NavList:

Thank you, Frank, for your comments.

You wrote:

The other method of Mendoza y Rios was apparently his pride and joy which he spent years compiling and setting in type. It was a well-constructed "direct method". I have a copy of those tables in their original binding in excellent condition sitting on my bookshelf a few feet from here. Their popularity is unclear, but a fairly large print run was supposedly funded by subscription. Unfortunately it's a big book whose only function is to clear lunars. Many navigators preferred the series methods because those volumes were compact, and the tables were often general purpose.

It is very interesting. May I ask you to look what is the name of the book in your bookshelf and where and when it was printed.

I am little confused abut number of methods under the name of Mendoza y Rios. At least I am sure that these two methods from 1801 and 1809 are printed with book's author name Mendoza Rios and with his preface. It is not just published name in other author's work.

I suspect that none of the books I had mentioned (printed in 1801. and 1809. accordingly) is the same in your bookshelf.

I saved in pdf format lunar clearing instructions and examples both from 1801. and 1809. books (see attached files).

The method from 1801 uses tables on 72 pages (full editions contains  about 350pages, but these 72 are devoted to lunar clearing specially). I think that this is quite compact version. For example, Thomson tables for clearing lunars occupies 86 pages.

The method from 1809 (this was used by Franklin) has tables for lunar clearing on 345 pages (with other tables the full book contains 660 pages). But althogh tables are more extensive, the process of clearing lunars is faster than by method from 1801.  

I am not sure, but it seems that method from 1809 is one of series methods (or “approximate” as it was called in 18. and 19. century), because tables contain values with 4 digits (index plus 4 digits). Besides calculations are very short. Maybe method from 1801 is direct (or as it was called historically—rigirous), because tables contains values with more digits. Maybe you can look at attached pdf to evaluate both metods just by visual arrangement of calculation examples (pdf contains two pages).

You wrote:

One little clarification: the altitudes are required to the nearest few minutes (+/-5' is generally acceptable) in all methods of clearing lunars. It's a property of the geometry of lunars rather than any specific clearing method.

In fact I did not mean the accuracy needed in altitudes. In NavList archive there can be found wonderful posts from you about this theme including 90 degrees miracle. On this subject I will post individual message (yesterday just for fun I made tables based on formulas you published, that allows to evaluate impact of errors in altitudes on the final accuracy of cleared distance).

I was talking about Mendoza y Rios method from 1809, where altitudes are summed (values to nearest minutes) and then Objects complementary correction and Moons correction is to be added (but these correction values are to be taken out from tables to second of arc). I think that use of seconds in altitudes makes additional error in this particular case, because precise correction values determine how much the Moon and the Object are shifted. But may be I am wrong. It is just my assumption.   

Modris Fersters

PS:

1)Link to Mendoza's book from 1801:

https://books.google.lv/books?id=FzIAAAAAQAAJ&redir_esc=y

2) Link to Mendoza's book from 1809: 

https://archive.org/details/b22007659