TRACT 21.
' 437
x. This same property has also been noticed by many other-authors since Napier’s time. And the same, or a similar pro-'perty, is evidently true in all systems of logarithms whatever,namely, that the modulus of the system is to any number, asthe fluxion of its logarithm is to the fluxion of the number.
Now from this property, by means of the doctrine ef flux-ions, are derived other ways for making logarithms, whichhave been illustrated by many writers on this branch, as Craig,John Bernoulli , and almost all the writers on fluxions. Andthis method chiefly consists in expanding the reciprocal ofthe given quantity in an infinite series, then multiplying eachterm by the fluxion of the said quantity, and lastly taking thefluents of the terms; by which there arises an infinite seriesof terms for the logarithm sought. So, to find the logarithmof any number n; put any .compound quantity for n, as
n + a:
suppose —\
then the flux, of the log. or - being —— ■
the fluents give log. of n or log. of
0 ° °nn 2n a 1 3?i3 <in*
-t, ~ —“i
And writing —-.r for x gives lo
Also, because ;
= 1
n + xn + xn
n—xn
xx x!*x x^x 0
-J- — —
mi 7). 4 7
n 3
2 n* 1
fl
2 Hr
f3 X' Q
N 3 - 4ttA C -
, or loo
tlieref. log. —= — - -f
t 5 a + X n
ȣ *
a! _
2n»
-a
n
= 0 — log.-hr- &c,
Sti 3 1 4n4 ’
&C.
< “d 1 og. r :-=+i + 5i + £ 1 **
And by adding and subtracting any of these series, to orfrom one another, and multiplying or dividing their corre-sponding numbers, various other series for logarithms maybe found, converging much quicker than these do.
In like manner, by assuming quantities otherwise com-pounded, for the value of K, various other forms of logarith-mic series may be found by the same means.