NavList:
A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
Re: Emergency Navigation
From: Tom Sult
Date: 2012 Jul 14, 23:00 -0500
From: Tom Sult
Date: 2012 Jul 14, 23:00 -0500
I just remember 3.1415926. Seems easier that dividing those big numbers. Plus for this exercise remembering 3 is probably good enough.
Thomas A. Sult, MD
Thomas A. Sult, MD
Sent from iPhone
or remember 113355. split inthe middle, 355/113 = 3.1415929 the last digit should be a six so 355/113ths gets it almost right to 8 significant figures.
gl
--- On Sat, 7/14/12, eremenko@math.purdue.edu <eremenko@math.purdue.edu> wrote:
From: eremenko@math.purdue.edu <eremenko@math.purdue.edu>
Subject: [NavList] Re: Emergency Navigation
To: NavList@fer3.com
Date: Saturday, July 14, 2012, 1:32 AM
Computing the sines of few angles is easy.
The only thing which I recommend to memorize is pi.
To memorize pi, I recommend this:
http://www.math.purdue.edu/~eremenko/dvi/pi.pdf
(for English, French or Russian speakers:-)
Then you proceed as follows: if x is your angle in degrees,
then y=2.pi.x/360 is your angle in radians.
Then
sin(y)=y-y^3/6+y^5/120-y^7/(1.2.3.4.5.6.7)+ etc.
This is better to use for small angles. If your angle is not small
use the doubling formulas like sin(2t)=2sin(t)cos(t),
cos(t)=1-t^2/2+t^4/24-t^6/1.2.3.4.5.6+ etc.
Dividing a 1-foot ruler into 1/10 of an inch divisions,
without any instruments, is a much
harder task:-)
Alex.
P.S. This is a late XVII century math. Ptolemy had to use a different
method, much more sophisticated, starting from few angles (30, 45, 60, 90)
for which sine and cosine can be found precisely,
and then using division and addition formulas. To be more precise, he
did not use our modern functions, sine and cos, but used the chord instead,
chd(x)=2sin(x/2). His book contains the first table of chords that
survived to out time. He also had to compute without having our decimal
positional system, poor guy:-)