Of the classical astronomers, Copernicus is my favorite. He combines the best traits of the theorist and the observer. In his treatise that in 1543 presented to the broad scientific public his Sun-centric theory of planetary motion, On the Revolutions of Heavenly Spheres, Copernicus carefully develops his theory mathematically, and he skillfully marshals all of the available data in support of his theory and in the criticism of the Ptolemaic epicycle theory of planetary motion that then dominated.
Today we are so accustomed to the Copernican Solar System that we reflexively think of Copernicus's predecessors as naive. We forget the numerous difficulties that the astronomers at that time had to overcome to arrive at this simple theory, and we forget that Ptolemy and other proponents of an Earth-centric system had persuasive arguments in support of their theories.
Recall that the Ptolemaic solar system attempts to describe the orbital motion on the sky of the planets with sets of circular motion. The Sun and Moon move on circles around a stationary Earth. The three outer planets, Mars, Jupiter, and Saturn, move around Earth on two circular orbits, each described by the rotation of a sphere; the larger sphere, the eccentric, was centered on the Earth and rotated at a constant rate. The second sphere, the epicycle, is centered on a point attached to the eccentric, and rotates at a constant rate . The epicycle explains the retrograde motion—the eastward motion of a planet against the stars—that one sees when one of these planets is at opposition. Two spheres are used to describe the motion of Venus and Mercury, but in this case, the eccentric sphere centered on Earth rotates at the same rate as the Sun, and the center of the epicycle sphere is centered on a point that is in line with the Sun. Because the theory only explains the motion of the planets projected against the sky, there is no information about the distance of a planet from Earth. In fact, some variations of the theory placed Venus and Mercury farther away from Earth than the Sun, while others places them closer than the Sun. While the theory successfully describes the motion of the planets against the stars, it cannot explain why the brightness of a planet changes with position, or why a planet is brightest at opposition.
It is against this theory that Copernicus casts his own theory. He makes his arguments without the modern mathematics we now take for granted. Large portions of this book are devoted to geometric proofs of theorems that allow Copernicus to calculation angles with cords. I suspect most readers will approach these theorems in the manner that I did: read the theorem and either recall the result from trigonometry or derive the result with trigonometry. One does not appreciate until he reads this book the high mathematical hurtles that Copernicus jumps with only Euclidean geometry. With his geometric proofs and his tables of cords, he is able to compare his theory directly to the observations of planetary motion.
A key part of Copernicus's reasoning is that his theory can explain the variety of observed phenomena with a minimum of motions. We forget that Copernicus explains not only the orbital motion of the planets on the sky with his theory, but also the phases of the moon and the change in the brightness of a planet as it orbits the Sun. We also forget that in arguing for this system, he must explain why the arguments of Ptolemy based on the contemporary understanding of physics are invalid.
Our own physics leads us to a different perception from Copernicus's of the planetary motions. In describing the motion of the Sun across the sky over a year, Copernicus talks about the three motions that Earth must make. My first reaction to this statement is what is the third motion? We have the rotation of the Earth, and we have the motion of the Earth around the Sun. What else is there? But I've been trained to think in terms of the conservation of angular momentum, so Earth's axis points in one direction; for Copernicus, this pointing in one direction is a third motion, because he thinks that it is natural for the direction of the pole to move with the Sun. (To be more precise, Copernicus uses this rotation to account for both the seasons and the drift of the equinoxes, while today we think of only motion associated with the drift of the equinoxes).
Copernicus would not be out of place in today's scientific world. While we have a better understanding of physics than he did, the current methods of science are no different than those that he used. Because of this kinship, a reader can gain insights into how science is conducted by reading Copernicus.
This book takes some effort to read, but most of the difficulty comes from our unfamiliarity with the way that astronomers of the 16th century think about and describe the planets. The mathematics is high school level geometry; many of the most interesting sections contain no mathematics. The paperback version published in English by Prometheus Books contains footnotes that explains Ptolemy's theory for the Solar System and help illuminate the more opaque parts of Copernicus's arguments.