A Source Book in Mathematics, 1200-1800 [Paperback]

This is an excellent source book. One of the many benefits of reading original sources is that it frees us from the often dogmatic single-mindedness of modern textbooks. I would like to illustrate this by looking at some excerpts from this book on the topic of power series. The modern approach to power series representations of functions is of course to find the coefficients by repeated differentiation. However, history teaches us that this approach is backwards and obscures some important insights.

The first published derivation of the so-called general Maclaurin series (by Taylor in 1715, excerpted here on pp. 329-333) was based on entirely different ideas than that of repeated differentiation, namely Newton's forward-difference formula. It may be summarised as follows. An infinite polynomial A+Bx+Cx^2+Dx^3+... has infinite degrees of freedom. Therefore we expect to be able to construct an infinite polynomial passing through an infinite number of given points, just as a parabola of the form y=Ax^2+Bx+C can be constructed going through essentially any three points, but not any four, owing to its having three coefficients. Newton's forward-difference formula constructs such a polynomial, namely a polynomial which takes the same values as a given function at the x-values 0,b,2b,3b,.... Taylor's derivation of his series consists in letting b go to zero is this formula. The nowadays more popular method of finding the series by repeated differentiation was not published until decades later by Maclaurin in 1742 (pp. 338-340). Thus history alerts us to the fact that the blind-computation approach favoured today robs us of an opportunity to "see" the infinitely many degrees of freedom of a power series in an illuminating way that is based on open-minded reasoning.

A related lesson from history concerns the binomial series. Today it is popular to derive the general binomial series by finding its Maclaurin series through repeated differentiation. Particular binomial expansions are then obtained by plugging numbers into this general series. Again this textbook approach based on blind computations has a much more vivid and interesting counterpart in history. When we read Newton we realise that the binomial series is not a "theorem" to be "proved." Although we often say that we "use a binomial expansion" to find some integral or other, this is really just a time-saving device, not a fundamental and substantial reliance on some profound theorem in all its generality. The classical applications of binomial series expansions do not by any means require that we "prove" the binomial series in general. When Newton wants to find the power series for sqrt(1+x^2) he does not say "I apply the binomial theorem with the exponent 1/2." Rather he simply says "I extract the root." Indeed, it is very straightforward to "extract the root" without knowing anything about the binomial series: what we want is a power series such that
sqrt(1+x^2)=A+Bx+Cx^2+Dx^3+...
or in other words,
1+x^2=(A+Bx+Cx^2+Dx^3+...)(A+Bx+Cx^2+Dx^3+...).
Very well, we simply multiply out the right hand side and identify coefficients with the left hand side, and there's our root. The benefit of calling this a binomial expansion is that we can shortcut out algebraic labours by using the quick and easy way of thinking about the binomial series by analogy with the integer-exponent case. We can do so with good conscience knowing that we could always fall back on the direct algebraic way to "extract the root" if pressed for justification. Here are Newton's own words:

"Fractions are reduced to infinite series by division; and radical quantities by extraction of the roots, by carrying out those operations in the symbols just as they are commonly carried out in decimal numbers. These are the foundations of these reductions: but extractions of roots are shortened by this theorem [the binomial theorem]." (pp. 285-286)