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The History of Marine Navigation from a Mathematical Perspective

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The History of Marine Navigation from a Mathematical Perspective The history of marine navigation from a mathematical perspective Erik I Paling 32 I must go down to the seas again, To the lonely sea and the sky. And all I ask is a tall ship And a star to steer her by, (John Masefield, 1902, Sea Fever. 18) Introduction The term ‘navigation’ generally refers to any skill or study that involves the determination of position and direction24. More specifically, finding one’s way on land, at sea and in the air. The word ‘navigate’ however is relatively recent in human history (15th century), was associated with the ocean from the outset and was derived from the Latin concatenated word navigare ‘to sail, sail over, go by sea, steer a ship’ derived from the words for ‘ship’ (navis) and ‘to drive’ (agere). A navigator needs practical judgment to make good decisions with incomplete or overly complex data based upon mathematics, astronomy, physics, oceanography, meteorology, earth sciences and hydrodynamics. The mathematics required may include arithmetic, algebra, trigonometry, logarithms and geometry22. The story of navigation is very important for the development of the relationship between art and science within the discipline of mathematics. In the later 15th and especially throughout the 16th centuries, a number of mathematicians took up the theme of introducing geometry into a range of ‘practical arts’ where a potential benefit seemed possible20. Geometry could provide shared principles, whose truth was demonstrable, that would underpin a reformed and regularised practice, delivered through the use of mathematical instruments adapted to particular needs6. In other words, these disciplines would become mathematical arts underpinned by mathematical science. This had long been the situation in astronomy but one of the first arts where it was shown that such a pattern of development was possible elsewhere, was navigation5. The generation of practical navigational charts in particular involved a great deal of conceptual mathematical thought in relation to mapping the Earth’s sphere on to a plane27. Scope of this resource The history of navigation, particularly by observing celestial objects, stretches back from 800 BC with Homer’s description of Odysseus finding his way through the Greek islands in the Iliad and Odyssey. Mythology takes us even earlier to the Argonauts helping Jason in finding the Golden Fleece (~1300 BC). Like geometry itself, humans were always examining relationships when practical problems needed to be solved13. There are three basic regions where its development and use can be traced: the Mediterranean (e.g. the Mycenaeans and Phoenicians); the Indian Ocean (Arabs, Indians and Chinese17); and the Pacific Ocean (the Melanesians were accomplished Wayfinders). Particularly notable was the circumnavigation of Africa in 600 BC by the Phoenicians – but it should not be forgotten that navigating desert or steppe regions without landmarks was also a great (and for survival, a necessary) achievement. Its story is also peppered with delightful tales of skulduggery, as navigational charts and various instruments (e.g. the astrolabe and compass) were ‘traded’, stolen or indeed plundered under the definition of ‘well-deserved booty’. Additionally, in a world of male-celebrated scientists and mathematicians, it is gratifying to observe that women such as Janet Taylor (1804–1870) played a major role. Unfortunately such a vast topic cannot be covered properly in the limited space available here, so some limitations and caveats are necessary. A good starting point for this reduction in scope is to examine the two basic navigational requirements; where you are (your ‘position’), and in which direction you need to travel in order to arrive at where you want to be (your ‘bearing’). Your current position on the planet can be simply described by your latitude and longitude. Your bearing can also be straightforwardly determined by the changes in both of these coordinates over time, as can your speed of travel. Regrettably, measuring time at sea is fraught with difficulty and it took the invention of an accurate, seaworthy timepiece (something thought impossible by Sir Isaac Newton at the time), along with the establishment of a standard clock at Greenwich by King Charles II in 1675, to allow the accurate determination of longitude. Until that time, mariners either did not use longitude or calculated it using other, often complex, means. The story of the marine chronometer is fascinating but less relevant here and the reader is directed to follow it up if interested14. One should not assume before this invention however that ship navigators could not keep time. Various instruments (sand glasses, candles) and experience were available to measure time’s passage, but few were accurate enough to give more than a rough estimate of longitude. Latitude, however, could be measured fairly ‘simply’ and this is where this resource is focused. Basically a navigator would know the latitude of their starting position (e.g. a port) and then, when wishing to return after a voyage, would use instrumentation to both determine and then return their ship to the port’s latitude, after which they would sail east or west (or often in those days ‘left’ or ‘right’) knowing that at some point they would run into their start point. The term ‘running down the line’ was the descriptor used for this practice. The earliest ‘device’ (other than finger widths) used for sailing by latitude was the Kamal developed by Arabs in the 9th century and this was followed by the astrolabe, quadrant and cross-staff (Figure 1). Each of these devices measured latitude by determining the angle either of the sun in the day or stars like Polaris (the north star) at night. Polaris is closely positioned at the north celestial pole and so was also very useful for determining north – at least in the northern hemisphere. While the development of navigational instruments is both fascinating and relevant to the history of mathematical practical arts, the following analysis and resource details the development of the geometrical awareness that latitude could be calculated from the apparent angles of stars, including the sun. Detailed below is a critical analysis of the historical background of angle geometry, its cultural context, the people who were important in the development of the related mathematics, how it relates to other topics and themes that are part of the mathematics curriculum, and how it relates to other subject areas. The resource included in this document is a lesson plan that details the construction and use of a simple quadrant in or outside the classroom. Figure 1: Timeline of the development of major latitudinal instruments used for celestial navigation9. Historical background How did geometry come to be associated with latitude? There were two historical figures pivotal in this process: Eratosthenes, born around 270 BC; and Hipparchus (of Nicaea) who came somewhat over 100 years afterward. The following description focuses on both them and the era in which they lived. Third century BC saw two ‘kingdoms’ dominating the Mediterranean, The Hellenistic (Greek) kingdoms in the east, and Carthage in the west. Carthage had been, according to legend, founded around 810 BC by a Phoenician queen (Elissa) and developed into a great mercantile port. After the defeat of Tyre by Alexander the Great in 323 BC, refugees fled to Carthage with whatever wealth they had. This wealth proved to be not insubstantial and the city under their influence, along with banishment, enslavement or tribute extraction of the native Africans, grew into the richest in the Mediterranean. Expansionist activities and their riches led the city to become a target for Roman ire and a series of wars ensued, starting with the First Punic War in 264–241 BC and ending with the Third (149–146 BC) when it was sacked and burnt to the ground. It would not rise into prominence again until Julius Caesar ordered it rebuilt 100 years later as a colony, which it remained until the fall of the Roman empire. Around Eratosthenes’ time, the Greek cities of Egypt and further east were flourishing both materially and culturally, and most of them were at their peak. Greek was firmly established as a common tongue, as was cultural unity along most of the eastern Mediterranean. Every educated person was familiar with the language and it was used in diplomacy, literature and science. Thus a book written in Greek could not only be understood by native Greek speakers but also by virtually every educated non-Greek in the eastern Mediterranean. It has been estimated that there were literally hundreds and thousands of books being produced at the time and in 290 BC, the Greek Pharaoh (Ptolemy I) established a museum of which the famous Library of Alexandria was a part. A part which, due to its scholarship and research activities, eventually overshadowed the museum itself and to which we shall return below. The list of influential Greeks from this century are quite familiar to many. Among them, apart from Eratosthenes, were the mathematicians, astronomers and physicists such as Apollonius of Perga, Archimedes, Aristarchus of Samos, Aristyllus, Conon of Samos, Euclid, and Philo of Byzantium. Famous philosophers included Demetrius of Phalerum, Epicurus, Pyrrho, Theophrastus, Timon of Philus, and Zeno of Citium. Cultural and historical context In order to place our two characters into a historical/cultural context and help bring them to life, it is worth describing them and their activities as well their mathematical contributions. Several sources have been used to weave together an account of the lives of these two men, predominantly Arabella Buckley’s historical work on the sciences7. Much use has also been made of several other authors5,11,16. (a) (b) Figure 2: (a) Eratosthenes teaching in the Alexandrian Library26. (b) Hipparchus holding his celestial globe (an artistic impression by Raphael in his School of Athens (1510)3.
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