Converging Series for the Circle. 4 89- 
Here it is evident, that none of thefe latter- cafes afford 
any numbers that are fit for this purpofe. And to try 
any other fractions lefs than ± for the value of t, docs- 
not feem likely to anfwer any good purpofe, efpecially, as 
the divifors, after 1 2, become too large to be managed in 
the eafy way of fnort divifion in one line. 
By the foregoing means it appears, that we have dif- 
fered five different forms of the value of a, or ith of 
the femi-circumference, all of which are very proper 
for readily computing its length; viz. three forms in the 
fil'd: cafe and its corollary, one in the third cafe, and one 
in the fourth cafe. Of thefe, the firft and laft are the 
fame as thofc invented by euler and machin refpec- 
tively, and the other three are quite new, as far as I 
know. 
But another remarkable excellency attending the firft 
three of the before mentioned feries is, that they are 
capable of being changed into others which not only 
converge ftill fafter, but in which the converging quan- 
tity fhall be or fome multiple or fub-multiple of it, 
and fo the powers of it raifed with the utmoft eafe. The 
feries’, or theorems, here meant are thefe three; 
ift, A = ' 
2dly, A—' 
I I 
+ — x ; I 
1 
4" r — 
2 3-4 
5-4 
1 1 
1 
+ — -x : 1 — — 
4- — 2 — 
■ 3 3-9 
5-9 
1 
4 I 
' 4* - — - 
3*4 
5-4 
1 1 
1 
x : 1 
7 3-49 
5-49* 
tV + 9 ^>’ Scc -’ 
+ 7 ?’ &c - 
7-49 1 + 949*’ ^ :c •■ 
3 <%> , 
