I can not stop to be amused and amazed by the way John Wallis contempories are corresponding with each other. I do not intend to read the book from start to end, but I am regularly studying one chapter, to immerse myself in the reasoning at that time. The text also illustrates, some of the shortcuts taken in that period of history, such as the concept of proof by induction. The book is a translation from latin and is further complemented with an introduction by Dr Stedall, helping a lot to put the work in the right frame of its importance, that a Geometrical problem is reduced to a pure arithmetic problem. If you want to study mathematics, in particular rational numbers, concepts of infinity and relation with geometry, this book is not very effective, be it only for the tedious style of formulation.īut when you look at the book from its historical importance and understanding of the mathematical reasoning in the 17th century, this book is to be considered as a masterpiece, or (why not) a collectors item. So we do this for a bunch of integers p and q to get an enlarged Pascal's triangle with fractional entries and then use ordinary properties of Pascal's triangle extended to the fractional setting to deduce the value for the entry we want. We don't know what that entry should be, but remember that we can integrate (1-x^(1/p))^q, which can be interpreted as a fractional entry in Pascal's triangle.
The binomial series expansion for an integer exponent is of course given by Pascal's triangle, but to expand things like (1-x^2)^(1/2) we need a corresponding "fractional entry" sitting between two lines in Pascal's triangle. Today we would feel like using the binomial series expansion, but this is not available to Wallis so he more or less has to invent it. So what was this "one real stroke of genius"? Well, to find pi we need to integrate (1-x^2)^(1/2). In what was perhaps the one real stroke of genius in Walli's long mathematical career, he saw how to complete his by a method now set out in Proposition 191, and so arrived at his infinite fraction for 4/pi", namely 3*3*5*5*7*7*./2*4*4*6*6*8*. But because his ultimate aim was the quadrature of the circle, the curve he was really interested in was y=(1-x^2)^(1/2). From his startingpoint of simple powers, he could easily handle sums (or differences) of sequences, and hence eventually quadratures of any curve of the form y=(1-x^(1/p))^q provided p and q were integers. Many of the results demonstrated by Wallis were already well known but, as he repeatedly pointed out, his aim was to establish a method by which those results, and others, could be systematically obtained. Wallis's major contribution to the development of seventeenth-century mathematics was perhaps, as he himself recognized, the transformation of geometric problems to the summation of arithmetic sequences. His purpose in doing so was ultimately to find a general method of quadrature (or cubature) of curved spaces, a promise held out in De sectionibus conicis and taken up at length in the Arithmetica infinitorum. From Stedall's introduction: "In De sectionibus conicis, Wallis found algebraic formulae for the parabola, ellipse and hyperbola, thus liberating them, as he so aptly expressed it, from 'the embranglings of the cone'.
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