# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Re: H.R. Mills on setting/rising bodies**

**From:**Christian Scheele

**Date:**2010 Jan 25, 21:05 +0200

Hello Robert, thank you very much for your quick reply and your elegant solution. I didn't go "Duh", if only for the reason that I don't think I would have made your sharp observation. Here's the rest of Mills deriviation, should the aforegoing have made you curious, that is, assuming you are not already familiar with it - admittedly a meagre reward for your effort, but I don't know what else I can offer right now. Departing from where we left of at equation(4), quoting Mills: "By the sine relationship:" (Mills is referring to the sine rule relating to pherical triangles) "sin(HA)/cos(alt) = sin(Az)/cos(dec) Therefore sin x = {cos(dec).sin(Az)}/cos(dec) (5) From...we know that cos(Az) = sin(dec)/cos(lat) when the (alt) = O (6)" [Mills is rearranging cos(Az) = {sin(dec) - sin(lat).sin(alt)}/cos(lat).cos(alt)] "so we can eliminate Az from (5) and (6) and so obtain an expression for x in terms of lat and dec. ..... by squaring and adding (5) and (6) we have sin^2(Az) + cos^2(Az) = 1" ( i.e. Mills applies trig. identities) "and therefore, {sin^2(x).cos^2(dec)}/cos^2(lat) + sin^2(dec)/cos^2(lat) = 1 therefore cos^2(lat) = sin^2(x).cos^2(dec) + sin^2(dec) sin^2(x) = {cos^2(lat) - sin^2(dec)}/cos^2(dec) = [cos^2(lat) - {1 - cos^2(dec)}]/ cos^2(dec) = {cos^2(dec) - 1 + 1}/ cos^2(dec)" The above line appears to contain another mistype: I'd say it should read = [{cos^2(dec) - 1}/ cos^2(dec)] + 1 Mills proceeds: "therefore 1 - sin^2(x) = cos^2(lat)/ cos^2(dec)" In my opinion there is another mistype in this line. I am referring to "cos^2(lat)". It can't possibly have been put in deliberately. I'd say that the term should in fact be (1 - cos^2(lat) so that the entire line of the equation reads: 1 - sin^2(x) = {1 - cos^2(lat)}/ cos^2(dec) Mills continues: "cos x = sin(lat)/ cos(dec)" I agree (most humbly). Mills then closes the section with this observation: "so the angle of rising or setting of a body depends only on lat and dec...." Kind regards and many thanks once again, Christian Scheele ----- Original Message ----- From: berneckywr-mail@yahoo.com To: NavList@fer3.com Sent: Monday, January 25, 2010 2:09:24 AM GMT +02:00 Harare / Pretoria Subject: [NavList] Re: H.R. Mills on setting/rising bodies Hi Christian, You are right, Mills did drop a minus sign in equation (3). Getting to step 4 is not hard (I'm thinking you are going to think, "Duh"). But it's just not easy to always be clear when writing about math. Here's one way to see what he did: cos(alt).d(alt)= cos(lat).cos(dec).sin(HA).d(HA) (3) Divide both sides of the equation by d(HA).cos(dec): cos(alt).d(alt)/(d(HA).cos(dec))= cos(lat).sin(HA) so that we can replace d(alt)/(d(HA).cos(dec)) on the left-hand side with sin(x) (from equation 2). This gives us cos(alt).sin(x)= cos(lat).sin(HA) or, solving for sin(x): sin(x)= cos(lat).sin(HA)/cos(alt) and, by equation 2, sin(x)= d(alt)/(d(HA).cos(dec)) so we have two different expressions for sin(x), and that is what Mills is indicating with (4). Robert Bernecky Mystic CT ---------------------------------------------------------------- NavList message boards and member settings: www.fer3.com/NavList Members may optionally receive posts by email. To cancel email delivery, send a message to NoMail[at]fer3.com ----------------------------------------------------------------