NavList:

We have been discussijng why negative numbers were rarely used in navigation. I will go further and ask why complex numbers or quaterions were not used. I will suggest that  the traditional methods worked so complex numbers were not needed. Also the masters thesis I am looking at was not written until 2020!

 "The application of complex numbers and quarternions in navigation" -  Auburn University Alabama,  Alason Lakhani.

There are two problems. First I do not know what a quarternian is. Secondly what letter would navigators use for the complex number? Mathematicians use "i" and electrical engineers use"j".  "k" is the next letter of the alphabet but it would be confusing  as "k" is a factor used in several short method tables.

The above was written half in jest. To be serious perhaps we should consider Edward J Willis. I discussed this southern gentleman on navlist some years ago. I have attached a text file with some notes I wrote at the time. Note my comment that IMHO 19th century navigation texts are easier to read than Willis. Maybe his lack of communication skills are why he did not become well known.

Edward J Willis was an interesting fellow. I have searched this forum 
archives but have not found any discission about him. He was born in 1866 and 
died in 1941.

His father was a lieutenant in the old USN and later a captain in the 
Confederate States navy. He was dishonourably discharged along with certain 
other Southern officers whose resignations were not accepted.

Willis wrote two books on navigation - in 1921 and 1925 - and constructed and 
patented a navigating machine.

Willis claims to have devised a superior CN method using differential calculus 
rather than spherical trigonometry.  However his method appears to use the 
standard spherical trigonometry equations. He recommends using Crelle's tables 
but will accept Martelli to reduce the sights. The differential calculus part of 
his method is the calculation of latitude and longitude corrections (which he 
calls differentials) to determine position. This is exactly what Cloudy Weather 
had done nearly 50 years earlier.

Wllis does not like the haversine formula - see p17 Mathematics of Navigation.

I have a problem with his writing style - dense text with few paragraphs or sub 
headings.  19th century texts are sometimes difficult to read because algebra 
is avoided but I would rather try and read a 19th century text than Willis. 

Bowditch 1958 devotes several pages to Willis.

I am having  difficulty fully understanding Willis' method.  Has anyone studied 
Willis?  If so it will save me lots of work!

**** comments on haversines******

============================================================================
Links to  relevant documents follow:

1921 The Mathematics of Navigation
https://archive.org/details/mathematicsnavi00willgoog/page/n7/mode/1up

1925 The Methods of ModerN Navigation
Hathitrast (cannot download pdf)  
https://hdl.handle.net/2027/mdp.39015038721869

Google books
https://books.google.co.nz/books/about/The_Methods_of_Modern_Navigation.html?id=
GMmEAAAAIAAJ&redir_esc=y

Hard copies are for saleon Abe Books.

Willis Navigating Machine 
https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&ved=
2ahUKEwiyoc6XyOXsAhXaZSsKHZAKCCIQFjABegQIARAC&url=https%3A%2F%2Fcatalogs.
marinersmuseum.org%2Fmedia%2FMedia%2F00002%2FNQ32-The%2520Willis%2520Navigating%
2520Machine-A%2520Forgotten%2520Inv00636640696068796211.pdf&usg=AOvVaw3gS6l-
qg0DY5YWA1qoUeoA


Patent
https://patents.google.com/patent/US2064236