Module

Where am I?  (the problem of determining longitude at sea)

Topics:  geography, trigonometry, geometry, time

For millennia, sailors have taken to sea and lost sight of land.  How did they know where they were?  Good weather, and experience, took them to their destinations if they were lucky because they couldn't know where they were with certainty.  If they weren't so lucky, they were lost for a long time or forever, or arrived to a shore far from their destination.

In the fifteenth century, European sailors ventured far beyond their familiar Mediterranean Sea into the Atlantic Ocean and from there to the Indian and Pacific Oceans and in the course of their explorations opened our globe to the European powers.  Shipwrecks were frequent, often caused by sailing off course.  So the problem of reckoning a ship's position at sea had important implications for exploration and trade.

A fascinating book by Dava Sobel (Longitude) tells the story of the efforts to solve the problem of accurately determining the location of a ship at sea, and of "a lone genius who solved the greatest scientific problem of his time";  her book has inspired this module.  We highly recommend it; it reads like a great novel.

Some geography and geometry

Look at, or imagine, a globe representing the earth.  Twenty four lines (meridians) run from pole to pole and tell us how far east or west of the Greenwich meridian a point lies (these 'longitude' lines are drawn every 15o relative to the center of the earth).  The equator separates the northern and southern hemispheres, and five similar lines (parallels: 'latitude' lines) draw smaller and smaller circles in the direction of both poles. These are also drawn every 15o relative to the center of the earth.  (Usually the tropics are also drawn, but that's for another day.)  We give our location on the planet by these angles, for example, Chicago is 42oN and 88 oW.  Maps going as far back as those by Ptolemy show lines of latitude and longitude.

Latitude at sea

Navigators determined their latitude (position north or south of the equator) by using the length of the day, or sighting the sun or the stars above the horizon.  Instruments such as the cross-staff or the backstaff were used to measure the altitude of the sun or a star at noon or at midnight.  Observation of the sun could lead, in time, to blindness in one eye!

Longitude at sea

Longitude is a different story:  no direct astronomical observation gives the east / west position easily.  Sailors used 'dead reckoning' (estimating the ship's speed by throwing something overboard, trying to account for currents, and regularly charting the estimated distances traveled), but the errors were large and sometimes catastrophic.

Galileo devised a method, which turned out to be impractical at sea, to determine longitude by observing of the eclipses of the moons of Jupiter. Three quarters of a century later (1675-76), the Royal Observatory was established in Greenwich, England to compile astronomical observations that would allow mariners to use the stars to locate their longitude.  And thirty nine years after that the British Parliament created a prize equal to a 'king's ransom' for a "Practicable and Useful" means of determining longitude:  the Longitude Act of 1714.

Dava Sobel tells the story of this prize, and how it was finally won by a clock maker.  Why a clock?

Let's do a mental experiment.  Suppose you live in Chicago and you use a sundial (another link ) to set your watch.  Now you drive about 770 miles west to somewhere near Scottsbluff, Nebraska with your sundial, without resetting your watch.  You check your sundial in Scottsbluff and observe that the sundial time is about one hour earlier than your watch.  As long as you travel directly east or west of Chicago, a 770 mile trip will change the local time by about an hour relative to Chicago's solar time.  So if you know the time difference, you know the distance!

The earth rotates completely in 24 hours, so it rotates 15o in an hour.  Thus, noon (or any other time you choose) occurs with an hour difference at points on meridians 15o apart.  An angle of 30o would give a two hour difference, and two points 45o apart would have a three hour difference.  So the time difference between two points at the same latitude is proportional to the angle between their meridians.  Now, how far apart are two 15o meridians?  At the pole:  0 miles;  at the equator:  a bit more than 1030 miles;  and everything in between!

So here is the problem:  if my latitude is given, what is the distance between two meridians 15o apart on that parallel (that is, what is the distance between two points with one hour difference in solar time)?

Problems

1) Take the challenge:  can you figure out how to calculate the distance between parallels given a latitude?  Click here.

The radius of the earth (which we assume spherical, but really isn't) is approximately 3950 miles.

2) Verify the calculations given above for the distance between to 15o meridians (at the equator: 0o;  Chicago: 42 o;  North Pole : 90o ).

3)  The prize promised by the Longitude Act had the following monetary rewards:

20,000 pounds sterling for a method that determined longitude to an accuracy of half a degree;
15,000 pounds for 2/3 of a degree; and
10,000 pounds for a method accurate to within one degree.

A time was not specified, but the technique was to be tested between Great Britain and a port in the 'West Indies'.  Assume a six week voyage, and calculate how many seconds per day a clock could lose (or gain, assume loss or gain was equal for the whole trip) and still satisfy the prize conditions for each of the prizes.

References

Longitude by Dava Sobel,  Walker and Co.

The Illustrated Longitude by Dava Sobel and William J.H. Andrewes. Walker and Co.

Answers
 

Link to NASA Project Home Page

David Rutschman and A.Lise Jensen, 2002