In the end, the error turned out to be a trifle: less than 1% deviation on an order of magnitude of 250,000.
To be precise, 257,142 stadia. The stadion, a unit of measurement in the Hellenic world, was equivalent to our 157.5 meters.
The protagonist of this ‘brilliant mistake’ was Eratosthenes, mathematician, astronomer, geographer and poet, originally from Cyrene, who lived between 276 and 194 BC.
Eratosthenes was the head of that formidable sanctuary of knowledge which was the library of Alexandra of Egypt, the nerve centre of that epic of ante-litteram (ahead of its time) globalization which was Hellenism.
And in the name of universal knowledge, the best brains of the time found themselves in Alexandria, Egypt. Eratosthenes was precisely one of these.
He was responsible for the almost exact calculation of the inclination of the ecliptic (in other words, the inclination of the Earth’s axis), the drafting of a sort of stellar inventory, in addition to the so-called “riddle of Eratosthenes“, a technique to compile the table of prime numbers.
But his fame is linked rather to one aspect: the calculation of the radius of the Earth, which he deduced using truly primitive tools.
In particular one, banal yet effective: the gnomon.
Nothing more than a stick that, placed vertically on a flat ground, allowed one to study, through the shade generated, the movements of the Sun during the day and during the year.
The idea of how to calculate the radius of the Earth came to Eratosthenes after noting that in Siene (today’s Aswan, Egypt), on the day of the summer solstice (June 21), the Sun illuminated the bottom of the wells .
It was an epiphenomenon (by-product) generated by the fact that, on that day and in that hour, the sun’s rays in Siene were perpendicular to the ground.
Thus, what technology could not solve, trigonometry could.
In fact, Eratosthenes used it to measure the immeasurable.
He therefore relied on that part of mathematics, precisely trigonometry, which allows one to determine the length of the sides and the width of the corners of a triangle, once three of its elements are known, including at least one side.
For this Eratosthenes made sure that on the same day the shadow of the gnomon (no more than a stick stuck in the ground) was measured in Alexandria, the city which, according to his information, was located north of Siene, on the same meridian, at a distance of 5000 stadia.
Thanks to this measurement he was able to establish that the direction of the sun’s rays formed an angle of 7.2 degrees with the vertical, that is 1/50 of a round angle.
With this data available, Eratosthenes even obtained the circumference of the Earth: it was equivalent to 50 times the distance between Alexandria and Siene. So 250 000 stadiums, or more or less 39 000 km.
Let’s face it though. This enterprise reconciles us with one of the most demanding topics of our schooling: trigonometry. This fundamental element of mathematics has always enjoyed a bad reputation among generations of students who consider it a black, useless, boring, complicated and difficult beast.
It was 240 BC when Eratosthenes of Cyrene measured the radius and therefore the circumference of the Earth with an accuracy equal to the simplicity of the method.
The calculation model was none other than the application of the theorem of angles originating from a transverse line that cuts two parallel lines. In the case of the observation of Eratosthenes, the angle ‘a’ formed between the shadow and the stick in Siene is equal to the angle ‘b’ which has the centre of the Earth as its apex, while its sides pass respectively through Siene and Alexandria.
Angle b corresponds to the arc that represents the real distance between Siene and Alessandria (at that time already known with a certain precision: 5,000 stadia).
Having these data available, Eratosthenes actually carried out a calculation in these terms:
[Siene-Alessandria distance (HK arc)]: (circumference of the Earth) = 7°20 ‘: 360 ° (full angle)
5,000: (circumference of the Earth) = 7 ° 20 ‘: 360 °
(circumference of the Earth) = (5,000 x 360) / 7,2 = 250,000 stadia.
[distanza Siene-Alessandria (arco HK)]: (circonferenza della Terra) = 7°20’ : 360° (angolo giro)
5.000: (circonferenza della Terra) = 7° 20’ : 360°
(circonferenza della Terra) = (5.000 x 360) / 7,2 = 250.000 stadi.
Given that an Egyptian stadion was 157.5 meters, multiplying by the unit of measurement we obtain that the circumference of the Earth is equal to:
250,000 x 157.5 = 39,375,000 meters = 39,375 kilometers
Assuming that the Earth is perfectly spherical, today we know that its circumference is about 40,075 km.
How then to explain the small mistake of Eratosthenes? He slightly underestimated the distance between Siene and Alessandria …
You will understand !!..