ous 3.14159..., which insists
on calling itself the circumference to a unit of diameter. A reader who is
competent to follow processes of arithmetic may be easily satisfied that
such methods do actually exist. I will give a sketch, carried out to a few
figures, of three: the first two I never met with in my reading; the third
is the old method of Vieta.[354] [I find that both the first and second
methods are contained in a theorem of Euler.]

What Mr. James Smith says of these methods is worth noting. He says I have
given three "_fancy_ proofs" of the value of [pi]: he evidently takes me to
be offering demonstration. He proceeds thus:--

"His first proof is traceable to the diameter of a circle {211} of radius
1. His second, to the side of any inscribed equilateral triangle to a
circle of radius 1. His third, to a radius of a circle of diameter 1. Now,
it may be frankly admitted that we can arrive at the same result by many
other modes of arithmetical calculation, all of which may be shown to have
some sort of relation to a circle; but, after all, these results are mere
exhibitions of the properties of numbers, and have no more to do with the
ratio of diameter to circumference in a circle than the price of sugar with
the mean height of spring tides. (_Corr._ Oct. 21, 1865)."

I quote this because it is one of the few cases--other than absolute
assumption of the conclusion--in which Mr. Smith's conclusions would be
true if his premise were true. Had I given what follows as _proof_, it
would have been properly remarked, that I had only exhibited properties of
numbers. But I took care to tell my reader that I was only going to show
him _methods_ which end in 3.14159.... The proofs that these methods
establish the value of [pi] are for those who will read and can understand.

200000000 31415 3799
66666667 2817
26666667 1363
11428571 661
5079365 321
2308802 156
1065601 76
497281 37
234014 18
110849 9
52785 5
25245 2
12118 1
5834
-------- -------- -------
314153799 31415 9265

{212}

1. Take any diameter, double it, take 1-3d of that double, 2-5ths of the
last, 3-7ths of the last, 4-9ths of the last