Rung 1: The size of the Earth

Question 1: What lead Eratosthenes and other Greek Astronomers to believe each of these statements?

Question 2: What is the signifigance of assuming the Sun to be very far away?

Eratosthenes knew that at noon on June 21 (the summer solstice - longest day), the Sun would shine into the bottom of a deep vertical pit in the city of Syene, meaning the Sun was at the zenith in the sky at this time. Being in Alexandria, which was located more or less due north of Syene, he could measure the length of a shadow cast by an obelisk at the same time, noon on the summer solstice. This measurement, along with the height of the obelisk, gave him the angle that the Sun appeared south of the zenith in Alexandria, an angle he determined to be 7.2 degrees. The Sun was not as high in the sky in Alexandria as in Syene at the same time of day and the same time of year. This is in fact a measurement of the curvature of the Earth's sphere between Syene and Alexandria, and once distance of this small arc is known, the circumference of the entire Earth is determined. To this end (Astronomers having considerably more political clout in those days!) he ordered some soldiers to march off the distance between Alexandria and Syene, a distance of 5000 stadia (thought today to be equivalent to about 500 miles). Click on the picture to the right to see the geometry of this experiment.

The circumference of the Earth could be found by using the following proportionality:

              Syene to Alexandria          7.2 degrees
            -----------------------    =    ---------
          circumference of the Earth       360 degrees

Question 3: What value for the circumference of the Earth would Eratosthenes have determined from his measurements? (Convert your answer to kilometers.)

Question 4: What value for the radius of the Earth did he determine? (Recall the formula for the circumference of a circle.)

Using a similar triangles construction like the one shown in the above figure, we can directly determine the radius of the Earth. The length of the shadow is to the distance between Syene and Alexandria as the height of the obelisk is to the radius of the Earth. We can write this in equation form as:

            Syene to Alexandria     Length of the Shadow      
            -------------------  =  --------------------
            Radius of the Earth     Height of the Obelisk

Question 5: Using the above equation and your measurements from the sketch, calculate the radius of the Earth in kilometers.

• Continue to the second rung: The size and distance to the Moon
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