"As [Wallis] was unacquainted with the he could not effect the quadrature of the circle, whose equation is y = (1 - x^2)^\frac{1}{2}, since he was unable to expand this in powers of x. He laid down however the principle of interpolation. He argued that as the ordinate of the circle is the geometrical mean between the ordinates of the curves y = (1 - x^2)^0 and y = (1 - x^2)^1, so as an approximation its area might be taken as the geometrical mean between 1 and \frac{2}{3}. This is equivalent to taking 4\sqrt{\frac{2}{3}} or 3.26... as the value of \pi. But, he continued, we have in fact a series 1, \frac{2}{3}, \frac{8}{15}, \frac{16}{35},... and thus the term interpolated between 1 and \frac{2}{3} ought to be so chosen as to obey the law of this series. This by an elaborate method leads to a value for the interpolated term which is equivalent to making\pi = 2\frac{2\cdot2\cdot4\cdot4\cdot6\cdot6\cdot8\cdot8...}{1\cdot3\cdot3\cdot5\cdot5\cdot7\cdot7\cdot9...}The subsequent mathematicians of the seventeenth century constantly used interpolation to obtain results which we should attempt to obtain by direct algebraic analysis."