lunar distance tables
Precomputed lunar distance tables
This page is not intended as a comprehensive history of lunar
distances, but rather as an account of some of its
details. Quite some knowledge of the history of longitude determination
Principle of lunar distances
The lunar distance method is a way of finding one's
longitude without an accurate clock. It is based on the movement
of the moon relative to the sun and stars. The moon completes its
orbit around the earth in a little less than a month: 27.3 days
relative to the stars, 29.5 days relative to the sun. Therefore the
position of the moon relative to the stars or the sun is a measure
of time, just like the position of the hands of a clock relative
to the dial.
That is the principle of the lunar distance method. It was
known in the 16th century (e.g, Johann Werner described it in
1514), and possibly even earlier. But it took until the middle of
the 18th century before the motion of the moon could be predicted
accurately enough for the method to be of any practical value.
Tobias Mayer and the longitude prize
In 1753, the German astronomer Tobias Mayer published lunar tables
of outstandig accuracy. For the first time, predictions of the
position of the moon were so accurate that longitude
determination by lunar distances was possible within the limits
set by the Longitude Act.
Mayer's tables were used by Nevil Maskelyne on his
St. Helena journey (1761, to observe the transit of Venus from that
island). Maskelyne later adopted these tables for publication in the
British Mariner's Guide.
Mayer hoped to
be rewarded according to the rules stipulated by the
Longitude Act. Therefore he had an improved version of
the 1753 tables sent to England in 1755, together with a specially
designed measuring circle and a description of how to find
longitude at sea. This version of his tables has never been
published. James Bradley, the Astronomer Royal at that time,
confirmed the accuracy of these tables.
When Mayer died in 1762, no decision had been
reached in England concerning the
longitude problem. New, further improved lunar
tables (on which he had been
working over the last 7 years) were sent to England after
his death, according to his will. Maskelyne tested these newer
tables on his famous journey to
Barbados in 1763, where he went to check the going of Harrison's H4 watch,
which arrived there on another ship.
Finally, on Feb 9, 1765, the Board of Longitude advised
Parliament that Mayer (posthumous) and Harrison should both be
rewarded for their contributions to the solution of the
longitude problem, but not to the extent that
both men had hoped. The Board signaled serious deficiencies in
each of the two methods. Harrison's method was not
general, because only one watch is clearly insufficient to
determine the longitude of a whole fleet, and it was not yet
possible to produce accurate watches (chronometers, as they
were later called) in sufficient quantity.
And Mayer's method was
not practical because it entailed too much work, in the form of
many hours of calculations. Besides, his measuring device was
brushed aside in
favour of the sextant, which is a development of Hadley's octant,
conceived by Capt. Campbell, meeting
the requirements of high accuracy and large angles set by the
lunar distance method.
There is a remarkable parallellism in the history of longitude
The principles of both (lunar and watch) methods were
published for the
first time early in the sixteenth century: longitude by watch in 1530 by
Gemma Frisius, longitude by lunars in 1514 by Johann Werner. Both
methods became practicable even more simultanuous:
Mayer's lunar tables were handed over to George Anson, First Lord
of the Admiralty, in 1755, whereas
John Harrison's H4 watch was shown to the Board of Longitude in
1760. The two-and-half centuries between idea and realisation were
necessary to learn, in one case, how to preserve the
even motion of a pendulum or spring,
and in the other case, how to predict the
uneven motion of the moon. I find it fascinating that
both quests begin and end so close together in time.
The Nautical Almanac
Nevil Maskelyne is often portrayed as one who
actively opposed Harrison and his clocks (Dava Sobel's book is a
bestselling example). Whether that was his attitude or not,
I think that Maskelyne has done more than anybody else to give
the mariner his longitude. By the time the
Board of Longitude reached its
decision in 1765, Maskelyne was an ex officio member of
the board, because he had just taken up the important post of
Astronomer Royal. He was one of a handful of people who had
actually employed the lunar distance method successfully at sea
(among the others were Carsten Niebuhr, Capt. Campbell, and Nicolas
Louis de Lacaille, but Campbell left the calculations to
Bradley onshore), and Maskelyne had been entrusted with the
task of verifying the quality of H4 at Barbados. He was in an
excellent position to realise that the
deficiencies signalled by the Board (mentioned above) were much easier to
overcome in the case of lunars, than in the case of watches.
His plan was to take as much as possible of the dreadful
calculations ashore, away from the mariner.
Maskelyne's plan materialised
in the publication of the Nautical Almanac, containing
(among other data) pre-computed lunar distances for three-hourly
In the first ten almanacs, for the years 1767-1776, all lunar
positions were computed
from Mayer's 1762 tables. Later almanacs (until the first years
of the 19th century) used Mayer's tables as improved by Charles
Additional tables for data that did not change from year to year
separately in the Tables Requisite.
Mayer's lunar tables proper, adjusted to the meridian of the
Greenwich observatory, were published by Maskelyne in 1770 under the title
Tabulae Motuum Solis et Lunae novae et correctae
All computations for the Nautical Almanac were performed by
various human computers in different locations in England. The
computers were directed and controlled from Maskelyne's office in
the Greenwich Observatory. That was really an appropriate place,
since in 1675 the observatory had been established with the
explicit assignment to its astronomer, John Flamsteed,
to apply himself with the most exact care and
diligence to the rectifying the tables of the motions of the
heavens, and the places of the fixed stars, so as to find out the
so much desired longitude of places for the perfecting the art of
Precomputed lunar distances were published in the
Nautical Almanac from the first edition (1767) until
Since about the 1840s, chronometers were produced in
sufficient quantities and at sufficiently low prices that they
could be afforded on board
many ships, and lunars gradually faded out. Lunars lingered
around partly because they were part of officers' curriculae,
partly because they provided a means for checking
chronometers, and partly because some ships simply didn't carry
chronometers. It should be noted that with one chronometer, you
get no warning when its rate is off, and with two, you have no
way to figure out which one is wrong, so you need three
chronometers at least if you intend to rely on them.
By 1905, radio time signals were available as an independent reference
and lunars were definitely obsolete. According to Lecky, they had
been "as dead as Julius Ceasar" for quite some time already.
But Lecky could never have foreseen the current state of affairs.
A century ago, radio revolutionised communication, spelling the
end of the precomputed lunar distance tables in the Nautical
Almanac. Now, the internet revolutionises communication again, and
it brings together a set of individuals highly interested in the
history of navigation in general, and of lunars in particular.
The renewed interest in lunars brings about
an equally renewed purpose for precomputed lunar distance tables.
Nowadays, the length of the computations involved in the reduction
of a lunar distance observation need not deter anybody. The whole
procedure can be easily programmed in an electronic computer
(somewhat similar to Maskelyne programming his human computers..?).
Still, there are two possible uses for precomputed distance
tables. First, they are very nice to have for planning purposes,
and for presetting your sextant so as to get moon and star in view
without much difficulty. (I admit that the latter purpose could
be catered for by a well-designed computer programme.) Second,
precomputed lunar distances serve a purpose for anyone interested
in historical reduction
methods, because almost all those methods presuppose the
availability of precomputed distances.
New precomputed distances
On these pages I provide precomputed lunar distances in
approximately the same format as they were in the 19th century
Nautical Almanac. The format in the oldest almanacs was
different and less familiar to a modern user, and there were
other calendar conventions in force, too.
The tables are in PDF format only. I discontinued the text
format of the tables because they take much time to prepare.
The pdf files are typeset by LaTeX.
Format of the tables
Each month is contained in a separate file. For each day of a
month, you will
find the times (in UT) on the left hand side, and geocentric distances to
five selected bodies (star, planet, or sun) next to it. Distances
are tabulated in
degrees, minutes, and tenths of a minute, which I consider
sufficiently accurate even for lunars. It also resembles more
closely the data format employed in the modern Nautical
The five daily bodies in the tables were automatically selected, on a
day by day
basis, by the
generating program, details are mentioned below. Next to the name
of the body is a sign indicating increasing (+) or decreasing (-)
The distance increases when the moon is to the east of
the body, and it decreases when the moon is to
the west. In the old days it was advertised that more accurate
results could be obtained by averaging longitude computed from an
easterly and a westerly
distance, since that would nullify any error in the predicted position
of the moon as well as any systematic error in the observation of
Although predicting errors nowadays are negligible, the effect
of systematic observing
errors might still be significantly reduced by this averaging technique.
The columns marked "P.L." contain the Proportional Logarithms of
the tabulated distances. The proportional logarithm was a mathematical
device conceived by Nevil Maskelyne to aid in the necessary
interpolation, but it was also instrumental in selecting the
bodies most suitable for observation. By definition,
prop.log x = log 10800 - log x.
Note that there are 10800 seconds in 3 hours (the time
interval between the tabulated distances).
Now suppose you have a lunar distance measurement of d degrees,
and suppose this d falls between the adjacently tabulated distances
D1 and D2,
valid for times t1 and t2 respectively,
which are 3 hours or 10800 seconds apart.
Assuming that the moon travels with uniform speed over the 3-hour
interval from t1 to t2, the time t
of the actual observation
follows from simple linear interpolation:
10800/(t-t1) = (D2-D1)/(d-D1).
Taking logarithms, we obtain
prop.log(t-t1) = prop.log(d-D1) - prop.log(D2-D1).
The second term on the right hand side is (and was) provided by the
lunar distance tables. The first term on the right hand side is easily
looked up in a prop.log table, which was
originally supplied in the
Tables Requisite. Entering the prop.log table again with
the difference of those two terms, you pick out the value
t-t1, i.e., the time elapsed between the first tabulation and your
actual lunar distance measurement.
Apart from its utility in interpolation, prop logs come in handy
in another way. From the defining formula it follows that a low
prop.log is associated with a fast changing
distance between the moon and the other body. So by studying the
tabulated prop.logs, you are able to pick the body with the
fastest changing distance, thus maximising the accuracy of the
derived time and longitude. Of course the availability of the
observation of your choice is still subject to your position on
the globe (i.e., either the moon or the preferred body might be
too close to, or even below, your horizon).
Computation of the tables
I am on Un*x and I love to write data filters. Steve
Moshiers aa program appears to be trustworthy; and it can even be
driven by a script because it takes all its input from the command
line. Therefore I opted to use aa rather than write and debug my
own computations for the positions of celestial bodies. In case you
want to know more about the filter chain, look at the
(sketchy) notes that I drew up while writing
the routines and filters. In short, I draw heavily on Steve's aa
and on computing power. When I wrote these routines I was not yet
aware that Maskelyne c.s. had a 'short list' of selected stars.
In my algorithm, selection of the bodies to tabulate is governed
by the following rules. Decisions are made for Greenwich noon.
Bright planets (Venus to Saturn) are possible candidates, as well
as bright stars not too far from the zodiac, and, of course, the sun.
Since the 2006 tables I use aa version 5.6.
There are slight differences between tables generated by versions
2.4g (formerly used) and 2.6: as far as I checked not larger than
- If the sun is in reasonable distance, select it. If its distance
is too low, this date is too near to new moon and tabulation for
this day is senseless.
- Distances over 120 deg are considered impractical.
- Bodies too close to the sun are rejected because they are not
visible at all (diff RA < 3h, but 1h for Venus).
- Bodies on the 'other side' of the sun are rejected because
they are not visible simultanuous with the moon.
- Remaining bodies are weighted by an empirical formula encorporating
distance, rate of change of distance, and magnitude.
- Select the best weighted bodies, making sure that at least one
easterly and one westerly body is included.
At the bottom of every page in my tables is a copyright
notice. You don't owe me
anything since I produce and publish these tables on equipment
owned by my
employer. I just hate the idea of somebody publishing them
commercially. As long as you do not publish these tables commercially,
you can ignore the copyright. I guess it's a form of copyleft. Actually I
do not believe there is any commercial value in lunar distances.
William Andrewes (ed), The quest for longitude: The Proceedings of
the Longitude Symposium, Harvard University Press, Cambridge, Mass
Mary Croarken, Providing Longitude for all: The eighteenth century
computers of the Nautical Almanac, in: Journal for Maritime
Research, sep 2002.
Eric Gray Forbes, Tobias Mayer (1723-62) : pioneer of enlightened
science in Germany, Vandenhoeck & Ruprecht, G"ottingen 1980
Greenwich time and the discovery of the longitude,
Oxford University Press, Oxford 1980.
Squire T S Lecky,
Wrinkles in practical navigation,
George Philip and Son, London 1881.
The British Mariner's Guide containing ... Instructions for the
Discovery of the Longitude ... by observations of the distance of
the moon from the sun and stars, taken with Hadley's Quadrant,
Tobias Mayer, Novae Tabulae motuum Solis et Lunae, in:
Commentarii Societatis Regiae Scientiarum Gottingensis, Vol II,
Tabulae Motuum Solis et Lunae / New and concice Tables of the Motions
of the Sun and Moon, by Tobias Mayer, to which is added The Method of
Finding Longitude Improved by the Same Author,
Commisioners of Longitude, London 1770.
Dava Sobel, Longitude.
Steve Moshier's programs can be found at
The archive of the Navigation-L list (where the modern
lunartics meet) can be found at
Last modified: Sun Dec 23 14:15:26 CET 2012