NavList:

Randy, you wrote:
"I understand there are tables that can correct an altitude for noon."

There's an awful lot of confusion over these historical sights. An ex-meridian solution serves no purpose if you have modern sight reduction tools available. It adds no information. It does not tease out any extra navigational information from a sight. It does not "correct" an altitude. If you never heard of ex-meridian sights, or indeed time sights, which are the other side of the same coin, you would not be a worse navigator. On the other hand, ex-meridian sight reductions and time sight reductions can be interesting from a historical point of view and in some circumstances, see below, they have an interesting practical advantage.

To place an ex-meridian sight in perspective from the point of view of "standard" line of position celestial navigation, draw an LOP on a chart for a noon observation. It is a horizontal line in the usual north-south orientation of plotting charts. Now imagine taking a sight 20 minutes before local noon. The LOP for this observation will be "tilted" on the chart, running from somewhere around WSW to ENE. All of the navigational information available from that sight is contained within that LOP. But unlike the sight at true noon, since the LOP is not quite horizontal on the chart, it does not immediately tell us latitude. Instead, we have to separately determine longitude (by crossing another LOP, or by knowing our location) or we have to pick an arbitrary longitude. Given some longitude, there is only one point on that nearly horizontal LOP that passes through it. You can then read off a latitude from that given longitude. And that is all that an ex-meridian sight does! In fact, this chart-based procedure of picking off the latitude from the plotted line of position, will frequently produce better results than the standard ex-meridian tables. I must emphasize that there is no other navigational information in a sight that is not already analyzed by the standard line of position. It may seem as if we are improving the sight, correcting it for noon, but all we're doing is picking off a single point from the LOP based on an assumed longitude (the assumption that we know how many minutes we are away from LAN at the time the observation was taken is exactly equivalent to assuming a known longitude). 

Now for some math. Let's consider the most basic spherical trig equation for the altitude of a celestial body:

  sin h = sin δ · sin L + cos δ · cos L · cos(GHA - λ)

where L, latitude, and λ, longitude, are the things we're looking for, and δ, declination, and GHA, Greenwich Hour Angle, are data that we look up in an almanac or other database for the given date and time, and finally, of course, h, the altitude (corrected) is the quantity that we measure. Supposing we have measured an altitude (at any time of day) and also looked up the correct almanac data, we are left with a single equation with two unknowns in it. Only latitude and longitude are unknown. Or, in words, with known values for GHA and declination and with a known (observed) altitude, there is a one-to-one relationship between latitude and longitude, which is identical to a string of points in a line drawn across a chart (ignoring the bigger picture where the LOP eventually curves into a circle).

In the traditional "time sight", the navigator would use an estimated latitude, and then would solve the above standard equation for longitude. That's not difficult since longitude appears in only one place in the equation. Solving:

  λ = GHA + cos^-1[(sin h - sin δ · sin L) / (cos δ · cos L)]

But notice that this gives us a longitude for one specific assumed/estimated latitude. Unless the celestial body is obsrerved exactly due west or due east, implying that the corresponding LOP is "vertical" (running north-south) on the chart, a different choice of latitude would yield a different longitude. That's why navigators traditionally were advised to take longitude sights (or local "time sights") when the Sun was relatively close to true east or west. And in fact, for any azimuth (not necessarily east/west), we can generate as many points on the actual LOP as we want by repeating this calculation for various latitudes. On a calculator or any modern computing device, this is very fast. This was Sumner's procedure for generating a line of position, though in the 19th century it was a lot of work.

Now consider the same process when the Sun is nearly due south or due north so that the corresponsing LOP is nearly "horizontal" on the chart. The basic equation still gives us a one-to-one relationship between latitude and longitude --in other words, the mathematical relationship is equivalent to an LOP. But how do we solve it for latitude? The basic equation:

  sin h = sin δ · sin L + cos δ · cos L · cos(GHA - λ)