Johannes Kepler

How to use this course

Johannes Kepler is remembered for three laws of planetary motion. Those laws were the end of a long argument he had with the sky, recorded in books that show every false start. He was also the founder of modern optics, a pioneer of sphere packing and of the calculus of volumes, and the first astronomer to insist that planetary orbits need physical causes. This course follows that work alongside a life spent in schools, courts and exile during the Counter-Reformation and the Thirty Years’ War.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements in modern notation, derivations and exercises with folded solutions. Kepler wrote without algebraic notation or calculus. Where the course uses modern forms, it says so, and it tries to show what Kepler himself could and could not prove.

Timeline

Weil der Stadt and the cloister schools, 1571–1589

Johannes Kepler was born on 27 December 1571 in Weil der Stadt, a small free imperial city near Stuttgart. His grandfather Sebald had been the town’s mayor, but the family was in decline. His father Heinrich served as a mercenary and left home for the last time when Johannes was about five; he is thought to have died in the wars in the Netherlands. His mother Katharina, an innkeeper’s daughter, practised as a herbal healer. The family moved to nearby Leonberg in 1577.

Two memories from these years mattered to him later. In 1577 his mother took him up a hill to see the Great Comet, and in 1580 he was called outside to see the reddened Moon of a lunar eclipse. A childhood bout of smallpox left him with poor eyesight and weakened hands, which made him a calculator more than an observer.

Württemberg’s Lutheran dukes ran a scholarship system to train clergy, and it was Kepler’s way out of poverty. He attended the Latin school at Leonberg, then the cloister schools at Adelberg from 1584 and Maulbronn from 1586. He matriculated at Tübingen in 1587 and passed the baccalaureate examination there on 25 September 1588 while still a Maulbronn pupil. The schools taught Latin, Greek, rhetoric and theology with rigour, and Kepler later wrote harsh, self-mocking horoscopes of his own anxious and quarrelsome adolescence.

Tübingen and Graz, 1589–1600

In 1589 Kepler entered the Tübinger Stift, the theological seminary of the University of Tübingen. The professor of mathematics and astronomy, Michael Maestlin, taught the geocentric system in lectures but explained Nicolaus Copernicus’s heliocentric system to able students. Kepler became a convinced Copernican, defending the system in student disputations, and argued for it on physical as well as mathematical grounds. He took his master’s degree on 11 August 1591 and began theology.

In 1594, before he finished, the Tübingen faculty recommended him for a vacancy as mathematics teacher at the Protestant seminary school in Graz, in Catholic Styria. He arrived on 11 April 1594. The post included the office of district mathematician, which meant producing annual calendars with astrological forecasts. His first calendar predicted a cold winter and a Turkish incursion, both of which came, and his local reputation rose.

The inspiration of 1595

On 19 July 1595, drawing a figure in class, Kepler noticed that a triangle inscribed between two circles fixed a ratio close to that of the orbits of Saturn and Jupiter. Plane figures did not generalize, but the five regular solids did. His first book, Mysterium cosmographicum, appeared at Tübingen in 1596 with Maestlin’s help; Lesson 04 treats its geometry. He sent copies to leading astronomers, including Tycho Brahe and Galileo.

On 27 April 1597 he married Barbara Müller, a twice-widowed miller’s daughter with a young daughter. Their first two children died in infancy. Religious pressure in Styria grew under Archduke Ferdinand. The Protestant school was closed in 1598, and on 17 July 1600 a decree ordered all Lutherans to convert or leave. Kepler refused to convert and left Graz with his family on 30 September 1600.

Prague: Tycho and the Emperor, 1600–1612

Tycho Brahe, the Danish nobleman who had built the finest observatory in Europe at Uraniborg, had moved to Bohemia in 1599 as Imperial Mathematician to Rudolf II. His observations of planetary positions were accurate to about two minutes of arc, several times better than anything before. Kepler met him on 4 February 1600 at Benátky castle, north-east of Prague. Relations were tense: Tycho guarded his data, and Kepler needed both the data and a salary. Tycho set him to work on the orbit of Mars, the planet whose motion had defeated Tycho’s assistant Longomontanus.

Tycho died on 24 October 1601. Two days later Kepler was appointed his successor as Imperial Mathematician, and he kept the observations despite disputes with Tycho’s heirs. The Prague years were his most productive. He published his optics in 1604, a book on the new star that appeared in October 1604 (De stella nova, 1606), and in 1609 the Astronomia nova, whose first two laws are the subject of Lessons 06–07.

In 1610 Galileo announced the moons of Jupiter. Kepler replied in support within weeks, before he had looked through a telescope, and then observed the moons himself with a borrowed instrument. In his Narratio of 1611 he introduced the word satellite for them. His Dioptrice (1611) gave the first theory of the telescope.

The year 1611 broke the Prague household. Rudolf was forced to abdicate in favour of his brother Matthias. Kepler’s six-year-old son Friedrich died of smallpox, and his wife Barbara died shortly afterwards. Rudolf died in January 1612, and Kepler left for Linz.

The cosmic mystery: polyhedra and planets

For the general reader. Why are there six planets, and why are they spaced as they are? Kepler’s first answer was that God built the solar system around the five perfect solids of Greek geometry, nested between the planetary spheres. The answer is wrong, but the question was new. Kepler asked for the cause of the planetary distances, and he asked that the cause be checked against measurement.

The five solids

Proof. Each interior angle of a regular \(p\)-gon is \(\pi(1-2/p)\). At a convex vertex the face angles must total less than \(2\pi\), so \(q\pi(1-2/p)<2\pi\), equivalently \((p-2)(q-2)<4\) with \(p,q\ge3\). The solutions are the five pairs listed, and each is realized. ∎

The nesting

Kepler placed the solids, from the inside out, in the order octahedron, icosahedron, dodecahedron, tetrahedron, cube. Each solid is inscribed in the sphere of the planet outside it and circumscribes the sphere of the planet inside it:

The model therefore predicts that each ratio of neighbouring planetary distances equals \(r/R\), the ratio of inradius to circumradius of the solid between them.

For the cube with vertices \((\pm1,\pm1,\pm1)\), \(R=\sqrt3\) and \(r=1\). Duality exchanges face centres and vertices: the polar of a solid with radii \(r,R\) about the unit sphere has radii \(1/R,1/r\), with the same ratio. For the icosahedron with vertices at cyclic permutations of \((0,\pm1,\pm\varphi)\), \(\varphi=\tfrac{1+\sqrt5}2\), we have \(R^2=1+\varphi^2\), and the inradius is the distance from the origin to the centroid of a face, which gives the stated value.

Copernican distances fit only roughly. Kepler gave each sphere a thickness to contain the planet’s eccentric orbit and made a special adjustment for Mercury, but discrepancies remained. He blamed Copernicus’s numbers, and the desire for better data led him to Tycho. In a second edition of 1621 he kept the model, now subordinated to the harmonies of Lesson 08.

The first physical idea

Chapter 20 of the Mysterium contains the germ of celestial physics. Outer planets move more slowly, and Kepler proposed that a moving power, issuing from the Sun and weakening with distance, drives them. He tested a relation between periods and distances and found it did not fit. The search for the right relation took him another 22 years.

Optics: light, the eye and the telescope

For the general reader. To measure the sky one must understand the instrument, and the first instrument is the eye. Kepler’s optics of 1604 explained how a pinhole forms an image, showed that the eye focuses an inverted picture on the retina, and stated that light weakens with the square of distance. In 1611 he explained how lenses combine in a telescope and designed the form that astronomers still use.

The pinhole and the eclipse

Observers of solar eclipses through a pinhole saw the Moon looking smaller than it should. Kepler, puzzled by the eclipse of July 1600, reasoned by tracing rays with threads. Each point of the source sends a pencil of rays through the aperture and makes a small copy of the aperture on the screen. The image is the superposition of these copies: the source shape blurred by the aperture shape. An aperture of finite size enlarges a bright image and shrinks a dark intruder such as the Moon’s disc. This settled the eclipse paradox.

The retinal image

Medieval optics placed vision in the crystalline lens. Kepler, using anatomical drawings of Felix Platter, argued that the lens refracts the cone of rays from each object point to a point on the retina, so the eye forms a real, inverted image there, like a camera. How an inverted picture is seen upright he left to the natural philosophers, a clear division between optics and perception.

Kepler argued that the same quantity of light passes through every sphere about the source, and the area of a sphere grows as the square of its radius. He did not apply the square law to the Sun’s moving force, which he took to weaken as \(1/d\) because it acted only in the plane of the orbits.

The Dioptrice

Kepler did not know the sine law of refraction, found later by Willebrord Snell and René Descartes. He used the small-angle rule that in glass the angle of refraction is about two thirds of the angle of incidence, equivalent to an index of 1.5. With this he worked out the foci of convex and concave lenses, explained real and virtual images, described total internal reflection, and proposed a telescope of two convex lenses. In thin-lens form, with object and image distances \(u,v\) and focal length \(f\),

The Keplerian telescope gives an inverted image but a wider field than Galileo’s design, and it allows a crosshair or micrometer at the common focus. It became the standard astronomical instrument by the mid-seventeenth century.

The war on Mars and the area law

For the general reader. Since antiquity, planets were assumed to move on circles at uniform speed, viewed from a suitable point. Kepler, working on Tycho’s observations of Mars, found that the best such model still missed by eight minutes of arc, a quarter of the Moon’s width. He refused to accept the error. Instead he looked for a physical rule, and found that a line from the Sun to a planet sweeps out equal areas in equal times.

The equant and the eight minutes

In Ptolemy’s scheme a planet moves on a circle whose centre \(C\) is offset from the observer, and it moves uniformly in angle as seen from a third point, the equant \(Q\). Ptolemy placed \(C\) midway between the observer and \(Q\), which is called bisecting the eccentricity. Kepler, working with the true Sun and not the mean Sun of Copernicus and Tycho, first freed the ratio and fitted the positions of \(C\) and \(Q\) to four of Tycho’s oppositions of Mars. This vicarious hypothesis reproduced Tycho’s longitudes at opposition to within about two minutes, but it predicted wrong distances. When he forced the physically natural bisected ratio, the errors in longitude rose to about eight minutes elsewhere in the orbit. Since Tycho’s data were good to about two minutes, Kepler concluded that the circle itself had to go.

Earth first

To get distances from the Sun to Mars, Kepler needed Earth’s orbit. He used observations of Mars made one Martian period apart, 687 days, when Mars is at the same point of its orbit: the changing direction to it from Earth then traces Earth’s orbit by triangulation. He found that Earth’s orbit also needs a bisected eccentricity, and that Earth’s speed varies inversely with its distance from the Sun, at least near the apsides.

From the distance law to the area law

Kepler’s physical idea was that the Sun pushes the planets with a force that weakens as \(1/r\), so speed is inversely proportional to distance. Summing distances along the orbit to compute the time was laborious. As a short cut he replaced the sum of distances by the area swept out, and he found the area version predicted positions well. In modern form:

Kepler reached this law first, around 1602, before he had the ellipse. His distance law is correct only at perihelion and aphelion, where the velocity is perpendicular to the radius. The area law is exact, and Isaac Newton showed why.

Newton’s derivation

If the force on the planet points toward the Sun, then the torque about the Sun vanishes and the angular momentum \(\mathbf L=m\,\mathbf r\times\dot{\mathbf r}\) is constant. Its magnitude is \(mr^2\dot\theta=2m\,dA/dt\). So the area law holds for every central force, whatever its dependence on distance. Newton’s Principia proved this geometrically in Book I, Proposition 1.

The ellipse and Kepler’s equation

For the general reader. With the area law in hand, Kepler still needed the shape of the orbit. He tried egg-shaped ovals for years. In 1605 he found that Mars moves on an ellipse with the Sun at one focus. Combining the ellipse with the area law gives an equation for where a planet is at a given time. That equation has no solution in closed form, and it has occupied mathematicians ever since.

The clue

Kepler found that the orbit of Mars falls inside the circle on the line of apsides, leaving a crescent whose greatest width is about 0.00429 of the radius. The largest angle between the directions of Mars seen from the Sun and from the orbit’s centre, the “optical equation”, was about \(5^\circ18'\), and \(\sec 5^\circ18'\approx1.00429\). The coincidence led him to a distance formula that, after one more false turn, gave the ellipse. Chapter 58 of the Astronomia nova describes the moment.

The auxiliary circle

Draw the circle of radius \(a\) about the centre of the ellipse. Through the planet \(P\) draw a perpendicular to the major axis, meeting the circle at \(Q\). The angle \(E\) at the centre from perihelion to \(Q\) is the eccentric anomaly. The ellipse is the circle compressed by \(b/a\), with \(b=a\sqrt{1-e^2}\), so

with coordinates measured from the focus.

Kepler’s equation

In the circle, the area swept from the focus between perihelion and \(Q\) is the circular sector at the centre, \(\tfrac12a^2E\), minus the triangle with the centre and the focus, \(\tfrac12a\cdot ae\sin E\). Compressing by \(b/a\) gives the area swept on the ellipse, \(\tfrac12ab(E-e\sin E)\). By the area law this grows uniformly in time, reaching \(\pi ab\) after one period \(T\). Define the mean anomaly \(M=2\pi(t-t_p)/T\).

Kepler could not invert the equation and said so, inviting geometers to try. Two methods work well. The fixed-point iteration \(E_{k+1}=M+e\sin E_k\) converges because the map is a contraction with constant \(e\). Newton’s method,

converges quadratically. Joseph-Louis Lagrange gave a power series in \(e\), and Friedrich Bessel expanded the solution in the functions that now bear his name: \(E=M+\sum_{n\ge1}\frac{2}{n}J_n(ne)\sin nM\).

The harmony of the world and the third law

For the general reader. Kepler believed the world was built on harmony, in the musical sense. In Harmonices mundi (1619) he looked for musical ratios in geometry, astrology and the planets, and found them in the planets’ speeds. In the same book, almost in passing, he stated the law that links every planet’s period to its distance from the Sun. That law later led Newton to the inverse-square law of gravity.

Book II: congruence of figures

Kepler studied how regular polygons fit together in the plane and in space. He identified the eleven tilings of the plane by regular polygons with all vertices alike, described the thirteen Archimedean solids as a complete family, and introduced two new regular star polyhedra, the small and great stellated dodecahedra. Louis Poinsot later added two more, and the four are now the Kepler–Poinsot polyhedra.

Book V: the harmonies

Kepler compared each planet’s angular speed, as seen from the Sun, at aphelion and perihelion. For Saturn the ratio is close to \(4:5\), a major third; for Jupiter, close to \(5:6\), a minor third; for Earth about \(15:16\), a semitone. Ratios between extreme speeds of neighbouring planets also give consonances. Kepler imagined each planet singing a range of notes and the whole system sounding chords at rare moments. The theory has not survived, but it made him compare periods and distances across all planets.

Kepler first saw the relation on 8 March 1618, rejected it after a computing error, and confirmed it on 15 May. He stated it as a ratio of periods equal to the ratio of distances raised to the power \(\tfrac32\). In the Epitome he showed it also held for the four moons of Jupiter, using observations by Simon Marius, with its own constant.

Newton’s form

For a circular orbit under gravity, \(GMm/a^2=m(2\pi/T)^2a\), so

after including the motion of the Sun about the centre of mass (two-body problem). The same holds for ellipses with \(a\) the semi-major axis. Read backwards, the third law for circles forces the force to fall as \(1/a^2\), which is how Newton, Robert Hooke, Christopher Wren and Edmond Halley each saw the inverse-square law around 1684.

Snowflakes, packings and wine barrels

For the general reader. Two small books show Kepler as a pure mathematician. A New Year’s gift to a patron in 1611 asked why snowflakes have six points and how equal spheres can be packed most tightly. His answer about packing became a conjecture that was proved only in this century. A dispute with a wine merchant in 1613 led him to compute the volumes of curved solids by slicing them into infinitely many thin pieces, a step toward integral calculus.

The six-cornered snowflake (1611)

In Strena seu de nive sexangula Kepler compared snowflakes with honeycombs and pomegranate seeds. Spheres packed in square layers, each layer sitting in the hollows of the one below, give an arrangement where each sphere touches twelve others. Squeezed together, the spheres become rhombic dodecahedra, as pomegranate seeds do. Kepler asserted that this packing is the tightest possible.

David Hilbert included the problem in his eighteenth problem in 1900. Thomas Hales, partly with Samuel Ferguson, announced a computer-assisted proof in 1998, published in the Annals of Mathematics in 2005. Doubts about checking the computations led Hales to organize the Flyspeck project, which completed a formal machine-verified proof in 2014.

The new stereometry of wine barrels (1615)

In Linz, buying wine for his second wedding, Kepler saw the merchant measure every barrel by pushing a rod through the bung hole to the far edge of the bottom, whatever the barrel’s shape. Kepler set out to see whether this could be fair. He first recomputed Archimedes’ results by treating a circle as infinitely many thin triangles with apex at the centre, so its area is half the radius times the circumference. He then found volumes of many solids of revolution, which he named after fruit (apple, lemon, olive), by slicing.

Kepler cut the ring into thin wedges by planes through the axis. Each wedge is thicker on the outside than the inside, but the excess and the deficit cancel about the centre circle, so each acts like a cylinder of length equal to the arc of the centre. This anticipates the centroid theorem of Pappus and Guldin. Kepler also noted that near a maximum a quantity changes very little, an observation that Pierre de Fermat later made into a method.

Logarithms and the Rudolphine Tables

For the general reader. An astronomical theory is useful only if it predicts. Kepler’s duty as Imperial Mathematician was to turn Tycho’s observations into tables from which anyone could compute planetary positions. The tables took him 26 years. Along the way he adopted John Napier’s new logarithms, rebuilt their theory and used them throughout. The tables were far more accurate than any before, and they predicted transits of Mercury and Venus across the Sun.

Logarithms

John Napier published his logarithms in 1614. Kepler saw the book in 1617 and grasped its value in 1618, the year of the third law. Distrusting Napier’s kinematic definition, he gave his own derivation from the theory of proportions, published as Chilias logarithmorum (1624) with a supplement in 1625. The key property, in modern form, is

which turns multiplication into addition and roots into division. The third law becomes linear: \(\log T=\tfrac32\log a+\text{const}\).

The tables

The Tabulae Rudolphinae, named for Rudolf II, were finished in 1623 but delayed by war and money. Kepler printed 1,000 copies at his own expense at Ulm in 1627. They contained Tycho’s star catalogue, the planetary theory based on ellipses and the area law, and logarithm tables. They remained the standard for about a century.

A transit occurs when an inferior conjunction happens near a node, where the planet’s orbit crosses the ecliptic. Using his tables, Kepler predicted in 1629 a transit of Mercury on 7 November 1631 and one of Venus a month later. Pierre Gassendi saw Mercury cross the Sun from Paris on 7 November 1631, within about ten arc-minutes of Kepler’s position; older tables had been off by degrees. The Venus transit happened at night in Europe. In 1639 Jeremiah Horrocks, correcting the Rudolphine Tables, predicted and observed a transit of Venus that Kepler had missed.

Linz, the witch trial and the war, 1612–1630

In 1612 Kepler became mathematician to the estates of Upper Austria at Linz, keeping his title of Imperial Mathematician under Matthias and then Ferdinand II. On 30 October 1613 he married Susanna Reuttinger, an orphan raised in a noble household; they had several children, some of whom died young. He chose her methodically after considering eleven candidates, and the marriage proved happy. In Linz the Lutheran pastor barred him from communion because he would not accept every article of the Formula of Concord. Kepler held to his own Lutheranism for life while refusing to condemn Calvinists.

In 1615 his mother Katharina was accused of witchcraft in Leonberg, in the middle of a wave of trials in Württemberg. Kepler took charge of her defence. She was arrested in August 1620 and held for over a year, shown the instruments of torture, and refused to confess. Kepler’s legal writing helped secure her release on 4 October 1621. She died about six months later.

The Linz years produced the Epitome astronomiae Copernicanae (1617–1621), the first textbook of astronomy built on the three laws, and the Harmonices mundi (1619). The first part of the Epitome was placed on the Catholic Index of prohibited books in 1619. In 1626 Linz was besieged during a peasants’ revolt, and the press printing the Rudolphine Tables was damaged. Kepler left for Ulm, finished the tables there, and in 1628 entered the service of the imperial general Albrecht von Wallenstein at Sagan in Silesia, partly in hope of the unpaid imperial salary. He set up a printing press there in 1629.

In October 1630 he set out for Regensburg, where the imperial diet was meeting, to recover money owed him. He fell ill soon after arriving and died there on 15 November 1630. He was buried in a Protestant churchyard outside the city, and the grave was lost when the churchyard was destroyed during the war. He had written his own epitaph: having measured the heavens, he now measured the shadows of the Earth.

From Kepler to Newton and beyond

Kepler’s laws were not accepted at once. Many astronomers used his tables without his ellipses, and Galileo never adopted them. The tables’ success, confirmed by the 1631 transit, persuaded the computers first. Jeremiah Horrocks applied elliptical theory to the Moon. Ismaël Boulliau, who first suggested an inverse-square law for a solar force, kept ellipses but replaced the area law by a geometric substitute.

Isaac Newton’s Principia (1687) turned the three laws into consequences of the laws of motion and universal gravitation. The area law follows from any central force, the ellipse from the inverse square, and the third law fixes the constant. Newton also showed where Kepler’s laws fail: the planets disturb each other, and the Sun moves. Kepler’s demand for a physical cause, set out in the subtitle of the Astronomia nova, “a celestial physics”, was the programme that Newton completed.

Legacy

• Astronomy: the three laws, the first accurate planetary tables, and the methods of orbit determination that run through Gauss to modern celestial mechanics. Kepler’s equation is solved billions of times a day in navigation software.
• Optics: the retinal image, the inverse-square law of illumination, and the astronomical telescope.
• Mathematics: the Kepler–Poinsot polyhedra, the classification of Archimedean tilings, infinitesimal volume methods, and the sphere-packing conjecture settled by Hales.
• Literature: the Somnium, a dream of a journey to the Moon that describes how the Earth would look from there, written around 1609, annotated in 1621 and printed in 1634. It is often called the first work of science fiction.
• Honours: the NASA Kepler space telescope (2009–2018), which found thousands of exoplanets by transit photometry, and many craters, an asteroid and a supernova remnant (SN 1604) bear his name.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Johannes Kepler, MacTutor History of Mathematics.
2. O. Gingerich, Kepler, Johannes, Dictionary of Scientific Biography, vol. 7 (1973).
3. Johannes Kepler, Wikipedia (with extensive references).
4. M. Caspar, Kepler, trans. C. D. Hellman, Dover, 1993.
5. J. Connor, Kepler’s Witch, HarperCollins, 2004.
6. U. Rublack, The Astronomer and the Witch, Oxford University Press, 2015.

Physics and mathematics

1. J. Kepler, Mysterium cosmographicum, Tübingen, 1596; trans. A. M. Duncan, The Secret of the Universe, Abaris, 1981.
2. J. Kepler, Astronomia nova, Heidelberg, 1609; trans. W. H. Donahue, Green Lion Press, 2015.
3. J. Kepler, Optics: Paralipomena to Witelo and Optical Part of Astronomy, trans. W. H. Donahue, Green Lion Press, 2000.
4. J. Kepler, The Harmony of the World, trans. E. J. Aiton, A. M. Duncan, J. V. Field, American Philosophical Society, 1997.
5. J. Kepler, The Six-Cornered Snowflake, trans. C. Hardie, Clarendon Press, 1966.
6. Epitome Astronomiae Copernicanae, Wikipedia.
7. C. Wilson, Kepler’s derivation of the elliptical path, Isis 59 (1968).
8. J. V. Field, Kepler’s Geometrical Cosmology, University of Chicago Press, 1988.
9. P. Colwell, Solving Kepler’s Equation over Three Centuries, Willmann-Bell, 1993.
10. T. C. Hales, A proof of the Kepler conjecture, Ann. of Math. 162 (2005).
11. T. Hales et al., A formal proof of the Kepler conjecture, Forum Math. Pi 5 (2017), arXiv:1501.02155.