Monday, February 16, 2009
Sticks and Shadows to Measure the Earth
The size of the Earth was determined for the first time some 2,200 years ago. At the time it was already known that our world is a sphere, but nobody had as yet come up with a way of accurately measuring its circumference.
That the earth is round was clear from several easily observable facts: when ships put out to sea their hulls always sink below the horizon before their masts; during a lunar eclipse the Earth’s shadow on the moon is always round. And so on.
One day the mathematician Erathostenes, head of the famed Alexandria library, learned about a curious fact while “leafing through” a papyrus book (presumably part of the library’s huge collection). Every year at noon on June 21 the columns of the temples in the distant city of Syene (present day Aswan), in Egypt, ceased to cast a shadow. As Erathostenes later verified, this was not the case in Alexandria, where vertical columns cast definite shadows at noon on the summer solstice.
Erathostenes knew that on a round Earth columns in Alexandria and columns in Syene do not point in the same direction. He reasoned that at noon on June 21 the sun came directly overhead in Syene so that temple columns were parallel to its rays, while at the same time vertical columns in Alexandria (or vertical sticks, or vertical whatever) were at an angle to the sun’s rays. Erathostenes saw how he could use this fact to determine the Earth’s circumference.
He planted a vertical stick on the ground in Alexandria and waited for the summer solstice. He then computed the angle formed by his stick and the sun’s rays by measuring the length of the stick’s shadow and comparing it to its height, a method involving math taught in highschool today. Erathostenes hired someone to walk all the way from Alexandria to Syene (located exactly due south from Alexandria, near the first cataract of the Nile) and measure the distance between the two cities. It is not too hard to see that the angle formed by the sun’s rays and the stick in Alexandria must be the same as the angle that the vertical of Syene and the vertical of Alexandria would form if extended to the center of the earth. So the clever mathematician now had an angle and the length of arc it traced on the surface of the earth. The angle turned out to be one fifitieth of 360 degrees, so the distance between Alexandria and Syene must equal one fiftieth of the Earth’s circumference. The figure Erathostenes came up with is equivalent to some 40,000 kilometers --remarkably close to present-day measurements.
Many centuries later a Genoese seaman by the name of Cristoforo Colombo was trying to prove that the Earth was small enough for him to reach China by sailing westward from Europe in a reasonably short time. His critics, who were probably aware of Erathostenes’s figure, claimed that the ocean separating Europe from Asia to the west was too vast, that Columbus’s proposed voyage could not be done. And they were on to something. In his eagerness to prove himself right the studious future Admiral of the Ocean Sea had rejected all ancient measurements which were incompatible with his claim, including Erathostenes’s. Had our continent not been in the way, Columbus would have sailed from Palos into oblivion.