64 PART 1^ |^ EXPLORING THE SKY
Danish astronomer. Tycho’s sudden death in 1601 left Kepler, the
new imperial mathematician, in a position to use the observations
from Hveen to analyze the motions of the planets and complete Th e
Rudolphine Tables. Tycho’s family, recognizing that Kepler was a
Copernican and guessing that he would not follow the Tychonic
system in completing Th e Rudolphine Tables, sued to recover the
instruments and books of observations. Th e legal wrangle went on
for years. Tycho’s family did get back the instruments Tycho had
brought to Prague, but Kepler had the books, and he kept them.
Whether Kepler had any legal right to Tycho’s records is
debatable, but he put them to good use. He began by studying
the motion of Mars, trying to deduce from the observations how
the planet moved. By 1606, he had solved the mystery, this time
correctly. Th e orbit of Mars is an ellipse and not a circle, he real-
ized, and with that he abandoned the 2000-year-old belief in the
circular motion of the planets. But even this insight was not
enough to explain the observations. Th e planets do not move at
uniform speeds along their elliptical orbits. Kepler’s analysis
showed that they move faster when close to the sun and slower
when farther away. With those two brilliant discoveries, Kepler
abandoned uniform circular motion and fi nally solved the puzzle
of planetary motion. He published his results in 1609 in a book
called Astronomia Nova (New Astronomy).
In spite of the abdication of Rudolph II in 1611, Kepler
continued his astronomical studies. He wrote about a supernova
that appeared in 1604 (now known as Kepler’s supernova) and
about comets, and he wrote a textbook about Copernican astron-
omy. In 1619, he published Harmonice Mundi (Th e Harmony of
the World), in which he returned to the cosmic mysteries of
Mysterium Cosmographicum. Th e only thing of note in Harmonice
Mundi is his discovery that the radii of the planetary orbits are
related to the planets’ orbital periods. Th at and his two previous
discoveries are so important that they have become known as the
three most fundamental rules of orbital motion.Kepler’s Three Laws of Planetary
Motion
Although Kepler dabbled in the philosophical arguments of his
day, he was at heart a mathematician, and his triumph was his
explanation of the motion of the planets. Th e key to his solution
was the ellipse.
An ellipse is a fi gure that can be drawn around two points,
called the foci, in such a way that the distance from one focus to any
point on the ellipse and back to the other focus equals a constant.
Th is makes it easy to draw ellipses using two thumbtacks and a loop
of string. Press the thumbtacks into a board, loop the string about
the tacks, and place a pencil in the loop. If you keep the string taut
as you move the pencil, it traces out an ellipse (■ Figure 4-14).
Th e geometry of an ellipse is described by two simple num-
bers. Th e semimajor axis, a, is half of the longest diameter, as
you can see in Figure 4-14. Th e eccentricity, e, of an ellipse is
half the distance between the foci divided by the semimajor axis.expedition. Kepler’s mother was apparently an unpleasant and
unpopular woman. She was accused of witchcraft in later years, and
Kepler had to defend her in a trial that dragged on for three years.
She was fi nally acquitted, but died the following year.
In spite of family disadvantages and chronic poor health,
Kepler did well in school, winning promotion to a Latin school and
eventually a scholarship to the university at Tübingen, where he
studied to become a Lutheran pastor. During his last year of study,
Kepler accepted a job in Graz teaching mathematics and astronomy,
a job he resented because he knew little about the subjects.
Evidently he was not a good teacher—he had few students his fi rst
year and none at all his second. His superiors put him to work
teaching a few introductory courses and preparing an annual alma-
nac that contained astronomical, astrological, and weather predic-
tions. Th rough good luck, in 1595 some of his weather predictions
were fulfi lled, and he gained a reputation as an astrologer and seer.
Even in later life he earned money from his almanacs.
While still a college student, Kepler had become a believer in
the Copernican hypothesis, and at Graz he used his extensive spare
time to study astronomy. By 1596, the same year Tycho arrived in
Prague, Kepler was sure he had solved the mystery of the universe.
Th at year he published a book called Th e Forerunner of Dissertations
on the Universe, Containing the Mystery of the Universe. Th e book,
like nearly all scientifi c works of that age, was written in Latin and
is now known as Mysterium Cosmographicum.
By modern standards, the book contains almost nothing of
value. It begins with a long appreciation of Copernicanism and
then goes on to speculate on the reasons for the spacing of the
planetary orbits. Kepler assumed that the heavens could be described
by only the most perfect of shapes. Th erefore he felt that he had
found the underlying architecture of the universe in the sphere plus
the fi ve regular solids.* In Kepler’s model, the fi ve regular solids
became spacers for the orbits of the six planets which were repre-
sented by nested spheres (Figure 4-13). In fact, Kepler concluded
that there could be only six planets (Mercury, Venus, Earth, Mars,
Jupiter, and Saturn) because there were only fi ve regular solids to act
as spacers between their spheres. He provided astrological, numero-
logical, and even musical arguments for his theory.
Th e second half of the book is no better than the fi rst, but it
has one virtue—as Kepler tried to fi t the fi ve solids to the plan-
etary orbits, he demonstrated that he was a talented mathemati-
cian and that he was well versed in astronomy. He sent copies of
his book to Tycho on Hveen and to Galileo in Rome.
Joining Tycho
Life was unsettled for Kepler because of the persecution of
Protestants in the region, so when Tycho Brahe invited him to
Prague in 1600, Kepler went readily, eager to work with the famous
- Th e fi ve regular solids, also known as the Platonic solids, are the tetrahedron,
cube, octahedron, dodecahedron, and icosahedron. Th ey were considered perfect
because the faces and the angles between the faces are the same at every corner.