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    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
    
    
    

       
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