Copernicus and Kepler

Nilay Patel

Ancient astronomers gave considerable thought to the structure of the universe and to the laws governing the motions of heavenly bodies, looking both for convenient models for the purpose of calculation, as well as deeper explanations of the underlying p rinciples. One debate concerned the determination of the center of the universe. The "universe," or cosmos, of the ancients was similar to what we now think of as the solar system, except that the "fixed stars" were supposed to lie at its outer edge. The Greek philosopher Aristotle (ca. 350 BC) attempted to supply a scientific foundation for the generally accepted, common-sense theory of a geocentric, or earth-centered, universe. A Greek astronomer, Aristarchos of Samos (ca. 330 BC), put forth the first heliocentric, or sun-centered, theory of the universe [4, p.85]. Accompanying this debate over a sun-centered vs. an earth-centered universe was the problem of understanding or predicting the actual movements of the planets around the center, or from the perspective of the earth: the shapes of the paths they followed, their velocities, and their distances from the center. The adoption of a geocentric theory considerably complicated the task of understanding the motion of the planets around this supposed center. Aristotle believed the universe to be built in concentric spheres over a spherical Earth-the sphere being the most "perfect" solid figure, and circular motions being a distinguishing characteristic of perfection and, therefore, befitting descriptions of the perfect heavens.

The Greek astronomer Ptolemy (AD 100-170) was concerned with both understanding the observed planetary motions and allowing them to be calculated accurately and ensured the survival of Aristotle's universe for centuries by fitting to it a sophisticated mathematical model. Today, planetary orbits are known to be elliptical, but for a long time, largely due to Aristotle's influence, there existed a dogma that everything must be explained in terms of uniform circles. Ptolemy, though he built his system o n a foundation of interlocking circles, still provided planetary astronomy with its first oval, in his geocentric model: an oval somewhat pulled in at the waist, which approximated the desired orbit quite well, at least for small deviations from a circula r path. However, this oval was not an ellipse, which is a very special curve. Whereas an oval can be any vaguely circular closed curve, the ellipse is a particular oval: a conic section, whose shape depends on only one numerical parameter, called the e ccentricity (basically a measure of how much the curve deviates from a circle). Nor was his model able to account for all the available data. That point was not reached until the period between 1500 and 1543 A.D., when Polish astronomer Nicolaus Copernicus revolutionized the world of astronomy with his model of a heliocentric universe. He is renowned as an astronomer for having been able to work out a specific heliocentric system of planetary motion whose accuracy matched that of the observations at his disposal. Of equal importance was his notion that a quantitative understanding of the universe was as necessary as a qualitative one. By int roducing the sun into the theory of the motion of the planets, Copernicus made it possible to represent all in a single system, explaining many phenomena better than a geocentric model could, and explaining some phenomena that a geocentric model could not . However, as correct as his model was in many different areas, to explain motion he, too, philosophically maintained old ideas of circular motions and uniform velocities. Then, on the heels of Copernicus, with the understandings of the center of the universe and the motions of the planets still at large, came a man who "would change the course of man's quest to understand the nature of the physical world" [1, p.45]. The German astronomer Johannes Kepler (1571-1630) had many accomplishments: he did some work in optics, devised a method for finding volumes of solids of revolution, and helped calculate the Rudolphine Tables, which were the most accurate astronomical tables known for a long time, and which helped establish the utility of heliocentric astronomy. However, his greatest influence was on our understanding of the motions of the planets. He held firmly to the belief in fundamental laws (harmonies, in his phrase) w hich would explain the universe's design. This idea was based on the belief that the physical universe had a divine origin, and that any mathematical relationships in it were relections of this. Convinced that the complex motions of the planets must rel ate to simple underlying laws that can, and should, be expressed mathematically, corresponding to the intentions of the Creator, Johannes Kepler transformed astronomy by showing that certain regularities were demonstrated in planetary motions and stating these regularities in what are now known as Kepler's Laws of Planetary Motion, which led to a new fusion of physics and geometry The Copernican System Nicolaus Copernicus was born in 1473 in Torun, in Poland, and died in 1543. He was the youngest of four children. His father was a prosperous immigrant Polish merchant and his mother was a rich bourgeois woman. The family belonged to the typically Germ an aristocracy of the time. His father died when Copernicus was 10 years old, but his uncle on his maternal side, a bishop, saw to it that Copernicus continued his education. At 18, he registered as a student of liberal arts in the Natio Germanorum of t he University of Cracow, where he studied law and Aristotelian astronomy. At 22, his uncle had him named a canon, an ecclesiastical position that provided a generous lifetime income. Though heavily weighted with duties, rules, and regulations, the position allowed him much time for study and contemplation. In 1496, Copernicus obtained a leave of absence to study law at the Natio Germanorum of the University of Bologna. Here, he also studied astrology and Greek, and became an avid humanist, studying Greek philosophy and reading Ptolemy's major work Almagest in the original. Four years later he visited Rome and lectured on mathematics and registered as a medical student in the University of Padua-the leading medical school in Europe at the time-the following year. In 1503, he returned home and became secretary and special ph ysician to his ailing uncle. Three years later, Copernicus started the first serious work on his heliocentric system of the planets. In 1510, he moved to nearby Fraunberg. It was here, after his friends urged him to publish something of his astronomical research on a heliocentric system, that he wrote and circulated a few copies of a pamphlet called Commentariolus, which presented the early features of his system. Around this time, Copernicus also started working on his greatest work De revolutionibus orbium caelestium (On the Rev olutions of the Celestial Spheres), but it would not be published until the year of his death. By 1529, he had made enough observations to enable him to recalculate the major components of the supposed orbits of the Sun, moon, and planets around the Eart h; and by 1535 he had prepared new and much more accurate planetary tables to form the basis for the computation of almanacs. In 1539, Copernicus's disciple, Georg Rheticus, persuaded him to let his great work be published. Rheticus himself published a description summarizing the new cosmology in 1540 and again in 1541. It was called Narratio Prima and in it Copernicus was lef t anonymous, probably to see how the public would receive it before he acknowledged it [5, p.71]. Finally, in 1541, Copernicus quit stalling and prepared his manuscripts for publication-preparations which included writing a preface in which he briefly an d qualitatively described his planetary system, with the Sun at the center of the world and the Earth as one of the moving planets, and also dedicating his work to Pope Paul III in hopes that it would help avert ridicule of his system by the public. Interestingly, there was some controversy surrounding the publication. During the proofreading process, Andreas Osiander ( a mathematician and Lutheran pastor ( modified Copernicus's preface, supposedly after obtaining his permission to do so, by anonymo usly stating, "For these hypotheses need not be true or even probable. On the contrary, if they provide a calculus consistent with the observations, that alone is enough" [1, p.32]. He implies that the work is simply a mathematical device for predicting planetary positions, and should not be considered a physical system of cosmogony. At the time this preface was generally attributed to Copernicus. Rheticus found the statement cowardly. However, in the face of possible persecution for heresy, the stat ement served as a disclaimer for Copernicus, giving him a "welcome escape from the necessity of having to give too positive a statement of his personal viewpoints," even though he believed that his system probably was a physical reality [3, 108]. Historically, Copernicus is said to have caused a revolution, the so-called Copernican Revolution-"the gradual acceptance of the heliocentric idea of planetary cosmology and the abandonment in general of an anthropocentric viewpoint in nearly all human relations" [3, p.103]. He revived the model of the solar system with the Sun at the center and released Earth from its static (Aristotelian), central position and set it in motion, both around the Sun and on its own axis. "For Copernicus astronomy was an intellectual hobby" [1, p.28]. He decided to study the then known theories of planetary motion, and it was upon studying these that he endeavored to elaborate a new theory of planetary motions, which he discusses in De Revolutionibus. He states: I was led to think of a method of computing the motions of the spheres by nothing but the knowledge that Mathematicians [i.e. astronomers] are inconsistent in these investigations...I pondered long upon this uncertainty of the mathematical traditions in e stablishing the motions of the systems of the spheres...I therefore took pains to re-read the works of all the philosophers on whom I could lay my hands to find out whether any of them had ever supposed that the motions of the spheres were other than thos e demanded by the mathematical schools [5, p.69]. Copernicus noticed a great variety and multiplicity of astronomical systems and how great the differences of opinion were among the astronomers. He found the complexity of the orbital theories, their internal contradictions, their disagreements with each other, and the disagreement with the available data unacceptable. All these systems were unable to simultaneously reproduce the apparent motions of the planets exactly and remain faithful to the principle of uniform circular motion of celestial bodies, which Copernicus believed must be upheld. This led him to conclude that earlier astronomers either had overlooked some basic principles or introduced some false assumption into their systems. Furthermore, he found that some earlier astronomers had attri buted some motion to the earth. "Hence [he] thought that [he] too would be readily permitted to ascertain whether explanations sounder than those of [his] predecessors would be found for the revolution of the celestial spheres on the assumption of some m otion of the earth" [1, p.29]. Perhaps a similar assumption would help matters.

Copernicus revolutionized astronomy but was by no means of a revolutionary attitude. "Although he was capable of doing observational astronomy, he did not aspire to producing a complete overhaul of the science. He was primarily a philosopher who used his observations only to check certain constants of his system" [3, p.105]. He was a conservative, quiet thinker, somewhat timid and non-aggressive. He kept De Revolutionibus in abeyance for about thirty years, partly because he felt that advanced scient ific ideas should be discussed only among scientists, but mostly because he feared ridicule, unfair criticism, and even the possibility of persecution. "Copernicus was born into a world in which astronomers were groping for reform, and educated in a worl d in which only the first step-mastery of Ptolemy-had been taken" [5, p.68]. Ironically, though, he was a part of the establishment in terms of the Church, the Aristotelian orthodoxy, and the Ptolemaic tradition. However, as his astronomical studies pro gressed, he became increasingly dissatisfied with the Ptolemaic system of planetary motion. To fit observations of planetary motions, Ptolemy had been forced to offset the centers of regular motion of the planets slightly away from the Earth. Copernicus believed this was in conflict with the basic rule of true circular motion required by the Aristotelian doctrine-the failure of the system to meet this requirement and the incoherent and irrational picture of the universe that it provided being Copernicus 's major objections to it.

Still, he believed Ptolemy's system was right in conception, but simply wrong in details and needed a reinterpretation. He realized that up to that point there was no system of astronomy comparable to that of Ptolemy because none offered a computational scheme and method that could replace his. Thus, "If one genuinely wanted to supersede Ptolemy (as Copernicus did) one needed to offer more than a qualitative cosmology; one must, in addition, present a thoroughly worked out mathematical system, capable of giving results at least as good as those derived from the Ptolemaic system when used for computing planetary tables" [5, p.75]. Indeed, De Revolutionibus is identical to Ptolemy's Almagest in framework-written in careful parallel, book by book and sec tion by section, with the mathematical and computational methods revised for a different concept of planetary motion.

Copernicus paints his picture of the universe in De Revolutionibus. His system attributes motion to the Earth: two motions, in fact - a daily rotation and an annual revolution. The idea of a moving Earth, though, was thought to be absurd, dangerous, and contrary to common sense and, thus, required carefully reasoned arguments. One argument for why the Earth should not be assumed to move was that to move was contrary to its nature. According to Aristotle, the natural motion of the terrestrial elements w as rectilinear, whereas the natural motion of the celestial element was circular. Copernicus argued, however, that it was easier to imagine that the relatively small Earth moved than that the great heavens hurled themselves around every twenty-four hours -motions that would require tremendous speeds. Further, it was easier to imagine that the apparent motion of the heavens was really the result of the motion of the Earth, turning on its axis every twenty-four hours. Another argument against the motion o f the Earth was something that Ptolemy feared: that a rotating Earth would imply motions so violent that the Earth would fragment and be dispersed throughout the heavens. To this concern, Copernicus simply argued that everything within the reach of the e arth's attraction takes part in its rotation.

The motion of the Earth explained several motions at once. Earth's diurnal rotation produced the apparent rising and setting of the Sun, planets, and fixed stars. Earth's annual revolution about the Sun produced the apparent annual motion of the Sun, a nd the apparent retrogradations (back-and-forth movements) of certain planets. Explaining retrograde motion was one of the greatest problems faced by the geocentric system, but Copernicus's scheme did it immediately. In his system, retrograde motions we re only apparent motions, relative to the fixed stars. The actual motion of each planet was always in the same direction about the Sun, but did not seem to be so because of the Earth's motion. The Earth's motion causes the people on it to see a planet f rom different points of view against the background of fixed stars as the retrograde motion occurs, due to both planets' movements in the same direction around the Sun, but at different speeds. Copernicus continues, describing several more features of his system. The world has the form of a sphere, which is the most perfect and most roomy figure. He also makes the old arguments for the superiority of circular motion. For example, the motion of the heavenly bodies is uniform and circular since only a circle makes endless repetition possible, continuously bringing a body back to its original position. In his system, there was no one center of motion for all the heavenly bodies-even if the plan ets all revolved around the sun, the moon clearly revolved around the earth. Furthermore, Copernicus removed the Earth from the center of the universe and assumed the stationary Sun to be at the center instead. A point near the Sun, which was the center of the Earth's orbit, actually served as the center of motion of the planetary system. To Copernicus, this special positioning of the Sun explained and recognized the special properties of the Sun: it had always been given special consideration in the m inds of people, and it alone sheds the light and warmth that foster life. Copernicus also assumed a very large universe-finite but vast, consisting mostly of empty space. This allowed him to explain the problem of stellar parallax (any apparent shift in the positions of fixed stars as the Earth moves in its orbit around the Sun) inherent in a Sun-centered universe. He reasoned that the stars were at such enormous distances that their parallax was immeasurably small. Lastly, he arranged the universe as follows: the Sun at the center, Mercury, Venus, Earth, Mars, Jupiter, Saturn, and the sphere of fixed stars, which is the boundary and limit to the universe. He was able to arrange the planets in the correct order based on their speeds (the faster the p lanet, the closer it was to the center of the world) and the different sizes of their retrograde arcs (the smaller the arc, the farther the planet is from the Earth). Copernicus could not introduce any real proof for his opinions, but The Copernican would replace the Ptolemaic system, he believed, because it was simpler, more harmonious, more ingenious and more in keeping with the underlying philosophical basis, which demanded that the motions of the heavenly bodies, being perfectly circular, be represented by mathematical curves that were as nearly perfect circles as mig ht be [5, p.77]. His arguments in favor of his system were that it adhered religiously to the principle of uniform motion on a circle and that it provided a much more systematic and ordered picture of the universe than did the Ptolemaic system. However, his insistence on uniform circular motion for the planets caused his model to be as complicated as Ptolemy's in the sense that it employed upwards of thirty circles (compared to Ptolemy's seventy or so) to reproduce the observed motions of the planets. Nonetheless, his system was truer.

Kepler's Harmonies

Copernicus was a great inspiration to another astronomer: Kepler. Kepler made serving him his life's task, stating, "I deem it my duty and task to advocate outwardly also with all the powers of my intellect the Copernican theory, which I in my innermost have recognized as true and whose loveliness fills me with unbelievable rapture when I contemplate it" [7, p.384]. And Kepler was fascinated by the Copernican system for the superior order and harmony it seemed to display. His fascination serves to rev eal his own character. By order and harmony Kepler meant two slightly different aspects of the cosmos: one, a reflection of the properties of the divine creator, the other, a set of mathematico-physical relationships: a mystic harmony and a mathematical one [5, p.294]. Kepler was deeply mystical and fervently religious, as evidenced in his belief that the Divine Mind only creates in simple, perfect patterns. He was also a passionately devoted mathematical computer who only cared for those mathematical representations of the heavens which offered the possibility of interpretation in physical terms and who used mathematics as a tool to solve problems. To him, everything in nature was arranged in a mathematical way, and these harmonies could be discovered through calculat ion and should conform to accurate observation. Kepler was born in Weil der Stadt, near Stuttgart, in south-west Germany in 1571 and died on November 15, 1630 in Regensburg, Germany. His grandfather was a mayor of the town, and his father was a mercenary soldier who abandoned his family. Kepler recei ved scholarships offered by the dukes of the nearby town of Wurttemburg in order to get an education. He studied at schools and seminaries, and eventually attended the University of Tubingen. Here, he studied theology, and he also learned astronomy from a firm adherent of the Copernican system of the universe, Michael Mästlin. Kepler also adopted the system enthusiastically. He was recommended by the university to fill the position of mathematician at the Lutheran school at Graz, in Austria, and left in his final year of training in theology to take up the post in 1594.

Kepler's great experience at Graz was one he had while trying to discern a pattern that could explain why the numbers, sizes, and motions of the planetary orbits had their observed values. Kepler was a firm adherent of mathematical harmonies in the heavens [1, p.43], and while studying and teaching geometry in Graz, he was inspired to propose what was his first unified theory to explain the structure of the universe: a geometrical scheme for the relationships in the orbits of the planets that fit the me asured radial distances of the Copernican system to within about 5%. He saw that there were only five intervals between the six known planets and only five regular solid figures. Each of these solids can perfectly contain a sphere and be perfectly circu mscribed by a sphere. Thus, he proposed that the five regular solids could be fitted between the spheres of the six known planets and published his arrangement in his Mysterium Cosmographicum (Cosmographic Mysteries) in 1596. It reads: The Earth's orbit is a measure of all things; circumscribe around it a dodecahedron [12-sided figure] and the circle containing it will be Mars; circumscribe around Mars a tetrahedron [4-sided figure], and the circle containing it will be Jupiter; circums cribe around Jupiter a cube, and the circle containing it will be Saturn. Now inscribe within the Earth an icosahedron [20-sided figure], and the circle containing it will be Venus; inscribe within Venus an octahedron [8-sided figure], and the circle con taining it will be Mercury. You now have the reason for the number of the planets [1, p.43]. The scheme was of a nested cube, tetrahedron, dodecahedron, icosahedron, and octahedron.

Kepler's attitude in dealing with the scheme of the universe separates him from many of his predecessors because in his search for the harmonies of the heavens, "he was matching his geometry against measured quantities" [4, p.317]. He relied on the numbers to point the way for him rather than blindly follow any beliefs he may have had. This is important because in mathematics and science, credibility often lies with support or proof in the form of numbers.

In 1600, Kepler went to assist observational astronomer Tycho Brahe in his new observatory at Prague. They had a difficult working relationship. For one thing, Kepler expected free access to Tycho's data, but Tycho considered them his personal treasure s, guarding them heavily and releasing them to Kepler only in small amounts. Furthermore, Kepler's interests lied in the problems of Mars' orbit and the structure of the universe, but Tycho generally kept him working on other problems. All of this chang ed, however, when Tycho died in 1601 and Kepler was appointed his successor. Kepler finally had full access to Tycho's astronomical observations, which comprised an unparalleled collection because of its size and accuracy, an accuracy in which Kepler had the greatest faith. And above all, Kepler now found himself free to devote all his efforts to the determination of the orbit of Mars.

Using Tycho's observations, Kepler struggled with Mars' orbit. In his search for harmonies, he not only wanted to know the harmonies of scale in the universe but also their causes. He started by assuming circular orbits. After extensive calculations, Kepler realized that the motion of Mars could not be explained with the Earth at the center. The motion had to be relative to the position of the Sun and concluded that there was one force from the Sun that moved the planets around it, with the planets c loser to the Sun moving faster than the ones farther out. He decided that the central position of the Sun must be the key to understanding the causes of the planetary motions-evidence of Copernicus's influence on him. However, after eight years, he coul d not fit his ideas to Tycho's data, and boldly decided to abandon not the heliocentric theory but circular orbits-something Copernicus was unable to do. His decision was a testament to his faith in Tycho and his data, his faith in mathematics, his faith in himself, and his scientific integrity. He was willing to abandon his own belief in circular motion rather than question the observational data. He realized that Mars' orbit seemed to be noncircular, but the accuracy of the data depended on the preci se form of the orbit-how much it deviated from an actual circle. Only after much floundering with various ovals did Kepler finally realize that planets travel in elliptical orbits with the Sun at one focus. Finding the shape of the orbit was by no means an easy task. Kepler performed calculation upon calculation before finally arriving at the answer. He writes, "with extremely laborious demonstrations and by the handling of exceedingly many observations, I have discovered that the path of a Planet in the Heavens is not a circle, but an Oval route, perfectly Elliptical" [5, p.306]. This would become what is now his first law. "The law of ellipses represents a submission to facts far removed from a dream of perfection" [2, p.207]. Abandoning circula r orbits must have been a difficult step for Kepler because having the planets travel non-circular orbits went against his notions of simplicity and perfection in the world, the circle being perfect and simpler than the ellipse. Kepler was not quite sure he understood the physical meaning of the ellipse either because although the Sun was at one focus, the other focus and the center were empty. To Kepler, however, observation and computation did not lie, and in the end, circular perfection had to be alt ered in order to account for the reality of planetary orbits-another example of Kepler matching his geometry against measured quantities.

Besides discovering the shapes of planetary orbits, Kepler also tried to relate the velocity of a planet with its distance from the center of the universe. He became disturbed by the erratic way in which the planet moved through its orbit-its velocity varied according to no obvious law. The planet moved faster when approaching the Sun, and slower when receding from it, but there was no uniformity in the change of motion. Kepler wanted to find a mathematical expression for this variation as well as an explanation for its existence. He sought to derive the true motions of the planets from physical causes-a governing physical force that caused the planets to move the way they did. Books on the magnet by William Gilbert gave him the idea that magnetic f orces emanating from the Sun might explain planetary motions-that the planets were kept in their orbits by a force that decreased with distance from the sun. This force, if it spread like light in three dimensions, would vary as the inverse square of the distance. However, this did not work because if the force were confined to the orbit's plane, which is two dimensional, then its effect should diminish in proportion to the distance rather than the square of the distance. With this reasoning, Kepler ca me up with a distance law in which orbital velocity is inversely proportional to the distance from the Sun [8, p.393]. Also, the dogma of circular orbits required planetary motion to be uniform, meaning a planet must traverse equal arcs in equal intervals of time (i.e. planets move along their orbits with constant velocities). If the distance law holds, however, than this could not be true of noncircular elliptical orbits because, unlike circles, they are not uniform in shape. Their eccentricity demands that the distance from the solar focal point of a given planet in its orbit varies as the planet moves to different points along the orbit. Thus, with a distance-dependent force continuously acting on it, the planet's velocity at different points in its orbit must also vary because the strength of the force that moves it varies. Kepler, however, found another form of uniformity that would become what is now his second law of planetary motion. He showed that as a planet revolves in its orbit, the radi us vector from the Sun to the planet sweeps out equal areas of the ellipse in equal intervals of time. The real implication here is that a planet speeds up as it approaches the Sun and slows down at a regular rate as it moves farther away from it. Provi ding a physical explanation seems to be a bit of a weakness for Kepler. His strength is in analyzing complex astronomical data to find any patterns hidden in them. Still, "Kepler's explanation of the planetary motions is the first serious attempt to int erpret the mechanism of the solar system" [8, p.397]. What are today known as Kepler's first two laws of planetary motion were published in 1609 in his Astronomia Nova (New Astronomy): 1) Planets move round the Sun not in circles but in ellipses, the Sun being at one focus. 2) A planet does not move with a uniform speed but moves such that a line drawn from the Sun to the planet sweeps out equal areas of the ellipse in equal time. [1, p.47] Kepler also knew that planetary periods lengthened with distance from the sun and was interested in finding a possible relation between the two quantities. Looking at the data, he realized that there was a problem with the known distances of planets; the y were too inaccurate to be useful in providing the relationship he was searching for. Thus, he first dealt with the problem of the measurement of the distances of celestial bodies and developed geometric methods to measure the distances of the planets f rom the Sun. It took him ten more years to determine the distances to the other five known planets, measured relative to the distance from the Earth to the Sun. Although his measurement of the distance between the earth and Sun was not accurate, Tycho's data was accurate enough for Kepler to determine the relative distances to the planets precisely enough to find the relation between the period and the distance of a planet. This was to be what is now called his third law of planetary motion, which he p ublished in 1618 in his Harmonice Mundi (The Harmony of the World). He writes: 3) But it is absolutely certain and exact that the proportion between the periodic times of any two planets is precisely the sesquialterate proportion of their mean distances, that is, of the actual spheres [9, p.411] In other words, the square of the period of revolution of a planet round the Sun is proportional to the cube of the average distance of the planet from the Sun [1, p.48]. Theodor Jacobsen provides Kepler's "proof" for this law: " In the 'Harmony of the Universe' section of the Epitome Kepler says: 'In order to dispel an unbearable state of monotony in the Universe, the Creator made the following choices of the relations between distances and velocities, masses, and densities of the planets:

v1/v2 = a2/a1 , m1/m2 = a1/ a2 , d1/d2 = a2/a1 .'
Consider any two planets moving in circular orbits of radii a1 and a2. Assume that the 'mass' of any planet is proportional to the square root of the radius of its orbit, i.e., let m ~ a1/2. According to Aristotelian dynamics, resistance to motion ('slu ggishness') is proportional to 'mass,' and the speed was instantly conferred to a body acted upon by a 'projectile' force. Its magnitude would instantly attain and preserve a value inversely proportional to the resistance to motion. Thus v ~ 1/m, or equivalently v ~ 1/a1/2.

Now consider two planets m1 and m2. For these we have accordingly v1/v2=a21/2/a11/2. Of course s, the distance covered, is proportional to a, and the period will be P = 2 a/v. Combining these statements, we have

P2/P1 = (2 a2/v2)(v1/2 a1) = (a2/a1)( v1/v2) = (a2/a1)(a21/2/a11/2) = a23/2/a13/2.
Or (P2/P1)2 = (a2/a1)3, which 'proves' Kepler's Third Law for circular orbits" [3, p.252]. This can also be expressed as P2/a3 = k, where k is constant. Notice Jacobsen put quotation marks around the word "proof" and "proves." This suggests that it possibly is not Kepler's actual proof for his law, or that Jacobsen himself has his doubts abou t the logic of it. It does seem a little unfounded to have proportions between a planets mean distance from the Sun and its mass or density. Kepler called his third law "the harmonic law because it represented to him the 'divine harmony of the world.' T his law convinced him that the 'ultimate secret of the universe' would be found in an 'all embracing synthesis of geometry, music, astrology, astronomy and epistemology'" [1, p.48]. Interestingly, Kepler did not call his laws "laws," nor were these three laws the only ones. In fact, in The Harmony of the World, Kepler has thirteen "laws", of which his three famous ones are smaller parts. He never distinguished these three from the others; he saw all of them as equally important. Most of the other "law s" are now forgotten. The three that survived probably did so because they actually worked and served some importance to the progress of astronomy.

Nicolaus Copernicus and Johannes Kepler both revolutionized the world of astronomy. Copernicus not only showed that the assumption of the annual motion of the earth round the sun would explain in a very simple manner the most glaring irregularities in t he motions of the planets, but he built up a complete system of astronomy thereon. Kepler's laws of planetary motion revealed a planetary system capable of being described mathematically. Equally important to figuring out the motions of the planets was his attempt to provide physical reasoning for the geometry of their motions, attributing their cause to a magnetic force from the Sun. Thus, Kepler's laws provided a fusion of physics and geometry based on astronomical observations. In the centuries to come, mathematical expression becomes the only acceptable means of expressing scientific ideas because of the ease and clarity in which the universality of mathematics allows their communication.


Editor's Remarks

Back Back to the papers