Question for the Astro-Navvies

[requiem]

Haversines make an appearance in the 1916 edition...


The 1888 version describes the Sumner method, but it had not yet advanced to the intercept method. Instead only the circles of equal altitude are introduced and portions of those are what is plotted. For the plotting, two latitudes are assumed for each observation, giving two points on the circle, and a line of position is drawn between them. In the figure below, Case I illustrates using observations of separate objects and Case II illustrates using the same object at different times (your basic running fix).

 

[Frank Holden]

Haversines make an appearance in the 1916 edition...
View attachment 186999

The 1888 version describes the Sumner method, but it had not yet advanced to the intercept method. Instead only the circles of equal altitude are introduced and portions of those are what is plotted. For the plotting, two latitudes are assumed for each observation, giving two points on the circle, and a line of position is drawn between them. In the figure below, Case I illustrates using observations of separate objects and Case II illustrates using the same object at different times (your basic running fix).

View attachment 187002

Haversines and a description of the Marc St Hilaire method only appeared in Bowditch in about 1916.
No point having one without the other as you couldn't use logs with the Cosine formula - logs can't handle negative values..
Another minor point is that haversines are more accurate than cosines at small angles.

 

[Frank Holden]

Enjoying the thread.

Interesting that Bowditch uses air tables / intercept method and doesn't describe the Haversine method as an alternative (my copy is 2002 Bicentennial Edition).

I've been Googling, trying to keep up with some of the very knowledgeable posters on here, and come across Bygrave Slide Rule.

http://tinas-sliderules.me.uk/Slide Rules/Tina's Bygrave.pdf

Interesting that the author of that slide rule link doesn't seem to have graduated beyond the Sumner method - scroll down to 'Latitude and Longitude'.
Re Bowditch - it is aimed at the American market, the same people as gave us 'H for height' instead of 'Altitude'. I guess that is what you have to do when training Iowa farm hands to navigate their B-17s across the Atlantic.

I have all three volumes of the Air tables on my boat along with Nories and a 2016 almanac. I keep the almanac as some of the auxiliary tables are handy to have.
So all I need to do before embarking on a long sea voyage is download the required daily pages of the almanac.
The person who doesn't own their own hard copy of the air tables has a lot more downloading to do.
I think I have already mentioned that I tend to use the haversine formula for the sun but the air tables for stars. The benefit with the tables for the stars is mainly in the planning.