**Johannes Kepler** was a German mathematician and astronomer who discovered that
the Earth and planets travel about the sun in elliptical orbits.
He is known for his first, second, and third planetary law of motion.
The possibility of being able to describe the laws of heavenly motions by earthly mathematics positively influenced the boldness of contemporary mathematicians and finally led to the invention of Newton's theory of gravitation.
He also did important work in optics and geometry.

- 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'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 Tübingen, then (as now) a bastion of Lutheran orthodoxy.
- 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 Tübingen Kepler was taught astronomy by one of the leading astronomers of the day, Michael Mästlin (1550 - 1631).
- Kepler did not take this attitude.
- At Tübingen, 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 Mästlin 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 Mästlin 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 Tübingen and formulated in the Confessio Augustana Ⓣ(Augsburg Confession).
- 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.
- In his writings, Kepler is given to laying his opinions on the line - which is very convenient for historians.
- These may explain why Mästlin persuaded Kepler to abandon plans for ordination and instead take up a post teaching mathematics in Graz.
- Kepler was excommunicated in 1612.
- Kepler's answer to these questions, described in his Mysterium cosmographicum Ⓣ(Mystery of the Cosmos), Tübingen, 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 Ⓣ(Mystery of the Cosmos).
- Kepler carried out this work while he was teaching in Graz, but the book was seen through the press in Tübingen by Mästlin.
- 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).
- 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.
- Ⓣ(New Astronomy), 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.
- 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.
- 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).
- This design, in which the final image is inverted, was so successful that it is now usually known not as a Keplerian telescope but simply as the astronomical telescope.
- Kepler had to leave Prague.
- Ⓣ(New Stereometry of wine barrels), Linz, 1615, in which Kepler, basing himself on the work of Archimedes, used a resolution into 'indivisibles'.
- 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.
- 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.
- 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.
- 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.

Born 27 December 1571, Weil der Stadt, Württemberg, Holy Roman Empire (now Germany). Died 15 November 1630, Regensburg (now in Germany).

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Analysis, Astronomy, Geometry, Origin Germany, Physics, Special Numbers And Numerals

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive