2023-07-31 10:50:09
Literate as he was, Eratosthenes had been nourished by the ideas of his predecessors which he considered relevant. Thus, as early as the 6th or 5th century BC, Pythagoras had suggested that the Earth is round. Quite simply because this esthete did not imagine that the Earth could display a geometry less perfect than that of the sphere. A cubic or potato-shaped planet would certainly have been the fruit of gods falling on their heads!
Fortunately for the quest to know that drives humanity, Aristotle had put logic into all this, two centuries later, raising his eyes to heaven without looking for many saints to devote himself to. First by noting that the shadow of the Earth draws an arc on its satellite during lunar eclipses. Which, the grumpy will rightly object, is completely compatible with the idea of a discal earth sketched out by Thales in the 7th-6th century BC. J.-C. But Aristote had also noted that the sky makes appear and disappear stars when one moves. Think: this time it doesn’t work with a flat Earth!
Aristotle’s calculations, far from the account
This talented philosopher had also clumsily tried to quantify the size of our alcove. In his Treaty of Heaven, he estimated his circumference – with a wet finger? No one knows – at 440,000 stadia, roughly 70,000 km if we consider a value of 158 meters for this unit of length adopted by the Greeks since the epic of Alexander the Great. In short, Aristotle was far off the mark since the Earth measures “only” 40,075 km at the equator.
Eratosthenes cannot content himself with the pifometer: an inconceivable “instrument” for the “Cartesian” that he was, well before his time. He prefers to construct a reasoning on the basis of a very audacious hypothesis – and since proven: given the considerable distance from the Sun in relation to its size and the presumed smallness of the Earth, its rays are necessarily parallel when they reach us.
During a visit to Syene (now Aswan), our genius had noticed that at the summer solstice, in June, the Sun is perfectly reflected at the bottom of a well. Event that he interpreted as the fact that the star is, exactly and at this moment, vertical to the place since it is not yet customary to dig wells at an angle. However, endowed with memory, Eratosthenes knew that when the Sun rises to the height of the summer solstice in Alexandria, the buildings there still cast a small shadow on the ground. The sun is therefore not totally vertical. Do you see where he was coming from?
A calculation in five acts
Act I Since the Earth is round, thought Eratosthenes, imagine two straight lines. One extends the axis of the well of Syene to the center of the Earth. Similarly, if we plant a gnomon vertically in Alexandria – this wooden stake used in sundials on the ground – its extension towards the bowels of the globe will define a second virtual straight line which also joins the center of the Earth, since this is a sphere. “It’s geometry, stupid!” There, under our feet, these lines of thought form a certain angle.
Acts II Eratosthenes knows that Syene and Alexandria lie more or less on the same meridian, because what is now called true solar time – time in the sun, to put it simply – is roughly the same there.
Act III Eratosthenes studied his geometry, notably Pythagorean, and knows what internal alternate angles are. A barbarous but vital term for his enterprise: it signifies, in this case, that the angle between the sunbeam which skims over the top of the gnomon of Alexandria and the latter is identical to that formed by the two imaginary straight lines near the center of the Earth. It remains to be determined, in Alexandria, for which our philosopher will recall his trigonometry lessons, by deducing it from the measurement of the length of the gnomon and that of its shadow on the ground.
Act IV One day of the summer solstice, Eratosthenes and an accomplice, whose name is – alas for him – not remained in posterity, patiently wait for the course of the Sun to reach the zenith (which, in Syene, means that the shadow no longer decreases). At this instant, the desired angle is 7.2°. That is exactly the fiftieth of the 360° of the complete turn of a meridian. A real stroke of luck which avoids any calculating recourse to the “pascaline”, still unavailable because created by Blaise Pascal in 1642. Understand: once the distance between Alexandria and Syene is known, it will suffice to multiply it by 50 to obtain the length of the meridian, and therefore the circumference of the Earth.
Act V Eratosthenes turned to the bematists, those tireless surveyors known to measure distances by counting their steps — or those of their camels, who knows. Unless they already have an odometer, which remains to be demonstrated for a device supposedly invented in Rome around the year 15 BC. J.-C. In short, Syene and Alexandria are distant of 5000 stages, which roughly gives 39 500 kilometers of circumference for the sphere, that is to say an error of 575 km on what one knows from now on of it. Impressive, isn’t it?
For the record, historians still wonder if Christopher Columbus would have started his journey west in 1492 – or at least increased the stocks of food – if he had known of this value and not of the reference in force at his time, the 30,000 kilometers calculated by Ptolemy in the 2nd century AD. Like what geometry can change the course of the world!
An exhausted but not exhausting source!
We should pay tribute to the painstaking work of Arkan Simaan, historian, whose thrilling work – out of print but available second-hand – inspired us.
Science at the risk of its life Arkan Simaan, Editions Vuibertà-Adapt, 2001.
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#Eratosthenes #stroke #genius