Eratosthenes knew that on the summer solstice at local noon in the Ancient Egyptian city of Swenet (known in Greek as Syene, and in the modern day as Aswan) on the Tropic of Cancer, the sun would appear at the zenith, directly overhead. He also knew, from measurement, that in his hometown of Alexandria, the angle of elevation of the Sun would be 1/50 of a full circle (7°12') south of the zenith at the same time. Assuming that Alexandria was due north of Syene he concluded that the distance from Alexandria to Syene must be 1/50 of the total circumference of the Earth. His estimated distance between the cities was 5000 stadia (about 500 geographical miles or 950 km). He rounded the result to a final value of 700 stadia per degree, which implies a circumference of 252,000 stadia. The exact size of the stadion he used is frequently argued. The common Attic stadium was about 185 m, which would imply a circumference of 46,620 km, i.e. 16.3% too large. However, if we assume that Eratosthenes used the "Egyptian stadium"[7] of about 157.5 m, his measurement turns out to be 39,690 km, an error of less than 1%.[8]

Although Eratosthenes' method was well founded, the accuracy of his calculation was inherently limited. The accuracy of Eratosthenes' measurement would have been reduced by the fact that Syene is slightly north of the Tropic of Cancer, is not directly south of Alexandria, and the Sun appears as a disk located at a finite distance from the Earth instead of as a point source of light at an infinite distance. There are other sources of experimental error: the greatest limitation to Eratosthenes' method was that, in antiquity, overland distance measurements were not reliable[*citation needed*], especially for travel along the non-linear Nile which was traveled primarily by boat. Assuming enough difficulties in measurements, the accuracy of Eratosthenes' size of the earth is surprising.[*original research?*]

Eratosthenes' experiment was highly regarded at the time, and his estimate of the Earth’s size was accepted for hundreds of years afterwards. His method was used by Posidonius about 150 years later.[*citation needed*]

**Radius of the Earth**

In this assignment we will investigate one of the first known methods to calculate of the radius of the earth. This method was discovered by Erastothenes of Cyrene (third century B.C.). You may have heard his name in connection with a method for finding prime numbers (the Sieve of Erastothenes). The ancient Greeks believed that the earth was round, and, as we will see, Erastothenes was able to estimate the size of the earth quite well. This knowledge was lost to Europe during the dark ages, when people believed that the earth was flat. This belief lasted up to the voyages of Columbus.

Erastothenes observed that at noon on the day of the summer solstice the sun shone directly down a deep well at Syene (present day Aswan). At the same time in Alexandria, the sun was found to cast a shadow corresponding to an angle of 1/50 of a full circle (about 7 degrees in our current measurement of angles). Erastothenes knew that the distance between Alexandria and Syene was approximately 5000 stades (1 stade is about 1/10 mile). He then estimated the radius of the earth from this information. How did he do this, and what value for the radius did he obtain?

To help you figure out how Erastothenes figured this out, consider how the sun's rays shine on the earth.

Because the sun is so large compared to the size of the earth, the sun's rays are practically parallel to each other as they hit the earth. Draw a portion of a large circle to represent the surface of the earth. Then draw a stick placed at Syene and one at Alexandria, and then draw a line representing how the shadow of the stick is formed. Finally compare the angle made by the shadow at Alexandria with the angle formed by connecting the center of the earth with Alexandria and Syene.

There are a couple of geometric facts that you should find useful in this assignment. First, there is a close relationship between an angle and the corresponding arc length on a circle. This relation states that the fraction of a full circle taken up by an angle is equal to the fraction of the circumference taken up by the corresponding arc length. In terms of an equation using proportions, we have

The following picture can help to understand this formula. For example, an angle of 36 degrees is 1/10 of a full 360 degree angle. The corresponding arc length is 1/10 of the full circumference of the circle. The formula says the fraction of the full 360 degree angle made up by an angle is equal to the fraction of the circumference made up by the arc length corresponding to the angle.

Second, because the circumference of a circle is equal to 2 times pi times the radius, if we know the circumference, we can find the radius, and vice-versa. For the third and final fact, if two parallel lines are crossed by a diagonal line, then the angles marked as a in the picture below are all equal.

For some geographical information, Syene (present day Aswan) and Alexandria are marked in the following map of Egypt.

or thisa360rad_1**Erastothenes' method for determining the radius of the Earth**

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.