geometry of the P
LATONIC SOLID
s, for his discovery of
two nonconvex regular polyhedra, and for his mathe-
matical treatment of the close-packing properties of
SPHERE
s. (He also explained why the honeycomb shape
is the most efficient design for dividing a planar region
into separate cells.) More importantly, Kepler devised
a method for computing the volumes of many
SOLID
s
OF REVOLUTION
with the aid of
INFINITESIMAL
s. Today
this is seen as a significant contribution to the develop-
ment of
CALCULUS
.
Kepler studied astronomy and theology at the Uni-
versity of Tübingen, Germany. At the time, only six
planets were known to astronomers, and all were
assumed to be in circular orbit about the Sun. In 1596
Kepler published Mysterium cosmographicum (Mys-
tery of the cosmos), in which he presented a mathemat-
ical theory explaining the relative sizes of the planets’
orbits. Convinced that God had created the universe
according to a mathematical plan, Kepler posed that if
a sphere were drawn about the path of Saturn and a
CUBE
inscribed in this sphere, then the orbit of Jupiter
lies on a sphere inscribed in this cube. Moreover,
inscribing a
TETRAHEDRON
in this second sphere and a
sphere within the tetrahedron captures the orbital path
of Mars. Continuing in this way, with a dodecahedron
between Mars and Earth, an icosahedron between
Earth and Venus, and an octahedron between Venus
and Mercury, Kepler produced a model for orbit sizes
that is accurate to within 10 percent of observed val-
ues, well within experimental error. As there are only
five Platonic solids, this model also explained why, sup-
posedly, there were only six planets. Of course Kepler’s
Platonic model of the solar system is not correct—three
more planets were later discovered and, as Kepler him-
self later established, no orbit of a planet is circular.
Kepler moved to Prague near the turn of the cen-
tury to work with one of the foremost astronomers of
the time, Tycho Brahe (1546–1601). When Brahe died,
Kepler succeeded him as imperial mathematician.
Brahe had kept extensive records on the orbit of
Mars, and from them, Kepler was forced to conclude
that its orbit was an ellipse. He also noted, from the
data, that the velocity of the planet altered in such a
way that the line connecting the Sun to the planet swept
out equal areas in equal times. These two laws, when
extended to all planets, are today called Kepler’s first
two laws. He published them in his 1609 piece Astrono-
mia nova (New astronomy). Ten years later he added a
third law: the squares of the times taken by the planets
to complete an orbit are proportional to the cubes of
the lengths of the major axes of their elliptical orbits.
Kepler had no explanation as to why these laws
were true other than the compelling evidence of the
data. It was not until S
IR
I
SAAC
N
EWTON
(1642–1727)
formulated his famous law of gravitation that Kepler’s
laws could be mathematically deduced.
During his marriage ceremony in 1613 to his sec-
ond wife Susanna (his first wife Barbara died in 1611),
Kepler noticed that the servants would measure the vol-
ume of a wine barrel by slipping a rod diagonally
through the bunghole and measuring the length that fit.
He began to wonder why this method worked. This led
him to his study of the solids of revolutions and the
292 Kepler, Johannes
Johannes Kepler, an eminent astronomer of the 17th century, is
noted for his introduction of three laws of planetary motion. These
laws provided Sir Isaac Newton the inspiration to develop his
inverse-square law of gravitation. (Photo courtesy of the Science
Museum, London/Topham-HIP/The Image Works)