During his career, Kepler was a mathematics teach at a seminary school in Graz, Austria, where he became an associate to astronomer Tycho Brahe, and eventually the imperial mathematician to Emperor Rudolf II and his two successors Matthias and Ferdinand II. He was also a mathematics teacher in Linz, Austria. He also did several works in the field of optics, invented an improved version of refracting telescope.

Kepler lived in an era where there was no clear distinction between astronomy and astrology, but there was a strong division between astronomy and physics. Astronomy was considered as a branch of mathematics within the liberal arts, while the physics was considered as a branch of natural philosophy. Kepler incorporated religious arguments and reasoning into his work, motivated by the religious conviction and beleif that God has created the world according to an intelligible plan that is accessible through the natural light of reason. Kepler described his new astronomy as "celestial physics" transforming the ancient tradition of physical cosmology by treating astronomy as part of a universal mathematical physics.

**Childhood**

Kepler was born in the small town of Weil der Stadt in Swabia and moved to nearby Leonberg with his parents in 1576. His father was a mercenary soldier and his mother the daughter of an innkeeper. Johannes was their first child. His father left home for the last time when Johannes was five, and is believed to have died in the war in the Netherlands. As a child, Kepler lived with his mother in his grandfather's inn. He tells us that he used to help by serving in the inn. One imagines customers were sometimes bemused by the child's unusual competence at arithmetic.

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.

**Kepler's opinions**

**University education**

*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).

*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. In real life, it seems likely that a similar tendency to openness led the authorities at Tübingen to entertain well-founded doubts about his religious orthodoxy. These may explain why Mästlin persuaded Kepler to abandon plans for ordination and instead take up a post teaching mathematics in Graz. Religious intolerance sharpened in the following years. Kepler was excommunicated in 1612. This caused him much pain, but despite his (by then) relatively high social standing, as Imperial Mathematician, he never succeeded in getting the ban lifted.

**Kepler's first cosmological model (1596)**

*Mysterium cosmographicum*.

**The 'War with Mars'**

*New Astronomy ...*(

*Astronomia nova, ...*, 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.

**Observational error**

**Optics, and the New Star of 1604**

*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. These results were published in

*Supplements to Witelo, on the optical part of astronomy*(

*Ad Vitellionem paralipomena, quibus astronomiae pars optica traditur*, Frankfurt, 1604). 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. 257, line 23).

*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.

**Leaving Prague for Linz**

**Kepler's laws of planetary motion**

1. The orbit of each planet is an ellipse with the sun occupying the focus (center object).

2. The line joining the sun to a planet sweeps out equal areas in equal intervals of time

3. A planets orbital period is proportional to the mean distance between the Sun and the planet raised to the power 3/2.

**Marriage and wine barrels**

*New Stereometry of wine barrels ...*,

*Nova stereometria doliorum ...*, Linz, 1615) in which Kepler, basing himself on the work of Archimedes, used a resolution into 'indivisibles'. This method was later developed by Bonaventura Cavalieri (c. 1598 - 1647) and is part of the ancestry of the infinitesimal calculus.

*The Harmony of the World**The Harmony of the World*, planned since 1599 as a development of his

*Mystery of the Cosmos*. This second work on cosmology (

*Harmonices mundi libri V*, Linz, 1619) presents a more elaborate mathematical model than the earlier one, though the polyhedra are still there. The mathematics in this work includes the first systematic treatment of tessellations, a proof that there are only thirteen convex uniform polyhedra (the Archimedean solids) and the first account of two non-convex regular polyhedra (all in Book 2).

*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. In fact, although the Third Law plays an important part in some of the final sections of the printed version of the

*Harmony of the World*, it was not actually discovered until the work was in press. Kepler made last-minute revisions. He himself tells the story of the eventual success:

...and if you want the exact moment in time, it was conceived mentally on 8^{th}March in this year one thousand six hundred and eighteen, but submitted to calculation in an unlucky way, and therefore rejected as false, and finally returning on the 15^{th}of May and adopting a new line of attack, stormed the darkness of my mind. So strong was the support from the combination of my labour of seventeen years on the observations of Brahe and the present study, which conspired together, that at first I believed I was dreaming, and assuming my conclusion among my basic premises. But it is absolutely certain and exact that "the proportion between the periodic times of any two planets is precisely the sesquialterate proportion of their mean distances ..."

(Harmonice mundiBook 5, Chapter 3, trans. Aiton, Duncan and Field, p. 411).

**Witchcraft trial**

*Harmony of the World*, his mother was charged with witchcraft. He enlisted the help of the legal faculty at Tübingen. 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. The surviving documents are chilling. However, Kepler continued to work. In the coach, on his journey to Württemberg to defend his mother, he read a work on music theory by Vincenzo Galilei (c.1520 - 1591, Galileo's father), to which there are numerous references in

*The Harmony of the World*.

**Astronomical Tables**

*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. All astronomical tables that made use of new observations were accurate for the first few years after publication. What was remarkable about the

*Rudolphine Tables*was that they proved to be accurate over decades. 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**

*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).

**Death**

*Rudolphine Tables*. He was buried in the local church, but this was destroyed in the course of the Thirty Years' War and nothing remains of the tomb.

**Historiographic note**

*Sleepwalkers*the late Arthur Koestler made Kepler's battle with Mars into an argument for the inherent irrationality of modern science. There have been many tacit followers of these two persuasions. 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. Closer study shows Koestler was simply mistaken in his assessment.