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of
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Calculations based on
FO = .9999999999 and
r = 1
FO = .9999999999
; i.e. 10 digits of 9
's
to the right of the decimal point.
r = 1
A=
(r**2 + FO**2)**1/2 =
1.4142135623377397097419194062514e

5
B =
A
/ FO =
1.4142135624791610659898355128504e

5
C
=
ArcTan
{
B
}
=
8.1028468454814783497100949580609e

4
D =
C
/ 180
=
4.5015815808230435276167194211449e

6
E
=
pi * r**2 *
D
=
1.414213562384880161821729681326e

5
Therefore,
E

A =
4.7140452079810275074595111990254e

16
; i.e. a
positive
quantity.
Admittedly, the
ArcTan
value here is not less than 1/10**6 (10 to the minus 6

th power),
however, this is the best I can do using my handheld calculator.
But even here, one can
appreciate the greatening affect the
contributing
factor
pi
has in raising t
he magnitude of E over
that of A.
This should be taken into account when
forming a
rough estimat
e
.
Insofar as your suspicion
go
es
,
lebesque,
you will have to offer a
specific
counter

example that
shows S will be
negative
for when ArcTan has a value t
hat is less than 10 to the minus 6

th
power.
Perhaps, we can decip
her an answer to your suspicion
, if you
or a friend
have access to a
university
computer
which uses a high precision computational program that accepts inputted
values of, say, r = 1 and F
O = .9999999999999999
(to 16 decimal places or more).
That ought to do it.
I once wrote
fou
r
computational
programs (in PL1 language) that expand
enormously on
the
inherent limitations of the IBM
mainframe
computer to give
exact
results on the operat
ions of
addition, subtraction, multiplication and division.
Needless to say,
IBM quickly
became
enchanted
with
the
se
programs
.
Unfortunately, I presently
do not have
access to one
,
because
I
am
no longer in
volved with
th
at
field
.
Other than trying to
il
legitimize
my equation by nibbling away at it
s
structural edges, as you
have attempted here,
I can't see any
other way of doing it except for delving head

on into
the
matter
, whereby either disproving it or vindicating it
through logic
.
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