In our look at the beginnings of our solar system we see just how involved we are in circles and things which are circular (I have left out ellipses for a good reason) We can turn on the T.V. and view the Earth from the view point of the astronauts or we can take a trip to an IMAX theater (Maritime Center in Norwalk, Connecticut), and see the Earth in all of its beauty and magnitude. The ancients had no such tools (neither space ships nor satellites). How then did they come to know the planet‘s dimensions.
There was a man who was in the right place at the right time. His name was Eratosthenes (276-194 B.C.) he was an excellent mathematician and he was in charge of the library, in Alexandria, Egypt. At that time this library was the best in the world. At his finger tips were the works of the great minds of the world to that day.
One of the great problems of the day was how large is the Earth, and how do we set about to measure it? Several men were working on this problem. The results of their labors had been published. Eratosthenes was not satisfied with their “results” so he set about to set the record straight. His experiment was a simple one based on a good understanding of basic geometric principles. Erastosthenes’ approached the problem this way.
In Egypt there were two well known cities which lay pretty much on the same meridian (imaginary line of longitude) they were Alexandria and Syene (today known as Aswan). The road between these two cities had been well traveled and well measured. These cities were 5000 stadia apart. It was also common knowledge that at certain day at noon, in Syene, when the sun was at its solstice (directly over head), a very deep well was filled with the reflection of the sun. At the same time the Sun was overhead at Syene. Measurements were taken on a gnomon (a column erected perpendicular to the earth) in Alexandria. The angle which the Sun made with the column as well as the lengths of the column and its shadow were recorded.
The Sun’s rays were assumed to be traveling in parallel lines so all he had to have was an accurate measure of the angle which was formed on the gnomon at Alexandria. It is here that we must speculate. One thought is that some primitive trigonometry was in play (remember if he was the librarian he was privy to all the new ideas of the day) and he could have quickly computed the angle of depression, and its complement which the Sun made on the pole. Sine Alpha = opposite side/hypotenuse would have done the trick. Or the angle could have been measured with an accurate protractor used in combination with a plumb bob. This is not as far fetched as it first seems. I recall reading about the accuracy of Tycho Brahe’s (1546Ð1601) quadrant (half a protractor with sights) which had a radius of 19 feet, not an item to be carried in one‘s pocket. On this scale one degree was 4 inches on the circumference. One millimeter corresponded to 1/100 of a degree. While a good many years separated these men, the idea of constructing large model was an “in thing”.
I suggest that Eratosthenes could have made an accurate large scale drawing on paper and taken an accurate reading from it. I like to tell my classes a story about the great astronomer, Johannes Kepler (1571Ð1630). In Johannes’s day the scientific community had assumed that the Earth, and all of the other planets, followed circular paths in their flight around the Sun. This was a problem for the data which had been collected did not support this theory. Kepler spent years working on this incongruity and got no where with the problem until one day he decided to plot the points (given by the data) onto a floor, and it was not until he stepped back to view his work that he saw that the Earth was traveling in an elliptical orbit around the Sun. And the Sun was located at one of the focal points of this ellipse. An accurately drawn picture and great perseverance gave Johannes the incentive he needed to continue. Today every physics book contains this three laws on the planets.
In the “accurately” drawn sketch which follows we see the Earth as it is struck by parallel light rays from the Sun. The well at Syene and the vertical pole and its shadow at Alexandria are shown, along with the angles (alternate interior) which are located at the Earth’s center and at the pole. It should be noted that this angle has been listed as both 7 and 1/2 degrees and as 7 and 1/5 degrees. In my calculations I have chosen to use the latter. A stadia is listed as 516.73 feet which I will round off to 517 feet.
When we set up the following proportion we will get:
Total number of degrees in a circle = Circumference of Earth
Angle of shadow of rod at Alexandria Distance between Alexandria and
Syene
360° = X(Circumference of Earth)
7.2° 5000 stadia
250,000 stadia = X
250,000 stadia x 517 feet/stadia = 12950000 feet
12950000 feet/ 5280 feet/mile= 24,479.16667 miles
By anyone’s standards this is a great calculation. The sketch which follows is the geometric representation of his work.
Figure available in printed form