# NavList:

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

**Easy Lunars in 1790**

**From:**Ken Muldrew

**Date:**2006 Apr 26, 15:33 -0600

One of the real gems among the gold mine of 18th century navigation documents that everyone has been downloading is Margetts' Longitude Tables by George Margetts, published in 1790. "Tables" was probably a poor choice for a title as the collection is really a series of graphs allowing one to clear a lunar distance by interpolating the necessary corrections. Before describing the work, let me start by describing my experience of using it for the first time last night. I took a couple of lunars that David Thompson had taken in 1801. Using his data (apparent altitudes of the bodies, uncleared distance, and horizontal parallax) I was able to clear both lunars in about 5 minutes for each one (that was the first time using these tables, I have no doubt that practice would cut that time down substantially). I used a pocket calculator to clear the distance exactly and the table versions were off by 6" and 8" respectively. I then proceeded to clear the first lunar using Witchell's method. I've done about 20 or so lunars using this method, so I'm not a complete beginner (I know my way around Raper's log tables, which is what I use (since I have a print copy)). It took me an hour to clear the lunar using this method (I was slowed down by an error part way through; but then I always make at least one mistake that slows me down using this method). I didn't bother clearing the second lunar using Witchell's method; it's just too slow and tedious, and for the purpose of this experiment, there was no point in doing so. The lunar cleared using Witchell's method was 4" off from my calculator's value, but there was really no value in an extra 2" accuracy in 1801. In short, the "problem" of clearing lunar distances was just as fast and easy to solve in 1790 (if you had Margetts' book) as it is in 2006 using a pocket calculator. That brings up a question or two but let me describe the tables first before wondering what kind of spell was placed on 18th century navigators that caused them to reject such a gift. The easiest way to understand what's in these tables is to download a copy from http://trials.galegroup.com/nlw2006/history.html Basically, there are 100 graphs on 70 plates, each graph showing the necessary corrections for integral lunar distances from 20? to 120?. The x- axis of the graphs gives lunar altitudes from 5? to 90? and the y-axis has corrections from about -50 to +50 minutes (different for each case) to correct for parallax and refraction. There are up to 85 lines on each graph, one for each degree of star (or sun) altitude that give the necessary corrections for that star altitude as a function of the moon's altitude. Each of these lines is for a horizontal parallax of 53'. There are also dotted lines on the graphs (maybe 10-20 per graph) that give the parallactic correction for a horizontal parallax of 62' over 53'. To use the tables, you look up the graph corresponding to the whole degree of uncleared distance below the measured distance. Find the intersection of the moon's apparent altitude with the star's apparent altitude (interpolating between whole degrees) and then get the first correction from the y-axix (each line on the graph corresponds to one minute, so seconds are interpolated). An index for the second correction is interpolated from the dotted lines and then found on another graph of parallactic interpolation (where the horizontal parallax is on the y-axis, the second correction is on the x-axis (12 seconds per vertical line) and some straight angled lines correspond to the index). The same thing is done for the whole degree above your uncleared lunar distance and then the correction is proportioned according to the minutes and seconds of your uncleared lunar (using a proportioning graph or, if they're close, by just guessing). The final correction is applied and the distance is cleared. To explain how these graphs are made, I'll take a particular example as that is probably the simplest way to illustrate what they look like if you don't have them at hand. The graph for 30 degrees of lunar distance has 75 lines (not straight, but each adjacent line is near-parallel (over a short distance) to its neighbors) representing star altitudes from 5 to 80 degrees (with a short line showing the limit of 90 degrees). Let's consider the line for a star altitude of 50?. Since the distance is 30?, the line goes from a lunar altitude of 20? (where the moon is directly below the star) to a lunar altitude of 80? (where the moon is directly above the star). For the part in between these limits, one can think of the star fixed at 50? and the moon on a rod representing an angular distance of 30? rotating in a half circle about the star. There will be a unique correction for every angle in that half circle that removes the components of parallax and refraction that act along the arc connecting the two bodies. Since the moon is assumed to have a horizontal parallax of 53', the vertical components of the corrections are functions of altitude for both the moon and the star. One only needs to calculate the corner cosines to find out the relative contributions of correction for each value of the moon's altitude (see Frank Reed's posts in the archives on "Easy Lunars" for a full explanation of this operation). Basically, one needs to solve the following two equations: dM=[sin(s_alt)-cos(d)sin(m_alt)]/cos(m_alt)sin(d) dS=[sin(m_alt)-cos(d)sin(s_alt)]/cos(s_alt)sin(d) where dM and dS are the corrections for the moon and star(sun) respectively, s_alt is the altitude of the star(sun), m_alt is the altitude of the moon, and d is the distance between them. Since the distance is fixed, and for this 50? line the star's altitude is fixed, the terms involving d and s_alt and be precomputed. Then the corrections can be calculated for every 5? of lunar altitude (so 13 points of calculation for this particular line) and a fair curve drawn through the points. One could do this for every 5? of star altitude (16 lines with between 8 to 18 points of calculation for each line) and then use a ruler to interpolate the curves for integral degrees of star altitude (looking at the graphs, I think this would be quite reasonable, though in points of higher curvature, more points might be calculated). If the correction was calculated for each degree of star and lunar altitude, the job would involve hundreds of thousands of points of calculation - but graphically, it is quite easy to fill in the missing points so the computational task would have been greatly simplified. The dotted lines for the second correction are nearly horizontal and would be easier to calculate. Perhaps 100 - 200 points of calculation per graph. With 100 graphs, the task would not be trivial, but the reward - allowing anyone to clear lunar distances almost effortlessly - clearly made it worthwhile. So the question is, why is the history of navigation utterly silent on this brilliant method to clear lunar distances? In 1790, clearing the lunar distance could have been made the most trivial part of finding one's longitude, yet navigators persisted in flipping through log tables and following arcane recipes, torturing themselves to clear the distance (well, not exactly torture, but certainly an unpleasant half hour even at the best of times). Googling Margetts shows much of his clock and watchmaking, but very little about his tables (aside from one reference where Matthew Flinders says in a letter that he has no great opinion of Margetts' tables but recommends Mendoza's). Could it be that this was another poor watchmaker who was badgered into submission by the arch-villain Nevile Maskelyne? Will Dava Sobel write another bestseller by just changing the name of the hounded watchmaker? But seriously, how could these tables have been ignored? I realize that to us, analytic geometry is as plain as simple arithmetic, so that interpolating from a graph is as natural as tying one's shoes. And that using tables of logarithms to perform calculations in these days of miracles, when machines have rendered arithmetic utterly trivial, is completely foreign to us. But even taking that into account, after having done both I cannot believe that anyone could prefer the log table recipes to graphical interpolation on any grounds whatsoever aside from prejudice. But such a prejudice would have to be so strong and irrational that it really would deserve the Dava Sobel treatment to understand it. Has anyone else ever read about Margetts' tables? Ken Muldrew