VOL. XLVIII,] PHILOSOPHICAL TRANSACTIONS. 397 



co-efficients may be obtained, by simple equations ; and these co-efficients will, 

 by the process annexed, appear to be the reciprocals of the several indexes of the 

 powers of that excess, affected alternately with the signs -|- and — , as was before 

 found, by the quadrature of the hyperbola, and by Dr. Halley in the above- 

 cite<l Phil. Trans., and by many who have used a fiuxional process. 



But there is another logarithmic series, equally simple with the former, con- 

 sisting of the same terms, but all affirmative. This has been demonstrated to 

 be the log-arithm of that fraction, whose numerator is unity, and denominator a 

 number as much less than unity, as the former number exceeded it. Now if an 

 infinite series be formed from that fraction, by actual division, it will consist of unity 

 and all the jKDwers of that defect ; and if the several powers of the excess of this 

 infinite series above unity, be multiplied by the co-efficients above found, and 

 ranged according to the powers of that defect, their sums will exhibit the above- 

 described series for the logarithm of that fraction, as appears by the operation 

 subjoined. 



Secondly, the terms of one of the best series, for the rectification of the circle, 

 are composed of the odd powers of the tangent of any arc, not exceeding 43°, 

 severally dividetl by their respective Indexes; of which the 1st, 3d, 5th, &c. 

 terms are affirmative ; and the 2d, 4th, 6th, &c. terms are negative ; and the 

 difference between the sums of the affirmative and negative terms, is the length 

 of that arc, of which the tangent and its powers constitute the series. 



Now a mathematician, who understands the nature and management of series, 

 though wholly ignorant of fluxions, might investigate this series in the following 

 manner: It has been geometrically demonstrated that, the radius of a circle 

 being unity, if double the tangent of any arc, be divided by the difference be- 

 tween unity and the square of that tangent, the quotient will be the tangent of 

 twice the arc. Now if an infinite series be formed by actual division, its terms 

 will consist of the doubles of the odd powers of" the tangent, and will be all affir- 

 mative ; which series will express the length of the tangent of the double of that 

 arc whose tangent and its powers constitute the same. 



If a series, consisting of the tangent and its powers, with unknown co-effi- 

 cients, be assumed, as in the former case, to express the length of the arc ; then 

 the length of double that arc may be expressed 1 ways ; viz. either by multiplying 

 each term of the series assumed by the number 2 ; or by finding the powers of 

 the series above described, which exhibits the length of the tangent of the double 

 arc, multiplying each power by its proper co-efficient, ranging the products under 

 each other, according to the powers of the tangent of the single arc, and finding 

 their sum. Now, since the length of the double arc may be thus expressed by 2 

 infinite series, each constituted of the tangent of the single arc and its powers ; 

 therefore the co-efficients of the like powers of that tangent, in each series, will 



