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Navigation

Mathematics

Navigation

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In the broadest sense, navigation is the act of moving about from place to place on land, sea, in air, or in outer space. Navigation, with its primitive beginnings, has evolved to become a sophisticated science.

Early Navigation

Prior to the fifteenth century, European mariners were reluctant to sail out of sight of land, partly because they feared getting lost and partly because they did not know what lay beyond the horizon. Thus, sailing voyages by Europeans were largely confined to the Mediterranean Sea or close to shore in the Atlantic Ocean. The high and broad continental shelf of Northern Europe, where the continent ends and the ocean begins, allowed for shallow sailing waters within sight of land from the Iberian Peninsula (Portugal and Spain) to Scandinavia (Norway, Sweden, and Denmark).

The Vikings of Scandinavia were renowned coastal navigators. Not only did the Vikings sail the coast of Europe, but they also followed the continental shelf into the Northern Atlantic to Iceland, Greenland, and ultimately to North America. Although such extended voyages were remarkable accomplishments, they involved no sophisticated navigational techniques.

In about the year 1000, the Norseman Leif Ericson made a transatlantic voyage to North America with the midnight sun lighting his way. Using the pole star as his only navigational guide, he followed the North Atlantic's generous continental shelf to the northeastern coast of mainland North America.

While ambitious open sea voyages such as Ericson's were possible in the extreme northern latitudes, the South Atlantic was not as accommodating. Africa's continental shelf was narrow, and left very little room for navigational error before a ship could be swept into the deep currents and unfamiliar winds off the African coast. These currents and winds were unpredictable and tended to flow to the north and east, exactly the opposite direction from that in which sailors wanted to go.

The Europeans, including the Vikings, remained essentially coastal navigators until the first half of the fifteenth century. The situation was the same in all parts of the world at that time. All navigation was local rather than global. Sailing on the open sea was possible only where there were predictable winds and currents or a wide continental shelf to follow.

Medieval Navigation

In the early part of the fifteenth century, Portuguese sailors began to sail farther out into the Atlantic using favorable winds, currents, and the paths of birds as guides. By the 1440s, they had reached as far as the Azores, an archipelago of small islands some 800 miles west of Portugal. To venture farther than this would require the beginnings of a more scientific and mathematical type of navigation. This more scientific approach took two forms. The first was a type of navigation known as "dead reckoning" and the second was the application of astronomy and mathematics to what is known as "celestial navigation," or navigation by the stars.

In the process of dead reckoning, a triangular wooden slab, called a chip log, attached to a rope with evenly spaced knots along its entire length, was tossed into the ocean from the stern of the ship. Sailors would then count the number of knots pulled out by the log in a given amount of time, usually measured by sand glasses calibrated for one minute or less. From this observation, an approximation of the speed of the ship could be calculated. Such measurements were taken each time the ship changed course due to a change in wind direction.

This was an early attempt to measure what we now call the longitude of the ship at a given moment. The method was not very accurate, but it was the best that could be done at the time. The captain's log of Christopher Columbus's 1492 journey to the Americas suggests that Columbus relied almost exclusively on dead reckoning to navigate to the New World. Truly accurate measures of longitude would have to wait until the invention of the chronometer in the eighteenth century.

Celestial navigation could help in estimating a ship's latitude . In the Northern Hemisphere, mariners could use the pole star as a reference point. At the north pole the star would be directly overhead at all times, but as one moves farther south it appears lower and lower in the sky until, at the equator, it dips below the horizon.

An instrument called a quadrant could be used to measure the angle of the pole star above the horizon. The quadrant was a quarter circle with degree markings from 0 to 90 along its arc. A plumb line hung from the point at the center of the circle and the observer would then line up the edge of the quadrant with the pole star. The plumb line would then cross the arc of the circle at the position that would indicate the number of degrees above the horizon at which the pole star was located. In this way latitude could be approximately determined.

Of course this method worked only at night, but an alternative method for determining latitude in the daytime made use of the astrolabe, a heavy brass disk with degrees marked around its edge. An observer would move a rotating arm attached at the center of a disk until sunlight shone through a hole at one end of the arm and fell on a hole at the other end. The arm would indicate the altitude of the Sun by the degrees marked around the edge of the disk.

In 1473, the astronomer Abraham Zacuto created a book of tables called Rules for the Astrolabe that allowed mariners to determine the latitude for any day of the year. Use of the tables depended upon knowing in which constellation of stars the Sun rose on the day of the measurement. An observer would view the eastern horizon before sunrise and note the constellation in which the Sun rose. Later in the day, when the Sun reached its highest point in the sky, the observer would take a reading with the astrolabe. Zacuto's Rules for the Astrolabe could then be used to look up the latitude with a degree of accuracy never before possible.

Zacuto constructed this extensive set of tables using mathematics, specifically trigonometry, developed between the ninth and thirteenth centuries by Judeo-Arab mathematicians and astronomers in Portugal and Spain. To produce these tables, Zacuto needed, in addition to trigonometry, an accurate solar calendar giving the location of the Earth with respect to the Sun at any time during the year. Such a calendar had been constructed in the eleventh century by Muslim astronomers in Spain. Making use of this calendar, the Sun's position relative to the constellations, and the height of the midday Sun above the horizon, Zacuto produced the first scientifically accurate method for determining latitude. This method was used by European navigators for more than a century.

By the 1520s, the ability to determine latitude at sea with reasonable accuracy was well established, but the problem of finding longitude with an acceptable degree of precision remained intractable for another 300 years. Whereas latitude measures positions north and south of the equator, longitude uses imaginary "great circles" passing through the north and south poles to measure positions east and west of a predetermined great circle called the Prime Meridian .

The first prime meridian was established by the Portuguese map-maker Pedro Reinel in 1506. It passed through the Portuguese Madeira Islands. Reinel's prime meridian would remain the world's standard for more than 300 years, but with the decline of Portuguese sea power and the rise of England in the seventeenth century, a British prime meridian was established passing through Greenwich, England. In 1884, a conference of European nations ratified the new prime meridian as the world's standard. It remains so to this day.

The problem of determining longitude involves knowing the time at the prime meridian and the time aboard the ship on which one is traveling. Earth rotates on its axis once every 24 hours. One revolution is 360 degrees of longitude, so 360 ÷ 24 gives 15 degrees per hour. Thus if the ship has a clock which accurately gives the time at the prime meridian and the time on board the ship, then the longitude of the ship can be calculated.

This may seem a trivial matter to people of the twenty-first century who possess incredibly stable and accurate time-pieces, but such was not the case for navigators of the fifteenth, sixteenth, and early seventeenth centuries. Clocks of that time period were of the pendulum type and were useless on the deck of a rocking ship. An obscure English clockmaker, John Harrison, would finally solve the longitude problem in 1764 with the invention of a clock that could keep time to within less than a second of accuracy per day and could withstand the rocking and temperature extremes experienced aboard a ship on the open sea. Harrison's invention was the forerunner of the modern chronometer that is present on all ocean-going vessels today.

At about the same time that Harrison was creating his chronometer, a more stable and accurate version of the astrolabe, called the sextant, was invented. Together, these two inventions ushered in a new, more scientifically based era of navigation.

Modern Navigation

In the cold-war era of tension between the United States and the former Soviet Union, the U.S. Department of Defense authorized about $12 billion for research and development to devise and perfect a navigational system that could provide an almost instantaneous and accurate reading for the location of any point on the surface of the Earth. The military's purpose was to allow pinpoint accuracy in the launch of its missiles from submarines in the ocean. Yet in the mid-1990s, this Global Positioning System technology was made available to the civilian population.

GPS Technology. The Global Positioning System (GPS) utilizes satellites in orbit around Earth to send signals to Earth-based devices for the purpose of calculating the exact latitude and longitude of the Earth-based unit. The ability of computer-chip makers to pack more memory onto smaller and smaller chips has resulted in GPS devices that can be held in the palm of a hand and are reasonably priced.

The mathematics behind GPS is essentially the same as that used by Abraham Zacuto to develop his tables for use with the astrolabe except that the calculations are done by computer through the trigonometric idea of triangulation . The distances from a handheld GPS receiver to three of the orbiting satellites is determined by the time-encoded signals traveling at the speed of light from each satellite to the receiver. Then using the familiar "rate × time = distance" equation, the GPS device calculates the distance to each satellite from the device's position on the ground. With these three measurements, the GPS can calculate this position to within a few meters of accuracy. Essentially, the distances to the three satellites can be thought of as the radii of three imaginary spheres. These three spheres will intersect in two points, only one of which will be a reasonable position on Earth's surface. The GPS device will give a reading of the latitude and longitude of this position.

With the introduction of the Global Positioning System, the age-old problem of knowing where you are on Earth's surface at any given time has essentially been solved, assuming that you are carrying a GPS receiver at all times. That is now the case for most ocean vessels and airplanes, both commercial and military. Many of the latest model cars come equipped with navigation systems powered by GPS technology. This may well become standard equipment on all vehicles in the near future.

Maps and Planning. Even without sophisticated technology, it is still possible to plan trips on land using a map of the area you are interested in navigating. Using a well-marked map, you can decide whether you want to take a scenic route or a more direct and quicker route. Using the mileage markings on the map or the legend that gives the scale of the map, you can determine how far you must travel using each route. With a little mathematics, you can determine the approximate length of time required to reach the destination.

If you know that you can average about 60 miles per hour on the direct route, which is 240 miles long, then you calculate 240 miles divided by 60 miles per hour to get 4 hours as the approximate time to make the trip. If the scenic route is 280 miles and you can only average 40 miles per hour then it will take 7 hours to travel the scenic route, by calculating 280/40.

You can also estimate the gasoline cost for each route. If your car gets about 24 miles to the gallon when traveling at 60 miles per hour on the open highway, and if gasoline is $1.50 per gallon, then you can calculate the gasoline cost for the direct route as approximately 240 miles divided by 24 miles per gallon times $1.50 per gallon = $15.00. Of course if you are coming back by the same route, you could double this to $30.00 for the round trip. Similar calculations would allow you to compare the cost of this route to that of the scenic route, taking into account that your car may get poorer gas mileage on the scenic route due to frequent starts and stops, climbing hills, and the like. Perhaps future generations of GPS devices will do these calculations as well as letting you know where you are at each second of your trip.

see also Angles of Elevation and Depression; Angles, Measurement of; Distance, Measuring; Flight, Measurements of; Geometry, Spherical; Global Positioning System; Mile, Nautical and Statute.

Stephen Robinson

Bibliography

Andrews, William, ed. The Quest for Longitude: The Proceedings of the Longitude Symposium. Harvard University, Cambridge, Massachusetts, November 4–6, 1993. Cambridge, MA: Collection of Historic Scientific Instruments, Harvard University, 1996.

Ferguson, Michael. GPS Land Navigation. Boise, ID: Glassford Publishing, 1997.

Sobel, Dava. Longitude: The True Story of a Lone Genius Who Solved the Greatest Scientific Problem of His Time. New York: Walker and Company, 1995.

Toghill, Jeff. Celestial Navigation. New York: W. W. Norton and Company, 1998.

Internet Resources

All About GPS. Timble. <http://www.trimble.com/gps/index.htm>.

"Early Navigation Methods." The Mariner's Museum—Newport News, Virginia. August 1997. <http://www.mariner.org/age/earlynav.html>.

Latitude: The Art and Science of Fifteenth-Century Navigation. <http://www.ruf.rice.edu/~feegi>.

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