This is a technique that was suggested by the Greek Astronomer Erastothenes. He did not have trigonometric functions available to him, but the essential idea is to measure the lengths of shadows at noon at different points, p1 and p2, on a north-south line. Suppose we take take poles of height H which cast shadows of length  L1 and L2 at the two locations p1 and p2. p1 and p2 are separated by a distance D. We assume that the sun is so far away, compared to the radius R of the earth, that the rays of sunlight r1 and r2 are parallel. The zenith angles z1 and z2 can be determined by measuring the shadow lengths L1 and L2 and the height H of the poles. From geometry it can be shown that the angle A subtended by the two points p1 and p2 is related to the zenith angles by

                        A = z2 - z1 .

Also we have the relationship that connects the distance D to the radius R and angle A.

                      D/(2piR) = A/360 .

Where we measure the zenith angles and angle A in degrees. Since D and A can be measured
we can solve this equation for the radius R of the earth.

When we actually attempt to measure a shadow cast by a pole we encounter a fuzzy edge to the shadow. The reason for this is the finite size of the sun, about 1/2 degree in width. There is then an ambiguity in deciding where the shadow ends. Ideally we want to locate the center of the sun. An ingenious technique for defining the shadow was developed by Guo Shoujing
(A.D. 1231-1316). It is essentially an application of the principle of a pin-hole camera. The Imperial Chinese astronmers built a special tower for determining the elevation of the sun at noon. At Dengfeng, in Honan Province, there is a tower about nine meters high. A horizontal bar is suspended at the top of the tower about waist high. The shadow of the bar is measured on a scale at the base of the tower, leveled with water. The shadow of the bar is too diffused to be easily identified. However, Guo Shoujing used a thin metal plate with a hole in it, about 2 mm in diameter. The plate is installed on a carriage which can be slid along a track on the scale. When the center of the sun, the hole in the plate and the horizontal bar are on a line an image of the sun can be seen on the scale with the bar's image splitting the sun's image exactly in half. This is an alternate technique for us to use in our shadow measurements. It will require two people. One to hold the plate with the hole in it in the path of the bar's shadow, and the other to mark the image of the bar on the paper attached to the ground. In practice this is more difficult than the Chinese astronomers had to face. The person holding the pin-hole needs to have a rather steady hand, otherwise the bar's image moves about on the paper on the ground. Nevertheless, it is an interesting procedure to try. It illustrates the ingenuity people had to employ in making precise measurements. Precision measurements reveal long term slow changes in astronomical parameters. A comparison of measurements of the length of the year are in the table below from Hugh Thurston's book, "Early Astronomy",1994, Springer-Verlag, New York. The discussion of the ying fu ( shadow finder ) above comes from Thurston's book.

Length of the tropical year as measured through the ages.

Modern values:
Tropical  year for  2001 ( equinox to equinox ) : 365.2421898 (0.0000012) days
Sidereal year for 2001 (fixed star to fixed star)   : 365.2563634(0.0000012) days