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Frank Reed quoted the following interesting letter (which had little
connection with Jefferson, except as the recipient).

=============

>From William Dunbar to Thomas Jefferson in 1804 (Library of Congress web
>site):
>  "I mentioned in my last that one very simple method had occured to  me of
>ascertaining in certain Circumstances the Longitude of places, which is  ;
>the method  is
>such that a Single observer with a good altitude instrument, altho' deprived
>of the use of a time keeper, may still make useful observations for the
>advancement of geographical Knowledge. I shall now just mention the
>principles  &
>shall hereafter Send you some examples of the Calculation. The excellence  of
>the usual lunar method of determining the Longitude depends (supposing her
>theory to be perfect) upon her quick change of place from west to east; but it
>cannot be denied that it requires great dexterity to make good observations,
>which is evident from the disproportion of the times to the distances in the
>hands of the best Observers, and this arises from the slow progress of the moon
>which Causes the Contact to appear to be continued for many seconds of time;
>were this observation similar to a meridian altitude, it might certainly be
>taken to any desireable accuracy, that is, were the motion of the moon from
>North to South in place of from West to east, the moon's altitude when brought
>upon the meridian by the rotation of the earth would furnish an easy & very
>Correct mode of ascertaining the Longitude: Now altho' the proper motion of
>the  moon is from West to East, yet her orbit makes so considerable an angle
>with the  equinoctical circle, that there are two portions of each
>lunation when
>the  moon's change of declination is very rapid, exceeding 6 in 24 hours, that
>is 5"  of a degree in one minute of time; if therefore under favorable
>Circumstances we  take the moon's greatest altitude near the meridian, we shall
>thence be enabled  to ascertain the moon's declination at the moment of her
>passing our meridian;  we must then find the time at Greenwich when the
>moon had
>that declination and  also the time when the moon passed the meridian of
>Greenwich, from which data  the Longitude is easily found: this method
>will require
>the use of some  interpolations and an equation for the Correction of the
>Moon's
>altitude on the  Meridian, because her greatest altitude will not be on the
>meridian, but to the  East or West according as She is increasing or
>diminishing her North polar  distance. I have communicated this method to
>my Worthy
>friend Mr. Briggs who is  pleased with the idea & intends giving it
>consideration.

===================

This accords with a suggestion Frank made recently, (in a posting that I've
inadvertently deleted), that changes in Moon declination, rather than in
lunar distances, might in theory be used to determine Greenwich Time .

I haven't come across Dunbar's name before, but he has raised an
interesting idea which is at least worth examining to see if there's
anything in it.

But first, there seems to be an error on Dunbar's part, or a transcription
error since, that makes Dunbar's proposal seem more impractical than it
really was.

Dunbar is quoted as saying- "there are two portions of each lunation when
the  moon's change of declination is very rapid, exceeding 6 in 24 hours,
that is 5"  of a degree in one minute of time." No doubt that "6 in 24
hours" refers to 6 degrees in 24 hours, a rate of change of declination
which the Moon can achieve under certain extreme circumstances. But that
would give rise to a change of 15", not 5" as stated, in a minute of time.
The resulting motion of the Moon in dec. would then be just half the speed,
not one-sixth of the speed, of the change in lunar distance. Where has the
erroneous 5" figure come from?

The big limitation of the lunar distance method for determining time is
that it's so damned inaccurate, because the Moon moves so slowly past the
stars, at only about 30 arc-minutes in an hour. Dunbar's proposal would
involve measuring declination changes which are at best 15 arc-minutes in
an hour, so if declinations could be measured as precisely as lunar
distances can, the resulting time would have twice the error of a lunar
distance. (If the 5" per hour figure had been correct, the error would have
been 6 x the error in a lunar distance, which would exclude it from any
further consideration.)

That doubling of the error of a lunar distance observation is a serious
handicap for Dunbar's method, which on its own would be enough to explain
why it was never adopted. But we will see that other factors come in which
degrade his proposed method further.

The Sun goes round the ecliptic once a year, and in that time its dec.
changes over a range of ? 23 and-a-bit degrees, the tilt of the Earth's
axis.

The Moon goes round in a month, in a plane near to that of the ecliptic: if
it was exactly on the ecliptic, the dec. of the Moon, like that of the Sun,
would also swing through ? 23 and-a-bit degrees, each month.

But the Moon's plane is tilted off that of the ecliptic, by about 5
degrees, and the direction of that tilt gradually shifts, over a period of
about 18 or 19 years, before it comes back to where it started. This is
known as the precession of the Moon's orbit, which I won't go into further
unless someone asks.

Over part of this 18-year cycle, the tilt of the Moon's plane adds to that
of the Earth to the ecliptic, so the Moon's dec. swings through ? (23 + 5)
deg, or ?28 deg over each month. Nine years later, the Moon's tilt
subtracts, so it's dec swings through ? (23 - 5) deg, or ?18 deg, each
month. So you can see that, from Dunbar's point of view, there's a
succession of "good" years for applying his proposed method, followed 9
years later by a succession of "bad" years, in which its sensitivity is
considerably reduced. It's no great surprise that the numbers Dunbar quoted
apply only to those good years. In bad years for his method, the resulting
time-error would increase to more than three times the error that a lunar
distance would provide.

And remember, these declination changes are far from constant over a lunar
month, and we have only been considering those times in the month when the
declination is changing at its fastest, twice a month near when the
declination passes through zero. Twice in the month the declination stops
changing, and anywhere near those times Dunbar's method would be quite
useless.

Next, Dunbar's proposal involves measuring the Moon's altitude. If measured
at sea, that involves observing the horizon, with all the horizon's
uncertainties: waves and swell, haze, dip and anomalous dip. Lunar
distances at sea can be very precise because the horizon doesn't enter the
measurement at all. Dunbar accepts that his proposal is "much
better calculated for travellers by land than Voyagers by Sea", and he is
right, because use of a liquid-horizon on land avoids all those problems of
a sea horizon.

Dunbar suggests- "if therefore under favorable Circumstances we  take the
moon's greatest altitude near the meridian, we shall thence be enabled  to
ascertain the moon's declination at the moment of her passing our
meridian." He doesn't go into the details of this calculation; with the
Moon's dec. changing so fast, there can be a large interval between the
moment of max altitude and that of meridian passage, which has to be
carefully allowed for, in a way as yet unexplained.

Well, it seems clear, with hindsight, why Dunbar's proposal failed to
"catch on". One of those ingenious methods which might be all right in
concept but which fails to do the job in practice.

George.


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contact George Huxtable by email at george@huxtable.u-net.com, by phone at
01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy
Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
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