Of the contemporaries of Newton on of the most prominent was John Wallis 1. He studied theology at Emmanuel College, Cambridge, and took the degree of B.A. in 1637 and that of M.A. in 1640, the year in which he was ordained. He became a fellow of Queens' College in 1644. His tastes, however, were in the line of physics and mathematics, and in 1649 he was elected to the Savilian professorship of geometry at Oxford, a positi on which he held until his death. He was awarded the degree of doctor of divinity in 1653 2, became chaplain to Charles II in 1660, and was one of the founders if the Royal Soc iety (1663).
Wallis was a voluminous writer, and not only are his writings erudite, but they show a genius in mathematics that would have appeared the more conspicuous had his work not been so overshadowed by that of his great Cambridge contemporary. He was on of the first to recognize the significance of the generalization of exponents to include negative and fractional as well as positive and integral numbers. He recognized also the importance of Cavalieri's method of indivisibles, and employed it 3 in the quadrature of such curves as y = x^n, y = x^(1/n), and y = x^0 + x^1 + x^2 + . . .. he failed in his efforts at the approximate quadrature of the circle by means of series because he was not in possession of the general form of the binomial theorem. He reached the result, however, by another method. he also obtained the equivalent of ds = dx * (1 + (dx/dy)^2)^(1/2) for the length of an element of a curve, thus connecting the pro blem of rectification with that of quadrature. In 1673 he wrote his great work De Algebra Tractatus; Historicus & Practicus, of which an English edition appeared in 1685 4. In this there is seen the first serious attempt in England to write on the history of mathematics, and the result shows a wide range of reading of the classical literature of the science. This work is also noteworthy because it contains the first re cord of an effort to represent the imaginary number graphically by the method now used. The effort stopped short of success but was an ingenious beginning. Wallis was in sympathy with the Greek mathematics and astronomy, editing parts of the works of Ar chimedes, Eutocius, Ptolemy, and Aristarchus; but at the same time he recognized the fact that the analytic method was to replace the synthetic, as when he defined a conic as a curve of the second degree instead of a section of a cone, and treated it by t he aid of coordinates. His writings include works on mechanics, sound, astronomy, the tides, the laws of motion, the Torricellian tube, botany, physiology, music, the calendar (in opposition to the Gregorian reform), geology, and the compass,--a range to o wide to allow of the greatest success in any of the lines of his activity. He was also an ingenious cryptologist and assisted the government in deciphering diplomatic messages 5< /A>.
Among his interesting discoveries was the relation
4 3 * 3 * 5 * 5 * 7 * 7 . . .
_ = ___________________,
pi 2 * 4 * 4 * 6 * 6 * 8 . . .
one of the early values of pi involving infinite products 6.
(Smith, pp. 406-409.)