Around the year 1600, a rare but most fortunate constellation occurred. Not in the sky this time, but in the city of Prague. It consisted of three men whose destinies by chance crossed there and then. They could not be more different as to their origins, life stories, and careers, as well as to their characters, temperaments, and expectations. But they all played their parts in what would almost thirty years later result in a longawaited publication, the Tabulae Rudolphinae, or the Rudolphine Tables. The three men were:
• Rudolph II, the Holy Roman Emperor and patron whose name is in the
title;
• Tycho Brahe, who had performed and collected the best astronomical observations the world ever saw before the telescope;
• Johannes Kepler, the author of the book, who out of these observations
managed to put an end to the old obvious truth that planets move uniformly in circles.
The Danish Protestant nobleman Tycho Brahe had been awarded the island of Ven in the strait separating today’s Sweden and Denmark. There he erected a castle together with the best equipped observatories available. He understood that systematically higher precision was necessary to comprehend how the planets moved. After twenty years of recorded observations on an almost industrial scale, the haughty aristocrat fell into political disfavour with the young, rising Danish king and had to abandon his island.
In this period the Holy Roman Emperor resided in Prague. Rudolph II was
senior to all the other Roman Catholic monarchs. He was a patron of the arts and showed an interest towards science, mostly in the form of astrology and alchemy, but he also showed growing symptoms of mental instability. Now Rudolph was in need of a new imperial mathematician. After swallowing his aristocratic feelings, Tycho Brahe accepted the job with a sense of relief and became an employee for the first time.
Tycho had by this time realized that neither
he nor his assistants could find coherence in all the observations he had
collected. He thought he had seen something special in the speculative mind
of a young Protestant school teacher and district mathematician in Graz by
the name of Johannes Kepler. In the Roman Catholic city of Graz, the last
Protestants were finally ordered to leave town. Kepler travelled to Prague
in hopes of a secure position and the most precious thing in his
world—Tycho’s observations. After some personal friction and scientific
disagreement, Tycho decided to introduce the disillusioned Kepler to Emperor
Rudolph. At an audience in September 1601 the three men met and Tycho
suggested a glorious new project, to be entitled the Rudolphine Tables. Everyone was pleased, and so
it happened that the two Protestant scientists were engaged in the Roman
Catholic imperial epicenter of Prague. But just a month later Tycho Brahe
died unexpectedly. It was promptly decided that Kepler was to take over
Tycho’s position, and his main task was to compile the Tables. He successfully managed to achieve this
goal, but the book was not published until 1627. In the meantime Kepler had
several problems and conflicts to manage: fighting with the Brahe heirs who
watched over their interests; inventing a new planetary theory and other
methods to improve accuracy and facilitate the reading of the Tables; procuring a salary from the imperial
treasury for his family to survive; dealing with many personal
disappointments; carefully maneuvering both in war conflicts and around three
successive emperors; and curbing his love for new mathematical speculations.
The Rudolphine
Tables were the first set of astronomical tables that allowed one to
calculate the position of the planets even in the far future with an accuracy
sufficient to make reliable predictions. To be sure, there were earlier
tables much resorted to by navigators, astronomers, and astrologers, but
regardless of what model they were based on, their predictions gave errors of
up to five degrees. (For comparison, the angular diameter of the moon is
around half a degree.) The predictions made from the new Tables, however, proved to be 30 to 50 times more
accurate. The astronomical tables themselves did not state any positions at a
certain day, but they served as the basis for annual ephemerides which
provided calculated daily positions of celestial objects.
This is the only book by Kepler to carry an
elaborate frontispiece, and he designed it himself (see Figure 1). By taking a tour around the picture, we may see how
Kepler pays tribute to his astronomical predecessors. At first sight the
impression is that he portrays himself in a very unobtrusive or even
submissive way. It is true that he wanted to give prominence to Tycho, and
that he was under pressure from the Brahe family to do so. However, after a
closer look we will discover that Kepler, with subtle methods, often manages
to emphasize his own achievements and hardships. After the title pages of the
Tables there is a long hexameter poem by
Kepler’s Latinist friend Johannes Hebenstreit. It is a complimentary idyll,
steeped in mythology, but does not always follow the frontispiece, to put it
mildly.
The frontispiece nicely matches the original
French architectural term designating a decorated front of a building. A frontispiece should set the character of the interior of a building or a book, soon to be
disclosed to the eyes of the viewer or reader. In a book, it also gave a
pleasant contrast to the often long and nested text following the title page.
We stand in front of a classical temple
dedicated to the muse of astronomy, Urania. She is seen at the top of the
dome rushing forward in a carriage. Carrying a starred crown, she seems to be
searching for someone to crown with a laurel wreath. Well, there are some
good candidates further down.
There are twelve columns on the high
foundation, two of them hidden from our view. This corresponds to a temple
shaped as a dodecagon. The twelve sides represent the signs of the Zodiac,
some of them visible as sections on the floor.
The increasing solidity and elegance of the
columns (proceeding from the back to the front of the temple) describes the
development of astronomy from ancient times until Copernicus and Tycho Brahe.
Close to the columns, some historic figures are absorbed in work and
discussion. At the back, we spot a Chaldean astronomer in a Phrygian cap
making observations between roughhewn log pillars. It is in the Babylonian
and Chaldean civilizations that we find many of the earliest traces of
Western mathematical astronomy. Their priestscribes compiled catalogues,
recognized many cyclic occurrences, and tried to make predictions
arithmetically. They observed that the movements of the planets were not
uniform. To describe and explain this behaviour became the central
theoretical problems for astronomers ever since the Greek era. Kepler also
knew well that the sexagesimal number system had its origins in ancient
Mesopotamia, and sexagesimal coordinates were something that he made
frequent use of in his Tables.
The next two columns are portrayed as fragile
piles of stone blocks, but proceeding to the brick pillars, we find our
first two names. On the right, the name Meton sticks out. Active at the time
of Socrates, he was a wellknown Athenian geometer and astronomer. He tried
to improve the calendar and gave name to the Metonic cycle, which is based on
the observation that 19 years are (coincidentally) very close to 235 months
counted as the period of the moon’s phases. Many civilizations used this cycle
to form lunisolar calendars. On Meton’s column hangs a tribute to them: a
disc divided into 19 sections, decorated with sun and moon. In Kepler’s days
Gregorian calendar reform was fiercely debated. Although criticized by other
Protestants, both Tycho and Kepler stood up for it.
On the left side, it may at first seem odd
that Kepler chose the Greek poet Ara tus as one of the pillars of astronomy.
Aratus is mainly known for a long didactic hexameter poem called Phaenomena (“Appearances”). This work, a blockbuster
in antiquity, is actually a commissioned verse setting of treatises by
Eudoxus, one of the finest Greek mathematicians and astronomers. None of his
works have survived, however, and we have to rely on others to get glimpses
of his contributions and brilliance. In Aratus’s poem we find the first
description of our familiar star constellations. Eudoxus combined a number of
nested rotating spheres to represent a planet’s motion around the Earth.
With this clever geometric model he could qualitatively account for
apparently irregular motions seen in the sky, not only deviation from the
ecliptic, but also retrograde motion (where he needed four spheres). The
first mathematical model combining the Pythagorean and Platonic ideas of
uniform circular motions was born, the starting point of a long tradition.
Nailed to Aratus’s column is an old
astronomical instrument, the armillary sphere. The Earth in the middle is
surrounded by the equator, the ecliptic, and other great circles of interest. It was widely
used as a quick and simplified way to determine celestial positions in
various frames of reference. Look at the flag of Portugal and you will see
another example honouring a seafaring nation.
While the last two brick columns were
associated with the foundations of astronomy in the Greekspeaking world,
the next two represent the culmination.
The armillary sphere can be seen as a skeleton
version of the celestial globe displayed on the neighbouring column of
Hipparchus. The smooth surface could show the stars and the constellations
(seen from outside), all often designed as a work of art. Being a skilled
observer, Hipparchus compiled a famous catalogue around 130 BC with hundreds
of star positions, which was not surpassed, in its essence, until Tycho
Brahe’s observations. Kepler made the lost catalogue appear again in the
picture, where Hipparchus carries in his right hand a book with the Latin
title Catalogus stellarum fixarum. In his
left hand, we see a document labelled Test.
with two affixed seals. The title could be short for Testimonium or Testamentum, a testimony or legacy not only of the influential
contributions made by Hipparchus himself, but also of observations and views
by his predecessors. These led Hipparchus to the discovery of precession, which we would now describe as a slow
change in the orientation of the Earth’s axis of rotation. We should also honour
him as the founder of trigonometry: the first trigonometric table listed
chord length as a function of the central angle.
Around the middle of the second century AD,
the most important astronomical book that survived the turmoils of antiquity
and its aftermath appeared. With Ptolemy’s Almagest
we have for the first time a table and parameters that allowed the reader to
calculate the heavenly motions. The author also presented a coherent
geocentric model of the known universe based on his own and earlier
observational data.
Ptolemy, like a true mathematician, is sitting
writing on the tablecloth, with his book close at hand, and we see one
version of the Greek title, Megale syntaxis.
Our current English title, Almagest, goes
back to the mixed Arabic name meaning “the greatest”. A small figure can be
discerned on the table: a quadrilateral inscribed in a circle. Apparently
Ptolemy has just discovered his powerful theorem saying that the product of
the two diagonals equals the sum of the products of the opposite sides. With
its help he would generate a famous trigonometric chord table from a few
basic chord values known since Euclid.
On a stone tablet leaning against the
pedestal, there is an illustration of the three mathematical
“e”constructions used by Ptolemy and his forerunners: The planet moved
uniformly on a small circle called an epicycle,
the center of which revolved along a larger circle. If the Earth was outside
the center of this larger circle, its position was called eccentric. The most controversial trick used to
account for the variation in a planet’s angular speed was to make the motion
along the large circle uniform only when seen from another point, called the equant, different from the center and the Earth.
Much of Ptolemy’s impressive, but incorrect, conception of the world was
adopted by scholars and the Christian Church, but with the decline of the
Hellenistic culture and loss of many of its texts, these ideas fossilized
during the early Middle Ages.
The handy flat instrument on Ptolemy’s column
is a version of the celebrated astrolabe. It was a versatile predecessor of
the simple didactic planisphere, or (why not) an analog smartphone in
antiquity and the Middle Ages. Ptolemy used and described the astrolabe, which was basically a
plane model of the sky through stereographic projection, with nested
revolving plates attached.
Continuing on to the two most elegant columns
at the front, there are but a few steps, though it calls for a giant leap in
time. Copernicus is sitting close to a Roman Doric column. Right above him
hangs an instrument that goes back at least to Ptolemy, but which Copernicus
described in his famous book De revolutionibus
orbis coelestium, lying in his lap. It was called a triquetrum, a practical design for reading the
altitude of objects from the horizon. Further up we find a long cross.
Although Copernicus had official positions at the cathedral in Frombork for
most of his life, he never became a priest. So we should rather understand
the cross as another instrument to measure altitudes and angles, known as a
crossstaff or a Jacob’s staff. The simplicity made it popular with
navigators.
Copernicus conducted many observations from
his Frombork tower, still with quite primitive instruments. But as recorded
through the manuscript at the base of the pillar, late in his life he was
also presented with a large stock of unpublished planetary observations made
by the two fifteenth century Nuremberg astronomers Regiomontanus and Bernhard
Walther.
For more than thirty years Copernicus
developed his new model with the Earth both rotating around its axis and
revolving around a static sun. In 1543, the last year of his life, he was
finally persuaded to print the book De
revolutionibus. Even though it is clear that he meant to describe the
physical reality, there were several reasons why the theory was treated as
another convenient hypothesis for mathematicians to work out the planetary
motions. This was the tradition for all earlier models, and the church
authorities were satisfied as long as it remained so. But even most
scientists were not yet convinced. Theoretically, it was not much simpler
than the more intuitive geocentric model. Copernicus had still felt the need
to employ a system of circles and a few epicycles, as did Ptolemy. And
practically, the observations available at this time, which should confirm a
correct model, were simply not accurate and systematic enough.
Tycho Brahe had advocated his own model to
account for the planetary motions, a kind of compromise between the Ptolemaic
and Copernican ideas, known as the Tychonic system. The Earth is still at the
center with the sun revolving around it, but the other planets move around
the sun. In the picture we can see Tycho pointing at the model adorning the
temple ceiling while trying to persuade Copernicus, Quid si sic? (“What about this?”) This question
appears less innocently in a popular emblem book from 1611 by Gabriel
Rollenhagen (familiar to Kepler). An emblem was a picture accompanied by a
short aphoristic text. In this particular one, a dwarf on stilts looks at his
mirror image, and the text where he poses the question ends with the morality
“art can never improve on the wisdom of nature”.
But the scene in the Tables frontispiece actually went back to Tycho
himself. There are descriptions of the interior of his halfsubterranean
observatory Stjerneborg on Ven. There in an ornamented, heated, central room,
the Tychonic model could be seen from the cupolaceiling. Much in that room,
such as allegorical figures, paintings, and inscriptions, must have inspired
Kepler to the frontispiece, even though he never saw the room. Beyond seven
paintings of astronomers from antiquity to Tycho himself, a yet unborn,
messianic figure called Tychonides was portrayed, with the Latin inscription
“that he might be worthy of his great ancestor”!
In the frontispiece Tycho Brahe is leaning
against the most decorated Corinthian column. After complaints from the Brahe
family, he is now dressed in his wellknown ermine robe with the collar of
the Order of the Elephant around his neck (the highest order in Denmark). We
see one of his books, the Progymnasmata,
which described his planetary theories and contained the new star catalogue.
Kepler completed and published the book in 1602 after Tycho’s death.
On the column, we see two instruments which
Tycho had developed and improved in order to attain better precision and
stability. A breakthrough was the MediumSized Quadrant from 1580 made of
solid brass, just above his head. He also constructed a number of sextants
with 60° arcs which were mainly made of wood. They were often light and
portable, but later he made them larger and fixed, like the Triangular
Sextant from 1582 further up on the column. (It looks rather like a 45°
octant here, but we look at it slightly from the side!)
These instruments were used on the isle of
Ven, which is prominently displayed in the central panel of the foundation
with the castle in the middle. According to Hebenstreit’s poem, the important
base meridian used by Tycho and Kepler is shown. The dashed line in the
picture, however, is if anything perpendicular to this meridian, since the
north is directed towards the right, as the compass rose correctly shows.
In the panel to the left we finally catch
sight of Kepler himself. By candlelight, tired and dressed in nightwear, the
protagonist looks at us, perhaps asking for pity. The long hours have taken
their toll on him. Only a few coins from the Habsburg eagle have reached his
table, but he is the proud architect of the temple. On the table lies a
replica of the dome, where he displayed his skills as the culmination of the
tradition below, as we shall see. On the wall are four titles of his major
works and four city coats of arms, representing Bohemia, Prague, Upper
Austria, and Linz (where Kepler moved in 1612).
In the adjoining panel, one of Tycho Brahe’s
heirs points upwards at Tycho’s observational records and to the right at
Kepler. The two panels at the other end refer to the printing of the book (in
Ulm). In between them the doubleheaded Habsburg eagle grants the printing
privilege. Below we are told that Georg Coler engraved the frontispiece in
Nuremberg.
laws, were wrong, the argument of a
suncentered force had put him on the right track.
As a consequence Kepler also insisted that the
Earth in his model should behave just like the other planets. Some
traditional Earthspecific features could still be seen in Copernicus’s
theory, but Kepler now exploited all aspects of heliocentrism. In this way
the assumptions became simple, natural, and uniform, though it turned out
that he had to invent new techniques and go through long and complicated
calculations in order to truly describe the orbits. The focus was on the
planet Mars. Kepler’s “warfare on Mars” engaged him between 1600 and 1605,
and the process has been called “the birth of the scientific method”.
In his first attempts he tried to keep a
circular orbit for Mars around the sun. To account for the observations, he
put the sun a little off center, and on the opposite side he reinserted an
equant; i.e., seen from this point Mars moved uniformly. Even though he varied
the relative distances of the sun and the equant from the center of the
orbit, the model refused to come sufficiently close to Brahe’s trusted
observations. If the distances were equal, however, he noted that Mars’
linear velocity seemed to be inversely proportional to its distance from the
sun. This distance law was strictly valid only when the planet was furthest
from or nearest to the sun, but he hypothesized the rule for the whole orbit.
Later he found the correct rule by substituting the linear velocity with its
component perpendicular to the radius vector from the sun. From the distance
law it was theoretically possible to determine the position of Mars in its
orbit at a given time. With no calculus available, but inspired by ideas from
Archimedes and Nicholas of Cusa, he first resorted to numerical summation to
compute an elliptic integral and then solved the inverse problem by
interpolation. The calculations became exceedingly laborious, but he found a
shortcut by summing areas instead of distances: the
line joining a planet to the sun sweeps out equal areas in equal time.
It has become known as the area law or Kepler’s second law. Sector areas in an eccentric
circle were known, and the calculations could be simplified. Typical is
Kepler’s use of conjectures which, even if not completely accurate, led him
forward to new discoveries and relations, from which he could go back and
modify the conjectures and their connections. The level of proof was set by
Tycho Brahe’s observations and the requirement that all motions should be
physically natural.
The next muse, Stathmica
(second from right), is meant to illustrate the area law. She holds a
steelyard balance. The sun is at the fulcrum and (instead of a fish and some
weights) two planets are hanging so that the system is in equilibrium. This
is a simplified version of a figure in Book IV of Epitome
astronomiae Copernicanae (1621), the full and mature outline of
Kepler’s achievements and the basis for the Tables.
His idea of how the motion of a planet is balanced under the force from the
sun is symbolized in this simple figure. We may also see both the law of the
lever and Kepler’s area law as expressing the principle of conservation of
angular momentum. Newton later showed the area law to be valid under any
central force.
Implicit in Kepler’s models thus far had been
the traditional assumption that the Earth moved in an eccentric circular
orbit, but uniformly with respect to the center (i.e., no need for a distinct
equant). He set as his next task to find out whether this was really true.
The determination of a planet’s orbit around the sun required of course a
proper description of how the Earth moved. By a famous new method in which he
used specific Mars observations from Tycho’s abundant data, Kepler
found that also in the case of the Earth the
equant point had to differ (slightly) from the center of the orbit. Now he
used this better approximation of the Earth’s orbit, together with the area
law, to continue his attack on Mars.
But with its larger eccentricity, the circular
orbit model for Mars still refused to agree with Tycho’s observations. It
took Kepler two more years to finally arrive at the correct elliptic orbit.
It was clear that the circle had to be compressed to an oval of some kind. He
first tried with the boundary of an asymmetric twodimensional egg, since
this path could be generated by an (oldstyle) epicycle. He approximated this
oval with an ellipse in order to more simply use his area law; sector areas
were explicitly known for ellipses since antiquity. Finally, he found a close
fit when he chose a new ellipse halfway between the circle and the
approximating ellipse.
He might have stopped here and triumphed. But
just as when he abandoned the circle, he still did not understand how an
elliptic orbit could be physically motivated. As a fortunate consequence of
the further pursuit, he arrived at an explicit construction of the orbit. An
opening came when he noticed a striking geometric expression for the distance
between the sun and a specific position of Mars. Again he could make a
natural generalization of this expression around the whole orbit,
representing the distance as a function of the orbital angle he had used
before, the socalled eccentric anomaly.
This angle was measured at the center along the eccentric auxiliary circle.
He was disappointed to find that the orbit (called the via buccosa) did not comply with the observations
sufficiently well, and it was not even an ellipse. Struck by a last brilliant
idea, Kepler now realized that he could represent the suntoMars direction
by another generalization, and still keep the earlier relation between
distance and eccentric anomaly. In technical terms he had redefined the
relation between the eccentric anomaly and the true anomaly (the angle of the
Mars position seen from the sun). And this time the calculated orbit was in
accordance with Tycho’s data. He could also prove that the curve was an
ellipse with the right eccentricity, and he claimed that, as described, the
ellipse had physical grounds. (It was shown much later that also the via buccosa actually fulfilled all of Kepler’s
criteria. He seems to have made an incorrect implicit assumption which caused
the discrepancy. Perhaps Kepler would have attacked and resolved this problem
as well, had he been aware of it! At least one can say that he had taken his
theory down to the very limits of Tycho’s observations.)
Kepler was ready in 1605 to publish the new
results. In the exceptional book Astronomia nova
he included also the setbacks and failures along the road to the ellipse.
Conflicts with the Brahe family and financial problems delayed the printing
until 1609.
In the right center position on the dome, the
muse known as Doctrina triangulorum (the
science of triangles), i.e., trigonometry, reminds us of Kepler’s first law, as it was later called: planets move in ellipses with the sun at one focus.
In a firm grip she holds the traditional square and compasses. In front of
her we see a figure which Kepler used to describe and determine his last
ellipse. It is shown dashed inside its auxiliary circle. The sun S is at a focus, and some positions of Mars are
designated by F, H, and L along
the ellipse. Later in the book, in Chapter 20 of the explanatory notes before
the actual tables, Kepler refers to this tiny figure found only in the
frontispiece!
Just above, you cannot miss the Habsburg eagle
hovering over the dome. Rudolph II and the two succeeding emperors here got
their due share. The bird of prey generously drops coins from its beak,
which we may follow down through the temple.
The fourth muse in the left center position
bears the name of Logarithmica. For Kepler
it seemed as though the heavens repaid him for his efforts when he heard of
Napier’s invention of the logarithms in 1617. He understood at once their
importance and how much time and effort would be saved in the computations
necessary for his Tables, and he became the
first to apply logarithms on a large scale. Napier’s second book, which
explained the construction, had not yet appeared, so Kepler began to develop
his own theory and logarithmic tables. He derived the independent heavy
construction from Euclid’s theory of proportions. Theory and tables were
published in two books a few years later. For the convenience of the reader,
Kepler cleverly included two logarithmic tables in the Rudolphine Tables as well. Take a look at the
halo above Logarithmica’s head where the number 69314.72 shines. We recognize
it today as a multiple of the basic log_{e} 2. Actually, Kepler’s
first table gives —10^{5} log_{e} x, while the second
tabulates —10^{5} log_{e} sin y. So x = 2 and y =
30° give the logarithmic value in the halo. The lengths of the two rods that
muse Logarithmica carries are in proportion 1 : 2. In music theory, this
ratio corresponds to the basic octave interval. Such harmonies always played
a vital role in Kepler’s thoughts.
It is interesting to note that it was when
Kepler learned and thought about the new logarithms that he also discovered
his socalled third law: for any two planets the squares of their orbital periods
are to each other as the cubes of their mean distances from the sun.
Kepler’s mind was now open to thinking in terms of geometric sequences.
The last two muses both reflect the fact that
Kepler wrote two books which laid the foundation for theoretical optics. You
do not hear much about this since he neither invented the telescope (he had
bad eyes) nor discovered the exact law of refraction (he only needed a good
approximation). Descartes and others later acknowledged Kepler as the master
of optics. Early questions regarding eclipses and refraction led Kepler to
write the first modern book on optics, Astronomiae
pars optica (1604). This was done in the midst of his intense Mars
studies. He also came up with novel ideas about conic sections when he
treated curved mirrors.
On the far left, the radiating head of the
muse Physica lucis et umbrarum (“the
physics of light and shadows”) seems to cast a shadow cone behind the sphere
in her right hand. The sphere and shadow also bear some resemblance to a
comet with a tail. Among the several comets Kepler observed and described
during his life, one was sighted in 1607. It was to reappear 75 years later
and became known as Halley’s comet.
The muse Optica
(second from left) exhibits a telescope of an odd rectangular form. In 1610
Kepler first heard of Galileo’s telescope and what it had revealed about the
heavens. Again he had immediately recognized and acknowledged a new
revolutionary invention. As with the logarithms he wrote an important
treatise, Dioptrice, explaining
mathematically how light passed through systems of lenses, especially in
Galileo’s magical tube. He went on to design a new type of telescope with a
convex lens also for the eyepiece instead of Galileo’s concave one, which
allowed for higher magnification.



At the top we see six scientific muses
bordering the dome, all at the service of Queen Urania above them. Such
allegorical figures were abundant in medieval and Renaissance illustrations,
typically personifying the seven liberal arts which formed the educational
canon. Kepler here instead invokes a different set of sciences covering
important discoveries, inventions, and ideas in his career.
We start at the right side of the dome with
the charged figure of Magnetica holding a
lodestone and a compass. Already in his first work Kepler had insisted on a
physical cause of planetary motion. Earlier it was argued that you could not,
or at least should not try to, figure out the physical grounds. Kepler, with
his firm belief that all the planets orbited the sun in almost the same
plane, thought that such a force must emanate from the sun. And without the
unifying theory of gravity and the conception of inertial motion, he turned
to magnetism. This was the force in vogue at a time when the Earth had
recently been found to act as a giant magnet. The rotation of the sun and the
magnetic interaction between sun and planet caused the planet to orbit the
sun, Kepler proposed. Even though these physical speculations, and other
neoPlatonic ideas based on simplicity of natural found that also in the case of the Earth the
equant point had to differ (slightly) from the center of the orbit. Now he
used this better approximation of the Earth’s orbit, together with the area
law, to continue his attack on Mars.
But with its larger eccentricity, the circular
orbit model for Mars still refused to agree with Tycho’s observations. It
took Kepler two more years to finally arrive at the correct elliptic orbit.
It was clear that the circle had to be compressed to an oval of some kind. He
first tried with the boundary of an asymmetric twodimensional egg, since
this path could be generated by an (oldstyle) epicycle. He approximated this
oval with an ellipse in order to more simply use his area law; sector areas
were explicitly known for ellipses since antiquity. Finally, he found a close
fit when he chose a new ellipse halfway between the circle and the
approximating ellipse.
He might have stopped here and triumphed. But
just as when he abandoned the circle, he still did not understand how an
elliptic orbit could be physically motivated. As a fortunate consequence of
the further pursuit, he arrived at an explicit construction of the orbit. An
opening came when he noticed a striking geometric expression for the distance
between the sun and a specific position of Mars. Again he could make a
natural generalization of this expression around the whole orbit,
representing the distance as a function of the orbital angle he had used
before, the socalled eccentric anomaly.
This angle was measured at the center along the eccentric auxiliary circle.
He was disappointed to find that the orbit (called the via buccosa) did not comply with the observations
sufficiently well, and it was not even an ellipse. Struck by a last brilliant
idea, Kepler now realized that he could represent the suntoMars direction
by another generalization, and still keep the earlier relation between
distance and eccentric anomaly. In technical terms he had redefined the
relation between the eccentric anomaly and the true anomaly (the angle of the
Mars position seen from the sun). And this time the calculated orbit was in
accordance with Tycho’s data. He could also prove that the curve was an
ellipse with the right eccentricity, and he claimed that, as described, the
ellipse had physical grounds. (It was shown much later that also the via buccosa actually fulfilled all of Kepler’s
criteria. He seems to have made an incorrect implicit assumption which caused
the discrepancy. Perhaps Kepler would have attacked and resolved this problem
as well, had he been aware of it! At least one can say that he had taken his
theory down to the very limits of Tycho’s observations.)
Kepler was ready in 1605 to publish the new
results. In the exceptional book Astronomia nova
he included also the setbacks and failures along the road to the ellipse.
Conflicts with the Brahe family and financial problems delayed the printing
until 1609.
In the right center position on the dome, the
muse known as Doctrina triangu lorum (the
science of triangles), i.e., trigonometry, reminds us of Kepler’s first law, as it was later called: planets move in ellipses with the sun at one focus.
In a firm grip she holds the traditional square and compasses. In front of
her we see a figure which Kepler used to describe and determine his last
ellipse. It is shown dashed inside its auxiliary circle. The sun S is at a focus, and some positions of Mars are
designated by F, H, and L along
the ellipse. Later in the book, in Chapter 20 of the explanatory notes before
the actual tables, Kepler refers to this tiny figure found only in the
frontispiece!
Just above, you cannot miss the Habsburg eagle
hovering over the dome. Rudolph II and the two succeeding emperors here got
their due share. The bird of prey generously drops coins from its beak,
which we may follow down through the temple.
The fourth muse in the left center position
bears the name of Logarithmica. For Kepler
it seemed as though the heavens repaid him for his efforts when he heard of
Napier’s invention of the logarithms in 1617. He understood at once their
importance and how much time and effort would be saved in the computations
necessary for his Tables, and he became the
first to apply logarithms on a large scale. Napier’s second book, which
explained the construction, had not yet appeared, so Kepler began to develop
his own theory and logarithmic tables. He derived the independent heavy
construction from Euclid’s theory of proportions. Theory and tables were
published in two books a few years later. For the convenience of the reader,
Kepler cleverly included two logarithmic tables in the Rudolphine Tables as well. Take a look at the
halo above Logarithmica’s head where the number 69314.72 shines. We recognize
it today as a multiple of the basic log_{e} 2. Actually, Kepler’s
first table gives —10^{5} log_{e} x, while the second
tabulates —10^{5} log_{e} sin y. So x = 2 and y =
30° give the logarithmic value in the halo. The lengths of the two rods that
muse Logarithmica carries are in proportion 1 : 2. In music theory, this
ratio corresponds to the basic octave interval. Such harmonies always played
a vital role in Kepler’s thoughts.
It is interesting to note that it was when
Kepler learned and thought about the new logarithms that he also discovered
his socalled third law: for any two planets the squares of their orbital periods
are to each other as the cubes of their mean distances from the sun.
Kepler’s mind was now open to thinking in terms of geometric sequences.
The last two muses both reflect the fact that
Kepler wrote two books which laid the foundation for theoretical optics. You
do not hear much about this since he neither invented the telescope (he had
bad eyes) nor discovered the exact law of refraction (he only needed a good
approximation). Descartes and others later acknowledged Kepler as the master
of optics. Early questions regarding eclipses and refraction led Kepler to
write the first modern book on optics, Astronomic
pars optica (1604). This was done in the midst of his intense Mars
studies. He also came up with novel ideas about conic sections when he
treated curved mirrors.
On the far left, the radiating head of the
muse Physica lucis et umbrarum (“the
physics of light and shadows”) seems to cast a shadow cone behind the sphere
in her right hand. The sphere and shadow also bear some resemblance to a
comet with a tail. Among the several comets Kepler observed and described
during his life, one was sighted in 1607. It was to reappear 75 years later
and became known as Halley’s comet.
The muse Optica
(second from left) exhibits a telescope of an odd rectangular form. In 1610
Kepler first heard of Galileo’s telescope and what it had revealed about the
heavens. Again he had immediately recognized and acknowledged a new
revolutionary invention. As with the logarithms he wrote an important
treatise, Dioptrice, explaining
mathematically how light passed through systems of lenses, especially in
Galileo’s magical tube. He went on to design a new type of telescope with a
convex lens also for the eyepiece instead of Galileo’s concave one, which
allowed for higher magnification.
With unflagging enthusiasm Hebenstreit reveals
in the poem that behind the dome, out of our sight, there are six more muses
personifying other sciences which had engaged Kepler’s thoughts.
When you look at a page in the Rudolphine Tables, give a thought to the mathematical
and astronomical symbols. These special printing types were precious to
Kepler. He only had confidence in his own set, which he carried around from
city to city in the 1620s when the war came closer.
Shortly before his death in 1629 Johannes
Kepler published a little pamphlet where he pointed out that two years later
at specified times both of the inner planets, Mercury and Venus, were to pass
across the face of the sun. He also described how to observe these events for
the first time. The prediction of the Mercury transit in 1631 could be seen
in Europe, and it turned out to be correct within a few hours.
More and more people accepted that at last
someone had figured out how our planets move in the sense that meaningful
predictions could be made. On planet Earth the Thirty Years’ War still raged.
In a few years, a young man would be seen sitting under an apple tree,
perhaps in the orchard around Kepler’s temple, pondering over how to further
embellish the temple and explain its construction. But that was a
constellation the Rudolphine Tables could
not predict.
FIGURE
1. Clear copy of the
cover: Frontispiece designed by Johannes Kepler for the Tabula Rudolphine
(the Rudolphine Tables). The Habsburg eagle is flying above the temple,
and directly below it is Urania, the muse of astronomy. Arrayed around her
above the front columns of the temple are six more muses: Physica lucis (at
left), Optica, Logarithmica, Doctrina triangulorum, Stathmica, and Magnetica.
Within the temple we see an ancient astronomer (back center), then (from left
to right) Hipparchus, Copernicus, Brahe, and Ptolemy. Kepler is modestly
positioned on the base of the temple in the secondtoleft panel.
Institut MittagLeffler, Aurav¨agen 17, SE18260 Djursholm, Sweden
Email address: ragstedt@mittagleffler.se
In: BULLETIN (new series) OF THE
AMERCICAN MATHEMATICAL SOCIETY
Volume 50, Number 4, October 2013, Pages 629–639
S 02730979(2013)014162
Article electronically published on June 10, 2013
Under the flag of war: Mapping during 600 years anniversary of the astronomical tower clock situated at Old Town Square in center of Prague.









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