How far the moon?

Anyone coming to the history of the search for a method to accurately determine longitude through Dava Sobel’s Longitude might be forgiven for thinking that the lunar distance method was just some sort of excuse dreamed up by Neville Maskelyne to prevent John Harrison receiving his just deserts. This is far from being the case. The lunar distance method first explicated by Johannes Werner in Nürnberg at the beginning of the 16th century was the method supported by nearly all astronomers since at least Newton as they were of the opinion that it would not be possible to construct a clock sturdy enough to survive a rough sea voyage and extreme changes of temperature and accurate enough to keep its time over several weeks in the foreseeable future. It should be remembered that it was the astronomers, responsible for keeping track of time since the dawn of civilisation, who had invented and developed the mechanical clock and it was the astronomical instrument makers who were the leading clock makers of the period so they really did know what they were talking about. For these highly knowledgeable men the lunar distance method genuinely seemed to offer more hope of a solution to the problem.

The lunar distance method is one of several ‘astronomical clock’ methods of determining longitude. The theory says that if one has a set of tables detailing the position of the moon respective to a given star or group of stars for accurately determined time intervals for a given fixed position then by observing the moon’s distance from said star or stars locally and noting the local time it should be possible to calculate the time difference of the two observations, real and tabular, and thus determine the longitude of the current position relative to the fixed position in the tables. Four minute of time difference equal one degree of longitude difference.

Two lunar observations taken at the same true local times

with the observers separated by 90 degrees of longitude

To make this system viable one needs two things, an instrument capable of accurately determining the lunar distance on a moving ship and accurate lunar distance tables. The first problem was solved by the English mathematician and instrument maker John Hadley who I’ve written about before and the second by the German cartographer and astronomer Tobias Mayer who died two hundred and fifty years ago today on 20th February 1762, aged just 39.

Tobias Mayer

Mayer was born on in 17th February 1723 in Marbach but his family moved to Esslingen less than two years later.

Mayer’s place of birth in Marbach

Now the Mayer Museum

He grew up in comparative poverty and when his father died in 1731 Tobias was placed in an orphanage. He received only very basic schooling and was a mathematical autodidact. However at the age of eighteen he had already published a book on geometry and a town plan of Esslingen. In 1743 he moved to Augsburg where he worked for the Pfeffel publishing house and where he published a Mathematical Atlas and a book on fortification. His publications led to him being appointed to a senior position at the Homanns Erben cartographical publishing house in Nürnberg, one of the leading cartography companies in Europe, in 1745.

Mayer’s workplace in Nürnberg home of the Homanns Erben publishing house

Today the museum of the history of the city of Nürnberg Fembo Haus

In his six years in Nürnberg Mayer published about thirty maps and numerous astronomical papers with a special emphasis on lunar research. It was during his time in Nürnberg that Mayer laid the foundations of his lunar tables for the lunar distance method.

By 1751 Mayer enjoyed a reputation as one of the leading European astronomers and he was offered the chair of mathematics at Göttingen and the directorship of the university observatory. During the next ten years Mayer would publish extensively on astronomy, mathematics, geodetics, mensuration and the design and construction of scientific instruments. He died on 20th February 1762 of typhoid.

Although the moon obeys Kepler’s laws of planetary motion, because it is fairly large and lays between the earth and the sun it gets pulled all over the place by the force of gravity and as a result its orbit is a very ragged and irregular affair. In both the systems of Ptolemaeus and of Copernicus the models for the moon’s orbit are less than successful. Kepler ignored the problem and did not supply a lunar model in his system. This omission was corrected by the young English astronomer Jeremiah Horrocks who proved that the moon also has, at least in theory, an elliptical Keplerian orbit and delivered the best lunar model up till that time. Even the great Newton had immense difficulties with the moon and although he based his efforts on Horrocks’ work he was unable to show that the moon really conforms to his gravitation theory. It would have to wait for Simon Laplace to tame the moons orbit at the end of the eighteenth century. Before Laplace none of the mathematical models of the moons orbit was accurate enough to deliver tables that could be used for the lunar distance method.

Mayer took a novel approach, he argued that what was needed was not a new model but more accurate observations and more accurate calculations based on those observations and set to work to deliver and deliver he did. Over several years Mayer made very exact observations of the moons positions and very accurate calculations for his tables and thus he succeeded where others had failed. In 1752 he published his first set of lunar tables and in 1755 he submitted them to the Board of Longitude in London. With his tables it was possible to determine the position of the moon within five seconds of arc making it possible to determine longitude to within half a degree.

I am the more unwilling my tables should lie any longer concealed; especially as the most celebrated astronomers of almost every age have ardently wished for a perfect theory of the Moon … on account of its singular use in navigation. I have constructed theses tables … with respect to the inequalities of motions, from that famous theory of the great Newton, which that eminent mathematician Eulerus first elegantly reduced to general analytic equations.

Mayer’s preface to the 1760 edition of his tables.

Combined with Hadley’s quadrant now modified to a sextant the problem of longitude was effectively solved, although only for days when the moon was visible. After trials and a new improved set of tables, published posthumously, the Board awarded Mayer’s widow a prize of £3000 a very large sum of money in the eighteenth century although only a fraction of the sum awarded to Harrison. The calculations necessary to determine longitude having measured the lunar distance proved to be too complex and too time consuming for seamen and so Neville Maskelyne produced the Nautical Almanac containing the results pre-calculated in the form of tables and published for the first time in 1766.

Neville Maskelyne

The Tables of the Moon had been brought by the late Professor Mayer of Göttingen to a sufficient exactness to determine the Longitude at Sea to within a Degree, as appeared by the Trials of several Persons who made use of them. The Difficulty and Length of the necessary Calculations seemed the only Obstacles to hinder them from becoming of general Use.

Maskelyne’s preface to the first edition of the Nautical Almanac

Contrary to the impression created by Sobel the two methods, lunar distance and marine chronometer, were not rivals but were employed together as a double safety system. Thanks to Sobel’s highly biased book John Harrison’s efforts have become well known and Harrison has become a household name. Mayer however whose services to navigation are just as important remains largely unknown, an anonymity that he does not deserve.

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16 Comments

Filed under History of Astronomy, History of Navigation, History of science, Local Heroes

16 responses to “How far the moon?

  1. Pingback: How far the moon? | Whewell's Ghost

  2. Rebekah Higgitt

    Many thanks for this admirable outline – a perfect anniversary celebration for Tobias!

    I had a couple of thoughts to add. One, right up front, is that not only were most people sceptical of the idea that a clock could be made that ran adequately on a moving ship, through changes of temperature etc, but they were also keenly aware that, as Newton said, a timekeeper could never find longitude, it could only keep it. Thus if the clock stopped, or ran irregularly even for a fairly brief period, it was essential that an astronomical solution should be available by which to check it. The early sea watches certainly were unreliable and it was only in the 19th century, when chronometers were affordable enough for each ship to carry at least three, that the astronomical back-up plan or, indeed, old-fashioned dead-reckoning, declined in importance. (By then, as well, there were time signals and small observatories in most ports, where chronometers could be checked and rated against astronomical observation.)

    Secondly, I wanted to highlight the use of Mayer’s tables beyond the observatory. On the voyages of Cook and Phipps at least, Mayer’s tables were taken out as well as Nautical Almanacs. On the first and third Cook voyages, they ran out of Nautical Almanac predictions and would have had to fall back on lunar distance done the very long way (as practised by Maskelyne himself on his St Helena and Barbados voyages) if they required a check on longitude or, later, on their timekeeper.

    • Thank you for those useful addition to my post.

      • Rebekah Higgitt

        I’ve just located a quote that makes my point rather more succinctly: near the end of his second voyage (1772-75) he wrote to the Secretary of the Admiralty to say: “Mr Kendals Watch has exceeded the expectations of its most zealous Advocate and by being now and then corrected by lunar observations has been our faithfull guide through all the vicissitudes of climates” (my emphasis). This is quoted in full in Sobel, but it’s amazing how little she does to help readers understand the complementary nature of the two methods!

      • Rebekah Higgitt

        When I say “he” I mean Cook, of course! (Clearly a God for us at the NMM, who needs no name ;-) )

  3. Pingback: zum 250. Todestag von Tobias Mayer « mathematik, bücher & meer

  4. Pingback: On this day: the role of anniversaries | teleskopos

  5. DCA

    Just to pick a nit, I’m not sure that the statement aboujt astronomers inventing the clock is right; though if you mean “precision clock” tthere would be no argument, given Huygen’s invention of the pendulum and balance spring. (I leave Burgi aside as having no influence). Also, I may be misremembering, but I thought one of the earliest examples of nautical application of lunars was on Wallis’ expedition, allowing him to produce a longitude for Tahiti — which gave Cook a definite target to be directed to.

    • Given that almost all the earliest mediaeval mechanical clocks, whether European, Chinese or Islamic, where astronomical clocks I think that the claim is more than justified.

    • Wallis did, more or less, use lunar distances to establish longitude at Tahiti: “Taking the Distance of the Sun from the Moon and Working it according to Dr. Masculines Method [as published in the British Mariner’s Guide] which we did not understand”. They didn’t understand it, but got a reasonable measurement and showed that Tahiti was ideal for observing the transit of Venus in 1769.

      However, this was in 1766, and Maskelyne had already used lunars successfully on his 1761 voyage to St Helena (also for a transit of Venus) and in 1763 to Barbados.

  6. The differences in position are pretty small. I’ve been thinking that showing what these measurements look like would be an interesting demonstration. I think it might be possible to do it in something like the Stellarium software.

    • “The differences in position are pretty small..” but if you look at the example above visible to the naked eye. However this is exactly the reason that the tables and the measuring instrument both had to be very accurate.

    • Ian – think that would be really interesting. We will try to do something similar come our big longitude exhibition in 2014, perhaps as an interactive, and also in the planetarium.

      • As the idea was first published on my blog I’m claiming 50% of the copyright and acting as Ian’s agent for the licence fee! Ian’s 50% is to be paid in nappies for @babycasserole ;)

      • I think it’s a patent rather than copyright you’re looking for ;-)

        I reckon Jovian moons as clocks, lunar distances and zenith measurements for latitude would be good. With added ship’s deck simulator / freezer for veracity.

  7. Pingback: The “first” Atlas | The Renaissance Mathematicus

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