Planetary motion and mathematics

Summary: It took mathematicians thousands of years to accurately describe planetary motion.

For millennia, the shape of the paths in which the planets orbited was dominated by metaphysical concerns and assumed, almost without question, to be circular. It was not until the seventeenth century that science discovered the actual shape of planetary orbits, the ellipse.

Early Conceptions

In ancient Greek astronomy, it was assumed that the Earth was the center of the universe, and all of the known planets (including the sun and the moon) as well as the stars revolved around it. Furthermore, at least from the time of Pythagoras (c. 569–475 b.c.e.), these orbits were assumed to be circular. This assumption was a metaphysical one.

The Pythagoreans believed in the perfection of mathematics and held the view that the circle was perfect because of its symmetry and continuity. Therefore, the universe must surely be constructed to reflect this perfection by requiring the planets to revolve around the Earth in perfect circular motion. That influential philosophers such as Plato and Aristotle accepted the perfection of circular motion contributed to the fact that the idea went almost unchallenged for nearly 2000 years.

With the increasing ability to make accurate observations of the movements of the heavens and mathematical calculations to predict those movements, the simple assumption of perfect circular motion became more problematic. The predictions of the planetary positions did not match the actual observed locations. Eudoxus (408–355 b.c.e.) addressed this discrepancy by devising a complicated system of nested spheres in which each planet moved, maintaining circular motion of each sphere while more accurately predicting the location of the planets.

For many centuries, one man’s work dominated European thinking on planetary motions. The Greek mathematician and astronomer Ptolemy (85–165 c.e.) compiled all that was known about the movements of heavenly bodies into one work that came to be known as The Almagest. This book employed an array of very complex geometric and trigonometric theories to describe the movement of the planets, with the Earth remaining at the center. In order for the observations to be as close as possible to the calculations, Ptolemy used epicycles (small circles revolving upon bigger circles as they revolve around the Earth) and moved the Earth away from the center of revolution of the planets.

The new center of revolution was an imaginary point some distance away from the Earth. Ptolemy’s influence on Western astronomy was partially because of its general agreement with Christian doctrine. As the center of God’s creation, the Earth must rest at the center of the cosmos. Furthermore, a perfect Creator would use the perfect circle to put His creation in motion.

Challenges

The most serious challenge to Ptolemaic cosmology came from the Polish church official, Nicolaus Copernicus (1473–1543), whose revolutionary work De Revolutionibus placed the sun, not the Earth, at the center of the universe, relegating the Earth to mere planethood. Copernicus, however, remained adamant in his belief that the planets orbited the sun in a composite of perfect circular motions. The doctrine of perfect circular motion in the heavens was finally challenged by the German astronomer Johannes Kepler (1571–1630). Kepler, after many years of tedious and painstaking calculations involving the orbit of Mars, finally determined that Mars actually orbited the sun in an elliptical orbit, not a circular one. This revolutionary idea was based in part on another discovery by Kepler that the speed of the planets varied as they orbited the sun. Later, the great British mathematician and scientist, Isaac Newton (1643–1727), used his universal law of gravitation and laws of motion to provide a mathematical explanation for Kepler’s claim of elliptical orbits, finally putting an end to the ancient doctrine of circular motion in the heavens.

Mathematics continues to play an important role in modeling planetary orbits. For example, Mercury’s orbit is more accurately represented with hyperbolic geometry than with Euclidean geometry. Further, the orbit of Mercury allows researchers to see the impact of the sun’s gravitational field on the curvature of space.

Bibliography

Danielson, Dennis Richard. The Book of the Cosmos: Imagining the Universe From Heraclitus to Hawking. Cambridge, MA: Perseus Publications, 2000.

Gingerich, O. The Eye of Heaven: Ptolemy, Copernicus, Kepler. New York: American Institute of Physics, 1993.

Heath, Thomas L. Greek Astronomy. New York: E. P. Dutton, 1932.

Kopache, Gerald. “Planetary Motion: Also Featuring Some Stars, Some Comets and the Moon.” Ceshore Publishing Company, 2004.

Montenbruck, Oliver, and Gill Eberhard. Satellite Orbits: Models, Methods and Applications. Berlin: Springer, 2000.

Pannekoek, Anton. A History of Astronomy. New York: Dover Publications, 1989.

Sagan, Carl. Cosmos. New York: Random House, 1980.