# 1.1 Eratosthenes Measures the Earth (6 pages)

6. If the value of 252,000 stades were indeed the true value of the circumference of

the earth, and Eratosthenes' angle measurement was correct, how many stades are

there from Alexandria to Syene?

7. The original definition of a nautical mile was the length of arc of the earth's

cicumference subtended by an arc of an angle at the center of the earth of one

(ordinary) minute. Find the circumference of the earth in nautical miles.

8. Use a modern value of the circumference of the earth in (ordinary) statute miles

to calculate the difference between a nautical mile and a statute mile in feet.

9. Is the number 252,000 really divisible by 60?

Now we turn to Eratosthenes' measurement of the angle between the sun's rays and a

vertical pole at Alexandria at the summer solstice. One source we have for Eratosthenes'

measurement is Cleomedes who probably wrote in the middle of the first century B.C -

over 150 years after the measurement. According to this account, Eratosthenes measured

the angle using a sundial in the shape of a bowl.

If a sundial is shaped like a hemisphere and has a vertical pointer (called a gnomon -

pronounced "no–mahn") from the bottom to the center of the sphere, then the inside of the

bowl represents an inverted map of the dome of the sky and the position of the tip of the

shadow of the gnomon shows the position of the sun in the sky. Eratosthenes could easily

have used such a device to measure the angle,

α

, between the sun and the vertical at

Alexandria.

Sun

Hemispherical Bowl

α

Arc of angle

α

of circle

1

50

__

=

A similiar, but very large, hemispherical sundial was constructed as part of the Jantar

Mantar, a stone observatory constructed in 1724 by Maharaja Jai Singh at Delhi in India.

Today the Jantar Mantar (Time Device) is a city park and tourist attraction in the modern

capital, New Delhi. Jai Singh constructed similar Jantar Mantars at other spots in India.

(The one in Ujain is in particularly good condition and still "works".)

The hemispherical "sundial" is but one of several large structures in the Jantar Mantar built

for observing the heavens. It is called the Jayaprakash Yantra. The Jayaprakash Yantra

actually consits of two complementary bowls.