Mathematics and the Real World: The Remarkable Role of Evolution in the Making of Mathematics (2014)
CHAPTER III. MATHEMATICS AND THE VIEW OF THE WORLD IN EARLY MODERN TIMES
17. ELLIPSES VERSUS CIRCLES
An important step in presenting a picture of the world as we know it was made by Johannes Kepler. We will first state Kepler's three laws of planetary motion, which constitute his main contribution to the mathematical description of the world. We will then review the tortuous path by which he reached these understandings.
Kepler's first law: The path of the planets about the Sun is elliptical in shape, with the Sun being located at one focus of the ellipse.
Kepler's second law: The speed of each planet on its orbit is such that the line from its location on the ellipse to the Sun covers equal areas in equal periods of time.
Kepler's third law: The square of the time taken by a planet for a complete orbit around the Sun is proportional to the cube of its average distance from the Sun. This can be written as
T2 = kD3,
where T is the time of the planet to complete one round, D is the average distance of the planet from the Sun, and k is the same constant for all planets.
It is hard to overstate the mental difficulties Kepler overcame in presenting his three laws, and in particular the first one. The first law was formulated in accordance with the classical Greek approach describing the world with the help of geometry. At a remove of four hundred years, the transition from a circle to an ellipse seems simple. Today we actually consider a circle as a special instance of an ellipse in which the two foci converge. Kepler's first law, however, represented a contradiction to the deep-rooted tradition of two thousand years that as the firmament is perfect, the orbits of the planets must be perfect and, hence, circular. Kepler changed circles for ellipses and did not offer a reason or alternative purpose. His law was based on observations and measurements. At that time, those observations did not constitute clear proof of his claim, as the ellipses along which the planets moved were very close to circles, and the means for measurement in those days made it very difficult to differentiate between them. Kepler's ellipses did however enable the use of epicycles revolving around deferents to be abandoned, thereby offering a great simplification of the system.
Kepler's second law was also based on observations and measurements, without a reason or a purpose. This law also contradicted a two-thousand-year-old convention that was also based on the purpose of divine perfection, according to which the planets move at a uniform speed on their circular orbits, and the fact that we see the planets moving at varying speeds is the result of an optical illusion caused by their moving on epicycles whose center revolves on a deferent. Again, Kepler did not offer a reason or an explanation for the mathematical relation he found. He did not ignore divine wisdom, but he claimed that just as previously it had been thought that the Creator's wisdom had made the planets move at a uniform speed, it was the Creator's wisdom that made the planets depict equal angular areas in equal time intervals.
The second law compares a planet's speed at different locations on its path but does not refer to the actual speed itself. The third law does, and it too is based on observations. Kepler went as far as discovering the mathematical relation, which also is not precise, as it was not clear how to calculate the average distance of a planet from the Sun, but here again he did not explain or justify why his law was correct. The empirical confirmation of the law was possible only because the ellipses traced by the planets were almost circles, almost with fixed radii.
Kepler's laws constituted a crucial turning point in the relation between mathematics and the description of the solar system in at least two aspects. One was the abandonment of the unchallenged assumption that the purpose of the planets was to move along circular orbits. The second was the presentation of the laws of nature in the form of mathematical relations that were based purely on measurements and observations. These two aspects represented a sharp break with the general tradition of Greek thought, and in particular with Aristotelian thought, that had been the undisputed dominant system of thought until then. Kepler himself was greatly influenced by the Greek tradition, and the break with it caused him to do a lot of soul-searching, as we shall see.
Johannes Kepler was born in 1571 in the German town of Weil der Stadt, and he died in 1630. His family was a problematic one for various reasons: His father was a drunkard, violent toward his family and those around him, in trouble with the law, drawn to adventure, serving as a volunteer in the czar's army to fight against the rebel Protestants in Holland (although he himself was a Protestant). In doing so, he abandoned his wife, his firstborn Johannes, and the latter's younger brother for long periods without arranging for their livelihood, and eventually he simply did not return from one of his journeys. Johannes was apparently born prematurely and was a sickly child who remained sickly all his life. His mother also had a reputation as a wild, unconventional, rebellious woman. Her mother had been hanged as a witch, and she herself almost met with the same fate. Johannes was not provided with any formal basic education. When he started learning in an orderly fashion at the University of Tübingen, he set out to study theology, but when he graduated at the age of twenty, he was offered a position as a teacher of mathematics and astronomy in Graz, in the Austro-Hungarian Empire. For reasons that are not very clear, he was recommended for the post by his teachers in Tübingen, although he had only scanty knowledge of mathematics and astronomy. It would seem that they wanted to get rid of Kepler, who was not a popular student. They may have wanted to remove him because he had publicly expressed support for Copernicus, although he did not have much knowledge of Copernicus's work. His support of Copernicus's views was due to the influence of classical Greek literature that he read, particularly Pythagorean, according to which the Sun was the source of heat in the world and was therefore at its center. Kepler accepted the position in Graz, among other things because he had no other economic alternatives, and in his new job he immersed himself in the study of astronomy and mathematics, which in the course of time resulted in his contribution via the findings we have described. This was one of the first instances of such a major contribution by a scientist who came from a disadvantaged section of the population. The great majority of successful scientists and philosophers prior to Kepler came from the well-established social strata.
From his youth, Kepler was fascinated by mathematics in general and specifically by astrology, and a large part of his income over the years derived from astrological tables he compiled. He became famous because of his successful astrological prediction of the Turkish invasion of Austria. It is not clear whether he himself actually believed in astrology. Some of his writings contain derisive references to it while, on the other hand, he expressed the view that there was a foundation for the belief that the planets have a direct effect on our lives and that astrology could be developed on a scientific basis.
His leanings toward the mystical together with his support of the Pythagorean and Platonic approach to nature led him to search for links between numbers and geometric shapes and astronomic structures. Kepler knew of Plato's attempt to relate the existence of five perfect geometric solids to five elements of nature. At an early stage of his career it occurred to Kepler that there was a link between those five perfect shapes and the six planets known since ancient times. That happened long before the idea of elliptical orbits came to him. The assumption was that the planets move along circles that are on spheres, that is, that envelope balls. Kepler's idea was that each such ball surrounds and supports a perfect geometric body and is also supported by a perfect body; in total five perfect bodies that separate the paths of the six planets. Complex calculations that took many years resulted in his proposal of a precise model of the orbits and the perfect bodies that separated them. In that model, Mercury was on the ball that was closest to the Sun. The ball is supported by the octagonal body, which is itself supported by Venus's ball, which is in turn supported by the icosahedron (the twenty-sided figure) that is supported by the Earth, which itself is supported by the dodecahedron (i.e., the twelve-sided figure), which is surrounded by the sphere of Mars, which is contained within the pyramid that is supported by Jupiter's ball. The latter is within the cube that is surrounded by Saturn's ball. Kepler took the trouble to construct a tin model of this structure and later succeeded in interesting his patron, the prince, to build a model of the universe out of pure silver, but the model was never completed.
Along with his attempts to construct and improve the accuracy of his geometric model, Kepler sought more, and more-accurate, data on the movement of the celestial bodies. These data were in the possession of the most famous astronomer of his time, Tycho Brahe. Brahe was a Danish nobleman who became famous through his observational proof that the star that “was born” suddenly in 1577 was located among the other “fixed,” distant stars (which do not appear to move relative to each other), that is, it was not a lower body, such as a comet. At that time it was believed that the comets were closer to the Earth than was the Moon, and they were considered to be some sort of atmospheric disturbances. The appearance of the new star, followed by another new star that appeared in 1604 (these are known today as supernovae, stars that suddenly become far brighter than normal), were seminal events in astronomy in those days and constituted an important step toward undermining the Aristotelian approach that the firmament of the stars is fixed and unchanging. Ferdinand II, who became King of Denmark, built a sophisticated observatory for Tycho Brahe on a tower on a private island; there the Danish nobleman gathered a vast collection of accurate astronomical data, an effort unparalleled since the days of Ptolemy. Brahe's relationship with Prince Christian IV, Ferdinand's successor, was somewhat less cordial, and the astronomer departed for Prague with all his data, where, under the sponsorship of Emperor Rudolf II, he served as the imperial mathematician. Brahe did not believe in Copernicus's ideas but thought that the planets revolved around the Sun, which itself orbited the stationary Earth. Kepler, follower of Copernicus, set out for Prague to ask for Brahe's permission to use his data to support his theory of perfect bodies and to make it more accurate.
Kepler had another objective in mind in studying Brahe's data. The Pythagoreans had already noted that the relationships between musical harmonies are like those of natural numbers. Kepler, who was familiar with and influenced by this part of Greek mathematical writings, tried to match those harmonies with the orbits of the planets and the geometric bodies that separate them. In his book Harmonices Mundi (Harmony of the World), Kepler presents such an adaptation, in which Saturn and Jupiter are the basses, Mars is the tenor, the Earth and Venus are contraltos, and Mercury is the soprano. This description is accompanied by precise calculations with the support of astronomical readings that Kepler hoped to improve with the help of the information gathered by Tycho Brahe.
A few months after Kepler joined Brahe, the latter died and Kepler was appointed imperial mathematician. He continued with his calculations with the aid of Brahe's data, and after great effort and much soul-searching he arrived at the mathematical discoveries that we described at the beginning of this section. His strong loyalty to Copernicus's model aroused opposition from sections of the Church; in his later years he had to leave his post and return to Bavaria, where he died. Although he was convinced that the model he proposed was correct, he did not abandon his mystical notions about the harmonies of the heavens and the perfect bodies. In his last book he presented side by side his laws—Kepler's laws—that are still accepted today and his interpretation of the Pythagorean concepts of the celestial music in the hope that one day the two may be reconciled.
Kepler's laws did not provide an answer to the question that had arisen repeatedly regarding the heliocentric model: Why do people not fall off the Earth that is spinning so fast? The idea that the Earth somehow exerts a force that attracts people to it was raised in the time of the Greeks, but in itself the idea was not enough to provide a basis for the assertion. Without a quantitative theory—in other words, without a mathematical basis—the claim regarding the Earth's “attraction” lacked persuasive power. It is interesting to note that Kepler's answer to this difficulty indicates a thought process in terms of evolution. He did not seek a reason for the fact that people were not thrown off the Earth as a result of its rotation, but he claimed that a world in which people were thrown off Earth could not maintain humans. Hence, as humans do exist, the force of attraction also exists. With regard to the nature of the Earth's force, Kepler made do with the use of the term magnetic connection without any further explanation. It was Newton who invested mathematical content into the concept of gravity.
Viewed from the aspect of the motivation that drove them to their discoveries, the processes followed by Copernicus and Kepler were complete opposites. Copernicus rejected Ptolemy's model not because of its lack of precision but because of its lack of simplicity, and he preferred a less-accurate but more-aesthetic model. Kepler replaced circles with ellipses because the circles lacked accuracy, despite the fact that since the Greeks, ellipses were regarded as imperfect and thus unaesthetic. The dilemma of choosing between accuracy and simplicity has accompanied the development of science and continues even today.
The three giants, Galileo, Descartes, and Kepler, lived at the same time. Nothing is known of any meetings they may have had. Descartes supported Copernicus's astronomy but did not acknowledge that publicly, apparently because of concern that he would meet the same fate as Galileo. Descartes opposed Galileo's law of motion because it was based on observations and measurements and not on intellectual analysis. Kepler, as imperial mathematician, did not hesitate before complying with Galileo's request for a letter expressing support for his (Galileo's) findings. Galileo had no qualms about using Kepler's name and support for his findings, but he ignored Kepler's request for one of the telescopes he had built so that he, Kepler, could improve his data. Instead, Galileo opted to give the telescopes he had built to noblemen with no mathematical aptitude but with political influence. Kepler managed to obtain one of the telescopes through a mutual acquaintance. Galileo also opposed Kepler's law of ellipses and supported the Aristotelian axiom of planetary motion on perfect circles, while he himself had developed a different law of motion for earthly bodies, a law that was compatible with Kepler's law. Galileo did not mention Kepler in his scientific writings, presumably because he disdained his mystical Pythagorean arguments. Despite the lack of “harmony” between these three giants of scientific development, each made a crucial contribution to the imminent revolution in the relation between science and mathematics.