Mastering the Seas: Advances in Trigonometry and Their Impact upon Astronomy, Cartography, and Maritime Navigation

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Mastering the Seas: Advances in Trigonometry and Their Impact upon Astronomy, Cartography, and Maritime Navigation


Until the advent of modern navigational tools in the sixteenth century, mariners had since ancient times used similar methods of navigating, largely by instinct. Even as late as the earliest voyages to the New World by Spanish and Portuguese explorers, mariners who embarked on voyages across open waters, out of the sight of land, could primarily only navigate by keeping a daily record of the general distances and directions they traveled, surrounding currents, wind patterns, hazards, and sightings of land. These journals, or ship logs, were used to notice "landmarks" at sea and retrace one's path back to their port of origin. Though pin-point navigation from these journals was difficult, the body of information collected over numerous voyages was immensely useful and often later incorporated into more mathematically accurate charts.

Practical inventions of the sixteenth through the nineteenth centuries, and innovations of existing instruments, were largely responsible for the modernization of navigation on the high seas. With increased ability to accurately plan voyages, trade boomed, transforming forever the shape of Europe and the Americas. Though early European exploration sparked interest in the lands across the Atlantic, advancements in navigation made colonial settlement and international trade a perceptible reality. These marvels, however, would have been impossible without the significant developments in the field of mathematics—most especially in trigonometry—that underscored scientific and engineering advancements of the seventeenth and eighteenth centuries.


Trigonometry, the subdivision of mathematics concerned with the unique functions of angles and their applications in geometry calculations, was primarily developed and advanced from a need to compute measurements in various scientific fields such as map making, navigation, astronomy, and surveying. Incorporating elements of geometry, algebra, and simple arithmetic, plane trigonometry covers problems involving angles and distances on one plane. Angle and distance problems in three-dimensional space, which occupy more than one plane, are the subject of spherical trigonometry. The latter branch of trigonometry was almost always applied to questions in astronomy and navigation and thus was mastered and developed, most especially by the Arab and Chinese civilizations, faster than plane trigonometry. In fact, trigonometry did not separate itself as a unique discipline from astronomy until the later thirteenth century.

The computation of trigonometric problems is reputed to have developed following the calculation of a table of chords by Hipparchus (fl. c. 100 b.c.) in the second-century b.c. However, to what extent he developed and applied the uses of the tables are unknown as the complete original work is lost. The oldest extant work on trigonometry, dating to the middle of the second-century, is contained in Ptolemy of Alexandria's (fl. a.d. 127-145) voluminous work on astronomy, The Almagest. The cornerstone equations of trigonometric problems, those involving right triangles (both plane and spherical), were most likely derived by Hindu astronomers and mathematicians, translated into Arabic around 750 a.d., and then slowly filtered into Western Europe through contact with Arab civilizations in the Middle East and in Spain. Moorish astronomers in Spain improved upon these basic formulas over the course of the next several hundred years, adding the laws of cosines, and the use of tangents and cotangents. In the mid-tenth century, Arab astronomer and mathematician Abu al-Wafa' (940-998) discovered a more accurate method for computing sines, one of the primary mathematical operations used in navigational calculations, and also introduced secant and cosecant functions (assigning them both the more familiar terms of sine and cosine.)

Western Europe, through repeated contact with the Arab world—mostly during the Crusades (1095-1228)—became more familiar with advanced mathematics, including trigonometry. In Europe, Prussian astronomer Johann Müller (1436-1476), known as Regiomontanus, was responsible for systematizing plane and spherical trigonometry and establishing it as a discipline separate of astronomy. Later developments in algebra allowed Regiomontanus's successors to unify some elements of plane and spherical trigonometry (substituting ratio for a trigonometric line, and the angle for the arc), thus simplifying the lengthy equations that the original author had previously scripted.

In the years following the initial explorations to the New World, the need for creating accurate maritime charts pushed for a deeper understanding of the relationships between spherical trigonometry (used in collecting data and navigating) and plane trigonometry (until the late seventeenth century, the predominate mathematical system used to create charts and maps). Sixteenth-century French mathematician François Viète worked to figure out properties of plane triangles similar to those that were known for arc-triangles, such as the cosine and tangent laws. Viète's success inspired others to find similar formulas and properties. Similar means of computing plane triangle properties, as well as the formula for half-angles, appeared in the work of Austrian mathematician Georg Rheticus (1514-1574) in 1568, and nearly 90 years later in a publication by English mathematician William Oughtred (1574-1660).

English mathematician John Napier (1550-1617) invented logarithms, which he called "analogies," in 1619. Logarithms, defined as the powers of a base such that a certain numerical result is obtained as a power of the base, challenged other mathematicians to develop formulas suitable to their use. The eventual outcome of this line of inquiry was the understanding of calculus, which in turn enabled English physicist Sir Isaac Newton (1642-1727) to set forth the first modern physical theories, transforming the predominant modes of scientific and theoretical inquiries in physics, mathematics, and astronomy.


Developments in trigonometry aided navigation not only through astronomy, but also in the development of more systematized and accurate methods of cartography, the science (and arguably, art) of graphically representing a certain geographic region on a map or chart. Many, but certainly not all, ancient and medieval maps in Europe represented more stylized depictions of land forms, rarely endeavored to identify distances, and often depicted mythological creatures or reflected prevailing religious thought. Contact with the New World provided a need for more scientifically sophisticated maps, which could represent landforms with a greater degree of accuracy and aid in the establishment of location at sea.

On longer journeys, as became standard during the Age of Exploration, the ancient method of dead reckoning did not work. Over great distances, the approximated rhumb lines of the Mediterranean chart could not be taken as straight, and the equations devised by astronomers and mathematicians failed to approximate location. In other words, longer voyages required some means of taking into account the curvature of Earth. To this end, the Mercator projection was developed in 1569 by Gerardus Mercator (1512-1594) to represent sections of the spherical Earth on flat charts. Instead of bearings and distances, location was defined by the larger and more defined latitude and longitude. Mercator's charts featured equally spaced representations of the lines of longitude, or meridians, and compensated for the distortions in distances that appear in attempting to illustrate a curved surface by representing the lines of latitude, or parallels, closer together at the Equator and further apart at the Poles. Mercator guarded the secret of calculating his maps, but mariners soon realized that east-west distances were slightly distorted at some latitudes. Not until 1599, when English mathematician Edward Wright provided an explanation of the trigonometry involved in Mercator's map projections, could the distorted distances be corrected.

Innovations in cartography continued to be made in the seventeenth and even into the eighteenth centuries, but practical applications of spherical trigonometry in astronomy and planar trigonometry in chart making did not yield the solution to perhaps the most problematic piece of the puzzle of navigation: determining longitude while at sea. Latitude, one's north-south position, could be determined by measuring the altitude of the Sun at noon or the altitude of any star, provided it was tabulated in one of the various astronomical almanacs of the time, when it crossed the local meridian. In higher latitudes, most marine navigators determined latitude by observing the altitude of the polestar, or the angle between its direction and the horizontal; as ships approached the equatorial latitudes, stars were often below or too close to the observable horizon and navigators had to rely on measurements of the position of the Sun. Knowledge of one's east-west position—longitude—was critical, especially on long transatlantic voyages on which water and foodstuffs were prone to ruin or were in short supply. Calculating longitude required determining the precise time as one element of the equation—a measurement that could not readily be determined at sea. The calculation of longitude using astronomical observations and trigonometry was a mathematical possibility first introduced in 1474 by Regiomontanus. Nevil Maskelyne (1732-1811), who was appointed British Astronomer Royal in 1765, also proposed that complex trigonometry could be used in conjunction with a voluminous catalog of observed and predicted star positions to calculate longitude. The mathematically intensive task was cumbersome and impractical, and the problem of determining longitude was eventually settled with the invention and acceptance of a portable timepiece by English inventor John Harrison (1693-1776) in 1773.

Besides the chronometer, other practical mechanical devices were employed in aiding navigation. The patent log, designed by English inventor Humphry Cole in 1688, aided in approximating the speed at which a ship was traveling. Innovations to the original design of the patent log made it more reliable, and an 1861 design by another Englishman, Thomas Walker, remains in use. Telescopes were employed in the eighteenth century to determine the length of a degree of longitude. The principle of the magnetic compass, long known in China and the Arab world, was introduced in Europe with the basic modern design emerging in the West in the seventeenth century. The advent of ironclad and steel ships required the addition of magnetic stabilizers to keep compass readings of magnetic north accurate. The determination of true directional north still required mathematical computation.

As the problems of navigating the open waters came to be solved, there was an increased need to refocus attention back to the careful navigation of those waters closer to port. The enormous increase of commercial vessel traffic, a result of the discovery of the New World and the establishment of large-scale maritime mercantile trade, made navigating ships into harbor precarious. A growing concern among ship pilots was the avoidance of other vessels. The largest of the merchant and military ships were easy to see, but were also more difficult to maneuver. Sail power and varying winds close to shore complicated matters further. Simple trigonometric formulas could be employed to predict the velocity and path of another ship, but a more practical solution, the designation and division of incoming and outgoing shipping lanes, was also needed to manage traffic.

Rising investments in merchant endeavors and bitter rivalries over colonial territories led some European nations to begin to build strong navies. Designing military vessels was a special challenge, as they needed to be sturdy, yet also quick and maneuverable. Simple trigonometry advanced sail designs and allowed for design and implementation of heavy weaponry. Relatively new advances in war weaponry, especially cannons and mortars, required a basic knowledge of the principles of trigonometry to aim the implements and determine the trajectory of their projectiles. Thus, new mathematical understanding facilitated nearly all aspects of the European maritime and colonial endeavors.


Further Reading

Sobel, Dava. Longitude: The True Story of the Lone GeniusWho Solved the Greatest Scientific Problem of His Time. New York: Penguin, 1996.

Turner, Gerald L'estrange. Scientific Instruments 1500-1900: An Introduction. Berkeley: University of California Press, 1998.

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Mastering the Seas: Advances in Trigonometry and Their Impact upon Astronomy, Cartography, and Maritime Navigation

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