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to Hypotenuse Contents| The Art of Navigation is to be perfected by the Solution of this Problem. To find, at any Time, the Longitude of a Place at Sea. A Public Reward is promised for the Discovery. Let him obtain it who is able. |
Even at the time when Christopher Columbus sailed across the Atlantic and discovered the West Indies instead of arriving in the Far East, the uneducated still believed the earth to be flat and that sailing too far was to risk falling off its edge. The notion that the earth was not flat can be traced back to the philosophers of ancient Greece though, as is so often the case, not to any one particular philosopher. It would seem that Aristotle and his pupils believed the earth to be a perfect sphere which rotated about an axis through its centre. They also subscribed to the common view that the earth was the centre of the universe, but the best recorded attempt to measure the size of the earth was that by Eratosthenes in the third century BC. The method that he used supports the belief that the early Greek philosophers had already established a system of latitude for the world as they knew it, and it is also interesting that his method was still regarded as effective 2000 years later by French cartographers. Eratosthenes had noticed that at noon on the day of the summer solstice the sun shone directly down a deep well in Syene in Upper Egypt. His method was to measure the shadow cast by the sun at the same time in Alexandria which was 5000 stadia north of Syene, and then calculated that the angular distance between the two places was one-fiftieth of a circle, resulting in a global perimeter of 250,000 stadia. What is not clear is how this distance translates into modern measures.
The belief that the earth was a perfect sphere persisted among astronomers and mathematicians until the beginning of the 18th century. In the meantime, the Roman Empire had risen and fallen, so too had the Islamic Empire, Marco Polo had travelled to China, printing had been invented and Columbus had crossed the Atlantic. The known world was a bigger place but it was easier for mathematicians, astronomers, and scientists to communicate with each other and to become aware of the needs of society and of necessary demands for improved technology. No longer could mathematicians indulge in mathematics only as an academic study for its own sake when pressing practical problems had to be solved. Two fields which needed urgent attention were the difficulties of drawing accurate maps and the problem of navigating with a reasonable degree of certainty. Progress in these two fields depended on knowing the true shape of the earth and being able to pinpoint any place on earth with accuracy by a combination of latitude and longitude. If the true shape of the earth were known latitude presented no problem but longitude was much more difficult, particularly at sea and out of sight of land.
By the end of the 15th century, Spain and Portugal had become the leading seafaring nations in Europe and inevitably in dispute about lands discovered in their various maritime expeditions. Pope Alexander VI settled the dispute by drawing a meridian line 100 leagues west of the Azores on a chart of what was then known as the Western Ocean. Any land not already belonging to a Christian prince which was discovered west of the line would belong to Spain and, similarly, land to the east of the line would belong to Portugal. The trouble was that the line was meaningless because no-one knew how to find any meridian line at sea. Furthermore, the inability of the navigators to find their objectives with sufficient accuracy in the now frequent voyages to bring back the treasures of the West Indies meant that too many ships were lost. In 1598, over a hundred years of navigational uncertainty since the first voyage by Columbus, King Philip III of Spain offered a substantial monetary prize and pension to the "discoverer of longitude". The inevitable result was the submission of hundreds of totally unworkable ideas, nor was this to be the last time that a European country made such an offer to find a solution to the problem of longitude. By the time the court of Spain received a proposed method from Galileo in 1616 Philip had lost interest in the project and after protracted and intermittent correspondence Galileo gave up trying to arouse interest in his ideas.
It was the invention of the telescope that accelerated Galileo’s theories about the nature of the solar system and his study of the movement of planets. He was particularly fascinated by Jupiter and one of the high points of excitement was when the use of the telescope showed that there appeared to be four moons in orbit around Jupiter. This was in January 1610 and he studied and recorded the movements of Jupiter’s satellites every night for the next eight weeks. During the following two years he drew up tables of the positions of these satellites for different times of night and concluded that, with such tables, Jupiter could be used as a celestial time-keeper. When he also realised that such tables could be worked out months in advance he devoted more time to perfecting them, so when Holland became yet another country to offer a reward for the invention of a solution to the longitude problem, Galileo was ready to submit his ideas again. Knowing the exact time was vital to the calculation of longitude, but being able to tell time accurately in minutes and seconds was still a dream rather than a reality. Because of his brush with the Inquisition, Galileo submitted his plan to Holland through a friend in Paris and the commissioners in Holland who assessed the proposed solutions were impressed. Unfortunately Galileo died before the negotiations were completed and his ideas were put on hold for a while. What he never knew was that another aspect of his work would contribute to the solution of this still unresolved problem, and this was his study of the behaviour of pendulums.
Among many other things, Christiaan Huygens is known in mathematics for his work on the radius of curvature of a plane curve and for his short treatise "On reasoning in games of chance", in science for his work on the wave theory of light, in astronomy for his observation of Saturn’s rings and in mechanics, having read Galileo’s work on pendulums, for his invention of the pendulum clock in 1657. On land, at least, accurate time-keeping was possible but a pendulum clock would be of no use at all on a ship negotiating rough seas. The French took advantage of the adjustability of pendulum clocks, and of Galileo’s method of finding longitude by observing the moons of Jupiter, to embark upon an ambitious programme to produce accurate maps of European and other known countries, but in some of the more distant parts of the world the pendulums needed unexpected adjustment. Initially the surveyors were suspected of carelessness but the problem did not go away.
When Isaac Newton published his Principia in 1686 another piece of the longitude jigsaw was put in place. Newton established his Laws of Motion and developed them in the first two books of Principia , and then in the third book proceeded to apply them to the motions of the planets, thereby confirming mathematically the astronomical observations made by the mathematician and astronomer Johannes Kepler in his study of conic sections. Newton’s laws with regard to circular motion led him to argue that by virtue of the earth’s diurnal rotation its shape must be that of a spheroid flattened at the poles rather than a perfect sphere. Applying his gravitational theory he went on to conclude that a standard pendulum would beat faster in latitudes further away from the equator than in those nearer to the equator. The French were sceptical. Eventually, in the 1730’s, two French mathematicians proposed to resolve the question of the shape of the earth by leading an expedition to Lapland and arranging for others to go to Peru and put Newton’s theory about the pendulum to the test. Not only did their findings prove Newton to be right but they also explained why the French cartographers had experienced so much difficulty when using a pendulum, so the seekers of longitude had one less detail to worry about.
The final breakthrough came when the longitude problem was linked to the diurnal rotation of the earth which completes a 360° turn in 24 hours, or 15° in one hour. If it were possible to transport a time-keeper set to the time at a fixed meridian and which would be unaffected by the motion of the sea, and also changes in temperature, humidity and other conditions associated with life on board ship, the difference between the time on the time-keeper and and the time read by the sun at any point on the voyage would give the angular distance travelled -- in a word, longitude. But no such watch had ever been made.
In the early 18th century England became the last of the countries to offer a reward which was to be:
The first chronometer was built by John Harrison who was largely self-taught but who was fascinated by clocks and watches. In the course of his work, he studied the art of navigation and the expansion and contraction of metals, and experimented on how to reduce friction to a minimum. His first effort was hardly elegant in appearance and weighed 72 pounds but it was sufficient for the Board of Longitude to advance £500 to help with expenses arising from building chronometer number two. In all, he built five chronometers, each more accurate than its predecessors, and sea trials showed number five to be accurate to within less than half a degree of longitude, which qualified him for the reward of £20,000. At last, nearly three hundred years after that first transatlantic crossing, longitude could be found accurately. John Harrison had spent 48 years perfecting the key to the solution, but he had to wait 10 years before the reward was paid and that was only after the intervention of King George III.