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The Cornell “Friends of Astronomy”

The next event is:

Frontiers of Cornell AstronomySaturday August 16th, 2014

Ask Martha, Riccardo or Chuck Mund

We’ll be happy to add you to the email list for future events.

Join us on Facebook!

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The Birthdate of George Washington

The Washington Bible preserved in Mt Vernon reads:

“George Washington, Son to Augustine and Mary his Wife was Born ye 11th Day of February 1731/2 about 10 in the Morning & was Baptiz’d on the 30th of April following.”

However, historically his birthday is celebrated on February 22nd.

* In 1732, Virginia was still a British colony, thus using the Julian calendar (until 1752). The day of February 11th in the Julian calendar is also February 22nd in the Gregorian one.

* On the meaning of 1731/2: the initial date of the year was not universally agreed. The most common practices were December 25th, January 1st, March 1st and March 25th. The “style” used in Great Britain at the time was March 25th, while most of Europe already used January 1st. Hence “1731/2” meant “in the year 1731, March 25th style, and in the year 1732, January 1st style”.

Ancient Greek World

Ithaca, sometime before Cornell…

MiletusSamos

Athens

Crete

Crotone

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Pythagoras, born in Samos ~580 BCE, founded school in Crotone

A “sect” of men and women who sought to liberate their souls through religious asceticism and practice of intellectual studies [vegetarians, but no beans and no fruit that had fallen from tree; no walking on highways].

A selected few in the school were “mathematicians”: they partook in the most advanced studies

mathema=“that which has been learned” through study

Mathematics, theorems, e.g. a2 + b2 = c2

Harmony: harmonics sounds of harp related to lengths of strings with integer number ratios

Numerology: 1+2+3+4=10 where 1=point, 2=line, 3=surface, 4=volume

number 10 encompasses all forms

number 224 written on wood and swallowed= aphrodisiac

Pythagorean “quadrivium” : arithmetic, geometry, music, astronomy

Astronomy

-“kosmos” = the “good array” of numbers

-Knew that some constellations are circumpolar far North, but are not when one moves South inferred sphericity of Earth

-Sphere is purest of forms fundamental structure of Universe each “planet” (including Sun and Moon) has associated sphere

that moves independently on others; so do the “fixed stars” and beyond that there is yet another sphere (they struggled to have 10). They also speculated that the spheres’ motions and sizes would

be in the ratios of small numbers “Harmony of the Spheres”

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Plato (428-348 BCE): perfection in the conceptual form is key to the understanding of the world. Perfect form exists, if it can be conceived. The idea of a circle is conceivable, even if a perfect circle is not achievable in practice.Plato also thought that uniform circular motion, i.e. at constant angular speed, is perfect and could go on forever.Exemplified by the motions of the stars [motions of planets, Sun and Moon more difficult to explain with circular motions, perhaps because they are closer to the messy Earth]

The idea of explaining celestial motions through combination of uniform circular motions lasted 2000 years.

Eudoxos of Cnidos (406-350 BCE) proposed a geometric model to describe the motions of Sun, Moon and planets, which uses a set of concentric spheres. It took 27 spheres to satisfactorily (at the time) reproduce the observed motions. While concentric, the spheres have different rotational axes and speeds. Messy messy…

Aristotle (384-322 BCE), a platonist, eventually formed his own school. Enormous influence, which lasted longer than that of any thinker.

Because Eudoxos sphere model did not work well for Venus and Mars, Aristotle expanded it from 27 to 56 spheres!

He adopted a geocentric view; due to his intellectual authority later the Christian Church carried geocentrism into dogma, well into the XVII century

Aristotle wondered about the Physics behind geometrical models, i.e. about how and what made a model work.

Five elements: earth, air, water and fire. And the substance of which unchanging bodies – the stars – are made of: quintessence

Two completely different theories of motion required: one for things on Earth, one for celestial bodies.

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• Aristotle, unlike Plato, paid a lot of attention to the results of observation and the experience of others (*).

• Proposed 4 observational proofs that the Earth is a sphere:1. Ships disappear slowly over the horizon.2. The shape of the Earth at lunar eclipse appears circular.

• Rejected the heliocentric model because of the absence of observable parallax. Parallax was not first proved until 1838 by Bessel.

(*) “Experience is the foundation of all knowledge”

• Aristotle, unlike Plato, paid a lot of attention to the results of observation and the experience of others (*).

• Proposed 4 observational proofs that the Earth is a sphere:1. Ships disappear slowly over the horizon.2. The shape of the Earth at lunar eclipse appears circular.3. Different stars are visible at northern and southern latitudes.4. Elephants are found in both India (to the East) and Morocco

(to the West). Hence Earth is spherical.

• Rejected the heliocentric model because of the absence of observable parallax. Parallax was not first proved until 1838 by Bessel.

(*) “Experience is the foundation of all knowledge”

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Alexandria was founded in 330 BCE. In 290 BCE, a museum (i.e. a building dedicated to the muses, demigodesses of poetry, dance, history, drama, astronomy (Urania), nine in all) was created. The idea was promoted by Demetrius of Phaleron, a disciple of Aristotle, on the model of the Aristotelic Lyceum of Peripathetics in Athens. The Museum of Alexandria came to be during the kingdom of Ptolemy II Philadelphus.

The Museum was run as a “research institution”. It had the richest library of antiquity. It became the center of most astronomical learning – and writing – for several centuries.

Aristarchus of Samos [310-230 BCE]Eratosthenes of Cyrene [275-194 BCE]Ptolemy [100-165 CE]

were among its resident astronomers.

-Librarian at Alexandria, nicknamed -First to call himself a “philosopher”-Mathematician (prime numbers, Geometry) and Geographer

sketched the course of the Nile up to Khartoum; explainedconnection between flooding downriver and rains far upriver

-In “On the Measuring of Earth” (lost, reported by Cleomedes), he described a geometrical technique to estimate the size of Earth.

-According to Ptolemy, he obtained a very accurate estimate of the tilt angle of Earth’s axis: (11/83)x360 = 23deg 51’

-He worked out a calendar which included leap years, and tried to apply systematic chronography to historical events, starting with the fall of Troy.

-Reportedly, he produced a catalog of 675 stars.

And the Size of Earth

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Eratosthenes’

Technique to measure the size of Earth

7.2/360=1/50

i.e. the circumference of Earth is 50 times the distance between Alexandria and Syene, or

250,000 stadia

We think a stadium was either 157.7 m or 166.7m, so he was right within a few %

Used noontime altitude of the Sun and geometry to estimate the circumference of the Earth

(310-230 BCE)

•Proposed a heliocentric model for the Solar System, in which the Earth rotates about its own axis, as it revolves about the Sun

•The Moon, in turn, revolves about Earth; its light is reflected solar light, hence the lunar phases

•In one of his writings,

“On the Sizes and Distances of the Sun and Moon” (extant),

he proposed an ingenious method to carry out a measurement of such parameters.

The airport of the island of Samos is named after Aristarchos

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M

M

M’

E

SFB

Assume the Moon revolves at uniform speed on a circular orbit Then the arc between the two quarter phases MFM’ < MBM’

By measuring the time from quarter M to M’ through F and that of M’ to M through B, we can determine the angle M’EM=1/2(angle MES)Since the angle EMS is 90deg ME/ES=(Earth-Moon)/(Earth-Sun)=cos(MES)

By timing the succession of Moon phases, Aristarchos measured the ratio of distances Earth-Moon to Earth-Sun

Problem: the angle MSE is very small and Aris’ accuracy was poor. He estimated MSE=3deg, while in reality MSE=10’. He measured the ratio ME/ES=1/20, while real value is 1/390. But method was great!

While the accuracy of the measurement(*) was poor, the method devised was very clever and the assumptions made were correct.

Most important, Aristarchos’ experiment showed the Sun to be much farther, and thus much larger than the Moon.

Aware of the non-detection of stellar parallaxes, he also inferred that the “fixed stars” had to be much farther away than most thought.

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Aristarchos’ idea only yields the ratiobetween the Earth-Sun and the Earth-Moon distances. It does not yield an independent value for either of them…

However…

E

A

B

Earth’s shadow

Orbit of the Moon

Because the Sun is so much farther than the Moon, the width of the Earth’s shadow at the Moon’s distance, AB, is about equal to the Earth’s diameter.

The ratio of the length AB to the circumference of the Moon’s orbit, 2DM, is equal to that between the duration of a total lunar eclipse and the duration of the synodic month.

Thus, by timing the duration of a total lunar eclipse, Aristarchos could relate the Earth’s size to the distance to the Moon.

Another way: compare the curvature of the Earth’s shadow to that of the lunar disk.

DM

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The curvature of the shadow of Earth’s on the surface of the Moon shows that Earth is a much larger body than the Moon

The duration of totality for a lunar eclipse (i.e. the time the Moon spends within Earth’s umbra, sets a lower limit to the ratio between the Earth’s size and the distance to the Moon

“Aristarchos of Samos brought out a book consisting of some hypotheses, of which the premises lead to the result that the cosmos is many times greater than that now so called. His hypotheses are that the fixed stars and the Sun remain unmoved, that the Earth revolves around the Sun in the circumference of a circle, the Sun lying in the middle of the orbit, and that the sphere of the fixed stars, situated about the same center as the Sun, is so great that the circle in which he supposes the Earth to revolve bears such a proportion to the distance of the fixed stars as the center of the sphere bears to its surface.”

Archimedes of Syracuse (287-212 BCE)

Cleanthes of Assos: “Aristarchos should be indicted on a charge of impiety for putting into motion the hearth of the Universe”

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Consideration:

•Earth is a few times bigger than the Moon(by observing lunar eclipses)•The Sun is many times bigger than the Moon

Is it sensible to maintain “allegiance” to the geocentric model, as it appears that Earth is not the outstanding body in the Solar System?

(ca. 190-125 BCE)

Perhaps most outstanding of Greek astronomers

-Lived in Rhodes, studied extensively Babylonian astronomy

-Built instruments

-Created catalogs of “fixed” stars

-Introduced concept of magnitude (still used)

-Discovered precession

-He was a “geocentrist”, and introduced

epicycles, eccentrics and deferents

important aids in facilitating understanding of planetary motions within the geocentric framework

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[100-165 CE]

Prodigious author, great synthesizer. He established the geocentric model of the cosmos for the next 1500 year, through the Classic, Middle Age European, Arabic and Indian cultures.

“The 13 Books of the Mathematical Composition of Klaudios Ptolemaios”

“Great Composition” or “Megale Syntaxis”

(in Greek: “megale”=great, the superlative of which is “megiste”; early Arabic astronomers added the article “al” “al Megiste”, corrupted by medieval Latin writers to “Almagest”(The Greatest), as it is now commonly known.

The wide breath and influence of this book were so great that earlier works by other authors tended to be ignored and many are lost, known only in references by others and in the Almagest itself.

See also ptolcoper.mov animation

Plato (428-348 BCE): perfection in the conceptual form is key to the understanding of the world. Perfect form exists, if it can be conceived. The idea of a circle is conceivable, even if a perfect circle is not achievable in practice.Plato also thought that uniform circular motion (*), i.e. at constant angular speed, is perfect and could go on forever.Exemplified by the motions of the stars [motions of planets, Sun and Moon more difficult to explain with circular motions, perhaps because they are closer to the messy Earth]

(*) the only kind of motion “in agreement with the Divine Beings” (Ptolemy)

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E

pE

p

eccentric

A planet moving at uniform speed on a circular but eccentric orbit is seen by Earth as moving at variable angular speed

A planet moving at uniform speed on an epicycle, the center of which orbits at uniform speed on a circular deferent, itself eccentric, can be seen by Earth to temporarily invert its motion.

epicycle

deferent

How to combine circular orbits and uniform speeds in order toplatonically “save the phenomena”[and make your life difficult…]

A non-eccentric deferent

The Equant Point, opposite Earth w.r.t. the center of the deferent.

In order to better describe the motions of superior planets, Ptolemy proposed that the center of the planet’s epicycle move at constant angular speed as seen from the Equant point, rather than from the center of its deferent.

A number of astronomers later found fault with idea, a breach with the platonic postulate of uniform circular motion of a planet w.r.t. to the center of its orbit. Among those was Copernicus.

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The administrative structure of the Roman Empire was the glueof the Mediterranean and western European world, provided conduit to ideas, commerce, science, technology: “pax romana”

V century CE: Fall of Western Roman Empire, dominance of the Christian Church interpretation of social and natural history seen through optics of Scriptures

Science or “Natural Philosophy” should not be in conflict with teachings of the Church, but rather be subservient to it

Many of the Classic Greek texts were lost in Europe, although some were preserved in monastic and cathedral centers.

Production of scientific knowledge ceased.

Dark Ages

… yet darkness was not ubiquitous

Eastern Roman Empire (Constantinopolis) lasted through mid XV Century, remained repository of ancient texts.

More important, a young, dynamic culture rose in the VII century, providing political and cultural cohesion from India to the Atlantic and fostering the growth of science, with 2 important unifying characteristics: the Arabic language and religion: Islam.

After foundation of Baghdad (762), Arab kaliphs start taking interest in science and learning. In Baghdad, the House of Learning (Academy, Library and Observatory), and similar institutions spread through the Arab world, Cairo to Cordoba

In the XI century, the library of Cairo had 100,000 volumes, a large monastic library in Europe would have 100.

Islam provided the “new glue and conduit” for ideas, old and new, e.g. bringing Indian Astronomy (Siddanthas) and Trigonometry to Mediterranean + creation of new knowledge

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Some Arabic achievements:

*Advanced trigonometry: cos, tan, sec, cosec, law of sines*Introduced zero, rudiments of Algebra*Refined values of astronomical parameters, quality of timekeeping*Questioned the validity of circular orbits•Measured the angle and the eccentricity of the Sun’s (i.e. Earth’s) orbit [they remained geocentrists]•Built superb astrolabes, great & accurate computational tools•Improved enormously Geodesy and geodetic tools•Produced tables of latitude/longitude for sites through the Arab world•Produced astronomical tables of great accuracy, the paradigm of the age later imitated by Europeans, starting with king of Castille Alphonse X the Wise in 1252 (“Alphonsine Tables”)•Climate of relative cultural tolerance

•XII century: European travellers start copying texts from Arab libraries and translate them into Latin• “cultural transfer”• 1126? Adelard of Bath• 1170? Gerardo of Cremona

• Greek Science gets re-incorporated into Europe and adapted to Christian dogma

“scholasticism” (Thomas Aquinas 1225-1274)an amalgam at the core of European culture for next 3 centuries

Cusanus (Nicolaus of Cues 1401-1464), bishop of Brixen:- stressed the necessity of Mathematics in understanding nature- natural philosophy must be based on experience & experiment- Universe has no boundaries, no fixed, immovable center- Earth is neither center of the Universe nor immovable- If observers could be placed on Sun, Moon, other planets, theywould see Universe revolving about themselves precursor of Cosmological Principle (of Mediocrity).

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N. Copernicus (1473-1543)

•b. in Torun, son of merchant, nephew of bishop

•U of Crakow, 1491

•U of Bologna, Law and Astronomy (GMaria di Novara)

•U of Ferrara, Canon Law

•Ambassador, Diocese administrator

•Published book on monetary policy 1528

“De Revolutionibus Orbium Coelestium” 1543

Heliocentric model

Circular orbits*De Revolutionibus was placed in the Index of Forbidden Books by the Catholic Church

*He did not fare better with Protestants:“An upstart astrologer, this fool wishes to reverse the whole science of Astronomy; but sacred Scripture tells us that Joshua commanded the Earth to stand still, and not the Sun”wrote Martin Luther

Planet Copernican value True Value

Mercury 0.38 AU 0.387

Venus 0.72 0.723

Earth 1 1

Mars 1.52 1.52

Jupiter 5.22 5.20

Saturn 9.17 9.54

Sizes of Planetary Orbits

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(1571-1630)

-b. near Stuttgart, family hitting hard times-1589 attends U of Tuebingen-Teacher at Graz Seminary School-1599 Assistant to Tycho Brahe in Prague-Court Mathematician in Prague-1604 Observes supernova (last seen in MW)-Mathematics teacher in Linz-1617 Mother tried for witchcraft-1619 Third Law of Planetary Motion-1627 Completes Rudoplhine Tables-Mystic, Astrologer, Pythagorean (Mysterium Cosmographicum, 1593)

Kepler never discovered the deeper physical understanding behind his laws of planetary motion, which remain an accurate mathematical description of the physical phenomenon.

That understanding was achieved decades later, as Newton showed them to be the consequence of the laws of Mechanics and universal Gravitation.

Kepler’s Laws of Planetary Motion

1. The planets revolve around the Sun following elliptical orbits. The Sun lies at one focus of the ellipse of each planet’s orbit. The 2nd focus is empty.

2. The line connecting the planet to the Sun sweeps out an equal area along the orbit in any given amount of time. To do so, the planet must travel at different speeds when it is located at different parts of its orbit: fastest at perihelion, slowest at aphelion.

3. The time it takes a planet to complete one revolution of the Sun, that is, its orbital period P, is related to its mean distance from the Sun, that is, the semimajor axis a of its elliptical orbit via

P2 a3

where P is in years and a is in A.U.

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About Ellipses…

About Kepler’s Second Law…

+ ++

C ABP

Eccentricity=CA/CP

(1564-1642)

• b. in Pisa, son of mathematician/musician• Attends U of Pisa, becomes lecturer in 1589• 1590 joins faculty of U of Padua, until 1610• 1609 builds the first telescope

• 1610 publishes Sidereal Messenger, discovers satellites of Jupiter• 1612 joins Accademia dei Lincei in Rome; announces phases of Venus• 1616 admonished by Church to not advocate Copernican system• 1624 builds the first known microscope• 1632 publishes Dialog Concerning the Chief World Systems• Condemned by Holy Office on 22Jun1633 – cancelled on 31Oct1992

2009: U.N. “Year of Astronomy”

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Galileo & the Stars

Wherever he pointed his telescope, he found more and more stars; the Milky Way is a complex of stars unresolved by the naked eye.

Galileo & the Moon

Found craters, mountains, valleys on the Moon.So the Moon is not so different from Earth.

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Galileo and the Sun

Discovered spots (imperfections) on the SunThe heavenly bodies are not perfect

What is a Sun spot?

Galileo and Jupiter

Discovered four Moons of Jupiter:They orbit Jupiter, not Earth Earth is not

the center of all cosmic motions

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Galileo and VenusIn 1612, Galileo announced that ,

with the aid of the telescope, Venus showed phases, like the Moon.

These phases were found to be consistent with:

(a) Venus shines with reflected light (Thales, 600 BCE), and

(b) Venus moves around the Sun, not Earth

Galileo & the Church

1616 – Galileo instructed to stop discussing his observations1623 – Cardinal Barberini (his friend) becomes Pope Urban VIII1632 – Galileo publishes “Dialog Concerning the Two Chief World

Systems”• Written in Italian (accessible)• Three characters

1. Salviati – brilliant; mouthpiece of Galileo2. Sagredo – skeptical, but quick to understand Salviati3. Simplicio – not-very-smart Aristotelian who raises

objections to Copernican system and is easily refuted1633 - Galileo tried and found guilty; spends last 10 years of his life

under house arrest in Arcetri1835 – “Dialog” removed from list of “forbidden books”.1992 – Galileo formally exonerated by Catholic Church

“The Navy is never wrong, but occasionally it takes a while in being right.”[Henry Fonda in the movie “In Harm’s Way”]

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The Easter Problem

The day of Easter is defined to be the first Sunday after the first full Moon on or after the Sun crosses the northern Spring Equinox. Early calendars produced look-up tables for events such as Easter which – given their lack of precision – could differ from each other (or from the accurate time) by weeks.

Regiomontanus: the Church should measure the equinox transit, rather than rely on tables

The Obliquity of the Ecliptic

The obliquity of the ecliptic, i.e. the angle between the Celestial Equator and the plane of the Earth’s orbit around the Sun is not constant; it varies due to perturbations from other planets in the solar system:

ε = 23° 27′ 08.26″ − 46.845″ T − 0.0059″ T2 + 0.00181″ T3

where ε is the obliquity and T is the number of centuries from 1900.0 to the date in question. The current value is about 23:26:20 and it is decreasing by about 47” per century.

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How a Meridiana Works

1. South-facing wall2. N—S nave: maximize

length3. Perfectly flat floor line4. Clear line of sight5. Well-calibrated pin hole

height

The Meridiana of Egnatio Danti

In 1574, Egnatio Danti arranged for a meridiana to be built in Florence, at Santa Maria Novella. It wasn’t very precise; Danti reported a values of 23deg 26’ 49”.55, off by about 2’,belying the excessive claim for accuracy.

He also reported the length of the tropical year as 365 days, 5 hrs, 45 min, 36 sec, which is 3 min 36 sec shorter than the Gregorian year.

After losing the support of the Grand Duke of Tuscany, he moved to Bologna, where he succeeded in obtaining support in designing and building a meridiana in San Petronio. From the fabriccieri of the city.

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The Meridiana of Egnatio Danti

His choice for the location of the hole in the wall – through which the Sun light would enter San Petronio – prevented him from laying a meridian line precisely along a N-S direction. His meridiana was off by 9deg.

Readers of the meridiana would differ by large amounts from each other: the equinox transit would be no more accurate than half a day.

Enter Giovanni Domenico Cassini:The Master of the Grand MeridianaInaugurated in 1655

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The Grand Meridiana

It does clear the pillars!

The Grand Meridiana

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• The Obliquity of the Ecliptic• The Latitude of San Petronio and the Asinelli

Tower • The Length of the Tropical Year: 365d 5h 49m 0s

(12s by Riccioli & Grimaldi)(after 60 yrs: down to 1 sec of time)

• Solar parallax

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The “orbit” of the Sun: Kepler was Right

Elliptical orbit can be mimicked with Ptolemaic circle + equant

However direct emasurement of variation in solar image diameter across the year indicated Kepler was right

At apogee: 31’ 8” (Cassini) 31’ 0” (Riccioli & Grimaldi)At perigee: 32’ 10” (C) and 32’ 4” (R & G)

E

pE

p

eccentric

A planet moving at uniform speed on a circular but eccentric orbit is seen by Earth as moving at variable angular speed

A planet moving at uniform speed on an epicycle, the center of which orbits at uniform speed on a circular deferent, itself eccentric, can be seen by Earth to temporarily invert its motion.

epicycle

deferent

How to combine circular orbits and uniform speeds in order toplatonically “save the phenomena”[and make your life difficult…]

A non-eccentric deferent

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The Equant Point, opposite Earth w.r.t. the center of the deferent.

In order to better describe the motions of superior planets, Ptolemy proposed that the center of the planet’s epicycle move at constant angular speed as seen from the Equant point, rather than from the center of its deferent.

A number of astronomers later found fault with idea, a breach with the platonic postulate of uniform circular motion of a planet w.r.t. to the center of its orbit. Among those was Copernicus.

Geocentrism vs. Heliocentrism and the Church

Cassini moves to Paris (1669)

Negotiations