Who was Eratosthenes?

Eratosthenes was born in Cyrene in the year 276 BC; he was educated in Athens and later called to Alexandria (Egypt) by the king Ptolemaios III Evergetes to be the director of the great library of Alexandria. He received the best education available at the time and being the chief librarian of the greatest library of antiquity he had direct access to the most prestigious thinkers and to the results of exciting scientific research. He became a first-rate geographer and excelled in many fields including mathematics, philosophy, astronomy and the letters.

The Shape and Size of the Earth

From the time of Pythagoras (570-496 BC) it has been considered that the Earth has spherical shape, however its size was not well known at that time except for assuming that it was "very big"; later on other philosophers came up with a numeric value of 400,000 stadia for the size of the Earth, but it is not clear how that value was computed and to make matters worse historians have a hard time pining down the conversion from those strange units called stadia into something we can understand, like meters. One of Eratosthenesí main achievements was precisely to establish the mathematical geography of the spherical Earth (this work by the way is what makes possible to construct accurate maps); another outstanding achievement was to measure the circumference of the Earth with a precision of 1%. Caveat: because of the uncertainty in the length of the stadium historians argue endlessly about the actual accuracy achieved by Eratosthenes, but historians aside Eratosthenes was a man of genius and his method to determine the size of the Earth is brilliant.

Eratosthenes measured the circumference of the Earth putting a stick on the ground in Alexandria and measuring the length of the shadow when the Sun reached the point of highest elevation (which is near noon). Eratosthenes knew that at exactly this time the Sun was directly overhead (no shadow projected on the floor) in a place (Syene) 5,000 stadia directly south of Alexandria. Using elementary geometry and the results of measuring the shadow in Alexandria he determined that the difference in latitude between Alexandria and Syene was 7.2 degrees. Now, since 7.2 degrees is a fraction of 1/50 of the full circle (which is 360 degrees) then it follows that the length of the circumference is 50 times the Alexandria-Syene distance = 50 x 5,000 stadia = 250,000 stadia. How did Eratosthenes measure the Alexandria-Syene distance? Again, historians argue about this, but some think that Eratosthenes used the standard technique of the time to measure long distances consisting of hiring a team of bematistes (surveyors trained to walk with equal steps and to count them).

There is a deep well (the "well of Eratosthenes") in the Elephantine island near Aswan (Egypt) that exhibits the amazing property that in the day of summer solstice the sun at noon lights up the well right down to the water and cast no shadow on the walls. In other words, the sunís rays enter the well perfectly parallel to the walls of the well. This is an important fact that facilitated the (already straightforward) geometric computations performed by Eratosthenes. The map (below) shows a Google Earth view of the places in Egypt where the measurements were made (Aswan and Alexandria).


Whatís the deal with the stadia?

Stadia, stadium, stade refer to the same unit of length. Stadia is the plural for the word in Latin for stadium which has been anglicized as stade. The Greek stadium was not a fix quantity. Military, merchants, builders, and athletes used different versions of stadia and to make things worse neighboring cities also have their own standards that differ from each other. The historical record indicates that Eratosthenes result was 252,000 stadia but things got a little messy few years later when a new measurement was made by a philosopher called Posidonius which reported the circumference to be 180,000 stadia. How can such a big difference be explained? The difference of course is not due to measurement errors (those ancient Greek guys were very smart and careful with their measurements). The problem here again is due to the differences in the length of the stadia from place to place. According to a study by Aubrey Diller, there were two stadia in common use: in one, a stadium converts to 198.4 meters in the other to 148.8 meters. The latter (148.8 meters) being the most probable value that applies to Eratosthenesí measurement implies an error of 6% for his measurement. Not bad! On the other hand, if one applies the conversion factor of 198.4 meters to the 180,000 stadia of Posidonius the difference with Eratosthenesí is not that large. Bottom line is that Posidonius used the larger stadia and Eratosthenes used the shorter stadia.

At this point the story of the stadia brings us to a fascinating chapter in world history. It turns out that Ptolemy, who wrote the most popular book on geography up to the time of Columbus, reported the size of the Earth as 180,000 stadia (instead of 252,000 stadia). This created a lot of confusion, so much so that it deceived Columbus into underestimating the distance which one would have to sail westward to reach India! In other words, if Ptolemy had quoted the size of the Earth as 252,000 stadia probably the King and Queen of Spain would have not been persuaded as to the viability of Columbus plan and the discovery of America would have to wait for many more years!!!

The sources that I used for this short note on Eratosthenes are:
1) Sarton, G., A History of Science, Vol. 2, "Geography and chronology in the third century -- Eratosthenes of Cyrene", Chapter 6, p. 99, Harvard University Press, Cambridge Massachusets, 1959
2) Aubrey Diller, "The Ancient Measurements of the Earth", ISIS, 40, pp. 6-9 (1949)
3) Dutka, J., "Eratosthenesí Measurement of the Earth Reconsidered", Archive for History of Exact Sciences, Volume 46, No. 1, March 1993