Eratosthenes

(276-196 B.C.)

Compiled and illustrated by D. Newsome

 

-Nicknamed ÒBetaÓ [second to whom, I donÕt knowÉ perhaps Aristotle???

-He is generally described as having been the Òhead of the library at AlexandriaÓ  What it means to be a head librarian in an ancient library is hard to tell.  I imagine he was sort of a resident researcher. 

His claim to eternal fameÉ     He calculated the circumference of the earth.

 

How he did itÉ

   Idealized version firstÉ.

 

1. On the summer solstice, at noon, the apogee of the sunÕs motion in relation to the horizon, he observed that there was no shadow of a perfectly vertical pole at Syene, which I am locating at 23.9¼ N in order to make the math as clean as possible.

 

2. At noon, a vertical pole at Alexandria, located by me for easy math at 31.1¼N, casts a shadow amounting to 7.2¼. 

 

3. Recall EuclidÉ

 

4. Eratosthenes knew that Syene was 5000 stades from Alexandria.

 

5. Do the mathÉ

The math gives you 25,000 stades around.  WeÕll look at stades in a minute.

EratosthenesÕ assumptions and measurements: 

 

1) Distance from Alexandria to Syene was 5,000 stades.  It seems that he is assuming that Syene and Alexandria are on the same meridian (constant longitude), which they are not. This doesnÕt have to matter so long as it is factored into the final answer.  Judging from this GoogleEarth experiment, the latitudinal distance is 96% of the actual distance.  So we may want to keep that in mind when we get final results.

    

 

2) Sun rays are, for all practical purposes, parallel. The assumption here is that the sun is really, really far away.

 

3) A line intersecting parallel lines has equal alternate angles.  [Euclid]

 

4) The shadow cast was 1/50 of a circle, or 7.2¼

Now here are some nitty gritty details that make the calculation a bit messier.

 

Eratosthenes noticed that at midday, the sun at its apogee,  of the solstice at Syene (almost at the Tropic of Cancer, from bet. 24¼07ÕN to about 23¼52ÕN,   One source claims 23¼26Õ) (actually 60 km north and not quite on the same meridian) a vertical rod (or well) casts no shadow. Tropic of Cancer is 23¼27ÕN.

 

In Alexandria (bet 31¼05ÕN and 31¼17ÕN) (to the north of Syene (modern Aswan)) on the same day a vertical rod always cast a shadow.  The minimum shadow cast (when the sun at its apogee) was determined to be 1/50 of a circle, 7.2¼.  On summer solstice the sun will be directly overhead.

 

The difference range is from 7¼10Õ to  7¼25Õ.  That is 7.16667¼ to 7.41667¼, which yields 1/50.23 to 1/48.21. 

The result is that he figured the circumference to be 250,000 stades.

 

The next big questionÉ

 

            What is a stade? or stadia? Éor stadium?  I have found many answers to this question.  ItÕs a bit like asking what the distance to right field is in baseball.

 

1 mile = 7.5 stades approx.

1 stade = 157m =  0.1333 miles approx. = 515 ft.

1 stade = 185 meters (607 feet) [this one gets lots of votes]

      

       also have seen the stade to be

1 stade = 600 = 183m

and

1 stade = 500 ft. = 152m

      

       also saw

1 stade = 528 ft. = 161m, which makes 10 stadia = 5280 ft = 1 mile.  Thus 5000 stadia = 500 miles.

 

Étrack at Olympia measures 210 yards (630 ft.)

 

from N.M.U web site

1 stade=177.6 m (Attic-Roman)=185 m (Std. Roman)=192 m (Olympic)=197.3 m (Philetaeran).

 

So, the stade estimates range from 152m (500ft.) to 197.3m  (647ft.) É thatÕs a 30% increase from 152.

 

Modern circumference numbers 24,901.55 miles (40,075.16 kilometers).

 

If a stade is the low estimateÉ that gives a result of               38,000km  (5.2% low error)

If a stade is the high estimateÉ that gives a result of  49,325km  (23% high error)

 

If we want a stade estimate that cheats to the modern circum,

            use the stade = 160.3m = 526ft.  (0% error)

 

Earlier estimates for the circumference of the earth:

            Eudoxus of Cnidus (d. ca. 355 B.C.)              400,000 stades

            Dicaearchus (student of Aristotle)                   300,000 stades

            Aristarchus of Samos (d. 230 B.C.)                300,000 stades

 

A later librarian of Alexandria, Ptolemy, É.His estimate was 180,000 stadia (about 28,800 km.)  (about 28% low)

 

Combined with this was a popular belief in the late 15th c. that Asia was wider than it really is due to trying to incorporate Marco PoloÕs descriptions of JapanÉ(Wilford, p. 61) and É

ÉColumbus was expecting Asia to be a quick trip.

 

Many reasons for ColumbusÕs voyage usually stated as ÒGod, gold, and glory.Ó : economic (spices), exploration, conquest, imperial ambitionsÉ

The ÒGodÓ reason is more interesting that it at first seems.  Why would God be a reason? He wrote to his benefactors Ferdinand and Isabella that he wanted to find a quick root to the Orient (East, China, India, etc.) in order to facilitate the conversion of those non-Christians into good Catholic people.  WhatÕs the rush?  Time was running out.  Columbus believed that the end-of-days was going to happen in 155 years. The world was going to be destroyed and saved all at the same time.  But in order for this to happen, all on the earth had to be converted to Christianity.  Columbus was on a mission of God, missionizing the peoples of the earth in the Òlast days.Ó   He called himself ÒChristo-ferens.Ó [Christ-bearing]

Where did he get this figure, 155 years?  The end was going to come in the 1650s.

He got this number in the same place he got his circumference of the earth figure, in a book by Pierre dÕAilly (1350-1420).  It was a book on geography and astrology and the end of the world.  DÕAilly later revised his date to 1789 (the French Revolution coincidentally). 

 

Cheating historians of science from the Web:

Circumference = 50 x 5000 stades = 250,000 stades


250,000 stades x 157 meters/stade = 39,250 kilometers

The modern value: 40,070 kilometers.

This is a difference of only 2%!!

Ptolemy made a similar geometric estimate based on stellar (rather than solar) measurements made earlier by Marinus of Tyre (by way of Posidonius). This value gave a circumference of 28,800 kilometers. This is 72% of the correct value.

 

By about 300AD, the idea of a Flat Earth was revived:

Early Christian rejection of the "pagan absurdity" of a spherical earth.

This view was held sporadically until about 1300 AD.

By 1300, the works of Ptolemy and others arrived in Europe by way of Islamic Spain, and fully restored the Spherical Earth to respectability.

Contrary to popular myth, few educated people after about 300 BC doubted that the Earth was a sphere. While a few early Christian thinkers did try to reject the idea, there is nothing in Christian beliefs that dictates a Flat Earth, in fact is says nothing at all on the matter.

 

Eratosthenes' work was lost, except for a description of his method in an obscure source (Cleomedes Meteora, written around 370 AD).  (I think Martianus Cappella also mentions itÉ.)

Ptolemy's estimate survived in his influential writings on geography. An interesting consequence of this transmission was:

It makes the eastern tip of Asia closer to the western tip of Europe than it would be otherwise.

This fact (along with some other finagling) convinced Columbus that he might be able to reach Japan by sailing West from the Canaries.

Unlike many others of his time, however, Columbus not only argued for a smaller Earth, he also convinced the Spanish government to provide him the means to put his claims to the test.