Longitude: How to Navigate using Time and the Sky
Michael Kauper , 2008
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
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
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
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
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
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,
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.
(in hours and
(In miles and feet)
One hour =>
15 degrees =>
=> 66.7 miles
1/4th degree =>
1 arc-minute =>
(1/60th of a degree)
.25 arc minutes =>
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.
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
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
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
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
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