Eratosthenes

D Vautier
1/2023

Eratosthenes was one of several early Greek and Egyptian scientists (In this discussion I call him Era for convenience).  The guy lived in Alexandria Egypt and died in 194 BCE which makes him a close contemporary of the famous Archimedes of Syracuse (287 – 212 BCE) who lived just across the way and did all kinds of cool stuff.  I’m not sure how much the two geniuses communicated but there were so many things to invent and discover in those days that they probably didn’t have a lot of time to exchange notes.

One thing these two brilliant men seemed to share was the teachings of Aristotle and Plato who happen to live some 100 years earlier and both of these two earlier scientists believed that the earth was round.  So, based on this belief, Era began thinking about how to determine just how big the earth was.

That’s when another famous guy entered the picture, Euclid who is popularly known as the father of geometry.  Euclid was a student of the Plato school and a lot of his geometry was in use when Era was trying to solve his Earth size problem.

The city of Aswan Egypt was a little more than 500 miles south of Alexandria and was very close to the tropic of cancer.  This means that once a year on the summer solstice, (June 20 or 21), the sun passes through the ecliptic and is directly overhead.  This was pretty common knowledge because somebody had erected a perpendicular pole and it soon became known that on a certain day every year, (June 20 or 21)  the pole’s shadow would disappear.  Now the Egyptians were quite aware of seasonal changes and kept close track of dates, in fact they were the first civilization to develop a practical calendar.

So Era decided to erect another perpendicular pole at Alexandria where he lived and make note of the angle on the summer solstice.  He got an angle of about 7.2 degrees.  Degrees of arc did not exist then because the Babylonians hadn’t gotten around to popularizing their 360 degree system.  But Era was able to calculate the fractional size of the angle at 1/50th of a circle and that worked out fine for his purposes.

Still the biggest problem was finding an accurate distance from Alexandria to Aswan.  Both cities are about on the same longitude.  They are also on the Nile River which is far from straight and changes its path every year much like the Mississippi does.  There were professional walkers available that were employed to settle boundary disputes but to accurately calculate long distances over very difficult terrain was next to impossible.  Era probably spent a lot of time consulting many merchants and sailors and seamen who had frequently navigated those Nile waterways.  He came up with an estimation of the distance between these two cities at 5000 stades which was a nice round figure and also a pretty good guess.  His choice of a nice round figure had to do with the numbering system then in use.

So what is a stade?  There were at least three types of stadia in use during those times.  Some notable historians claim that Era probably used the 185 meter Roman Stade which would bring his earth circumference to 250,000 stadia, an incomprehensible size.

Era added an extra 2000 stadia bringing his earth circumference up to 252,000 stadia.  The reason for this extra 2000 may have something to do with the numeric counting system he used.  The Acrophonic system was much like the Roman System which uses symbols to represent groups of numbers.  So I=1, V=5, X=10, C=100 in the Roman system.  These number groups can be converted, added together, and multiplied (successive addition).  Although tedious, these systems can still do basic arithmetic operations as I explain in this article.  So 252,000 seems a lot easier to work with than 250,000 because it has more primes and the number groups can be moved around easier.  

This result means Era was off by an astonishingly mere +0.8%.  But if we use some of the other stades in existence at the time the error gets larger or rather greater, perhaps as bad as +17%.  In any case he over-estimated the size of the earth.

Era’s huge achievement was pretty much ignored at the time as totally preposterous. It was even ignored for the next 17 hundred years. Nobody could possibly imagine that the earth was so big.  Era must have made a huge mistake somewhere.  Columbus sailed west for china.  He didn’t believe earth was that big either.  Boy was he surprised because Era totally nailed it.  The world gradually awoke to the idea that it was a big big planet.