Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/50

CHAPTER III

The Second Period

The new Analysis

The foundations of the new Analysis were laid in the second half of the seventeenth century when Newton (1642—1727) and Leibnitz (1646—1716) founded the Differential and Integral Calculus, the ground having been to some extent prepared by the labours of Huyghens, Fermat, Wallis, and others. By this great invention of Newton and Leibnitz, and with the help of the brothers James Bernouilli (1654—1705) and John Bernouilli (1667—1748), the ideas and methods of Mathematicians underwent a radical transformation which naturally had a profound effect upon our problem. The first effect of the new Analysis was to replace the old geometrical or semigeometrical methods of calculating $\pi$ by others in which analytical expressions formed according to definite laws were used, and which could be employed for the calculation of $\pi$ to any assigned degree of approximation.

The work of John Wallis

The first result of this kind was due to John Wallis (1616—1703), Undergraduate at Emmanuel College, Fellow of Queens' College, and afterwards Savilian Professor of Geometry at Oxford. He was the first to formulate the modern arithmetic theory of limits, the fundamental importance of which, however, has only during the last half century received its due recognition; it is now regarded as lying at the very foundation of Analysis. Wallis gave in his Arithmetica Infinitorum the expression

${\frac {\pi }{2}}={\frac {2}{1}}.{\frac {2}{3}}.{\frac {4}{3}}.{\frac {4}{5}}.{\frac {6}{5}}.{\frac {6}{7}}.{\frac {8}{7}}.{\frac {8}{9}}\ldots$

for $\pi$ as an infinite product, and he shewed that the approximation obtained by stopping at any fraction in the expression on the right is in defect or in excess of the value ${\frac {\pi }{2}}$ according as the fraction is proper or improper. This expression was obtained by an ingenious method