# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

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Re: Haversine- how to derive it?
From: John Brown
Date: 2015 May 19, 01:12 -0700

Thanks and much appreciation to all who responded to my versine vs haversine question.

Greg, you wrote: Haversines eliminate rules for versine of meridian angles larger than 90° and versines of latitude + declination sums larger than 90°.

Using one example of the above conditions without any special rules:

Lat 56N, dec S35, LHA 91 and 5-figure logs with haversines, compared to 4-figure logs with versines, I get the same ZD of 118d 55'.  (Negative Hc, so only really useful for GC distance!).  I used Burton's Nautical Tables with log and natural haversines (5-figure) and Burton's 4-figure tables with log and natural versines.

Christophe, you wrote: The versine is bounded by [0, 2] therefore its log values can be both positive and negative - which can lead to errors when reading the log versine value in a table and then when working with that value.

I don't see this as a significant drawback.  We used log and natural versines without problems as a practical navigators, using tables designed for the job.  BTW, I love your drawing and although I am struggling to understand it, it seems like the kind of thing that we should be burying in time capsules or firing off into deep space. :-)

Lars, thanks for the interesting historical snapshot!

Hanno/Greg, I agree that Doniol works best with haversines.

Another plug for Burton's 4-figure Tables, a slim volume sadly out of print:

In the preface to the 1963 edition of the tables the author, SM Burton says this:

"....By the device of using versines instead of haversines in these tables and tabulating the naturals to five decimal places for the smaller angles (0-17) it will be found that solutions of the astronomical triangle will be correct within one minute of arc.  Whether this degree of accuracy is satisfactory for navigational purposes at sea will doubtless long remain a matter of dispute, but it is certain that a considerable body of both British and foreign competent opinion considers that it is so.  It may be advisable to mention here that the use of versines in place of haversines does not in any way affect the process of calculation.  Indeed, were the fact not mentioned, the change would probably not be noticed by the majority of users."

For calculating intercepts, Burton's 4-figure tables are fast and easy to use. There are no complicated rules to remember. Speed of use is somewhat enhanced and table entry errors minimised if whole degree LHAs and Latitudes from an assumed position are used instead of the EP or DR.

Finally, versine aficianados can always seek comfort in Reed's Nautical Almanac, still going strong!

Regards to all

John

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