Search Results for Kepler

Biographies

  1. Kepler biography
    • Johannes Kepler .
    • Johannes Kepler is now chiefly remembered for discovering the three laws of planetary motion that bear his name published in 1609 and 1619).
    • A large quantity of Kepler's correspondence survives.
    • In consequence, we know rather a lot about Kepler's life, and indeed about his character.
    • It is partly because of this that Kepler has had something of a career as a more or less fictional character (see historiographic note below).
    • Kepler was born in the small town of Weil der Stadt in Swabia and moved to nearby Leonberg with his parents in 1576.
    • As a child, Kepler lived with his mother in his grandfather's inn.
    • Kepler's early education was in a local school and then at a nearby seminary, from which, intending to be ordained, he went on to enrol at the University of Tubingen, then (as now) a bastion of Lutheran orthodoxy.
    • Kepler's opinions .
    • Throughout his life, Kepler was a profoundly religious man.
    • Man being, as Kepler believed, made in the image of God, was clearly capable of understanding the Universe that He had created.
    • Moreover, Kepler was convinced that God had made the Universe according to a mathematical plan (a belief found in the works of Plato and associated with Pythagoras).
    • Since some authors have given Kepler a name for irrationality, it is worth noting that this rather hopeful epistemology is very far indeed from the mystic's conviction that things can only be understood in an imprecise way that relies upon insights that are not subject to reason.
    • Kepler does indeed repeatedly thank God for granting him insights, but the insights are presented as rational.
    • At Tubingen Kepler was taught astronomy by one of the leading astronomers of the day, Michael Mastlin (1550 - 1631).
    • Kepler did not take this attitude.
    • At Tubingen, Kepler studied not only mathematics but also Greek and Hebrew (both necessary for reading the scriptures in their original languages).
    • At the end of his first year Kepler got 'A's for everything except mathematics.
    • Probably Mastlin was trying to tell him he could do better, because Kepler was in fact one of the select pupils to whom he chose to teach more advanced astronomy by introducing them to the new, heliocentric cosmological system of Copernicus.
    • It was from Mastlin that Kepler learned that the preface to On the revolutions, explaining that this was 'only mathematics', was not by Copernicus.
    • Kepler seems to have accepted almost instantly that the Copernican system was physically true; his reasons for accepting it will be discussed in connection with his first cosmological model (see below).
    • It seems that even in Kepler's student days there were indications that his religious beliefs were not entirely in accord with the orthodox Lutheranism current in Tubingen and formulated in the 'Augsburg Confession' (Confessio Augustana).
    • Kepler's problems with this Protestant orthodoxy concerned the supposed relation between matter and 'spirit' (a non-material entity) in the doctrine of the Eucharist.
    • This ties up with Kepler's astronomy to the extent that he apparently found somewhat similar intellectual difficulties in explaining how 'force' from the Sun could affect the planets.
    • In his writings, Kepler is given to laying his opinions on the line - which is very convenient for historians.
    • These may explain why Mastlin persuaded Kepler to abandon plans for ordination and instead take up a post teaching mathematics in Graz.
    • Kepler was excommunicated in 1612.
    • Kepler's first cosmological model (1596) .
    • Instead of the seven planets in standard geocentric astronomy the Copernican system had only six, the Moon having become a body of kind previously unknown to astronomy, which Kepler was later to call a 'satellite' (a name he coined in 1610 to describe the moons that Galileo had discovered were orbiting Jupiter, literally meaning 'attendant').
    • Kepler's answer to these questions, described in his Mystery of the Cosmos (Mysterium cosmographicum, Tubingen, 1596), looks bizarre to twentieth-century readers (see the figure on the right).
    • Kepler did not express himself in terms of percentage errors, and his is in fact the first mathematical cosmological model, but it is easy to see why he believed that the observational evidence supported his theory.
    • Kepler saw his cosmological theory as providing evidence for the Copernican theory.
    • Kepler asserts that its advantages over the geocentric theory are in its greater explanatory power.
    • Kepler lists nine such questions in the first chapter of the Mysterium cosmographicum.
    • Kepler carried out this work while he was teaching in Graz, but the book was seen through the press in Tubingen by Mastlin.
    • The agreement with values deduced from observation was not exact, and Kepler hoped that better observations would improve the agreement, so he sent a copy of the Mysterium cosmographicum to one of the foremost observational astronomers of the time, Tycho Brahe (1546 - 1601).
    • Kepler got the job.
    • Naturally enough, Tycho's priorities were not the same as Kepler's, and Kepler soon found himself working on the intractable problem of the orbit of Mars [(See Appendix below)].
    • He continued to work on this after Tycho died (in 1601) and Kepler succeeded him as Imperial Mathematician.
    • Tycho had made a huge number of observations and Kepler determined to make the best possible use of them.
    • Kepler concluded that the orbit of Mars was an ellipse with the Sun in one of its foci (a result which when extended to all the planets is now called "Kepler's First Law"), and that a line joining the planet to the Sun swept out equal areas in equal times as the planet described its orbit ("Kepler's Second Law"), that is the area is used as a measure of time.
    • ., Heidelberg, 1609), Kepler found orbits for the other planets, thus establishing that the two laws held for them too.
    • Both laws relate the motion of the planet to the Sun; Kepler's Copernicanism was crucial to his reasoning and to his deductions.
    • The actual process of calculation for Mars was immensely laborious - there are nearly a thousand surviving folio sheets of arithmetic - and Kepler himself refers to this work as 'my war with Mars', but the result was an orbit which agrees with modern results so exactly that the comparison has to make allowance for secular changes in the orbit since Kepler's time.
    • It was crucial to Kepler's method of checking possible orbits against observations that he have an idea of what should be accepted as adequate agreement.
    • Kepler may have owed this notion at least partly to Tycho, who made detailed checks on the performance of his instruments (see the biography of Brahe).
    • Meanwhile, in response to concerns about the different apparent diameter of the Moon when observed directly and when observed using a camera obscura, Kepler did some work on optics, and came up with the first correct mathematical theory of the camera obscura and the first correct explanation of the working of the human eye, with an upside-down picture formed on the retina.
    • He also wrote about the New Star of 1604, now usually called 'Kepler's supernova', rejecting numerous explanations, and remarking at one point that of course this star could just be a special creation 'but before we come to [that] I think we should try everything else' (On the New Star, De stella nova, Prague, 1606, Chapter 22, KGW 1, p.
    • Following Galileo's use of the telescope in discovering the moons of Jupiter, published in his Sidereal Messenger (Venice, 1610), to which Kepler had written an enthusiastic reply (1610), Kepler wrote a study of the properties of lenses (the first such work on optics) in which he presented a new design of telescope, using two convex lenses (Dioptrice, Prague, 1611).
    • Kepler's years in Prague were relatively peaceful, and scientifically extremely productive.
    • Kepler wrote to a friend that this death was particularly hard to bear because the child reminded him so much of himself at that age.
    • Then Kepler's wife died.
    • Kepler had to leave Prague.
    • Kepler seems to have married his first wife, Barbara, for love (though the marriage was arranged through a broker).
    • Kepler's new wife, Susanna, had a crash course in Kepler's character: the dedicatory letter to the resultant book explains that at the wedding celebrations he noticed that the volumes of wine barrels were estimated by means of a rod slipped in diagonally through the bung-hole, and he began to wonder how that could work.
    • ., Linz, 1615) in which Kepler, basing himself on the work of Archimedes, used a resolution into 'indivisibles'.
    • Kepler's main task as Imperial Mathematician was to write astronomical tables, based on Tycho's observations, but what he really wanted to do was write The Harmony of the World, planned since 1599 as a development of his Mystery of the Cosmos.
    • The Harmony of the World also contains what is now known as 'Kepler's Third Law', that for any two planets the ratio of the squares of their periods will be the same as the ratio of the cubes of the mean radii of their orbits.
    • From the first, Kepler had sought a rule relating the sizes of the orbits to the periods, but there was no slow series of steps towards this law as there had been towards the other two.
    • Kepler made last-minute revisions.
    • While Kepler was working on his Harmony of the World, his mother was charged with witchcraft.
    • Katharina Kepler was eventually released, at least partly as a result of technical objections arising from the authorities' failure to follow the correct legal procedures in the use of torture.
    • However, Kepler continued to work.
    • Kepler was accordingly delighted when in 1616 he came across Napier's work on logarithms (published in 1614).
    • (Similar comments were made about computers in the early 1960s.) Kepler's answer to the second objection was to publish a proof of how logarithms worked, based on an impeccably respectable source: Euclid's Elements Book 5.
    • Kepler calculated tables of eight-figure logarithms, which were published with the Rudolphine Tables (Ulm, 1628).
    • The astronomical tables used not only Tycho's observations, but also Kepler's first two laws.
    • And as the years mounted up, the continued accuracy of the tables was, naturally, seen as an argument for the correctness of Kepler's laws, and thus for the correctness of the heliocentric astronomy.
    • Kepler's fulfilment of his dull official task as Imperial Mathematician led to the fulfilment of his dearest wish, to help establish Copernicanism.
    • By the time the Rudolphine Tables were published Kepler was, in fact, no longer working for the Emperor (he had left Linz in 1626), but for Albrecht von Wallenstein (1583 - 1632), one of the few successful military leaders in the Thirty Years' War (1618 - 1648).
    • Wallenstein, like the emperor Rudolf, expected Kepler to give him advice based on astrology.
    • Kepler naturally had to obey, but repeatedly points out that he does not believe precise predictions can be made.
    • Like most people of the time, Kepler accepted the principle of astrology, that heavenly bodies could influence what happened on Earth (the clearest examples being the Sun causing the seasons and the Moon the tides) but as a Copernican he did not believe in the physical reality of the constellations.
    • Kepler died in Regensburg, after a short illness.
    • Much has sometimes been made of supposedly non-rational elements in Kepler's scientific activity.
    • In his influential Sleepwalkers the late Arthur Koestler made Kepler's battle with Mars into an argument for the inherent irrationality of modern science.
    • Both are, however, based on very partial reading of Kepler's work.
    • In particular, Koestler seems not to have had the mathematical expertise to understand Kepler's procedures.
    • The truly important non-rational element in Kepler's work is his Christianity.
    • Kepler's extensive and successful use of mathematics makes his work look 'modern', but we are in fact dealing with a Christian Natural Philosopher, for whom understanding the nature of the Universe included understanding the nature of its Creator.
    • Preface to a translation of Kepler's Foundations of modern optics .
    • Kepler's planetary laws .
    • Kepler's elliptical orbit for Mars .
    • Honours awarded to Johannes Kepler .
    • Lunar featuresCrater Kepler .
    • Planetary featuresCrater Kepler on Mars .
    • Pass Magazine (Kepler's proofs) .
    • Kevin Brown (Kepler and the third law) .
    • AMS (The Kepler sphere packing problem) [registration required] .
    • George W Hart (Kepler's polyhedra) .
    • http://www-history.mcs.st-andrews.ac.uk/Biographies/Kepler.html .

  2. Horrocks biography
    • Since Horrocks left Cambridge with a deep knowledge of the latest ideas in astronomy due to Copernicus and Kepler, as well as the expertise in mathematics to further develop their ideas, this tells us that he studied mathematics and astronomy in his own time.
    • He turned to Kepler's Tabulae Rudolphinae which had been published in 1627.
    • He therefore accepted Kepler's theory of elliptical orbits for the planets and tested Kepler's laws by direct observation.
    • However he rejected Kepler's theory of why the planetary orbits were ellipses, which was based on alternate attraction and repulsion of a planet by the sun.
    • Now with his greater understanding, Horrocks set to work improving Kepler's tables.
    • Kepler had predicted a transit of Venus would occur in 1631, and that another would occur in 1761.
    • Kepler had died in 1630 but even if he had lived he would not have seen the transit of 1631 since it was not visible in Europe as the sun was below the horizon during the transit.
    • Horrocks, after correcting Kepler's tables realised that a transit of Venus would occur on 24 November 1639, and that it would be visible from England.
    • 18 (2) (1987), 77-94.',16)">16] where Wilson traces the origin of Horrocks' theory in Kepler's work on the motion of the moon, as transformed and calibrated by further data, in particular critical data concerning the duration of lunar eclipses.

  3. Harriot biography
    • He began to develop a theory for the rainbow and, by 1606, Kepler had heard of the remarkable results on optics achieved by Harriot.
    • Kepler wrote to Harriot, but the correspondence never really achieved any significant exchange of ideas.
    • Perhaps Harriot was too wary of the difficulties that his work had nearly brought on him, or perhaps he did (as he claimed to Kepler) still intend to publish his results if his health permitted.
    • Kepler had discovered the comet six days earlier but it would be the observations of Harriot and his friend (and student) William Lower which eventually were used by Bessel to compute its orbit.
    • Later, in his correspondence with Kepler about atomic theory, Harriot mentioned the packing problem.
    • Kepler could not solve the problem but he believed that the densest packing of spheres would be attained if in each layer the centres of the spheres were above the centres of the holes in the layer below.

  4. Mastlin biography
    • As was the case with many young scholars including Kepler, his most famous student, [Mastlin] did his undergraduate studies at a preparatory school and came to the university to take his final exams and pick up his baccalaureate degree.
    • Mastlin had several students who became noted mathematicians, the most famous being Kepler.
    • Perhaps his greatest achievement (other than being Kepler's teacher) is that he was the first to compute the orbit of a comet, although his method was not sound.
    • However for the more advanced lectures he adopted the heliocentric approach - Kepler credited Mastlin with introducing him to Copernican ideas while he was a student at Tubingen (1589-94).
    • Michael Mastlin is regularly ascribed a firm if minor place in accounts of the Astronomical Revolution, as the teacher of Johannes Kepler and as an early believer in the physical reality of the Copernican system.
    • Another first for Mastlin is an accurate calculation of the golden ratio as "approximately 0.6180340" stated in a letter he wrote to Kepler in 1597.

  5. Brahe biography
    • Thus, as was also the case in the earlier study of fixed stars, Kepler's belief that Tycho's observations could be trusted to better than two minutes is amply confirmed.
    • Johannes Kepler joined him as an assistant, to help with mathematical calculations.
    • He received support from Rudolph for Kepler and himself to compile a new set of astronomical tables based on Tycho's recorded observations over 38 years.
    • Kepler describes his death (see for example [The nobleman and his housedog (London, 2002).',5)">5]):- .
    • When Tycho died, Kepler succeeded him as Imperial Mathematician.
    • They allowed Kepler, who (unlike Tycho) was a convinced follower of Copernicus, to deduce his three laws of planetary motion (1609, 1619) and to construct astronomical tables, the Rudolphine Tables (Ulm, 1627), whose enduring accuracy did much to persuade astronomers of the correctness of the Copernican theory.

  6. Wren biography
    • He discussed Kepler's theory of elliptical orbits for the planets and looked forward to the day when this could be properly explained.
    • He was the first to resolve Kepler's Problem on cutting a semicircle in a given ratio by a line through a given point on its diameter.
    • This problem had a basis in astronomy for it had arisen in Kepler's work on elliptical orbits.
    • Kepler reduced finding the mean motion of a planet to that of cutting an ellipse in a given ratio with a line through a focus.
    • In addition to solving Kepler's Problem, Wren independently proved Kepler's third law and, as we noted above, formulated the inverse-square law of gravitational attraction.

  7. Mercator Nicolaus biography
    • Again he gives an astronomical example where rational ratios corresponds to Kepler's structure of the planetary system given in terms of regular polyhedra, while irrational ratios correspond to the observed motions.
    • Boulliau, although rejecting Kepler's area rule and his vision of a celestial dynamics, became known in the middle decades of the seventeenth century as the leading proponent of elliptical orbits; whereas his younger correspondent, Mercator, author of several mathematical and astronomical texts, argued the superiority of the Keplerian ellipse and area rule over rival elliptical hypotheses in the latter half of the century.
    • His first publication in over ten years, and his first in England, was Hypothesis astronomica nova (published in London in 1664) in which he combined Kepler's theory of elliptical orbits with other ideas of his own.
    • It was probably through reading this text that Newton learned about Kepler's claim that the orbits of the planets were ellipses.

  8. Newton biography
    • The mechanics of the Copernican astronomy of Galileo attracted him and he also studied Kepler's Optics.
    • From his law of centrifugal force and Kepler's third law of planetary motion, Newton deduced the inverse-square law.
    • After his 1679 correspondence with Hooke, Newton, by his own account, found a proof that Kepler's areal law was a consequence of centripetal forces, and he also showed that if the orbital curve is an ellipse under the action of central forces then the radial dependence of the force is inverse square with the distance from the centre.
    • This discovery showed the physical significance of Kepler's second law.

  9. Cunitz biography
    • She examined Kepler's Rudolphine Tables (1628) which were based on Tycho Brahe's observations and Kepler's first two laws.
    • Cunitz found errors in the Kepler tables and also found that his use of logarithms made his tables difficult to use.
    • The Urania propitia certainly has the merit of being simpler to use than Kepler's tables and Cunitz was able to correct a number of errors in the Rudolphine Tables.

  10. Ward Seth biography
    • On the astronomy side Ward disputed with Ismael Boulliau in what has become known as the Boulliau-Ward controversy over Kepler's laws.
    • In 1645 Boulliau, although accepting elliptical orbits for planets, argued strongly against Kepler's laws, claiming that the planets were self-moved and totally dismissed Kepler's mathematics as "a-geometric".
    • He called Kepler a "mediocre geometer".

  11. Burgi biography
    • Burgi took a serious interest in mathematics, and it was to him that Johannes Kepler (1571 -1630), then Imperial Mathematician, was indebted for his introduction to algebra.
    • In exchange (as it were) it seems to have been Kepler who persuaded Burgi into writing up his original and interesting work on logarithms (the manuscript is largely in Kepler's handwriting), printed in 1620.

  12. Taylor biography
    • For example Taylor wrote to Machin in 1712 providing a solution to a problem concerning Kepler's second law of planetary motion.
    • It was, wrote Taylor, due to a comment that Machin made in Child's Coffeehouse when he had commented on using "Sir Isaac Newton's series" to solve Kepler's problem, and also using "Dr Halley's method of extracting roots" of polynomial equations.
    • Taylor initially derived the version which occurs as Proposition 11 as a generalisation of Halley's method of approximating roots of the Kepler equation, but soon discovered that it was a consequence of the Bernoulli series.

  13. Schickard biography
    • It was during his time as a Lutheran minister that he first met Johannes Kepler who came to Tubingen to support his mother who had been charged with witchcraft.
    • Kepler was working on his Harmony of the World at this time and, after meeting Schickard, he was so impressed with his abilities that he asked him to do some engravings and woodcuts for the book and also asked him to assist in calculating some tables.
    • Yet Kruger [Kepler's publisher], always ready to interfere with Kepler's plans, stipulated that the carving had to be done in Augsburg.
    • In June 1621 Kepler was in Frankfurt [arranging for the publication of books 5-7].
    • It was his work with Kepler which prompted him to think about making a machine to mechanise the astronomical calculations he was doing.
    • He wrote to Kepler on 20 September 1623:- .
    • Kepler clearly showed an interest in having one of Schickard's calculators since Schickard gave instructions for one to be built for him.
    • However, the half-built computer was destroyed by fire as he explained in another letter to Kepler written on 25 February 1624.
    • Schickard used the abridged multiplication for his machine which, Kistermann points out, was unknown to most of the scientific community in 1600, with only a handful of scientists (but including Jost Burgi, Kepler and Schickard) having knowledge of this technique.
    • Sketches of the calculator have been preserved in the manuscripts left by Schickard and Kepler.
    • These however, were not rediscovered until 1935 when they were found during research into Kepler's life.
    • In fact we know that Schickard also wrote to Kepler suggesting a mechanical means to calculate ephemerides.
    • We have mentioned above Schickard's correspondence with Kepler but he corresponded with many other astronomers including Ismael Boulliau and Pierre Gassendi.

  14. Galileo biography
    • However, Galileo argued against Aristotle's view of astronomy and natural philosophy in three public lectures he gave in connection with the appearance of a New Star (now known as 'Kepler's supernova') in 1604.
    • In a personal letter written to Kepler in 1598, Galileo had stated that he was a Copernican (believer in the theories of Copernicus).
    • Galileo's theory of the tides was entirely false despite being postulated after Kepler had already put forward the correct explanation.

  15. Gassendi biography
    • In 1632 Gassendi published Mercury seen on the face of the sun, which described his observations of the transit of Venus which he observed from Paris in November 1631 following the prediction of the event by Kepler in 1629 (the transit actually occurred a month before Kepler's predicted date).

  16. Lansberge biography
    • However he did not accept Kepler's ellipse theories and he published astronomical tables which he hoped would support Copernicus over Kepler.

  17. Von Dyck biography
    • Another important project which von Dyck undertook was one to publish the complete works of Kepler, including all Kepler's letters.

  18. Whiteside biography
    • He embarked on another major project, this time making an in depth study of the papers of Johannes Kepler held in the Soviet Academy of Sciences in Leningrad.
    • Once again, Whiteside's remarkable combination of memory, tenacity and ability to rework the calculations of every line of hundreds of pages of notebooks, and otherwise unconnected manuscript pages, allowed him to follow Kepler's mathematical and astronomical journey as he digested the planetary data of Tycho Brahe.

  19. Machin biography
    • Machin had explained to Taylor in Child's Coffeehouse how to use Newton's series to solve Kepler's problem and also how Halley's method finds roots of polynomial equations.
    • One other publication by Machin is worth noting, namely The solution of Kepler's problem which was published in the Philosophical Proceedings of the Royal Society in 1738.

  20. Stewart biography
    • In 1756 Stewart wrote on Kepler's second law of planetary motion using geometrical methods.
    • In this he solved Kepler's problem which involved determining the area of a focal sector of an ellipse.

  21. Gregory biography
    • The reader may not understand Gregory's reference to "the elliptic inequality" which in fact refers to Kepler's discoveries.
    • However, we now summarise these and other contributions in the hope that, despite his reluctance to publish his methods, his remarkable contributions might indeed be more widely understood: Gregory anticipated Newton in discovering both the interpolation formula and the general binomial theorem as early as 1670; he discovered Taylor expansions more than 40 years before Taylor; he solved Kepler's famous problem of how to divide a semicircle by a straight line through a given point of the diameter in a given ratio (his method was to apply Taylor series to the general cycloid); he gives one of the earliest examples of a comparison test for convergence, essentially giving Cauchy's ratio test, together with an understanding of the remainder; he gave a definition of the integral which is essentially as general as that given by Riemann; his understanding of all solutions to a differential equation, including singular solutions, is impressive; he appears to be the first to attempt to prove that π and e are not the solution of algebraic equations; he knew how to express the sum of the nth powers of the roots of an algebraic equation in terms of the coefficients; and a remark in his last letter to Collins suggests that he had begun to realise that algebraic equations of degree greater than four could not be solved by radicals.

  22. Halley biography
    • He proposed using transits of Mercury (and even better of Venus) to determine the distance of the Sun and therefore the scale of the solar system using Kepler's third law.
    • He had shown that Kepler's third law implied the inverse square law of attraction and presented the results at a meeting of the Royal Society on 24 January 1684 .

  23. Wilkins biography
    • gain acceptance for the new science, to bring the work of Copernicus, Kepler, Galileo, Gilbert, Mersenne, and others, to the attention of his countrymen.
    • The work is addressed to the general reader rather than experts in the subject, and its aim is to popularise the view of the universe due to Copernicus, Kepler and Galileo.

  24. Torricelli biography
    • He had also read almost everything that the contemporary mathematicians Brahe, Kepler and Longomontanus had written and, he told Galileo, he was convinced by the theory of Copernicus that the Earth revolved round the sun.
    • The method was a development of Archimedes' method of exhaustion incorporating Kepler's theory of infinitesimally small geometric quantities.

  25. Copernicus biography
    • The letter was unsigned and the true author of the letter was not revealed publicly until Kepler did so 50 years later.
    • Its notable defenders included Kepler and Galileo while theoretical evidence for the Copernican theory was provided by Newton's theory of universal gravitation around 150 years later.

  26. Wallis biography
    • He studied the works of Kepler, Cavalieri, Roberval, Torricelli and Descartes, and then introduced ideas of the calculus going beyond that of these authors.
    • His interpolation used Kepler's concept of continuity, and with it he discovered methods to evaluate integrals which were later used by Newton in his work on the binomial theorem.

  27. Boulliau biography
    • He claimed that if a planetary moving force existed then it should vary inversely as the square of the distance (Kepler had claimed the first power):- .
    • The Astronomia philolaica represents the most significant treatise between Kepler and Newton and it was praised by Newton in his Principia, particularly for the inverse square hypothesis and its accurate tables.

  28. Privat de Molieres biography
    • The problem was Kepler's laws, easily explained by Newton, but the cause of real problems for Descartes' vortex theory of planetary motion.

  29. Maseres biography
    • Of the reprints that Maseres made at his own expense, the most significant is the "Scriptores logarithmici" (1791-1807), six volumes devoted to the subject of logarithms, including works of Kepler, Napier, Snell, and others, interspersed with original tracts on related subjects.

  30. Clerke biography
    • She wrote famous biographies of Galileo, Huygens, Kepler, Lagrange, Laplace, and other scientists for the ninth edition of Encyclopaedia Britannica.

  31. Seifert biography
    • It was a quote from Kepler which reads in translation:- .

  32. Ford biography
    • Following his contributions to the war effort, Ford joined the faculty at the Rice Institution, Houston, Texas and while there he published papers such as On the closeness of approach of complex rational fractions to a complex irrational number (1925), The Solution of Equations by the Method of Successive Approximations (1925), On motions which satisfy Kepler's first and second laws (1927/28), and The limit points of a group (1929).

  33. Siguenza biography
    • a short book of great significance for its sound mathematical background, anti-Aristotelian outlook, and familiarity with modern authors: Copernicus, Galileo, Descartes, Kepler, and Tycho Brahe.

  34. Bryant biography
    • Bryant's conclusion that elongated rhombic semi-dodecahedra are the natural form of honeycomb cells had been observed by Kepler.

  35. Cassini Dominique biography
    • If Galileo, Newton or Kepler were to descend from heaven and appear at the Academy, they would not comprehend a word in the presentation of Citizen Lalande when he told them that on 20 brumaire, the moon, in a 200 degree opposition to the sun, passed the meridian at five hours ..

  36. Stewart Dugald biography
    • On the other hand, he was sensitive to the realistic interpretations of Kepler and Galileo, who saw that hypotheses were not simply instrumental devices.

  37. Roomen biography
    • In 1600 Roomen visited Prague where he met Kepler and told him of his worries about the methods employed in Rheticus's trigonometric tables.

  38. Doppelmayr biography
    • Besides star charts and a selenographic map, the Atlas includes diagrams illustrating the planetary systems of Copernicus, Tycho Brahe, and Riccioli; the elliptic theories of Kepler, Boulliau, Seth Ward, and Nicolaus Mercator; the lunar theories of Tycho Brahe, Horrocks, and Newton; and Halley's cometary theory.

  39. Bouguer biography
    • By moving the candle and using Kepler's inverse square law he was able to measure brightness.

  40. Cassini biography
    • He worked on this as part of a study of the relative motions of the Earth and the sun and proposed this as the curve for planetary orbits rather than the ellipse as proposed by Kepler.

  41. Brill biography
    • At age 87 he wrote a book on Kepler's astronomy.

  42. Cavalieri biography
    • Cavalieri's theory of indivisibles, presented in his Geometria indivisibilibus continuorum nova quadam ratione promota of 1635 was a development of Archimedes' method of exhaustion incorporating Kepler's theory of infinitesimally small geometric quantities.

  43. Libri biography
    • These two sales of books imported from France contain a magnificent series of manuscripts and books by Galileo, Copernicus, Kepler, Cardan, etc., many with long notes pointing out their significance, and we must not allow ourselves to be blinded to the showmanship and originality of Libri's catalogue by his unenviable reputation as a forger and a thief.

  44. Faulhaber biography
    • Among the scientists with whom Faulhaber collaborated were Kepler and van Ceulen.

  45. Chisini biography
    • For example he published in this journal: Sul principio di continuita (1956) which is an expository lecture on the principle of continuity in algebraic geometry, beginning with the ideas of Kepler; La superficie cubica (1957) which gives a clear and original treatment of the principal properties of cubic surfaces, presenting it as a preliminary introduction to the study of algebraic geometry; and Isoperimetri (1960) which contains elementary thoughts on the plane isoperimetric problem.

  46. Mansion biography
    • He wrote on the history of Greek mathematics and on Copernicus, Galileo and Kepler.

  47. Francesca biography
    • (All these modern names are due to Johannes Kepler (1619).) Piero appears to have been the independent re-discoverer of these six solids.

  48. Napier biography
    • It was through the use of logarithms that Kepler was able to reduce his observations and make his breakthrough which then in turn underpinned Newton's theory of gravitation.

  49. Stevin biography
    • Many pioneers of the Scientific Revolution, such as Galileo, Kepler, Stevin, Descartes, Mersenne, and others, wrote extensively about music theory.

  50. Kirch biography
    • The ephemeredes which Gottfried produced each year from 1681 were based on Kepler's Rudolphine Tables.

  51. Aaboe biography
    • This slim, elegantly written volume by Asger Aaboe might have been more accurate titled 'Highlights of Planetary Theory from Babylon to Kepler' since that is in fact its subject.

  52. Siegel biography
    • They studied the works of mathematicians including Euclid, Archimedes, Fibonacci, Cardan, Stevin, Viete, Kepler, Desargues, Descartes, Fermat, Huygens, Barrow, and Gregory.

  53. Archimedes biography
    • gave birth to the calculus of the infinite conceived and brought to perfection by Kepler, Cavalieri, Fermat, Leibniz and Newton.

  54. Al-Battani biography
    • Al-Battani is important in the development of science for a number of reasons, but one of these must be the large influence his work had on scientists such as Tycho Brahe, Kepler, Galileo and Copernicus.

  55. Infeld biography
    • He published two papers with A Schild, one of his doctoral students, in 1945, namely A note on the Kepler problem in a space of constant negative curvature, and A new approach to kinematic cosmology.

  56. Dee biography
    • For example Brahe firmly believed in alchemy and astrology as did Cavalieri and Kepler while Newton, like Dee, was obsessed with studying alchemy.

  57. Bradwardine biography
    • They were later investigated more thoroughly by Kepler.

  58. Poinsot biography
    • He wrote an important work on polyhedra in 1809 (already mentioned above), discovering four new regular polyhedra, two of which appear in Kepler's work of 1619 but Poinsot was unaware of this.

  59. Guldin biography
    • Guldin corresponded with Kepler, but on religious topics not mathematics or astronomy.

  60. Roberval biography
    • However, he did have some books in his rooms by authors such as Euclid, Archimedes, Viete, Torricelli, Gassendi, Descartes, Mersenne, Kepler, Vitruvius, Herodotus, Cicero and Quintillian.

  61. Bessel biography
    • He introduced this in 1817 in his study of a problem of Kepler of determining the motion of three bodies moving under mutual gravitation.

  62. Fincke biography
    • His other books on astronomy and astrology are of much less interest despite the fact that he was in touch with Brahe and Kepler.

  63. Leibniz biography
    • In this work he claimed, as had Kepler, that movement depends on the action of a spirit.

History Topics

  1. Kepler's Laws
    • KEPLER'S PLANETARY LAWS .
    • This account of Kepler's mathematical astronomy may well challenge some cherished and long-held beliefs, since most of what has been written about Kepler has either been based on secondary or tertiary sources, or has concentrated on his astronomical background and techniques.
    • But Kepler was a highly-talented geometer, and until now has there been no investigation of his work (derived from the original Latin) which has highlighted the mathematical aspect of his brilliance.
    • The greatest achievement of Kepler (1571-1630) was his discovery of the laws of planetary motion.
    • Kepler followed the ancients in always starting to measure at the point furthest from the Sun.) Almost certainly Kepler was responsible for introducing the term 'orbit', in Astronomia Nova Ch.1, and on his behalf we shall precisely define an orbit as possessing a pair of independent constituents: the path or curve, together with a (geometrical) way of representing time.
    • Kepler's principal aim was to find a solution that would satisfy observations - and in that respect he possessed the outlook of a modern scientist.
    • Strongly influenced both by Plato and by his underlying belief in God, Kepler believed more intensely than his contemporaries in the power of mathematics to expose the order in the universe that lay behind apparent complication, and he applied this criterion of simplicity with great effect in his astronomy.
    • There is much additional information, both on the circumstances of Kepler's life, and the context in which he worked, in the MacTutor biography.
    • Kepler originally investigated the orbit of Mars because that was the task allocated to him by Tycho Brahe (1546-1601), when Kepler joined him in Prague around 1600.
    • Kepler was introduced to Copernicanism as a student at the University of Tubingen by his teacher, Michael Maestlin (1550-1631).
    • Though his contemporaries were in general slow to recognize any advantages in this new idea, Kepler adopted the Copernican theory enthusiastically, because of its greater simplicity - which allowed him to abandon the set of (five) large and cumbersome epicycles that occurred in the Ptolemaic theory (they accounted for what we now recognize as the actual motion of the Earth).
    • In fact, Kepler gave Copernican theory a new, mathematical precision by specifying two fundamental properties that were consistent with his conviction that the Sun was metaphorically the place of God: .
    • This was bounded by the fixed stars and consisted of the six known (primary) planets, now including the Earth, with the Moon downgraded as its satellite (a term coined by Kepler himself); .
    • Thus Kepler's interpretation of heliocentricity provided him with an origin from which to determine the Sun-planet distances and so discover the actual path of the planet.
    • Kepler's new astronomy was, indeed, founded on circles, but there was a different reason for this, as we shall explain in Section 5.
    • In the earlier chapters of Astronomia Nova Kepler embarked on a programme of 'reducing the observations' (this term means removing, as far as possible, all effects due to the observer's position in time and space).
    • Kepler carried out the reduction to heliocentricity, and further simplifying procedures, in a series of steps: .
    • Bearing in mind that the observations contained no distance-measurements (as explained in Section 2), this involved expressing all the Mars-Sun distances in terms of the Earth-Sun distance, regarded as a standard unit or 'baseline' (since the path of the Earth is very nearly circular, this approximation happened to be accurate enough for Kepler's purpose); .
    • Thus Kepler knew the angular amount he should allow as compensation for them.
    • To provide the foundation for his new approach to astronomy, Kepler adopted the simplest geometrical structure consistent with observations.
    • Such a structural simplification allowed Kepler to examine the orbit of each individual planet in isolation, because all mutual interactions between planets had been eliminated.
    • In Kepler's day modern algebraic notation and techniques were just being developed, but for his approach to astronomy Kepler depended exclusively on the traditional geometry of Euclid in which he had been trained at the University of Tubingen, as part of the standard preparation for the ministry.
    • (This was originally Kepler's intended profession, and all his life he remained a devout, though somewhat unorthodox, Lutheran).
    • (Moreover, the same principle is invoked in relation to planetary motion when Kepler based his investigation on what Aristotle had specified as the only two simple motions, circular and rectilinear, discussed in Section 9.) This principle has far-reaching ramifications, as we will demonstrate in connection with the complementary pairings that recur in Kepler's mature work in Epitome Book V (1621) - where the term 'complementary' is used in the everyday sense that the pair complete one another, and also with the mathematical connotation of being at right angles.
    • For Kepler, simplicity was the hallmark of his treatment, and contributed overwhelmingly to his success.
    • Kepler always showed the greatest respect for his Greek predecessors, and read their works thoroughly, selecting material that he could incorporate into his new astronomical synthesis.
    • 129-141 AD), Kepler made use of precisely three propositions from the work of Archimedes; one of these was vital in supplying the geometrical backing for Section 6 (the other two - one cited in Section 7, one in Section 11 - were concerned with an innovative approach to 'infinitesimal' considerations which went well beyond traditional geometry).
    • However, it will come as a surprise to some readers to find that Kepler did not rely on Apollonius anywhere in his astronomical work.
    • Sometime in the years 1594-1604, Kepler studied the Conics of Apollonius, and expressed great admiration for it, citing it throughout his optical and stereometrical work - yet he never referred to any of its propositions in connection with his astronomy.
    • This is because Conics is expressed in terms of an oblique (non-orthogonal) frame of reference (coordinate-system), which Kepler implicitly rejected as inappropriate for the study of astronomy (nor did he need any of its propositions, as we confirm in Section 6).
    • Meanwhile we reiterate Kepler's belief that Euclid's Elements encapsulated the only geometry that could properly be applied to the heavens, which after all was the realm of God.
    • In spite of his splendid inheritance from Tycho, Kepler knew that no amount of empirical observations, however numerous, could give him the theoretical structure he required.
    • Therefore, when he had compensated for the observational uncertainties as far as possible, Kepler switched to a geometrical investigation - see Figure (1).
    • Kepler started from the initial framework illustrated in Figure (1), which could be described as standard Ptolemaic, except that Kepler automatically transposed it from geocentric to heliocentric mode.
    • (Mathematicians may like to regard ABQZ as a 'parallelogram of circular motions'.) We shall specify the typical point of any of the three successive orbits proposed by Kepler just as he did - determined by the angle at the centre of the eccentric circle, which we shall denote by beta for distinctiveness.
    • The ordinate QH is also extremely significant in Kepler's reasoning, as we shall demonstrate, concentrating on the three main stages of Kepler's progress once he had adopted the approach which would provide a rational route to his goal.
    • At the first stage Kepler took Q on the eccentric circle as the typical point and so he tested the eccentric itself as a proposed path.
    • So he finally rejected the idea that each planet moved in a single circle, and set out to find the actual curve that was the planet's path - naturally, this had to be constructed from a combination of (arcs of) circles by the geometry of Euclid, since Kepler recognized nothing else as appropriate for the heavens.
    • The three-stage procedure that Kepler adopted was to take geometrically-defined points (K', K'', K) along AZ, one at each stage in turn, then with centre A to draw the corresponding circular arc (radius AK', AK'', AK), so that each arc would end at a geometrically-defined point (Q, V, P) respectively.
    • (Kepler tried many variations at this stage, but this is the only ovoid to be properly defined).
    • By careful comparison with Tycho's observations, as always, Kepler found that the first outcome (Q) was an overshoot, and the second (V) an undershoot.
    • The martial analogy - defeat of Mars, the god of war - was Kepler's own, as was the description of the proposed non-circular curves he found, and named 'ovoid', or egg-shaped (always symmetrical about the line of apsides, but never with any assumption of a second axis of symmetry).
    • So the resulting radius vector AP that finally satisfied Kepler (in Ch.58) was quantified geometrically from the constructed rectangle AKQR, by applying nothing more than a Euclidean - straightedge-and-compasses - construction, as shown in Figure (3): .
    • However, such a construction had never been invented before and Kepler did not have the slightest idea what curve the above relationship represented.
    • There is good reason to believe that this was the earliest plane definition of an ellipse [Mathematical Gazette, forthcoming July 2007.',1)" onmouseover="window.status='Click to see reference';return true">1], as well as the one most commonly, if not exclusively, used by Kepler's contemporaries: it is just the ratio-property of the ordinates.
    • The identification of the curve as an ellipse also depends on a relationship that Kepler established in Ch.
    • Kepler had already invented the term 'focus' in Astronomiae Pars Optica (1604) in connection with his work on vision, though he did not realize its connection with his astronomy at that juncture - in Astronomia Nova he simply referred to the point A as punctum eccentricum, or eccentric point.
    • Actually, Kepler's approach was successful just because the ellipse is simpler than any circle in this situation - an unlikely assertion which is proved in Planetary motion tackled kinematically.
    • XI, Kepler set out a rigorous geometrical proof that the typical point he had constructed satisfied the ratio-property which defines an ellipse.
    • Hence, there can be no suggestion that Kepler merely selected an ellipse and checked it against observations (as many readers may have been told).
    • Incidentally, this provides additional confirmation of my contention (in Section 5) that Kepler did not rely on the Conics of Apollonius for his discovery of the ellipse - in his astronomy he simply did not need anything so sophisticated.
    • Thus, as Kepler realized, a connection exists between a small (micro) interval of time and the corresponding distance of the planet from the Sun.
    • Kepler's practical problem in Astronomia Nova, however, was to discover a way of measuring the time taken to reach the typical position (P) of the planet at an intermediate point of the orbit.
    • In Ch.40, at the first of the three stages set out in Section 6, Kepler put this into practice, by citing Archimedes, Measurement of a Circle, Prop.3, to justify him in taking a sum of distances to be equivalent to the area of a sector of a circle.
    • Next, Kepler extended that proposition, and took the distance-sum from the eccentric point (A, the position of the Sun) to be (approximately) proportional to the area of the eccentric sector (the area QAC, shown in Figure (4)).
    • At the second stage, various doubts and distractions arose, but these were put aside by the third stage, when Kepler discovered that the path was an ellipse (as described in Section 6).
    • Using these two results, Kepler deduced (again from Figure (5): .
    • So, at the end of Astronomia Nova, Kepler had discovered that, when the planet's path was an ellipse, time in orbit appeared to be precisely proportional to the area swept out.
    • (Kepler regarded his initial suggestion, the distance-sum representation, as rather unsatisfactory because it did not give a geometrical interpretation, and so could not provide an exact result.
    • Section 8 Kepler's subsequent justification of the two laws .
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    • This is where the accounts of Kepler's work generally stop - but Kepler achieved much more.
    • Unfortunately, Kepler's investigation of the motions was little appreciated by his contemporaries, and largely ignored subsequently.
    • However, the overall success of his theory was confirmed in practice through the Rudolphine Tables (1627) - which, unlike other astronomical tables, remained observationally accurate and useful for many decades: see Kepler.
    • The mathematical treatment carried out in Planetary motion tackled kinematically demonstrates that this angle is the uniquely appropriate foundation for a structure which is simple because it depends on orthogonality and therefore is the only workable basis for Kepler's astronomy.
    • Accordingly, on this authority, Kepler was able to match each one of the pair of results (the curve, and the independently-determined representation of time) that he had discovered in Astronomia Nova, to one of these mutually perpendicular components of motion.
    • (It is called 'transverse motion' by some mathematicians.) Indeed, had Kepler realized that one of the motions attributed to the planet is strictly (though instantaneously) circular, he would surely have been pleased that the Platonic precept (see Section 3) had not been entirely abandoned after all.
    • (We shall therefore exclude discussion of the radically alternative approach invented by Newton around half a century later, which was concerned with what was happening at the perimeter of the path - something that Kepler could never have known about, nor would have been interested in.) .
    • Neither was Kepler's approach to the problem of causes of motion in any mathematical sense an anticipation of the work of Newton (despite the views of some previous commentators); it was, by contrast, governed by his background in the Aristotelian tradition.
    • Though this Aristotelian 'physics' was becoming outdated even in Kepler's day, people still believed that an object would not move unless there was a 'force' or cause of motion to make it do so.
    • Kepler could never have supposed that the Sun could exert an attractive force because that concept did not exist in Aristotelian terms.
    • Kepler accounted for that motion by inventing the rotation of the Sun on its axis.
    • Thus, Kepler envisaged that the rays emitted by the rotating Sun would 'hit' or impel each planet continuously round in a circle.
    • To account for radial motion, Kepler obviously needed a cause that would be individual to each planet, because every planetary ellipse is a different shape.
    • Because of his Copernican convictions, Kepler extended this idea to suppose that every planet possessed magnetism, and contained a set of 'fibres' fixed within its body which could be activated by the Sun's magnetism; and he further supposed that each set of fibres possessed a unique potential magnetic 'strength' that could be associated with the individual eccentricity of the particular planetary path.
    • Thus it is not correct - nor is it meaningful - to interpret Kepler's magnetism as a 'force', either in an Aristotelean or in a modern context.
    • Kepler was the first to introduce the concept of causation into astronomy, and in accordance with his Copernican convictions, he naturally believed that the Sun was the generator of all causes.
    • Moreover, it seemed common sense to suppose that the Sun could only act (or activate) continuously either in a radial direction or circularly round itself, and this consideration, for Kepler, determined the direction of the causes available and limited their number to two.
    • We summarize Kepler's final suppositions: .
    • With hindsight we can appreciate that Kepler would not have been able to identify these causes satisfactorily until he had discovered the associated motions, in his mature work.
    • (Unfortunately Kepler's successors have often failed to distinguish these causes clearly, because they have only considered his early opinions in Astronomia Nova, when he had not yet sorted things out.) Of course Kepler's final views on causes were entirely wrong in modern eyes, but they were eminently sensible: one traditional, the other making use of the most recent knowledge of the day.
    • Kepler was able to formulate a complete account of planetary motion using only elementary geometry, and accordingly we will highlight the two overriding reasons for his achievement, putting them in a historical context.
    • (That book also contained a sophisticated treatment of tangential velocity in orbit, as well as formulating a concept of acceleration to accompany the concept of attractive force.) However, in the early part of the seventeenth century, Kepler's thinking had already progressed beyond that of Copernicus, to realize that a fresh formulation of time was crucial - even though he had no reason to be aware of the dimension of mass in its application to the solar system.
    • Therefore, from a modern viewpoint, Kepler's work was purely kinematical, and he was entirely correct to treat each individual planet as if it were the only particle in the universe apart from the fixed Sun.
    • It is interesting that an analogous situation occurred in the work of Galileo, Kepler's contemporary, when he idealized the motion of a projectile (as a perfect parabola) by neglecting air resistance.
    • However, unlike Kepler, these components were horizontal and vertical, but like Kepler, Galileo never felt the need to investigate the existence of a single 'resultant' motion, nor to attempt to determine its direction.
    • (In all other respects, the methods of the two were quite different.) The table below shows the fundamentally orthogonal structure of Kepler's planetary astronomy.
    • (This accounts for the absence of any mention of gravity in Kepler's work on planetary orbits - it is introduced only in his separate discussion of the Earth-Moon system - while gravity would have been unacceptable to Kepler anyway because it involves action at a distance.) .
    • Summary of Kepler's orthogonal astronomy (for a single planet) .

  2. Kepler's Laws references
    • References for: KEPLER'S PLANETARY LAWS .
    • Most accessibly in A E L Davis, Kepler's unintentional ellipse - a celestial detective story, Mathematical Gazette 82, no.493 (1998), p.42.
    • Fortunately, the incompatibility continued to worry Kepler, and motivated him eventually to work out a more effective approach to the mathematics of an elliptic orbit: see Sect.6 of A E L Davis, The Mathematics of the Area Law: Kepler's successful proof in "Epitome Astronomiae Copernicanae" (1621) Archive for History of Exact Sciences 57(5), 355-393 .

  3. Kepler's Laws references
    • References for: KEPLER'S PLANETARY LAWS .
    • Most accessibly in A E L Davis, Kepler's unintentional ellipse - a celestial detective story, Mathematical Gazette 82, no.493 (1998), p.42.
    • Fortunately, the incompatibility continued to worry Kepler, and motivated him eventually to work out a more effective approach to the mathematics of an elliptic orbit: see Sect.6 of A E L Davis, The Mathematics of the Area Law: Kepler's successful proof in "Epitome Astronomiae Copernicanae" (1621) Archive for History of Exact Sciences 57(5), 355-393 .

  4. Orbits
    • In 1600 Kepler became assistant to Tycho Brahe who was making accurate observations of the planets.
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    • After Brahe died in 1601 Kepler continued the work, calculating planetary paths to unprecedented accuracy.
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    • Kepler showed that a planet moves round the Sun in an elliptical path which has the Sun in one of its two foci.
    • You can see a diagram from Astronomia Nova showing Kepler's elliptical path for Mars.
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    • However scientists certainly did not accept Kepler's first two laws with enthusiasm.
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    • The second of Kepler's laws suffered an even worse fate in being essentially ignored by scientists for around 80 years.
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    • Kepler's third law, that the squares of the periods of planets are proportional to the cubes of the mean radii of their paths, appeared in Harmonice mundi (1619) and, perhaps surprisingly in view of the above comments, was widely accepted right from the time of its publication.
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    • In the same year I began to think of gravity extending to ye orb of the Moon and (having found out how to estimate the force with wch globe revolving within a sphere presses the surface of a sphere) from Kepler's rule of the periodical times of the Planets being in sesquialternate proportion to their distances from the centres of their Orbs, I deduced that the forces wch keep the Planets in their Orbs must reciprocally as the squares of their distances from the centres about wch they revolve: and thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the Earth, and found them answer pretty nearly.
    • In the Principia Newton also deduced Kepler's third law.
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    • The problem of the orbits of Jupiter and Saturn had troubled astronomers and mathematicians from Kepler's first theory of elliptical orbits.
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  5. Christianity and Mathematics
    • For example four men who perhaps did as much as any to revolutionise the mathematical sciences in the 16th and 17th Centuries, Copernicus, Kepler, Galileo and Newton, were all deeply religious Christians who in many ways saw their scientific work as a religious undertaking.
    • Kepler, on the other hand, made a bolder step away from the beliefs of Aristotle than had Copernicus.
    • Unlike the Catholic mathematicians we have looked at previously in this article, Kepler was a Lutheran.
    • The Lutherans were worried that Kepler's theories would cause splits in their own Church at a time filled with difficulties as various different branches of the Christian Church argued with each other.
    • Kepler, however, saw no reason why the Copernican theory should be seen to oppose the Holy Scripture.
    • For Kepler, a devout Christian, mathematics was itself a religious undertaking.
    • Galileo had long believed in Copernicus's theory and had corresponded with Kepler on the issue.
    • Galileo used arguments similar to those that we have quoted from Kepler's Astronomia nova (1609).
    • Now Galileo's theory of the tides was entirely false despite being postulated after Kepler had already put forward the correct explanation.

  6. Planetary motion
    • They were validated in his later work: Epitome of Copernican Astronomy, Book V, Frankfurt 1621.',3)" onmouseover="window.status='Click to see reference';return true">3], under the kinematical circumstances described above: see Kepler's Planetary Laws.
    • This is Law I: the equation of the elliptic path with respect to the origin at one focus: see Kepler's Planetary Laws: Section 6.
    • See Kepler's Planetary Laws: Section 7.
    • [For a less precise version of equation (10) - simply that the transradial motion is proportional (inverse-linearly) to the distance - see Kepler's Planetary Laws: Section 10.] .
    • This is the radial variation of the distance with respect to β: see Kepler's Planetary Laws: Section 11.] .
    • On the other hand, for the removal of doubt, we should confirm that this treatment is compatible with the modern dynamical approach, by determining the acceleration that corresponds to this motion (as has been said, this concept was an anachronism in Kepler's day).
    • Moreover, subject to precise determination of the values of all the constants involved, Kepler's own treatment was entirely satisfactory, up to the level of first order differentiation.
    • In the event the value of K was compared empirically for all the planets in pairs, and was found to be constant (within observational limits) for every pair tested; K was then assumed to have a common value for the whole planetary system -and the relationship is known as Kepler's Third Law [The Harmony of the World, Linz 1619.
    • (The proof involves the extension of a method actually used by Kepler in his proof of the area law.) .

  7. Orbits references
    • E J Aiton, Kepler's path to the construction and rejection of his first oval orbit for Mars, Ann.
    • E J Aiton, How Kepler discovered the elliptical orbit, Math.
    • J Blum and W Helmchen, Von Kepler zu Newton : Von den Planetenbahnen zum Gravitationsgesetz, Praxis Math.
    • J B Brackenridge, Kepler, elliptical orbits, and celestial circularity : a study in the persistence of metaphysical commitment.
    • J B Brackenridge, Kepler, elliptical orbits, and celestial circularity : a study in the persistence of metaphysical commitment.
    • J T Cushing, Kepler's laws and universal gravitation in Newton's 'Principia', Amer.
    • W H Donahue, Kepler's first thoughts on oval orbits : text, translation, and commentary, J.
    • A Koyre, La gravitation universelle de Kepler a Newton, Arch.

  8. Orbits references
    • E J Aiton, Kepler's path to the construction and rejection of his first oval orbit for Mars, Ann.
    • E J Aiton, How Kepler discovered the elliptical orbit, Math.
    • J Blum and W Helmchen, Von Kepler zu Newton : Von den Planetenbahnen zum Gravitationsgesetz, Praxis Math.
    • J B Brackenridge, Kepler, elliptical orbits, and celestial circularity : a study in the persistence of metaphysical commitment.
    • J B Brackenridge, Kepler, elliptical orbits, and celestial circularity : a study in the persistence of metaphysical commitment.
    • J T Cushing, Kepler's laws and universal gravitation in Newton's 'Principia', Amer.
    • W H Donahue, Kepler's first thoughts on oval orbits : text, translation, and commentary, J.
    • A Koyre, La gravitation universelle de Kepler a Newton, Arch.

  9. Classical light
    • The first person to make a significant step forward after the time of al-Haytham, however, was Kepler at the beginning of the 17th century.
    • Kepler worked on optics, and came up with the first correct mathematical theory of the camera obscura.
    • Kepler's work was a nice piece of mathematics, but people did not believe that the eye created an upside-down image on the retina.
    • Only about five years after the publication of Kepler's work, Galileo constructed a telescope, following ideas of Hans Lippershey from the Netherlands who had constructed one in the previous year.
    • In 1611 Kepler published Dioptrice which was another important work on optics.
    • It also described total internal reflection but failed to give the correct law of refraction of light, Harriot's result being unknown to Kepler (or anyone else) although the two had corresponded.
    • Other contributions around this time by Descartes was his belief in the mathematical argument by Kepler which showed that the image formed on the retina of the eye should be upside-down.
    • Inspired by Kepler's discoveries on light, James Gregory had begun to work on lenses and in Optica Promota (1663) he described the first practical reflecting telescope now called the Gregorian telescope.

  10. Gravitation
    • Although Galileo and Kepler did not put forward theories of gravitation as such, nevertheless they each made significant contributions which set the scene for later developments in gravitational theory.
    • Their contributions which we now discuss were not thought at the time to be in any way connected, for Kepler's contribution concerned the orbits of the planets round the Sun while that of Galileo concerned motion and the acceleration of falling objects.
    • Kepler's first two laws of planetary motion are: (1) a planet moves round the Sun in an ellipse with the Sun at one focus, and (2) a line joining the planet to the Sun sweeps out equal areas in equal times as the planet describes its orbit.
    • Newton deduced Kepler's three laws of planetary motion from the inverse square law of attraction.
    • Newton showed that Kepler's first and second law could not both hold under Descartes' theory.

  11. Physical world
    • Like Galileo, Kepler believed in the Copernican system.
    • Now Kepler could claim that the Copernican system was real since it provided an explanation for the planetary motions while that of Ptolemy did not.
    • In Apologia written in 1600, but unpublished, Kepler argues that accuracy in "saving the phenomena" cannot distinguish which mathematical theory might correspond to reality.
    • It was a belief that a simple mathematical relationship must be physically significant which led Kepler to discover his third law of planetary motion.
    • Kepler believed that there must be some physical significance in this mathematical discovery - of course there is none.

  12. Christianity and Mathematics references
    • G Simon, Kepler-astronome, astrologue (Paris, 1979).
    • F Krafft, Astronomie als Gottesdienst : Die Erneuerung der Astronomie durch Johannes Kepler, in Der Weg der Naturwissenschaft von Johannes von Gmunden zu Johannes Kepler (Vienna, 1988), 182-196.
    • E Rosen, Kepler and the Lutheran attitude towards Copernicus, Vistas in Astronomy 18 (1975), 225-231.

  13. Christianity and Mathematics references
    • G Simon, Kepler-astronome, astrologue (Paris, 1979).
    • F Krafft, Astronomie als Gottesdienst : Die Erneuerung der Astronomie durch Johannes Kepler, in Der Weg der Naturwissenschaft von Johannes von Gmunden zu Johannes Kepler (Vienna, 1988), 182-196.
    • E Rosen, Kepler and the Lutheran attitude towards Copernicus, Vistas in Astronomy 18 (1975), 225-231.

  14. Golden ratio
    • The first known calculation of the golden ratio as a decimal was given in a letter written in 1597 by Michael Maestlin, at the University of Tubingen, to his former student Kepler.
    • The mystical feeling for the golden ratio was of course attractive to Kepler, as was its relation to the regular solids.
    • We have just seen that he was not the first give the result and indeed Albert Girard also discovered it independently of Kepler.

  15. Planetary motion references
    • The laws appeared in Johannes Kepler (1571-1630): New Astronomy, Heidelberg 1609.
    • Johannes Kepler: The Harmony of the World, Linz 1619.
    • A E L Davis: 'Kepler's potential proof of his Third Law' in Miscellanea Kepleriana, ed.

  16. Calculus history
    • Kepler, in his work on planetary motion, had to find the area of sectors of an ellipse.
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    • His method consisted of thinking of areas as sums of lines, another crude form of integration, but Kepler had little time for Greek rigour and was rather lucky to obtain the correct answer after making two cancelling errors in this work.
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    • Cavalieri was led to his 'method of indivisibles' by Kepler's attempts at integration.
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  17. Planetary motion references
    • The laws appeared in Johannes Kepler (1571-1630): New Astronomy, Heidelberg 1609.
    • Johannes Kepler: The Harmony of the World, Linz 1619.
    • A E L Davis: 'Kepler's potential proof of his Third Law' in Miscellanea Kepleriana, ed.

  18. Harriot's manuscripts
    • In the period when the Royal Society sought Harriot's papers in the 1660s one has the feeling it was more to do with showing that an Englishman was superior to Vite, Kepler and Galileo than it was to do with the importance in studying the development of mathematical thought! .
    • .[Zach's] perusal showed Harriot to anticipate and be greater in his accomplishments than either Kepler or Galileo.
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  19. Size of the Universe
    • Kepler's Third Law, published in 1619, made the relative distances within the solar system known, but until one distance was known accurately the size was still unknown.
    • However, all that was now needed, as Kepler had pointed out, was an accurate measurement of the distance to Mars and the scale was fixed.

  20. Cosmology

  21. Forgery 1
    • Copernicus had proposed a sun centred system while Kepler had discovered that the planets revolved round the sun in ellipses with the sun at one focus.

  22. Cosmology references
    • J V Field, Kepler's geometrical cosmology (Chicago, IL, 1988).

  23. Greek astronomy references
    • J L E Dreyer, A history of astronomy from Thales to Kepler (New York, 1953).

  24. Cosmology references
    • J V Field, Kepler's geometrical cosmology (Chicago, IL, 1988).

  25. General relativity
    • Kepler's laws of planetary motion and Galileo's understanding of the motion and falling bodies set the scene for Newton's theory of gravity which was presented in the Principia in 1687.
      Go directly to this paragraph

  26. Greek astronomy references
    • J L E Dreyer, A history of astronomy from Thales to Kepler (New York, 1953).

Famous Curves

  1. Ellipse
    • Kepler, in 1602, said he believed that the orbit of Mars was oval, then he later discovered that it was an ellipse with the sun at one focus.
    • In fact Kepler introduced the word "focus" and published his discovery in 1609.

Societies etc

  1. Planetary features
    • Kepler .

  2. Lunar features

  3. Lunar features
    • (W) (L) Kepler .

  4. Lunar features
    • Kepler .

References

  1. References for Kepler
    • References for Johannes Kepler .
    • A Armitage, John Kepler (1966).
    • C Baumgardt, Johannes Kepler : Life and Letters (New York, N.
    • M Caspar, Kepler (Germany, 1948).
    • M Dickreiter, Der Musiktheoretiker Johannes Kepler (Berne and Munich, 1973).
    • J L E Dreyer, A History of Astronomy from Thales to Kepler (New York, 1953).
    • J V Field, Kepler's Geometrical Cosmology (Chicago, 1988).
    • W Gerlach, Johannes Kepler zum 400.
    • W Gerlach and M List, Johannes Kepler : Leben und Werk (Munich, 1966, 1980).
    • W Gerlach and M List, Johannes Kepler (1571 Weil der Stadt - 1630 Regensburg) (Munich, 1971).
    • J Hemleben, Johannes Kepler in Selbstzeugnissen und Bilddokumenten (Reinbek, 1971).
    • J Hoppe, Johannes Kepler : Biographien Hervorragender Naturwissenschaftler, Techniker und Mediziner (Leipzig, 1987).
    • J Kepler (translated A M Duncan, commentary E J Aiton), Mysterium cosmographicum.
    • J Kepler (translated W Donahue), Astronomia nova: New Astronomy Cambridge, 1992) .
    • J Kepler (translated E J Aiton, A M Duncan, J V Field), The Harmony of the World, Memoirs of the American Philosophical Society, 209, (Philadelphia, 1997).
    • A Koestler, The Watershed: A Biography of Johannes Kepler (1984).
    • E Oeser, Kepler (German) (Gottingen-Zurich, 1971).
    • G Simon, Kepler - astronome, astrologue (Paris, 1979).
    • B Stephenson, Kepler's physical astronomy.
    • K Walter, Johannes Kepler und Tubigen (Tubingen, 1971).
    • H Wussing, Kepler, in H Wussing and W Arnold, Biographien bedeutender Mathematiker (Berlin, 1983).
    • E J Aiton, Kepler's path to the construction and rejection of his first oval orbit for Mars, Ann.
    • E J Aiton, Johannes Kepler in the light of recent research, Hist.
    • E J Aiton, How Kepler discovered the elliptical orbit, Math.
    • E J Aiton, Kepler and the 'Mysterium Cosmographicum', Sudhoffs Arch.
    • W Applebaum, Donne's meeting with Kepler: a previously unknown episode, Philological Quarterly 50 (1) (1971), 132 - 4 .
    • E Badolati, On the history of Kepler's equation, Vistas Astronom.
    • B S Baigrie, The justification of Kepler's ellipse, Stud.
    • B S Baigrie, Kepler's laws of planetary motion, before and after Newton's 'Principia' : an essay on the transformation of scientific problems, Stud.
    • P Barker and B R Goldstein, Distance and velocity in Kepler's astronomy, Ann.
    • A Beer and P Beer (editors), Johannes Kepler.
    • Four hundred years: Proceedings of conferences held in honour of Johannes Kepler, Vistas in Astronomy 18 (1975) .
    • A Beer and J V Field, Kepler - his times, his work, his life and 400 years, Scientia (Milano) 114 (9-12) (1979), 585-599.
    • Ju A Belyi, Johannes Kepler (on the 400th anniversary of his birth) (Russian), Voprosy Istor.
    • V Bialas and E Papadimitriou, Materialien zu den Ephemeriden von Johannes Kepler, Bayer.
    • W Blaschke, Galilei und Kepler, Hamburger Math.
    • J Blum and W Helmchen, Von Kepler zu Newton.
    • S Bochner, Kepler : a personal footnote, Vistas Astronom.
    • J B Brackenridge, Kepler, elliptical orbits, and celestial circularity : a study in the persistence of metaphysical commitment.
    • J B Brackenridge, Kepler, elliptical orbits, and celestial circularity : a study in the persistence of metaphysical commitment.
    • G Buchdahl, Methodological aspects of Kepler's theory of refraction.
    • Johannes Kepler : b.
    • C de Buzon, Remarques sur l'interpretation de l'oeuvre de Kepler, Arch.
    • G Cifoletti, Kepler's 'De quantitatibus', Ann.
    • II, Kepler and Descartes, Stud.
    • Yu A Danilov and Ya A Smorodinskii, Johannes Kepler : from the mystery to the harmony, Soviet Physics Uspekhi 16 (1973), 136-155.
    • A E L Davis, Kepler's physical framework for planetary motion, Centaurus 35 (2) (1992), 165-191.
    • A E L Davis, Kepler's road to Damascus, Centaurus 35 (2) (1992), 143-164.
    • A E L Davis, Grading the eggs (Kepler's sizing-procedure for the planetary orbit), Centaurus 35 (2) (1992), 121-142.
    • A E L Davis, Kepler's 'distance law' - myth not reality, Centaurus 35 (2) (1992), 103-120.
    • A E L Davis, Kepler's resolution of individual planetary motion, Centaurus 35 (2) (1992), 97-102.
    • A E L Davis, Kepler, the ultimate Aristotelian, Acta historiae rerum naturalium necnon technicarum 2 (1998), 65 - 73.
    • W H Donahue, Kepler's approach to the oval of 1602, from the Mars notebook, J.
    • W H Donahue, Kepler's invention of the second planetary law, British J.
    • W H Donahue, Kepler's first thoughts on oval orbits : text, translation, and commentary, J.
    • W H Donahue, Kepler's fabricated figures : covering up the mess in the 'New astronomy', J.
    • S Drake, Galileo, Kepler, and the phases of Venus, Journal for the history of astronomy 15 (1984), 198-208.
    • S Drake, Galileo's 'Platonic' cosmogony and Kepler's 'Prodromus', J.
    • A M Duncan, Kepler on light and sound, Acta historiae rerum naturalium necnon technicarum 2 (1998), 98 - 103.
    • A Elena, On the different kinds of attractive forces in Kepler, Arch.
    • J Evans, The division of the Martian eccentricity from Hipparchos to Kepler : a history of the approximations to Kepler motion, Amer.
    • J V Field, Kepler's rejection of solid celestial spheres, Vistas Astronom.
    • J V Field, Kepler's star polyhedra, Vistas Astronom.
    • J V Field, Kepler's cosmological theories: their agreement with observation, Quarterly Journal of the Royal Astronomical Society 23 (1982), 556 - 68.
    • J V Field, Kepler's Rejection of Numerology in B W Vickers (ed.), Occult and Scientific Mentalities in the Renaissance (Cambridge, 1984), 273-296.
    • J V Field, A Lutheran Astrologer : Johannes Kepler, Archive for History of Exact Science 31 (1984), 189-.
    • J V Field, Two mathematical inventions in Kepler's 'Ad Vitellionem paralipomena', Stud.
    • J V Field, The relation between geometry and algebra: Cardano and Kepler on the regular heptagon in E.Kessler (ed.), Girolamo Cardano: Philosoph, Naturforscher, Arzt (Wiesbaden, 1994), 219 - 242.
    • J V Field, Rediscovering the Archimedean polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Durer, Daniele Barbaro, and Johannes Kepler, Archive for History of Exact Sciences 50 (1997), 241 - 289.
    • J V Field, Kepler's mathematization of cosmology, Acta historiae rerum naturalium necnon technicarum 2 (1998), 27 - 48.
    • A Franklin and C Howson, Newton and Kepler, a Bayesian approach, Stud.
    • W Gerlach, Johannes Kepler und die Copernicanische Wende, Nova Acta Leopoldina (N.F.) 37 (210) (1972).
    • W Gerlach, Humor und Witz in Schriften von Johannes Kepler, Bayer.
    • O Gingerich, Johannes Kepler, in Planetary astronomy from the Renaissance to the rise of astrophysics Part A (Cambridge, 1989), 54-78.
    • D Gronau, Johannes Kepler und die Logarithmen, Berichte der Mathematisch-Statistischen Sektion in der Forschungsgesellschaft Joanneum 284 (Graz, 1987).
    • J Halbronn, Les historiens des sciences face a l'activite astrologique de Kepler, in Proceedings of the 104th National Congress of Learned Societies (Paris, 1979), 135-145.
    • F Hallyn, Kepler, Snell and the law of refraction (Dutch), Med.
    • F Hammer, Die Astrologie des Johannes Kepler, Sudhoffs Arch.
    • J E Hofmann, Johannes Kepler als Mathematiker.
    • J E Hofmann, Johannes Kepler als Mathematiker.
    • S H Hollingdale, A note on Kepler's solar system, Bull.
    • G Holton, Johannes Kepler's universe : its physics and metaphysics, Amer.
    • G Hon, On Kepler's awareness of the problem of experimental error, Ann.
    • U Hoyer, Kepler's celestial mechanics, Vistas Astronom.
    • K-N Ihmig, Tragheit und Massebegriff bei Johannes Kepler, Philos.
    • U Klein, Keplers Bemuhungen um die Harmonieschriften des Ptolemaios und Porphyrios, Johannes Kepler Werk und Leistung (Linz, 1971), 51 - 60.
    • S A Kleiner, A new look at Kepler and abductive argument, Stud.
    • A Koyre, La gravitation universelle de Kepler a Newton, Arch.
    • Die Erneuerung der Astronomie durch Johannes Kepler, in Der Weg der Naturwissenschaft von Johannes von Gmunden zu Johannes Kepler (Vienna, 1988), 182-196.
    • V Maeyama, Kepler's hypothesis vicaria, Arch.
    • A Malet, Gregorie, Descartes, Kepler, and the law of refraction, Arch.
    • A Malet, Keplerian illusions : geometrical pictures vs optical images in Kepler's visual theory, Stud.
    • J Marek, Kepler's inventions in physical optics, in 1971 Actes XIIe Congres Internat.
    • A A Mikhailov, Kepler and modern science (Russian), Voprosy Istor.
    • C-U Moulines, Intertheoretic approximation : the Kepler-Newton case, Synthese 45 (3) (1980), 387-412.
    • G Nador, Kepler's world view and role in the development of the notion of law of nature (Hungarian), Magyar Tud.
    • O Neugebauer, Notes on Kepler, Comm.
    • A Postl, Correspondence between Kepler and Galileo, Vistas Astronom.
    • E Rosen, Galileo and Kepler: Their First Two Contacts, Isis 57 (1966), 262-264.
    • I Schneider, Wahrscheinlichkeit und Zufall bei Kepler, Philos.
    • O B Sheynin, J Kepler as a statistician, Bull.
    • B Stephenson, Kepler's physical astronomy, Studies in the History of Mathematics and Physical Sciences 13 (New York-Berlin, 1987).
    • S Straker, Kepler, Tycho, and the 'optical part of astronomy' : the genesis of Kepler's theory of pinhole images, Arch.
    • N M Swerdlow, Shadow measurement : the 'Sciametria' from Kepler's 'Hipparchus' - a translation with commentary, in The investigation of difficult things (Cambridge, 1992), 19-70.
    • S Thorvaldsen, Did Kepler 'prove' the planetary laws? (Norwegian), Normat 33 (2) (1985), 76-87, 100.
    • S Thorvaldsen, Kepler's path to the planetary laws (Norwegian), Normat 31 (2) (1983), 49-58.
    • P Urban and N Pucker, Johannes Kepler : sa contribution au progres des sciences naturelles, Ontologie und Logik/Ontology and logic (Berlin, 1979), 247-265.
    • J Vuillemin, La methodologie de Kepler, in Traditionen und Perspektiven der analytischen Philosophie (Vienna, 1989), 24-34.
    • B Wandt, Die geplante Berufung von Johannes Kepler an die Universitat Rostock (1629/30), NTM Schr.
    • R S Westman, Kepler's theory of hypothesis and the 'realist dilemma'.
    • Johannes Kepler : b.
    • R S Westman, The comet and the cosmos : Kepler, Mastlin and the Copernican hypothesis, in Etudes sur l'audience de la theorie heliocentrique (Wroclaw, 1972), 7-30.
    • D T Whiteside, Kepler, Newton and Flamsteed on refraction through a 'regular aire' : the mathematical and the practical, Centaurus 24 (1980), 288-315.
    • http://www-history.mcs.st-andrews.ac.uk/References/Kepler.html .

  2. References for Newton
    • J B Brackenridge, Newton's mature dynamics and the 'Principia' : a simplified solution to the Kepler problem, Historia Math.
    • J T Cushing, Kepler's laws and universal gravitation in Newton's 'Principia', Amer.
    • C Ferrini, On Newton's demonstration of Kepler's second law in Hegel's 'De orbitis planetarum' (1801), Philos.
    • A Franklin and C Howson, Newton and Kepler, a Bayesian approach, Stud.
    • D T Whiteside, Kepler, Newton and Flamsteed on refraction through a "regular aire" : the mathematical and the practical, Centaurus 24 (1980), 288-315.

  3. References for Brahe
    • J Kepler, Keplers Elegie in obitum Tychonis Brahe, Bayer.
    • S Straker, Kepler, Tycho, and the 'optical part of astronomy' : the genesis of Kepler's theory of pinhole images, Arch.

  4. References for Galileo
    • S Drake, Galileo, Kepler, and the phases of Venus, Journal for the history of astronomy 15 (1984), 198-208.
    • S Drake, Galileo's 'Platonic' cosmogony and Kepler's 'Prodromus', J.
    • A Postl, Correspondence between Kepler and Galileo, Vistas Astronom.

  5. References for Ptolemy
    • J L E Dreyer, A history of astronomy from Thales to Kepler (New York, 1953).
    • O Gingerich, The Eye of Heaven: Ptolemy, Copernicus, Kepler (1993).

  6. References for Descartes
    • Kepler and Descartes, Stud.
    • A Malet, Gregorie, Descartes, Kepler, and the law of refraction, Arch.

  7. References for Laplace
    • C Wilson, The great inequality of Jupiter and Saturn : from Kepler to Laplace, Arch.

  8. References for Gregory
    • A Malet, Gregorie, Descartes, Kepler, and the law of refraction, Arch.

  9. References for Durer
    • J V Field, Rediscovering the Archimedean polyhedra : Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Durer, Daniele Barbaro, and Johannes Kepler, Arch.

  10. References for Pacioli
    • J F Field, Rediscovering the Archimedean polyhedra : Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Durer, Daniele Barbaro, and Johannes Kepler, Arch.

  11. References for Herigone
    • A Malet, Gregorie, Descartes, Kepler, and the law of refraction, Arch.

  12. References for Bacon
    • G Simon, Roger Bacon et Kepler lecteurs d'Alhazen, Arch.

  13. References for Copernicus
    • J L E Dreyer, A History of Astronomy from Thales to Kepler (1953).

  14. References for Einstein
    • D J F Nonnenmacher, T F Nonnenmacher and P F Zweifel, Kepler, Einstein, and Ulm, Math.

  15. References for Hipparchus
    • J L E Dreyer, A History of Astronomy from Thales to Kepler (1953).

  16. References for Flamsteed
    • D T Whiteside, Kepler, Newton and Flamsteed on refraction through a 'regular aire' : the mathematical and the practical, Centaurus 24 (1980), 288-315.

  17. References for Poincare
    • G Holton, The thematic origins of scientific thought : Kepler to Einstein (Cambridge, MA, 1974).

  18. References for Bernoulli Johann
    • D Speiser, The Kepler problem from Newton to Johann Bernoulli, Arch.

  19. References for Mastlin
    • R S Westman, The comet and the cosmos: Kepler, Mastlin and the Copernican hypothesis, in Etudes sur l'audience de la theorie heliocentrique, Sympos.

  20. References for Burgi
    • M List and V Bialas (eds.), Die Coss von Jost Burgi in der Redaktion von Johannes Kepler.

  21. References for Hobbes
    • G De Lorenzo, Influsso di Galileo e di Kepler su Hobbes e Kant, Rend.

  22. References for Snell
    • F Hallyn, Kepler, Snell and the law of refraction (Dutch), Med.

Additional material

  1. Kepler's Planetary Laws
    • Kepler's Planetary Laws .
    • This account of Kepler's mathematical astronomy may well challenge some cherished and long-held beliefs, since most of what has been written about Kepler has either been based on secondary or tertiary sources, or has concentrated on his astronomical background and techniques.
    • But Kepler was a highly-talented geometer, and until now has there been no investigation of his work (derived from the original Latin) which has highlighted the mathematical aspect of his brilliance.
    • The greatest achievement of Kepler (1571-1630) was his discovery of the laws of planetary motion.
    • Kepler followed the ancients in always starting to measure at the point furthest from the Sun.) Almost certainly Kepler was responsible for introducing the term 'orbit', in Astronomia Nova Ch.1, and on his behalf we shall precisely define an orbit as possessing a pair of independent constituents: the path or curve, together with a (geometrical) way of representing time.
    • Kepler's principal aim was to find a solution that would satisfy observations - and in that respect he possessed the outlook of a modern scientist.
    • Strongly influenced both by Plato and by his underlying belief in God, Kepler believed more intensely than his contemporaries in the power of mathematics to expose the order in the universe that lay behind apparent complication, and he applied this criterion of simplicity with great effect in his astronomy.
    • There is much additional information, both on the circumstances of Kepler's life, and the context in which he worked, in the MacTutor biography.
    • Kepler originally investigated the orbit of Mars because that was the task allocated to him by Tycho Brahe (1546-1601), when Kepler joined him in Prague around 1600.
    • Kepler was introduced to Copernicanism as a student at the University of Tubingen by his teacher, Michael Maestlin (1550-1631).
    • Though his contemporaries were in general slow to recognize any advantages in this new idea, Kepler adopted the Copernican theory enthusiastically, because of its greater simplicity - which allowed him to abandon the set of (five) large and cumbersome epicycles that occurred in the Ptolemaic theory (they accounted for what we now recognize as the actual motion of the Earth).
    • In fact, Kepler gave Copernican theory a new, mathematical precision by specifying two fundamental tenets that were consistent with his conviction that the Sun was metaphorically the place of God: .
    • This was bounded by the fixed stars and consisted of the six known (primary) planets, now including the Earth, with the Moon downgraded as its satellite (a term coined by Kepler himself); .
    • Thus Kepler's interpretation of heliocentricity provided him with an origin from which to determine the Sun-planet distances and so discover the actual path of the planet.
    • Kepler's new astronomy was, indeed, founded on circles, but there was a different reason for this, as we shall explain in Section 5.
    • In the earlier chapters of Astronomia Nova Kepler embarked on a programme of 'reducing the observations' (this term means removing, as far as possible, all effects due to the observer's position in time and space).
    • Kepler carried out the reduction to heliocentricity, and further simplifying procedures, in a series of steps: .
    • Bearing in mind that the observations contained no distance-measurements (as explained in Section 2), this involved expressing all the Mars-Sun distances in terms of the Earth-Sun distance, regarded as a standard unit or 'baseline' (since the path of the Earth is very nearly circular, this approximation happened to be accurate enough for Kepler's purpose); .
    • Thus Kepler knew the angular amount he should allow as compensation for them.
    • To provide the foundation for his new approach to astronomy, Kepler adopted the simplest geometrical structure consistent with observations.
    • Such a structural simplification allowed Kepler to examine the orbit of each individual planet in isolation, because all mutual interactions between planets had been eliminated.
    • In Kepler's day modern algebraic notation and techniques were just being developed, but for his approach to astronomy Kepler depended exclusively on the traditional geometry of Euclid in which he had been trained at the University of Tubingen, as part of the standard preparation for the ministry.
    • (This was originally Kepler's intended profession, and all his life he remained a devout, though somewhat unorthodox, Lutheran).
    • (Moreover, the same principle is invoked in relation to planetary motion when Kepler based his investigation on what Aristotle had specified as the only two simple motions, circular and rectilinear, discussed in Section 9.) This principle has far-reaching ramifications, as we will demonstrate in connection with the complementary pairings that recur in Kepler's mature work in Epitome Book V (1621) - where the term 'complementary' is used in the everyday sense that the pair complete one another, and also with the mathematical connotation of being at right angles.
    • For Kepler, simplicity was the hallmark of his treatment, and contributed overwhelmingly to his success.
    • Kepler always showed the greatest respect for his Greek predecessors, and read their works thoroughly, selecting material that he could incorporate into his new astronomical synthesis.
    • 129-141 AD), Kepler made use of precisely three propositions from the work of Archimedes; one of these was vital in supplying the geometrical backing for Section 6 (the other two - one cited in Section 7, one in Section 11 - were concerned with an innovative approach to 'infinitesimal' considerations which went well beyond traditional geometry).
    • However, it will come as a surprise to some readers to find that Kepler did not rely on Apollonius anywhere in his astronomical work.
    • Sometime in the years 1594-1604, Kepler studied the Conics of Apollonius, and expressed great admiration for it, citing it throughout his optical and stereometrical work - yet he never referred to any of its propositions in connection with his astronomy.
    • This is because Conics is expressed in terms of an oblique (non-orthogonal) frame of reference (coordinate-system), which Kepler implicitly rejected as inappropriate for the study of astronomy (nor did he need any of its propositions, as we confirm in Section 6).
    • Meanwhile we reiterate Kepler's belief that Euclid's Elements encapsulated the only geometry that could properly be applied to the heavens, which after all was the realm of God.
    • In spite of his splendid inheritance from Tycho, Kepler knew that no amount of empirical observations, however numerous, could give him the theoretical structure he required.
    • Therefore, when he had compensated for the observational uncertainties as far as possible, Kepler switched to a geometrical investigation - see Figure (1).
    • Kepler started from the initial framework illustrated in Figure (1), which could be described as standard Ptolemaic, except that Kepler automatically transposed it from geocentric to heliocentric mode.
    • (Mathematicians may like to regard ABQZ as a 'parallelogram of circular motions'.) We shall specify the typical point of any of the three successive orbits proposed by Kepler just as he did - determined by the angle at the centre of the eccentric circle, which we shall denote by β for distinctiveness.
    • The ordinate QH is also extremely significant in Kepler's reasoning, as we shall demonstrate, concentrating on the three main stages of Kepler's progress once he had adopted the approach which would provide a rational route to his goal.
    • At the first stage Kepler took Q on the eccentric circle as the typical point and so he tested the eccentric itself as a proposed path.
    • So he finally rejected the idea that each planet moved in a single circle, and set out to find the actual curve that was the planet's path - naturally, this had to be constructed from a combination of (arcs of) circles by the geometry of Euclid, since Kepler recognized nothing else as appropriate for the heavens.
    • The three-stage procedure that Kepler adopted was to take geometrically-defined points (K', K'', K) along AZ, one at each stage in turn, then with centre A to draw the corresponding circular arc (radius AK', AK'', AK), so that each arc would end at a geometrically-defined point (Q, V, P) respectively.
    • (Kepler tried many variations at this stage, but this is the only ovoid to be properly defined).
    • By careful comparison with Tycho's observations, as always, Kepler found that the first outcome (Q) was an overshoot, and the second (V) an undershoot.
    • The martial analogy -- defeat of Mars, the god of war -- was Kepler's own, as was the description of the proposed non-circular curves he found, and named 'ovoid', or egg-shaped (always symmetrical about the line of apsides, but never with any assumption of a second axis of symmetry).
    • So the resulting radius vector AP that finally satisfied Kepler (in Ch.58) was quantified geometrically from the constructed rectangle AKQR, by applying nothing more than a Euclidean - straightedge-and-compasses - construction, as shown in Figure (3): .
    • However, such a construction had never been invented before and Kepler did not have the slightest idea what curve the above relationship represented.
    • There is good reason to believe that this was the earliest plane definition of an ellipse, (because it can be derived directly from a section of a cone in three easy steps [Mathematical Gazette, forthcoming July 2007.',1)" onmouseover="window.status='Click to see reference';return true">1]), as well as the definition most commonly, if not exclusively, used by Kepler's contemporaries: it is just the ratio-property of the ordinates.
    • The identification of the curve as an ellipse also depends on a relationship that Kepler established in Ch.
    • Kepler had already invented the term 'focus' in Astronomiae Pars Optica (1604) in connection with his work on vision, though he did not realize its connection with his astronomy at that juncture - in Astronomia Nova he simply referred to the point A as punctum eccentricum, or eccentric point.
    • Actually, Kepler's approach was successful just because the ellipse is simpler than any circle in this situation - an unlikely assertion which is proved in Planetary motion tackled kinematically.
    • XI, Kepler set out a rigorous geometrical proof that the typical point he had constructed satisfied the ratio-property which defines an ellipse.
    • Hence, there can be no suggestion that Kepler merely selected an ellipse and checked it against observations (as many readers may have been told).
    • Incidentally, this provides additional confirmation of my contention (in Section 5) that Kepler did not rely on the Conics of Apollonius for his discovery of the ellipse - in his astronomy he simply did not need anything so sophisticated.
    • Kepler set out a rigorous geometrical proof that the typical point he had constructed satisfied the ratio-property which defines an ellipse.
    • Hence, there can be no suggestion that Kepler merely selected an ellipse and checked it against observations (as many readers may have been told).
    • Incidentally, this provides additional confirmation of my contention (in Section 5) that Kepler did not rely on the Conics of Apollonius for his discovery of the ellipse - in his astronomy he simply did not need anything so sophisticated.
    • Thus, as Kepler realized, a connection exists between a small (micro) interval of time and the corresponding distance of the planet from the Sun.
    • Kepler's practical problem in Astronomia Nova, however, was to discover a way of measuring the time taken to reach the typical position (P) of the planet at an intermediate point of the orbit.
    • In Ch.40, at the first of the three stages set out in Section 6, Kepler put this into practice, by citing Archimedes, Measurement of a Circle, Prop.3, to justify him in taking a sum of distances to be equivalent to the area of a sector of a circle.
    • Next, Kepler extended that proposition, and took the distance-sum from the eccentric point (A, the position of the Sun) to be (approximately) proportional to the area of the eccentric sector (the area QAC, shown in Figure (4)).
    • At the second stage, various doubts and distractions arose, but these were put aside by the third stage, when Kepler discovered that the path was an ellipse (as described in Section 6).
    • Using these two results, again from Figure (5), Kepler deduced: .
    • So, at the end of Astronomia Nova, Kepler had discovered that, when the planet's path was an ellipse, time in orbit appeared to be precisely proportional to the area swept out.
    • (Kepler regarded his initial suggestion, the distance-sum representation, as rather unsatisfactory because it did not give a geometrical interpretation, and so could not provide an exact result.
    • This is where the accounts of Kepler's work generally stop - but Kepler achieved much more.
    • Unfortunately, Kepler's investigation of the motions was little appreciated by his contemporaries, and largely ignored subsequently.
    • However, the overall success of his theory was confirmed in practice through the Rudolphine Tables (1627) - which, unlike other astronomical tables, remained observationally accurate and useful for many decades: see Kepler.
    • The mathematical treatment carried out in Planetary motion tackled kinematically demonstrates that this angle is the uniquely appropriate foundation for a structure which is simple because it depends on orthogonality and therefore is the only workable basis for Kepler's astronomy.
    • Accordingly, on this authority, Kepler was able to match each one of the pair of results (the curve, and the independently-determined representation of time) that he had discovered in Astronomia Nova, to one of these mutually perpendicular components of motion.
    • (It is called 'transverse motion' by some mathematicians.) Indeed, had Kepler realized that one of the motions attributed to the planet is strictly (though instantaneously) circular, he would surely have been pleased that the Platonic precept (see Section 3) had not been entirely abandoned after all.
    • (We shall therefore exclude discussion of the radically alternative approach invented by Newton around half a century later, which was concerned with what was happening at the perimeter of the path - something that Kepler could never have known about, nor would have been interested in.) .
    • Neither was Kepler's approach to the problem of causes of motion in any mathematical sense an anticipation of the work of Newton (despite the views of some previous commentators); it was, by contrast, governed by his background in the Aristotelian tradition.
    • Though this Aristotelian 'physics' was becoming outdated even in Kepler's day, people still believed that an object would not move unless there was a 'force' or cause of motion to make it do so.
    • Kepler could never have supposed that the Sun could exert an attractive force because that concept did not exist in Aristotelian terms.
    • Kepler accounted for that motion by inventing the rotation of the Sun on its axis.
    • Thus, Kepler envisaged that the rays emitted by the rotating Sun would 'hit' or impel each planet continuously round in a circle.
    • To account for radial motion, Kepler obviously needed a cause that would be individual to each planet, because every planetary ellipse is a different shape.
    • Because of his Copernican convictions, Kepler extended this idea to suppose that every planet possessed magnetism, and contained a set of 'fibres' fixed within its body which could be activated by the Sun's magnetism; and he further supposed that each set of fibres possessed a unique potential magnetic 'strength' that could be associated with the individual eccentricity of the particular planetary path.
    • Thus it is not correct - nor is it meaningful - to interpret Kepler's magnetism as a 'force', either in an Aristotelean or in a modern context.
    • Kepler was the first to introduce the concept of causation into astronomy, and in accordance with his Copernican convictions, he naturally believed that the Sun was the generator of all causes.
    • Moreover, it seemed common sense to suppose that the Sun could only act (or activate) continuously either in a radial direction or circularly round itself, and this consideration, for Kepler, determined the direction of the causes available and limited their number to two.
    • We summarize Kepler's final suppositions: .
    • With hindsight we can appreciate that Kepler would not have been able to identify these causes satisfactorily until he had discovered the associated motions, in his mature work.
    • (Unfortunately Kepler's successors have often failed to distinguish these causes clearly, because they have only considered his early opinions in Astronomia Nova, when he had not yet sorted things out.) Of course Kepler's final views on causes were entirely wrong in modern eyes, but they were eminently sensible: one traditional, the other making use of the most recent knowledge of the day.
    • Kepler was able to formulate a complete account of planetary motion using only elementary geometry, and accordingly we will highlight the two overriding reasons for his achievement, putting them in a historical context.
    • (That watershed book also contained a sophisticated treatment of tangential velocity in orbit, as well as formulating a concept of acceleration to accompany the concept of attractive force.) Thus it is clear that Kepler could not have been aware of the modern implication of (the dimension of) mass in the solar system, though his interpretation of an orbit certainly involved the dimensions both of length and of time, as we have demonstrated.
    • Hence we can now recognize that Kepler's work was entirely kinematical, and acknowledge that he was, then, absolutely justified in treating each individual planet as if it were the only particle in the universe apart from the fixed Sun.
    • It is interesting that an analogous situation occurred in the work of Galileo, Kepler's contemporary, when he idealized the motion of a projectile (as a perfect parabola) by neglecting air resistance.
    • However, unlike Kepler, these components were horizontal and vertical, but like Kepler, Galileo never felt the need to investigate the existence of a single 'resultant' motion, nor to attempt to determine its direction.
    • (In all other respects, the methods of the two were quite different.) The table below shows the fundamentally orthogonal structure of Kepler's planetary astronomy.
    • (This explains the absence of mention of 'gravity' anywhere in this analysis - it is tentatively introduced only in some separate discussions of the Earth-Moon system: it would have been altogether redundant in Kepler's work on planetary orbits.) .
    • We accordingly conclude that Kepler's work on planetary motion was satisfactorily complete, and moreover justifiable, with the possible exception of the radial cause.
    • Summary of Kepler's orthogonal astronomy (for a single planet) .
    • A E L Davis: 'Kepler's Road to Damascus', Centaurus 35 (1992) pp.156-157; .
    • 'Kepler's unintentional ellipse - a celestial detective story', Mathematical Gazette 82, no.493 (1998), p.42; .
    • A E L Davis: 'The Mathematics of the Area Law: Kepler's successful proof in Epitome Astronomiae Copernicanae (1621)', Archive for History of Exact Sciences 57, 5 (2003), especially Section 6.

  2. Kepler's Planetary Laws
    • Kepler's Planetary Laws .
    • This account of Kepler's mathematical astronomy may well challenge some cherished and long-held beliefs, since most of what has been written about Kepler has either been based on secondary or tertiary sources, or has concentrated on his astronomical background and techniques.
    • But Kepler was a highly-talented geometer, and until now has there been no investigation of his work (derived from the original Latin) which has highlighted the mathematical aspect of his brilliance.
    • The greatest achievement of Kepler (1571-1630) was his discovery of the laws of planetary motion.
    • Kepler followed the ancients in always starting to measure at the point furthest from the Sun.) Almost certainly Kepler was responsible for introducing the term 'orbit', in Astronomia Nova Ch.1, and on his behalf we shall precisely define an orbit as possessing a pair of independent constituents: the path or curve, together with a (geometrical) way of representing time.
    • Kepler's principal aim was to find a solution that would satisfy observations - and in that respect he possessed the outlook of a modern scientist.
    • Strongly influenced both by Plato and by his underlying belief in God, Kepler believed more intensely than his contemporaries in the power of mathematics to expose the order in the universe that lay behind apparent complication, and he applied this criterion of simplicity with great effect in his astronomy.
    • There is much additional information, both on the circumstances of Kepler's life, and the context in which he worked, in the MacTutor biography.
    • Kepler originally investigated the orbit of Mars because that was the task allocated to him by Tycho Brahe (1546-1601), when Kepler joined him in Prague around 1600.
    • Kepler was introduced to Copernicanism as a student at the University of Tubingen by his teacher, Michael Maestlin (1550-1631).
    • Though his contemporaries were in general slow to recognize any advantages in this new idea, Kepler adopted the Copernican theory enthusiastically, because of its greater simplicity - which allowed him to abandon the set of (five) large and cumbersome epicycles that occurred in the Ptolemaic theory (they accounted for what we now recognize as the actual motion of the Earth).
    • In fact, Kepler gave Copernican theory a new, mathematical precision by specifying two fundamental properties that were consistent with his conviction that the Sun was metaphorically the place of God: .
    • This was bounded by the fixed stars and consisted of the six known (primary) planets, now including the Earth, with the Moon downgraded as its satellite (a term coined by Kepler himself); .
    • Thus Kepler's interpretation of heliocentricity provided him with an origin from which to determine the Sun-planet distances and so discover the actual path of the planet.
    • Kepler's new astronomy was, indeed, founded on circles, but there was a different reason for this, as we shall explain in Section 5.
    • In the earlier chapters of Astronomia Nova Kepler embarked on a programme of 'reducing the observations' (this term means removing, as far as possible, all effects due to the observer's position in time and space).
    • Kepler carried out the reduction to heliocentricity, and further simplifying procedures, in a series of steps: .
    • Bearing in mind that the observations contained no distance-measurements (as explained in Section 2), this involved expressing all the Mars-Sun distances in terms of the Earth-Sun distance, regarded as a standard unit or 'baseline' (since the path of the Earth is very nearly circular, this approximation happened to be accurate enough for Kepler's purpose); .
    • Thus Kepler knew the angular amount he should allow as compensation for them.
    • To provide the foundation for his new approach to astronomy, Kepler adopted the simplest geometrical structure consistent with observations.
    • Such a structural simplification allowed Kepler to examine the orbit of each individual planet in isolation, because all mutual interactions between planets had been eliminated.
    • In Kepler's day modern algebraic notation and techniques were just being developed, but for his approach to astronomy Kepler depended exclusively on the traditional geometry of Euclid in which he had been trained at the University of Tubingen, as part of the standard preparation for the ministry.
    • (This was originally Kepler's intended profession, and all his life he remained a devout, though somewhat unorthodox, Lutheran).
    • (Moreover, the same principle is invoked in relation to planetary motion when Kepler based his investigation on what Aristotle had specified as the only two simple motions, circular and rectilinear, discussed in Section 9.) This principle has far-reaching ramifications, as we will demonstrate in connection with the complementary pairings that recur in Kepler's mature work in Epitome Book V (1621) - where the term 'complementary' is used in the everyday sense that the pair complete one another, and also with the mathematical connotation of being at right angles.
    • For Kepler, simplicity was the hallmark of his treatment, and contributed overwhelmingly to his success.
    • Kepler always showed the greatest respect for his Greek predecessors, and read their works thoroughly, selecting material that he could incorporate into his new astronomical synthesis.
    • 129-141 AD), Kepler made use of precisely three propositions from the work of Archimedes; one of these was vital in supplying the geometrical backing for Section 6 (the other two - one cited in Section 7, one in Section 11 - were concerned with an innovative approach to 'infinitesimal' considerations which went well beyond traditional geometry).
    • However, it will come as a surprise to some readers to find that Kepler did not rely on Apollonius anywhere in his astronomical work.
    • Sometime in the years 1594-1604, Kepler studied the Conics of Apollonius, and expressed great admiration for it, citing it throughout his optical and stereometrical work - yet he never referred to any of its propositions in connection with his astronomy.
    • This is because Conics is expressed in terms of an oblique (non-orthogonal) frame of reference (coordinate-system), which Kepler implicitly rejected as inappropriate for the study of astronomy (nor did he need any of its propositions, as we confirm in Section 6).
    • Meanwhile we reiterate Kepler's belief that Euclid's Elements encapsulated the only geometry that could properly be applied to the heavens, which after all was the realm of God.
    • In spite of his splendid inheritance from Tycho, Kepler knew that no amount of empirical observations, however numerous, could give him the theoretical structure he required.
    • Therefore, when he had compensated for the observational uncertainties as far as possible, Kepler switched to a geometrical investigation - see Figure (1).
    • Kepler started from the initial framework illustrated in Figure (1), which could be described as standard Ptolemaic, except that Kepler automatically transposed it from geocentric to heliocentric mode.
    • (Mathematicians may like to regard ABQZ as a 'parallelogram of circular motions'.) We shall specify the typical point of any of the three successive orbits proposed by Kepler just as he did - determined by the angle at the centre of the eccentric circle, which we shall denote by beta for distinctiveness.
    • The ordinate QH is also extremely significant in Kepler's reasoning, as we shall demonstrate, concentrating on the three main stages of Kepler's progress once he had adopted the approach which would provide a rational route to his goal.
    • At the first stage Kepler took Q on the eccentric circle as the typical point and so he tested the eccentric itself as a proposed path.
    • So he finally rejected the idea that each planet moved in a single circle, and set out to find the actual curve that was the planet's path - naturally, this had to be constructed from a combination of (arcs of) circles by the geometry of Euclid, since Kepler recognized nothing else as appropriate for the heavens.
    • The three-stage procedure that Kepler adopted was to take geometrically-defined points (K', K'', K) along AZ, one at each stage in turn, then with centre A to draw the corresponding circular arc (radius AK', AK'', AK), so that each arc would end at a geometrically-defined point (Q, V, P) respectively.
    • (Kepler tried many variations at this stage, but this is the only ovoid to be properly defined).
    • By careful comparison with Tycho's observations, as always, Kepler found that the first outcome (Q) was an overshoot, and the second (V) an undershoot.
    • The martial analogy - defeat of Mars, the god of war - was Kepler's own, as was the description of the proposed non-circular curves he found, and named 'ovoid', or egg-shaped (always symmetrical about the line of apsides, but never with any assumption of a second axis of symmetry).
    • So the resulting radius vector AP that finally satisfied Kepler (in Ch.58) was quantified geometrically from the constructed rectangle AKQR, by applying nothing more than a Euclidean - straightedge-and-compasses - construction, as shown in Figure (3): .
    • However, such a construction had never been invented before and Kepler did not have the slightest idea what curve the above relationship represented.
    • There is good reason to believe that this was the earliest plane definition of an ellipse [Mathematical Gazette, forthcoming July 2007.',1)" onmouseover="window.status='Click to see reference';return true">1], as well as the one most commonly, if not exclusively, used by Kepler's contemporaries: it is just the ratio-property of the ordinates.
    • The identification of the curve as an ellipse also depends on a relationship that Kepler established in Ch.
    • Kepler had already invented the term 'focus' in Astronomiae Pars Optica (1604) in connection with his work on vision, though he did not realize its connection with his astronomy at that juncture - in Astronomia Nova he simply referred to the point A as punctum eccentricum, or eccentric point.
    • Actually, Kepler's approach was successful just because the ellipse is simpler than any circle in this situation - an unlikely assertion which is proved in Planetary motion tackled kinematically.
    • XI, Kepler set out a rigorous geometrical proof that the typical point he had constructed satisfied the ratio-property which defines an ellipse.
    • Hence, there can be no suggestion that Kepler merely selected an ellipse and checked it against observations (as many readers may have been told).
    • Incidentally, this provides additional confirmation of my contention (in Section 5) that Kepler did not rely on the Conics of Apollonius for his discovery of the ellipse - in his astronomy he simply did not need anything so sophisticated.
    • Thus, as Kepler realized, a connection exists between a small (micro) interval of time and the corresponding distance of the planet from the Sun.
    • Kepler's practical problem in Astronomia Nova, however, was to discover a way of measuring the time taken to reach the typical position (P) of the planet at an intermediate point of the orbit.
    • In Ch.40, at the first of the three stages set out in Section 6, Kepler put this into practice, by citing Archimedes, Measurement of a Circle, Prop.3, to justify him in taking a sum of distances to be equivalent to the area of a sector of a circle.
    • Next, Kepler extended that proposition, and took the distance-sum from the eccentric point (A, the position of the Sun) to be (approximately) proportional to the area of the eccentric sector (the area QAC, shown in Figure (4)).
    • At the second stage, various doubts and distractions arose, but these were put aside by the third stage, when Kepler discovered that the path was an ellipse (as described in Section 6).
    • Using these two results, Kepler deduced (again from Figure (5): .
    • So, at the end of Astronomia Nova, Kepler had discovered that, when the planet's path was an ellipse, time in orbit appeared to be precisely proportional to the area swept out.
    • (Kepler regarded his initial suggestion, the distance-sum representation, as rather unsatisfactory because it did not give a geometrical interpretation, and so could not provide an exact result.
    • Section 8 Kepler's subsequent justification of the two laws .
      Go directly to this paragraph
    • This is where the accounts of Kepler's work generally stop - but Kepler achieved much more.
    • Unfortunately, Kepler's investigation of the motions was little appreciated by his contemporaries, and largely ignored subsequently.
    • However, the overall success of his theory was confirmed in practice through the Rudolphine Tables (1627) - which, unlike other astronomical tables, remained observationally accurate and useful for many decades: see Kepler.
    • The mathematical treatment carried out in Planetary motion tackled kinematically demonstrates that this angle is the uniquely appropriate foundation for a structure which is simple because it depends on orthogonality and therefore is the only workable basis for Kepler's astronomy.
    • Accordingly, on this authority, Kepler was able to match each one of the pair of results (the curve, and the independently-determined representation of time) that he had discovered in Astronomia Nova, to one of these mutually perpendicular components of motion.
    • (It is called 'transverse motion' by some mathematicians.) Indeed, had Kepler realized that one of the motions attributed to the planet is strictly (though instantaneously) circular, he would surely have been pleased that the Platonic precept (see Section 3) had not been entirely abandoned after all.
    • (We shall therefore exclude discussion of the radically alternative approach invented by Newton around half a century later, which was concerned with what was happening at the perimeter of the path - something that Kepler could never have known about, nor would have been interested in.) .
    • Neither was Kepler's approach to the problem of causes of motion in any mathematical sense an anticipation of the work of Newton (despite the views of some previous commentators); it was, by contrast, governed by his background in the Aristotelian tradition.
    • Though this Aristotelian 'physics' was becoming outdated even in Kepler's day, people still believed that an object would not move unless there was a 'force' or cause of motion to make it do so.
    • Kepler could never have supposed that the Sun could exert an attractive force because that concept did not exist in Aristotelian terms.
    • Kepler accounted for that motion by inventing the rotation of the Sun on its axis.
    • Thus, Kepler envisaged that the rays emitted by the rotating Sun would 'hit' or impel each planet continuously round in a circle.
    • To account for radial motion, Kepler obviously needed a cause that would be individual to each planet, because every planetary ellipse is a different shape.
    • Because of his Copernican convictions, Kepler extended this idea to suppose that every planet possessed magnetism, and contained a set of 'fibres' fixed within its body which could be activated by the Sun's magnetism; and he further supposed that each set of fibres possessed a unique potential magnetic 'strength' that could be associated with the individual eccentricity of the particular planetary path.
    • Thus it is not correct - nor is it meaningful - to interpret Kepler's magnetism as a 'force', either in an Aristotelean or in a modern context.
    • Kepler was the first to introduce the concept of causation into astronomy, and in accordance with his Copernican convictions, he naturally believed that the Sun was the generator of all causes.
    • Moreover, it seemed common sense to suppose that the Sun could only act (or activate) continuously either in a radial direction or circularly round itself, and this consideration, for Kepler, determined the direction of the causes available and limited their number to two.
    • We summarize Kepler's final suppositions: .
    • With hindsight we can appreciate that Kepler would not have been able to identify these causes satisfactorily until he had discovered the associated motions, in his mature work.
    • (Unfortunately Kepler's successors have often failed to distinguish these causes clearly, because they have only considered his early opinions in Astronomia Nova, when he had not yet sorted things out.) Of course Kepler's final views on causes were entirely wrong in modern eyes, but they were eminently sensible: one traditional, the other making use of the most recent knowledge of the day.
    • Kepler was able to formulate a complete account of planetary motion using only elementary geometry, and accordingly we will highlight the two overriding reasons for his achievement, putting them in a historical context.
    • (That book also contained a sophisticated treatment of tangential velocity in orbit, as well as formulating a concept of acceleration to accompany the concept of attractive force.) However, in the early part of the seventeenth century, Kepler's thinking had already progressed beyond that of Copernicus, to realize that a fresh formulation of time was crucial - even though he had no reason to be aware of the dimension of mass in its application to the solar system.
    • Therefore, from a modern viewpoint, Kepler's work was purely kinematical, and he was entirely correct to treat each individual planet as if it were the only particle in the universe apart from the fixed Sun.
    • It is interesting that an analogous situation occurred in the work of Galileo, Kepler's contemporary, when he idealized the motion of a projectile (as a perfect parabola) by neglecting air resistance.
    • However, unlike Kepler, these components were horizontal and vertical, but like Kepler, Galileo never felt the need to investigate the existence of a single 'resultant' motion, nor to attempt to determine its direction.
    • (In all other respects, the methods of the two were quite different.) The table below shows the fundamentally orthogonal structure of Kepler's planetary astronomy.
    • (This accounts for the absence of any mention of gravity in Kepler's work on planetary orbits - it is introduced only in his separate discussion of the Earth-Moon system - while gravity would have been unacceptable to Kepler anyway because it involves action at a distance.) .
    • Summary of Kepler's orthogonal astronomy (for a single planet) .

  3. Kepler's 'Foundations of modern optics' Preface to a translation
    • Kepler's Foundations of modern optics Preface to a translation .
    • In 1980 Catherine Chevalley, Claude Chevalley's daughter, published a French translation from the Latin of Johannes Kepler's The foundations of modern optics: Paralipomena to Vitellius which was originally published in 1604.
    • It was with this work that Kepler in 1604 made his public entry into the domain of the theory of light and optics and thus effectively created the conditions of a radical renovation that would determine the development of this branch of physics in the seventeenth century.
    • We can only advise the reader to follow her in the progression leading to the French translation of Kepler's text.
    • But Kepler's text is not only a Latin text of the late sixteenth century written by an author from the Germanic cultural sphere, it is a technical text in which numerous passages - notably Chapter IV - testify more to a pure and simple transcription of personal notes than to a patiently executed draft.
    • Only a work of translation such as the one undertaken here can point out this state of affairs, and the translator, confronted with undeniable instances of Kepler's carelessness, had to resist the temptation of rewriting some hasty or allusive sentences.
    • http://www-history.mcs.st-andrews.ac.uk/Extras/Kepler_optics.html .

  4. Planetary motion tackled kinematically
    • They were validated in his later work: Epitome of Copernican Astronomy, Book V, (Frankfurt ,1621).',3)" onmouseover="window.status='Click to see reference';return true">3], under the kinematical circumstances described above: see Kepler's Planetary Laws.
    • It was discovered by Kepler, in 1609, where it was of course expressed in geometrical terms: .
    • see Kepler's Planetary Laws: Section 6.
    • This is Law II: the time expressed in angular measure, discovered by Kepler in 1609, where he established that (when the dimensional constant is specified, for the Sun-focused ellipse alone) time is proportional to area.
    • See Kepler's Planetary Laws: Section 7.
    • [Later, in 1621, Kepler demonstrated a less precise version of equation (10) -- simply that the transradial motion is proportional (inverse-linearly) to the distance.
    • See Kepler's Planetary Laws: Section 10.] .
    • [Kepler got no further than this: see Kepler's Planetary Laws: Section 11.] .
    • On the other hand, for the removal of doubt, we should confirm that this treatment is compatible with the modern dynamical approach, by determining the acceleration that corresponds to this motion (as has been said, this concept was an anachronism in Kepler's day).
    • Moreover, subject to precise determination of the values of all the constants involved, Kepler's own treatment was entirely satisfactory, up to the level of first order differentiation.
    • The laws appeared in Johannes Kepler (1571-1630): New Astronomy, Heidelberg 1609.
    • Johannes Kepler, The Harmony of the World (Linz, 1619).
    • A E L Davis, "Kepler's potential proof of his Third Law" in Miscellanea Kepleriana, ed.
    • http://www-history.mcs.st-andrews.ac.uk/Extras/Kepler_planetary_motion.html .

  5. Finlay Freundlich's Inaugural Address
    • Now celestial bodies cannot be placed and weighed on scales; their masses, therefore, in Astronomy have to be measured according to other principles, for which Kepler laid the foundation.
    • Strictly speaking, Kepler's third law is not quite correct.
    • But, since the solar mass is, at the least, a thousand times greater than the mass of the largest planet, Jupiter, Kepler's law works as a good first approximation.
    • Applying Kepler's law the mass of the Sun can be determined, when we know the mean distance of the planet from the Sun and the period of its revolution around the Sun.
    • However, applying Kepler's third law, one would find that during that time-interval the Moon would have lagged behind in its orbit by half of the apparent diameter of its disc.
    • Accurate observations extending at present over more than a century, disclose that the orbit, which if the Sun's gravitation were alone acting, should be a closed ellipse - remaining as a whole fixed relative to the stellar system, each revolution being completed in a time strictly prescribed by Kepler's third law - that this orbit appears actually to be closed about one-half of a second of time later than predicted by Kepler's law.

  6. Finlay Freundlich's Inaugural Address, Part 2
    • But due to the changed geometrical conditions of space, arising from the gravitational field produced by the Sun, this shortest connection is no longer a straight line of Euclid's geometry, but the arc of a Kepler orbit.
    • It is perhaps one of the most astounding discoveries, that a theory of motion, based on these new ideas, which do not seem to have any relationship whatsoever with the ideas on which Newton's theory was based, should lead to practically the same laws of motion for two gravitating bodies; namely, the laws which Kepler embodied in his three laws of planetary motion.
    • And, what is of even greater importance, the coincidence between the Kepler orbit and the orbit of a planet moving according to the theory of relativity in a quasi non-Euclidean space, is not complete.
    • The right attitude in this respect was pronounced by Kepler 350 years ago; let me cite therefore a passage from Kepler's Mysterium Cosmographicum, from 1596:- .

  7. Andrew Forsyth addresses the British Association in 1905, Part 2
    • It then occurred to Halley to calculate similarly the elements of the comet which Kepler and others had seen in 1607, and of which records had been made; the Newtonian theory gave elements in close accord with those belonging to the comet calculated from the latest, observations, though a new regret is expressed that the 1607 observations had not been made with more accuracy.
    • Halley, constant to his faith in the Newtonian hypothesis, used that hypothesis to calculate the elements of the orbit of the Apianus comet; once more regretting the uncertainty of the data and discounting a very grievous error committed by Apianus himself, Halley concluded that the Apianus comet of 1531, and the Kepler comet of 1607, and the observed comet of 1682 were one and the same.
    • Yet an aggregate of facts is not an explanatory theory any more necessarily than a pile of carefully fashioned stones is a cathedral; and the genius of a Kepler and a Newton is just as absolutely needed to evolve the comprehending theory as the genius of great architects was needed for the Gothic cathedrals of France and of England.

  8. Mathematicians and Music 2.2
    • Returning to the beginning of the Harmonic period let us consider the musical writings which were issued in the seventeenth century by such mathematicians as Kepler, Wallis, Mersenne, Desargues, Descartes and Christian Huygens.
    • Pythagorean ideas on the ratios of numbers and of proportions applied to the constitution of the universe seem to have been the point of departure of Kepler in his famous work Harmonices Mundi published in 1619.
    • Markedly contrasted to Kepler in abilities and habits of thought was John Wallis, the notably able Savilian professor at Oxford University, where a brilliant mathematical school was developed under his direction.

  9. Andrew Forsyth addresses the British Association in 1905
    • The simple laws of planetary motion were not formulated, for Kepler had them only in the making.
    • Kepler was gradually elucidating the laws of planetary motion, of which such significant use was made later by Newton; and Descartes, by his creation of analytical geometry, was yet to effect such a constructive revolution in mathematics that he might not unfairly be called the founder of modern mathematics.

  10. George William Hill's new theory of Jupiter and Saturn
    • This, in the main, is the character of all the tables of the planets until the publication of Kepler's Tabulae Rudolphinae in 1627, where, for the first time, the equation of the centre is derived from an elliptic formula, and we pass from heliocentric to geocentric positions in the modern way.
    • From Kepler onwards the fact of the deviation of Jupiter and Saturn from a purely elliptic theory was recognized.

  11. Von Neumann: 'The Mathematician
    • Kepler's first attempts at integration were formulated as "dolichometry" - measurement of kegs - that is, volumetry for bodies with curved surfaces.
    • Of this, Kepler was fully aware.

  12. George William Hill's new theory of Jupiter and Saturn
    • This, in the main, is the character of all the tables of the planets until the publication of Kepler's Tabulae Rudolphinae in 1627, where, for the first time, the equation of the centre is derived from an elliptic formula, and we pass from heliocentric to geocentric positions in the modern way.
    • From Kepler onwards the fact of the deviation of Jupiter and Saturn from a purely elliptic theory was recognized.

  13. Hevelius: fire
    • (4) all Kepler's MSS.

  14. James Jeans: 'Physics and Philosophy' I
    • Even when Kepler discovered the true shapes of these planetary orbits sixty years later, he still postulated a 'power' or influence to keep the planets moving; he thought they would all stop dead if a material emanation from the sun did not continually urge them on.

  15. Edward Sang on his tables
    • Kepler's celebrated problem has ever since his time exercised mathematicians, and, sharing the ambition of many others, I also sought often, and in vain, for an easy solution of it.

  16. The Tercentenary of the birth of James Gregory
    • We can picture Gregory at the age of 30, coming a stranger to St Andrews on a late Autumn day, to a University already 250 years old and wrapped in mediaeval tradition - this young firebrand, who was so ready to seize on a new mathematical idea with avidity, bursting into the cloistered calm where the new learning of Kepler, Galileo and Descartes was still unknown.

  17. The St Andrews Schmidt-Cassegrain Telescope
    • The theory of the telescope had, by this time, become the subject of scientific study and, in 1611, Kepler gave the theory of a telescope consisting of two converging lenses; the practical development of this form referred to later as the refractor is largely due to Huygens.

  18. Edmund Whittaker: 'Physics and Philosophy
    • Kepler had insisted that the chief aim of all investigations of the external world should be to discover the rational order and harmony which had been impressed on it by God and which He revealed to us in the language of mathematics; and Descartes saw that this doctrine had a wider application, that it could in fact be extended to general philosophy.

  19. H Weyl: 'Theory of groups and quantum mechanics' Introduction
    • Our generation is witness to a development of physical knowledge such as has not been seen since the days of Kepler, Galileo and Newton, and mathematics has scarcely ever experienced such a stormy epoch.

  20. Gibson History 10 - Matthew Stewart, John Stewart, William Trail
    • He undoubtedly obtained many important successes in this way; his solution of Kepler's problem being one of the most remarkable.

  21. Wolfgang Pauli and the Exclusion Principle
    • Sommerfeld, however, preferred, in view of the difficulties which blocked the use of the concepts of kinematical models, a direct interpretation, as independent of models as possible, of the laws of spectra in terms of integral numbers, following, as Kepler once did in his investigation of the planetary system, an inner feeling for harmony.

  22. W H Young addresses ICM 1928
    • In earlier times Kepler might boldly state: plurimum namque amo analogias fidelissimos meos magistros, omnium naturae areanorum conscios (For above all I love analogies, my most faithful teachers, acquainted with all the secrets of Nature),but, since then, imagination in Science has gone somewhat out of fashion: and, in fear of losing caste, the scientist has grown proverbially matter of fact.

  23. EMS obituary
    • The Edinburgh Mathematical Society vividly recall his lecture at a Colloquium upon the generation of the thirteen semi-regular figures of Archimedes constructed by short circuiting the plane repeated patterns of space filling regular polygons which he had found among the engravings in a book by Kepler.

  24. The South-Troughton quarrel
    • (Drinkwater Bethune wrote lives of Galileo and Kepler in the Library of Useful Knowledge, and with Sir John Lubbock a little book On Probability, in the same series.) Maule at once insisted that Troughton & Simms should be allowed to finish their work according to the plan proposed by Sheepshanks, but only to be paid for if successful.

Quotations

  1. Quotations by Kepler
    • Quotations by Johannes Kepler .
    • Abbreviation: KGW: Johannes Kepler gesammelte Werke, ed.
    • Kepler to Galileo Galilei, 28 March 1611, Letter 611, ll.
    • Epitaph (by Kepler, for himself) .
    • http://www-history.mcs.st-andrews.ac.uk/Quotations/Kepler.html .

  2. Quotations by Galileo
    • I wish, my dear Kepler, that we could have a good laugh together at the extraordinary stupidity of the mob.
    • Among the great men who have philosophized about [the action of the tides], the one who surprised me most is Kepler.

  3. A quotation by Lie
    • were introduced into science by the investigations of Kepler, .

Chronology

  1. Mathematical Chronology
    • Kepler publishes Astronomia nova (New Astronomy).
    • The work contains Kepler's first and second law on elliptical orbits, but only verified for the planet Mars.
    • Kepler publishes Nova stereometria doliorum vinarorum (Solid Geometry of a Wine Barrel), an investigation of the capacity of casks, surface areas, and conic sections.
    • He writes to Kepler suggesting using mechanical means to calculate ephemeredes.
    • The method incorporates Kepler's theory of infinitesimally small geometric quantities.
    • Bessel discovers a class of integral functions, now called "Bessel functions", in his study of a problem of Kepler to determine the motion of three bodies moving under mutual gravitation.
    • Thomas Hales proves Kepler's problem on sphere packing.

  2. Chronology for 1600 to 1625
    • Kepler publishes Astronomia nova (New Astronomy).
    • The work contains Kepler's first and second law on elliptical orbits, but only verified for the planet Mars.
    • Kepler publishes Nova stereometria doliorum vinarorum (Solid Geometry of a Wine Barrel), an investigation of the capacity of casks, surface areas, and conic sections.
    • He writes to Kepler suggesting using mechanical means to calculate ephemeredes.

  3. Chronology for 1625 to 1650
    • The method incorporates Kepler's theory of infinitesimally small geometric quantities.

  4. Chronology for 1990 to 2000
    • Thomas Hales proves Kepler's problem on sphere packing.

  5. Chronology for 1810 to 1820
    • Bessel discovers a class of integral functions, now called "Bessel functions", in his study of a problem of Kepler to determine the motion of three bodies moving under mutual gravitation.

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JOC/BS August 2001