Solar System Geometry, History of

Humanity's understanding of the geometry of the solar system has developed over thousands of years. This journey towards a more accurate model of our planetary system has been marked throughout by controversy and misconception. The science of direct observation would ultimately play the most important role in bringing order to our understanding of the universe.

Ancient Conceptions and Misconceptions

Archaeologists have found artifacts demonstrating that ancient Babylonians and Egyptians made numerous observations of celestial events such as full moons, new moons, and eclipses , as well as the paths of the Sun and Moon relative to the stars. The ancient Greeks drew upon the work of the ancient Babylonians and Egyptians, and they used that knowledge to learn even more about the heavens.

As far back as the sixth century b.c.e., Greek astronomers and mathematicians attempted to determine the structure of the universe. Anaximander (c. 611 b.c.e.–546 b.c.e.) proposed that the distance of the Moon from Earth is 19 times the radius of Earth. Pythagoras (c. 572 b.c.e.–497 b.c.e.), whose name is given to the Pythagorean Theorem , was perhaps the first Greek to believe that the Earth is spherically shaped.

The ancient Greeks largely believed that Earth was stationary, and that all of the planets and stars rotated around it. Aristarchus (c. 310 b.c.e.–230b.c.e.) was one of the few who dared question this hypothesis, and is known as the "Copernicus of antiquity." His bold statement was recorded by the famous mathematician Archimedes (c. 287 b.c.e.–212 b.c.e.) in the book The Sand Reckoner. Aristarchus is quoted as saying that the Sun and stars are motionless, Earth orbits around the Sun, and Earth rotates on its axis. Neither Archimedes nor any of the other mathematicians of the time believed Aristarchus. Consequently, Aristarchus's remarkable insight about the movements of the solar system would go unexplored for the next 1,800 years.

Eratosthenes (c. 276 b.c.e.–194 b.c.e.), a Greek mathematician who lived during the same time as Archimedes, was another key contributor to the early understanding of the universe. Eratosthenes was the librarian at the ancient learning center in Alexandria in Egypt. Among his many mathematical and scientific contributions, Eratosthenes used an ingenious system of measurement to produce the most accurate estimate of Earth's circumference that had been computed up to that time.

At Syene, which is located approximately 5,000 stades from Alexandria, Eratosthenes observed that a vertical stick had no shadow at noon on the summer solstice (the longest day of the year). In contrast, a vertical stick at Alexandria produced a shadow a little more than 7° from vertical (approximately 5¹0 th of a circle) on the same day. Using the property that two parallel lines (rays of the Sun) cut by a transversal (the line passing through the center of Earth and Alexandria) form congruent alternate interior angles, Eratosthenes deduced that the distance between Syene and Alexandria must be of the circumference of Earth. As a result, he computed the circumference to be approximately 250,000 stades. Thus, Eratosthenes's estimate equates to about 25,000 miles, which is remarkably close to Earth's actual circumference of 24,888 miles. Apparently, Eratosthenes's interest in this calculation stemmed from the desire to develop a convenient measure of longitude to use in the circumnavigation of the world.

The most far-reaching contribution during the Greek period came from Claudius Ptolemy (c. 85 c.e.–165 c.e.). His most important astronomical work, Syntaxis, (better known as Almagest ) became the premier textbook on astronomy until the work of Kepler and Galileo over 1,500 years later. Mathematically, Almagest explains both plane and spherical trigonometry , and includes tables of trigonometric values and square roots .

Ptolemy's text firmly placed Earth at the center of the universe without any possibility of its movement. His work also reflected the Greek belief that celestial bodies generally possess divine qualities that have spiritual significance for humans. Ptolemy synthesized previous explanations for planetary motion given by the Greek mathematicians Apollonius, Eudoxus, and Hipparchus into a grand system capable of accounting for almost any orbital phenomenon. This scheme included the use of circular orbits (cycles), orbits whose center lies on the path of a larger orbit ("epicycles"), and eccentric orbits, where Earth is displaced slightly from the center of the orbit. Despite its explanatory power, however, the sheer complexity of Ptolemy's model of the universe ultimately drove Nicolas Copernicus (1473–1543) to a completely different conclusion about Earth's place in the solar system.

The Dawn of Direct Observation

By the first half of the sixteenth century, Ptolemy's geocentric model of the universe had taken on an almost mystical quality. Not only were scientists unquestioning in their devotion to the model and its implications, but the leadership of the Catholic Church in Europe regarded Ptolemy's view as doctrine nearly equal in status to the Bible. It is ironic, therefore, that the catalyst for the scientific revolution would come from Copernicus, a Catholic trained as a mathematician and astronomer.

Copernicus's primary goal in his astronomical research was to simplify and improve the techniques used to calculate the positions of celestial objects. To accomplish this, he replaced Ptolemy's geocentric model and its complicated system of compounded circles with a radically different heliocentric view that placed the Sun at the center of the solar system. He retained Ptolemy's concept of uniform motion of the planets in circular orbits, but now the orbits were concentric rather than overlapping circles.

Copernicus's ideas were very controversial given the faith in Ptolemy's model. Consequently, despite the fact that his model was completed in 1530, his theory was not published until after his death in 1543. The book containing Copernicus's theory, De Revolutionibus Orbium Caelestium ("On the Revolutions of the Heavenly Spheres"), was so mathematically technical that only the most competent astronomer could understand it. Yet its effects would be felt throughout the scientific community. Copernicus's new theory was published at a critical juncture, at the intersection of the ancient and modern worlds. It would set the stage for the new science of direct observation known as empiricism , which would follow less than a century later.

Johannes Kepler (1571–1630) was a product of the struggle between ancient and modern beliefs. A teacher of mathematics, poetry and writing, he made remarkable breakthroughs in cosmology as a result of his work with the astronomer Tycho Brahe (1546–1601). Brahe had hired Kepler as an assistant after reading his book Mysterium Cosmographicum. In it, Kepler posits a model of the solar system in which the orbits of the six known planets around the Sun are shown as inspheres and circumspheres of the five regular polyhedra: the tetrahedron, hexahedron (cube), octahedron, dodecahedron and icosahedron.

Kepler's fascination with these geometric solids reflected his devotion to ancient Greek tradition. Plato had given a prominent role to the regular polyhedra, also known as Platonic solids, when he mystically associated them with fire, air, water, earth and the universe, respectively. When Brahe died suddenly and Kepler inherited his fastidious observations of the planets, Kepler set about to verify his own model of the solar system using the data. This began an eight-year odyssey that, far from confirming his model, completely changed the way Kepler looked at the universe. To his credit, unlike the vast majority of mathematicians and scientists of his time, Kepler recognized that if the model and the data did not agree, then the data should be assumed to be correct. Little by little, Kepler's romantic notions about

the structure of the solar system gave way to new, more scientific ideas, resulting in Kepler's Three Laws of Planetary Motion:

First Law The planets move about the Sun in elliptical orbits with the Sun at one focus .

Second Law The radius vector joining a planet to the Sun sweeps over equal areas in equal intervals of time.

Third Law The square of the time of one complete revolution of a planet about its orbit is proportional to the cube of the orbit's semi-major axis .

The first two laws were published in Astronomia Nova in 1609, with the third law following ten years later in Harmonice Mundi. Kepler's calculations were so new and different that he had to invent his mathematical tools as he went along. For example, the primary method used to confirm the second law involved summing up triangle-shaped slices to compute the elliptical area bounded by the orbit of Mars. Kepler gives one account in Astronomia Nova of computing and summing 180 triangular slices, and repeating this process forty times to ensure accuracy! This work with infinitesimals , which was adapted from Archimedes's method of computing circular areas, directly influenced Isaac Newton's development of the calculus about a half-century later.

Interestingly, Kepler's work with the second law led him to the first law, which finally convinced him that planetary orbits were not circles at all (as Copernicus had assumed). However, as Kepler's account in the Astronomia Nova makes clear, once he had deduced the equation governing Mars's orbit, due to a computation error he did not quite realize that the equation was that of an ellipse . This is not as strange as it might first appear, considering that Kepler discovered his laws prior to Descartes's development of analytical geometry , which would occur about a decade later. Thus, with the elliptical equation in hand, he suddenly decided that he had come to a dead end and chose to pursue the hypothesis that he felt all along must be true—that the orbit of Mars was elliptical! In short order Kepler realized that he had already confirmed his own hypothesis. In the publication of his second law, Kepler coined the term "focus" in referring to an elliptical orbit, though the importance of the foci for defining an ellipse was well understood in Greek times.

No discussion of either the geometry of the solar system or the science of direct observation is complete without noting the key contributions of Galileo Galilei (1564–1642). Like Kepler, Galileo's science was closely tied to mathematics. In his investigations of the velocity of falling objects, the parabolic flight paths of projectiles, and the structures of the Moon and the planets Galileo introduced new ways of mathematically describing the world around us. As he wrote in his book Saggiatore, "The Book [of Nature] is … written in the language of mathematics, and its characters are triangles, circles, and geometric figures without which it is … impossible to understand a single word of it; without these one wanders about in a dark labyrinth."

Using the latest technology (which included telescopes), Galileo's observations of Jupiter and its moons as well as the phases of Venus left no doubt about the validity of Copernicus's heliocentric theory. His close inspection of the surface of the Moon also convinced him that the ancient Greek assumption about the Moon and the planets being perfect spheres was simply not tenable . Galileo paid dearly for his findings, incurring the wrath of the Catholic Church and a skeptical scientific community. But as a result of his struggle, mathematics was no longer viewed as either the servant of philosophy or theology as it had been for millennia. Instead, mathematics came to be regarded as a means of validating theoretical models, ushering in a revolutionary era of scientific discovery.

Through the work of Copernicus, Kepler and Galileo, the mistaken beliefs, pseudoscience, and romanticism of the ancients were gradually replaced by modern observational science, whose goal was and continues to be the congruence of theory with available data. Though the scientific view of the solar system continues to evolve to this day, it does so resting firmly on the foundation of the breakthroughs of the sixteenth and seventeenth centuries.

see also Galilei, Galileo; Solar System Geometry, Modern Understandings of.

Paul G. Shotsberger

Bibliography

Boyer, Carl B. A History of Mathematics, 2nd ed., rev. Uta C. Merzbach. New York: John Wiley and Sons, 1991.

Callinger, Ronald. A Contextual History of Mathematics. Upper Saddle River, NJ: Prentice-Hall, 1999.

Cooke, Roger. The History of Mathematics: A Brief Course. New York: John Wiley and Sons, 1997.

Drake, Stillman. Discoveries and Opinions of Galileo. Garden City, NY: Doubleday, 1957.

Eves, Howard. An Introduction to the History of Mathematics. New York: Saunders College Publishing, 1990.

Fields, Judith. Kepler's Geometrical Cosmology. Chicago: University of Chicago Press, 1988.

Heath, Thomas L. Greek Astronomy. New York: Dover Publishing, 1991.

Koestler, Arthur. The Watershed. New York: Anchor Books, 1960.

Taub, Liba C. Ptolemy's Universe. Chicago: Open Court, 1993.

HOW FAR IS THE MOON FROM EARTH?

In contrast to Anaximander's smaller estimate, Claudius Ptolemy would later prove that the distance of the Moon from Earth is actually approximately 64 times the radius of Earth.