353 
1888,] Dr E. Sang on Diameter of a Circle. 
wherefore the difference between PC and PD is less than that 
between PD and PO ; it is therefore less than the half of CO ; but 
CO is half the difference between OA and OB ; so that at each 
duplication of the number of the sides, the outer and inner radii of 
the polygon are brought rather more than four times closer. Now 
the fifth power of 4 exceeds 1000, wherefore five successive dupli- 
cations will carry the precision three steps farther on the decimal 
scale. 
The toilsome parts of the works are, the extraction of the root of 
E^, and the squaring of r. In the case first mentioned, we have 
the simple expedient of retaining on the slate ail those figures 
Circumference = 8 *00000 00000 
r 
^2 
4 
1-00000 00000 
1*00000 00000 
1*00000 00000 
8 
1*20710 67812 
1*45710 67812 
*25000 00000 
16 
1*25683 48730 
1*57963 38981 
6250 00000 
32 
1*26914 62985 
1*61073 23269 
1562 50000 
64 
1*27221 67266 
1*61853 53993 
390 62500 
128 
1*27298 38710 
1*62048 79358 
97 65625 
256 
1*27317 56282 
1*62097 61803 
24 41406 
512 
1*27322 35657 
1*62109 82483 
6 10352 
1024 
1*27323 55500 
1*62112 87658 
1 52588 
+ 1 01725 
1*27323 95447 
1*62113 89383 
Circumference = 8 *00000 00000 
R2 
R 
NO 
2*00000 00000 
1*41421 35624 
4 
1*70710 67812 
1*30656 29649 
*00247 28831 
8 
1*64213 38981 
1*28145 77239 
15 15712 
16 
1*62635 73269 
1*27528 71547 
94275 
32 
1*62244 16493 
1*27375 10154 
5885 
64 
1*62146 44983 
1*27336 73855 
368 
128 
1*62122 03209 
1*273*27 15032 
23 
256 
1*62115 92835 
1*27324 75343 
1 
512 
1*62114 40246 
1*27324 15421 
1024 
- 50863 
1*62113 89383 
1*27323 95447 
which are available for the next extraction ; and as the work 
proceeds the figures to be effaced become very few. 
For the square of the new r, we take advantage of the law that 
