# NavList:

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

**Re: Haversines with Logarithms**

**From:**Matthew Gianelloni

**Date:**2020 Oct 11, 18:10 -0700

Greg,

I totally concur on both points. These particular tables are 6 years old. I have a condensed version that is six digits instead of 9. Also, at the angles close to 0° and 180°, the closer you get, the less differentiation there is between haversine values per minute of arc using 5 or 6 digits. Thus the original 9 digits.

Example:

On the Natural Haversine table you posted, the haversine values from 0°1' to 0°15' are all .00000 whereas on the 9 decimal CoValues table they each have a different value from .000000021 to .000004760. The next angle, 0°16' is .00001 on the Natural Haversine table, which is rounded a good bit from .000005415 on the CoValues table.

But really, how often will we actually encounter a situation that uses those values close to 0 or 180? Statistically not much. 5 digits is great for practical navigation.

The Daniol method is totally beautiful, and I wonder why it's not taught as the standard sight reduction method instead of the H.O. 229 tables. It does, however, produce a result (Hc and Z) for use in the intercept (St. Hilaire) method to get a fix.

The intercept method requires precalculation. My goal in this was to dispense with precalculations for specific target stars and planets, and just walk outside and shoot the stars I knew, walk back inside and kick out a position in 10 minutes. Laziness has driven most of what I do on a daily basis.

That being said, I am lazy about math too. I'd rather subtract two 9 digit decimals than divide them. On page 3 of the Daniol Method document you posted, it says "The draw back to this azimuth formula is a division step." The logarithm CoValues for each haversine and angle allow a subtraction to be done instead of that division.

Next, the math will show how sometimes you will end up with angles in the spherical triangle calculations that need a haversine for angles greater than 180. This occurs when additions or subtractions of logarithms yield doubled angles. I had to list those coresponding angles with each other for the algorithms to work.

Example:

The angles 43°29' and 316°31' both have a haversine value of .137212713

I haven't worked on these algorithms for Hc and Z for 6 years, so they're relatively untested compared to the one for Meridian Angle, but I've attached them here for use in the Daniol method.

Criticism welcome! Being wrong is how I learn!

Matt