# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Position-Finding

**Re: Yet another 4-place Nat-Haversine Table project?**

**From:**Greg Rudzinski

**Date:**2018 Dec 4, 13:05 -0800

Roger,

I will suggest adding or subtracting 1° to the assumed latitude for the few cases where the sums of latitude and declination are less than 1° when performing a longhand haversine sight reduction. For cases where the meridian angle (t) is less than 1° I will suggest doing an ex-meridian rather than a 4 place hav-doniol reduction.

Greg Rudzinski

**From: **Roger W. Sinnott**Date: **2018 Dec 3, 16:38 -0500

Tony,

Your haversine table looks quite nice, with no wasted white space!

However, one problem, as it stands, is that 4 decimal places isn’t enough to preserve significant digits in very small angles. So, for example, when you look up the haversine of 0° 48ʹ, the answer you get from the table is zero (not very useful).

Some time back I made a set of 3-place tables, each the size of a business card and laminated, that I keep in my wallet. I haven’t made one for haversines (yet), but I encountered a similar problem with significant digits in my 3-place sine table (picture attached). My solution was to invent a new notation, where a negative exponent precedes the decimal value, so that the sine of 1° is

given as ^{-1}.175 and means the same as 0.175 x 10^{-1}, or 0.0175. (I hope the superscripts I’ve typed survive this transmission by email!)

Note that this exponent could be anywhere from -1 up to -9 and not take up any more space in the table, so even a number like 0.000000000397 could be accommodated in a 3-place table.

Roger