Kepler took his new position in When Brahe died the following year, Kepler was appointed to replace him. His first job was to prepare Brahe's collection of studies in astronomy for publication, which came out between and Kepler was also left in charge of Brahe's records, which forced him to make an Johannes Kepler.
A difference between his theory and Brahe's data could be explained only if the orbit of Mars was not circular but elliptical oval-shaped.
This meant that the orbits of all planets were elliptical Kepler's first law. This helped prove another of his statements. It is known as Kepler's second law, according to which the line joining a planet to the sun sweeps over equal areas in equal times in its elliptical orbit. Kepler published these laws in his discussion of the orbit of the planet Mars, the Astronomia nova The two laws were clearly spelled out in the book's table of contents.
They must have been seen by any careful reader alert enough to recognize a new idea of such importance. Still, the Italian astronomer Galileo Galilei — failed to use the laws in his printed works—although they would have helped his defense of Copernicus's ideas. In Rudolph II stepped down from the throne, and Kepler immediately looked for a new job.
He obtained the post of province mathematician of Linz, Austria. By the time he moved there in with his two children, his wife and his favorite son, Friedrich, were dead. Kepler's fourteen years in Linz were marked by his second marriage to Susan Reuttinger, and by his repeated efforts to save his mother from being tried as a witch. Kepler also published two important works while in Linz.
In the Harmonice mundi his third law was announced. It stated that the average distance of a planet from the sun, raised to the third power, divided by the square of the time it takes for the planet to complete one orbit, is the same for all planets. Kepler believed that nature followed numeric relationships since God created it according to "weight, measure and number.
In Linz Kepler worked still on the Rudolfinian tables. In his world harmony with which he also worked in Linz Kepler searched for a harmonic model of the universe.
He worked out the theoretical basics of music. In July he published his results concerning the world harmony in a five-volume book. It was called Harmonices mundi. In Linz he completed two more works: the Epitome astronomiae copernicanae and the Tabulae Rudolphinae. Kepler himself defended her. But it took six years until she was absolved. Kepler left Linz with his family in because of the pressure of the counter-reformation and went to Ulm.
There he finished his work on the Rudolfinian tables which were printed in Tabulae Rudolphinae. To facilitate the commerce he received in Ulm also the order to determine new measuring units.
On November 15, when being in Regensburg Johannes Kepler died. He was next to Galileo Galilei and Isaac Newton — one of the most important natural scientists of modern times. Kepler's First Law. The orbits of the planets are ellipses, with the sun at one focus of the ellipse. Kepler has as the first one rejected the complete circular motion in his tests on planetary motion.
Kepler published this law in his work Astronomia nova. Kepler's Second Law. The line joining the planet to the sun sweeps out equal areas in equal times as the planet travels around the ellipse.
That means the planet moves faster when it is near the sun than away from the sun — its speed is not constant, it changes with its distance from the sun. Also this law Kepler published in his work Astronomia nova. The ratio of the squares of the revolutionary periods for two planets is equal to the ratio of the cubes of their semimajor axes. At the same time, it should be pointed out that the third law is not necessarily the best point of departure for a dynamical, causal approach to motion, as Kepler intends here; for, in comparison with the previous causal approaches, the question of the location of the cause of power responsible for the production of motion remains relevant.
The spheres, which in the traditional view transported the planets, had been abolished since the time of Tycho. The fact that the orbits are elliptical and not circular, shows that the motions are not caused by a spiritual power but rather by a natural one, which is internal to the composition of matter. The motive power vix motrix comes indeed from the Sun, which sends its rays of light and power in all directions.
These rays are captured by the planets. Kepler, however, tries to explain this behavior of the planets less through astrology and much more through magnetism a physical phenomenon which was by no means clearly understood in his time. Firstly, the Sun rotates and, by so doing, sets in motion the planets around it. Secondly, since the planets are poles of magnets and the Sun itself acts with magnetic power, the planets are, at different parts of their orbits, either attracted or repelled; in this way the elliptical path is causally produced.
Kepler partially gives up the mechanical approach by postulating a soul in the Sun which is responsible for its regular motion of rotation, a motion on which, finally, the entire system depends. In addition to astronomy and cosmology Kepler expanded his causal approach to include the fields of optics see Section 6 below and harmonics Section 7 below.
At least officially, his positions at Graz, Prague, Linz, Ulm and Sagan can be characterized as the typical professional occupations of a mathematician in the broadest sense, i. Besides the field of astronomy and optics, where mathematics is ordinarily applied in different ways, Kepler produced original contributions to the theory of logarithms and above all within his favorite field, geometry especially with his stereometrical investigations.
For the medieval proportions theory as a background for Kepler's logarithms, see Rommevaux-Tani, Thus, on account of his natural predilection and talent and the importance of mathematics, particularly of geometry, for his thought, it is not surprising to find many different passages in his works where he articulated his philosophy of mathematics.
According to Kepler, each branch of knowledge must, in principle, be reducible to geometry if it is to be accepted as knowledge in the strong sense although, in the case of the physics, this condition is, as the AN emphasizes, only a necessary and not a sufficient condition. Thus, the new principles he was elaborating over the years in astrology were geometrical ones. A similar case occurs with the basic notions of harmony, which, after Kepler, could be reduced to geometry.
Of course, not every geometrical statement is equally relevant and equally fundamental. For Kepler, the geometrical entities, principles and propositions which are especially fundamental are those that can be constructed in the classical sense, i.
Once again, Kepler understood this within the framework of his cosmological and theological philosophy: geometry, and especially geometrically constructible entities, have a higher meaning than other kinds of knowledge because God has used them to delineate and to create this perfect harmonic world. From this point of view, it is clear that Kepler defends a Platonist conception of mathematics, that he cannot assume the Aristotelian theory of abstraction and that he is not able to accept algebra, at least in the way he understood it.
The best example of this is perhaps the heptagon. This figure cannot be described outside of the circle, and in the circle its sides have, of course, a determinate magnitude, but this is not knowable. Kepler himself says that this is important because here he finds the explanation for why God did not use such figures to structure the world.
Consequently, he devotes many pages to discussing the issue KGW 6, Prop. Certainly for a geometer like Kepler, approximations constitute — as mathematical theory—a painful and precarious way to progress.
The philosophical background for his rejection of algebra seems to be, at least partially, Aristotelian in some of its basic suppositions: geometrical quantities are continuous quantities which therefore cannot be treated with numbers that are, in the inverse, discrete quantities.
For, despite his mainly theoretical approach in the natural sciences, Kepler often emphasized the significance of experience and, in general, of empirical data. In his correspondence there are many remarks about the significance of observation and experience, as for instance in a letter to Herwart von Hohenburg from KGW 13, let.
In MC chapter 18 he quotes a long passage from Rheticus for the sake of rhetorical support when, as was the case here, the data of the tables he used did not fit perfectly with the calculated values from the polyhedral hypothesis.
In this passage, the reader learns that the great Copernicus, whose world system Kepler defends in MC, said one day to Rheticus that it made no sense to insist on absolute agreement with the data, because these themselves were surely not perfect. In part 2 chap. This hypothesis represents the best result which can be reached within the limits of traditional astronomy.
This works with circular orbits and with the supposition that the motion of a planet appears regular from a point on the lines of apsides. Against the traditional method, here, Kepler does not cut the eccentricity into equal parts but leaves the partition open. To check his hypothesis, he needs observations of Mars in opposition, where Mars, the Earth, and the Sun are at midnight on the same line.
In chapters 17—21, Kepler carries out an observational and computational check of his vicarious hypothesis. On the one hand, he points out that this hypothesis is good enough, since the variations of the calculated positions from the observed positions fall within the limits of acceptability 2 minutes of arc.
On the other hand, this hypothesis can be falsified if one takes the observations of the latitudes into consideration. Further calculations with these observations produce a difference of eight minutes, something that cannot be assumed because the observations of Tycho are reliable enough. Kepler also gave an important place to experience in the field of optics.
As a matter of fact, he began his research on optics because of a disagreement between theory and observation, and he made use of scientific instruments he had designed himself see, for instance, KGW For astrology, he uses meteorological data, which he recorded for many years, as confirmation material. This material shows that the Earth, as a whole living being, reacts to the aspects which occur regularly in the heavens see Boner , pp.
In his musical theory Kepler was a modern thinker, especially because of the role he gave to experience. As has been noted Walker, , p. Kepler does not accept that this limitation is founded on arithmetical speculations, even if this was already assumed by Plato, whom he often follows, and by the Pythagoreans. On the basis of his experiments, Kepler found that there are other divisions of the string that the ear perceives as consonant, i. Today Kepler is remembered in the history of sciences above all for his three planetary laws, which he produced in very specific contexts and at different times.
Figure 2. The first two laws were published initially in AN , although it is known that Kepler had arrived at these results much earlier. His first law establishes that the orbit of a planet is an ellipse with the Sun in one of the foci see Figure 2. The planet P is therefore faster at perihelion, where it is closer to the Sun, and slower at aphelion, where it is farther from the Sun.
The first law abolishes the old axiom of the circular orbits of the planets, an axiom which was still valid not only for pre-Copernican astronomy and cosmology but also for Copernicus himself, and for Tycho and Galileo.
The second law breaks with another axiom of traditional astronomy, according to which the motion of the planets is uniform in swiftness. Copernicus, for his own part, insisted on the necessity of the axiom of uniform circular motion. Kepler, on the contrary, affirms the reality of changes in the velocities of the planetary motions and provides a physical account for them. After struggling strenuously with established ideas which were located not only in the tradition before him but also in his own thinking, Kepler abandoned the circular path of planetary motion and in this way initiated a more empirical approach to cosmology though see Brackenridge In his Epitome , he provided a more systematic approach to all three laws, their grounds and implications see Davis ; Stephenson In Book 5, chapter 3, as point 8 of 13 KGW 6, p.
As a consequence of the third law, the time a planet takes to travel around the Sun will significantly increase the farther away it is or the longer the radius of its orbit. The background for his investigation into optics was undoubtedly the different particular questions of astronomical optics see Straker In this context he concentrated his efforts on an explanation of the phenomena of eclipses, of the apparent size of the Moon and of atmospheric refraction.
Kepler investigated the theory of the camera obscura very early and recorded its general principles see commentary by M.
Hammer in KGW 2, pp. Besides these impressive contributions, Kepler expanded his research program to embrace mathematics as well as anatomy, discussing for instance conic sections and explaining the process of vision see Crombie and especially Lindberg b.
Following—but also inverting—the Aristotelian argument for the temporality of motion, he affirms that the motion of light takes place not in time but in an instant in momento. Light is propagated by straight lines rays , which are not light itself but its motion.
It is important to note that although light travels from one body to another, it is not a body but a two-dimensional entity which tends to expand to a curved surface. The two-dimensionality of light is probably the main reason why it is incorporeal. For Straker, the supposed link between optics and physics especially in Prop.
Two questions are intensively discussed by modern specialists. Firstly, to what extent is the attribution of a mechanistic approach to Kepler justified? There are well—grounded arguments for different positions on both questions. For Crombie , and Straker, Kepler develops a mechanical approach, which can be particularly appreciated in his explanation of vision using the model of the camera obscura.
In addition, the concept of motion and the explanations using the model of the balance are indicative of a commitment to mechanicism Straker , pp. From a philosophical point of view, Kepler considered the HM to be his main work and the one he most cherished.
Containing his third planetary law, this work represents definitively a seminal contribution to the history of astronomy. Of course, music is involved and plays a determining role — and along with it the corresponding mathematical concepts stemming from the Pythagorean tradition. Social and political aspects are also included. The first is to be found among natural, sensible entities, like sounds in music or rays of light; both could be in proportion to one another and hence in harmony.
He resolves this matter by combining three of the Aristotelian categories: quantity, relation and, finally, quality. Through the function of the category of relation Kepler passes over to the active function of the mind or soul. It turns out that two things can be characterized as harmonic if they can be compared according to the category of quantity. This process takes place through the comparison of different sensible things with an archetype archetypus present in the mind.
The next central question directly concerns gnoseology, for Kepler gives a psychological account of the path followed by sensible things into the mind. They arrive at the imagination and from there go over to the sensus communis , so that, according to the traditional teaching, the sensible information received is now able to be processed and used in statements. KGW 6, p. How do they come into the soul? His discussion lies at the origin of the classical debate between empiricism and rationalism which was to dominate the philosophical scene for generations to come.
A connection with idealism is, of course, apparent see, for instance, Caspar , Engl. Historically, however, it seems to be more accurate to link his position with the philosophical tradition of St. Rather, both the Earth and human beings, ultimately, like all other living entities, are provided with a soul in which the geometrical archetypes are present. By the formation of an aspect in the heavens, symmetry arises and stimulates the soul of the Earth or of human beings.
Emmanuel Bury , where the author was a Prof. The author also wishes to thank David T.
0コメント