TRACT 18.

FOR THE CIRCLE.

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ing the required proportion ; because, of the two series ofwhich it consists, the several terms of the one are found fromthe like terms of the other, by dividing these latter by 10,and its several successive powers, 100,1000, &c ; that is, theterms of the one consist of the same figures as the terms ofthe other, only removed a certain number of places farthertowards the right hand, in the decuple scale of numbers;and the number of places by which they must be removed,is the same as the distance of each term from the first termof the series, viz, in the 2d term the figures must be movedone place lower, in the 3d term two, in the 4th term three,&c; so that the latter series will consist of but about half thenumber of the terms of the former. Thus then this methodmay be said to effect the business by one series only, inwhich there is little more to do, than to divide by the severalnumbers 1, 3, 5, 7, ike; for as to the multiplications by thenumbers in the numerators of the terms, after they becomelarge, they are easily performed by barely multiplying bythe number 2, and subtracting one number from another: forsince every numerator is less by 2 than the double of its de-nominator, if d denote an}' denominator, exclusive alwaysof the powers of 10, then the coefficient of that term is

Qj/— 2 2

—j— or 2 — —, by which the preceding term is to be mul-tiplied ; to do which therefore, multiply it by 2, that isdouble it, and divide that double by the divisor d, and sub-tract the quotient from the said double.