Measuring The Unmeasurable

 

“The heart of science is measurement.” - Erik Brynjolfsson

    More than  2,000 years ago,   Eratosthenes  calculated  the size of the Earth  with   reasonable accuracy.  Calculating the Earth’s circumference  (the distance around a circle  or sphere) was his most lasting achievement. He computed this by using simple geometry and trigonometry and by recognizing Earth as a sphere in  space.  Most Greek scholars at the time agreed that   Earth was a sphere,  but none knew how big it was.

   How did Greek scholars know the Earth was a sphere? They observed that departing ships disappeared over the horizon while their masts were still visible. They saw the curved shadow of the Earth on the Moon during lunar eclipses. And they noticed the changing positions of the stars in the sky.

     Eratosthenes heard about a famous well in the Egyptian city of Swenet (Syene in Greek, and now known as Aswan), on the Nile River. At noon one day each year — the summer solstice (between June 20 and June 22) — the Sun’s rays shone straight down into the deep pit. They illuminated only the water at the bottom, not the sides of the well as on other days, proving that the Sun was directly overhead. Syene was located very close to what we call the Tropic of Cancer, about 23.5 degrees north, the northernmost latitude at which the Sun is ever directly overhead at noon.

     Eratosthenes erected a pole in Alexandria, and on the summer solstice he observed that it cast a shadow, proving that the Sun was not directly overhead but slightly south. Recognizing the curvature of the Earth and knowing the distance between the two cities would enable Eratosthenes to calculate the planet’s circumference. To determine the distance between Syene and Alexandria, Eratosthenes hired bematists, professional surveyors trained to walk with equal length steps. They found that Syene lies about 5000 stadia from Alexandria.

     Knowing the vertical height of the pole and the horizontal distance of its shadow as two legs of a right triangle, Eratosthenes determined the angle of the sun’s rays to be 7.2 degrees. That angle is one-fiftieth (1/50) of 360 degrees---the full angle of a circle. Multiplying the distance of the two cities by 50 (5000 x 50)  gave him the circumference of the Earth to be 250,000 stadia or 25,000 miles or 40,000 kilometers!

     Today, orbiting satellites routinely measure the earth’s circumference to be 24,901 miles. Eratosthenes’ ancient calculation  is only 00.4% off the modern measurement which is a remarkable feat considering the crude technology of his time.

A diagram showing how Eratosthenes measured the Earth