NavList:

Well, perhaps I exaggerate in order to get your attention. :) We don't need to kill the intercept method, but in the modern world it is much more reasonable to calculate lines of position by other means, despite the near-religious adherence to the intercept method.

The most transparent alternate method for calculating points on a line of position is a "modern time sight". Working time sights by logarithms was bread-and-butter navigation for 150 years, through the middle of the twentieth century. This calculation is not obsolete, not somehow wrong or mathematically inferior. It fell by the wayside because the intercept method was slightly faster for longhand computation and could be implemented in somewhat shorter mathematical tables. This doesn't matter anymore, so I highly recommend a time sight for calculator work. How do we work a "modern time sight"?

Here's the sequence of steps that I teach:
Note: h is the corrected latitude, Dec is the body's declination, and Lat is the assumed latitude (DR lat if accurate).

Get A = sin(h) / cos(Dec) / cos(Lat)
Get B = tan(Dec) · tan(Lat)
Get C = A - B. ...not that A is always positive except in an exotic case, but B may be positive or negative.
Take inverse cosine ("shift cos"): D = cos^-1(C)
Longitude = GHA +/- D, add if Sun or star is east of you (rising in altitude), else subtract.

This procedure yields one point on the line of position. If the latitude is accurate already, then you're done. If not, then the initial assumed latitude on the first pass should be the nearest tenth of a degree to your DR latitude. The calculation for longitude is then repeated with the initial latitude bumped up by 0.1°. Then the points are plotted on common graph paper and the line of position is drawn through them (recommended spacing: each square on graph paper should be 0.02° and no distinction is made for latitude and longitude --it's a pure "square" plot of points). Two sights yield two lines of position as usual, and you are where the lines cross. 

Advancing lines of position on common graph paper is slightly different, easiest by calculation rather than the plotting procedure you learn with the intercept method. Suppose we want to advance an earlier line of position, LOP1, which was calculated for some early time, UT1, in order to plot it with a later line of position taken at some time UT2. Then:
dT = UT2 - UT1 (express as hours, so 3:45 is 3.75 hours)
Dist = speed · dT / 60
dLat = Dist · cos(course)
dLon = Dist · sin(course) / cos(Lat)   ...note that the cos(Lat) here takes care of longitude scaling.

Adjusted LOP1 (modifies both points):
Lat1mod = Lat1 + dLat
Lon1mod = Lon1 - dLon
Lat2mod = Lat2 + dLat
Lon2mod = Lon2 - dLon

Then plot the adjusted points and draw a line through them. Where it crosses the second line of position, that's your fix. It's easy! 

This "modern time sight" methodology has no fuss with inverse tangents, few rules about special cases (I've omitted little details like east longitude but you can add those later ...or stay out of the eastern hemisphere), and best of all it does not depend on special plotting charts which are one of the defects of the intercept method as commonly taught. Note also that this process can be repeated as many times as desired, and the points as plotted represent points on the true circle of position, not merely poinst on some straight line that locally approximates the circle. For the experts, there is a technical detail to be worked out; an additional rule to be applied for the latitude spacing when the body is close to the local meridian. Anyone care to work it up?

This is exactly how I teach "Modern Celestial Navigation" and it's what I'll be doing in the workshop tomorrow and Sunday at Mystic Seaport.

Frank Reed
ReedNavigation.com
Conanicut Island USA