# NavList:

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

**Lunar tables - use of prop logs**

**From:**Jeff Gottfred

**Date:**1995 Jan 1, 00:00

Once one has cleared the distance, one then needs to determine what GMT that distance corresponds to. The Nautical Almanac from those days listed the lunar distances for the moon and various objects at three hour intervals. The cleared distance would lie within one of those intervals. The basic idea is to use interpolation to find the time corresponding to the cleared distance. If D is the difference between the tabulated distances which bracket your cleared distance, and d is the difference between the earlier tabulated distance and your cleared distance, and t is the elapsed time between the earlier tabulated distance and your cleared distance, then: D 3 hours - = ------- d t or, 3 * d t(hours) = ----- D or, d t (seconds) = 10800 * - D O.K., this is pretty straightforwards, but Nevil Maskelyne recognized that for the sailors of the time, this calculation was rather tedious. It would require four entries into log tables to solve. Maskelyne came up with a way to simplify this calculation. He define the "proportional logrithm" of time t as being the common log of 10800 minus the common log of t (seconds). Maskelyne then created a table of proportional logs (to four digits) and included them in his famous "Tables Requisite". Here is how you use them: Because: D * t = 3 * d we can write: log D + log t = log 3 + log d or, log t = log 3 + log d - log D or, (subtracting each term from log of 3 hours): (log 3 - log t) = (log 3 - log 3) + (log 3 - log d) - (log 3 - log D) or, (log 3 - log t) = (log 3 - log d) - (log 3 - log D) This is therefore: plog t = plog d - plog D where plog is the proportional log as defined above. This trick means that the sailor only has to make two entries into the prop log table (because prop log D was given in the almanac). Now, only one problem remains, over three hours the non-uniform motion of the moon means that simple interpolation will not yield enough accuracy. To correct this the "correction for second differences" was applied. If the values a,b,c, & d are the results of some varying function, then the first differences are the difference between the values. I.E. a - b, b - c, c - d, etc. The second differences are the differences in the differences, I.E. (a-b) - (b-c), (b-c) - (c-d), etc. Note that: (a-b) - (b-c) = a - b - b + c = a - 2b + c If the interpolation is carried to second differences, then you get a result accurate enough for navigation. The Nautical Almanac used to contain a list of differences in the prop logs for each interval of t. I.E., the prop log for each D is given, and the mean difference in the prop logs for each pair of D's. (that is the second difference value) which is basically the velocity of the moon relative to the object at that time. One of the neat features of this method of tabulating the distances is not only does it shorten the calculations required by the navigator, but the differences in prop logs (the second difference numbers) give you an indication of how fast the moon is moving relative to the tabulated objects. Obviously you want to pick the fastest rate of change to get the best lunar, so the table tells you which is the best object to select! Those guys were no dummies eh?! ^ | | (Certified genuine Canadianism) Cheers! Jeff. ------------------------------------------------------------------------ This mail list is managed by the majordomo program. To unsubscribe from this list, send the following message to majordomo@ronin.com: unsubscribe navigation For help, send the following message to majordom@ronin.com: help Do NOT send administrative requests to navigation@ronin.com. Thanks. -ben ------------------------------------------------------------------------ ------------------------------------------------------------------------ This mail list is managed by the majordomo program. To unsubscribe from this list, send the following message to majordomo@ronin.com: unsubscribe navigation For help, send the following message to majordom@ronin.com: help Do NOT send administrative requests to navigation@ronin.com. Thanks. -ben ------------------------------------------------------------------------