, 5-11ths of the last, and so
on. The sum of all is the circumference of that diameter. The preceding is
the process when the diameter is a hundred millions: the errors arising
from rejection of fractions being lessened by proceeding on a thousand
millions, and striking off one figure. Here 200 etc. is double of the
diameter; 666 etc. is 1-3rd of 200 etc.; 266 etc. is 2-5ths of 666 etc.;
114 etc. is 3-7ths of 266 etc.; 507 etc. is 4-9ths of 114 etc.; and so on.

2. To the square root of 3 add its half. Take _half_ the third part of
this; half 2-5ths of the last; half 3-7ths of the last; and so on. The sum
is the circumference to a unit of diameter.

Square root of 3.... 1.73205081
.86602540
------------
2.59807621
.43301270
.08660254
1855768
412393
93726
21629
5047
1188
281
67
16
4
1
------------
3.14159265

3. Take the square root of 1/2; the square root of half of one more than
this; the square root of half of one more {213} than the last; and so on,
until we come as near to unity as the number of figures chosen will permit.
Multiply all the results together, and divide 2 by the product: the
quotient is an approximation to the circumference when the diameter is
unity. Taking aim at four figures, that is, working to five figures to
secure accuracy in the fourth, we have .70712 for the square root of 1/2;
.92390 for the square root of half one more than .70712; and so on, through
.98080, .99520, .99880, .99970, .99992, .99998. The product of the eight
results is .63667; divide 2 by this, and the quotient is 3.1413..., of
which four figures are correct. Had the product been .636363... instead of
.63667..., the famous result of Archimedes, 22-7ths, would have been
accurately true. It is singular that no cyclometer maintains that
Archimedes hit it exactly.

A literary journal could hardly admit as much as the preceding, if it stood
alone. But in my present undertaking it passes as