A historical sketch is given of WALLIS'S infinite product for 4/~, and of the attempts which have been made, over more than three centuries, to find the method by which BROUNCKER obtained his equivalent continued fraction. A derivation of BROUNCKER'S formula is given. Early results obtained by Indian mathematicians for the series for ~/4, later named for LEmNIZ, are reviewed and extended. A conjecture is made concerning BROUNCKER'S method of obtaining close bounds for ~.
1. W~dlis's Product
In 1656, the largely self-taught Oxford mathematician, JOHN WALLIS (1616-1703) published his greatest work, the Arithmetica Infinitorum. By reformulating and systematizing the largely geometric methods of his predecessors, particularly J. KEVLER, R. DESCARTES and B. CAVALIER1, in (what would now be called) more analytic terms, he was able to develop techniques which permitted the quadrature and cubature of certain classes of curves and surfaces. His general mode of procedure, perhaps stemming from his experience in cryptanalysis (as a practitioner of which he was one of the most outstanding in history), was to rely upon analogy and induction. From particular numerical examples, supplemented perhaps by analogical extensions, heuristic rules would be developed which would later be formalized as propositions, without deductive proofs. His bold inductive approach, coupled with his generally correct mathematical intuition, led to numerous interesting results, and had considerable influence on his successors, including ISAAC NEWTON and LEONHARD EULER.
The last part of his book is devoted to the millenia-old problem of the quadrature of the circle, 1 and culminates in an expression for the reciprocal of the ratio 1 Some extracts, in English, are given in A Source Book in Mathematics, 1200-1800, edited by D. J. STRUIK, Cambridge, Mass., 1969.