Mathematical Astronomy in the
Renaissance
The Copernican Revolution
Math 1700 – Renaissance mathematical astronomy
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Nicholas Copernicus
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{
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{
1473-1543
Studied medicine at
University of Crakow
Discovered math and
astronomy.
Continued studies at
Bologna, Padua,
eventually took
degree in Canon Law
at University of
Ferrara.
Appointed Canon of
Cathedral of Frombork
(Frauenberg).
Math 1700 – Renaissance mathematical astronomy
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Copernicus' interests
{
{
{
A “Renaissance Man”
Mathematics, astronomy, medicine,
law, mysticism, Hermeticism
Viewed astronomy as a central
subject for understanding nature.
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Viewed mathematics as central to
astronomy
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1
The Julian calendar
{
The Julian Calendar, introduced in
45 BCE, was a great improvement
over previous calendars, but by the
16th century, it was registering 10
days ahead of the astronomical
events it should have tracked.
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The Julian Calendar had 365 days per
year and one extra “leap day” every 4
years.
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Copernicus’ Task
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{
The Julian calendar was associated
with Ptolemy.
Copernicus believed that Ptolemy’s
system was at fault and need a
(perhaps minor) correction.
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E.g. Mars' orbit intersects orbit of Sun.
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On the Revolutions of the Heavenly
Spheres
{
Studied astronomy over 30 years,
culminating in publication of On the
Revolutions in 1543
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It marks the beginning of the Scientific
Revolution.
In the same year, Tartaglia published
translations of both Euclid and
Archimedes. Two years later, Cardano
published his Ars Magna.
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2
The conflicting views of ancient
philosophy
{
{
{
Plato: the Forms (e.g. mathematics)
were reality.
Aristotle: the Forms only describe
an underlying physical reality.
This led to conflicting
interpretations in astronomy
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In particular, the problem of the
planets.
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Solutions to the problem of the planets
{
Aristotelian:
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The spheres of
Eudoxus
The superlunar
realm is filled with
crystalline shells.
{ A physical reality
Poor accuracy
Math 1700 – Renaissance mathematical astronomy
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Solutions to the problem of the planets
{
Platonic:
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The Ptolemy
Epicycle/Deferent
system
{ A formal
mathematical
system only
No physical
meaning
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3
Medieval reconciliation of Aristotle and
Plato
{
{
{
Epicycles and
concentric spheres.
Epicycles like ball
bearings running in
carved out channels.
Ptolemaic
mathematical
analysis, with a
physical
interpretation.
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The Problem of the Equant
{
{
Physically impossible
to rotate evenly
around a point not at
the geometric centre.
Could dispense with
the equant if planets
revolved around sun
(while sun revolved
around the stationary
Earth).
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The Problem of Mars' Orbit
{
{
Mars' orbit would cut into orbit of
Sun around Earth.
Solution: Leave the Sun stationary
and make the Earth move.
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4
The Copernican System
{
{
{
{
The Earth is a planet, circling the
Sun.
The Moon is not a planet, but a
satellite circling the Earth.
The “Fixed” stars truly are fixed, not
just fixed to the celestial sphere.
The Equant point is not required.
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The Three Motions of the Earth
{
1. Daily rotation on its axis
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z
Replaces the movement of the
celestial sphere.
Though counter-intuitive,
Copernicus argued that it was
simpler for the relatively small
Earth to turn on its axis every
day from west to east than for
the gigantic heavens to make
a complete revolution from
east to west daily.
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The Three Motions of the Earth
{
2. Annual revolution around the Sun.
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Accounts for retrograde motion of the
planets—makes them an optical illusion.
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5
The Three Motions of the Earth
{
3. Rotation of
Earth's NorthSouth axis, once a
year, around an
axis perpendicular
to the ecliptic.
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S
Provides the
seasons, and
incidentally
accounts for the
precession of the
equinoxes.
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The Calendar and the Church
{
{
For the Christian Church, it was
vitally important to know what day
it was.
The segments of the church year
required different prayers, different
rituals, and different celebrations.
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E.g. Easter is the first Sunday after the
first full moon after the vernal equinox.
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The Council of Trent
{
{
{
The Council of Trent was set up in
1545 to deal with the Protestant
threat to Catholicism.
It also undertook to repair the
calendar.
The Council used Copernicus’ new
system to reform and reset the
calendar.
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6
The Gregorian Calendar
{
{
In 1582, Pope Gregory adopted a new
calendar to replace the Julian calendar.
The Gregorian calendar, which we use
today, has 365 days per year, with one
extra day every fourth year.
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z
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But not if the year is a century year.
Unless it is divisible by 400.
Hence it adds 100-3=97 days every 400 years
– three less than the Julian calendar.
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Copernicus' Style of Argument
{
Pythagorean/Platonic
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z
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{
Mathematics is for mathematicians.
The reality is in the mathematical
elegance; other considerations
secondary.
Secretive and/of uninterested in the
riff-raff of popular opinion.
Ad hoc argument
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Solutions to problems found by logic
without supporting evidence.
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Problems Remaining in Copernicus
{
1. The moving Earth.
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Why can we not detect the motion of
the Earth, which is very fast at the
surface?
Why do the clouds not all rush off to
the west as the Earth spins toward the
East?
Why is there not always a strong East
wind?
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Problems Remaining in Copernicus
2. Phases of Venus
{
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If Venus is sometimes on the same side of the sun as the Earth
and sometimes across from the sun, it should appear different at
different times. It should show phases, like the moon
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Problems Remaining in Copernicus
{
3. Stellar parallax
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Because the Earth
moves around the
sun, it gets
sometimes closer
and sometimes
farther from certain
stars.
The Earth at position
1 is farther from
stars 1 and 2 than at
position 2.
The angle between
the stars at a should
be smaller than the
angle at b
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Copernicus’ ad hoc answers
{
{
{
1. We don’t notice movement because the
Earth carries everything around with it
(the air, the clouds, ourselves).
2. Venus does not show phases because it
has its own light (like the Sun and the
stars).
3. We do not see stellar parallax because
the entire orbit of the Earth around the
Sun is as a point compared to the size of
the celestial sphere.
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8
It does not matter if it is true….
{
The "Calculating Device“ viewpoint.
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Typical of the way
Phythagorean/Platonic conceptions are
presented to the public.
{
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That they are really just convenient
fictions.
For example, the preface to On the
Revolutions by Andreas Osiander.
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From Osiander’s Preface
{
There have already been widespread reports about
the novel hypotheses of this work, which declares
that the earth moves whereas the sun is at rest in the
center of the universe.… [I]t is the duty of an
astronomer to compose the history of the celestial
motions through careful and expert study. Then he
must conceive and devise the causes of these motions
or hypotheses about them. Since he cannot in any
way attain to the true causes, he will adopt whatever
suppositions enable the motions to be computed
correctly from the principles of geometry for the
future as well as for the past.… For these hypotheses
need not be true nor even probable. On the contrary,
if they provide a calculus consistent with the
observations, that alone is enough.
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The Copernican system
{
An illustration
from On the
Revolutions.
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Note the
similarity to
Ptolemy’s
system.
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9
Tycho Brahe
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{
{
{
1546-1601
Motivated by
astronomy's
predictive powers.
Saw and reported
the Nova of 1572.
Considered poor
observational data
to be the chief
problem with
astronomy.
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Tycho Brahe at Uraniborg
{
Established an
observatory-Uraniborg on
Hven, an island off
Denmark.
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Worked there 20
years.
Became very
unpopular with the
local residents.
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Tycho’s Observations
{
{
Made amazingly precise
observations of the
heavens with nakedeye instruments.
Produced a huge globe
of the celestial sphere
with the stars he had
identified marked on it.
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Tycho, the Imperial
Mathematician
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{
{
Left Uraniborg to become the
Imperial Mathematician to the Holy
Roman Emperor at the Court in
Prague.
Tycho believed that Copernicus was
correct that the planets circled the
Sun, but could not accept that the
Earth was a planet, nor that it moved
from the centre of the universe.
He developed his own compromise
system.
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Tycho’s System
{
{
{
{
Earth stationary.
Planets circle Sun.
Sun circles Earth.
Problem:
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Could not get Mars
to fit the system.
Note the
intersecting paths
of the Sun and
Mars that bothered
Copernicus.
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Johannes Kepler
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{
{
{
{
{
1571-1630
Lutheran
Mathematics
professor in Austria
(Graz)
Sometime astrologer
Pythagorean/NeoPlatonist
One of the few
Copernican converts
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Pythagorean/Platonic regularities in the
Heavens
{
Why are there precisely 6 planets in the
heavens (in the Copernican system)?
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Mercury, Venus, Earth, Mars, Jupiter, Saturn
With a Pythagorean mindset, Kepler was sure
there was some mathematically necessary
reason.
He found a compelling reason in Euclid.
{ A curious result in solid geometry that was
attributed to Plato.
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Euclidean Regular Figures
A regular figure is a closed linear figure with
every side and every angle equal to each other.
•For example, an equilateral triangle, a square, an
equilateral pentagon, hexagon, and so forth.
There is no limit to the number of regular
figures with different numbers of sides.
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Inscribing and Circumscribing
{
{
All regular figures can be inscribed within a
circle and also circumscribed around a circle.
The size of the figure precisely determines the
size of the circle that circumscribes it and the
circle that is inscribed within it.
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Regular Solids
{
{
In three dimensions, the comparable
constructions are called regular solids.
They can inscribe and be circumscribed
by spheres.
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The Platonic Solids
{
Unlike regular figures, their
number is not unlimited.
There are actually only
five possibilities:
Tetrahedron, Cube,
Octahedron, Dodecahedron,
Icosahedron
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{
{
This was discussed by Plato. They are traditionally
called the “Platonic Solids.”
That there could only be five of them was proved by
Euclid in the last proposition of the last book of The
Elements.
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Kepler’s brainstorm
{
{
Kepler imagined that (like Eudoxean
spheres), the planets were visible
dots located on the surface of
nested spherical shells all centered
on the Earth.
There were six planets, requiring six
spherical shells. Just the number to
be inscribed in and circumscribe the
five regular solids.
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Six Planets, Five Solids
{
{
Like Pythagoras, Kepler believed that neat
mathematical relationships such as this
could not be a coincidence. It must be the
key to understanding the mystery of the
planets.
There were six planets because there
were five Platonic solids. The “spheres” of
the planets were separated by the
inscribed solids. Thus their placement in
the heavens is also determined.
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The Cosmographical Mystery
In 1596, Kepler
published a short book,
Mysterium
Cosmographicum, in
which he expounded his
theory.
The 6 planets were
separated by the 5
regular solids, as
follows:
{
{
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Saturn / cube / Jupiter /
tetrahedron / Mars /
dodecahedron / Earth /
icosahedron / Venus /
octahedron / Mercury
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What was the mystery?
{
{
The cosmographical mystery that Kepler
“solved” with the Platonic solids was the
provision of reasons why something that
appeared arbitrary in the heavens
followed some exact rule. This is classic
“saving the appearances” in Plato’s sense.
The arbitrary phenomena were:
z
z
{
The number of planets.
Their spacing apart from each other.
Both were determined by his
arrangement of spheres and solids.
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Kepler and Tycho Brahe
{
{
{
{
Kepler's cosmic solution didn't
exactly work, but he thought it
would with better data.
Tycho had the data.
Meanwhile Tycho needed someone
to do calculations for him to prove
his system.
A meeting was arranged between
the two of them.
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Kepler, the Imperial Mathematician
{
{
{
Kepler became Tycho's assistant in
1600.
Tycho died in 1601.
Kepler succeeded Tycho as Imperial
Mathematician to the Holy Roman
Emperor in Prague, getting all of
Tycho's data.
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Kepler's “Discoveries”
{
{
Kepler found many magical and
mysterious mathematical relations in the
stars and planets.
He published his findings in two more
books:
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{
The New Astronomy, 1609
The Harmony of the World, 1619
Out of all of this, three laws survive.
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The first two involve a new shape for
astronomy, the ellipse.
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Conic Sections
In addition to Euclid, Kepler would have known of the work of the
Hellenistic mathematician Apollonius of Perga, who wrote a
definitive work on what are called conic sections: the intersection
of a cone with a plane in different orientations. Above are the
sections Parabola, Ellipse, and Hyperbola.
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The Ellipse
{
{
The Ellipse is formed by a plane cutting completely through
the cone.
Another way to make an ellipse is with two focal points (A
and B above), and a length of, say, string, longer than the
distance AB. If the string is stretched taut with a pencil and
pulled around the points, the path of the pencil point is an
ellipse. In the diagram above, that means that if C is any
point on the ellipse, AC+BC is always the same.
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Kepler's first law
{
1. The planets travel
in elliptical orbits with
the sun at one focus.
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z
All previous
astronomical theories
had the planets
travelling in circles,
or combinations of
circles.
Kepler has chosen a
different geometric
figure.
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A radical idea, to depart from circles
Kepler’s ideas were very different and unfamiliar to
astronomers of his day.
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What was the mystery?
{
{
{
Kepler’s “first law” gives some account of the
actual paths of the planets (i.e., “saves” them).
All of the serious astronomers before him had
found that simple circular paths didn’t quite work.
Ptolemy’s Earth-centered system had resorted to
arbitrary epicycles and deferents, often offcentre. Copernicus also could not get circles to
work around the sun.
Kepler found a simple geometric figure that
described the path of the planets around the sun.
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Kepler's second law
{
2. A radius vector from the sun to a
planet sweeps out equal areas in equal
times.
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What is the mystery here?
{
{
{
The second law provides a mathematical
relationship that accounts for the
apparent speeding up of the planets as
they get nearer the sun and slowing down
as they get farther away.
Kepler had no explanation why a planet
should speed up near the sun. (He
speculated that the sun gave it some
encouragement, but didn’t know why.)
But in Platonic fashion he provided a
formula that specifies the relative speeds.
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Kepler's third law
{
3. The Harmonic Law: d 3/t 2 = k
z
z
The cube of a planet’s mean distance d
from the sun divided by the square of
its time t of revolution is the same for
all planets.
That is, the above ratio is equal to the
same constant, k, for all planets.
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The mystery cleared up by the third law
{
{
{
{
Kepler noted that the planets all take different
times to complete a full orbit of the Sun.
The farther out a planet was from the Sun, the
longer was its period of revolution.
He wanted to find a single unifying law that would
account for these differing times.
The 3rd law gives a single formula that relates the
periods and distances of all the planets.
z
As usual, Kepler did not provide a cause for this
relationship.
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Kepler's three laws at a glance
{
1. The planets travel in elliptical orbits
with the sun at one focus.
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{
z
{
Accounts for the orbital paths of the planets.
2. A radius vector from the sun to a
planet sweeps out equal areas in equal
times.
Accounts for the speeding up and slowing
down of the planets in their orbits.
3. The Harmonic Law: d 3/t 2 = k
z
Accounts for the relative periods of revolution
of the planets, related to their distances from
the sun.
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Why these are called Kepler’s laws
{
Kepler did not identify these three
statements about the behaviour of the
planets as his “laws.”
z
{
We call these Kepler’s laws because Isaac
Newton pulled them out of Kepler’s works and
gave Kepler credit for them.
Kepler found many “laws”—meaning
regularities about the heavens—beginning
with the cosmographical mystery and the
5 Platonic solids.
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Most of these we ignore as either coincidences
or error on his part.
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What did Kepler think he was doing?
{
Kepler has all the earmarks of a
Pythagorean.
z
z
A full and complete explanation is
nothing more nor less than a
mathematical relationship describing
the phenomena.
In Aristotle’s sense it is a formal cause,
but not an efficient, nor a final cause.
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The Music of the Spheres
{
{
{
As a final example of Kepler’s frame
of mind, consider the main issue of
his last book, The Harmony of the
World.
Kepler’s goal was to explain the
harmonious structure of the
universe.
By harmony he meant the same as
is meant in music.
Math 1700 – Renaissance mathematical astronomy
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Music of the Spheres, 2
{
{
Since Pythagoras it has been known
that a musical interval has a precise
mathematical relationship. Hence all
mathematical relations, conversely,
are musical intervals.
If the planets’ motions can be
described by mathematical formula,
the planets are then performing
music.
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Music of the Spheres, 3
{
{
In particular, the orbits of the
planets, as they move through their
elliptical paths, create different
ratios, which can be expressed as
musical intervals.
The angular speeds at which the
planets move determine a pitch,
which rises and falls through the
orbit.
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Music of the Spheres, 4
{
{
{
{
As follows:
Mercury, a scale
running a tenth from
C to E
Venus—almost a
perfect circular orbit—
sounds the same
note, E, through its
orbit.
Earth, also nearly
circular, varies only
from G to A-flat
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Music of the Spheres, 5
{
{
{
{
Mars, which has a
more irregular path
than Venus or the
Earth, goes from F to
C and back.
Jupiter moves a mere
minor third from G to
B-flat.
Saturn moves a major
third from G to B.
The Moon too plays a
tune, from G to C and
back.
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