Venus
0.006820
Earth
0.016750
Mars
0.093312
Jupiter
0.048332
Saturn
0.055890
This is why simplified versions of the Copernican and Ptolemaic theories (with no epicycles in the Copernican theory and only one epicyle for each of the five planets in the Ptolemaic theory) would have worked pretty well.*
The replacement of circles with ellipses had another far-reaching implication. Circles can be generated by the rotation of spheres, but there is no solid body whose rotation can produce an ellipse. This, together with Tycho’s conclusions from the comet of 1577, went far to discredit the old idea that planets are carried on revolving spheres, an idea that Kepler himself had assumed in the Mysterium Cosmographicum. Instead, Kepler and his successors now conceived of planets as traveling on freestanding orbits in empty space.
The calculations reported in Astronomia Nova also used what later became known as Kepler’s second law, though this law was not clearly stated until 1621, in his Epitome of Copernican Astronomy. The second law tells how the speed of a planet changes as the planet moves around its orbit. It states that as the planet moves, the line between the Sun and the planet sweeps out equal areas in equal times. A planet has to move farther along its orbit to sweep out a given area when it is near the Sun than when it is far from the Sun, so Kepler’s second law has the consequence that each planet must move faster the closer it comes to the Sun. Aside from tiny corrections proportional to the square of the eccentricity, Kepler’s second law is the same as the statement that the line to the planet from the other focus (the one where the Sun isn’t) turns at a constant rate—that is, it turns by the same angle in every second. (See Technical Note 21.) Thus to a good approximation, Kepler’s second law gives the same planetary velocities as the old idea of an equant, a point on the opposite side of the center of the circle from the Sun (or, for Ptolemy, from the Earth), and at the same distance from the center, around which the line to the planet turns at a constant rate. The equant was thus revealed as nothing but the empty focus of the ellipse. Only Tycho’s superb data for Mars allowed Kepler to conclude that eccentrics and equants are not enough; circular orbits had to be replaced with ellipses.15
The second law also had profound applications, at least for Kepler. In Mysterium Cosmographicum Kepler had conceived of the planets as being moved by a “motive soul.” But now, with the speed of each planet found to decrease as its distance from the Sun increases, Kepler instead concluded that the planets are impelled in their orbits by some sort of force radiating from the Sun:
If you substitute the word “force” [vis] for the word “soul” [anima], you have the very principle on which the celestial physics in the Commentary on Mars [Astronomia Nova] is based. For I formerly believed completely that the cause moving the planets is a soul, having indeed been imbued with the teaching of J. C. Scaliger* on motive intelligences. But when I recognized that this motive cause grows weaker as the distance from the Sun increases, just as the light of the Sun is attenuated, I concluded that this force must be as it were corporeal.16
Of course, the planets continue in their motion not because of a force radiating from the Sun, but rather because there is nothing to drain their momentum. But they are held in their orbits rather than flying off into interstellar space by a force radiating from the Sun, the force of gravitation, so Kepler was not entirely wrong. The idea of force at a distance was becoming popular at this time, partly owing to the work on magnetism by the president of the Royal College of Surgeons and court physician to Elizabeth I, William Gilbert, to whom Kepler referred. If by “soul” Kepler had meant anything like its usual meaning, then the transition from a “physics” based on souls to one based on forces was an essential step in ending the ancient mingling of religion with natural science.
Astronomia Nova was not written with the aim of avoiding controversy. By using the word “physics” in the full title, Kepler was throwing out a challenge to the old idea, popular among followers of Aristotle, that astronomy should concern itself only with the mathematical description of appearances, while for true understanding one must turn to physics—that is, to the physics of Aristotle. Kepler was staking out a claim that it is astronomers like himself who do true physics. In fact, much of Kepler’s thinking was inspired by a mistaken physical idea, that the Sun drives the planets around their orbits, by a force similar to magnetism.
Kepler also challenged all opponents of Copernicanism. The introduction to Astronomia Nova contains the paragraph:
Advice for idiots. But whoever is too stupid to understand astronomical science, or too weak to believe Copernicus without [it] affecting his faith, I would advise him that, having dismissed astronomical studies, and having damned whatever philosophical studies he pleases, he mind his own business and betake himself home to scratch in his own dirt patch.17
Kepler’s first two laws had nothing to say about the comparison of the orbits of different planets. This gap was filled in 1619 in Harmonices mundi, by what became known as Kepler’s third law:18 “the ratio which exists between the periodic times of any two planets is precisely the ratio of the 3/2-power of the mean distances.”* That is, the square of the sidereal period of each planet (the time it takes to complete a full circuit of its orbit) is proportional to the cube of the longer axis of the ellipse. Thus if T is the sidereal period in years, and a is half the length of the longer axis of the ellipse in astronomical units (AU), with 1 AU defined as half the longer axis of the Earth’s orbit, then Kepler’s third law says that T2 / a3 is the same for all planets. Since the Earth by definition has T equal to 1 year and a equal to 1 AU, in these units it has T2 / a3 equal to 1, so according to Kepler’s third law each planet should also have T2 / a3 = 1. The accuracy with which modern values follow this rule is shown in the following table:
(The departures from perfect equality of T2 / a3 for the different planets are due to tiny effects of the gravitational fields of the planets themselves acting on each other.)
Never entirely emancipated from Platonism, Kepler tried to make sense of the sizes of the orbits, resurrecting his earlier use of regular polyhedrons in Mysterium Cosmographicum. He also played with the Pythagorean idea that the different planetary periods form a sort of musical scale. Like other scientists of the time, Kepler belonged only in part to the new world of science that was just coming into being, and in part also to an older philosophical and poetic tradition.
The Rudolphine Tables were finally completed in 1627. Based on Kepler’s first and second laws, they represented a real improvement in accuracy over the previous Prutenic Tables. The new tables predicted that there would be a transit of Mercury (that is, that Mercury would be seen to pass across the face of the Sun) in 1631. Kepler did not see it. Forced once again as a Protestant to leave Catholic Austria, Kepler died in 1630 in Regensburg.
The work of Copernicus and Kepler made a case for a heliocentric solar system based on mathematical simplicity and coherence, not on its better agreement with observation. As we have seen, the simplest versions of the Copernican and Ptolemaic theories make the same predictions for the apparent motions of the Sun and planets, in pretty good agreement with observation, while the improvements in the Copernican theory introduced by Kepler were the sort that could have been matched by Ptolemy if he had used an equant and eccentric for the Sun as well as for the planets, and if he had added a few more epicycles. The first observational evidence that decisively favored heliocentrism over the old Ptolemaic system was provided by Galileo Galilei.
With Galileo, we come to one of the greatest scientists of history, in a class with Newton, Darwin, and Einstein. He revolutionized observational astronomy with his introduction and use of the telescope, and his study of motion provided a paradigm for modern experimental physics. Further, to an extent that is unique, his scientific career was attended by high drama, of which we can h
ere give only a condensed account.
Galileo was a patrician though not wealthy Tuscan, born in Pisa in 1564, the son of the musical theorist Vincenzo Galilei. After studies at a Florentine monastery, he enrolled as a medical student at the University of Pisa in 1581. Unsurprisingly for a medical student, at this point in his life he was a follower of Aristotle. Galileo’s interests then shifted from medicine to mathematics, and for a while he gave mathematics lessons in Florence, the capital of Tuscany. In 1589 Galileo was called back to Pisa to take the chair of mathematics.
While at the University of Pisa Galileo started his study of falling bodies. Some of his work is described in a book, De Motu (On Motion), which he never published. Galileo concluded, contrary to Aristotle, that the speed of a heavy falling body does not depend appreciably on its weight. It’s a nice story that he tested this by dropping various weights from Pisa’s Leaning Tower, but there is no evidence for this. While in Pisa Galileo published nothing about his work on falling bodies.
In 1591 Galileo moved to Padua to take the chair of mathematics at its university, which was then the university of the republic of Venice and the most intellectually distinguished university in Europe. From 1597 on he was able to supplement his university salary with the manufacture and sale of mathematical instruments, used in business and war.
In 1597 Galileo received two copies of Kepler’s Mysterium Cosmographicum. He wrote to Kepler, acknowledging that he, like Kepler, was a Copernican, though as yet he had not made his views public. Kepler replied that Galileo should come out for Copernicus, urging, “Stand forth, O Galileo!”19
Soon Galileo came into conflict with the Aristotelians who dominated the teaching of philosophy at Padua, as elsewhere in Italy. In 1604 he lectured on the “new star” observed that year by Kepler. Like Tycho and Kepler he drew the conclusion that change does occur in the heavens, above the orbit of the Moon. He was attacked for this by his sometime friend Cesare Cremonini, professor of philosophy at Padua. Galileo replied with an attack on Cremonini, written in a rustic Paduan dialect as a dialogue between two peasants. The Cremonini peasant argued that the ordinary rules of measurement do not apply in the heavens; and Galileo’s peasant replied that philosophers know nothing about measurement: for this one must trust mathematicians, whether for measurement of the heavens or of polenta.
A revolution in astronomy began in 1609, when Galileo first heard of a new Dutch device known as a spyglass. The magnifying property of glass spheres filled with water was known in antiquity, mentioned for instance by the Roman statesman and philosopher Seneca. Magnification had been studied by al-Haitam, and in 1267 Roger Bacon had written about magnifying glasses in Opus Maius. With improvements in the manufacture of glass, reading glasses had become common in the fourteenth century. But to magnify distant objects, it is necessary to combine a pair of lenses, one to focus the parallel rays of light from any point on the object so that they converge, and the second to gather these light rays, either with a concave lens while they are still converging or with a convex lens after they begin to diverge again, in either case sending them on parallel directions into the eye. (When relaxed the lens of the eye focuses parallel rays of light to a single point on the retina; the location of that point depends on the direction of the parallel rays.) Spyglasses with such an arrangement of lenses were being produced in the Netherlands by the beginning of the 1600s, and in 1608 several Dutch makers of spectacles applied for patents on their spyglasses. Their applications were rejected, on the ground that the device was already widely known. Spyglasses were soon available in France and Italy, but capable of magnification by only three or four times. (That is, if the lines of sight to two distant points are separated by a certain small angle, then with these spyglasses they seemed to be separated by three or four times that angle.)
Sometime in 1609 Galileo heard of the spyglass, and soon made an improved version, with the first lens convex on the side facing forward and planar on the back, and with long focal length,* while the second was concave on the side facing the first lens and planar on the back side, and with shorter focal length. With this arrangement, to send the light from a point source at very large distances on parallel rays into the eye, the distance between the lenses must be taken as the difference of the focal lengths, and the magnification achieved is the focal length of the first lens divided by the focal length of the second lens. (See Technical Note 23.) Galileo was soon able to achieve a magnification by eight or nine times. On August 23, 1609, he showed his spyglass to the doge and notables of Venice and demonstrated that with it ships could be seen at sea two hours before they became visible to the naked eye. The value of such a device to a maritime power like Venice was obvious. After Galileo donated his spyglass to the Venetian republic, his professorial salary was tripled, and his tenure was guaranteed. By November Galileo had improved the magnification of his spyglass to 20 times, and he began to use it for astronomy.
With his spyglass, later known as a telescope, Galileo made six astronomical discoveries of historic importance. The first four of these he described in Siderius Nuncius (The Starry Messenger),20 published in Venice in March 1610.
1. On November 20, 1609, Galileo first turned his telescope on the crescent Moon. On the bright side, he could see that its surface is rough:
By oft-repeated observations of [lunar markings] we have been led to the conclusion that we certainly see the surface of the Moon to be not smooth, even, and perfectly spherical, as the great crowd of philosophers have believed about this and other heavenly bodies, but on the contrary, to be uneven, rough, and crowded with depressions and bulges. And it is like the face of the Earth itself, which is marked here and there with chains of mountains and depths of valleys.
On the dark side, near the terminator, the boundary with the bright side, he could see spots of light, which he interpreted as mountaintops illuminated by the Sun when it was just about to come over the lunar horizon. From the distance of these bright spots from the terminator he could even estimate that some of these mountains were at least four miles high. (See Technical Note 24.) Galileo also interpreted the observed faint illumination of the dark side of the Moon. He rejected various suggestions of Erasmus Reinhold and of Tycho Brahe that the light comes from the Moon itself or from Venus or the stars, and correctly argued that “this marvelous brightness” is due to the reflection of sunlight from the Earth, just as the Earth at night is faintly illuminated by sunlight reflected from the Moon. So a heavenly body like the Moon was seen to be not so very different from the Earth.
2. The spyglass allowed Galileo to observe “an almost inconceivable crowd” of stars much dimmer than stars of the sixth magnitude, and hence too dim to have been seen with the naked eye. The six visible stars of the Pleiades were found to be accompanied with more than 40 other stars, and in the constellation Orion he could see over 500 stars never seen before. Turning his telescope on the Milky Way, he could see that it is composed of many stars, as had been guessed by Albertus Magnus.
3. Galileo reported seeing the planets through his telescope as “exactly circular globes that appear as little moons,” but he could not discern any such image of the stars. Instead, he found that, although all stars seemed much brighter when viewed with his telescope, they did not seem appreciably larger. His explanation was confused. Galileo did not know that the apparent size of stars is caused by the bending of light rays in various directions by random fluctuations in the Earth’s atmosphere, rather than by anything intrinsic to the neighborhood of the stars. It is these fluctuations that cause stars to appear to twinkle.* Galileo concluded that, since it was not possible to make out the images of stars with his telescope, they must be much farther from us than are the planets. As Galileo noted later, this helped to explain why, if the Earth revolves around the Sun, we do not observe an annual stellar parallax.
4. The most dramatic and important discovery reported in Siderius Nuncius was made on January 7, 1610. Training his telescope on Jupiter, Galileo sa
w that “three little stars were positioned near him, small but very bright.” At first Galileo thought that these were just another three fixed stars, too dim to have been seen before, though he was surprised that they seemed to be lined up along the ecliptic, two to the east of Jupiter and one to the west. But on the next night all three of these “stars” were to the west of Jupiter, and on January 10 only two could be seen, both to the east. Finally on January 13, he saw that four of these “stars” were now visible, still more or less lined up along the ecliptic. Galileo concluded that Jupiter is accompanied in its orbit with four satellites, similar to Earth’s Moon, and like our Moon revolving in roughly the same plane as planetary orbits, which are close to the ecliptic, the plane of the Earth’s orbit around the Sun. (They are now known as the four largest moons of Jupiter: Ganymede, Io, Callisto, and Europa, named after the god Jupiter’s male and female lovers.)*
This discovery gave important support to the Copernican theory. For one thing, the system of Jupiter and its moons provided a miniature example of what Copernicus had conceived to be the system of the Sun and its surrounding planets, with celestial bodies evidently in motion about a body other than the Earth. Also, the example of Jupiter’s moons put to rest the objection to Copernicus that, if the Earth is moving, why is the Moon not left behind? Everyone agreed that Jupiter is moving, and yet its moons were evidently not being left behind.
Though the results were too late to be included in Siderius Nuncius, Galileo by the end of 1611 had measured the periods of revolution of the four Jovian satellites that he had discovered, and in 1612 he published these results on the first page of a work on other matters.21 Galileo’s results are given along with modern values in days (d), hours (h), and minutes (m) in the table below:
Jovian satellite
Period (Galileo)
To Explain the World: The Discovery of Modern Science Page 18
