NavList:

Hanno,

I received a copy of Jospeh Breed's "Navigation without Numbers". His method is not quite what I had anticipated in that it is not the straightforward application of the gnomonic projection and its properties. In 3D, Breed constructs a triangular pyramid whose apex is at the centre of the sphere. This is immediately recognized as the usual construction used in textbooks to derive the standard formulas of spherical trigonometry and he states as much, "Fig. 9 and Fig. 10 exhibit the geometrical form from which the mathematical formulas of navigation are derived". Breed's method involves constructing the four triangular sides of the pyramid on paper and measuring their angles directly. The approach is sound but, as with all graphical methods is limited by the accuracy with which one can draw and measure the results. In Breed's case this involves constructing plane triangles and measuring their vertex angles using a protractor. He says, "In some special cases there may be an error of as much as half a degree (30 minutes of arc or 30 nautical miles)". Actually to me it would seem to be challenging to do very much better than that. His construction is strictly linear in the sense that the error in constructing and measuring the relevant angle translates directly into the error in the position on the Earth's surface. Not discussed by Breed is the possible application for plotting great circle sailings. Still it is probably a useful method to have in one's naviational box of tricks possibly for emergency use.

Coming back to to the problem of double altitude sight reduction; in the same way that spherical trigonometry does, Breed's method deals with great circles. It is not clear to me that would offer any special advantages where small circles are needed. (Of course it would probably be possible to replicate the 5 step spherical trigonometric method for reducing double altitude sights - see John Karl's "Celestial Navigation in the GPS Age", graphically by Breed's method but I would expect that the accumulation of errors would be prohibitive.)

The graphical solution to the double altitude sight I described does not involve the direct measurement of angles using a protractor but constructs the position directly on the stereographic chart. It is also non-linear in the sense that the scale changes across the chart. In the example I treated it can be seen that since the geographic position (GP) of the Sun is in the Southern Hemisphere where the chart scale is condensed, errors in locating it have a reduced effect on the determination of the observer's position in the mid-Northern hemisphere where the scale is expanded,

Thanks again for pointing out this work,
Regards,
Robin Stuart

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