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Oct
4th
Tue
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Richard Feynman on Beauty, Honours and Curiosity



Richard Feynman, American physicist known for his work in the path integral formulation of quantum mechanics, Nobel Prize in Physics.
The Feynman Series is a companion project with The Sagan Series (Full playlist)

See also:

Richard Feynman and Jirayr Zorthian on science, art and beauty
Richard Feynman on the likelihood of Flying Saucers
Richard Feynman on how we would look for a new law (the key to science)
Richard Feynman on the way nature work: “You don’t like it? Go somewhere else!”

Mar
4th
Fri
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Timeline of Major Western Philosophers

Source: Department of Philosophy College of Liberal Arts Rochester Institute of Technology

See also:

Timeline of Western philosophers, Wiki
Timeline of Western philosophers, Opentopia
Philosophy timeline (interactive), Thomson Wadsworth
History of Western philosophy - Brief timeline, Global Oneness
The Chronology, Squashed Philosophers Abridged Editions
Philosophers, Timeline Index
Timeline of Early Chinese Philosophy
Hellenistic Philosophy (300BCE-200CE)
Timeline of Philosophy, The Radical Academy

Apr
29th
Thu
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Kurt Gödel (1906-1978) was an Austrian-American logician, mathematician and philosopher. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when many, such as Bertrand Russell, A. N. Whitehead and David Hilbert, were pioneering the use of logic and set theory to understand the foundations of mathematics.
Both Ludwig Wittgenstein (1889-1951) and Kurt Gödel  advanced theories of ‘incompleteness,’ though their theories were  opposed.  Gödel had a Platonic view of numbers and mathematics, whereby  numbers have their own reality that humans attempt, in their imperfect way, to grasp.  Wittgenstein, on the other hand, had a linguistic, pragmatic view of numbers, and he saw mathematics simply as a highly formal language, with its own rigorous syntax.  For Gödel, the ‘incompleteness’ was an aspect of our ability to describe numbers and their relationships, in a systematic, consistent manner.  Wittgenstein, on the other hand, saw mathematical incompleteness as an absurd concept, like a language that would be somehow ‘incomplete’ (in Wittgenstein’s view, language says what we need it to say, to the extent our needs can be expressed).  Wittgenstein, however, felt that there was another, perhaps more fundamental kind of ‘incompleteness’—our inability to express in language, to reduce to linguistic form, those things that are most important to us.  
Here is how philosopher, essayist, and novelist Rebecca Goldstein succinctly compares Gödel and Wittgenstein: 
"Gödel would most likely not have known that, on some level, he and (the early) Wittgentstein shared a profound conviction of incompleteness, a shared rejection of the logical positivists’ endorsement of the Sophist’s “measure of all things.”.(…) Of course, Gödel and Wittgenstein located the escaped parts of reality in irreconcilably different ways.  Gödel’s conviction, the mathematical interpretation he gave his incompleteness theorems (as well as his work on the continuum hypothesis), was that it was aspects of mathematical reality that must escape our formal systematizing (although not our knowledge), and Wittgenstein’s view on the foundations of mathematics would not countenance this conviction.  For Wittgenstein, at least early Wittgenstein, all of knowledge, a fortiori mathematical knowledge, is systematizable; what systematically escapes our systems is the unsayable, which includes all that is important. Gödel believed our expressible knowledge, demonstrably our mathematical knowledge, is greater than our systems.  Whereof we cannot formalize, thereof we can still know, the mathematician might have said, had he any inclination toward the oracular.”
— Rebecca Goldstein, Incompleteness: The Proof and Paradox of  Kurt Gödel, (2005) (via mhsteger)

"All the limitative theorems of metamathematics and the theory of computation suggest that once the ability to represent your own structure has reached a certain critical point, that is the kiss of death: it guarantees that you can never represent yourself totally. Gödel’s Incompleteness Theorem, Church’s Undecidability Theorem, Turing’s Halting Theorem, Tarski’s Truth Theorem — all have the flavour of some ancient fairy tale which warns you that “To seek self-knowledge is to embark on a journey which … will always be incomplete, cannot be charted on any map, will never halt, cannot be described."
— Douglas Hofstadter, 1979 via Vinod K. Wadhawan, Complexity Explained: 17. Epilogue, Nirmukta

Kurt Gödel (1906-1978) was an Austrian-American logician, mathematician and philosopher. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when many, such as Bertrand Russell, A. N. Whitehead and David Hilbert, were pioneering the use of logic and set theory to understand the foundations of mathematics.

Both Ludwig Wittgenstein (1889-1951) and Kurt Gödel advanced theories of ‘incompleteness,’ though their theories were opposed.  Gödel had a Platonic view of numbers and mathematics, whereby numbers have their own reality that humans attempt, in their imperfect way, to grasp.  Wittgenstein, on the other hand, had a linguistic, pragmatic view of numbers, and he saw mathematics simply as a highly formal language, with its own rigorous syntax.  For Gödel, the ‘incompleteness’ was an aspect of our ability to describe numbers and their relationships, in a systematic, consistent manner.  Wittgenstein, on the other hand, saw mathematical incompleteness as an absurd concept, like a language that would be somehow ‘incomplete’ (in Wittgenstein’s view, language says what we need it to say, to the extent our needs can be expressed).  Wittgenstein, however, felt that there was another, perhaps more fundamental kind of ‘incompleteness’—our inability to express in language, to reduce to linguistic form, those things that are most important to us.  

Here is how philosopher, essayist, and novelist Rebecca Goldstein succinctly compares Gödel and Wittgenstein: 

"Gödel would most likely not have known that, on some level, he and (the early) Wittgentstein shared a profound conviction of incompleteness, a shared rejection of the logical positivists’ endorsement of the Sophist’s “measure of all things.”.
(…)
Of course, Gödel and Wittgenstein located the escaped parts of reality in irreconcilably different ways.  Gödel’s conviction, the mathematical interpretation he gave his incompleteness theorems (as well as his work on the continuum hypothesis), was that it was aspects of mathematical reality that must escape our formal systematizing (although not our knowledge), and Wittgenstein’s view on the foundations of mathematics would not countenance this conviction.  For Wittgenstein, at least early Wittgenstein, all of knowledge, a fortiori mathematical knowledge, is systematizable; what systematically escapes our systems is the unsayable, which includes all that is important. Gödel believed our expressible knowledge, demonstrably our mathematical knowledge, is greater than our systems.  Whereof we cannot formalize, thereof we can still know, the mathematician might have said, had he any inclination toward the oracular.”

Rebecca Goldstein, Incompleteness: The Proof and Paradox of Kurt Gödel, (2005) (via mhsteger)

"All the limitative theorems of metamathematics and the theory of computation suggest that once the ability to represent your own structure has reached a certain critical point, that is the kiss of death: it guarantees that you can never represent yourself totally. Gödel’s Incompleteness Theorem, Church’s Undecidability Theorem, Turing’s Halting Theorem, Tarski’s Truth Theorem — all have the flavour of some ancient fairy tale which warns you that “To seek self-knowledge is to embark on a journey which … will always be incomplete, cannot be charted on any map, will never halt, cannot be described."

Douglas Hofstadter, 1979 via Vinod K. Wadhawan, Complexity Explained: 17. Epilogue, Nirmukta

Mar
20th
Sat
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(Pythagoras reading the book, Hypatia and Parmenides? to the right of her - fragment of The School of Athens by Raffaello Santi (Raphael), 1509-1510)
Hypatia (Greek: Ὑπατία, Hypatía, pronounced /haɪˈpeɪʃə/ in English; born between AD 350 and 370; died March 415) was a Greek scholar from Alexandria in Egypt, is the earliest woman scientist whose life is well documented; she was also the last scientist of the Golden Age of Pericles, considered the first notable woman in mathematics, who also taught philosophy and astronomy. She lived in Roman Egypt, and was killed by a Christian mob who falsely blamed her for religious turmoil. Some suggest that her murder marked the end of what is traditionally known as Classical antiquity, although others such as Christian Wildberg observe that Hellenistic philosophy continued to flourish until the age of Justinian in the sixth century.
A Neoplatonist philosopher, she belonged to the mathematical tradition of the Academy of Athens represented by Eudoxus of Cnidus; she followed the school of the 3rd century thinker Plotinus, discouraging empirical enquiry and encouraging logical and mathematical studies. The name Hypatia derives from the adjective ὑπάτη, the feminine form of ὕπατος (upatos), meaning “highest, uppermost, supremest”.
Hypatia was the daughter of Theon, who was her teacher and the last known mathematician associated with the Museum of Alexandria. She traveled to both Athens and Italy to study, before becoming head of the Platonist school at Alexandria in approximately 400. According to the 10th century Byzantine encyclopedia the Suda, she worked as teacher of philosophy, teaching the works of Plato and Aristotle. It is believed that there were both Christians and foreigners among her students. Although Hypatia was herself a pagan, she was respected by a number of Christians, and later held up by Christian authors as a symbol of virtue. The Suda controversially declared her “the wife of Isidore the Philosopher" but agreed she had remained a virgin. Hypatia rebuffed a suitor by showing him her menstrual rags, claiming they demonstrated that there was "nothing beautiful" about carnal desires.  Hypatia maintained correspondence with her former pupil Synesius of Cyrene, who in AD 410 became bishop of Ptolemais. Together with the references by Damascius, these are the only writings with descriptions or information from her pupils that survive. The contemporary Christian historiographer Socrates Scholasticus described her in his Ecclesiastical History:
 “There was a woman at Alexandria named Hypatia, daughter of the philosopher Theon, who made such attainments in literature and science, as to far surpass all the philosophers of her own time. Having succeeded to the school of Plato and Plotinus, she explained the principles of philosophy to her auditors, many of whom came from a distance to receive her instructions. On account of the self-possession and ease of manner, which she had acquired in consequence of the cultivation of her mind, she not unfrequently appeared in public in presence of the magistrates. Neither did she feel abashed in going to an assembly of men. For all men on account of her extraordinary dignity and virtue admired her the more.”
(More: Hypatia, daughter of Theron, Librarian of Alexandria)
 Documentary on Hypatia and the city of Alexandria

This documentary shows footage of Alejandro Amenábar's last film “Agora" proving the historical accuracy of the movie. — Bettany Hughes’ TV Tour of the Ancient World, Channel 4, Full playlist

(Pythagoras reading the book, Hypatia and Parmenides? to the right of her - fragment of The School of Athens by Raffaello Santi (Raphael), 1509-1510)

Hypatia (Greek: Ὑπατία, Hypatía, pronounced /haɪˈpeɪʃə/ in English; born between AD 350 and 370; died March 415) was a Greek scholar from Alexandria in Egypt, is the earliest woman scientist whose life is well documented; she was also the last scientist of the Golden Age of Pericles, considered the first notable woman in mathematics, who also taught philosophy and astronomy. She lived in Roman Egypt, and was killed by a Christian mob who falsely blamed her for religious turmoil. Some suggest that her murder marked the end of what is traditionally known as Classical antiquity, although others such as Christian Wildberg observe that Hellenistic philosophy continued to flourish until the age of Justinian in the sixth century.

A Neoplatonist philosopher, she belonged to the mathematical tradition of the Academy of Athens represented by Eudoxus of Cnidus; she followed the school of the 3rd century thinker Plotinus, discouraging empirical enquiry and encouraging logical and mathematical studies. The name Hypatia derives from the adjective ὑπάτη, the feminine form of ὕπατος (upatos), meaning “highest, uppermost, supremest”.

Hypatia was the daughter of Theon, who was her teacher and the last known mathematician associated with the Museum of Alexandria. She traveled to both Athens and Italy to study, before becoming head of the Platonist school at Alexandria in approximately 400. According to the 10th century Byzantine encyclopedia the Suda, she worked as teacher of philosophy, teaching the works of Plato and Aristotle. It is believed that there were both Christians and foreigners among her students. Although Hypatia was herself a pagan, she was respected by a number of Christians, and later held up by Christian authors as a symbol of virtue. The Suda controversially declared her “the wife of Isidore the Philosopher" but agreed she had remained a virgin. Hypatia rebuffed a suitor by showing him her menstrual rags, claiming they demonstrated that there was "nothing beautiful" about carnal desires. Hypatia maintained correspondence with her former pupil Synesius of Cyrene, who in AD 410 became bishop of Ptolemais. Together with the references by Damascius, these are the only writings with descriptions or information from her pupils that survive. The contemporary Christian historiographer Socrates Scholasticus described her in his Ecclesiastical History:

“There was a woman at Alexandria named Hypatia, daughter of the philosopher Theon, who made such attainments in literature and science, as to far surpass all the philosophers of her own time. Having succeeded to the school of Plato and Plotinus, she explained the principles of philosophy to her auditors, many of whom came from a distance to receive her instructions. On account of the self-possession and ease of manner, which she had acquired in consequence of the cultivation of her mind, she not unfrequently appeared in public in presence of the magistrates. Neither did she feel abashed in going to an assembly of men. For all men on account of her extraordinary dignity and virtue admired her the more.”

(More: Hypatia, daughter of Theron, Librarian of Alexandria)

Documentary on Hypatia and the city of Alexandria


This documentary shows footage of Alejandro Amenábar's last film “Agora" proving the historical accuracy of the movie. Bettany Hughes’ TV Tour of the Ancient World, Channel 4, Full playlist

Mar
19th
Fri
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Armonia by Remedios Varo, 1956.
"María de los Remedios Varo Uranga was a Spanish-Mexican mamacita with a classic look and artistic creations that look like demented Disney settings and characters. She was born December 16, 1908 in Anglés, Girona, Spain. She had a financially comfy childhood because her father was a hydraulic engineer (and no, I don’t mean Snoop Dogg and Dr. Dre hydraulics, we’re taking water flow hydraulics). Thanks to Papa’s job, Varo had the opportunity to travel to Spain and South Africa quite often. Taking trips to places most will never see sparked a lifelong interest in math, mechanical drawing, and locomotor vehicles in young María." Source (via surrealism)

Armonia by Remedios Varo, 1956.

"María de los Remedios Varo Uranga was a Spanish-Mexican mamacita with a classic look and artistic creations that look like demented Disney settings and characters. She was born December 16, 1908 in Anglés, Girona, Spain. She had a financially comfy childhood because her father was a hydraulic engineer (and no, I don’t mean Snoop Dogg and Dr. Dre hydraulics, we’re taking water flow hydraulics). Thanks to Papa’s job, Varo had the opportunity to travel to Spain and South Africa quite often. Taking trips to places most will never see sparked a lifelong interest in math, mechanical drawing, and locomotor vehicles in young María." Source (via surrealism)

Feb
28th
Sun
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J. M. William Turner, Ovid Banished From Rome, 1838 
“Publius Ovidius Naso (20 March 43 BCE – 17 or 18 CE), known as Ovid in the English-speaking world, was a Roman poet who is best known as the author of the Heroides, Amores, and Ars Amatoria, three major collections of erotic poetry, the Metamorphoses a mythological hexameter poem, the Fasti, about the Roman calendar, and the Tristia and Epistulae ex Ponto, two collections of poems written in exile on the Black Sea. Ovid was also the author of several smaller pieces, the Remedia Amoris, the Medicamina Faciei Femineae, and the Ibis, a long curse-poem. He also authored a lost tragedy, Medea. (…)
In 8 CE, Ovid was banished to Tomis, on the Black Sea, by the exclusive intervention of the Emperor Augustus, without any participation of the Senate or of any Roman judge, an event which would shape all of his following poetry. Ovid wrote that the reason for his exile was carmen et error — “a poem and a mistake”, claiming that his crime was worse than murder, more harmful than poetry. The Emperor’s grandchildren, Agrippa Postumus and Julia the Younger, were banished around the time of his banishment; Julia’s husband, Lucius Aemilius Paullus, was put to death for conspiracy against Augustus, a conspiracy about which Ovid might have known. The Julian Marriage Laws of 18 BCE, which promoted monogamous marriage to increase the population’s birth rate, were fresh in the Roman mind. Ovid’s writing in the Ars Amatoria concerned the serious crime of adultery, and he may have been banished for these works which appeared subversive to the emperor’s moral legislation. However, because of the long distance of time between the publication of this work (1 BC) and the exile (8 CE), some authors suggest that Augustus used the poem as a mere justification for something more personal.”

J. M. William Turner, Ovid Banished From Rome, 1838 

Publius Ovidius Naso (20 March 43 BCE – 17 or 18 CE), known as Ovid in the English-speaking world, was a Roman poet who is best known as the author of the Heroides, Amores, and Ars Amatoria, three major collections of erotic poetry, the Metamorphoses a mythological hexameter poem, the Fasti, about the Roman calendar, and the Tristia and Epistulae ex Ponto, two collections of poems written in exile on the Black Sea. Ovid was also the author of several smaller pieces, the Remedia Amoris, the Medicamina Faciei Femineae, and the Ibis, a long curse-poem. He also authored a lost tragedy, Medea. (…)

In 8 CE, Ovid was banished to Tomis, on the Black Sea, by the exclusive intervention of the Emperor Augustus, without any participation of the Senate or of any Roman judge, an event which would shape all of his following poetry. Ovid wrote that the reason for his exile was carmen et error — “a poem and a mistake”, claiming that his crime was worse than murder, more harmful than poetry. The Emperor’s grandchildren, Agrippa Postumus and Julia the Younger, were banished around the time of his banishment; Julia’s husband, Lucius Aemilius Paullus, was put to death for conspiracy against Augustus, a conspiracy about which Ovid might have known. The Julian Marriage Laws of 18 BCE, which promoted monogamous marriage to increase the population’s birth rate, were fresh in the Roman mind. Ovid’s writing in the Ars Amatoria concerned the serious crime of adultery, and he may have been banished for these works which appeared subversive to the emperor’s moral legislation. However, because of the long distance of time between the publication of this work (1 BC) and the exile (8 CE), some authors suggest that Augustus used the poem as a mere justification for something more personal.”

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L’expérience de Pierre Testu-Brissy, le 16 octobre 1798
Pierre Testu-Brissy (1770? - 1829) was a pioneering French balloonist who achieved fame for making many flights astride animals, particularly horses.
Testu-Brissy made his first balloon ascent in 1785, and the first night ascent on June 18, 1786, in a hydrogen balloon. He made the world’s first electrical observations on June 18th, 1786, as he ascended into thunderclouds, and said that he drew remarkable discharges from the clouds by means of an iron rod, carried in the car.
Testu-Brissy’s first solo ascent was on September 18, 1791 from Paris. He subsequently undertook more than 50 flights in his lifetime, including the first ascent on horseback on October 16, 1798 from Bellevue Park in Paris. He and his horse made more than fifty of these documented flights.

L’expérience de Pierre Testu-Brissy, le 16 octobre 1798

Pierre Testu-Brissy (1770? - 1829) was a pioneering French balloonist who achieved fame for making many flights astride animals, particularly horses.

Testu-Brissy made his first balloon ascent in 1785, and the first night ascent on June 18, 1786, in a hydrogen balloon. He made the world’s first electrical observations on June 18th, 1786, as he ascended into thunderclouds, and said that he drew remarkable discharges from the clouds by means of an iron rod, carried in the car.

Testu-Brissy’s first solo ascent was on September 18, 1791 from Paris. He subsequently undertook more than 50 flights in his lifetime, including the first ascent on horseback on October 16, 1798 from Bellevue Park in Paris. He and his horse made more than fifty of these documented flights.

Feb
19th
Fri
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Anicius Manlius Severinus Boethius, De Musica, ca. 1120-1150. Manuscripts Collection, Alexander Turnbull Library (via National Library of New Zealand), (via marsiouxpial) 
Boëthius (ca. 480–524 or 525) was a Christian philosopher of the early 6th century. He was born in Rome to an ancient and important family which included emperors Petronius Maximus  and Olybrius and many consuls. His father, Flavius Manlius Boethius, was consul in 487 after Odoacer deposed the last Western Roman Emperor. Boethius, of the noble Anicius lineage, entered public life at a young age and was already a senator by the age of 25. Boethius himself was consul in 510 in the kingdom of the Ostrogoths. In 522 he saw his two sons become consuls. Boethius was executed by King Theodoric the Great, who suspected him of conspiring with the Byzantine Empire.
Boethius’ De institutione musica, was one of the first musical works to be printed in Venice between the years of 1491 and 1492. It was written toward the beginning of the sixth century and helped medieval authors during the ninth century understand Greek music.
In his “De Musica”, Boethius introduced the fourfold classification of music: 1. Musica mundana — music of the spheres/world; 2. Musica humana — harmony of human body and spiritual harmony; 3. Musica instrumentalis — instrumental music (incl. human voice); 4. Musica divina — music of the gods  During the Middle Ages
Boethius was connected to several texts that were used to teach liberal arts. Although he did not address the subject of trivium, he did write many treatises explaining the principles of rhetoric, grammar, and logic. During the Middle Ages, his works of these disciplines were commonly used when studying the three elementary arts.

Anicius Manlius Severinus Boethius, De Musica, ca. 1120-1150. Manuscripts Collection, Alexander Turnbull Library (via National Library of New Zealand), (via marsiouxpial)

Boëthius (ca. 480–524 or 525) was a Christian philosopher of the early 6th century. He was born in Rome to an ancient and important family which included emperors Petronius Maximus and Olybrius and many consuls. His father, Flavius Manlius Boethius, was consul in 487 after Odoacer deposed the last Western Roman Emperor. Boethius, of the noble Anicius lineage, entered public life at a young age and was already a senator by the age of 25. Boethius himself was consul in 510 in the kingdom of the Ostrogoths. In 522 he saw his two sons become consuls. Boethius was executed by King Theodoric the Great, who suspected him of conspiring with the Byzantine Empire.

Boethius’ De institutione musica, was one of the first musical works to be printed in Venice between the years of 1491 and 1492. It was written toward the beginning of the sixth century and helped medieval authors during the ninth century understand Greek music.

In his “De Musica”, Boethius introduced the fourfold classification of music: 1. Musica mundana — music of the spheres/world; 2. Musica humana — harmony of human body and spiritual harmony; 3. Musica instrumentalis — instrumental music (incl. human voice); 4. Musica divina — music of the gods During the Middle Ages

Boethius was connected to several texts that were used to teach liberal arts. Although he did not address the subject of trivium, he did write many treatises explaining the principles of rhetoric, grammar, and logic. During the Middle Ages, his works of these disciplines were commonly used when studying the three elementary arts.

Feb
8th
Mon
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Jan Matejko - Astronomer Copernicus - Conversation with God, 1872
Nicolaus Copernicus (19 February 1473 – 24 May 1543) was the first astronomer to formulate a comprehensive heliocentric cosmology, which displaced the Earth from the center of the universe.  Copernicus’ epochal book, De revolutionibus orbium coelestium (On the Revolutions of the Celestial Spheres), published just before his death in 1543, is often regarded as the starting point of modern astronomy and the defining epiphany that began the scientific revolution. His heliocentric model, with the Sun at the center of the universe, demonstrated that the observed motions of celestial objects can be explained without putting Earth at rest in the center of the universe. His work stimulated further scientific investigations, becoming a landmark in the history of science that is often referred to as the Copernican Revolution.

Jan Matejko - Astronomer Copernicus - Conversation with God, 1872

Nicolaus Copernicus (19 February 1473 – 24 May 1543) was the first astronomer to formulate a comprehensive heliocentric cosmology, which displaced the Earth from the center of the universe. Copernicus’ epochal book, De revolutionibus orbium coelestium (On the Revolutions of the Celestial Spheres), published just before his death in 1543, is often regarded as the starting point of modern astronomy and the defining epiphany that began the scientific revolution. His heliocentric model, with the Sun at the center of the universe, demonstrated that the observed motions of celestial objects can be explained without putting Earth at rest in the center of the universe. His work stimulated further scientific investigations, becoming a landmark in the history of science that is often referred to as the Copernican Revolution.

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Aristarchus’s 3rd century BC calculations on the relative sizes of the Earth, Sun and Moon, from a 10th century AD Greek copy
Aristarchus (Greek: Ἀρίσταρχος, Arístarchos; 310 BC – ca. 230 BC) was a Greek astronomer and mathematician, born on the island of Samos, in Greece. He is the first known person to present a heliocentric model of the solar system, placing the Sun, not the Earth, at the center of the known universe. He was influenced by the Pythagorean Philolaus of Croton, but, in contrast to Philolaus, he had both identified the central fire with the Sun, as well as putting other planets in correct order from the Sun. His astronomical ideas were rejected in favor of the geocentric theories of Aristotle and Ptolemy until they were successfully revived nearly 1800 years later by Copernicus and extensively developed and built upon by Johannes Kepler and Isaac Newton.
Distance to the Sun (Lunar Dichotomy) 
Aristarchus claimed that at half moon (first or last quarter moon), the angle between sun and moon was 87°. Possibly he proposed 87° as a lower bound since gauging the lunar terminator's deviation from linearity to 1° accuracy is beyond the unaided human ocular limit (that limit being about 3° accuracy). Aristarchus is known to have also studied light and vision.
Using correct geometry, but the insufficiently accurate 87° datum, Aristarchus concluded that the Sun was between 18 to 20 times farther away than the Moon. (The true value of this angle is close to 89° 50’, and the Sun’s distance is actually about 400 times the Moon’s.) The implicit false solar parallax of slightly under 3° was used by astronomers up to and including Tycho Brahe, ca. AD 1600 CE. Aristarchus pointed out that the Moon and Sun have nearly equal apparent angular sizes and therefore their diameters must be in proportion to their distances from Earth. He thus concluded that the diameter of the Sun was between 18 to 20 times larger than the diameter of the Moon; which, although wrong, follows logically from his data. It also leads to the conclusion that the Sun’s diameter is almost seven times greater than the Earth’s; the volume of Aristarchus’s Sun would be almost 300 times greater than the Earth. This difference in sizes may have inspired the heliocentric model.

Aristarchus’s 3rd century BC calculations on the relative sizes of the Earth, Sun and Moon, from a 10th century AD Greek copy

Aristarchus (Greek: Ἀρίσταρχος, Arístarchos; 310 BC – ca. 230 BC) was a Greek astronomer and mathematician, born on the island of Samos, in Greece. He is the first known person to present a heliocentric model of the solar system, placing the Sun, not the Earth, at the center of the known universe. He was influenced by the Pythagorean Philolaus of Croton, but, in contrast to Philolaus, he had both identified the central fire with the Sun, as well as putting other planets in correct order from the Sun. His astronomical ideas were rejected in favor of the geocentric theories of Aristotle and Ptolemy until they were successfully revived nearly 1800 years later by Copernicus and extensively developed and built upon by Johannes Kepler and Isaac Newton.

Distance to the Sun (Lunar Dichotomy)

Aristarchus claimed that at half moon (first or last quarter moon), the angle between sun and moon was 87°. Possibly he proposed 87° as a lower bound since gauging the lunar terminator's deviation from linearity to 1° accuracy is beyond the unaided human ocular limit (that limit being about 3° accuracy). Aristarchus is known to have also studied light and vision.

Using correct geometry, but the insufficiently accurate 87° datum, Aristarchus concluded that the Sun was between 18 to 20 times farther away than the Moon. (The true value of this angle is close to 89° 50’, and the Sun’s distance is actually about 400 times the Moon’s.) The implicit false solar parallax of slightly under 3° was used by astronomers up to and including Tycho Brahe, ca. AD 1600 CE. Aristarchus pointed out that the Moon and Sun have nearly equal apparent angular sizes and therefore their diameters must be in proportion to their distances from Earth. He thus concluded that the diameter of the Sun was between 18 to 20 times larger than the diameter of the Moon; which, although wrong, follows logically from his data. It also leads to the conclusion that the Sun’s diameter is almost seven times greater than the Earth’s; the volume of Aristarchus’s Sun would be almost 300 times greater than the Earth. This difference in sizes may have inspired the heliocentric model.

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Qutb al-Din, 13th century AD, discussed whether heliocentrism was a possibility. 
Picture taken from old manuscript of Qotbeddin Shirazi’s treatise (13th century). The image depicts an epicyclic planetary model.
Qotb al-Din Shirazi (1236 – 1311) (Persian: قطب‌الدین شیرازی) was a 13th century Persian Muslim polymath and Persian poet who made contributions astronomy, mathematics, medicine, physics, music theory, philosophy and Sufism.

Qutb al-Din, 13th century AD, discussed whether heliocentrism was a possibility.

Picture taken from old manuscript of Qotbeddin Shirazi’s treatise (13th century). The image depicts an epicyclic planetary model.

Qotb al-Din Shirazi (1236 – 1311) (Persian: قطب‌الدین شیرازی) was a 13th century Persian Muslim polymath and Persian poet who made contributions astronomy, mathematics, medicine, physics, music theory, philosophy and Sufism.

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