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H. Prinz wrote (of methods for correcting for the Earth's oblateness):

"Charles Borda had already given such a procedure..."

What did his correction look like? Does it work?

This is a very good question! Yes, it works, ...sort of.

Although at its core, Borda's method of clearing the distance is of the so called "rigorous" type, his correction for oblateness only considers linear terms. This is perfectly reasonable. The method for the additional correction is based on algebraic and geometric arguments, and so are the simplifications that are made along the way, such as loosing a second order term here or using a plane triangle there.

Conceptually, there are two steps:

1. Adjusting the tabulated parallax of the moon to the one corresponding to the actual earth radius at the latitude of observation; and
2. Finding the true corrected distance in triangle SZ'M* , where S is the true star, Z' is the geocentric zenith, and M* is the true Moon as per the adjusted parallax.

However, obviously in order to minimize computational effort at the user end, these steps are not implemented in a straightforward manner. Instead, one first computes from sZm in the traditional way the "true" distance in the triangle SZM', where S is the true star, Z is the geodetic zenith, and M' is a fictitious place of the moon derived from a fictitious parallax to be picked from a table (Table VII in op. cit. in previous message). Then one finds two corrections to SM', each requiring the addition of  4 logarithms. These corrections being added to SM' yield SM*.

The derivation is given in the appendix, with minor gaps to be filled in by the reader. To be able to follow it in detail, one needs the diagram in Plate III. If you are interested in the details, the easiest for me would be to e-mail you photocopies of the relevant pages (after I am done with my Christmas shopping). There is also a worked example.

So far so good. Here is the bad news.

The method was published in 1887. Back then, nobody knew what the shape of the Earth was. Borda assumes the flattening to be 1/200. This affects his auxiliary table for adjusting the parallax, as well as a constant to be used in the formula in step 2. To make matters worse, the table he provides is geared towards the Connoissance des temps. Back then, this publication tabulated the horizontal parallax not for the equatorial radius of the Earth, but for that at the latitude of Paris. (I could not find information on what this radius was believed to be.) This was of course an advantage for navigators in mid latitudes who did not want to bother with such subtleties for time sights, shots for latitude or even lunars. But it means that Borda's procedure cannot be used without modification unless it is in direct combination with a contemporary Connoissance.

Of course, it is possible to modify his procedure to bring it in accordance with the modern parameters.

Herbert Prinz