sign in sign up
<<
Home , History of navigation

The history of marine from a mathematical perspective

Erik I Paling

32 I must go down to the again, To the lonely and the sky. And all I ask is a tall ship And a 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 ’s description of finding his way through the Greek islands in the Iliad and . 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 (, Indians and Chinese17); and the Pacific Ocean (the were accomplished Wayfinders). Particularly notable was the 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 ) 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 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 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 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 like (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 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 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 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 from this century are quite familiar to many. Among them, apart from Eratosthenes, were the mathematicians, astronomers and physicists such as , 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. Eratosthenes It was Eratosthenes, the first true geographer, where all this starts7. Born at Cyrene (in present day Libya) around 276 BC, he went to Athens to further his studies. There he was

taught Stoicism by its founder (Zeno), exposed to the Cynic philosophical school (Ariston of Chios), and eventually ended up in the Platonic Academy. In fact his first piece of scholarly work was Platonikos, a discourse inquiring into the mathematical foundation of Plato’s philosophies. He produced works covering a vast area of knowledge such as geography, mathematics, philosophy, chronology, literary criticism, grammar, poetry and (apparently) even old comedies. A polymath, he also generated what came to be called the Sieve of Eratosthenes, an algorithm for finding prime numbers. However, it was in geography where he proved to be more skilful and he is considered one of the greatest of all ancient geographers. His love however was mathematics causing 200 years later to describe him as ‘a mathematician among geographers and a geographer among mathematicians’. Eratosthenes was also known by the nickname ‘beta’, at least among those people who disliked him7, due to the fact that he was exceedingly good at many facets of scholarship, but usually came in second. He was friends with Archimedes, whose surviving work The Method of Mechanical Theorems explained to Eratosthenes how mechanical experiments can help the understanding of geometry. By the age of 40, Eratosthenes’ wide knowledge was so highly regarded that in 245 BC the Egyptian Pharaoh Ptolemy III (Euergetes) offered him the role of Keeper of the Royal Library at Alexandria (Figure 2). Two of Eratosthenes’ great works were laying down the parallel of latitude and measuring the circumference of the Earth. As the former is the focus of this analysis, we shall examine that in more detail. He laid down the parallel by observing that, at all places on the equator, the length of each day was exactly the same all year round. He also noted that the length of days and nights varied more as you moved northwards. He reasoned therefore that if he could draw a line east and west through a number of places whose longest day was exactly the same length, those places would all be the same distance from the equator. He drew a line from the Straits of Gibraltar (longest day = 14.5 hours) and then, by observing all other places with the same day length drew a line through Sicily, south of Peloponnesus, the Island of Rhodes and across the Tigris and to the Indian mountains. In fact he drew the line of 36th parallel of north latitude. As he aged, he contracted ophthalmia (an inflammation resulting from prolonged exposure to UV radiation, no doubt common in both Egyptians and people observing the sun). He became blind around 195 BC and, plagued by his inability to read and observe nature, fell into a depression which led him to voluntarily starve himself to death. He died in 194 BC at Alexandria aged 82. Hipparchus Hipparchus of Nicaea (in modern day Turkey) was a mathematician, but like many at the time also an astronomer and geographer who pioneered a number of ideas taken for granted today, such as the division of a circle into 360 degrees and the creation of one of the first trigonometric tables for solving triangles. Less is known about his life but it is likely that he was born in Nicaea around 190 BC. He is considered the founder of trigonometry. His contribution to marine navigation, particularly the measurement of latitude, is derived from his extensive astronomic observational skills. By utilising information from his forbears – particularly the Babylonians, Meton of Athens, and, within a 100 years of his lifetime, Aristarchus of Samos, Aristyllus, Conon of Samos, Eratosthenes, and no doubt Euclid and Philo – he developed and solved such problems as spherical trigonometry. It is reputed that he invented both the astrolabe and its spherical version the armillary sphere, to assist him in developing the first comprehensive star catalogue of the known world. He measured the stellar positions with greater accuracy than

any observer before him and his star catalogue was used both by Ptolemy (c.a. 120 AD), and then 1800 years later by in his determination of the course of what became known as Halley’s comet. Hipparchus’ catalogue, completed in 129 BC, listed about 850 stars and included their apparent brightness on a scale of six magnitudes similar to that used today. There are no portraits of him from his present day (e.g. Figure 2). Little work survives of Hipparchus and apparently much information on him is derived from Strabo, a Greek geographer, philosopher and historian who lived 100 years later. It is Strabo who notes that Hipparchus often (and thoroughly) criticised Eratosthenes for internal contradictions and inaccuracies in determining the positions of geographic localities11. Most important for our discussion is that he insisted that a geographic map must be based only upon astronomical measurements of and longitudes and that triangulation should be used for finding unknown distances34. Among other achievements, he also was the first to comprehensively describe the method to determine latitude from star observations (which we will examine below). Information about his general life is scarce but his dedication to his work is illustrated by his completion of his star catalogue – undeterred by the fact that at the time such an undertaking was considered ‘religiously improper’7. He also bruised the Greek psyche by introducing the concept of the eccentric circle to astronomy, that is that the Earth is not at the centre but at some point offset from it. Having done so to explain peculiarities he had (accurately) observed in the motions of the sun and moon. An eccentric circle did not support the Aristotelian view that nature was beautiful and planetary motions were perfectly uniform. Only one of his less important works Commentary on and Eudoxus survives but his observations, calculations and ideas, along with the invention of trigonometry, allowed astronomers and geographers such as Ptolemy to build and advance the state of their science16,34. Hipparchus probably died on the Island of Rhodes at an approximate age of 70. People who were important in the development of the related mathematics As we have already described Eratosthenes and Hipparchus, the two figures most important for developing the related mathematics to determine latitude geometrically (along with some trigonometry) by celestial objects, it would be more valuable to examine how the process was undertaken mathematically. This section therefore shows a step-by- step sequence modified from Plus23. Suppose you are on the open ocean and desire to determine your latitude - and have an instrument capable of doing so. The sun and most stars change their position in the sky over time - but some stars always appear to be in the same place. One example is Polaris (the ‘North star’), which always appears to be sitting almost directly overhead the . Because of this attribute, it turns out that your latitude is the angle at which Polaris appears to sit above the horizon. If you had a star almanac, such as the one Hipparchus developed, you could use any star contained within it. To see why, examine Figure A below. Consider the plane that contains three points; the North pole (in this case Polaris), the point you are sitting at (X) and the centre of the Earth (O). Although Polaris doesn’t really sit vertically above X, as shown in the figure, it is so far away that the line of sight from X to Polaris is pretty much vertical and so we are going to pretend that it does.

A B The angle that Polaris sits above the horizon is indicated by theta (θ) in Figure A above and it is the angle that our line of sight to Polaris (l) forms with the line (t) that is tangential to the Earth at the point X and which forms our line of sight toward the horizon. Extending t and l, we see the angle θ occurring again on the other side of the crossing point X (as in Figure B above). Now the latitude of X can be defined to be the angle phi (ϕ) that the line from O to X (lets call it r) makes with the plane containing the equator (see Figure C below). In our two- dimensional picture, that ‘equatorial plane’ is just a horizontal line (e) which passes through O. It then meets the vertical line l at point L and the tangent line t at point T.

C D

Because r is really the radius of the circle and t is a tangent, we know that r and t form a right angle at X. Now, since t and l form the angle θ, we know that the angle between l and r must be 90 - θ degrees, as shown in Figure D above.

Now consider the triangle with vertices O, X and L. As we have just seen, the angle at X is 90 - θ. Since l is vertical and e horizontal, the angle at L is 90°. Because angles in a triangle always add up to 180, the angle ϕ, which is our latitude, is: ϕ = 180 - 90 – (90 - θ) = θ. As shown in Figure E below.

E

And there you have it! Your latitude is therefore given by the angle θ that Polaris sits above the horizon and Hipparchus would be impressed. While there is no equivalent of Polaris in the southern hemisphere, the constellation termed ‘Crux’ – also known as the ‘Southern Cross’ and illustrated on the Australian flag and the Commonwealth Bank logo (Figure 3), along with two bright stars in the constellation Centaurus will help you. These two stars, alpha Centauri and beta Centauri are also known as ‘The Pointers’. As described in the accompanying resource, these two constellations can be used to find both the South Celestial pole and therefore due South.

(a) (b) Figure 3: (a) The Southern Cross constellation represented on the Australian flag37 and (b) the stylised version on the Commonwealth Bank logo8. The smaller star, Epsilon Crucis, is quite dim and not often visible due to haze and/or eyesight issues.

Navigators over the millennia have used different devices to measure the angle at which a star appears above the horizon, some of which are represented in Figure 1. The accompanying resource includes a simple quadrant that students could construct themselves. The quadrant was proposed by Ptolemy as a simpler version of the astrolabe but marine versions of this instrument arose much later around 1460. While the geometric

quadrant could be used to measure star angles, it was difficult to measure the sun’s altitude, as using it meant staring into the sun (remember Eratosthenes blindness?). While navigators could use this instrument for measuring the sun – by holding it in front of them and to the side, and measuring the sun’s shadow - it was eventually replaced by the back observation quadrant19, a device more practical and ‘eyesight-friendly’. This analysis should end with a cautionary note. Whatever instrument is used to measure your position, whether it be a Kamal, a mettang, an astrolabe, a quadrant, , or GPS, it will always be subject to some sort of error. As Ferdinand Hassler noted in the US Survey of the coast in 1825: “Absolute mathematical accuracy exists only in the mind of man. All practical applications are mere approximations, more or less successful. And when all has been done that science and art can unite in practice, the supposition of some defects in the instruments will always be prudent.”15

Teaching resource The following resource consists of a ‘generic’ lesson plan, constructed in a way that allows maximum flexibility by detailing a range of attention grabbers and worksheet ideas that could be utilised for various classes, year levels and student abilities.

Lesson Plan

Class: Year: Variable Date: Lesson X Time: ~ 60 min Lesson Topic: Variable depending upon curriculum element and discipline (Mathematics or Science) Main Maths Idea: Again variable

Australian Curriculum Maths Links: Strand: Geometry (although for some investigations in Statistics and probability may also enter) Thread: Variable Lesson Outcomes/Objectives: By the end of this lesson students will be able to: § Appreciate the contribution to maths knowledge by various historical figures. § Have converted physical investigations into symbolic representations. Students’ Prior Knowledge in this thread: Variable depending upon Year taught Lesson resources needed Teacher § Resources (pictures etc. from 9) § Worksheets. Student § Worksheets required Methods of achieving the outcomes 1. Introduction: (5-15 minutes) 1.1 Attention grabber (various depending upon timing): Video grab from ‘Shackleton’ (the Kenneth Branagh version) taking a sighting via sextant in a boat in the Antarctic and getting hit by a huge wave (see 28, diagram on right below from33).

Present pictures of vast landscapes (Desert, Steppe or mid-ocean). Pose question on how you would work out where you were if placed there.

Show Australian flag37 and discuss the stars. Pose question on who knows how to find south using the Southern Cross (there are usually 12 answers all mostly incorrect). Put up a sky picture on board and then use the following to show how it is done (see alternative exercise below).

Above left figure from36, above right from21. Homework can be all students practicing finding south using this method on the night after the lesson. If time permits, add historical description of both geometry and instruments (at least a picture). Explain that we will be making and using a quadrant (resources can be copied from9 e.g. page 23). 2. Body of Lesson: 2.1 Activity 1 – Constructing and using a quadrant (X minutes): Please see the following pages on constructing and using a quadrant. 2.2 Activity 2 – Determining the South Celestial pole (and South) using the Southern Cross (X minutes): This exercise can be used for lower years or less-abled students and involves using protractors and angles. 2.3 Activity 3 – Proof of Hipparchus’ observations (X minutes): Students can be scaffolded through the exercise with a prepared worksheet or work it out for themselves. Some excellent examples are also in38. Possible extensions An excursion to your local planetarium might also be possible. Scitech have one in Perth. Some excellent videos, along with data visualisation resources12, are also available on mathematics and navigation on the internet. 3. Lesson conclusion: (X minutes): Will vary depending upon the lesson.

Activity 1 – Information on the quadrant9.

Activity 1a – Constructing a quadrant Take the pattern for the quadrant (which you will recognize to be part of a simple protractor) and glue it to lightweight cardboard, a ruler or a manila folder. Cut it out and attach it to a drinking straw, being careful that the straight side of the quadrant template is aligned perpendicular to the length of the straw. Attach the thread or string through the “+” on the quadrant and tie the weight on the other end; you are now ready to begin. Teresa Coppen’s web page is also an excellent resource10.

The quadrant is a device for measuring the altitude of an object; that is, its angular distance above the horizon. To use the quadrant, sight along or through the straw at the object whose altitude you wish to measure.

Figure from35.

Let the weight hang down freely under the influence of gravity until it stops swinging. Then rotate the straw (keeping it pointed at the object) until the thread is lying against the quadrant. Then with your finger you can hold the thread against the quadrant while you move it away from your eye and read the altitude off the scale. To check that the quadrant template is properly mounted on the stick, you should find someplace with a clear horizon and sight the stick toward the distant horizon; it should then read zero, since 0˚ is indeed the altitude of the horizon. If it does not, reset the quadrant template. The altitude of the point overhead, the zenith, is 90˚.

Activity 1b – Using the quadrant outside Students can use the quadrant to measure the altitude of the sun but there are some safety issues. The better method might be to use it in front of them (like the old time mariners) and line the sun up so that its shadow runs down the ruler. Students will need to work in groups of two or three.

If they are measuring the altitude of the sun, you will need to download the appropriate azimuth and declination information for the sun times at your location. These are illustrated below for the Graduate School of Education at UWA. Information is available on the internet from sites such as SunEarthTools. Measuring the movement of the Sun also occurs in the movie Castaway with Tom Hanks.

An example of sun path and rays for a particular location (The University of Western Australia)29.

Example of azimuth and declination map, information can also be downloaded into a table29.

Activity 1c – Using the quadrant inside Should health and safety issues be a concern, the teacher can hang up a yellow balloon (with a small amount of water in it) at a central point on the ceiling of the classroom. Students can then move around the classroom and make readings. See also TeachEngineering31. An extension is to use a fan so that the balloon is always in motion. This re-creates conditions on the deck of a boat, which is never stationary. In this case students must take a number of readings and determine the average and possible other measures of central tendency – this allows statistics to be taught as well. Alternatively, various ‘stars’ can be placed around the room for student use. Note that some scaffolding may be needed to introduce the celestial globe and explain the sun’s passage over the Earth (as per below):

39 25 Other geometrical uses for the quadrant include measuring the height of objects (e.g. trees or buildings) outside by their angles.

Activity 2 – Determining the South Celestial pole (and South) using the Southern Cross Using a worksheet modelled on the images below4, it would be useful to have it to scale.

Students can then use protractors and rulers to do a perpendicular bisection of the pointers and physically draw the South Celestial Pole and then (when drawing the line perpendicular down to the horizon) due South. Extension can be the discussion that if the angle of the Southern Cross, or one of its stars, has been catalogued (like Hipparchus) then latitude could be calculated fairly easily as well (e.g. below30).

Activity 3 – Proof of Hipparchus’ observations23

How this analysis relates to topics and themes that are part of the mathematics curriculum It was intended that this analysis and resource be catered for Years 7 to 102. However there is much that could be carried out in Years 11 and 12, including spherical trigonometry using calculations for tracking drogues on the ocean – although this is a complex extension. In summary, it is considered, apart from human endeavour, that the following curriculum elements within ‘Measurement and Geometry’, Geometric Reasoning and Using Units and Measurement could be catered for; Year 7 § Identify corresponding, alternate and co-interior angles when two straight lines are crossed by a transversal (ACMMG163). § Classify triangles according to their side and angle properties and describe quadrilaterals (ACMMG165). Year 8 § Develop the conditions for congruence of triangles (ACMMG201). Year 9 § Use the enlargement transformation to explain similarity and develop the conditions for triangles to be similar (ACMMG220). § Solve problems using ratio and scale factors in similar figures (ACMMG221). § Apply trigonometry to solve right-angled triangle problems (ACMMG224). Year 10 and 10A § Formulate proofs involving congruent triangles and angle properties (ACMMG243). § Apply logical reasoning, including the use of congruence and similarity, to proofs and numerical exercises involving plane shapes (ACMMG224), § Use the unit circle to define trigonometric functions, and graph them with and without the use of digital technologies (ACMMG274). § Solve simple trigonometric equations (ACMMG275).

How this topic relates to other subject areas Clearly this topic relates to both HAAS Geography (across all years but particularly ATAR Geography) and Science1. In regard to the latter, there are extensive connections across the Science curriculum through all years. Most particular however this would be useful in Year 7, Science Understanding, Earth and space sciences: Predictable phenomena on Earth, including seasons and eclipses, are caused by the relative positions of the sun, Earth and the moon (ACSSU115). As historical information is provided, various aspects of the Science as Human Endeavour can be incorporated in any year group such as; Scientific knowledge has changed peoples’ understanding of the world and is refined as new evidence becomes available (ACSHE119); Science knowledge can develop through collaboration across the disciplines of science and the contributions of people from a range of cultures (ACSHE223); and People use science understanding and skills in their occupations and these have influenced the development of practices in areas of human activity (ACSHE121).

References 1. Australian Curriculum Assessment and Reporting Authority (ACARA). (2017a). Science Curriculum. Retrieved from http://www.australiancurriculum.edu.au/science/curriculum/f- 10?layout=1 2. ACARA. (2017b). Mathematics Curriculum. Retrieved from http://www.australiancurriculum.edu.au/mathematics/curriculum/f-10?layout=1 3. AncientHistoryEncyclopedia. (2017). Ancient History Encyclopedia Limited. Retrieved from https://www.ancient.eu/Hipparchus_of_Nicea/ 4. AnglesAndAcid. (2014). Finding South with the Southern Cross. Retrieved from https://www.youtube.com/watch?v=-qZaiE9n-Z8 5. Bennett, J. (2017). Navigation: A Very Short Introduction. Oxford: Oxford University Press. 6. Boyer, C. B., & Merzbach, U. C. (2011). A history of mathematics. John Wiley & Sons. 7. Buckley, A. B. (2009). A Short History of Natural Science and of the Progress of Discovery. General Books LLC. 8. CBA. (2017). Retrieved from https://www.commbank.com.au/ 9. Chan Wee Nee, W., Charmaine Ong Heng, L., Chong Jin Wei, M., Jordan Bai, B., Nah Weh, B., & Ng Bee Jing, W. (2017). Celestial Navigation: Heaven’s guide for mere mortals. Retrieved from http://www.math.nus.edu.sg/aslaksen/gem-projects/hm/0506-1-19- Celestial_Navigation.pdf 10. Coppens, T. (2016). The Astrolabe: How to make one and understanding its use. Retrieved from https://owlcation.com/stem/History-of-the-Astrolabe-and-How-to-Make-One 11. Dicks, D. R. (1960). The geographical fragments of Hipparchus (Vol. 1). University of London, Athlone Press. 12. Friendly, M., & Denis, D. J. (2001). Milestones in the history of thematic cartography, statistical graphics, and data visualization. URL http://www. datavis. ca/milestones, 32. 13. Gerdes, P. (2003). Awakening of geometrical thought in early culture. Mep Publications. 14. Gould, R. T. (1923). The marine chronometer, its history and development. London, JD Potter, 1923. 15. Hassler, F. R. (1825). Papers on Various Subjects connected with the Survey of the Coast of the United States (Vol. 2). 16. Hodgkin, L. (2005). A history of mathematics: from Mesopotamia to modernity. Oxford University Press on Demand. 17. Joseph, G. G. (2011). The crest of the peacock: Non-European roots of mathematics. Princeton University Press. 18. Masefield, J. (1923). The collected poems of John Masefield. London: Heinemann. 19. May, W. E., & Holder, L. (1973). A History of Marine Navigation. [With a Chapter on Modern Developments by Leonard Holder.] Foulis. 20. Neal, K. (2002). Mathematics and Empire, Navigation and Henry Briggs and the Voyages of 1631. Isis, 93(3), 435-453. 10.1086/374062 21. Pengqiao. (2017). How to find south using the Southern Cross. Retrieved from https://www.reddit.com/r/thegrandtour/comments/5lesui/in_case_anyone_wondered_how_to_fi nd_south_using/ 22. Penobscot. (2012). History of Navigation. Retrieved from http://www.penobscotmarinemuseum.org/pbho-1/history-of-navigation/history-navigation- introduction 23. Plus. (2017). Latitude by the stars. Retrieved from https://plus.maths.org/content/latitude-stars 24. Sobel, D. (1998). A brief history of early navigation. Johns Hopkins APL technical digest, 19(1), 11. 25. SolsticeDete. (2017). Le chemin sous les buis: Carnets païens. Retrieved from https://lecheminsouslesbuis.wordpress.com/2010/06/17/solstice-dete/ 26. Stewart, D. (2017). Eratosthenes. Retrieved from https://www.famousscientists.org/eratosthenes/

27. Stillwell, J. (2002). Mathematics and its History. Springer. 28. Sturridge, C. (Writer). (2002). Shackleton [Television]. In S. Roberts (Producer). London: Channel 4. Retrieved from www.channel4.com/programmes/shackleton 29. SunEarthTools. (2017). Tools for consumers and designers of solar. Retrieved from https://www.sunearthtools.com/index.php 30. SydneyObservatory. (2017). Retrieved from https://maas.museum/sydney-observatory/ 31. TeachEngineering. (2017). Hands-on Activity: The North (Wall) Star. Retrieved from https://www.teachengineering.org/activities/view/cub_navigation_lesson02_activity2 32. Thevenot, M. (1644). Melchisedech Thevenot (1620?–1692): Hollandia Nova detecta 1644; Terre Australe decouuerte l’an 1644, Paris: De l'imprimerie de Iaqves Langlois, 1663 Based on a map by the dutch cartographer Joan Blaeu. Langlois, 1663. Source: Thevenot’s Relations de divers voyages curieux, Paris. 33. Thomson, J., & Smith, M. (2009). Shackleton’s Captain: A Biography of Frank Worsley. Royal Foundation of the Blind. 34. Toomer, G. J. (1974). The chord table of Hipparchus and the early history of Greek trigonometry. Centaurus, 18(1), 6-28. 35. Utexas. (2002). The Quadrant: An exercise in error analysis. In T. U. o. T. a. A. M. Observatory (Ed.). 36. Ventrudo, B. (2017). Cosmic Pursuits. Retrieved from https://cosmicpursuits.com/550/constellation-crux-southern-cross/ 37. Wikipedia. (2017). Australian Flag. Retrieved from https://en.wikipedia.org/wiki/Flag_of_Australia 38. Willis, E. J. (1921). The mathematics of navigation. JW Fergusson & Sons. 39. WorldGlobeUniverse. (2017). Basic Celestial Globe. Retrieved from http://www.worldglobeu.com/product/310

Recommended further reading (for both teachers and students) Bennett, J. (2017). Navigation: A very short introduction. Oxford, UK: Oxford University Press. Sobel, D. (1998). A brief history of early navigation. Johns Hopkins APL Technical Digest, 19(1), 11–13.