900 Summation of the Jowly-converging 
bear very nearly the fame proportion to the lefler ordinate 
i, as the abfcifs of the axis intercepted between the two or- 
dinates, that is, as the logarithm of the ratio of the greater 
ordinate to the lefler, or the logarithm .000,000,000,001, 
bears to the difference of the laid ordinates. Say there- 
fore, as .434,294,481,9 (which is the fub-tangent of 
the logarithmick curve in briggs’s Syftem of Loga- 
rithms) is to 1 (or the lefler of the two ordinates) fa is 
.000,000,000,001 to a fourth number, which will be 
.000,000,000,002,302,585,093; and this fourth 
number will be the excefs of the greater of the faid 
two ordinates above the lefler, or of the billionth root 
of 1 o above 1 . Therefore the billionth root of 1 o 
will be = 1.000,000,000,002,302,585,093; which, 
being multiplied by 1,000,000,000,000, will be = 
1,000,000,000,002.302,585,093; from which if 
we fubtrafft 1,000,000,000,000, the remainder will be 
2.302,585,093. Therefore 2.302,585,093 is nearly 
v v x ^ x ■* x ^ 
equal to the infinite feries x + + t * “7 + t 
1 234507 
+ &c. when x is = q. e. i. 
4. This number 2.302,585,093 gives the value of 
PC 2C PC^ PC^ X s 
the feries x + -f + — + — + — + 8tc. exafft to nine places 
of figures, the error being only in the 10th figure 3, 
which ought to be a 2 inftead of a 3, the more accurate 
value of that feries (which is equal to the logarithm of 
4 the 
