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# 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 is assumed.

## 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 the British 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.

## Parallel

There is a remarkable parallellism in the history of longitude determination. 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

Maskelyne's plan materialised in the publication of the Nautical Almanac, containing (among other data) pre-computed lunar distances for three-hourly intervals. 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 Mason. Additional tables for data that did not change from year to year were published 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 navigation.

## Decay...

Precomputed lunar distances were published in the Nautical Almanac from the first edition (1767) until 1905.

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.

## ...and revival

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 Almanac.

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 (-) distances. 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 the distance. Although predicting errors nowadays are negligible, the effect of systematic observing errors might still be significantly reduced by this averaging technique.

## Proportional logarithms

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.
• 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.
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 0.1'.

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. Luckily.

## Literature

William Andrewes (ed), The quest for longitude: The Proceedings of the Longitude Symposium, Harvard University Press, Cambridge, Mass 1996.

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

Derek Howse, 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.

Nevil Maskeline, 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, London 1763.

Tobias Mayer, Novae Tabulae motuum Solis et Lunae, in: Commentarii Societatis Regiae Scientiarum Gottingensis, Vol II, Göttingen 1753.

Tobias Mayer, 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 http://www.moshier.net/

The archive of the Navigation-L list (where the modern lunartics meet) can be found at http://www.irbs.com/lists/navigation/

Steven Wepster