Of all the processes for computing to great precision this most important ratio, that by means of the series for the arc in terms of its tangent is the most rapid. Seemingly short, however, it is truly a very long process; to reach its beginning we must know the laws of differentials and integrals, must be familiar with the higher algehra, and with the advanced branches of trigonometry; must know how to divide an arc whose tangent is an aliquot part of the radius, into smaller arcs whose tangents have the same character; and while studying these chapters in mathematical science we shall have often used our knowledge of this very ratio of which we are in search. The real utility of this formula lies in its enabling us to extend the approximation to a great number of decimal places. Even although the labour be not overwhelming, we can hardly regard as other than useless, the toil of computing and verifying the value of π to upwards of five hundred places.