# NavList:

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

**Rhumb line by slide rule**

**From:**Paul Hirose

**Date:**2016 Nov 02, 14:41 -0700

Formulas for rhumb line course from point 1 to point 2: a = 45 + lat1 / 2 b = 45 + lat2 / 2 y = π * longitude difference x = 180 * (ln tan b - ln tan a) course = arc tan (y / x) The longitude difference is 0 to 180 degrees. A slide rule solution may be easier with some modifications to the x and y formulas. • Replace natural (base e) logarithms with common (base 10) logarithms. We need only multiply the latter by 2.303 (the natural log of 10). • Divide both equations by π to combine the constants in one place. That has no effect on the computed course. • Replace logarithm subtraction by division, since log a - log b = log (a / b). After those modifications, the equation for x contains the expression 180 / π * ln 10. To practical accuracy (better than 1 part in 1000), that equals 132. Then the equations for rhumb line course are a = 45 + lat1 / 2 b = 45 + lat2 / 2 y = longitude difference x = 132 * log (tan b / tan a) course = arc tan (y / x) In the equation for x, the log is negative if the expression in parentheses is less than 1. That's no problem. Read negative logs on the L scale as if it's numbered in reverse, i.e., with 0 at the right instead of the left. Negative natural logs are not so easy. In the original equation these occur when a latitude is negative (south). In that case, 45 + lat / 2 is less than 45, hence its tangent is less than 1 and its logarithm is negative. There's a way to read it, however. If Ln is on the slide, set a C index to the tangent on D, set the cursor to the D index, read the negative natural log at the cursor on Ln. For example, lat = -20. Then ln tan (45 + -20/2) = ln tan 35. Set a C index to tan 35 on D in the obvious way. Set the cursor to the D index and read -.357 on Ln. There's a similar trick if the Ln scale is on the body of the slide rule. If both points are south latitude, a simple dodge is to pretend the latitudes are north. The resulting course is measured from the south. But the following example is in north latitude, and I'll use the common log version of the formulas. Initial latitude = +30, final latitude = +50, longitude difference = 20°. Destination is to the east. a = 45 + 30 / 2 = 60 b = 45 + 50 / 2 = 70 y = 20 x = 132 * log (tan 70 / tan 60) = 26.4 course = arc tan (20 / 26.4) = 37.2 Compare that to 37.21717 per http://geographiclib.sourceforge.net/cgi-bin/RhumbSolve