[R] Value of 'pi'

[(Ted Harding)]

On 30-May-11 07:06:57, Peter Langfelder wrote:
> On Sun, May 29, 2011 at 11:53 PM,  <Bill.Venables at csiro.au> wrote:
>> There is an urban legend that says Indiana passed a law implying
>> pi = 3.
>>
>> (Because it says so in the bible...)
> 
> Apparently the Fortran language has a DATA statement just for this
> purpose. This is allegedly a quote from an early Fortran manual:
> 
> The primary purpose of the DATA statement is to give names to
> constants; instead of referring to pi as 3.141592653589793 at
> every appearance, the variable PI can be given that value with
> a DATA statement and used instead of the longer form of the
> constant. This also simplifies modifying the program, should
> the value of pi change.
> 
> Peter

My take on this discussion:

Take a nice-looking pie, say 113355, slice it, and put one
half on top of the other. Call it "pi":

  pi = 355/113

Compared with "pi = 22/7", which is not even pretty, it is
also a much closer approximation to the mathematical ideal:

To 20 decimal places (using 'bc' here)

"true pi"
= 3.14159265358979323844

355/113
= 3.14159292035398230088

22/7
= 3.14285714285714285714

so 355/113 is good to the 6th decimal place (3.141593),
while 22/7 breaks down at the 3rd (3.143 instead of 3.142).

In the back of my head is a memory of a passage I read
some 50 years ago. I write a paraphrase, since I don't
recall the exact words:

 "For an engineer, assuming that pi = 3.142 will
  probably enable him to build a very satisfactory
  bridge. Assuming that pi = 3.14159265358979323844
  will give the circumference of the Earth's orbit
  to one millionth of a millimetre. For a pure
  mathematician, however, either assumption leads to
  the conclusion that 1 = 0. It is necessary to
  preserve common sense in the application of
  mathematical deduction."

I suspect (from my context at the time) that it may
well have been by J.L. Synge (beautiful writer on
theoretical physics, especially Relativity Theory)
in one of his several writings on Ballistics.

However, the one possibly relevant printed item which
I still have from those days:

K.L. Nielsen and J.L. Synge,
"On the motion of a spinning shell"
Quarterly of Applied Mathematics, 4(3), Oct 1946,201-226.

discusses a very similar issue, but puts it quite
differently. If my "quotation" above reminds anyone
of the original, I would be very grateful to learn
of the reference to the source!

With thanks, and Many Happy Approximations to you all!
Ted.

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Date: 30-May-11                                       Time: 09:52:09
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