Measure the Planet

November 2nd, 2010

Columbus did not risk falling over the edge of the world, but his first voyage was far more risky. In fact, no one held the world to be flat. Those who dared oppose Columbus relied on a librarian who measured the world with a stick.

Pop quiz: for long have we known that the Earth is not flat? Or in more gen­eral terms: for how long have we as a spe­cies pondered the phys­ical shape of the ground beneath us? And in epi­stem­o­lo­gical terms: when did we real­ize that the shape of the phys­ical world is even a valid question?

…and some people need anger management.

Ships fall­ing off the Earth

My geo­metry stu­dents answered the Flat Earth ques­tion as I thought they would: Colum­bus, 1492, three ships and quasi-Indians. He sails west, doesn’t fall off the edge of the Earth. Accord­ing to legend, Colum­bus set out to find a shorter route to Asia by head­ing west and swinging round the back­side of the planet. Led by intu­ition to believe that the Earth is spher­ical, our brave cap­tain countered the belief that he would drop off the side of, well, everything. His crit­ics warned him off, queen Isa­bella sponsored him, and he hero­ic­ally set foot on the New World, to the score of an awe­some soundtrack. Fair enough, but we need to cor­rect one small detail here: Colum­bus was wrong; his crit­ics were right.

There really was no con­tro­versy regard­ing whether Earth has a back­side to tra­verse, but rather her size. Colum­bus did not stand up against Flat Earthers. He believed the Earth to be hand­ily small, while his oppon­ents warned that he and his crew would never last the long voy­age — excel­lent advice. To pro­ject a Flat Earth belief on 15th cen­tury schol­ars is but an arrog­ant mis­con­cep­tion of a dis­neyesque Dark Age.

Sci­ence.

Lit­er­at­ure has undeservedly ascribed the Cath­olic church with a belief in a Flat Earth and a sup­pres­sion of rational thought. Not so; for example Saint Augustine merely that the South­ern Hemi­sphere was unin­hab­ited. The equator was held as a scorch­ing desert, impossible to cross, and since

  1. all human­kind is uniquely des­cen­ded from Adam
  2. the Equator can­not be crossed
  3. human­kind is present on the north­ern hemisphere

it fol­lows that man­kind was spawned in the north exclus­ively, unable to visit South Side, and thus the South­ern Hemi­sphere is unin­ter­est­ing for trade and taxes. (though mod­ern read­ings of Augustine sug­gest that he did describe the Earth as being “the bot­tom of the uni­verse”, layered beneath water, air and fire. The Colum­bus con­tro­versy regarded the cir­cum­fer­ence of the Earth, which implies that not only was there a rough fig­ure for the size of the planet, but sev­eral con­flict­ing ones. The main­stream, non-Christopher meas­ure­ment was even quite good. Now, the ques­tion: how to meas­ure a planet?

Meas­ur­ing Earth’s Circumference

Enter, stage left, Erastothenes of Alex­an­dria, lib­rar­ian. In his work, he came across a passing ref­er­ence to a city in Egypt — Cyene, modern-day Aswan — where on a cer­tain day of the year, the sun shines straight into the bot­tom of a well. Being a rather sharp fel­low, he knew that meant the sun was at zenith at that par­tic­u­lar day. This enabled him to per­form an amaz­ing geo­met­rical stunt: estim­at­ing the size of the Earth.

Cranks turning 180, 90 and 23 degrees.

Half, quarter and one-seventh turns.

Turn­ing a crank half a turn makes the handle travel half the dis­tance around the full circle. A quarter turn makes the crank travel a quarter of the dis­tance. One sev­enth of a turn makes it travel one sev­enth of the dis­tance. So, there is a simple rela­tion between a par­tial angle and a par­tial arc length: if you turn the crank a part of the turn, the handle travels just as big a part of the dis­tance. This is true for any circle, for instance the one described by the Earth’s circumference.

Two cities separated by an angle

The dis­tance is known, the angle is not.

There is an angle between the two cit­ies from the Earth’s core, and there is a travel dis­tance between them meas­ured from Aswan to Alex­an­dria. The dis­tance is known, but the angle is not. Look­ing at the Earth as a whole, the situ­ation is the oppos­ite: we do not know the dis­tance around the entire planet, but we know the angle: 360º — just as with any circle. This is an amaz­ing jux­ta­pos­i­tion: we know the “local” dis­tance, but not the “local” angle. We know the “global” angle, but not the “global” dis­tance — just as with the crank handle, or any circle.

If we know how the local angle com­pares to the global angle (360º), we know how the local dis­tance com­pares to the global dis­tance, and vice versa. For Erastothenes: if he could fig­ure out the angle between the cit­ies and how it com­pares to 360º, he could also fig­ure out how the dis­tance between the cit­ies com­pares to the cir­cum­fer­ence of the Earth. If the angle between the cit­ies turned out to be, say, 1/8th of a full circle, then the cir­cum­fer­ence of the Earth must be eight times longer than the ground dis­tance between the cit­ies. He did fig­ure out that city-separating angle, and accord­ing to legend, he did it with a wooden stick.

Let’s start with a couple of assumptions:

  1. Sun­rays are par­al­lell. This is not abso­lutely true, but “true enough”. The sun is at a tre­mend­ous dis­tance, and the dif­fer­ence in sizes oblit­er­ates any angles.
  2. Aswan lies roughly 800 kilo­met­ers due south of Alex­an­dria. This estim­ate was sourced from many, many camel-driven trade cara­vans, numer­ous enough to aver­age a good guess. Thus, we have a good value for the local distance.

Erastothenes’ Exper­i­ment

On June 21st, the Sun’s rays shone straight into the well in Aswan (the sun hung dir­ectly over the city). They also shone upon the city of Alex­an­dria, but since Alex­an­dria lies fur­ther north, they entered at an angle (the sun appeared lower in the sky than in Aswan). A pole driven into the ground at Aswan would not cast a shadow, but it did in Alex­an­dria. By meas­ur­ing the length of the pole and the shadow, it is easy to cal­cu­late the Sun’s elev­a­tion above the hori­zon. In Erastothene’s case, he found that the sun hung roughly 83 degrees above the hori­zon, mak­ing it easy to cal­cu­late . Let’s take a look at the situ­ation now.

Calculation of sun's elevation from shadow length.We are still inter­ested in the sep­ar­a­tion between the cit­ies. Instead of cal­cu­lat­ing the angle of elev­a­tion of the Sun, we can cal­cu­late the angle marked in the dia­gram — the angle “under the pole”. It turns out that this angle is exactly the same as the city-separating angle! This is depend­ent on the assump­tion that the sun rays are par­al­lel, which, as noted above, is “true enough”. A slanted line cross­ing two par­al­lel lines will pro­duce two identical angles at each inter­sec­tion. Try it: tilt­ing the cross­ing line (the “trans­versal”) will make one angle nar­rower and widen the other by the same amount. The exact oppos­ite will hap­pen at the other inter­sec­tion, keep­ing the angles identical regard­less of the angle.

The angle at the top of the pole always matches the city-separating angle.

This is the exact same sys­tem as Erastothene’s set-up: the Sun’s rays form the two par­al­lel lines, and the line from the Earth’s core to Alex­an­dria pro­duces the trans­versal. So, by meas­ur­ing the angle at the top of the pole — a very mundane task — we find the city-separating angle at the cen­ter of the planet.

Then the deal is settled: we know the local angle between the cit­ies, the local dis­tance between the cit­ies, and the global angle (the full 360º), miss­ing only the global dis­tance (the cir­cum­fer­ence of the Earth). The dis­tance between Aswan and Alex­an­dria was then taken to be 5000 sta­dia (approx­im­ately 800 kilo­met­ers), the angle of sep­ar­a­tion (again, given by the Sun’s elev­a­tion in the sky at Alex­an­dria) turned out to be a little more than 7º, and the global angle is ever 360º.

Using the Wrong Measure

The only stone left unturned is the “sta­dia” meas­ure men­tioned above, the unit Erastothenes gave his res­ult as: the cir­cum­fer­ence of the world turned out to be 252000 sta­dia. The sta­dion is an ancient meas­ure of dis­tance, and in itself not prob­lem­atic. But, there were two sys­tems in use in Erastothenes’ time: the Attic (greek) and the Egyp­tian, and no-one knows which he used. If it was the Attic one, his cal­cu­la­tion was off by just about 15%, but if he used the Egyp­tian sta­dion, his value would imply that a trip around the Earth was 39690km, which is off by less than one percent.

Regard­less of accur­acy, his tri­umph was to devise a method of cal­cu­lat­ing the cir­cum­fer­ence, where earlier no-one knew any­thing about its size. Then, using this value, Erastothnes leap-frogged through the solar sys­tem: he cal­cu­lated the size and dis­tance of the moon and the Sun, but that’s another story. The cir­cum­fer­ence of the Earth is almost exactly four mil­lion meters, but the reason for that is also another story.

We return to Chris­topher Colum­bus. He set out west­ward to find a shorter route to Asia, spe­cific­ally to make the jump from the Canary Islands to Japan, but he was wrong regard­ing the dis­tance. Iron­ic­ally, his dis­tance estim­ate was cor­rect, but in the wrong unit: he mis­took the longer Arabic mile (1.8km) with the Italian mile (1.2km). All in all, he estim­ated the trip between the Canar­ies and Japan to be 3700km, while it in prac­tice is 19600km. This is the con­tro­versy he fought, not that he would fall off the Earth. Sail­ors and schol­ars alike knew that he’d never last the trip, but he per­sisted and sailed off into the sun­set. Luck­ily, Chris bumped into, uh, the Americas.

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