geodlab.htm

Villanova University, Department of Mathematical Sciences

Cartographiometry (MAT 1210/ GEO 1700)

Laboratory Exercise: Historical geodesy

(geodlab.htm)

In this lab, you will determine, mainly by direct measurement, the latitude and longitude of a particular place on earth.

You will need the following tools (at least):

these instructions;
a table of values of trigonometric functions;
an analemma;
a straight object of known length, such as a meter stick, to use as a gnomon;
a clock or watch set to Greenwich Mean Time (GMT);
a tape measure or other device for measuring lengths of things;
a protractor;
a piece of string (long enough to reach from the top of your gnomon to the tip of its shadow);
some chalk or other marker;
a calculator;
possibly a friend or two.

Now carry out the following steps. Do them in the order given. Be careful and patient -- it will improve your accuracy, understanding, and enjoyment. Have fun! Let your friends help and learn with you!

Place a gnomon vertically into the ground. An existing pole or tree may be used if you can reach the top of it and know its height. You must know how high the top of the gnomon is above the ground!
Set a reliable clock or watch to standard Greenwich Mean Time (GMT). (At Villanova, for instance, Greenwich Mean Time is just five hours ahead of our local official standard time; during daylight savings time, standard GMT is only four hours ahead of Villanova's official Daylight Savings Time.)
Calculate the time of your location's local solar noon by recording the time on the GMT watch at which the shadow cast by your gnomon is the shortest. Also, record the length of this shortest shadow. (For instance, mark the tip of the shadow with chalk and measure the distance of the mark from the base of the gnomon.)
Measure the angle between the gnomon and the sun's rays at local solar noon, as shown in the figure, by holding a string between the top of the gnomon and the tip of the gnomon's shortest shadow and using a protractor.
As a check on the angle measurement you made in the last step, you can compute the angle between the gnomon and the sun's rays at local solar noon like so. Let h be the height of the gnomon and s the length of its shadow at local solar noon. Then the tangent of the angle t° is given by tan(t° ) = s/h. Compute the ratio s/h, using a calculator if necessary, and then determine the value of t° using a table of values of the trigonometric functions.
Use an analemma to determine the latitude u° at which the sun is directly overhead for the day on which you are taking your measurements. The angle t° you computed in steps 4 and 5 is the number of degrees of latitude between your location and the latitude u° where the sun is directly overhead. Determine the latitude of your location. (Don't forget to take North/South into account.)
The analemma also records the difference between local solar time and local mean time for each day of the year. Use the analemma to determine this difference for the day of your measurements. (In late May and early June, local solar noon occurs shortly before local mean noon. In late June and July, local solar noon occurs shortly after local mean noon.) Now use this difference, together with the time at which your gnomon's shadow was shortest (this is the local solar noon you computed in step 3), to compute the correct Greenwich Mean Time of your location's local mean noon.

Finally, determine the longitude of your location}by computing the difference between 12:00 GMT and the Greenwich Mean Time of your location's local mean noon (which you just computed in step 7). Convert this time difference to a longitude angle by the relations shown in the table. Don't forget to include East/West.

Time unit	Longitude angle
1 hour of time	15° of longitude
1 minute of time	15' of longitude ( ¼ of a degree )

For extra credit, repeat this entire procedure at a second (significantly different) location. Have fun!