[ 2 7 6 ] 
Now, fince the logarithm of the fquare of the 
given number may be thus expreffed by two infinite 
feries, each conftituted of the excefs thereof, above 
unity, and its powers ; it follows, that the coefficients 
of the like powers of that excefs, in each feries, will 
be equal between themfelves ; and, confequently, 
the values of the unknown coefficients may be ob- 
tained, by fimple equations; and thefe coefficients 
will, by the procefs annexed, appear to be, the re- 
ciprocals of the feveral indexes of the powers of that 
excefs, affedted alternately with the figns -f- and — , 
as was before found, by the quadrature of the Hyper- 
bola, by Dr. Halley in the above-cited Philofophical 
Tranfattion, and by many who have ufed a fluxional 
But, there is another logarithmic feries equally fim- 
ple with the former, confiding of the fame terms, 
but all affirmative. This has been demonflrated to 
be the logarithm of that fraction, whofe numerator 
is unity, and denominator a number, as much lefs than 
unity, as the former number exceeded it. 
Now, if an infinite feries be formed from that 
fra&ion, by actual divifion, it will confift of unity, 
and all the powers of that defedt ; and if the feveral 
powers of the excefs of this infinite feries above unity, 
be multiplied by the coefficients above- found, and 
ranged according to the powers of that defedt, their 
fums will exhibit the above-defcribed feries for the 
logarithm of that fradtion, as appears by the opera- 
ration fubjoin’d. 
As to the application of thefe two feries, and their 
fum, to the finding the logarithms of numbers ; the 
fame, being copioufly treated of by Dr. Halley (in 
the j Philofophical Tranfaftion before quoted) there is 
