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How did Eratosthenes measure the circumference of the earth?

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(This article is a translation from the original spanish version).

The Greek mathematician Eratosthenes realized that on the day of the summer solstice (June 21) at noon, in the city of Siena (actual Aswan) sunlight casted no shadow on the bottom of a well, but in the city of Alexandria, north of Siena, on the same day at the same time it does casted a shadow on the bottom of a well.

The summer solstice is the longest day of the year and is produced by the tilt of Earth’s axis. At the summer solstice in the northern hemisphere the sun reaches the zenith at noon on the Tropic of Cancer, that is to say, in places located there, on June 21 the sun’s rays fall vertically on the earth, and of course as it is round, in other places fall leaning. The city of Siena is located very near of the Tropic of Cancer.

Observation of Eratosthenes confirmed something that other Greeks already suspected: that the earth was round, as if it were flat, in Alexandria should not cast a shadow over the well, as in Siena. Furthermore, because we can see the curvature in the sky, because the more one travels to the north, stars and constellations look increasingly above, like Polaris; and others that simply disappear into the horizon, as Canopus.

Doing these observations Eratosthenes came up with a brilliant idea. On June 21, at noon in Alexandría, he took a stick and measured the angle of the shadow casted on it and noted that it was one fiftieth of a circle (in those days there were no notions of degrees). The 50th part of a circle (360 degrees) equivalent to 7.2 degrees.

Thus, as the same day at the same hour the sun’s rays fell vertically on Siena casting zero degrees shadows on a vertical, thus between Syene and Alexandria there were a distance of 7.2 degrees or the 50th part of the circumference of the earth. (Eratosthenes assumed that the earth was perfectly circular).

Eratosthenes already knew from the caravans that traded between the two cities, that there was an estimated distance of 5,000 stades between them. Therefore, simply multiplied by 50. This is 250.000 stades. The stade was the Greek unit of length, which varied from one location to another between 157.5 meters to 184.8 meters. The stade used by Eratosthenes was the Italian attic of 184.8 meters. This is 46.200 kms.

In the first graph the mock up is tilted but toward the sun; there is no shadow. In the second graph the mock up was titled but keeping it straight and shadow that occurs in both mock up are the same. In the third and fourth graph the mock up is curved, leaving the first stick into the sun, and we see in the second stick the shadow is long but in the first stick there is no shadow.

                             

We should clarify something before continuing: How did Eratosthenes measure the angle of the shadow casted? Unfortunately the book written by Eratosthenes himself: “On the measurement of the earth”, which would give us details of their discoveries; was lost, as happened with many other writings of antiquity, who did not survive the destruction of the Library of Alexandria (by tsunami, burnings by invaders) of which Eratosthenes was the third director. The Greek astronomer Cleomedes in his book “On the circular motions of the celestial bodies” which is the main original source through which we know of the discovery of Eratosthenes, it only says that he used a gnomon- wich is a vertical rod or stylus that casts a shadow on a horizontal surface- but does not say how did he measure the shadow casted; but from this, there are only two possible ways to do it: The first and the one we all know, is using trigonometric functions. In this case would be finding the value of the tangent, that is to say, the measure of the shadow divided by the measure of the rod (opposite leg divided by leg adjacent), then we took the inverse tangent (tan-1) with the calculator to get the angle of the tangent, and optionally we can convert it to degrees with the button (.,,,) of the Casio. But the complexities and abstractionism of differential and integral calculus were not available at that time.

In my quest to solve this mystery, I discovered or rediscovered a simple but forgotten technique, just using a compass. Eratosthenes would draw to scale on a plane (or on papyrus) the measure of the rod and of the shadow on the floor and draw the hypotenuse, then would tumble the plane for comfort of course, leaving the opposite side to the hypotenuse (or the measure to scale of the gnomon) as a horizontal line and interpose a compass with one side at the vertex that is formed between the hypotenuse and the measure of the gnomon and would draw a circle around it. Then, at the point where the hypotenuse intersects the circumference, is placed one side of the compass and and in the other side of the compass is placed in the point where the circunference drawn touch the horizontal line. We left the compass with the same measure and began measuring on the circumference how many parts of it are equivalent. But, Eratosthenes neither had to use a compass necessarily; any vertical that is rotated around its own axle, form a perfect circle.

I used the software “ruler and compass” to build digital and exactly the circumference and angles. As we can see,  half of the elaborated circumference is divided into 25 equal parts of 7.2 degrees.

 

You can put a transparent protractor or a compass on the screen and check yourself.

Below there are compass of the Graeco-Roman age kept in the British Museum.

 

Now we have a simple gnomon Eratosthenes should have used and then three gnomons used to mark the time (sundials).

                             

 

The measure of the circumference of the earth made by advanced satellites is approximately 40.008 kilometers. Considering the simplicity and rudimentary but ingenious technique used by Eratosthenes, its calculus approximation was amazing. Only erred in 6.192 kms. This is 15%.

 

Now let’s use the powerful technological tools we have today, MapCrow and Google Map, and let’s remake Eratosthenes calculus with exact measurements to see if his reasoning was correct.

 

Latitude is the approximate angular distance between the equator and a given point on the planet. There are there horizontal lines on a map. They are expressed in angular measurements ranging from zero degrees at the equator to 90 degrees at the North Pole or 90 ° of the South Pole. If we draw a straight line running from any point of the earth to the center of it, the angle formed by that straight line and the equatorial plane, expresses the latitude of that point.

Before continuing, there are three assumptions that we must take into account:

1) We assume the earth is perfectly round. A degree of latitude does not measure exactly the same in each place, but varies slightly from 110.57 km at the equator till 111.70 at the poles, so we can not assume that 7 degrees between Alexandria and Syene have the same distance that 7 degrees between Alexandria and some city in Turkey. So our result could never be exactly the same as the made by advanced satellites.

2) If we do the subtraction of lengths (vertical lines of the map) there is a difference of 3 degrees (Eratosthenes assumed they were in the same length).

3) Another small error of Eratosthenes, is that Siena was not located exactly on the line of the Tropic of Cancer (the points where the sun’s rays fall vertically to the earth on June 21). Today is 72 kms (from downtown). But because changes in the earth’s axis fluctuate between 22.1 and 24.5 degrees over a period of 41,000 years; 2000 years ago was located at 41 kms distance-To calculate the coordinates of the Tropic i used the software of neoprogrammics.com and calculated values ​​for the year -200. –

Let’s see:

If we do the subtraction of the latitudes, get an angular distance of 7.1106 or 7 ° 6 ‘ between the two cities. This means that the distance between Alexandria and Aswan is a 50.6286 part of a circle (360 degrees). Eratosthenes got a 50th portion of a circumference which is 7.1997 or 7 ° 12 ‘.

The distance is not 924 kms, but 843 kms. 81 kms less -Air distance till center of cities. –

The corrected calculation of Eratosthenes results in 42, 662 kms. The error is only 6.6% or 2.654 kms.

Eratosthenes reasoning was quite correct. The assumptions he made did not affect the result too much, so it can be considered that were quite valid given constraints of the age.

Now let’s go to Daftlogic.com and calculate the distance between the city of Alexandria and a point on the map where there is the same length of Alexandria (29.9192) and is located exactly on the line of the tropic, that is latitude 23 ° 26 ‘or 23.4377.

 

If we subtract the coordinates of Alexandria and the Tropic of Cancer results in an angular distance of 7.7604, which means a 46.3894 part of a circle and multiplied by 863,876 kms, this results in 40,074. Impressive! Only 66 kms of difference (0.16%) from the calculus nowadays is aproximated for the earth.

If we adjust now the Tropic of Cancer to the position it was in the year -200 BC, then we will find the true measure of the shadow casted by the rod of Eratosthenes in Alexandria on June 21 at noon 2.200 years ago : 7.4815 which is 7 ° 29 ‘. If we trace the measure of 7.4815 in our software that makes 48.1 parts of a circle. 48 parts of a circle multiplied by the 863,876 km distance between Alexandria and the Tropic results in 41.561 kms. So among Eratosthenes mistakes we should add 0.2818 which is 0 ° 17 ‘ degrees of error in measuring the angle of the shadow. This is due to with a pencil, or a fine point, it is impossible to distinguish between 7.0 and 7.2 degrees, and very difficult between 7.0 and 7.5 degrees. It was expected that Eratostenes divisions of the circumference may have a margin of error of up to 4 parts of a circle. He was wrong in two parts, which is not bad considering the instruments.

150 years after Eratosthenes, the Greek mathematician Posidonius used a similar method to the one used by Eratosthenes (also described by Cleomedes in his book), but instead of using the sun as a reference, he used a star called Canopus (the second brightest star in the sky). He realized that in Rodhas, this star was visible just above the horizon, but being in Alexandria that star was upper in the sky. He measured the length of the arc that was drawn between the two positions of the star, probably using an astrolabe, and I guess subtracting angle measurement of the star in the sky of Alexandria less Rhodes sky angle, in order to find the angle distance between Rhodes and Alexandria. But Posidonius measurement was incorrect. He got a distance of 7. 5 ° or 7 ° 30 ‘, when in fact as we see on the map, is only 4.97. His error in measuring the angle should have due to the astrolabe was not very accurate really (worse primitive astrolabes), that’s why it was replaced by the sextant 1500 years later. The distance estimated by Posidonuis between Alexandria and Rodhas results in 72 a part of a circle and not the 48th part it real represents.

If we do calculation with correct data, results in 42.014 kms. Posidonius calculation resulted in 28.968 kilometers (28% error relative to the real circumference of the earth). It was this measurement and not the one of Eratosthenes which Ptolemy used in his famous work “Geography”. Columbus never read Ptolemy, but other authors of his time as Pierre d’Ailly, who based on the calculation of Posidonius used in Geographia, estimated the distance between the Canaries and Cipangu (Japan). But Columbus added another mistake to the matter, assuming that Ailly refered to Italian miles when actually refered to Arab miles (which are longer). Columbus believed that between the Canaries and Cipangu there were some 2,400 nautical miles, when actually there were 10,700. Luckily for him, he founded a continent in the way to Asia.

This error in measuring the circumference of the earth is probably the one has most influenced the history of mankind. If Columbus had known the length of the Earth’s circumference calculated by Eratosthenes, would never have made his journey, since for that age no ship could store enough water and provisions to stay so long at sea, and the discovery of a departure and return route to America would have been delayed perhaps hundreds of years.

 

Other sources:

Eratosthenes of Cyrene

How Long Is a Stade?

On the circular motions of the celestial bodies (Bilingual translation greek-spanish)

Eratosthenes Got the Circumference of the Earth: Physicist Klaus Kohl proposes that Eratosthenes could have ​​used compasses on a scaphe, which was an hemispherical sundial. The reasoning is the same, ie counting how many times the tilt angle fit the circumference, but Kohl did not realized that this can be built in a plane also.

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5 comentarios

  1. Roumiana Anguelova dice:

    Great!

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