Measuring the size of Earth

Greek astronomer and mathematician unravels the mystery of how an ancient Greek astronomer got the size of the Earth nearly right

Asweltering summer day, a sumptuous lunch and the scorching sun made Eratosthenes, a Greek astronomer, geographer, mathematician and the curator of the famous library of Alexandria in Egypt, dozy. A boastful, weary voice that he overheard jolted him awake. The traveller from Syene (modern-day Aswan), south of Alexandria, claimed, “On this day of summer solstice, vertical poles do not cast a shadow in my town.” He further asserted, the Sun illuminated the entire bottom of wells, without casting any shadow, indicating that the Sun was directly overhead at midday on the summer solstice at Syene.

We may chant “Sun rises in the East” because we learnt it in primary school, but the true fact is that only on two days in the year does the Sun rise in the true east. On one such day, March 21/22, we can note that the sunrise point is exactly at true east. If we continue to keep watch on subsequent days, we will notice that the sunrise point is gently shifting towards the north.

On June 21, summer solstice, the sunrise point will reach the maximum north of true east, when the daytime will be the longest in the northern hemisphere. From the next day, the sunrise point will move towards the south and in India it was called Dakshinayana.

On its southern journey, the sunrise point will again reach true east on September 21 and the maximum south-east sunrise point will be on December 21. Thereafter, the sunrise point will move towards the north compared to the previous day, called Uttarayana.

“Solstice” comes from the Roman word “solstitium”, Sol (Sun)+stitium (stoppage); and on June 21 and December 21, the sunrise point appears to “stay put” in the sky, neither making north or south movement.

The casual comment by the traveller triggered Eratosthenes’s interest. An idea struck him to use this to find out the size of the Earth. On summer solstice day, at Syene, located roughly on the Tropic of Cancer, at midday, as there are no shadows, the Sun should be exactly overhead; while in Alexandria, the vertical pole did cast shadows. Eratosthenes noted that given that the Earth is a sphere and the Sun is so far away, practically the rays are parallel as they reach Earth, and the angle of the shadow at Alexandria would be the same as the angle between Syene and Alexandria from Earth’s centre.

The legend has it that during the solstice in BCE 240, on June 19 (in modern times this occurs on June 21/22), Eratosthenes planted a vertical stick on the ground using a plumb-line. Exactly at local noon, when the Sun crossed the meridian overhead, he measured the size of the shadow cast by the stick. Dividing the height of the stick by the shadow length gave the angle of the shadow cast by a stick at noon on the day of the summer solstice in Alexandria. The angle was 7.2 degrees or about 1/50 of a complete circle.

This implied that the distance between Syene and Alexandria was 1/50th of the diameter of the Earth. In other words, the distance between the two locations multiplied by 50 will yield the circumference of the Earth. Travellers had already measured the distance as 5,000 stadia (a unit of measurement used in ancient Egypt), and thus Eratosthenes computed the circumference of Earth as 2,50,000 stadia. From the formula for the circumference of the circle, C=2TTr, he also computed the radius of the Earth to be 39,800 stadia.

INDIAN CALCULATIONS

The elevation of the Pole Star, Dhruv Tara, at a given location tells your latitude. It means if you stand at the equator, the pole star will be at horizon. Suppose you move north and reach Chennai, the pole star would be 13 degrees above horizon. At the North Pole, the pole star would be overhead, at 90 degrees.

Jnanaraja (c1500 CE) an astronomer from Rajasthan, in his Siddhanthasundara, observed that if one travelled exactly in a northward direction for a distance of 14 yojanas, then the angle of the pole star is elevated by one degree. From this he reasoned that if one travelled from equator to pole, the elevation would be 90 degrees, and that would imply the distance would work out to be 14x90=1,260 yojanas. Equator to pole is one-fourth of the circumference. From this, Jnanaraja computes the circumference of the Earth to be 5,040 yojanas. Ingenious, isn’t it?

Today we know that the radius and circumference of the Earth passing through poles is about 6,356.5 km and 40,007 km respectively. But how accurate was Eratosthenes? One is not sure what a stade — the unit he used — is in kilometres. Crucially, Syene is about 39 minutes north of the Tropic of Cancer and Alexandria’s longitude is off by 2 degrees from Syene. Nevertheless, Eratosthenes’s experiment yielded a reasonable estimate. If indeed long before Columbus we knew that the Earth was round and even how big it is, why were the masses ignorant of this during the 15th century?

Religious fanatics burnt the library of Alexandria charging that it contained blasphemous and lewd books, and in India, the orthodoxy was mired in the puranic Saptadwipa (in which the Earth was seen as one big island).

FANATICS BURNT THE LIBRARY OF ALEXANDRIA CHARGING THAT IT CONTAINED BLASPHEMOUS BOOKS