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I know of at least two ways to prove the cosine rule (one shown in
Nautical Astronomy by Smart and the other in the Kerns, Kell book I
mentioned) and there are many different ways of expressing it.  Working
through the proofs helps understand the different ways it can be
expressed.

Regards,
Bill Allen

-----Original Message-----
From: Navigation Mailing List
[mailto:NAVIGATION-L@LISTSERV.WEBKAHUNA.COM] On Behalf Of Patrick
Stanistreet
Sent: Wednesday, January 28, 2004 2:34 PM
To: NAVIGATION-L@LISTSERV.WEBKAHUNA.COM
Subject: Re: spherical right triangles, napiers circle

Thanks for the link to the online book.  I downloaded
the entire book as a single zip file only 685 K.
Just use the url without the chapter information.

In my original email I wasn't too sure what needed
to be included in the question.  The link provided
http://home.t-online.de/home/h.umland/Chapter10.pdf
does have the pertinent information regarding
napiers circle and the the two rules.
The Congleton book I am using has the same information.


The reason I am interested in a derivation or a
proof of Napiers rules is that the cosine rule
for spherical triangles is shown in the Congleton
book to follow.
Perhaps there are other ways to derive the cosine
rule in which case I would still like to read
through the proofs. Also I would guess that the
derivations of Napiers rules and the circle are
historically interesting.


Ex: These notes are from page 92-94 of Congletons book
Given an oblique sph. triangle with 3 points ABC.
A left bottom, B top middle and C right bottom
Now form 2 rt. sph. triangles by dropping a
perpendicular from B to segment AC. The point
of intersection to be called D. The segment
from B to D is called y and from A to D is called x
and from D to C is called  b - x.

In triangle DBC by napiers rule
Eq1:  sin(co - a) = cos(b - x)cos(y)
Eq2:  cos(a) = cos(b - x)cos(y)

using the trig identity for the difference of
two angles Eq2 becomes
cos(a) = cos(y)( cos(b)cos(x) + sin(b)sin(x) )

Eq3:  cos(a) = cos(y)cos(b)cos(x) + cos(y)sin(b)sin(x)

Using triangle ABD and napiers rules we can solve for
x and y in terms of c and A

Eq4:  sin(y) = sin(c)sin(A)
Eq5:  sin(x) = tan(y)cot(A)
Eq6:  cos(c) = cos(x)cos(y)

Using Equations 4,5 and 6 in Equation 3
we get the cosine rule.

cos(a) = cos(b)cos(c) + sin(b)sin(c)cos(A)

I like the proof but it bugs me about using
Napiers rules since I dont have any proof
for them.


Noyce, Bill wrote:
> George Huxtable asks:
>
>
>>Patrick does well to ask where those rules are derived from.
>>I haven't seen them before, and wonder if they are true.
>>Is Patrick really quoting from the book word-for-word?
>>Has he left something out? I'm not very conversant with "Napier's
>
> rules".
>
> Yes, there's something left out.  Napier's "parts" are typically
> arranged in a circle, with the right angle omitted.  The parts
> are the two sides adjacent to the right angle, and the *complements*
> of the other angles and other side.  See, for example, the nice
> tuorial at http://home.t-online.de/home/h.umland/Chapter10.pdf
>
> So for George's example, if we consider the right angle at
> Lat=0, Lon=0 to be "the" right angle to be omitted, then
> the parts are, in order:  The 90d length of the arc from
> there to the pole, 0d for the 90d angle at the pole, 0d for
> the 90d arc back to the equator, 0d for the angle there,
> and 90d for the arc along the equator.  With these fixes the
> formulas test out fine.
>
> I don't recall having seen a derivation of these rules.  I
> suspect that given time I could work them out myself.  The
> critical thing to notice is that an angle on the surface of
> the sphere is simply the angle between two planes defining
> the great circles, which can be measured in a plane normal
> to both of them.  Similarly, the "length" of an arc on the
> sphere is the angle between its endpoints at the center of
> the sphere.
>
>         -- Bill
>
>

   

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