Like the Sun, the Moon moves eastward relative to the stars but at a faster rate, completing its motion in one month. The apparent motion of the Moon relative to the Sun produces the cycle of lunar phases as well as both lunar and solar eclipses. Ancient Greek mathematicians devised ways of estimating the distances and sizes of the Sun and Moon from observational data, including the phenomenon of parallax. The planets, too, appear to move relative to the stars. They generally move eastward relative to the stars but occasionally they halt their eastward motion and move westward (in retrograde motion) before resuming their normal eastward trek. The planets can be classified into two groups, inferior and superior, each of which displays certain characteristics of motion.

The ancient Greek mathematician Eudoxus developed a model for the motion of the Sun, Moon, and planets in which each body was carried around on a series of nested spheres that were all centered on Earth. Eudoxus’ geocentric model was incorporated into the highly successful cosmology of Aristotle. However, this model was unable to account accurately for the observed motions of the planets. Later astronomers such as Hipparchus and Ptolemy developed a new set of models in which each planet is carried around a circular epicycle, which in turn is carried around a circular deferent with its center near the Earth. Ptolemy even used these models to estimate distances to each planet. Although these models were quite accurate, they did suffer from some problems and were criticized or modified by medieval scholars.

This appendix provides mathematical details to supplement the ideas presented in the main text. Topic covered include: angular measurement, apparent diameter, trigonometry, finding the Sun’s altitude from the length of a shadow, determining the relative distances of the Sun and Moon, and finding the distance to an astronomical object using parallax measurements. In addition, this appendix shows how to calculate the sizes of epicycles in the Ptolemaic theory and the periods and sizes of planetary orbits in the Copernican theory. Mathematical details are also provided for Kepler’s Laws of Planetary Motion, Galileo’s measurement of mountains on the Moon, Galileo’s studies of falling bodies and projectiles, Newton’s universal gravitational force, and Bradley’s theory of the aberration of starlight.

Although Newtonian physics provided a sensible explanation for why the Earth should rotate on its axis and orbit the Sun, there was still no direct evidence for Earth’s motion. The first such evidence was provided by James Bradley, who attempted to reproduce Hooke’s parallax measurements and instead discovered the aberration of starlight. This slight displacement of a star’s apparent position occurs because of the Earth’s orbital motion and the finite speed of light. It was not until the late 1830s that astronomers finally detected annual stellar parallax, again confirming Earth’s orbital motion. Astronomers also sought direct evidence for Earth’s rotation. French astronomers confirmed that the Earth bulges out slightly at the equator, an effect that Newton had predicted as a result of Earth’s rotation. Experiments on the deflection of falling bodies also seemed to confirm Earth’s rotation, but the results were clouded in uncertainty. It was Foucault’s famous pendulum that provided the best direct evidence for the rotation of the Earth. These and other successes helped to establish the validity of Newtonian physics and brought about the successful conclusion of the Copernican Revolution.

The stars move from east to west across the sky each night. The ancient Greeks realized that the apparent movement of the stars would make sense if the stars were stuck on the inner surface of a giant celestial sphere that rotated around the Earth once every sidereal day. The Sun also moves from east to west across the sky, but not quite in the same way as the stars. The Sun’s motion can be tracked using shadows, and it appears to move eastward relative to the sphere of the stars along a path that is tilted relative to the celestial equator. The Sun completes its motion around the celestial sphere in one year, traveling through the constellations of the zodiac along a path called the ecliptic. As an observer moves around on Earth the apparent motions of the stars and Sun change in a way that shows the Earth to be spherical. The stars also display a very slow motion known as the precession of the equinoxes with a period of about 26,000 years.

This chapter introduces the fundamental mystery of the night sky: the wandering planets and their unusual retrograde motion. The story of the Copernican Revolution is the story of how this puzzle was solved, and then solved again. There are many reasons to learn about the Copernican Revolution: it is one of the great human intellectual accomplishments, it has had a tremendous effect on our understanding of the universe and the place of humanity within it, and it is a story that is often misunderstood. Furthermore, the Copernican Revolution is an excellent example of how science progresses. Science is a bit like puzzle solving, like making and using a map, like cooking, or even like doing art. New scientific theories provide us with new ways of perceiving the world that may be radically different from our previous perceptions. Learning the scientific story of the Copernican Revolution will help readers to better understand the nature of science.

Johannes Kepler was working as a mathematics teacher in Austria when he had a vision of how the universe must be constructed. Using the Copernican system as his model, Kepler thought that between each planetary orbital sphere was nested a regular polyhedron. There are only five regular polyhedra, so there could be only six planets. The relative sizes of the planetary orbits were set by the shapes that lay between them. Kepler’s idea caught the attention of Tycho Brahe and eventually he became Tycho’s assistant. When Tycho died, Kepler inherited Tycho’s accurate planetary data and he used these data to propose a new theory of planetary motion. Kepler found that the planets move in elliptical orbits with the Sun at one focus of the ellipse. Furthermore, Kepler believed the motion of the planets was powered by a force from the Sun that caused the planets to speed up when closer to the Sun and slow down when farther away. Kepler also discovered a curious mathematical relationship between the orbital periods of the planets and the size of the planetary orbits.

In 1671 Robert Hooke thought he had detected an annual parallax for the star Gamma Draconis, thus proving that the Earth orbits the Sun. Setting aside the uncertainty of Hooke’s meagre measurements, there remained the problem of how the Earth could orbit the Sun. Hooke thought he knew: the planets orbited the Sun because of a combination of straight line inertial motion and an attraction toward the Sun. But it was left to Hooke’s rival, Isaac Newton, to work out the mathematical details. While working out these details Newton established an entirely new physics based on three fundamental laws of motion and a universal gravitational attraction between all massive objects. Newton’s physics explained not only the orbits of planets, but also the motion of projectiles, the orbits of the Moon and comets, the precession of the equinoxes, and the tides. Newton’s physics was hailed in England but many European natural philosophers initially dismissed universal gravitational attraction as an “occult quality.”

While the Copernican theory was endorsed by some writers such as Thomas Digges and Giordano Bruno, most astronomers remained skeptical of a moving Earth. One such skeptic was Tycho Brahe, who set out to reform astronomy through improved observational methods. His careful observations of a new star (or “nova”) and a comet indicated that change could occur in the heavens, in contrast to the teachings of Aristotle. However, his observations of the angular size of stars seemed to contradict Copernicus’ theory. Tycho proposed a new geoheliocentric model in which the Earth sits at rest at the center of the cosmos and the Sun orbits around the Earth, but the planets orbit around the Sun. This model retained the stationary Earth but included some of the best features of the Copernican model. In an attempt to find evidence for his model, Tycho made extraordinarily extensive and accurate measurements of planetary positions, particular of the planet Mars.

In 1543, Nicolaus Copernicus published a radical new theory of the heavens. He proposed that the Earth rotates on its axis while the celestial sphere remains stationary. He also placed the Sun at rest near the center of the celestial sphere, while the Earth and other planets orbited around the Sun. Copernicus’ heliocentric theory could account for the motions of the stars, Sun, and planets about as well as Ptolemy’s theory did. It also helped to explain certain features of planetary motion that were mysterious in Ptolemy’s model. However, the idea that the Earth moved was too revolutionary for most of Copernicus’ contemporaries. While Copernicus believed that his model represented the real motions of the universe, most of his readers denied the Earth’s motion and accepted Copernicus' theory as nothing more than a useful mathematical device.