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1 Math 1700 – Renaissance mathematical astronomy Mathematical Astronomy in the Renaissance The Copernican Revolution 2 Math 1700 – Renaissance mathematical astronomy Nicholas Copernicus 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). 3 Math 1700 – Renaissance mathematical astronomy Copernicus' interests A “Renaissance Man” Mathematics, astronomy, medicine, law, mysticism, Hermeticism Viewed astronomy as a central subject for understanding nature. Viewed mathematics as central to astronomy

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Mathematical Astronomy in the Renaissance

The Copernican Revolution

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Nicholas Copernicus1473-1543

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).

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Copernicus' interests

A “Renaissance Man”Mathematics, astronomy, medicine, law, mysticism, HermeticismViewed astronomy as a central subject for understanding nature.

Viewed mathematics as central to astronomy

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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.

The Julian Calendar had 365 days per year and one extra “leap day” every 4 years.

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Copernicus’ Task

The Julian calendar was associated with Ptolemy.Copernicus believed that Ptolemy’s system was at fault and need a (perhaps minor) correction.

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

It marks the beginning of the Scientific Revolution.In the same year, Tartaglia published translations of both Euclid and Archimedes. Two years later, Cardanopublished his Ars Magna.

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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

In particular, the problem of the planets.

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Solutions to the problem of the planets

Aristotelian:The spheres of EudoxusThe superlunarrealm is filled with crystalline shells.

A physical realityPoor accuracy

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Solutions to the problem of the planets

Platonic:The Ptolemy Epicycle/Deferent system

A formal mathematical system only

No physical meaning

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Medieval reconciliation of Aristotle and Plato

Epicycles andconcentric 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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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 axisReplaces 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. Accounts for retrograde motion of the planets—makes them an optical illusion.

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The Three Motions of the Earth

3. Rotation of Earth's North-South axis, once a year, around an axis perpendicular to the ecliptic.

Provides the seasons, and incidentally accounts for the precession of the equinoxes.

S

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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.

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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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.

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/PlatonicMathematics 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 argumentSolutions to problems found by logic without supporting evidence.

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Problems Remaining in Copernicus

1. The moving Earth.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 VenusIf Venus is sometimes on the same side of the sun as the Earth and sometimes across from the sun, it should appear different atdifferent times. It should show phases, like the moon

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Problems Remaining in Copernicus

3. Stellar parallaxBecause 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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It does not matter if it is true….

The "Calculating Device“ viewpoint.Typical of the way Phythagorean/Platonic conceptions are presented to the public.

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.

Note the similarity to Ptolemy’s system.

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Tycho Brahe

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.

Worked there 20 years.Became very unpopular with the local residents.

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Tycho’s ObservationsMade amazingly precise observations of the heavens with naked-eye 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 MathematicianLeft 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 SystemEarth stationary.Planets circle Sun.Sun circles Earth.Problem:

Could not get Marsto fit the system.Note the intersecting paths of the Sun and Mars that bothered Copernicus.

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Johannes Kepler1571-1630

LutheranMathematics professor in Austria (Graz)Sometime astrologerPythagorean/Neo-PlatonistOne 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)?

Mercury, Venus, Earth, Mars, Jupiter, SaturnWith 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 SolidsUnlike regular figures, their number is not unlimited. There are actually only five possibilities:

Tetrahedron, Cube, Octahedron, Dodecahedron, Icosahedron

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 brainstormKepler imagined that (like Eudoxeanspheres), 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 SolidsLike 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 MysteryIn 1596, Kepler published a short book, MysteriumCosmographicum, in which he expounded his theory.The 6 planets were separated by the 5 regular solids, as follows:

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:

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 BraheKepler'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:

The New Astronomy, 1609The Harmony of the World, 1619

Out of all of this, three laws survive.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 law1. The planets travel in elliptical orbits with the sun at one focus.

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 off-centre. 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 = kThe cube of a planet’s mean distance dfrom 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.

As usual, Kepler did not provide a cause for this relationship.

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Kepler's three laws at a glance1. The planets travel in elliptical orbits with the sun at one focus.

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 = kAccounts 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 lawsKepler did not identify these three statements about the behaviour of the planets as his “laws.”

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.

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.

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 SpheresAs 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.

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Music of the Spheres, 2Since 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, 3In 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, 4As follows:Mercury, a scale running a tenth from C to EVenus—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, 5Mars, 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.