Brian Doore

Math G

March 15, 2002

Midterm

                        Discoveringan Accurate Method for Determining Longitude

            Afterwatching 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 JohnHarrison used in the 1700’s to determine longitude.   People in the 1700’s knewthat the ability to measure longitude was the scientific problem of theday.  Sailors were lost at seabecause they could not measure their longitude.  Lives and fortunes were lost.  A solution was necessary.  England’s Parliament offered money for a successfulmethod.  John Harrison developed anaccurate clock, which is called a chronometer.  When you know your time, your location can bedetermined.  Today 24 atomic clocksorbit the earth in satellites.  Theglobal positioning system (GPS) constantly gives us the latitude and longitudewithin centimeters any place on the globe. 

            “Longitudedepends on an arbitrary fixed point on the earth’s surface.  The line on the earth’s surfaceformed by the intersection of a plane through this fixed point and the northand south poles is called the prime meridian.  A line formed similarly through any other point of theearth’s surface is called a meridian or line of  longitude.  Thelongitude 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 theprime meridian.  Longitude beginsat 0 degrees at the prime meridian and increases as one moves west, until, at360 degrees, one reaches the prime meridian again.” (1)

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

            The17^th century brought the age of exploration. Quickly trade andwealth flourished, but many ships, sailors, and goods were lost at sea becausethey 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 thesun and stars are found above the horizon at a certain fixed height.  This distance the mariners wouldmeasure with one’s fingers held at arms length.  The 17^th century was an age of exploring, and amore reliable method was necessary.  Others sailors were following the flight paths of birds.  The Polynesians watched the waves.  Many seamen thought that the compasswas black magic.  The sextant wasinvented in 1731.  It was used tofigure out longitude using lunar distance and astronomical tables.  This took hours. 

            “Realtime relates to the movements of heavenly bodies; earth, sun, moon andstars.  From the earliest times,astronomers studied the heavens to track these movements.  As they counted days, months, and yearsthey kept records of the positions of the sun, moon, and stars in theheavens.” (3)  Navigatorsstudied the positions of the stars and that is how they steered theirships. 

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

            In1581, in Italy, Galileo Galilei was experimenting with pendulums. He made amechanical clock.  Weeks, hours,minutes and seconds were now used.  The telescope was invented in 1608 and Galileo improved it in 1609. 

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

            Bythe 1666’s it was generally believed that mathematics and astronomy heldthe key to solve the longitude navigation problem of the day.  Many countries were offering rewardsfor research to solve the longitude problem.  Some of the solutions were very interesting.  One proposal “involved the useof…the powder of sympathy…the powder of sympathy was applied, notto the wound but to the weapon that inflicted it…before sailing everyship should be furnished with a wounded dog.  A reliable observer on shore, equipped with a standard clockand 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 onshipboard would yelp the hour.” (5)  Another solution was to anchor 20 or 30 lightships permanently in theprincipal shipping lanes in the Atlantic Ocean at regular intervals.  The lightship would fire a skyrocket atmidnight and the sea captains could calculate their distance by timing theinterval between the flash of the shell and the report time.  Who is on the lightship, how theysurvive, etc.  was notaddressed.  There were hundreds ofimpossible solutions presented to the various governments. 

            Theclock had been invented by now, but clocks did not work well on ships becausethe sea would disrupt their mechanism. There were variations in heat and cold, wet and dry, gravitydifferences.  In addition, the sunrose and set at different times depending on where the ship was at sea.  About 1700 navigators realized bycomparing the time difference between noon where they were and noon at homethey could figure out where they were at sea.  What was needed was a clock to keep accurate time on amoving ship. 

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

            Manyattempts were tried.  John Harrisonbuilt his first clock in 1715.  In1760 John Harrison was made an accurate clock impervious to motion.  He called it a chronometer.  It was a spring driven clock that kepttime so accurately that navigators could find their position at sea afterseveral months.  He spent nearly 50years working on his clock.  At theage of 78 he was awarded the prize money.

            JohnHadley in 1731 invented a quadrant called an octant.  It had an arc, which measured one-eighth of a circle, andtwo mirrors which allowed the Sun or a star and the sea horizon to be seensimultaneously.  The allowed formore accurate measurements at sea. John Bird who invented the sextant in 1757 further advanced thequadrant.   Now with JohnHarrison’s chronometer and a sextant a navigator was able to figure outhis longitude with the accurate clock. Simply one points a cross staff at the sun and checks a clock.  Observe the length and time the shadowof a pole of known height crosses a true north-south line.  From the length of the shadow, latitudeis found.  Longitude is found fromthe observed time.  Together yourposition can be determined. 

“By the 19thcentury 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 SouthPole).  At sea the ship had a clockthat told exactly what time it was at Greenwich.  Then they checked the position of the sun, moon, and starsand with the sea charts, which told them where the celestial bodies weresupposed to be they could get a fix on their location.  The lines of longitude called meridiansdivide the earth.  The distancebetween one degree of longitude and the next is about 69 miles.  Each meridian is about 15 degreesapart.    Each 15-degreesection marks how far the sun travels in one hour.  The sun stands still and it’s Earth that rotates onits axis.  24 hours times 15degrees is 360 degrees, the circumference of a circle and of a sphere likeEarth.  Earth rotates on its axis afull 360 degrees in 24 hours.” (6) “So if a sailor know when it was noon at the home port and thenhad to wait one hour until noon on board his ship, he would know that the shipwas 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 putthe ship 60 miles off course.” (7)  The chronometer stayedon Greenwich time and the sailor would compare local time with Greenwichtime. 

 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 aroundEarth.  Moving east, it would be 1p.m. 15 degrees from Greenwich, 2p.m.15 degrees farther east, and so on.  “GMT is Greenwich meantime.”  (8)  Day begins at Greenwich.  A ball is dropped every day in theMeridian Courtyard in Greenwich at 1 p.m. Even though ships today rely on radio and satellite signals the ceremonyis still performed daily as it was in 1833.

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

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

                                                Footnotes

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

(2)   Longitudeand 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)   EnglishAttack on the Longitude Problem.  http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Longitude2.html

(6)   Skurzynski,page 26.

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

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

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

(10)LongitudeHorizon Script.

Bibliography

Encyclopedia AmericanaInternational Edition.  Volume17, 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 LongitudeProblem.  http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Longitude2.html

Longitude and the AcademieRoyale.  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 andLongitude at all Remarkable Points. http://www.lib.virginia.edu/exhibits/lewis_clark/ch5.html