Converging Series for the Circle . ' 491 
are found from the like terms of the other, by dividing 
thefe latter by 10 and its feveral fucceftive powers 100, 
1000, Sec.; that is, the terms of the one confift of the 
fame figures as the terms of the other, only removed 
a certain number of places farther towards the right, 
in the decuple fcale of numbers; and the number of 
places, by which they muft be removed, is the fame 
as the diifance of each term from the firft term of the 
feries, viz. in the fecond term the figures muft be moved 
one place lower, in the third term two, in the fourth 
term three, 8ec. fo that the latter feries will confift of 
but about half the number of the terms of the for- 
mer. Thus, then, this method may be faid to effect the 
bufinefs by one feries only, in which there is little more 
to do, than to divide by the feveral numbers 1, 3, 5, 7, 
&:c.; for as to the multiplications by the numbers in the 
numerators of the terms, after they become large, they 
are eafily performed by barely multiplying by the num- 
ber two, and fubtradling one number from another : for 
fmce every numerator is lefs by two than the double of 
its denominator, if d denote any denominator (exclufive 
always of the powers of 10) then the co-efficient of that 
' id . — 2 2 
term is - d > or 2 — 7 ? by which the preceding term is to 
be multiplied ; to do which, therefore, multiply it by two, 
that is double it, and divide that double by the divifor d , 
and fubtra<ft the quotient from the faid double. 
Vol. LXVI. 
T t t 
