From Ptolemy to Copernicus

Since the time of the Greeks, all learned and cultured Europeans knew that the Earth was a stationary object at the center of the universe. The Sun and the stars, a category which included fixed stars as well as moving planets, traveled around the Earth in motions composed of uniform and circular parts. This worldview, in which crystalline spheres rotated around the Earth, dominated until the beginning of the modern era. Yet given the evidence, a geocentric view seemed perfectly reasonable. Look outside on a clear night, and you will see the orderly rotation of the stars around a seemingly immovable Earth. In order to reconcile theory with observations, however, especially the unwelcome phenomenon of retrograde motion of planets, the Greek astronomer Ptolemy had postulated a modification to this simple system. The Ptolemaic system maintained the essential circular motion of stars and planets around the Earth. However, to represent accurately the retrograde motion, Ptolemy introduced to the planet's orbit an epicycle, a second circular orbit that itself moved along the deference circle in an orderly fashion. Ptolemy also added a third element, an equant, which varied the speed of the deference rotation, but this will not be a factor in our discussion. In 1543, the Polish astronomer Nicholas Copernicus published his best-known work, De Revolutionibus. In it, he argued that the Sun is in fact the center of the universe rather than the Earth, that the Earth is a planet just like Jupiter or Venus, and that all of the planets travel around the immovable Sun in fixed orbits. Copernicus did not present any new data on which to base his claim. Indeed, the experimental confirmation of many of his proposals would have to wait for centuries. More interestingly, however, through the use of simple geometry, it was possible to prove that the two systems are very nearly equivalent, that the Copernican system could represent the current data as accurately as the reigning Ptolemaic system. The first thing that we need to know about the Ptolemaic system is that it offered no way to ascertain the distance to the planets. The Greeks had a figure for the distance to the Sun, albeit too small by a factor of 20. As for the planets, their only directly observable quantity was their position as seen from the Earth. Here we see an inner planet such as Venus with its epicycle indifferent. We also see the Sun, which doesn't have an epicycle. Note the angle of the Earth-Venus line. Now let's see what happens when we move Venus further away from the Earth. As we move Venus, the angle to the Earth never changes. This means that we can place Venus as far or as close to the Earth as we want without contradicting the data from observation. One of the attributes of the Ptolemaic system that had never been ascribed to a physical cause was that the center of Venus's epicycle always lay on the Earth-Sun line. With the Copernican system, however, this attribute becomes very significant. Since we can scale up the orbits as much as we like, let's scale up Venus's path so that its deferent is the same size as the Sun's orbit around the Earth. Now we see the importance of that previously ignored fact. If the center of the epicycle is on the Earth-Sun line, when the orbit is scaled up to the size of the Sun, the deferent becomes the Sun's orbit, and the epicycle becomes an orbit around the Sun. In other words, we have made our first step from a geocentric system, in which planets travel around the Earth, to a heliocentric system, in which planets travel around the Sun. When Ptolemy arranged his cosmos, since he didn't know the distance to the planets, he could pick an arbitrary figure. In fact, he made all the orbits just the right size so that the crystalline spheres that contained them would fit together perfectly. If Venus goes around the Sun, however, its distance is fixed and certain. The ratio of deferent size to epicycle size was known to Ptolemy by observation, and since the deferent radius is now the Earth-Sun distance, which was also known, the size of Venus's orbit could easily be calculated. The period of the planet around its epicycle was also known, which meant that the period of the orbit around the Sun was also fixed and certain. This ability to calculate and fix the distances and periods for any planet around the Sun was one of the reasons the system of Copernicus was so attractive to its followers. This scheme may have worked well for an inner planet such as Venus, but for the outer planets, the system was decidedly more complicated. The center of their epicycles did not lie on the Earth-Sun line, as can be seen here. However, they also had a special relationship with the Sun that was not accounted for by Ptolemy. If you look closely at the path of this outer planet, let's call it Jupiter, you'll find that it is always at the same angle from the center of its epicycle as the Sun is from the Earth. Again, because the distance is arbitrary, we can scale up the path. Let's scale up Jupiter's path until its epicycle is the same size as the Sun's orbit. Since Jupiter is moving on its epicycle in the same way that the Sun is moving around the Earth, we can notice that the Sun-Jupiter distance is constant. In fact, we could describe Jupiter as moving in a constant orbit around the Sun, rather than its complicated orbit around the Earth. This new orbit will be the exact same size as its deferent around the Earth, as the parallelogram created here shows. This parallelogram connects the Earth, the Sun, Jupiter, and the center of Jupiter's epicycle. Because the sides of a parallelogram are equal, the radii of the circles are equal, and Jupiter's new orbit is the same size as its old deferent. Because the angles of a parallelogram are equal, Jupiter will move along its new orbit in the same way that its epicycle moves along the deferent, with uniform circular motion. So now we have once more turned a Ptolemaic orbit around the Earth into a circular orbit around the Sun. As it did for inner planets, the new system allows us to fix the distance. Jupiter's new orbit is just the size of its old deferent. But because the old epicycle size was known, the Earth-Sun distance, and because the ratio of epicycle size to deferent size was known through Ptolemy's calculations, again the distance from Sun to Jupiter can be calculated. Using these methods, both sets of planets, inner and outer, could be placed in orbit around the Sun instead of the Earth. This system had numerous attractions. First, it locked all of the planets into orbits of defined periods at given distances from the Sun, with the fastest-traveling planets closest to the center. Second, it used fewer large circles than the Ptolemaic system and was aesthetically simpler. Most importantly, however, it explained the phenomena without having to change the basic premise that the Earth was at rest in the center of the universe. Copernicus had considered this system. He rejected it, however, because it caused the path of Mars and the Sun to cross, and therefore did not allow the planets to be encased in crystalline spheres. It was not until many years later that Tycho Brahe's observations of the comet smashed the notion of crystalline spheres and made the system possible. Also, Tycho's failure to find a parallax of Mars convinced him that the Earth was at rest. As a result, he championed this system, and it is called the teconic system in his honor. Luckily for Copernicus, a simple construction could achieve his goal and place the Sun at the Imagine an observer on the Earth in the teconic system. He would see all the planets going around the Sun, and the Sun going around a stationary Earth once a year. But what about an observer on the Sun? The observer on the Sun would see the inner and outer planets going around him, and the Sun would seem at rest. He would also see the Earth traveling around the Sun once a year with uniform motion. In this way, we have moved from the Ptolemaic system to a full-fledged Copernican system. The Copernican transition was not so simple as this. Because planets' orbits are not really circular, Copernicus had to put in a small epicyclic for each planet to keep the predictions accurate. Furthermore, outside of simple elegance, there was little to recommend the Copernican system over its predecessor. What's more important, however, is that the geometry of Ptolemy, which placed at the center of the Universe an immutable Earth, could be employed to rearrange the Universe, to move the Earth from its hallowed position, and make it one of many planets, all circling a motionless Sun.