wallis showed considerable ingenuity in reducing the equations of curves to the forms given above, but, as he was unacquainted with the binomial theorem, he could not effect the quadrature of the circle, whose equation is formula_11, since he was unable to expand this in powers of "x". he laid down, however, the principle of interpolation. thus, as the ordinate of the circle formula_11 is the geometrical mean of the ordinates of the curves formula_13 and formula_14, it might be supposed that, as an approximation, the area of the semicircle formula_15 which is formula_16 might be taken as the geometrical mean of the values of