4 Proceedings of the Royal Society 
The second artifice is founded on the well-known properties of 
chain-fractions. 
When we use Lord Brouncker’s process in approximating to the 
value of an irrational quantity, we obtain a series of fractions 
alternately too great and too small, having this property that the 
difference between two contiguous members of the series is a 
fraction having unit for its numerator, and for its denominator 
the product of the two denominators. Hence it follows, that the 
error of the last obtained fraction is much less than unit divided 
by the square of its denominator ; and thus a Brounckerian fraction 
having only three figures in its denominator will give as much 
precision as a decimal fraction of five places. 
On treating the above residue *6513585 by Brouncker’s method 
we get the fraction of which the value is *65135834:; hence 
if we use the addend 1 09 51 63 yf beginning with 00 we 
shall obtain the required table on merely rejecting the fractions. 
We also avoid the need for writing the denominator by placing 
on the lower edge of the card the equivalent expressions : — 
i / ft tit 
1 
09 
51 
63 
+ 1007 
and 
1 
09 
51 
64 
- 539 
0 
0 
00 
00 
00 
773 
1 
1 
09 
51 
64 
234 
2 
2 
19 
03 
27 
1241 
3 
3 
28 
54 
91 
702 
4 
4 
38 
06 
55 
163 
5 
5 
47 
58 
18 
1170 
6 
6 
57 
09 
82 
631 
7 
7 
66 
61 
46 
92 
8 
8 
76 
13 
09 
1099 
9 
9 
85 
64 
73 
560 
10 
10 
95 
16 
37 
21 
Here we observe that for the 773d day the fractional remainder 
will be 0, and that, therefore, we are in doubt whether to write 
03'" * 0 or 02"' 1546. From the rank of the fraction in the 
Brounckerian series we perceive that it is in defect ; wherefore we 
choose the former of the two. The same thing will recur at the 
2319th day. Had the fraction been in excess, we should have 
written 02" with 1546 over. 
When the denominator of the approximate fraction is small, the 
same series of remainders may recur often during the work. In 
