Mathematical evolution of marine navigation

Summary: With the help of mathematicians, sailors throughout history have been able to devise ingenious methods for navigation.

Marine navigation is the process of conducting a waterborne craft from one point on the surface of the Earth to another, using all the associated science and techniques. The primary activities required for marine navigation may be organized into two closely related components: the planning of the craft’s movement, including the determination of the course and speed needed to reach a chosen destination at a specific time, and determining and controlling the craft’s position at sea. Many problems in marine navigation are complex because the Earth is spherical. With the help of mathematicians and other scholars, sailors throughout history have been able to devise ingenious methods for approximating workable solutions, resulting in great voyages. Before the 1400s, many cultures sailed in the open ocean, including Pacific Islanders, Persians, Arabs, and inhabitants of some Indian Ocean islands. They used techniques such as poems or visual imagery to remember the positions of the stars, which were their primary guide. Polynesians and Micronesians created the most elaborate star maps and star compasses, as they sailed the longest distances. The Chinese developed a magnetic compass in the eleventh century, which then spread to India and Europe. However, many problems remained numerically and geometrically impractical until the development of modern computers, which are capable of resolving these problems.

Early Developments

Since the first nautical charts were produced in the Mediterranean approximately 800 years ago, the basic principles of marine navigation have remained unchanged. Still a dramatic improvement of efficiency, accuracy, and safety has occurred during this long period, largely as a result of new navigational techniques. These developments include astronomical methods for measuring latitude (c. 1450), the invention of the maritime chronometer (c. 1750), and the advent of electronic positioning systems (twentieth century). Mathematics has played a fundamental role in this evolution. Scientific navigation in Europe can be traced back to the first quarter of the thirteenth century, following the adoption of the decimal numeral system, the introduction of the magnetic compass in the Mediterranean, and the creation of the first nautical charts. Contrary to traditional medieval maps, which represented the world in some schematic or symbolic way, these charts were drawn to scale using distances and directions measured by pilots at sea. Taking into account the relatively crude methods used for estimating these quantities, the result is astonishingly accurate and detailed when compared to the terrestrial cartography of the time.

The mathematics of navigation is somewhat complicated by the fact that ships move on the spherical surface of the Earth, where the calculation of angles and distances is considerably more complex than on a plane. However, these complications only began to be relevant for the routine practice of navigation when Europe’s period of great explorations began in the middle of the fifteenth century and ships started to sail routinely in the open sea.

Although the spherical shape of the Earth was well known to most educated people of the time, including the cosmographers of the Middle Ages and the Renaissance, the fact could be ignored when sailing in the relatively confined waters of the Mediterranean and western Europe. This omission was possible because the geometric errors from assuming a flat Earth were usually smaller than those resulting from the crude navigational methods of the time. In these circumstances, the mathematics of navigation was largely reduced to estimating the distance sailed during a given period, based on simple practical rules and pilots’ experience, and determining the ship’s position as a function of the course steered and the distance sailed. This determination could be made graphically on a nautical char t using the graphical scale of distance and the mesh of colored lines radiating from chosen spots, representing the directions of the winds, given by the magnetic compass. Because it was not always possible to sail along the straight line connecting the point of departure to the point of destination, tables and abacuses were created to help determine a ship’s position relative to that line. These were called the toleta de marteloio and gave no more than the solution of the right triangle for some different angles and distances between the planned track and the present track.

By the middle of the fifteenth century, ships started sailing into the open sea on a regular basis and the traditional method for determining their position, based on distances estimated by the pilots and directions given by the compass, was no longer adequate because of the long periods of time ships went without sighting land. This problem was solved with the introduction of astronomical navigation, c. 1450, which permitted sailors to easily determine the latitude by observing the height of the sun and stars above the horizon. Before this important development could be possible, it was first necessary to construct adequate tables with the positions of the heavenly bodies for each day of the year (ephemerides), simplify the instruments of observation used on land (the quadrant and the astrolabe), and devise methods simple and accurate enough to be used on board a ship by uneducated people. The toleta de marteloio was then replaced by the regimento das léguas (regiment of the leagues), which solved the right triangle formed by the track of the ship along its course (the hypotenuse) and the arcs of meridian and parallel connecting the point of departure to the point of destination. Once again, no allowance was made for the spherical shape of the Earth, since these components were small enough to be considered planar and straight. Soon, though, it was necessary to establish a relation between the degree of latitude and the corresponding arc of meridian on the surface of the Earth so that the length of the degree could be expressed in distance units (leagues). Because this length was directly related to the size of the Earth (a longer degree implied a larger Earth), the problem had significant strategic and political implications. For example, Columbus always defended a degree smaller than the one used by the pilots of his time because this made the distance sailing west to the Indies—what he proposed to the Catholic Monarchs of Spain—considerably shorter. A similar reason was behind the dispute between Portugal and Spain over the location of the spice islands of the Moluccas, in the first quarter of the sixteenth century. The new astronomical methods were soon reflected in the geometry of the charts used for navigation. The new cartographic model, known as the “latitude chart (or “plane chart”), was based on observed latitudes and magnetic courses rather than on estimated courses and estimated distances. It replaced the old portolan chart of the Mediterranean for representing the newly discovered lands.

Though it may contradict common sense, a ship sailing with a constant course between two points on the Earth’s surface does not usually follow the shortest track. This paradox is because a line that makes a constant angle with all meridians (called a “rhumb line” or “loxodrome”) does not coincide with a great circle arc (or “orthodrome”). This discrepancy was first recognized in 1538 by the Portuguese mathematician Pedro Nunes (1502–1578), who showed that all loxodromes, except the meridians and the equator, are spirals asymptotically approaching the poles without reaching them. This knowledge was later used by Gerardus Mercator (1512–1594) in the construction of a new world map intended to be used in navigation (1569) in which all loxodromes were represented by straight segments making true angles with the meridians. This important map, known as the “Mercator projection,” is still used today to support marine navigation, though history has shown that the projection was developed well before it could be consistently put to practical use. Any flat map of the Earth must contain some type of distortion, since it must represent features of a spherical object in a flat surface. Despite its geometric inconsistencies, the latitude chart, based on magnetic courses and observed latitudes, continued in use throughout the seventeenth and eighteenth centuries. This longevity was because determining the longitudes at sea remained impossible, and the spatial distribution of the magnetic declination was still unknown—both necessary for the construction and use of the Mercator projection. Only in the second half of the eighteenth century, following the invention of the maritime chronometer and the development of practical methods for finding the longitude at sea, was the latitude chart finally abandoned by the pilots and replaced with the Mercator projection.

Knowing that the Earth completes one rotation around its axis every 24 hours, the longitude can be expressed as the difference between the local time and the time at the prime meridian (Greenwich Time), from where longitudes are reckoned. Thus, one day corresponds to 360 degrees of longitude, one hour to 15 degrees, and so on. In any method based on this principle, an error in the determination of the time is thus directly reflected as an error in the longitude. Two independent methods for solving the longitude problem were developed in the eighteenth century, encouraged by an important prize offered by the British Admiralty: the lunar distances method, based on measurements of the angular distance between the moon and the sun or a given star, from which the Greenwich Time was determined; and the chronometer method, where the Greenwich Time was given by a very accurate maritime chronometer kept on board. In both methods, local time could be determined by observing the position of the heavenly bodies in the sky. The second method proved to be the more practical of the two and is still used today. However, at the core of both were long and fastidious calculations, done by hand using tables of logarithms and trigonometric functions.

Significant improvements in the accuracy and efficiency of the astronomical navigation methods were made possible in the beginning of the twentieth century by the advent of telecommunications, which permitted ships to receive the exact time on board. The construction of more sophisticated tables in the second half of the century further simplified and significantly shortened the required calculations. Finally, the introduction and dissemination of handheld calculators and computers in the last quarter of the century permitted pilots and other users to easily solve the complex equations governing the heavenly bodies and to determine a ship’s position using the time and astronomical observations made on board.

The introduction of radio positioning systems, first relying exclusively on land stations (Loran, Omega, Decca) and later on artificial satellites (GPS, Galileo), represents the latest development in maritime navigation. At the beginning of the twenty-first century, it is possible to find the exact position of a ship in the middle of the ocean and to control its movements with unprecedented accuracy. Although the mathematics involved in all the components of the present navigational systems is vast and complex, the interface is usually transparent enough that the navigator can concentrate his attention on other aspects of the ship’s activity and safety.

Bibliography

Calder, Nigel. How to Read a Nautical Chart: A Complete Guide to the Symbols, Abbreviations, and Data Displayed on Nautical Charts. New York: McGraw-Hill, 2002.

Hollingdale, S. H. Mathematical Aspects of Marine Traffic. New York: Academic Press, 1979.

Molland, Anthony. The Maritime Engineering Reference Book: A Guide to Ship Design, Construction and Operation. Oxford, England: Butterworth-Heinemann, 2008.

Williams, Roy. Geometry of Navigation Cambridge, England: Woodhead Publishing, 1998.