Brian Doore

Math G

March 15, 2002

Midterm

 

                        Discovering an Accurate Method for Determining Longitude

 

            After watching a PBS mini-series entitled Lost at Sea: The Search for Longitude, I was surprised how mathematics was involved in celestial navigation.   Our GPS systems today employ the same calculations as John Harrison used in the 1700’s to determine longitude.   People in the 1700’s knew that the ability to measure longitude was the scientific problem of the day.  Sailors were lost at sea because they could not measure their longitude.  Lives and fortunes were lost.  A solution was necessary.  England’s Parliament offered money for a successful method.  John Harrison developed an accurate clock, which is called a chronometer.  When you know your time, your location can be determined.  Today 24 atomic clocks orbit the earth in satellites.  The global positioning system (GPS) constantly gives us the latitude and longitude within centimeters any place on the globe. 

            “Longitude depends on an arbitrary fixed point on the earth’s surface.  The line on the earth’s surface formed by the intersection of a plane through this fixed point and the north and south poles is called the prime meridian.  A line formed similarly through any other point of the earth’s surface is called a meridian or line of  longitude.  The longitude of a point on the earth’s surface is the angle, in degrees, between the plane forming the meridian of the point and the plane forming the prime meridian.  Longitude begins at 0 degrees at the prime meridian and increases as one moves west, until, at 360 degrees, one reaches the prime meridian again.” (1)

            Hipparchus (190-120 BC) proposed a zero meridian through Rhodes, with east/west distances from this line determined by comparing the local time of a place with an absolute time.  “He suggested the absolute time be determined using lunar eclipses, measuring the time when the lunar eclipse began and ended, and finding the difference between this absolute time and local time.” (2)  This method is sound but at this time in history there was no accurate clock.

            The 17^th century brought the age of exploration. Quickly trade and wealth flourished, but many ships, sailors, and goods were lost at sea because they were unable to determine their positions.  Early seafarers had to keep land in sight.  This didn’t always work.  There were no maps.  Navigators would look to the heavens, followed the sun in the day and the stars at night   At any one time in a grid at any point on a globe the sun and stars are found above the horizon at a certain fixed height.  This distance the mariners would measure with one’s fingers held at arms length.  The 17^th century was an age of exploring, and a more reliable method was necessary.   Others sailors were following the flight paths of birds.  The Polynesians watched the waves.  Many seamen thought that the compass was black magic.  The sextant was invented in 1731.  It was used to figure out longitude using lunar distance and astronomical tables.  This took hours. 

            “Real time relates to the movements of heavenly bodies; earth, sun, moon and stars.  From the earliest times, astronomers studied the heavens to track these movements.  As they counted days, months, and years they kept records of the positions of the sun, moon, and stars in the heavens.” (3)  Navigators studied the positions of the stars and that is how they steered their ships. 

            “Mathematicians and astronomers noted celestial positions precisely; their records were studied century after century by other astronomers who learned to predict, with reasonable accuracy, the eclipses of the sun and the moon.” (4)  People believed the earth stood at the center of the universe.  In 1543 Nicolaus Copernicus said that the earth is not the center of the universe.  Our earth goes around the sun, which is the center of our solar system.  This shook the world. 

            In 1581, in Italy, Galileo Galilei was experimenting with pendulums. He made a mechanical clock.  Weeks, hours, minutes and seconds were now used.   The telescope was invented in 1608 and Galileo improved it in 1609. 

            With the improvement of the telescope, the moons of Jupiter became used as a means of determining longitude.  The moons were used as a clock.  The idea was to compare the moon positions and formations observed on the same day at the same time from these to deduce the difference in local (solar) times between the two locations.  The time of each eclipse was compared to the local time at the reference location.  From the difference in times the longitude could be found. This method was slow and again not accurate enough. Galileo made tables of the moons movements that were accurate enough to allow him to predict their positions several months ahead.  It also required a high-powered telescope. 

            By the 1666’s it was generally believed that mathematics and astronomy held the key to solve the longitude navigation problem of the day.  Many countries were offering rewards for research to solve the longitude problem.  Some of the solutions were very interesting.  One proposal “involved the use of…the powder of sympathy…the powder of sympathy was applied, not to the wound but to the weapon that inflicted it…before sailing every ship should be furnished with a wounded dog.  A reliable observer on shore, equipped with a standard clock and a bandage from the dog’s wound…would every hour, on the dot, immerse the bandage in a solution of the powder of sympathy and the dog on shipboard would yelp the hour.” (5)   Another solution was to anchor 20 or 30 lightships permanently in the principal shipping lanes in the Atlantic Ocean at regular intervals.  The lightship would fire a skyrocket at midnight and the sea captains could calculate their distance by timing the interval between the flash of the shell and the report time.  Who is on the lightship, how they survive, etc.  was not addressed.  There were hundreds of impossible solutions presented to the various governments. 

            The clock had been invented by now, but clocks did not work well on ships because the sea would disrupt their mechanism.  There were variations in heat and cold, wet and dry, gravity differences.  In addition, the sun rose and set at different times depending on where the ship was at sea.  About 1700 navigators realized by comparing the time difference between noon where they were and noon at home they could figure out where they were at sea.  What was needed was a clock to keep accurate time on a moving ship. 

            Scientists were getting closer to determining the longitude.  Telescopes were better and pendulum clocks were more accurate.   The Board of Longitude was established in England in 1714 and offered 20,000 pounds (12 million dollars in today’s currency) to whoever would come up with a method for determining longitude with in a distance of 30- nautical miles during a voyage from England to the West Indies. 

            Many attempts were tried.  John Harrison built his first clock in 1715.  In 1760 John Harrison was made an accurate clock impervious to motion.  He called it a chronometer.  It was a spring driven clock that kept time so accurately that navigators could find their position at sea after several months.  He spent nearly 50 years working on his clock.  At the age of 78 he was awarded the prize money.

            John Hadley in 1731 invented a quadrant called an octant.  It had an arc, which measured one-eighth of a circle, and two mirrors which allowed the Sun or a star and the sea horizon to be seen simultaneously.  The allowed for more accurate measurements at sea.  John Bird who invented the sextant in 1757 further advanced the quadrant.   Now with John Harrison’s chronometer and a sextant a navigator was able to figure out his longitude with the accurate clock.  Simply one points a cross staff at the sun and checks a clock.  Observe the length and time the shadow of a pole of known height crosses a true north-south line.  From the length of the shadow, latitude is found.  Longitude is found from the observed time.  Together your position can be determined. 

“By the 19th century most of the world’s sailing ships used sea charts with Greenwich, England as the prime meridian.  This means 0 degrees longitude (imaginary line from North Pole to South Pole).  At sea the ship had a clock that told exactly what time it was at Greenwich.  Then they checked the position of the sun, moon, and stars and with the sea charts, which told them where the celestial bodies were supposed to be they could get a fix on their location.  The lines of longitude called meridians divide the earth.  The distance between one degree of longitude and the next is about 69 miles.  Each meridian is about 15 degrees apart.    Each 15-degree section marks how far the sun travels in one hour.  The sun stands still and it’s Earth that rotates on its axis.  24 hours times 15 degrees is 360 degrees, the circumference of a circle and of a sphere like Earth.  Earth rotates on its axis a full 360 degrees in 24 hours.” (6)  “So if a sailor know when it was noon at the home port and then had to wait one hour until noon on board his ship, he would know that the ship was 15 degrees west of Greenwich.  If the sailor had to wait two hours for the sun to reach its high point, he was 30 degrees west, but an error in the clock of just one minute would put the ship 60 miles off course.”  (7)  The chronometer stayed on Greenwich time and the sailor would compare local time with Greenwich time. 

 At noon in the city of Greenwich, England, it would be 11a.m. 15 degrees to the west, 10 a.m.  and this continues halfway around Earth.  Moving east, it would be 1 p.m. 15 degrees from Greenwich, 2p.m.15 degrees farther east, and so on.  “GMT is Greenwich mean time.”  (8)  Day begins at Greenwich.  A ball is dropped every day in the Meridian Courtyard in Greenwich at 1 p.m.  Even though ships today rely on radio and satellite signals the ceremony is still performed daily as it was in 1833.

“Measurements of longitude requires accurate timepieces in order to ascertain the exact time at a known position, such as the prime meridian.  Then if the sun is directly above a given observer, and the precise time of day at the prime meridian at that same instant is known, the observer can determine his longitude east or west of the prime meridian… For example, at noontime the observer looks at the chronometer and it says that it is 3:15 p.m. at the prime meridian.  Thus the observers longitude is 48 degrees 45 ’, since 3 ¼ times 15 degrees equals 48 degrees 45’.  The longitude is west, because the noon sun has already passed the prime meridian and has just arrived at the meridian of the observer.” (9)

Today 24 atomic clocks orbit the Earth 10,000 miles overhead.  “The satellites of the Global Positioning System (GPS) constantly monitor for almost perfect accuracy, provide navigators with their latitude and longitude to within centimeters anywhere on the globe.” (10)  In 1985 the first GPS receiver was used for marine navigation. Today the GPS receivers are very small and economical.  They can be found in cars, boats, planes, construction equipment, movie making gear, farm machinery and laptop computers.  Soon they will be a basic as the telephone.  In 2002, as in 1799, the secret of knowing where you are is knowing what time it is. 

 

                                                Footnotes

(1)   Observations of Latitude and Longitude at all Remarkable Points, Chapter 5.  http://www.lib.virginia.edu/exhibits/lewis­_clark/ch5.html

(2)   Longitude and the Academie Royal.  http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Longitude1.html

(3)    Skurzynski, Gloria.  On Time.  National Geographic Society, Washington, D.C., 2000, page 20.

(4)   Skurzynski, page 21.

(5)   English Attack on the Longitude Problem.  http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Longitude2.html

(6)   Skurzynski, page 26.

(7)   Longitude Horizon Script.  http://www.bbc.co.uk/science/horizon/longitudetrans.shtml

(8)   Sobel, Dava. Longitude. Walker and Company, New York. 1995, page 168.

(9)The Encyclopedia Americana International Edition.  Grolier Incorporated, Connecticut, 2000.  Volume 17, page 51.

(10)Longitude Horizon Script.

 

Bibliography

Encyclopedia Americana International Edition.  Volume 17, 2000.  Grolier Incorporated, Connecticut. 

Skurzynski, Gloria.  On Time.  Washington, DC:  National Geographic Society, 2000.

Sobel, Dava.  Longitude.  New York: Walker and Company, 1995. 

 

BBC – Horizon – Script.  http://www.bbc.co.uk/science/horizon/longitudetrans.shtml

English Attack on the Longitude Problem.  http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Longitude2.html

Longitude and the Academie Royale.  http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Longitude1.html

Longitude at Sea.  http://es.rice.edu/ES/humsoc/Galileo/Things/longitude.html

Observations of Latitude and Longitude at all Remarkable Points.  http://www.lib.virginia.edu/exhibits/lewis_clark/ch5.html