# Length problem

The expression longitude problem or longitude problem denotes the long unsolved problem of being able to determine the geographical longitude of a ship in the open sea, for example.

While the geographical latitude can be measured relatively easily using the starry sky with sufficient accuracy for seafaring , a comparably precise determination of the longitude is far more difficult. This is because the circles of latitude have a physical meaning due to the rotation of the earth, while the circles of longitude represent a purely arbitrary division. To determine the length of any location, the exact solar time at a location of known length is required as a reference . The difference in length results from the difference to the local time of the ship. The problem lay in determining this exact reference time as long as sufficiently accurate clocks were technically not feasible.

Without the possibility of precise determination of the geographical longitude, it was hardly possible to drive directly to a destination that was not yet visible. The risk of missing the target laterally was too great. As a reliable indirect navigation method, wide sailing was common, but this involved detours that could add weeks to the journey. That is why the King of Spain had already offered a price for a solution in 1600, at the time unsuccessful.

The length problem was only solved satisfactorily after 1750 with Harrison's very precise ship's clocks .

## A pan-European problem

The establishment of the Royal Observatory at Greenwich by Charles II in 1675 was England's first step in finding an accurate measure of length. Observatories were also established in Paris ( Paris observatory ) and St. Petersburg.

After Admiral Sir Cloudesley Shovell ran aground in 1707 on his return from a victorious battle in the Isles of Scilly , with four of the 21 ships sunk and around 1,450 men killed, and after a petition by William Whiston and Humphry Ditton (substantiated by Opinions by Isaac Newton and Edmond Halley ), in 1714 the Parliament of England awarded a high prize money for a practically usable solution to the problem of length: £ 20,000 for an accuracy of less than half a degree and £ 10,000 for an accuracy of up to one degree. One degree of length corresponds to 60 nautical miles (approx. 111 km) at the equator and decreases towards the poles. One degree of longitude corresponds to about 40 nautical miles (about 74 km) across the width of the English Channel. A clock must not display more than 4 minutes incorrectly, in order to ensure that the accuracy is still far too low for seafaring.

At that time, the prize money meant an impressive fortune, comparable to today's double-digit million amount. A medium-sized seagoing ship then cost around £ 1,500 to £ 2,500, a simple laborer lived on £ 10 a year. The "Length Commission", the Board of Longitude , to which the most important astronomers and mathematicians in England belonged, but also the President of the Royal Society , the Royal Society for the Advancement of Science, was founded to administer the prize money and to evaluate the submitted proposals .

## Possible solutions

The relative longitude to a position (Greenwich) can only be determined with a reference time (UTC / GMT).

All possible solutions are based on time measurements. The local solar time is determined on the ship, which is relatively easy based on the course of the sun. In addition, the current solar time of a location with a known geographic longitude is required as a reference time. The length of your own position can be calculated from the time difference, because it is related to 24 hours like the difference between the lengths to 360 °.

Example: It is noon on the ship while it is 3:30 p.m. in Greenwich (0 ° longitude). The difference in time is therefore 3.5 hours. Then you are on longitude, namely west longitude, since it is later in Greenwich than at your own position. ${\ displaystyle \ textstyle {\ frac {3 {,} 5h} {24h}} \ cdot 360 ^ {\ circ} = 52 {,} 5 ^ {\ circ}}$

The problem was knowing the exact reference time, in the example above, knowing what time it is in Greenwich. Useful methods for this consist either in observing astronomical events, the times of which could be precisely calculated in advance and listed in a table, or in carrying a length clock that shows the local time of the reference location during the entire journey - a procedure that Gemma R. Frisius already carried out in 1530 suggested. At the beginning of the 18th century, however, it was not technically possible to build a clock that would move with sufficient accuracy on a moving ship under changing climatic conditions, so astronomical observations were initially used.

### Jupiter moons

The moons of Jupiter, discovered by Galileo (1564–1642), orbit Jupiter so quickly that around 1000 times a year a precisely predictable appearance or disappearance of a moon occurs. These events can be observed practically at the same time for all observation locations on earth and are therefore suitable as global "time stamps". This solution, which Galileo propagated - probably also to advertise his telescopes - could not be used on ships, as the ship's movements did not allow sufficiently precise observation.

Nevertheless, scientists from different nations tried at least on the mainland by observing Jupiter's moons to determine the exact longitude. Among them were Giovanni Domenico Cassini , Erasmus Bartholin and Ole Rømer .

### moon

The earth's moon , on the other hand, is easy to observe, although its movement could only be predicted through complex calculations based on precise observations. However, lunar eclipses could be predicted much earlier .

#### Lunar eclipses

Many of these phenomena can be observed from almost half the surface of the earth, and by comparing the times of entry and exit of the earth's shadow on the moon, differences in the geographical lengths of the individual observers can be determined. This method has been shown to have been used by Pliny (approx. 23–79) and Ptolemy (approx. 100– approx. 175).

The eclipse on May 24, 997 was observed by Al-Biruni in Khiva and Abu l-Wafa in Baghdad to determine the differences in length.
Christopher Columbus had advance calculations from Regiomontanus available, but tried twice in the Caribbean to determine his geographical longitude by observing eclipses (1494, 1504).

#### Lunar distances

Within a good 27 days, the moon completes a full circle in front of the starry background and thus moves so quickly (about half a degree per hour) that an accurate measurement of its angular distance to a bright fixed star near its orbit gives a good time reference.

This method was first mentioned by Johannes Werner in his work "In hoc opere haec continentur Nova translatio primi libri geographiae Cl 'Ptolomaei ..." (Nuremberg 1514), but only gained attention when Peter Apian used it in his "Cosmographicus liber ..." (Landshut 1524) discussed in more detail. Many observatories, including the one in Greenwich, were founded specifically for the purpose of measuring the course of the moon so precisely that precise lunar distances could be calculated months in advance.

Edmond Halley , who had undertaken two journeys in connection with the length problem in 1698 and 1700, failed because of the inadequacy of his moon tables calculated by Thomas Street in 1661 and was one of the few scientists who supported John Harrison's approach with his ship's clocks , but did not live to see his breakthrough .

The lunar distance method was advanced by the German cartographer and mathematician Tobias Mayer (1723–1762), who, at the age of 28, was given a chair for mathematics in Göttingen without having completed regular studies. While working for a map publisher in Nuremberg , he developed the first usable moon tables based on mathematical calculations. These were translated and edited by Sir Nevil Maskelyne (1732–1811) and for a long time offered a cheap method of determining time.

James Cook had Maskelyne's editing of Mayer's tables available on his first South Sea voyage (1768–1771). In 1767, the Nautical almanac and astronomical ephemeris , which appeared annually afterwards, was published for the first time , in which moon tables were printed which listed the angular distances of the moon to seven fixed stars every full hour. Cook also had an astronomer on board on this trip.

Maskelyne has published explanations of its complex calculation method. In the decades that followed, numerous simpler approximation methods emerged. The method proposed by Nathaniel Bowditch (1773–1838) in particular found widespread use through his famous (still published) navigation manual American Practical Navigator . The "Bowditch" kept corresponding auxiliary tables until 1914, although the nautical almanac had not included lunar distances for years.

### Ship clock

The trained carpenter John Harrison took a completely different and ultimately successful path : A particularly precise clock on board would make it possible to "take" any reference time on the sea voyage and read it at any time.

This method made all time-consuming observations, forecasts and tables superfluous. The problem lay in the accuracy of the clock: until around 1700, clocks with a deviation of only one minute per day were considered highly precise and technically hardly feasible - that was true for stationary clocks on solid ground. Every movement makes a mechanical watch run less precisely, and a time deviation of ten minutes corresponds to 2.5 degrees of longitude or around 280 km on the equator. On a ship, the clock is constantly in motion and also exposed to changing climates and temperatures, which affects its accuracy. The construction of a clock that deviates by only a few minutes under the real conditions of a month-long voyage seemed impossible.

To solve this problem, Harrison devised various clock concepts with counter-rotating mechanisms, whose rate errors compensated for each other when the movement fluctuated, and presented a first concept to the Longitude Commission in 1728 and a first functioning clock in 1735. But the committee delayed the appraisal and recognition of its work for decades, partly for political and strategic reasons, but also because the leading scientists did not want to take the suggestion of a simple craftsman seriously. Newton fundamentally doubted the technical feasibility of a sufficiently accurate clock, although Harrison had made mechanical grandfather clocks with very low rate errors as early as 1725. The astronomer Sir Nevil Maskelyne , who himself propagated the lunar distance method, changed the interpretation of the tender to Harrison's disadvantage, especially after he became royal astronomer in 1765. Harrison was also hampered by his own ingenuity and perfectionism: each of his watches was vastly different from the earlier ones, so he kept coming up with new concepts, which damaged his credibility. He was asked, for example, to hand over his plans to other watchmakers and to have the models made by them in order to rule out cheating.

A pocket watch with a new type of drive, which Harrison had made for himself in 1753, finally moved him to a completely new fourth concept, which he worked on until 1759. This brought the breakthrough after more than three decades: During a test on a sea voyage to Jamaica that lasted several months, the rate error of the watch later known as "H4" added up to less than two minutes.

### Other suggestions

Due to the high prize money, unsuitable and absurd ideas for solving the length problem were also brought up and partly discussed publicly. A very absurd suggestion had been published in the Curious Inquiries leaflet in 1687 :

First, a dog is wounded with a knife before starting the journey. The dog goes with you on the journey, the knife stays in the home port. Then in the home port, weapon ointment is applied to this knife every day at lunchtime , which, due to a supernatural connection between the weapon and the wound, makes the dog on board the ship howl in pain and thus informs the ship's crew that noon is in the home port. ( Umberto Eco took up this method in his novel The Island of the previous day .)

The mathematicians William Whiston and Humphry Ditton suggested that ships be anchored in the sea at regular intervals, which should help determine their position several times a day by firing gunshots : the distance to the gunship could be calculated from the time difference between lightning and bang. Whiston, a student of Newton, had assumed a maximum depth of 600 m for the oceans. This method has not been found to be applicable at sea, but there have been applications on land.

The evaluation of irregularities of the earth's magnetic field, which Edmond Halley , William Whiston , Christoph Semler dealt with, was seriously considered . However, these suggestions soon turned out to be impractical.

## The solution

It was not until James Cook, after returning home from his second trip around the world, enthusiastically praised the usefulness of the time keeper , which Larcum Kendall had built on Harrison's order as an exact copy of the clock from 1759, that most astronomers considered the problem of length to be solved. In the logbook , the initially skeptical Cook Kendall's work called his “never-failing guide”: a clock “took the time of the departure port with it on the journey”. Three other watches, which Cook also had to test, were not up to the stresses of the trip.

After a long struggle, John Harrison received the last portion of the prize money due to him shortly before his death. Posthumously were also Tobias Mayer awarded £ 3000 and given to his widow.

In 1780, John Arnold coined the term chronometer for his further development of Harrison's clock . Maskelyne's moon tables continued to be used until every ship was equipped with one of the initially very expensive chronometers. First the British East India Company equipped their ships with chronometers, the conversion of the Royal Navy lasted until 1840. Captains of smaller merchant ships worked with lunar distances for a few decades until cheaper chronometers came onto the market.

After the problem was resolved , the Board of Longitude was dissolved in 1828 and replaced by the Resident Committee for Scientific Advice for the Admiralty (' Committee for Scientific Advice to the Admiralty').

## literature

• Dava Sobel and William JH Andrewes: Longitude - The Illustrated Edition . The true story of a lonely genius who solved the greatest scientific problem of his time. Berlin-Verlag, Berlin 2010, ISBN 3-8270-0970-7 (English: Longitude . Translated by Matthias Fienbork and Dirk Muelder).
• Johann Matthias Hassencamp: Brief history of the efforts to invent the length of the sea . Rinteln 1769
• Johann Samuel Traugott Gehler's physical dictionary . Vol. 6 Abth. 1, 1834
• Peter Boy Andresen: The history of the lunar distances . Marbach 1986 (reprint of the Hamburg 1924 edition)
• William JH Andrewes (ed.): The Quest for Longitude . Cambridge, Mass. 1996
• Erwin Roth: Tobias Mayer, 1723–1762. Surveyor of the sea, the earth and the sky. Esslingen 1985
• Umberto Eco : The island of the previous day .
• Joan Dash (translation from the American by Tamara Willmann): The hunt for longitude . Youth book published by C. Bertelsmann, ISBN 3-570-12717-6
• Felix Lühning: Longitude. Critical consideration of a bestseller . In: Contributions to the history of astronomy, Volume 10. Frankfurt a. M. 2010, pp. 104–186 (on Dava Sobel: longitude )
• Köberer, Wolfgang: Instrument unde Declinatie der Sünnen, The oldest Low German navigation manual by Jacob Alday from 1578. Facsimile, transcription and commentary volume , Edition Stiedenrod, Wiefelstede 2009.

## motion pictures

The longitude problem and its solution by John Harrison is also the subject of a feature film entitled “The Longitude” with Jonathan Coy , Christopher Hodsol and Jeremy Irons in the leading roles. The film was made based on the book Längengrad by Dava Sobel.