## Saturday, August 03, 2013

### Longitude

I coulda been rich if I had thought of this 250 years ago.

There once was the problem of measuring longitude which attracted the attention of the greedy in 1714 when Parliament promised a huge prize.  In 1714 they could tell their latitude by seeing how high the sun got at high noon (a minor cost was navigators all ended up blind in one eye from looking at the sun).  They couldn't tell their longitude by measuring when noon occured and contrasting it with the hour of noon at say Greenwich, because they didn't know what time it was.

The problem was "solved" (to the satisfaction of George III who personally intervened) by John Harrison who made clocks which kept good time for months at sea.  I use scare quotes, because when I say "clocks" I really mean "3 clocks" and a fourth would have cost a fortune to make (he worked on each for years).  The real practical solution was to figure out which stars are blocked by the moon as a function of time (and latitude ouch).  This produced a method based on ink on paper star charts which could be reproduced at low cost.

Once there were two huge clock problems.  The first is that clocks used pendulums so they didn't work on rocking  boats.  This was solved by Huygens et all who invented the main-spring (a coiled spring which oscilates horizontally and isn't much affected by gravity -- the way watches worked when I was a boy).  This lead to the second  problem which is that the period of the spring depended on the termperature.

Harrison had solved this problem for a pendulum clock.  He made a zig-zag pĂ¨ndulum of copper and zinc one horizontal and one diagonal.  One expands more with temperature.  The height of a zig (or zag) depends on (diagonal squared minus horizontal squared (citation of Pythagoros needed).  The derivative with respect to temperature is 2(horizontal length)^2(expansion constant of the horizontal) - 2(diagonal lenght)^2(expansion constant of the diagonal).  With the right angle of the diagonal that would be zero at room temperature.

This trick is much more useful than the pendulum clock which isn't affected (to first order) by temperature.  It is the way thermostats work (well again in my childhood -- I have no idea how they work now).  If you have zink and copper strips stuck together at one end then the direction the thing points depends on the temperature.  Harrison accidentally invented a new thermometer. More usefully that direction could determine whether, say, the copper touched another conductor and completed a circuit.  Back in the 18th century, it might be necessary to have a person look and turn a flame off and on.

But you see that's it.  The problem can be solved by keeping the clock a constant temperature.  They didn't have refrigerators back then so the temperature would have to be fairly hot (over 100 F if one wanted to navigate the tropics).  Now this could probably semi work with a constant flame which the copper zink thingy moved closer to and further from the clock (with adjustment by hand occasionally).

Now this approach would require a person basically to work full time as time keeper (ok 2 or 3 people in shifts).  but it was better than waiting a decade for Harrison to make another temperature invariant clock.

Also, by the way, they could have made decent maps. The problem always was keeping time when at sea, that's when they wanted to know their longitude.  On land, they already had a good clock based on the work of Galileo. They could tell what time it was in Greenwhich by looking when moons of Jupiter appeared from behind Jupiter.  They couldn't keep a telescope pointed at Jupiter when at Sea, but they could have used this to find out where they were when they landed.  They didn't.  Old maps don't look at all like the territory, because they often had longitude off by several degrees (so hundreds of miles).