From Geocentric to
Heliocentric Cosmology
Nicolaus Copernicus (1473-1543)
While he was in Italy, Copernicus visited Rome, and it seems to have
been for friends there that in about 1513 he wrote a short account of what has
since become known as the Copernican theory, namely that the Sun (not the
Earth) is at rest in the center of the Universe. A full account of the theory
was apparently slow to take a satisfactory shape, and was not published until
the very end of Copernicus's life, under the title On the revolutions of the
heavenly spheres (De revolutionibus orbium coelestium, Nuremberg, 1543).
Copernicus is said to have received a copy of the printed book for the first time
on his deathbed. Copernicus's heliostatic cosmology
involved giving several distinct motions to the Earth. It was consequently
considered implausible by the vast majority of his contemporaries, and by most
astronomers and natural philosophers of succeeding generations before the
middle of the seventeenth century. Its notable defenders included Johannes Kepler
(1571-1630) and Galileo Galilei (1564-1642). Strong theoretical underpinning
for the Copernican theory was provided by Newton's theory of universal
gravitation (1687). Tycho Brahe (1546-1601)
Tycho Brahe, who came from the nobility, was intended by his family for
a career as a lawyer and diplomat. He eventually became an astronomer. His
scientific interests included Alchemy and he was a convinced believer in
Astrology (celestial influences were believed to play a part in alchemical
processes).
With financial help from the King of Denmark, he went on to set up a
purpose-built observatory, on the island of Hveen in Copenhagen Sound. The
observatory, called Uraniborg, was equipped with exceptionally large and
accurate instruments (and with an alchemical laboratory in its basement). At
Uraniborg Tycho made twenty years' worth of astronomical observations. After
quarrelling with the new King, Tycho closed down his observatory and sought a
place for himself and his instruments at various courts. In 1599 he was appointed Imperial Mathematician to the Holy Roman Emperor,
Rudolph II, in Prague. Johannes Kepler
(1571-1630) joined him as an assistant, to help with mathematical calculations.
Tycho intended that this work should prove the truth of his cosmological model,
in which the Earth (with the Moon in orbit around it) was at rest in the center
of the Universe and the Sun went round the Earth (all other planets being in
orbit about the Sun and thus carried round with it). When Tycho died, Kepler
succeeded him as Imperial Mathematician. Tycho's observations of planetary
positions, which were made using instruments with open sights (a telescope was
not used for astronomy until about 1609), were much more accurate than any made
by his predecessors. They allowed Kepler,
who (unlike Tycho) was a convinced follower of Copernicus, to deduce his three
laws of planetary motion (1609, 1619) and to construct astronomical tables, the
Rudolphine Tables (Ulm, 1627), whose enduring accuracy did much to persuade
astronomers of the correctness of the Copernican theory. However, until at
least the mid-seventeenth century, Tycho's model of the planetary system was
that favored by most astronomers. It had the advantage of avoiding the problems
introduced by ascribing motion to the Earth. Galileo Galilei (1564-1642)
Galileo Galilei's father,
Vincenzo Galilei (c.1520 - 1591), who described himself as a nobleman of
Florence, was a professional musician. He carried out experiments on strings to
support his musical theories. Galileo studied medicine at the university of
Pisa, but his real interests were always in mathematics and natural philosophy.
He is chiefly remembered for his work on free fall, his use of the telescope
and his employment of experimentation. After a spell teaching mathematics, first privately in Florence and
then at the university of Pisa, in 1592 Galileo was appointed professor of
mathematics at the university of Padua (the university of the Republic of
Venice). There his duties were mainly to teach Euclid's
geometry and standard (geocentric) astronomy to medical students, who would
need to know some astronomy in order to make use of astrology in their medical
practice. However, Galileo apparently discussed more unconventional forms of
astronomy and natural philosophy in a public lecture he gave in connection with
the appearance of a New Star (now known as 'Kepler's
supernova') in 1604. In a personal letter written to Kepler
(1571 - 1630) in 1598, Galileo had stated that he was a Copernican (believer in
the theories of Copernicus).
No public sign of this belief was to appear until many years later. In the summer of 1609, Galileo heard about a spyglass that a Dutchman
had shown in Venice. From these reports, and using his own technical skills as
a mathematician and as a workman, Galileo made a series of telescopes whose
optical performance was much better than that of the Dutch instrument. The
astronomical discoveries he made with his telescopes were described in a short
book called Message from the stars (Sidereus Nuncius) published in Venice in
May 1610. It caused a sensation. Galileo claimed to have seen mountains on the
Moon, to have proved the Milky Way
was made up of tiny stars, and to have seen four small bodies orbiting Jupiter.
These last, with an eye on getting a job in Florence, he promptly named 'the
Medicean stars'. It worked. Soon afterwards, Galileo became 'Mathematician and [Natural]
Philosopher' to the Grand Duke of Tuscany. In Florence he continued his work on
motion and on mechanics, and began to get involved in disputes about
Copernicanism. In 1613 he discovered that, when seen in the telescope, the
planet Venus showed phases like those of the Moon, and therefore must orbit the
Sun not the Earth. This did not enable one to decide between the Copernican
system, in which everything goes round the Sun, and the Tychonic (Tycho Brahe)
one in which everything but the Earth (and Moon) goes round the Sun which in
turn goes round the Earth. Most astronomers of the time in fact favored the
Tychonic system. However, Galileo showed a marked tendency to use all his
discoveries as evidence for Copernicanism, and to do so with great verbal as
well as mathematical skill. He seems to have made a lot of enemies by making
his opponents look fools. Moreover, not all of them actually were fools. There eventually followed some expression of interest by the
Inquisition. Prima facie, Copernicanism was in contradiction with Scripture,
and in 1616 Galileo was given some kind of secret, but official, warning that
he was not to defend Copernicanism. Just what was said on this occasion was to
become a subject for dispute when Galileo was accused of departing from this
undertaking in his Dialogue concerning the two greatest world systems,
published in Florence in 1632. Galileo, who was not in the best of health, was
summoned to Rome, found to be vehemently suspected of heresy, and eventually
condemned to house arrest, for life, at his villa at Arcetri (above Florence).
He was also forbidden to publish. By the standards of the time he had got off
rather lightly. Galileo's sight was failing, but he had devoted pupils and amanuenses,
and he found it possible to write up his studies on motion and the strength of
materials. The book, Discourses on two new sciences, was smuggled out of Italy
and published in Leiden (in the Netherlands) in 1638. Galileo wrote most of his later works in the vernacular, probably to
distance himself from the conventional learning of university teachers.
However, his books were translated into Latin for the international market, and
they proved to be immensely influential. Johannes Kepler (1571-1630)
A large quantity
of Kepler's correspondence survives. Many of his letters are almost the
equivalent of a scientific paper (there were as yet no scientific journals),
and correspondents seem to have kept them because they were interesting. In
consequence, we know rather a lot about Kepler's life, and indeed about his
character. It is partly because of this that Kepler has had something of a
career as a more or less fictional character (see historiographic note). 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 enroll at the
University of Tübingen, then (as now) a bastion of Lutheran orthodoxy. Throughout his
life, Kepler was a profoundly religious man. All his writings contain numerous
references to God, and he saw his work as a fulfillment of his Christian duty
to understand the works of God. Man being, as Kepler believed, made in the
image of God, was clearly capable of understanding the Universe that He had
created. Moreover, Kepler was convinced that God had made the Universe
according to a mathematical plan (a belief found in the works of Plato and
associated with Pythagoras). Since it was generally accepted at the time that
mathematics provided a secure method of arriving at truths about the world
(Euclid's common notions and postulates being regarded as actually true), we
have here a strategy for understanding the Universe. Since some authors have
given Kepler a name for irrationality, it is worth noting that this rather
hopeful epistemology is very far indeed from the mystic's conviction that
things can only be understood in an imprecise way that relies upon insights
that are not subject to reason. Kepler does indeed repeatedly thank God for
granting him insights, but the insights are presented as rational. At this time, it
was usual for all students at a university to attend courses on
"mathematics". In principle this included the four mathematical
sciences: arithmetic, geometry, astronomy and music. It seems, however, that
what was taught depended on the particular university. At Tübingen Kepler was
taught astronomy by one of the leading astronomers of the day, Michael Maestlin
(1550 - 1631). The astronomy of the curriculum was, of course, geocentric
astronomy, that is the current version of the Ptolemaic system, in which all
seven planets - Moon, Mercury, Venus, Sun, Mars, Jupiter and Saturn - moved
round the Earth, their positions against the fixed stars being calculated by
combining circular motions. This system was more or less in accord with current
(Aristotelian) notions of physics, though there were certain difficulties, such
as whether one might consider as 'uniform' (and therefore acceptable as
obviously eternal) a circular motion that was not uniform about its own center
but about another point (called an 'equant'). However, it seems that on the
whole astronomers (who saw themselves as 'mathematicians') were content to carry
on calculating positions of planets and leave it to natural philosophers to
worry about whether the mathematical models corresponded to physical
mechanisms. Kepler did not take this attitude. His earliest published work
(1596) proposes to consider the actual paths of the planets, not the circles
used to construct them. Instead of the
seven planets in standard geocentric astronomy the Copernican system had only
six, the Moon having become a body of kind previously unknown to astronomy,
which Kepler was later to call a 'satellite' (a name he coined in 1610 to
describe the moons that Galileo had discovered were orbiting Jupiter, literally
meaning 'attendant'). Why six planets? Moreover, in
geocentric astronomy there was no way of using observations to find the
relative sizes of the planetary orbs; they were simply assumed to be in
contact. This seemed to require no explanation, since it fitted nicely with
natural philosophers' belief that the whole system was turned from the movement
of the outermost sphere, one (or maybe two) beyond the sphere of the 'fixed'
stars (the ones whose pattern made the constellations), beyond the sphere of
Saturn. In the Copernican system, the fact that the annual component of each
planetary motion was a reflection of the annual motion of the Earth allowed one
to use observations to calculate the size of each planet's path, and it turned
out that there were huge spaces between the planets. Why these particular
spaces? Kepler's answer to
these questions, described in his Mystery of the Cosmos (Mysterium
cosmographicum, Tübingen, 1596), looks bizarre to twentieth-century readers
(see the figure on the right). He suggested that if a sphere were drawn to
touch the inside of the path of Saturn, and a cube were inscribed in the sphere,
then the sphere inscribed in that cube would be the sphere circumscribing the
path of Jupiter. Then if a regular tetrahedron were drawn in the sphere
inscribing the path of Jupiter, the insphere of the tetrahedron would be the
sphere circumscribing the path of Mars, and so inwards, putting the regular dodecahedron between Mars
and Earth, the regular icosahedron
between Earth and Venus, and the regular octahedron between Venus
and Mercury. This explains the number of planets perfectly: there are only five
convex regular solids (as is proved in Euclid's Elements , Book 13). It also
gives a convincing fit with the sizes of the paths as deduced by Copernicus,
the greatest error being less than 10% (which is spectacularly good for a
cosmological model even now). Kepler did not express himself in terms of
percentage errors, and his is in fact the first mathematical cosmological
model, but it is easy to see why he believed that the observational evidence
supported his theory. Kepler saw his
cosmological theory as providing evidence for the Copernican theory. Before
presenting his own theory he gave arguments to establish the plausibility of
the Copernican theory itself. Kepler asserts that its advantages over the
geocentric theory are in its greater explanatory power. For instance, the
Copernican theory can explain why Venus and Mercury are never seen very far from
the Sun (they lie between Earth and the Sun) whereas in the geocentric theory
there is no explanation of this fact. Kepler lists nine such questions in the
first chapter of the Mysterium cosmographicum. Kepler carried out
this work while he was teaching in Graz, but the book was seen through the
press in Tübingen by Maestlin. The agreement with values deduced from
observation was not exact, and Kepler hoped that better observations would
improve the agreement, so he sent a copy of the Mysterium cosmographicum to one
of the foremost observational astronomers of the time, Tycho Brahe
(1546 - 1601). Tycho,
then working in Prague, had in fact already written to Maestlin in search of a
mathematical assistant. Kepler got the job. Naturally enough, Tycho's
priorities were not the same as Kepler's, and Kepler soon found himself working
on the intractable problem of the orbit of Mars. He continued to work on this
after Tycho
died (in 1601) and Kepler succeeded him as Imperial Mathematician.
Conventionally, orbits were compounded of circles, and rather few observational
values were required to fix the relative radii and positions of the circles. Tycho
had made a huge number of observations and Kepler determined to make the best
possible use of them. Essentially, he had so many observations available that
once he had constructed a possible orbit he was able to check it against
further observations until satisfactory agreement was reached. Kepler concluded
that the orbit of Mars was an ellipse
with the Sun in one of its foci (a result which when extended to all the
planets is now called "Kepler's First Law"), and that a line joining
the planet to the Sun swept out equal areas in equal times as the planet
described its orbit ("Kepler's Second Law"), that is the area is used
as a measure of time. After this work was published in 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. The actual process
of calculation for Mars was immensely laborious - there are nearly a thousand
surviving folio sheets of arithmetic - and Kepler himself refers to this work
as 'my war with Mars', but the result was an orbit which agrees with modern
results so exactly that the comparison has to make allowance for secular
changes in the orbit since Kepler's time. Following Galileo's
use of the telescope in discovering the moons of Jupiter, published in his
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. Kepler's years in
Prague were relatively peaceful, and scientifically extremely productive. In
fact, even when things went badly, he seems never to have allowed external
circumstances to prevent him from getting on with his work. Things began to go
very badly in late 1611. First, his seven year old son died. Kepler wrote to a
friend that this death was particularly hard to bear because the child reminded
him so much of himself at that age. Then Kepler's wife died. Then the Emperor Rudolf,
whose health was failing, was forced to abdicate in favor of his brother
Matthias, who, like Rudolf, was a Catholic but (unlike Rudolf) did not believe
in tolerance of Protestants. Kepler had to leave Prague. Before he departed he
had his wife's body moved into the son's grave, and wrote a Latin epitaph for
them. He and his remaining children moved to Linz (now in Austria). Kepler's main task
as Imperial Mathematician was to write astronomical tables, based on Tycho's
observations, but what he really wanted to do was write 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. Calculating
tables, the normal business for an astronomer, always involved heavy
arithmetic. Kepler was accordingly delighted when in 1616 he came across Napier's
work on logarithms (published in 1614). However, Maestlin promptly told him
first that it was unseemly for a serious mathematician to rejoice over a mere
aid to calculation and second that it was unwise to trust logarithms because
no-one understood how they worked. (Similar comments were made about computers
in the early 1960s.) Kepler's answer to the second objection was to publish a
proof of how logarithms worked, based on an impeccably respectable source: Euclid's
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 fulfillment of his dull official task as Imperial Mathematician led to
the fulfillment of his dearest wish, to help establish Copernicanism. Reference for
all biographies: John
J O'Connor and Edmund F Robertson, The MacTutor
History of Mathematics archive, http://www-history.mcs.st-andrews.ac.uk/history/index.html
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Nicolaus
Copernicus came from a middle class background and received a good standard
humanist education, studying first at the university of Krakow (then the
capital of Poland) and then traveling to Italy where he studied at the
universities of Bologna and Padua. He eventually took a degree in Canon Law at
the university of Ferrara. At Krakow, Bologna and Padua he studied the
mathematical sciences, which at the time were considered relevant to medicine
(since physicians made use of astrology). Padua was famous for its medical
school and while he was there Copernicus studied both medicine and Greek. When
he returned to his native land, Copernicus practiced medicine, though his official
employment was as a canon in the cathedral chapter, working under a maternal
uncle who was Bishop of Olsztyn (Allenstein) and then of Frombork (Frauenburg).
Johannes
Kepler is now chiefly remembered for discovering the three laws of planetary
motion that bear his name published in 1609 and 1619). He also did important
work in optics (1604, 1611), discovered two new regular polyhedra (1619), gave the
first mathematical treatment of close packing of equal spheres (leading to an
explanation of the shape of the cells of a honeycomb, 1611), gave the first
proof of how logarithms worked (1624), and devised a method of finding the
volumes of solids of revolution that (with hindsight!) can be seen as
contributing to the development of calculus (1615, 1616). Moreover, he
calculated the most exact astronomical tables hitherto known, whose continued
accuracy did much to establish the truth of heliocentric astronomy (Rudolphine
Tables, Ulm, 1627).