354 Proceedings of Royal Society of Edinhurgh. [june 4 , 
the square of the half sum is less than half the sum of the squares 
of two numbers, hy the square of the half difference : — 
fR + r\ 
2 -f ^ 
^K-r\ 
V 2 ) 
' 2 ' 
^ 2 ) 
Now the half difference 
K 
is small, and its square contains few 
effective figures, so that the labour of computing it is trifling. We 
therefore annex a sixth column 
r-?) 
to our scheme, as is done 
in the accompanying computation to ten places. Each half 
difference is less than the fourth part of the preceding, wherefore 
the numbers in this column decrease sixteen times at each step. 
In our example, when we have reached the number of sides 1024, 
the square of the half difference does not amount to unit in the 
eleventh place, and no longer affects the work, so that the sub- 
sequent are computed as among themselves directly; the 
being augmented by and the being diminished by 
Now these changes are quartered at each successive step, wherefore 
the whole augmentation of r'^ is (J + |-+ 3 V + * • •) f while 
the whole diminution of E^ is (J + tV + + • • •) ^2 i 
There is, then, no need for continuing further the details of the 
work. To the value of for the 1024 sided polygon, we add two- 
thirds of the corresponding value of ; from that of E^ we sub- 
tract one-third part of and so get the ultimate values of and 
E^, which values necessarily agree. The square root of this is then 
the radius of the circle whose circumference is 8*00000 00000 ; 
hence there results for the ratio of the diameter to the circumference 
of a circle 
1*27323 95447 : 4*00000 00000 ' 
or *31830 98862 : 1*00000 00000 
or 1*00000 00000 : 3*14159 26536 
Thus these very simple artifices serve to lessen the labour attend«r 
ing Leslie’s elegant solution of the problem, by nearly one-half. 
