**Geodesy** is the science that studies the shape and size of the Earth. As discussed in *Voyagers*, the shape of the earth was long known to be round. Aristotle and Pythagoras both argued that the earth was a sphere from the curved shadow it cast during lunar eclipses. About two thousand years later, Sir Isaac Newton suggested that the earth was not a perfect sphere, but rather somewhat flattened at its poles.

From mathematical considerations of the combined gravitational and centrifugal forces which the earth experiences, Newton computed that the Earth's shape should be an **oblate spheroid**, a solid formed when an ellipse is rotated about its axis (see Figure 3). Expeditions to Peru in 1735 and to Lapland in 1736 confirmed this theory. The difference in axes is about 1 part in 300. This means the earth's equatorial diameter is 7,926 miles, while its polar diameter is 7,900 miles.

Although the difference between equatorial and polar diameters is only 26 miles, the oblate shape of the earth complicates geographical matters.

### Activities:

### Have the class research the following geographical questions:

1.) How does the weight of a 10-lb bag of rice compare at the equator and at the North Pole? **Solution** *Since the bag of rice is farther from the earth's center at the equator, it weighs less at the equator than at the North Pole.*

2.) Which mountain is the tallest on earth: Mt. Everest, Mauna Kea or Chimborazo? Have the class consider the following information before answering.

• **Mount Everest** rises in the eastern Himalayas between Nepal and Tibet (latitude: 27.6^{o}N). This young, limestone mountain has two peaks, one of which reaches 29,108 ft above sea level This height was confirmed in December 1983 during a surveillance mission by the space shuttle *Columbia*. Everest is covered with snow except for its bare, gale-swept summits. The Tibetan name Qomolangma means "goddess mother of the world."

• **Mauna Kea** is a large dormant volcano on the Big Island of Hawaii (latitude: 19.5^{o}N). It was last active over 4,000 years ago. Its summit is covered in winter by snow, giving its name Mauna Kea or White Mountain. Measuring 13,796 ft high above sea level, it extends an additional 18,200 ft to the ocean floor. Several major astronomical observatories are now in operation at its summit.

• **Chimborazo** is a volcanic peak in the Andes of Ecuador (latitude: 1.3^{o}S). Its crater is buried under a thick ice cap. Tributaries of the Amazon River run down its eastern slopes. Chimborazo measures 20,561 ft above sea level.

**Solution**

The answer depends upon how **height** is defined (see Figure 3).

a.) When height is measured from the **base of the mountain**, Mauna Kea is the tallest mountain. Its total height as measured from the ocean bottom is 31,996 ft.

b.) When height is measured from **sea level**, Mt. Everest is the tallest. Its peak reaches higher into the atmosphere than any other mountain.

c.) When height is measured from the **center of the earth**, Chimborazo is tallest since it is located nearly on the equator, where the earth bulges the most.

### Size of the Earth

*Voyagers* details how the earth's size was computed at the Royal Library of Alexandria. This famous library, which was founded by Alexander the Great (a student of Aristotle, who proved the earth was round), had a collection of a half million scrolls. As the primary center of learning in ancient times, the Royal Library lured many great scholars to its halls. Besides Eratosthenes and Ptolemy, who measured the earth's size, there were other famous researchers, like Euclid, Archimedes and Hypatia (a famous female mathematician).

Eratosthenes became head librarian at the Royal Library around 235 BC. There, he made the first accurate measurement of the earth's circumference. His value (based on the altitude of the noontime sun as seen from Alexandria and Syene on the first day of summer) was approximately 25,000 miles. The actual equatorial circumference is 24,902 miles.

In 150 AD, Claudius Ptolemy recomputed this measurement based on the altitude shift of Canopus, second brightest star in the sky. His figure equaled 18,000 miles.

### Activity:

To understand the geometry behind Eratosthenes' calculation, distribute copies of Figure 4 to the class. Students are to determine the circumference of a pizza, given only one slice. Have the students measure the length of the left-over crust. (This represents the distance across the desert separating Alexandria and Syene.)

A single slice contains enough information to determine the pizza's circumference. (The angle made by this slice represents the angular shift of the noontime sun as measured by Eratosthenes.) In order to compute how many slices make up a full pizza, students can cut out copies of several slices and piece them together to form one pie. More advanced students can measure the slice's angle with a protractor and do the computations mathematically.

### Further Reading:

Sagan, Carl, *Cosmos*, Random House, NY, pp. 14-22, 1980.