Eratosthenes calculated the Earth's circumference and he was the first to attempt to produce a map of the World based on a system of lines of latitude and longitude. Hipparchus was the first to specify the positions of places on the Earth using latitude and longitude as coordinates. His work on spherical trigonometry led him to this system. He suggested measuring latitude, the distance north or south of the equator, by determining the ratio of the longest to the shortest day at that place.
For longitude, Hipparchus proposed a zero meridian through Rhodes, with east/west distances from this line determined by comparing the local time of a place with an absolute time. He suggested the absolute time be determined using lunar eclipses, measuring the time when the lunar eclipse began and ended, and finding the difference between this absolute time and local time. The method is theoretically sound but required an accurate clock which was not available at that time. The other problem was that lunar eclipses depend from where on the Earth one observed.
Ptolemy wrote Guide to Geography in eight books. It gave information on the construction of maps and listed places in Europe, Africa and Asia giving their latitude and longitude. This work was not accurate at all. Ptolemy used a value for the circumference of the Earth which was too small; much better estimates were known even at that time.
The 10th century saw Abu'l-Wafa and Mansur produce trigonometric results which were applied by, in particular, al-Biruni to the main problem in mathematical geography, namely the determination of latitude and longitude. Al-Biruni believed that the Earth rotated on its axis and made accurate calculations of latitude and longitude.
The age of exploration quickly exhibited the problems of navigation. Firstly there were no maps by which to navigate and a major task of the early explorers was to map the lands which they discovered. However if maps were to be produced it was necessary to be able to determine position on the Earth. How was a practical method to be found?
Spain and Portugal were two major countries involved in exploring. They were in dispute over the "new world" and, in 1493, Pope Alexander VI who was Spanish, issued the Bull of Demarcation which would settle the dispute. He drew a meridian one hundred leagues from the Azores and assigned to Spain all undiscovered land west of the line, while to Portugal he assigned all undiscovered land east of the line. A great solution if one were able to determine where land was relative to the line
100 leagues west of the Azores.
In 1514 Johann Werner published a translation of Ptolemy's Geography. In his commentary on Chapter IV, Werner put forward what became known as the lunar distance method for determining longitude. The method uses the fact that the Moon moves fairly quickly against the background stars, moving through its diameter in about one hour. If tables are given for the distance of the Moon from certain stars then it is possible in principle to determine an absolute time for the place of observation and thus to determine the longitude by comparing the absolute time with the local time in the same way as Hipparchus had proposed for his lunar eclipse method. Werner proposed using a cross-staff, an instrument derived from the Jacob staff of Levi ben Gerson, to measure the distance of the Moon from the chosen stars. He writes:-
Therefore the geographer goes to one of the given places and from there observes, by means of this observational rod at any known moment, the distance between the Moon and one of the fixed stars which diverges little or nothing from the ecliptic.
The method is theoretically correct but Werner had not solved the longitude problem since the cross-staff could not make accurate enough measurements, and more seriously there was no mathematical theory of the Moons orbit (and even when Newton gave his theory of gravitation 150 years later the Moon's motion, a three body problem, was beyond solution). Thus compiling tables of the Moon's position was only possible by collecting data and extrapolating to obtain predictions of the position which soon deviated from the actual position.
Trade flourished and great wealth was produced from ships returning with spices and other goods of great value. However many ships were lost as they were unable to determine their positions. Fine, around 1520, began to produce maps of France and World maps. He advocated a refinement of the Greek method of lunar eclipses to determine longitude.
The position of the Spice Islands was in dispute and Spain sought a solution to these costly problems. Nunes was appointed professor of mathematics in 1529 specifically to try to solve this and related problems. He devoted himself to problems of navigation as well as to producing maps and map projections. He became the leading expert in his day on the new discoveries of Spain and Portugal.
Gemma Frisius, in 1530, proposed a methods of finding the longitude using a clock. Basically the clock was set on departure and kept an absolute time which could be compared with the local time on arrival. The east/west distance travelled could then be calculated. He wrote:-
... while we are on our journey we should see to it that our clock never stops. When we have completed a journey of 15 or 20 miles, it may please us to learn the difference of longitude between where we have reached and our place of departure. We must wait until the hand of our clock exactly touches the point of an hour and at the same moment by means of an astrolabe ... we must find out the time of the place we now find ourselves.
Gemma Frisius then gives precise instructions to translate difference in time into east/west distance travelled. Of course this method was totally impractical as clocks were nor accurate enough. However it is worth noting that 250 years later Gemma Frisius was to be proved right as this became the eventual method to determine longitude at sea.
The problem of determining position at sea and producing accurate maps of the World was extremely important. The fact that no solution to this problem had been found was costing countries vast sums of money. A solution had to be found, so countries began to adopt the standard method, namely to offer money, prizes, pensions, wealth beyond belief to mathematicians and astronomers who could give a method to find the longitude at sea.
The first country to offer a prize was Spain. First Philip II offered a prize in 1567. Soon after Philip III of Spain came to the throne in 1598 he was advised to offer a large prize to
the discoverer of longitude.
A prize of 6000 ducats plus 2000 ducat's income for life with 1000 ducats expenses was offered. Philip however was rather indifferent to his responsibilities as king and the huge response to his prize offer left him with little enthusiasm for any of the schemes proposed. One scheme which was proposed was from Galileo. He wrote to the Spanish Court in 1616 proposing that the way to measure absolute time, which could be measured at any point on the Earth, was to use the moons of Jupiter. Galileo first observed the moons in 1610 and by 1612 he had tables of their movements which were accurate enough to allow him to predict their positions several months ahead.
A long correspondence over a period of 16 years failed to convince Spain of the virtues of the scheme so, when Holland offered a large prize in 1636
to the inventor of a reliable method of finding the longitude at sea
then Galileo tried to convince the States General, the body of delegates representing the United Provinces of the Netherlands, of his scheme involving the moons of Jupiter. A commission was set up and Galileo's proposal was taken much more seriously by Holland than by the Spanish. However by this time Galileo was essentially under house arrest at Arcetri near Florence and when one of the commissioners tried to visit Galileo, the Inquisition made sure that contact was impossible. The States General lost interest in the method a couple of years later when Galileo died.
Since there were many large prizes on offer, large numbers of people tried to win them. In fact several people were awarded smaller amounts of money to continue work on their particular method. A serious proposal, however, came forward from Jean-Baptiste Morin in 1634 and was made to his own country France. Cardinal Richelieu, chief minister to King Louis XIII of France from 1624 to 1642, set up a commission consisting of Étienne Pascal, Mydorge, Beaugrand, Hérigone, J C Boulenger and L de la Porte to investigate Morin's claims.
Morin did not believe in the transporting clock method first suggested by Gemma Frisius. In fact he distrusted clocks and said
... I do not know if the Devil will succeed in making a longitude timekeeper but it is folly for man to try.
He proposed a variation of the lunar distance method with some improvements such as better instruments and taking lunar parallax into account. However his method still was not practical and the commission were in dispute with Morin for the five years after he made his proposal. He did propose the setting up of an observatory to provide accurate lunar data in his attempts to convince the commissioners. Cardinal Richelieu died in 1642 and his successor, Cardinal Mazarin, gave Morin 2000 livres for his efforts in 1645.
In 1651 Cardinal Mazarin, who was then the chief political figure in France, was forced to leave Paris during the struggle between the King and the parliament. Jean-Baptiste Colbert became Mazarin's agent in Paris and Colbert was rewarded by Mazarin who, on his deathbed in 1661, recommended Colbert to the King, Louis XIV. From this time Colbert dedicated all his efforts to serve the King in all possible ways and he was soon in a position to achieve this, becoming minister for home affairs.
Colbert believed that science and sea power were the most important way of achieving great things for France. In 1666, at Colbert's instigation, the Académie Royale des Sciences was founded. By the spring of that year he had persuaded the King to commit himself to funding the new Society. It had the general purpose to study a broad range of scientific activities but its specific objectives were to improve maps, sailing charts and advance the science of navigation. It was firmly believed that mathematics and astronomy held the key to solve these outstanding problems of the day.
Colbert was determined that the Académie Royale des Sciences should have the best scientists in the world so he sent personal invitations, with offers of vast sums of money both for personal use and for research, to many top scientists and mathematicians including Huygens, Leibniz, Tschirnhaus, Hevelius, Viviani, Romer and Newton. Huygens and the Danish astronomer Romer accepted at once and they were joined by Jean Picard, Adrian Auzout and other French scientists. In fact Colbert selected fifteen top scientists and it was with that number that the Académie Royale opened on 22 December 1666.
Using large sums of money which was made available for research, the mathematicians and scientists of the Académie Royale began working on a wide range of mathematical and scientific problems many of them associated with solving the longitude problem.
Huygens was particularly important to the Académie Royale des Sciences as he had patented the pendulum clock in 1656 and several of his clocks had been tried, although not very successfully, at sea in an attempt to find the longitude. Howse, writing in , says:-
Having invented the pendulum clock in 1657, Huygens turned his attention to the longitude problem, convinced that the horological approach - to produce a marine timekeeper that would keep accurate and regular time for months on end in any climate, regardless of the motion of the ship - would soon make it possible to discover the longitude. He produced various marine timekeepers which were tried at sea between 1662 and 1687.
After beginning work for the Académie Royale in Paris, Huygens tried to perfect the operation of his pendulum clocks. The scientists were given a house near Cordeliers and set up astronomical instruments such as a quadrant, a sextant and a large sundial in the garden. They began making observations from the garden of the house near Cordeliers in January 1667, also making observations from the garden of the Louvre.
These sites were far from ideal for research purposes and Colbert was able to obtain a grant from the King to set up the Observatory of Paris in Faubourg, St Jacques, far enough from Paris to avoid lights and other problems. On 21 June 1667, the day of the summer solstice, the Observatory officially opened and observations were made to determine the exact location of the new Observatory. The meridian line through the Observatory became the official meridian line of Paris.
In its early days the Académie Royale des Sciences met in secret and the first publications by its members were anonymous. The Académie invited scientists other than their members to meeting only to evaluate their work and even then the visitors had to leave the sessions before the members of the Académie Royale discussed the quality of the work presented. One person who presented his ideas on the longitude was Jacques Graindorge, the prior of a Benedictine abbey in Fontenay near Caen. He had claimed to know the secret of the longitude as early as 1662 but refused to divulge his theories. Graindorge wrote to Colbert claiming that his method allowed mariners to determine lines of longitude by direct measurement as easily as they could calculate their latitude.
In November 1668 Colbert invited Graindorge to come to Paris and explain his methods. Graindorge was pleased to receive the invitation but said that he could not afford the expense of the journey. Colbert offered to pay all the expenses of the trip and finally, after two places were booked in a coach so that he could have a more comfortable journey, Graindorge agreed. In January 1669 he presented his methods to the Académie Royale des Sciences who set up a committee consisting of Huygens and Jean Picard to examine it. The committee declared the method useless but Colbert was still prepared to pay 1200 livres to Graindorge to cover all possible expenses for his trip. The Académie Royale was desperate to examine every chance for a solution and money was no problem.
The members of the Académie Royale des Sciences made observations of the Moon over the years 1667 to 1669 which convinced them that the mathematics of the position of the Moon was too difficult to make it useful as a solution to the longitude problem. During this time Huygens kept trying to perfect his clocks with sea trials. Howse, writing in , says:-
In 1668 one of his timekeepers, which had kept going during both gales and a sea battle, gave a difference of longitude between Toulon and Crete as 20 ° 30' as against the true value of 19 ° 13', an error of only 100 km or so. His early timekeepers were controlled by pendulums but, in anything but a flat calm, their going was most erratic.
However in 1668 Cassini, working in Italy, published tables of Jupiter's moons which he had compiled over a period of 16 years. The data was now better than when Galileo first proposed the method and so observations began at the Paris Observatory and Colbert set about bringing Cassini to Paris. With offers of large amounts of money Cassini came to Paris on 4 April 1669, although the Senate of Bologna, the Pope and Cassini himself believed it to be only for a short visit.
Cassini found that work at the Académie Royale des Sciences was progressing rapidly. Huygens and Auzout had been working on grinding lens and mirrors and had developed new telescopes which had enabled Huygens to compute Saturn's rotation period and to have discovered Saturn's ring system and one of Saturn's moons. The universal time keeper that Jupiter's moons provided seemed to provide part of the answer to the longitude problem but other difficulties remained. The size of the Earth was still not known with sufficient accuracy to allow precise conversion between linear distance on the surface and angular measures provided by comparing local and absolute times. In 1669 Picard was assigned the task of making precise measurements of the size of the Earth.
Picard used a triangulation method, the method first proposed by Gemma Frisius, choosing as base points the Pavilion at Malvoisine near Paris and the clock tower in Sourdon near Amiens. Thirteen large triangles were surveyed to give the precise distance between these base points. Observations of Jupiter's moons were made with three telescopes and Picard used two pendulum clocks to measure time, one with a pendulum beating once per second, the other clock beating every half second. He reported that his clocks
...marked the seconds with greater accuracy than most clocks mark the half hours.
After the measurements had all been taken and the results of the survey had been studied it was announced that the diameter of the Earth was about 12554 km, a good result compared with the equatorial diameter now known to be 12756 km.
Soon Cassini was in charge of the Paris Observatory and he began a project to use the moons of Jupiter method in conjunction with the new data available for the size of the Earth to map the World. He corresponded with scientists in many other countries and precise data was obtained for the locations of hundreds of towns and cities. On the third floor of the Paris Observatory Cassini had laid out a planisphere, a map of the World using an azimuthal projection with the North Pole at the centre. Although this greatly distorted land shapes it gave precise latitudes and longitude. A cord was attached to the centre with a movable pointer on it. The pointer was set to the correct latitude and the cord rotated to the correct longitude to locate position. The King, Colbert and the whole French Court came to view this wonderful creation of the Académie Royale des Sciences which was demonstrated by Cassini, Picard and La Hire.
Having completed his measurements of the size of the Earth, Picard was sent on an expedition to Cayenne in 1672. The main reason for the expedition was to observe the opposition of Mars and it was a successful expedition. However Picard took with him a pendulum clock which had been carefully calibrated in the Paris Observatory before he left. Once in Cayenne, however, the clock lost about 2.5 minutes a day. Picard had to shorten its pendulum by about 0.2 cm to get it to keep correct time. Cassini suspected that this was due to an error in the observations. Other expeditions which set out from Paris on longitude measurements were all told to watch out for any unexpected variations in the performance of their pendulum clocks.
In 1681 the Académie Royale des Sciences mounted an expedition to the island of Gorée in the West Indies. Varin and des Hayes were chosen to lead it and they were trained by Cassini in Paris before leaving so that they might perfect their skills in obtaining precise longitude measurements. This was an important task for there were few reliable longitude measurements from that part of the World. Although primarily an expedition to determine longitude it was also used for general scientific purposes and the scientists were instructed to take readings of temperature, pressure and gather scientific data.
Cassini wrote a detailed description of precisely how the longitude measurements were to be carried out. These instructions are contained in several articles and give an excellent account of the scientific methods of the time. Great care was taken with timing the eclipses of the moons. Io, the closest moon to Jupiter, was used and six phases of the eclipse were timed for increased accuracy. The first reading is taken when Io is a distance equal to its diameter from Jupiter, the next when it touches the planet, the third when it is completely eclipsed, the fourth when it fist appears from behind the planet, the fifth when it touches the planet and the sixth and final timing when Io is its own diameter from the planet.
Brown, in , explains Cassini's instructions for one man and two man observing:-
To observe and time these phases was a two man job: one to observe and one to keep a record of the time in minutes and seconds. If an observer had to work alone ... [he] begins to count out loud 'one-five-hundred, two-five-hundred, three-five-hundred' and so on, the instant the eclipse begins, and he continues to count until he can get to his clock and note the time. Then by subtracting his count from the clock reading, he has the time at which the observation was made.
Two clocks were used, one keeping mean time i.e. 24 hours a day, the other sidereal time of 23 hours, 56 minutes and 4 seconds to the day (the length of time until the stars reach the same position as the previous day). The clocks were calibrated taking observations of the Sun and of a star. The latitude of the place from which the observations were being made was found by calculating the height of the Pole Star and of the Sun at noon and consulting tables of declination. The longitude was calculated using the difference in local time and absolute time as found from the timing of the eclipses.
Varin and des Hayes found that, like those of Picard, their clocks did not run correctly and they had to shorten the length of the pendulums. Cassini was not convinced that the behaviour of the clocks was due to differences in gravity near the equator. Newton, who had predicted that the Earth was flattened at the poles, was pleased to accept this experimental evidence as proof of his claims and he quoted the results of these experiments in the third edition of his Principia.
The Académie Royale des Sciences had solved the problem of the longitude for places on land. Their data was accurate and, collected with Cassini's supervision, led to accurate knowledge of the Earth for the first time. However there was still the problem of finding the longitude at sea which was vital for trading ships and naval dominance. The method of Jupiter's moons was of no use for this because the movement of the ship made observations of the timings of the eclipses impossible. Many people designed platforms designed to remain fixed as the ship rolled but none were successful.
Other MacTutor references:
History of the Académie Royale des Sciences
Article by: J J O'Connor and E F Robertson
MacTutor History of Mathematics