the farther the multiplication is continued, and that the result always comes out too small if we stop at a proper fraction but too large if we stop at an improper fraction. Lord Brouncker, on the other hand, represents the value in question by a continued fraction in which all the denominators are equal to 2 and the numerators are odd square numbers. Wallis, to whom Brouncker had communicated his elegant result without proof, demonstrated the same in his "Arithmetic of Infinites." The computation of π could hardly be farther advanced by these results than Ludolf and others had carried it, though of course in a more laborious way. However, the series of powers derived by the assistance of the differential calculus of Newton and Leibnitz furnished a means of computing it to hundreds of decimal places. #Other calculations.# Gregory, Newton, and Leibnitz next found that the fourth part of π was equal exactly to 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 - ... if we conceive this series, which is called the Leibnitzian, indefinitely continued. This series is indeed wonderfully simple, but is not adapted to the computation of π, for the reason that entirely too many members have to be taken into account to obtain π accurately to a few decimal places only. The original formula, however, from which this series is derived, gives other formulas which are excellently adapted to the actual computation. This formula is the general series: α = _a_ - 1/3_a_^3 + 1/5_a_^5 - 1/7_a_^7 + ..., where α is the length of the arc that belongs to any central angle in a circle of radius 1, and where _a_ is the tangent to this angle. From this we derive the following: π/4 = (_a_ + _b_ + _c_ + ...) - 1/3(_a_^3 + _b_^3 + _c_^3 + ...) + 1/5(_a_^5 + _b_^5 + _c_^5 + ...) - ..., where _a_, _b_, _c_ ... are the tangents of angles whose sum is 45°. Determining, therefore, the values of _a_, _b_, _c_ ..., which are equal to small and easy fractions and fulfil the condition just mentioned, we obtain series of powers which are adapted to the computation of π. The first to add by the aid of series of this description additional decimal places to the old 35 in the number π was the English arithmetician Abraham Sharp, who following Halley's instructions, in 1700, worked out π to 72 decimal places. A little later Machin, professor of astronomy in London, computed π to 100 decimal places; putting, in the series given above, _a_ = _b_ = _c_ = _d_ = 1/5 and _e_ =-1/239, that is employing the following series: π/4 = 4. [1/5 - 1/3.5^3 + 1/5.5^5 - 1/7.5^7 + ...] - [1/239 - 1/3.239^3 + 1/5.239^5 - ...] #The computation of π to many decimal places.#