# NavList:

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

**Sight Reduction by the Cosine Haversine Method**

**From:**Chuck Taylor

**Date:**2004 Oct 2, 22:38 -0700

For many years the preferred method of reducing sights by the method of St. Hilaire was the "Cosine Haversine Method". It required tables of the logarithms of trigonometric functions, plus tables of natural and logarthmic haversines. For anyone who may not be familiar with haversines, the haversine of an angle t is defined as haversine(t) = (1/2)*(1 - cosine(t)) or, equivalently, haversine(t) = (sine(t/2))^2 Haversines are merely a vehicle for simplifying the computations. While sines and cosines range from -1 to + 1, haversines range only from 0 to + 1, and the haversine of a negative angle is the same as the haversine of the absolute value of that angle. In the example below, I show both the intercept and the azimuth computed by this method. Often the azimuth was obtained from azimuth tables such as H.O. 71 for the Sun and other bodies with declination 0 to 23 degrees, or H.O. 120 for bodies with declination 24 to 70 degrees. The quadrant in which the computed azimuth angle lies is not always obvious. In this case the declination is South and the meridian angle (t) is West, so the azimuth angle is S 55.8 W, or 236 degrees. At the time of the sight I noted that the bearing of the Sun was WSW, so this checks. Quoting from Hosmer's "Navigation" (1926), "In the case the sun is about east or west and the Lat. and Decl. are of the same name it may be difficult to tell whether the bearing is from the N or the S. To remove this doubt, add the log cosec. Lat to log sin Decl, obtaining the log sin Alt. when the sun is on the prime vertical, E or W. If the observed alt. is less than this the sun is on the side toward the [elevated] pole (N in N. hemisphere)." Here is a worked example, which is from the same sight as the Time Sight example I posted recently. This time I worked it both by computer and by tables. Again, I hope that someone will find this useful. Best regards to all, Chuck Taylor N of Seattle ============================================================== At anchor, position by GPS: Lat 48d 30.1' N Lon 122d 49.5' W 25 September 2004 Corrected UT (GMT): 23-17-12 Corrected Ho: 24d 54.3' Almanac data: GHA Sun: 171d 27.4' t(W) = 171d 27.4' - 122d 49.5' = 48d 37.9' Dec Sun: 1d 1.67' S GMT: 23-17-12 Altitude Azimuth t 048 37.9 log hav 9.22930 log sin 9.87534 Lat 048 30.1 log cos 9.82125 Dec 001 16.7S log cos 9.99989 log cos 9.99989 ------- log hav 9.05044 ------- nat hav 0.11232 L~d 049 46.8 nat hav 0.17714 ------- z 065 05.8 nat hav 0.28945 log csc 0.04238 -------- ------- Hc 024 54.2 log sin 9.91761 Ho 024 54.3 Z = S 55.8 W -------- a 000 00.1 Zn = 236 Checking the Azimuth (Z): Lat log csc 0.12553 Dec log sin 8.34849 ------- Alt log sin 8.47402 Alt on P.V. 1d 47.2' Since Hc is greater than the Alt on the Prime Vertical, the Sun was south of the Prime Vertical, and the Azimuth was indeed S 55.8 W, or 236d. In this case it was very near the equinox, so I knew that the altitude at the Prime Vertical would be very low, but I showed the computations anyway for the sake of completeness. From now until spring, the altitude of the Sun at the Prime Vertical will be negative (below the horizon). ============================================================ _______________________________ Do you Yahoo!? Declare Yourself - Register online to vote today! http://vote.yahoo.com