352 Proceedings of Royal Society of Edinhurgh. [june 4, 
more than some fifteen minutes; on dividing 5280 by 1680*7 we 
get the quotient 3*1415 + . This computation may also be made 
Circumference = 5280 
n 
r 
r2 
R2 
R 
4 
660*0 
43 5600 
43 5600 
87 1200 
933*4 
8 
796*7 
63 4731 
10 8900 
74 3631 
862*3 
16 
829*5 
68 8070 
2 7225 
71 5295 
845*8 
32 
837*7 
70 1741 
6806 
70 8547 
841*8 
64 
839*7 
70 5096 
1701 
70 6797 
840*7 
128 
840*2 
70 5936 
426 
70 6362 
840*5 
256 
840*3 
70 6104 
106 
70 6210 
840*4 
Diameter =1680 7 
by beginning with the hexagon whose base and circumscribing 
radius are each 880. By changing the 5280 into 5680, so as to 
make it divisible by 71, we get an easy verification of the well- 
known ratio first given by Metius ; thus : — 
Circumference = 5680 
n 
r 
7*2 
«2 
R2 
R 
4 
710*0 
50 4100 
50 4100 
100 8200 
1004*1 
8 
857*0 
73 4449 
12 6025 
86 0474 
927*6 
16 
892*3 
79 6199 
3 1506 
82 7705 
909*8 
32 
901*0 
81 1801 
7876 
81 9677 
905*4 
64 
903*2 
81 5770 
1969 
81 7739 
904*3 
128 
903*8 
81 6854 
492 
81 7346 
904*1 
256 
903*9 
81 7035 
123 
81 7158 
904*0 
Diameter 1808 
1808 :5680 :: 113 : 355 
Thus with very little labour we are able to compute the value of 
7 T with exactitude sufficient for almost all business purposes, and 
by a process requiring a knowledge only of the mere elements of 
geometry and arithmetic. 
When we address ourselves to the serious task of making the 
computations to a great number of places, we find that, as in all 
analogous cases, the labour increases in a much higher ratio than 
the number of the places. Imagining the line DO to be drawn, we 
see at once that PD is the mean proportional between PC and PO, 
