# NavList:

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

**Re: Traditional navigation by slide rule**

**From:**Brad Morris

**Date:**2016 Jan 20, 10:52 -0500

Hello Francis

You asked for a deeper level of explanation of my ah ha moment.

If we can multiply and divide sine and cosine, then we can directly compute using the law of cosines.

Similar to the Bygrave layout, we have the sine and cosine scales laid out in a logarithmic fashion. To multiply sine by cosine, we just add the logarithmic values like any slide rule. Since my ah ha has both sine and cosine on each cylinder, we can perform sine*sine, sine*cosine, cosine*sine and cosine*cosine.

Now examine the law of cosines

Cos(c)=cos(a)cos(b)+sin(a)sin(b)Cos(C)

With the exception of the simple addition, all other steps can be performed directly on the rule.

Provide the accuracy of the exceptionally long scales on a cylindrical rule and we can solve this equation, with accuracy, directly on the rule.

I won't provide an example, its entirely too obvious!!

Brad

Thanks Brad,

I love Ah Ha moments! Had a few (not enough!) during my career in medicine, but now with celnav, very cool!

I’m not quite sure what you mean by the suggested double sin/cos tubes arrangement. Could you send me a simple drawing and description of how it would work?

Currently, the long inner tube has scale A at top, then cos/sin marked in degrees and mins below, then the outer tube has another , identical scale A,(or is it B?) but not another cos/sin scale.

You find the sine or cosine by lining up the correct degrees and minutes and use the outermost curser tube to find the appropriate number on the A scale.( eg sine 30 or cosine 60 gives 0.5). Multiplication and division are done by lining up the inner and outer A scales as appropriate. But I think you are suggesting 2 cos/sin scales concentrically working directly against each other? Perhaps the outer curser tube could have another cos/sin, or even cotan for Bygrave formula?

I had a feeling this slide rule could go further.

Onward and upward!

Francis

From:NavList@fer3.com [mailto:NavList@fer3.com]On Behalf OfBrad MorrisSent:16 January 2016 21:27To:francisupchurch---.comSubject:[NavList] Re: Traditional navigation by slide rule

Francis

I have been following along but your last presented me with an Ah Ha moment.

With sine and cosine on the inner tube and sine and cosine on the outer tube, we can perform the standard equation of spherical trigonometry on a cylindrical rule.

Further, because the scale length is enormous, we have accuracy too!!

Now that was an Ah Ha moment for me!

Brad

On Jan 16, 2016 11:48 AM, "Francis Upchurch" <NoReply_Upchurch@fer3.com> wrote:

Bob,

photos of my latest miniFuller2 with sines and cosines marked in degrees and minutes and generally accurate to 1' except at the "squashed" ends of the scales.

The design format is my own, based on the Otis King which has 2 identical A/B scales(see attached of both for comparison) Mine has that plus sine scale in black with cosines in red going the other way.

This slide rule does all the celnav triangle calculations to 4-7 decimal places, so for me is the ideal general purpose CelNav slide rule.This one is robust and seaworthy too.

I find the simple cylindrical slide rules easy and cheap to build with plastic drain pipe off- cuts and very little precision engineering, other than printing, cutting and lining up the computer generated scales.

Build details on request.

Best wishes

Francis

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