Mathematics Emerging: A Sourcebook 1540 - 1900 Review

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Mathematics Emerging: A Sourcebook 1540 - 1900 ReviewFresh 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.)Mathematics Emerging: A Sourcebook 1540 - 1900 Overview

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