathematics Emerging: A Sourcebook 1540 - 1900, by Jacqueline Stedall
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Aimed at students and researchers in Mathematics, History of Mathematics and Science, this book examines the development of mathematics from the late 16th Century to the end of the 20th Century. Mathematics has an amazingly long and rich history, it has been practised in every society and culture, with written records reaching back in some cases as far as four thousand years. This book will focus on just a small part of the story, in a sense the most recent chapter of it: the mathematics of western Europe from the sixteenth to the nineteenth centuries. Each chapter will focus on a particular topic and outline its history with the provision of facsimiles of primary source material along with explanatory notes and modern interpretations. Almost every source is given in its original form, not just in the language in which it was first written, but as far as practicable in the layout and typeface in which it was read by contemporaries. This book is designed to provide mathematics undergraduates with some historical background to the material that is now taught universally to students in their final years at school and the first years at college or university: the core subjects of calculus, analysis, and abstract algebra, along with others such as mechanics, probability, and number theory. All of these evolved into their present form in a relatively limited area of western Europe from the mid sixteenth century onwards, and it is there that we find the major writings that relate in a recognizable way to contemporary mathematics.
- Sales Rank: #1394621 in Books
- Published on: 2008-11-15
- Original language: English
- Number of items: 1
- Dimensions: 7.30" h x 1.50" w x 9.70" l, 3.60 pounds
- Binding: Hardcover
- 680 pages
Review
"The book is based on a course given by the author for mathematics undergraduates to provide the historical background for some of their basic mathematics. Therefore the book is addressed to students, teachers, and persons interested in history who have some mathematical training. For all of them, it will provide a valuable resource."--Mathematical Reviews
About the Author
Jacqueline Stedall is at The Queens's College, University of Oxford.
Most helpful customer reviews
4 of 7 people found the following review helpful.
Yet another good but not great sourcebook
By Viktor Blasjo
Fresh translations of important mathematical texts are of course always a welcome and valuable resource, but this book can hardly be said to be superior to previous sourcebooks. The selections at times seem quite arbitrary and the editor's introductions are brief and superficial (incidentally, they are also typeset in a larger font size than the actual translations, which I find rather tasteless).
As an illustration of the general shortcomings of the book, I am going to consider the selection from Johann Bernoulli on the use of complex numbers in integration (pp. 468-472). Bernoulli's text itself is a very dry and factual list of rules for transforming integrals using complex substitutions. In particular, it includes the partial fraction decomposition of dx/(1+x^2) into "imaginary logarithmic differentials" (as Bernoulli calls them), i.e., in effect, the integration
arctan(x) = integral of dx/(1+x^2) = integral of (1/2)/(x+i)-(1/2)/(x-i) dx = (1/2)log(x+i/x-i)
"Such transformations were extremely helpful" (p. 468), claims the editor's introduction without further discussion. Such an attitude, I think, undermines the very purpose of a sourcebook, namely to see substantial mathematics with your own eyes instead of being fed vague and unsubstantiated and uncheckable assertions by some historian about which ideas were "extremely helpful." The novelist's adage "show, don't tell" should be an axiom for historians as well---above all for all sourcebook editors. Sadly, in the present case, it is not. Inquisitive readers will suffer accordingly. As later selections in this book illustrates, even the most rudimentary facts about the theory of complex integration were not understood until over a century later. So, then, how exactly was Bernoulli's algebraic hocus-pocus supposed to be "extremely helpful"? What did the logarithm of an imaginary number even mean at this time, if in fact it was a meaningful concept at all?
All of these concerns could have been addressed in a much more satisfactory manner by including Bernoulli's main application of his idea rather than this cryptic table of hocus-pocus "transformations". In 1712 Bernoulli wrote a short and beautiful paper on how to put his imaginary formula to "real" use, namely for finding multiple-angle formulas for tan(a). If we let y=tan(na) and x=tan(a) then arctan(y)=na=narctan(x), so by the above formula for the arctangent we get log(y+i/y-i) = log(x+i/x-i)^n. Complex logarithms may be mysterious but it does not take too much courage to "cancel the log's" in this equation, giving (y+i)(x-i)^n=(x+i)^n(y-i), which is an algebraic relationship between y=tan(na) and x=tan(a), as sought. Bernoulli admits this formula contains "quantitates imaginarias ... quae per se sunt impossibilia"---imaginary quantities which are by themselves impossible. But this, he says, is not a problem since they "in casu quolibet particulari evanescunt"---vanish in any particular case. For example, if n=3 the formula reduces to tan(3a)=(6*tan(a)-2*tan^3(a))/(1-3*tan^2(a). Reducing tan(na) to tan(a) is a hard problem. Bernoulli himself had previously tackled the problem using power series, but now he is quite proud to have carried out the derivation "sine serierum auxilio"---without the help of series. One benefit of this approach, he notes, is that it shows that the relationship is "semper algebraicum"---always algebraic---which is not clear from a series approach. Apparently, he considered working with "impossible quantities" a small price to pay for this added insight and simplicity. (I learned of this paper of Bernoulli's from Stillwell's excellent book, Mathematics and Its History, which I recommend far more highly than any sourcebook for readers who want a "show, don't tell" history of mathematics.)
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