# NavList:

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

**Noon by least squares.**

**From:**Herbert Prinz

**Date:**2002 Apr 11, 06:22 +0000

What I don't like about Walter Guinon's procedure is that it assumes the DR position as an absolute given and derives local noon as a function of it and some observed altitudes. What is the application of this to the real world? In the general case it forces an error into t (= local time) to compensate for any error in assumed latitude. The beauty of the combined altitude method, however, is that, correctly applied, it will give a simultaneous solution for local time AND latitude. If one really must have a least square fit, it still can be done. After all, solving for latitude and local time from two altitudes is the same problem as solving for latitude and longitude from two altitudes, except that the last step, i.e. converting local time to longitude via GMT is omitted. So we can use St. Hilaire on the celestial sphere instead of on the globe of the Earth. The least square algorithm is given in the explanatory pages of the N.A. Just replace all GHAs (and your longitude) by SHAs. Rod Deyo holds that "You can determine the circles of constant altitude for three or more celestial bodies (assuming a spherical earth - something quite reasonable for practical navigation) and find their best-fit intersection numerically. A problem arises if you need to plot the circles of constant altitude on a Mercator chart to find the intersection [...]. Then you really do want something like the Marcq St. Hilarie intercept method [...]." Passing over the mistake about having to assume a spherical earth, which George Huxtable has already pointed out, it must be emphasized that St. Hilaire is not just needed for plotting. It is vital to the best-fit solution. I would be curious to know how Rod finds the best fit intersection of three circles, if not by first linearizing the problem. This is exactly what the intercept method does. It enables us to formulate the problem of the n-overdetermined fix as a system of n+2 linear equations in 2 variables for the solution of which we have a well established procedure. The penalty for the linear approximation is that the procedure is iterative. Rod does not need to worry about changed azimuths or high altitudes: The algorithm will converge. In short, trying to somehow solve for a least square fit for noon and/or latitude while bypassing St. Hilaire would be re-inventing the wheel or, worse, solving the wrong problem. Herbert Prinz