# NavList:

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

**Re: Smart and Shearme Position Line Tables**

**From:**Lars Bergman

**Date:**2020 May 3, 09:22 -0700

The tables are defined by

cot(U)=tan(latitude)·sec(hour angle) and V=log_{10}[sin(latitude)·sec(U)]+10

Out of the 368 values on the page shown by David, 16% have an error of one unit in the last decimal, but never larger than that. Approximately equally distributed between U and V.

The tables are derived by letting v·cos(U)=sin(latitude) and v·sin(U)=cos(latitude)·cos(hour angle). This gives cot(U)=tan(latitude)·sec(hour angle). Then putting these expressions into the altitude formula

sin(altitude)=sin(latitude)·sin(declination)+cos(latitude)·cos(declination)·cos(hour angle) gives

sin(altitude)=v·cos(U)·sin(declination)+v·sin(U)·cos(declination)=v·sin(U+declination), or with logarithms log[sin(altitude)]=log(v)+log[sin(U+declination)]=V+log[sin(U+declination)]

Lars