Longitude:  How to Navigate using Time and the Sky
by Michael Kauper , 2008

     A ship's captain needs to know their boat's position, the latitude and the longitude, to arrive safely.  Latitude is how far north or south you are, the same as up or down on a map, and it is relatively easy to find.  Longitude is how far east-west you are, the same as right or left on a map, and it is very hard to find.

     For millennia the world's greatest military and economic quest was a way to determine longitude reliably, both on land and at sea.

     I will talk first about latitude, the easy one.

     One simple way to find latitude is to measure the distance of the North Star above the horizon. If the North Star is 30 degrees above the horizon, then the ship is at 30 degrees north latitude. (This only works in the northern hemisphere!)

      You can also get your latitude from the height of the sun above the horizon, at noon, providing you know the date. The height of the noon sun varies thru the year, higher in summer, lower in winter.  So all you need is a sextant (see below), plus a calendar, book, or instrument which tells how high the sun will be on any given day. So calculating your latitude, your "north-south", is relatively easy, anywhere in the world.

     Now the hard one, longitude. Longitude is how far east - west you are, the same as right or left on a map, and it's very hard to find. Just having a calendar and a book will not work.  You need to know the exact time in two places to calculate your longitude:  the time where you are plus the time at some known longitude line, such as Greenwich , zero degrees longitude.

     If you know the time back at Greenwich ( zero longitude), from the Moon and stars, or from an accurate sea-going clock, plus the time where you are, from the Sun, Moon, or stars, then you can calculate your longitude. Here is how that works (slightly simplified for explanation).

     The day has 24 hours, and the earth is 360 degrees all the way around. 360 degrees divided by 24 hours gives 15 degrees per hour. The earth turns 15 degrees every hour. So one hour west is the same as 15 degrees west. 

     If you know that the time at Greenwich is 1:00 PM and the time where you are is 12:00 noon, then you are 15 degrees (or one hour) west of Greenwich . This is also called 15 degrees wet longitude

     We can also do this for minutes.  If 60 minutes (one hour) equals 15 degrees, then 4 minutes of time equals one degree of longitude. (60 divided by 15 = 4)

     In the Twin Cities we are about 6 hours and 12 minutes behind Greenwich time, so our position is about 93 degrees west longitude.  The 6 hours gives us 6 x 15 = 90 degrees;  plus threes more degrees from the 12 minutes.  (12 minutes, at 4 minutes per degree, gives 3 more degrees, right?)  90 degrees + 3 degrees = 93 degrees west longitude.

     The big question for a sailor is how accurately can I know where I am? Will I get where I want to go safely? If I know the time to within 4 minutes, I know where I am to within 1 degree of longitude, or about 69 miles.

     Where did "69 miles = 1 degree" come from?  Let's say that the distance around the earth is roughly 25,000 miles, at the equator. Divide 25,000 miles by 360 degrees, and you get 69 miles per degree. (Less if you are north or south of the equator. See below.)

     So if you know the time within 4 minutes, you know your position within 69 miles. Not so great. You could easily miss a small island, or hit the shoals, and sink. If you know the time within 1 minute, then you know your position to 69 divided by 4, or 17.5 miles. Much better.  You would probably be able to see an island as you went by.

     So, to safely travel to a distant island or port, a sailor would really like to know the time to within 1 minute. Before the 18th century (the 1700's), no one knew how to tell time so accurately, especially not at sea. The best known clocks used a pendulum, which was completely messed up by the rocking and pitching of the ship.

     Not knowing longitude lead to thousands of deaths at sea. On October 22, 1707 over 1600 lives were lost when four naval vessels floundered off the coast of Sicily . In 1714 the English Board of Longitude offered a huge fortune, the Longitude Prize, to whoever could solve the problem. Two competing methods were developed, and each was eventually successful.

     A brilliant clockmaker named John Hamilton built four sea-going clocks, between 1730 and 1753, each better than the one before. These amazing clocks were immune to the tossing and turning of the ship, and kept time within a few seconds a day. Hamilton 's clocks still keep perfect time, over 250 years after they were created.

     The other great idea was called "The Lunar Distance Method", based on the accurate star maps created by the first Astronomer Royal, John Flamsteed.  The ships navigator could measure the distance from the Moon to 9 or 10 stars, and then using a huge book of moon position tables calculate the longitude. This gave the correct longitude to within one degree, but it required clear weather and took 2 to 3 hours to calculate.

     The English Board of Longitude hated the idea of giving the prize to a lowly clockmaker, a mere "mechanic". They wanted to award the money to an astronomer, so they kept changing the rules to keep John Hamilton from winning. However, everyone knew that Hamilton 's clocks were wonderful. Eventually, he appealed to King George, who finally paid him most of the prize money, just three years before Hamilton died.

     I note that the two men who were most important in solving the longitude problem, after thousands of years, led lives of struggle and poverty, and while they were eventually honored, they benefited little from their great contributions.

Time versus Longitude, a Table

     Below is a table showing the equivalence between time, longitude, and distance. For easy, round numbers we will use 24,000 miles for the circumference of the earth.

Time                                         Longitude                        Distance
(in hours and minutes)             (In degrees)                      (In miles and feet)

One hour             =>                  15 degrees          =>         1,000 miles

4 minutes             =>                 1 degree               =>        66.7 miles

1 minute              =>                  1/4th degree         =>      16.6 miles
                                                (15 arc-minutes)

4 seconds             =>                 1 arc-minute          =>      1.11 miles
                                               (1/60th of a degree)

1 second               =>               .25 arc minutes        =>       .28 miles
                                               (15 arc-seconds)                 (1,465 feet)

    So, if a navigator or ship's captain knows the time at Greenwich , from a Sea Going Clock, plus the local time, to within 1 minute, then he or she knows their location within about 16 miles. Good enough to find a small island or arrive safely at a port.

    Or, our sailor can use the Lunar Distance Method. In fact, both methods were used. The lunar distance method was in wide use until about 1850. Accurate sea going clocks are still used, supplemented by satellite GPS.

Three amazing instruments used to solve for longitude, plus Additional Information for the Advanced (or Curious) Student:

The wall mural quadrant, or transit instrument:   This is the telescope used by astronomers, such as John Flamsteed, to map the sky. It only moves up and down, or north - south. Draw a line on the sky from the south celestial pole to the north celestial pole, thru the very top of the sky -- the zenith -- and this is called the celestial meridian, the line covered by a mural quadrant.

     Watch the passing of the stars across this meridian, record the time of each passing, and you can make an accurate map of the sky. We need the up-down (north-south) plus the side-to-side (east-west) of a star to place it on our map.

     The up-down is easy. Just place the star on the cross-hairs in the eyepiece, and read the number off the engraved scale on your mural quadrant and you have the north - south position of the star in the sky. On earth we call this north - south position latitude;  in the sky it is called declination.

     The east- west position is harder, and requires an accurate clock, called an astronomical regulator. Here is how it works.

     Our astronomer watches each star go by, and notes the exact time when it crosses the meridian, the "middle" of the sky, the highest that the star will rise as it slowly crosses the sky from east to west. We can put this into numbers if we remember that the entire sphere of the sky is 360 degrees around, the earth goes all the way around in 24 hours, so 1 hour = 15 degrees; and 4 minutes = 1 degree, just like the navigation table, above.

     If we mark one special line in the sky as zero "longitude", then we can find out how far east a star is, in the sky, by timing when it passes directly over Greenwich .  If the star pases over one hour later than the zero line, then it is 15 degrees east of zero. If the star crosses exactly over Greenwich 4 minutes after the zero line, then it is 1 degree east.

     Astronomers call this one hour right ascension (analogous to longitude), and 4 minutes right ascension, because the stars move toward the right as they rise or ascend, for an observer facing south.

     Timing the stars passing overhead lets us map the sky. In 150 B.C., Greek astronomer Hipparchus mapped 1,000 stars, to an accuracy of just about 1/3rd of a degree. Pretty good. Nearly 2,000 years later, Flamsteed mapped 4,000 stars to an accuracy of 1/360th of a degree, or 10 arc-seconds, about 100 times better than Hipparchus.

     The Hipparchus satellite (named after you-know-who), mapped 120,000 stars to an accuracy of .001 arc-seconds, ten thousand times better than Flamsteed. And next, in 2012, the Gaia satellite will map 1 billion stars to an accuracy of about .00001 arc-seconds, or about 1 million times more accurately than Flamsteed. So it goes.

  The Double Mirror Reflecting  Quadrant was the next great instrument developed to solve the longitude problem. Measuring celestial angels accurately was impossible until the invention of a new type of sextant in 1731. A navigator needs to know the precise angel between the Moon and the Sun, or between the Moon and stars, or the distance of the Sun above the horizon.

     Fine sextants were available, for use on land, but (just like the fine clocks) they were useless when on a ship, swaying in the waves. Try to measure the distance from the Moon to a star, or the Sun to the horizon.  The navigator could point their sights directly at the Moon, but as they moved to sight a star, the ship would also move, spoiling their reading.

    In 1731 John Hadley in England and Thomas Godfrey in America both proposed a double mirror sextant which would sight two objects at the same time, superimposing one image over the other, and give a perfect reading of their angular separation.  This allowed sailors to measure the distance of the Sun to the Moon, or above the horizon, even from a pitching boat, and allowed use of the Lunar Distance Method for telling time, as described above.

John Hamilton's sea going clocks were the final great development of the three instruments used to solve for longitude. Unlike anyone before him, Hamilton built his beautiful clocks to be accurate at sea, regardless of the pitching and yawing of the boat, and despite changes in temperature.

     His amazing clocks used many innovations, including double pendulums, nearly frictionless bearings that used no lubrication, and the world's first bi-metal strips, to compensate for temperature changes.

     They still run perfectly today, and are all on display, under heavy guard, at the Royal Greenwich Observatory.

A Final Note, for the seriously curious or the very geeky:
1 degree of longitude decreases in size (length) as we move away from the equator. This happens because the distance -- traveling due east or west -- around the world gets less as you move north or south away from the equator.

    The distance around the world at zero degrees latitude is about 24,900 miles. If you travel due east or west from Minneapolis , you would only travel about 17,675 miles to go around the world. So, at 45 degrees north latitude, one degree of longitude equals 17,675 divided by 360 = 49 miles. At the north pole, all the longitude lines converge, and one degree of longitude is zero miles, no distance at all.

Michael Kauper, April 2008
Minnesota Astronomical Society

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