thegoodfellas/blogset-br huggingface dataset:
wrong, the argument of a sun-centered force had put him on the right track.As a consequence Kepler also insisted that the Earth in his model should behave just like the other planets. Some traditional Earth-specific features could still be seen in Copernicuss theory, but Kepler now exploited all aspects of heliocentrism. In this way the assumptions became simple, natural, and uniform, though it turned out that he had to invent new techniques and go through long and complicated calculations in order to truly describe the orbits. The focus was on the planet Mars. Keplers warfare on Mars engaged him between 1600 and 1605, and the process has been called the birth of the scientific method.In his first attempts he tried to keep a circular orbit for Mars around the sun. To account for the observations, he put the sun a little off center, and on the opposite side he reinserted an equant; i.e., seen from this point Mars moved uniformly. Even though he varied the relative distances of the sun and the equant from the center of the orbit, the model refused to come sufficiently close to Brahes trusted observations. If the distances were equal, however, he noted that Mars linear velocity seemed to be inversely proportional to its distance from the sun. This distance law was strictly valid only when the planet was furthest from or nearest to the sun, but he hypothesized the rule for the whole orbit. Later he found the correct rule by substituting the linear velocity with its component perpendicular to the radius vector from the sun. From the distance law it was theoretically possible to determine the position of Mars in its orbit at a given time. With no calculus available, but inspired by ideas from Archimedes and Nicholas of Cusa, he first resorted to numerical summation to compute an elliptic integral and then solved the inverse problem by interpolation. The calculations became exceedingly laborious, but he found a shortcut by summing areas instead of distances: the line joining a planet to the sun sweeps out equal areas in equal time. It has become known as the area law or Keplers second law. Sector areas in an eccentric circle were known, and the calculations could be simplified. Typical is Keplers use of conjectures which, even if not completely accurate, led him forward to new discoveries and relations, from which he could go back and modify the conjectures and their connections. The level of proof was set by Tycho Brahes observations and the requirement that all motions should be physically natural.The next muse, Stathmica (second from right), is meant to illustrate the area law. She holds a steelyard balance. The sun is at the fulcrum and (instead of a fish and some weights) two planets are hanging so that the system is in equilibrium. This is a simplified version of a figure in Book IV of Epitome astronomiae Copernicanae (1621), the full and mature outline of Keplers achievements and the basis for the Tables. His idea of how the motion of a planet is balanced under the
force from the sun is symbolized in this simple figure. We may also see both the law of the lever and Keplers area law as expressing the principle of conservation of angular momentum. Newton later showed the area law to be valid under any central force.Implicit in Keplers models thus far had been the traditional assumption that the Earth moved in an eccentric circular orbit, but uniformly with respect to the center (i.e., no need for a distinct equant). He set as his next task to find out whether this was really true. The determination of a planets orbit around the sun required of course a proper description of how the Earth moved. By a famous new method in which he used specific Mars observations from Tychos abundant data, Kepler found that also in the case of the Earth the equant point had to differ (slightly) from the center of the orbit. Now he used this better approximation of the Earths orbit, together with the area law, to continue his attack on Mars.But with its larger eccentricity, the circular orbit model for Mars still refused to agree with Tychos observations. It took Kepler two more years to finally arrive at the correct elliptic orbit. It was clear that the circle had to be compressed to an oval of some kind. He first tried with the boundary of an asymmetric two-dimensional egg, since this path could be generated by an (old-style) epicycle. He approximated this oval with an ellipse in order to more simply use his area law; sector areas were explicitly known for ellipses since antiquity. Finally, he found a close fit when he chose a new ellipse halfway between the circle and the approximating ellipse.He might have stopped here and triumphed. But just as when he abandoned the circle, he still did not understand how an elliptic orbit could be physically motivated. As a fortunate consequence of the further pursuit, he arrived at an explicit construction of the orbit. An opening came when he noticed a striking geometric expression for the distance between the sun and a specific position of Mars. Again he could make a natural generalization of this expression around the whole orbit, representing the distance as a function of the orbital angle he had used before, the so-called eccentric anomaly. This angle was measured at the center along the eccentric auxiliary circle. He was disappointed to find that the orbit (called the via buccosa) did not comply with the observations sufficiently well, and it was not even an ellipse. Struck by a last brilliant idea, Kepler now realized that he could represent the sun-to-Mars direction by another generalization, and still keep the earlier relation between distance and eccentric anomaly. In technical terms he had redefined the relation between the eccentric anomaly and the true anomaly (the angle of the Mars position seen from the sun). And this time the calculated orbit was in accordance with Tychos data. He could also prove that the curve was an ellipse with the right eccentricity, and he claimed that, as described, the
ellipse had physical grounds. (It was shown much later that also the via buccosa actually fulfilled all of Keplers criteria. He seems to have made an incorrect implicit assumption which caused the discrepancy. Perhaps Kepler would have attacked and resolved this problem as well, had he been aware of it! At least one can say that he had taken his theory down to the very limits of Tychos observations.)Kepler was ready in 1605 to publish the new results. In the exceptional book Astronomia nova he included also the setbacks and failures along the road to the ellipse. Conflicts with the Brahe family and financial problems delayed the printing until 1609.In the right center position on the dome, the muse known as Doctrina triangulorum (the science of triangles), i.e., trigonometry, reminds us of Keplers first law, as it was later called: planets move in ellipses with the sun at one focus. In a firm grip she holds the traditional square and compasses. In front of her we see a figure which Kepler used to describe and determine his last ellipse. It is shown dashed inside its auxiliary circle. The sun S is at a focus, and some positions of Mars are designated by F, H, and L along the ellipse. Later in the book, in Chapter 20 of the explanatory notes before the actual tables, Kepler refers to this tiny figure found only in the frontispiece!Just above, you cannot miss the Habsburg eagle hovering over the dome. Rudolph II and the two succeeding emperors here got their due share. The bird of prey generously drops coins from its beak, which we may follow down through the temple.The fourth muse in the left center position bears the name of Logarithmica. For Kepler it seemed as though the heavens repaid him for his efforts when he heard of Napiers invention of the logarithms in 1617. He understood at once their importance and how much time and effort would be saved in the computations necessary for his Tables, and he became the first to apply logarithms on a large scale. Napiers second book, which explained the construction, had not yet appeared, so Kepler began to develop his own theory and logarithmic tables. He derived the independent heavy construction from Euclids theory of proportions. Theory and tables were published in two books a few years later. For the convenience of the reader, Kepler cleverly included two logarithmic tables in the Rudolphine Tables as well. Take a look at the halo above Logarithmicas head where the number 69314.72 shines. We recognize it today as a multiple of the basic loge 2. Actually, Keplers first table gives 105 loge x, while the second tabulates 105 loge sin y. So x = 2 and y = 30 give the logarithmic value in the halo. The lengths of the two rods that muse Logarithmica carries are in proportion 1 : 2. In music theory, this ratio corresponds to the basic octave interval. Such harmonies always played a vital role in Keplers thoughts.It is interesting to note that it was when Kepler
learned and thought about the new logarithms that he also discovered his so-called third law: for any two planets the squares of their orbital periods are to each other as the cubes of their mean distances from the sun. Keplers mind was now open to thinking in terms of geometric sequences.The last two muses both reflect the fact that Kepler wrote two books which laid the foundation for theoretical optics. You do not hear much about this since he neither invented the telescope (he had bad eyes) nor discovered the exact law of refraction (he only needed a good approximation). Descartes and others later acknowledged Kepler as the master of optics. Early questions regarding eclipses and refraction led Kepler to write the first modern book on optics, Astronomiae pars optica (1604). This was done in the midst of his intense Mars studies. He also came up with novel ideas about conic sections when he treated curved mirrors.On the far left, the radiating head of the muse Physica lucis et umbrarum (the physics of light and shadows) seems to cast a shadow cone behind the sphere in her right hand. The sphere and shadow also bear some resemblance to a comet with a tail. Among the several comets Kepler observed and described during his life, one was sighted in 1607. It was to reappear 75 years later and became known as Halleys comet.The muse Optica (second from left) exhibits a telescope of an odd rectangular form. In 1610 Kepler first heard of Galileos telescope and what it had revealed about the heavens. Again he had immediately recognized and acknowledged a new revolutionary invention. As with the logarithms he wrote an important treatise, Dioptrice, explaining mathematically how light passed through systems of lenses, especially in Galileos magical tube. He went on to design a new type of telescope with a convex lens also for the eyepiece instead of Galileos concave one, which allowed for higher magnification. At the top we see six scientific muses bordering the dome, all at the service of Queen Urania above them. Such allegorical figures were abundant in medieval and Renaissance illustrations, typically personifying the seven liberal arts which formed the educational canon. Kepler here instead invokes a different set of sciences covering important discoveries, inventions, and ideas in his career.We start at the right side of the dome with the charged figure of Magnetica holding a lodestone and a compass. Already in his first work Kepler had insisted on a physical cause of planetary motion. Earlier it was argued that you could not, or at least should not try to, figure out the physical grounds. Kepler, with his firm belief that all the planets orbited the sun in almost the same plane, thought that such a force must emanate from the sun. And without the unifying theory of gravity and the conception of inertial motion, he turned to magnetism. This was the force in vogue at a time when the Earth had recently been found to act as a giant magnet. The rotation of the sun and the magnetic
interaction between sun and planet caused the planet to orbit the sun, Kepler proposed. Even though these physical speculations, and other neo-Platonic ideas based on simplicity of natural found that also in the case of the Earth the equant point had to differ (slightly) from the center of the orbit. Now he used this better approximation of the Earths orbit, together with the area law, to continue his attack on Mars.But with its larger eccentricity, the circular orbit model for Mars still refused to agree with Tychos observations. It took Kepler two more years to finally arrive at the correct elliptic orbit. It was clear that the circle had to be compressed to an oval of some kind. He first tried with the boundary of an asymmetric two-dimensional egg, since this path could be generated by an (old-style) epicycle. He approximated this oval with an ellipse in order to more simply use his area law; sector areas were explicitly known for ellipses since antiquity. Finally, he found a close fit when he chose a new ellipse halfway between the circle and the approximating ellipse.He might have stopped here and triumphed. But just as when he abandoned the circle, he still did not understand how an elliptic orbit could be physically motivated. As a fortunate consequence of the further pursuit, he arrived at an explicit construction of the orbit. An opening came when he noticed a striking geometric expression for the distance between the sun and a specific position of Mars. Again he could make a natural generalization of this expression around the whole orbit, representing the distance as a function of the orbital angle he had used before, the so-called eccentric anomaly. This angle was measured at the center along the eccentric auxiliary circle. He was disappointed to find that the orbit (called the via buccosa) did not comply with the observations sufficiently well, and it was not even an ellipse. Struck by a last brilliant idea, Kepler now realized that he could represent the sun-to-Mars direction by another generalization, and still keep the earlier relation between distance and eccentric anomaly. In technical terms he had redefined the relation between the eccentric anomaly and the true anomaly (the angle of the Mars position seen from the sun). And this time the calculated orbit was in accordance with Tychos data. He could also prove that the curve was an ellipse with the right eccentricity, and he claimed that, as described, the ellipse had physical grounds. (It was shown much later that also the via buccosa actually fulfilled all of Keplers criteria. He seems to have made an incorrect implicit assumption which caused the discrepancy. Perhaps Kepler would have attacked and resolved this problem as well, had he been aware of it! At least one can say that he had taken his theory down to the very limits of Tychos observations.)Kepler was ready in 1605 to publish the new results. In the exceptional book Astronomia nova he included also the setbacks and failures along the road to the ellipse. Conflicts with the Brahe family
and financial problems delayed the printing until 1609.In the right center position on the dome, the muse known as Doctrina triangu- lorum (the science of triangles), i.e., trigonometry, reminds us of Keplers first law, as it was later called: planets move in ellipses with the sun at one focus. In a firm grip she holds the traditional square and compasses. In front of her we see a figure which Kepler used to describe and determine his last ellipse. It is shown dashed inside its auxiliary circle. The sun S is at a focus, and some positions of Mars are designated by F, H, and L along the ellipse. Later in the book, in Chapter 20 of the explanatory notes before the actual tables, Kepler refers to this tiny figure found only in the frontispiece!Just above, you cannot miss the Habsburg eagle hovering over the dome. Rudolph II and the two succeeding emperors here got their due share. The bird of prey generously drops coins from its beak, which we may follow down through the temple.The fourth muse in the left center position bears the name of Logarithmica. For Kepler it seemed as though the heavens repaid him for his efforts when he heard of Napiers invention of the logarithms in 1617. He understood at once their importance and how much time and effort would be saved in the computations necessary for his Tables, and he became the first to apply logarithms on a large scale. Napiers second book, which explained the construction, had not yet appeared, so Kepler began to develop his own theory and logarithmic tables. He derived the independent heavy construction from Euclids theory of proportions. Theory and tables were published in two books a few years later. For the convenience of the reader, Kepler cleverly included two logarithmic tables in the Rudolphine Tables as well. Take a look at the halo above Logarithmicas head where the number 69314.72 shines. We recognize it today as a multiple of the basic loge 2. Actually, Keplers first table gives 105 loge x, while the second tabulates 105 loge sin y. So x = 2 and y = 30 give the logarithmic value in the halo. The lengths of the two rods that muse Logarithmica carries are in proportion 1 : 2. In music theory, this ratio corresponds to the basic octave interval. Such harmonies always played a vital role in Keplers thoughts.It is interesting to note that it was when Kepler learned and thought about the new logarithms that he also discovered his so-called third law: for any two planets the squares of their orbital periods are to each other as the cubes of their mean distances from the sun. Keplers mind was now open to thinking in terms of geometric sequences.The last two muses both reflect the fact that Kepler wrote two books which laid the foundation for theoretical optics. You do not hear much about this since he neither invented the telescope (he had bad eyes) nor discovered the exact law of refraction (he only needed a good approximation). Descartes
and others later acknowledged Kepler as the master of optics. Early questions regarding eclipses and refraction led Kepler to write the first modern book on optics, Astronomic pars optica (1604). This was done in the midst of his intense Mars studies. He also came up with novel ideas about conic sections when he treated curved mirrors.On the far left, the radiating head of the muse Physica lucis et umbrarum (the physics of light and shadows) seems to cast a shadow cone behind the sphere in her right hand. The sphere and shadow also bear some resemblance to a comet with a tail. Among the several comets Kepler observed and described during his life, one was sighted in 1607. It was to reappear 75 years later and became known as Halleys comet.The muse Optica (second from left) exhibits a telescope of an odd rectangular form. In 1610 Kepler first heard of Galileos telescope and what it had revealed about the heavens. Again he had immediately recognized and acknowledged a new revolutionary invention. As with the logarithms he wrote an important treatise, Dioptrice, explaining mathematically how light passed through systems of lenses, especially in Galileos magical tube. He went on to design a new type of telescope with a convex lens also for the eyepiece instead of Galileos concave one, which allowed for higher magnification. With unflagging enthusiasm Hebenstreit reveals in the poem that behind the dome, out of our sight, there are six more muses personifying other sciences which had engaged Keplers thoughts.When you look at a page in the Rudolphine Tables, give a thought to the mathematical and astronomical symbols. These special printing types were precious to Kepler. He only had confidence in his own set, which he carried around from city to city in the 1620s when the war came closer.Shortly before his death in 1629 Johannes Kepler published a little pamphlet where he pointed out that two years later at specified times both of the inner planets, Mercury and Venus, were to pass across the face of the sun. He also described how to observe these events for the first time. The prediction of the Mercury transit in 1631 could be seen in Europe, and it turned out to be correct within a few hours.More and more people accepted that at last someone had figured out how our planets move in the sense that meaningful predictions could be made. On planet Earth the Thirty Years War still raged. In a few years, a young man would be seen sitting under an apple tree, perhaps in the orchard around Keplers temple, pondering over how to further embellish the temple and explain its construction. But that was a constellation the Rudolphine Tables could not predict.FIGURE 1. Clear copy of the cover: Frontispiece designed by Johannes Kepler for the Tabula Rudolphine (the Rudolphine Tables). The Habsburg eagle is flying above the temple, and directly below it is Urania, the muse of astronomy. Arrayed around her above the front columns of the temple are six more muses: Physica lucis (at left), Optica, Logarithmica, Doctrina
triangulorum, Stathmica, and Magnetica. Within the temple we see an ancient astronomer (back center), then (from left to right) Hipparchus, Copernicus, Brahe, and Ptolemy. Kepler is modestly positioned on the base of the temple in the second-to-left panel. Institut Mittag-Leffler, Aurav agen 17, SE-18260 Djursholm, Sweden E-mail address: ragstedt@mittag-leffler.seIn: BULLETIN (new series) OF THEAMERCICAN MATHEMATICAL SOCIETY Volume 50, Number 4, October 2013, Pages 629639S 0273-0979(2013)01416-2Article electronically published on June 10, 2013Under the flag of war: Mapping during 600 years anniversary of the astronomical tower clock situated at Old Town Square in center of Prague.