Theories of Aristarchus

Theories of Aristarchus

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Aristarchus meaning

The name Aristarchus occurs five times in the New Testament. Aristarchus was a Macedonian of Thessalonica (Acts 27:2) and a traveling companion of Paul. He was manhandled together with Gaius when Demetrius stirred Ephesus into an uproar (Acts 19:29) and accompanied Paul on his subsequent missionary journey to Macedonia and Greece (Acts 20:4).

Aristarchus was with Paul when the ship they sailed on to Rome went down off the coast of Malta (Acts 27:2). Both men made it to Rome, where they both were imprisoned (Colossians 4:10, Philemon 1:24).

The name Aristarchus occurs 5 times in the New Testament see full concordance.


It’s funny, but not all the scientists we talk about on this website are actually famous. Some of them, like Aristarchus, deserve to be… but they’re not.

An artist’s view of how Aristarchus might have looked.

If you’re looking for an unsung hero of science, you could do worse than Aristarchus of Samos, or Aristarchus the Mathematician as some people called him. Today, a better name might be Aristarchus, who said the earth orbits the sun.


Aristarchus was born in about the year 310 BC, probably on the Greek island of Samos, the same island Pythagoras was born on 260 years earlier. We know very little about Aristarchus’s life, but we know enough to be astounded by his science. We know:

  • Aristarchus lived at about the same time as two of our other scientific heroes, Archimedes and Eratosthenes he was 20 to 30 years older than them.
  • His greatest work has been lost in the mists of time we know about it because Archimedes mentions it in The Sand Reckoner, more of which soon.

Lifetimes of Selected Ancient Greek Scientists and Philosophers

Copernicus Says Earth Orbits the Sun

To appreciate what Aristarchus did over 2,000 years ago, it’s worthwhile thinking about one of the greats of astronomy, Nicolaus Copernicus.

In 1543 Nicolaus Copernicus published his famous book: On the Revolutions of the Heavenly Spheres. He told us that Earth, and all the other planets, orbit the sun. In other words, he said the Solar System is heliocentric.

Until Copernicus published his work, people thought we lived in a geocentric Solar System – i.e. Earth was at the center of everything. They believed the moon, the planets, the sun, and the stars orbited the earth.

The geocentric idea was taught by the Catholic Church, and Copernicus was a Catholic. Copernicus’s book was suppressed by the Church, but gradually, his theory came to be accepted.

However, Copernicus was rather late coming to the heliocentric view.

Aristarchus beat him by 18 centuries.

Archimedes tells us about Aristarchus’s Book

Sadly, the book Aristarchus wrote describing his heliocentric Solar System has been lost – the fate of many great Ancient Greek works. Fortunately, we know a little about it, because it is mentioned by other Greeks, including Archimedes, who mentions it in a letter he addressed to a King named Gelon. This letter was The Sand Reckoner. Archimedes wrote:

“You know the universe is the name astronomers call the sphere whose radius is the straight line from the center of the earth to the center of the sun. But Aristarchus has written a book in which he says that the universe is many times bigger than we thought. He says that the stars and the sun don’t move, and that the earth revolves about the sun and that the path of the orbit is circular.”

Aristarchus must have used the concept of parallax to show that the stars are a very large distance from Earth. In doing so, he expanded the size of the universe enormously.

It would be marvelous if we could learn the details of Aristarchus’s observations, calculations, arguments, could read his notes and see his diagrams but, unless a copy of his ancient book can be discovered in some forgotten, dusty corner of an ancient library, that is a pleasure we shall never have.

A modern view of the bodies orbiting in our heliocentric Solar System. Aristarchus would have been thrilled to know what we know now. Image credit: NASA/JPL-Caltech (click for larger image).

Aristarchus also believed that, in addition to orbiting the sun, Earth spins on its own axis, taking one day to complete one revolution.


It’s sometimes said there was pressure for Aristarchus to be put on trial for daring to say the earth is not at the center of the universe. It turns out this was a mistranslation of a work by the Greek historian Plutarch.

There was no persecution of Aristarchus. His idea just didn’t find many fans. Most Ancient Greeks rejected his work, and continued to believe in a geocentric Solar System.

Thankfully, Archimedes was happy to use Aristarchus’s model of the universe in The Sand Reckoner, to discuss calculations using larger numbers than the Greeks had used before.

Only one of Aristarchus’s works has survived, in which he tried to calculate the sizes of the moon and sun and tried to figure out how far they were from Earth. He already knew the sun is much larger than Earth by observing Earth’s shadow on the moon during a lunar eclipse, and he also knew the sun is much farther away from us than the moon.

Although the optical technology of his time didn’t allow Aristarchus to know the finer details of our Solar System, his deductions were absolutely correct based on what he could actually see. What he lacked in technology, he made up for in deductive genius.

What Aristarchus got Right

23 centuries ago, Aristarchus’s proposed, with evidence, that the earth and the planets orbit the sun. He further deduced that the stars are much farther away than anyone else had imagined, and hence that the universe is much bigger than previously imagined. These were major advances in human ideas about the universe.

What did Copernicus know about Aristarchus’s Work?

Copernicus actually acknowledged in the draft of his own book that Aristarchus might have said the earth moved around the sun. He removed this acknowledgement before he published his work.

In Copernicus’s defense, he was probably unaware of The Sand Reckoner by Archimedes, because, after its rediscovery in the Renaissance, The Sand Reckoner only seems to have existed as a few hand-written copies until it was finally printed in 1544. By then Copernicus had published his own book and had died. What he knew of Aristarchus probably came from the following very brief words written by Aetius:

“Aristarchus counts the sun among the fixed stars he has the earth moving around the ecliptic [orbiting the sun] and therefore by its inclinations he wants the sun to be shadowed.”

Galileo knew that Aristarchus was the First Heliocentrist

Galileo Galilei, who most certainly had read The Sand Reckoner, and understood its message, did not acknowledge Copernicus as the discoverer of the heliocentric Solar System. Instead, he described him as the ‘restorer and confirmer’ of the hypothesis.

Clearly, Galileo reserved the word ‘discoverer’ for Aristarchus of Samos.

Aristarchus lived for about 80 years. If we could have built on his insights, rather than forgetting about them for so many centuries, how much further might we have come in our understanding the universe?

Our Cast of Characters

Aristarchus lived in Ancient Greece. He was born in about 310 BC and died in about 230 BC.
Pythagoras lived in Ancient Greece. He was born in about 570 BC and died in about 495 BC.
Archimedes lived in Ancient Greece. He was born in about 287 BC and died in 212 BC.
Nicolaus Copernicus lived in Poland. He was born 19 February 1473 and died 24 May 1543.
Galileo Galilei lived in Italy. He was born 15 February 1564 and died 8 January 1642.

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Further Reading
Sir Thomas Heath
Aristarchus of Samos: the ancient Copernicus
Oxford at the Clarendon Press, 1913

Lucio Russo
The Forgotten Revolution: How Science Was Born in 300 BC and Why it Had to Be Reborn
Springer, 2004

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This famousscientist website is a gem! Great summaries of the bodies of work that individuals made in their time- a Cliff Notes of intellect! I wish more people could read and understand the import of this website.

Aristarchus of Samos

Aristarchus, the famous ancient astronomer and mathematician born in Samos: Aristarchus (310 BC-230 BC) was a famous Greek mathematician and astronomer, popular for his theories regarding the heliocentrism of our solar system. He was the first to say that the Sun, and not the Earth, was the center of our universe. This theory brought him ridicule during his lifetime.

However, when his works were unearthed and studied around 1800 years later by Copernicus, the rightness of his theory was proven. Although his works were considered inferior to those of Aristotle and Ptolemy, he has made many significant contributions to science.

Aristarchus was born on Samos island. He probably studied in Alexandria, Egypt, under Strato of Lampsacus. His only surviving work is entitled On the Sizes and Distances of the Sun and Moon.

Aristarchus managed to place the Sun in the middle of the solar system and he also placed the planets in the right order from the Sun. He gave a model of the universe with a stationary Sun and planets rotating in circular orbits around the Sun. The stars, which are actually stationary, seemed to be rotating because the Earth rotates on its own axis.

Aristarchus was one of the first astronomers to calculate the relative sizes of the Sun, the Moon and the Earth. He did this by observing the Moon during a lunar eclipse and by estimating the angle and the size of the Earth. He understood that the Sun, the Moon and the Earth form a near right angle during the last and the first quarter of the Moon.

Based on this, he calculated that the Sun was nineteen times further away from Earth than the Moon. However, he made a mistake in his calculations: he took the angle as 87 degrees while the correct angle is 89° 50'. Thus, the actual distance is 390 times and not nineteen times, as proposed by Aristarchus. Although the geometric theory is current, the calculations were wrong due to lack of precise instruments rather than logic.

His theory that the diameters of the Moon and the Sun should be proportional to their distance from the Earth is also logical but gave wrong results. Today that the intelligence of Aristarchus and his contribution to science has been renowned, scientists have given his name to a crater on the Moon.


Thomas W. Africa, “Copernicus’ Relation to Aristarchus and Pythagoras,” in Isis, 52 (1961), 403–409 Angus Armitage, Copernicus, the Founder of Modern Astronomy (London, 1938) John L. E. Dreyer, A History of the Planetary Systems from Thales to Kepler (Cambridge, England, 1906 repr., New York, 1953) Pierre Duhem, Le système du monde, Vols. I-II (Paris, 1954) Sir Thomas Heath, Aristarchus of Samos (Oxford, 1913) and A History of Greek Mathematics, 2 vols. (Oxford, 1921) Otto Neugebauer, “Archimedes and Aristarchus,” in Isis. 39 (1942), 4–6 Giovanni V. Schiaparelli, “Origine del sistema planetario eliocentrico presso i Greci,” in Memorie del’Istituto lombardodi scienze e lettere, 18 (1898), asc. 5 and William H. Stahl, “The Greek Heliocentric Theory and Its Abandonment.” in Transactions of the Amerrican Philological Association, 77 (1945), 321–332.

Theories of Aristarchus - History

We have very little in the form of recorded information on early man's impression of the heavens, mostly some drawings of eclipses, comets, supernovae such as the Pueblo Petrograph (see below). However, early man was clearly frightened/overwhelmed by the sky. One of the earliest recorded astronomical observations is the Nebra sky disk from northern Europe dating approximately 1,600 BC. This 30 cm bronze disk depicts the Sun, a lunar crescent and stars (including the Pleiades star cluster).

The disk is probably a religious symbol as well as a crude astronomical instrument or calendar. In the Western hemisphere, similar understanding of basic stellar and planetary behavior was developing. For example, Native American culture around the same time were leaving rock drawings, or petroglyphs, of astronomical phenomenon. The clearest example is found below, a petroglyph which depicts the 1,006 AD supernova that resulted in the Crab Nebula.

Early man also believed that the heavens held power over earthy existence (psychology of the unknown) which is the origins of the pseudo-science astrology as an attempt to understand, predict and influence events

The earliest written records (i.e. history) were astronomical observations produced by the Babylonians (

1600 B.C.) who recorded positions of planets, times of eclipses, etc. There is also evidence of interest in astronomical phenomenon from early Chinese, Central American and North European cultures such as Stonehenge, which is a big computer for calculating the position of planets and the Sun (i.e. when to have that big blowout Solstice thing)

Thus, Astronomy was the 1st science, as it was the first thing we recorded observations for.

Later in history, 5,000 to 20,000 years ago, humankind begins to organize themselves and develop what we now call culture. A greater sense of permanence in your daily existences leads to the development of culture, where people develop narrative stories for cultural unity which we now call myths.

Most myths maintain supernatural themes, with gods, divine and semi-divine figures, but there was usually an internal logical consistence to the narrative. For example, myths are often attempts at a rational explanation off events in the everyday world, their goal is to teach. Even if we consider some of the stories to be ridiculous, they were, in some sense, our first scientific theories. They also, usually, follow a particular religion, and so this time is characterized by a close marriage of science and religion.

About 1,000 years later, the ancient Greeks inherited astronomical records from the Babylonians and applied the data to construct a cosmological framework. Data was not just used for practical goals, such as navigation, but also to think of new experiments, the origin of what we call natural philosophers.

Of the many natural philosophers before the time of Socrates (the Presocratics) was Thales (

480 B.C.). His combination of math and Babylonian data allowed him to predict eclipses.

Between the cosmological foundation set by the Presocratics and the world of Ideas introduced by Plato was a set of fundamental calculations on the size of the Earth, Moon, Sun and the distances between the nearby planets performed by Eratosthenes and Aristarchus (c. 250 BC). Using some simple geometry, these two natural philosophers were able to, for the first time, place some estimate of the size of the cosmos in Earth terms.

For a long time it was realized that the earth's surface was curved by people familiar with the behavior of incoming and outgoing ships. For it was obvious that as a ship passed over the horizon, the hull disappeared first, then the topmost sailing masts (although one could argue this is an effect of refraction in the atmosphere). Ancient astronomers could see with their eyes that the Sun and the Moon were round. And the shadow of the Earth, cast on the lunar surface during a lunar eclipse, is curved. A sphere is the simplest shape to explain the Earth's shadow (a disk would sometimes display a shadow shaped like a line or oval).

Eratosthenes used a spherical Earth model, and some simple geometry, to calculate its circumference. Eratosthenes knows that on a special day (the summer solstice) at noon in the Egyptian city of Syene, a stick placed in the ground will cast no shadow (i.e., it is parallel to the Sun's rays). A stick in the ground at Alexandria, to the north, will cast a shadow at an angle of 7 degrees. Eratosthenes realizes that the ratio of a complete circle (360 degrees) to 7 degrees is the same as the ratio of the circumference of the Earth to the distance from Alexandria to Swenet. Centuries of surveying by Egyptian pharaohs scribes gave him the distance between the two cities of 4900 stadia, approximately 784 kilometers. This resulting in a circumference of 40,320 kilometers, which is amazingly close to the modern value of 40,030 kilometers. With this calculation, Eratosthenes becomes the father of geography eventually drawing up the first maps of the known world and determining the size of the most fundamental object in the Universe, our own planet.

Hipparchus (100 B.C.) produced first star catalog and recorded the names of constellations.

During the times before the invention of the telescope, there were only seven objects visible to the ancients, the Sun and the Moon, plus the five planets, Mercury, Venus, Mars, Jupiter and Saturn. It was obvious that the planets were not on the celestial sphere since the Moon clearly passes in front of the Sun and planets Mercury and Venus can be seen to transit the Sun (the Sun passes in front of Mars, Jupiter and Saturn). Plato first proposed that the planets followed perfect circular orbits around the Earth (for the circle is the most perfect shape). Later, Heraclides (330 B.C.) developed the first Solar System model, placing the planets in order from the Earth it was is now called the geocentric solar system model and the beginning of the geocentric versus heliocentric debate.

Note that orbits are perfect circles (for philosophical reasons = all things in the Heavens are "perfect")

Slightly later, Aristarchus (270 B.C.) proposed an alternative model of the Solar System placing the Sun at the center with the Earth and the planets in circular orbit around it. The Moon orbits around the Earth. This model became known as the heliocentric model.

Aristarchus was the first to propose a "new" Sun centered cosmology and one of the primary objections to the heliocentric model is that the stars display no parallax (the apparent shift of nearby stars on the sky due to the Earth's motion around the Sun). However, Aristarchus believed that the stars were very distant and, thus, display parallax's that are too small to be seen with the eye (in fact, the first parallax will not by measured until 1838 by Friedrich Bessel). The Sun is like the fixed stars, states Aristarchus, unmoving on a sphere with the Sun at its center. For Aristarchus it was absurd that the "Hearth" of the sky, the Sun, should move and eclipses are easy to explain by the motion of the Moon around the Earth.

Problems for Heliocentric Theory:

While today we know that the Sun is at the center of the solar system, this was not obvious for the technology of the times pre-1500's. In particular, Aristarchus' model was ruled out by the philosophers at the time for three reasons:

  1. Earth in orbit around Sun means that the Earth is in motion. Before the discovery of Newton's law of motion, it was impossible to imagine motion without being able to `feel' it. Clearly, no motion is detected (although trade winds are due to the Earth's rotation).
  2. If the Earth undergoes a circular orbit, then nearby stars would have a parallax. A parallax is an apparent shift in the position of nearby stars relative to distant stars.

Of course, if all the stars are implanted on the crystal celestial sphere, then there is no parallax.

Ptolemy wrote a great treatise on the celestial sphere and the motion of the planets call the Almagest. The Almagest is divided into 13 books, each of which deals with certain astronomical concepts pertaining to stars and to objects in the solar system. It was, no doubt, the encyclopedic nature of the work that made the Almagest so useful to later astronomers and that gave the views contained in it so profound an influence. In essence, it is a synthesis of the results obtained by Greek astronomy it is also the major source of knowledge about the work of Hipparchus, who made a map of the heavens and named the constellations.

The Christian Aristotelian cosmos, engraving from Peter Apian's Cosmographia, 1524

In the first book of the Almagest, Ptolemy describes his geocentric system and gives various arguments to prove that, in its position at the center of the universe, the Earth must be immovable. Not least, he showed that if the Earth moved, as some earlier philosophers had suggested, then certain phenomena should in consequence be observed. In particular, Ptolemy argued that since all bodies fall to the center of the universe, the Earth must be fixed there at the center, otherwise falling objects would not be seen to drop toward the center of the Earth. Again, if the Earth rotated once every 24 hours, a body thrown vertically upward should not fall back to the same place, as it was seen to do. Ptolemy was able to demonstrate, however, that no contrary observations had ever been obtained.

Ptolemy accepted the following order for celestial objects in the solar system: Earth (center), Moon, Mercury, Venus, Sun, Mars, Jupiter, and Saturn. However, when the detailed observations of the planets in the skies is examined, the planets undergo motion which is impossible to explain in the geocentric model, a backward track for the outer planets. This behavior is called retrograde motion.

The solution to retrograde motion was to use a system of circles on circles to explain the orbits of the planets called epicycles and deferents. The main orbit is the deferent, the smaller orbit is the epicycle. Although only one epicycle is shown in the figure below, over 28 were required to explain the actual orbits of the planets.

In the Ptolemaic system, deferents were large circles centered on the Earth, and epicycles were small circles whose centers moved around the circumferences of the deferents. The Sun, Moon, and planets moved around the circumference of their own epicycles. In the movable eccentric, there was one circle this was centered on a point displaced from the Earth, with the planet moving around the circumference. These were mathematically equivalent schemes.

Although Ptolemy realized that the planets were much closer to the Earth than the "fixed" stars, he seems to have believed in the physical existence of crystalline spheres, to which the heavenly bodies were said to be attached. Outside the sphere of the fixed stars, Ptolemy proposed other spheres, ending with the Primum Mobile ("prime mover"), which provided the motive power for the remaining spheres that constituted his conception of the universe. His resulting solar system model looked like the following, although the planets had as many as 28 epicycles (not shown) to account for all the details of their motion.

This model, while complicated, was a complete description of the Solar System that explained, and predicted, the apparent motions of all the planets. The Ptolemic system began the 1st mathematical paradigm or framework for our understanding of Nature.

We know from history that the great library at Alexandria burns in 272 AD, destroying a great deal of the astronomical data for the time. Roman culture collapses and we enter the Dark Ages. But, the Roman Catholic Church absorbs Aristotle's scientific methods and Ptolemy's model into its own doctrine. Thus, preserving the scientific method and Ptolemy's Solar System. Unfortunately, the geocentric model was accepted as doctrine and, therefore, was not subjected to the scientific method for hundreds of years.

Until . the Renaissance, where new ideas were more important than dogma.

Copernicus (1500's) reinvented the heliocentric theory and challenged Church doctrine. Copernicus (c. 1520) was not the first astronomer to challenge the geocentric model of Ptolemy, but he was the first to successfully formulate a heliocentric model and publish his model. He was able to overcome centuries of resistance to the heliocentric model for a series of political and scientific reasons. Politically, the authority of the Church was weakening in Northern Europe in the 15th century allowing more diversity in scientific thinking (although the new Protestant faiths were also not quick to embrace the heliocentric model). Scientifically, a better understanding of motion (particularly inertia) was undermining the whole concept of an unmoving Earth. A rotating Earth is a much simpler explanation for the diurnal motion of stars, an Earth that rotates is only one step away from an Earth that revolves around the Sun. The heliocentric model had a greater impact than simply an improvement to solve retrograde motion. By placing the Sun at the center of the Solar System, Copernicus forced a change in our worldview = paradigm shift or science revolution.

Copernicus began his quest for an improved solar system model with some basic principles. Foremost was the postulate that the Earth was not the center of the Universe, only the center of local gravity and the Moon. Second, the postulate that the Sun was the center of the solar system, all planets revolved around the Sun. In this fashion, retrograde motion is not cause by the planets themselves, but rather by the orbit of the Earth.

While Copernicus includes a rotating Earth in his heliocentric model, he continues to cling to Aristotle's celestial motions, i.e. orbits that are perfect circles (rather than their true shape, an ellipse). This forces Copernicus to adopt a series of moving sphere's for each planet to explain longitude motion. While Copernicus has fewer sphere's, since more of the retrograde motion is accounted for, his system is still extremely complicated in a computational sense. It's two greatest advantages is that it places the inferior planets near the Sun, naturally explaining their lack of large eastern or western elongations, and removing any extreme motions, such as that needed to explain durnal changes.

Copernicus also changes the immovable empyrean heaven into a fixed sphere of stars, severing theology from cosmology. However, Copernicus fails to produce a mechanically simple scheme for astrologers to cast horoscopes or astronomers to produce almanacs, for ultimately the tables he produces are as complicated as Ptolemy's and he did not publish all his results in the final edition of his work, "On the Revolutions of the Heavenly Spheres".

However, Copernicus, like Ptolemy, also used circular orbits and had to resort to epicycles and deferents to explain retrograde motions. In fact, Copernicus was forced to use more epicycles than Ptolemy, i.e. a more complicated system of circles on circles. Thus, Copernicus' model would have failed our modern criteria that a scientific model be as simple as possible (Occam's Razor).

Tycho Brahe (1580's) was astronomy's 1st true observer. He built the Danish Observatory (using sextant's since telescopes had not been invented yet) from which he measured positions of planets and stars to the highest degree of accuracy for that time period (1st modern database). He showed that the Sun was much farther than the Moon from the Earth, using simple trigonometry of the angle between the Moon and the Sun at 1st Quarter.

The Earth's motion, as a simple matter of dynamics, was extremely perplexing to the medieval thinker. The size and mass of the Earth was approximately known since Eratosthenes had measured the circumference of the Earth (thus, the volume is known and one could simply multiple the volume with the mean density of rock to obtain a rough mass estimate). The force required to move the Earth seemed impossible to the average medieval natural philosopher.

Brahe had additional reason to question the motion of the Earth, for his excellent stellar positional observations continued to fail to detect any parallax. This lack of annual parallax implied that the celestial sphere was "immeasurably large". Brahe had also attempted to measure the size of stars, not understanding that the apparent size of a star simply reflects the blurring caused by the passage of starlight through the atmosphere. Brahe's estimate for the size of stars would place them larger than the current day estimate of the size of the Earth's orbit. Such "titanic" stars are absurd according to Brahe's understanding of stars at the time.

Beyond Tycho Brahe's accomplishments in the observational arena, he is also remembered for introducing two compromise solutions to the solar system model now referred to as the geoheliocentric models. Brahe was strongly influenced by the idea of Mercury and Venus revolving around the Sun to explain the fact that their apparent motion across the sky never takes them more than a few tens of degrees from the Sun (called their greatest elongation). The behavior of inner worlds differs from the orbital behavior of the outer planets, which can be found at any place on the elliptic during their orbital cycle.

Brahe proposed a hybrid solutions to the geocentric model which preserves the geocentric nature of the Earth at the center of the Universe, but placed the inner planets (Mercury and Venus) in orbit around the Sun. This configuration resolves the problem of Mercury and Venus lack of large angular distances from the Sun, but saves the key criticism of the heliocentric model, that the Earth is in motion. In other works, Brahe's geoheliocentric model fit the available data but followed the philosophical intuition of a non-moving Earth.

Neither successfully predicts the motion of the planets. The solution will be discovered by a student of Tycho's, who finally resolves the heliocentric cosmology with the use of elliptical orbits.

Kepler (1600's) a student of Tycho who used Brahe's database to formulate the Laws of Planetary Motion which corrects the problems of epicycles in the heliocentric theory by using ellipses instead of circles for orbits of the planets.

This is a key mathematical formulation because the reason Copernicus' heliocentric model has to use epicycles is due to the fact that he assumed perfectly circular orbits. With the use of ellipses, the heliocentric model eliminates the need for epicycles and deferents. The orbital motion of a planet is completely described by six elements: the semi-major axis, the eccentricity, the inclination, the longitude of the ascending node, the argument of the perihelion and the time of the perihelion.

The formulation of a highly accurate system of determining the motions of all the planets marks the beginning of the clockwork Universe concept, and another paradigm shift in our philosophy of science.

Kepler's laws are a mathematical formulation of the solar system. But, is the solar system `really' composed of elliptical orbits, or is this just a computational trick and the `real' solar system is geocentric. Of course, the answer to questions of this nature is observation.

The pioneer of astronomical observation in a modern context is Galileo. Galileo (1620's) developed laws of motion (natural versus forced motion, rest versus uniform motion). Then, with a small refracting telescope (3-inches), destroyed the the idea of a "perfect", geocentric Universe with the following 5 discoveries:

mountains and "seas" (maria) on the Moon

Milky Way is made of lots of stars

These first three are more of an aesthetic nature. Plato requires a `perfect' Universe. Spots, craters and a broken Milky Way are all features of imperfection and at odds with Plato's ideas on purely philosophical grounds. However, the laws of motion are as pure as Plato's celestial sphere, but clearly are not easy to apply in the world of friction and air currents etc. So these observations, by themselves, are not fatal to the geocentric theory. The next two are fatal and can only be explained by a heliocentric model.

Jupiter has moons (Galilean moons: Io, Europa, Callisto, Ganymede)

Notice that planets with phases are possible in a geocentric model. But for a planet to change in apparent size with its phases, like Venus is impossible if the planet orbits the same distance from the Earth. And, lastly, if all bodies orbit around the Earth, then the moons of Jupiter, which clearly orbit around that planet, are definitive proof that the geocentric model is wrong.

Newton (1680's) developed the law of Universal Gravitation, laws of accelerated motion, invented calculus (math tool), the 1st reflecting telescope and theory of light.

. off to the 18-20th century, with discovery of the outer planets and where astronomy moves towards discoveries in stellar and galactic areas, next paradigm shift occurs in early 1960's with NASA deep space probes

Facts about Aristarchus 3: The Sand Reckoner

The Sand Reckoner was a book written by Archimedes. This book attempted to describe the work of Aristarchus. He believed that the heliocentric model of Aristarchus could be the alternative for geocentrism.

Facts about Aristarchus 4: the rejection

There were many contemporaries of Aristarchus rejected his heliocentric view. The proof of other people’s rejection can be seen in Plutarch’s On the Apparent Face in the Orb of the Moon. Plutacrh reported that Aristarchus jokingly told his contemporary named Cleanthes to charge of impiety since he was the opponent of heliocentric model and a worshiper of the sun.

A Very Brief History of Heliocentric Theory

300 BC Greek Philosophers Plato and Aristotle models Geocentric Theory with Earth as a Sphere. Aristotle publishes in his book “On the Heavens”.

200 BC Greek Aristarchus of Samos placed Earth and other planets in motion around the central Sun but rejected by Aristotleans.

140 CE Cladius Ptolemy of Alexandria devised complex system of “epicycles” to account for retrograde (going backwards) motion of the planets. Published his theories in book called “Almagest”

1270 Roman Catholic church adopts priest Thomas Aquinas theory of a “God-ordained and man-centered” universe which declared the glory of God.

1453 Guttenberg Printing Press developed

1543 Nicholas Copernicus publishes “On the Revolution of Heavenly Spheres” in his last year of his life which postulates a heliocentric, Sun centered, solar system where Earth and planets are revolving around the Sun. “For who would place this lamp of a very beautiful temple in another or better place than this, wherefrom it can illuminate everything at the same time?”

1580 Tycho Brahe, A Danish Astronomer, claimed the most accurate measurement of planet and stars yet still was uncertain of a heliocentric or geocentric model. Was first to suggest a non-circular orbit of planets.

1582 Gregorian Calendar replaces Julian Calendar by Roman Catholic Church by Pope Gregory.

1609 Galileo Galilei grinds his own glass and makes telescope. Observes Venus moons going around Venus “proving” the Heliocentrism. Publishes his work in Italian so all laymen can read instead of scholar Roman Latin. Recants at Roman Inquisition and banished to _____.

1619 Johannes Kepler, German Astronomer student of Tycho Brahe “proved” Heliocentric theory by identifying planet orbits as elliptical and not circular

1687 Sir Isaac Newton, English Astronomer and President of the Royal Society of England, stated “Law of Universal Gravitation” which mathematically showed the force that kept the Earth and planets going around the Sun as well as what kept the oceans in and air from flying away. Wrote the book “Principia Mathematica.

1758 Edmund Halley successfully predicted, using Newton equations, the return of a comet last seen

1822 Congregation of the Holy Office remove heliocentric books from the Vatican banned book list.

1838 Friedrich Bessel measures “Stellar Parrallax” method to measure first distance of star, 61 Cygni.

in 1915, Albert Einstein (1879 – 1955) published the general theory of relativity, in which gravity is not a force but it is a consequence of the curvature of space-time.

From as far back as Man has records until the Age of Enlightenment, science and philosophy taught the the Earth was the center of the Universe with the exception of Aristarchus (310-230 BC) who was said to be the first to propose a sun-centered universe.

Plato and Aristotle (300 BC) postulated that the Earth was a sphere but still a geocentric, earth centered universe.

Claudius Ptolemy (85-165 AD) of Alexandria devised a complex system of “epicycles” to account for planets that appeared to go backward, or in retrograde.

The Bible taught that the Earth had four corners and was flat, a plane. Thomas Aquinas (1225-1274), as the Roman Catholic Church was coming to power, stated famously the heavens were “God-ordained and man-centered”.

“For who would place this lamp of a very beautiful temple in another or better place than this, wherefrom it can illuminate everything at the same time?” Copernicus defending his heliocentric theory.

Nicholas Copernicus (1473-1543), a priest at the University of Bologna realized that the rising and setting of the Sun, Moon, and stars could be accounted for by a daily revolution of the Earth. Also, he found that if he put the Sun at the center of the planet’s orbits he could simplify the number of epicycles from 80 in Ptolemy’s system to a mere 34.

Although epicycles do not exist, Copernicus was the first to set to prove that the earth, and all planets, rotated around a stable sun not earth. His idea that the Earth and planets orbited about the sun became know as the “heliocentric theory.” He wrote about it in his book “De Revolutionibus, ” which translates to “Concerning the Revolutions.”

However, for his contemporaries, the ideas presented by Copernicus were not markedly easier to use than the geocentric theory and did not produce more accurate predictions of planetary positions. Copernicus was aware of this and could not present any observational “proof”, relying instead on arguments about what would be a more complete and elegant system.

Tycho Brahe (1546-1601) was a Danish astronomer who made measurements of the planet and stars. His measurements were the most accurate that had yet been made. Tycho began his observations in Denmark but later moved to Prague to continue his work.

Tycho proposed a system in which all of the planets except for Earth orbited about the Sun. He claimed that the Sun still orbited about the Earth. As an astronomer, Tycho worked to combine what he saw as the geometrical benefits of the Copernican system with the philosophical benefits of the Ptolemaic system into his own model of the universe, the Tychonic system. Furthermore, he was the last of the major naked eye astronomers, working without telescopes for his observations.

After disagreements with the new Danish king Christian IV in 1597, he was invited by the Bohemian king and Holy Roman emperor Rudolph II to Prague, where he became the official imperial astronomer. He built the new observatory at Benátky nad Jizerou.

There, from 1600 until his death in 1601, he was assisted by Johannes Kepler who later used Tycho’s astronomical data to develop his three laws of planetary motion working without telescopes for his observations.

• Tycho was the first to suggest a non-circular orb it for a celestial body (a comet).
• Used calibrated and bigger instruments, new techniques to measure angles (similar to a sextant).
• Built an observatory (remember – no telescopes yet) and made accurate and continuous measurements for 20 years. His measurements helped to prove that planets orbited the sun.
• Accurate map of the stars with 777 stars.
• Measured length of the year to within 1 second.
• Was still unable to choose between the geocentric and heliocentric model. He had his own model with the Earth at the center, orbited by the sun and the moon, with planets orbiting the sun. Never worked out the mathematical details, and his model was never accepted.

Tycho’s observations of stellar and planetary positions were noteworthy both for their accuracy and quantity. His celestial positions were much more accurate than those of any predecessor or contemporary.

Interestingly, before the breakthrough by Galileo of direct observation the Roman Catholic Church led by Pope Gregory adopted the Solar Calendar in 1582. The Gregorian calendar, also called the Western calendar and the Christian calendar, is internationally the most widely used civil calendar.

The calendar was a refinement in 1582 to the Julian calendar amounting to a 0.002% correction in the length of the year.

The Gregorian reform contained two parts: a reform of the Julian calendar as used prior to Pope Gregory XIII’s time and a reform of the lunar cycle used by the Catholic Church, with the Julian calendar, to calculate the date of Easter. The reform was a modification of a proposal made by Aloysius Lilius.

His proposal included reducing the number of leap years in four centuries from 100 to 97, by making 3 out of 4 centurial years common instead of leap years. Lilius also produced an original and practical scheme for adjusting the epacts of the moon when calculating the annual date of Easter, solving a long-standing obstacle to calendar reform.

The Gregorian reform modified the Julian calendar’s scheme of leap years as follows:
Every year that is exactly divisible by four is a leap year, except for years that are exactly divisible by 100, but these centurial years are leap years if they are exactly divisible by 400. For example, the years 1700, 1800, and 1900 are not leap years, but the year 2000 is.

In addition to the change in the mean length of the calendar year from 365.25 days (365 days 6 hours) to 365.2425 days (365 days 5 hours 49 minutes 12 seconds), a reduction of 10 minutes 48 seconds per year, the Gregorian calendar reform also dealt with the accumulated difference between these lengths. (Source)

“Though the implications of the new science were not worked out immediately, it began to be suspected that if the theories were true, man had lost his birthright as the creature for whose sake all else existed, and had been reduced to the position of a puny and local spectator of infinite forces unresponsive to his wishes and unmindful of his purposes.” Preserved Smith

In the same era, Italian Galileo Galilei (1564-1642) saw a crude magnifying looking glass at a circus coming through town and got the idea to make a telescope. He learned to grind his own glass and made the first telescope to peer into the heavens.

He used the newly-invented telescope to make his own observations. He studied mountains and craters on the Earth’s moon, the phases of Venus, and the moons of Jupiter. Particularly he noted that Venus at times appears to be a crescent, just as the Earth’s moon does. All of these findings supported Copernicus’ heliocentric theory.

In 1610, Galileo observed that Venus has a full set of phases like the phases of the Moon. It was contradictory to geocentric model where Venus should not have a full lit from the perspective of the Earth.

Actually, Venus phases are result of the orbit of Venus around the Sun inside of Earth orbit so here we see he was incorrect in his theory. (That way, when Venus is between Sun and Earth, it is full shadowy. Then Venus is partially illuminated when it moves in its orbit until it becomes fully lit when it is on the opposite side of the Earth orbit. In sequence, the shadowy is covering Venus when it moving from the opposite side of Earth orbit to the position between Sun and Earth. Thus, Venus has a complete set of phase when complete its orbit around the Sun.)

In the same year, Galileo observed with his telescope four objects moving near the planet Jupiter. After analyzing data of their full period of moving, he concluded that actually these four objects are orbiting the Jupiter as moons. This was unacceptable by the geocentric model where all celestial body should just orbit around a stationary earth.

Galileo’s significance of what he saw:

  • Cast doubt on the view of the “perfection of the heavens” (of Aristotle and Plato)
  • Showed deficiencies of the geocentric (Ptolemaic) model
  • Rotation of sunspots around sun suggested that if the sun could rotate, perhaps the Earth could too.
  • Phases of Venus would be a natural consequence of the heliocentric model.
  • Jupiter’s moons showed that centers of motion other than Earth existed.

Galileo wrote about his observations and thus angered the Roman Catholic Church. The Church eventually placed him under house arrest. The Inquisition was the tribunal of the Roman Catholic Church at this time. The Inquisition made Galileo kneel before them and confess that the heliocentric theory was false.

Interestingly, Galileo (why do we use his first name?) published in the commoner language of Italian, not Roman Catholic scholarly Latin as to be widely disseminated by the commoner now that books were coming available to the more thanks to book publishing technology.

Johannes Kepler (1571 – 1630) was a German astronomer. Kepler was invited to live in Prague by Tycho Brahe. Tycho died a year after Kepler’s arrival. Kepler inherited a wealth of astronomical data from Tycho. In 1594 Kepler accepted an appointment as professor of mathematics at the Protestant seminary in Graz (in the Austrian province of Styria). He was also appointed district mathematician and calendar maker.

Kepler used this data to draw conclusions about the orbits of the planets.

  1. Why are there only 6 planets?
  2. How are their orbital periods related to their distance from the sun?

After trying many geometric curves and solids in Copernicus’s heliocentric model to match earlier observations of planetary positions, Kepler found that the model would match the observed planetary positions if the Sun is placed at one focus of elliptical planetary obits. This is Kepler’s First Law of Planetary Motion. Kepler’s three laws of planetary motion allow accurate matches and predictions of planetary positions.

Kepler hypothesized that a physical force moved the planets, and that the force diminished with distance. Planets closer to the sun feel a stronger force and move faster. The concept of a physical force was a monumental step. Kepler was on the verge of assigning physical causes to celestial motions.

Kepler later determined that the orbits were not circular but elliptical.

  1. PERIHELION = where a planet is closest to the sun
  2. APHELION = where a planet is farthest from the sun
  1. “The orbit of every planet is an ellipse with the sun at a focus.”
  2. “A line joining a planet and the sun sweeps out equal areas during equal intervals of time.”
  3. “The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.”

“Nature and nature’s laws lay hid in night God said “Let Newton be” and all was light. ” Alexander Pope

Sir Isaac Newton (1642 – 1727) lived in England. He was an English physicist and mathematician (described in his own day as a “natural philosopher”) who is widely recognised as one of the most influential scientists of all time and as a key figure in the scientific revolution. His book Philosophiæ Naturalis Principia Mathematica (“Mathematical Principles of Natural Philosophy”), first published in 1687, laid the foundations for classical mechanics.

In 1666, Newton observed that the spectrum of colours exiting a prism in the position of minimum deviation is oblong, even when the light ray entering the prism is circular, which is to say, the prism refracts different colours by different angles. This led him to conclude that colour is a property intrinsic to light—a point which had been debated in prior years. Replica of Newton’s second Reflecting telescope that he presented to the Royal Society in 1672.

He also showed that coloured light does not change its properties by separating out a coloured beam and shining it on various objects. Newton noted that regardless of whether it was reflected, scattered, or transmitted, it remained the same colour. Thus, he observed that colour is the result of objects interacting with already-coloured light rather than objects generating the colour themselves. This is known as Newton’s theory of colour

Newton moved to London to take up the post of warden of the Royal Mint in 1696, a position that he had obtained through the patronage of Charles Montagu, 1st Earl of Halifax, then Chancellor of the Exchequer. He took charge of England’s great recoining, somewhat treading on the toes of Lord Lucas, Governor of the Tower (and securing the job of deputy comptroller of the temporary Chester branch for Edmond Halley). Newton became perhaps the best-known Master of the Mint upon the death of Thomas Neale in 1699, a position Newton held for the last 30 years of his life.

These appointments were intended as sinecures, but Newton took them seriously, retiring from his Cambridge duties in 1701, and exercising his power to reform the currency and punish clippers and counterfeiters. As Warden, and afterwards Master, of the Royal Mint, Newton estimated that 20 percent of the coins taken in during the Great Recoinage of 1696 were counterfeit. Counterfeiting was high treason, punishable by the felon’s being hanged, drawn and quartered.

Newton was made President of the Royal Society in 1703 and an associate of the French Académie des Sciences. In his position at the Royal Society, Newton made an enemy of John Flamsteed, the Astronomer Royal, by prematurely publishing Flamsteed’s Historia Coelestis Britannica, which Newton had used in his studies.

In April 1705, Queen Anne knighted Newton during a royal visit to Trinity College, Cambridge. Newton was the second scientist to be knighted, after Sir Francis Bacon.

Newton was one of many people who lost heavily when the South Sea Company collapsed. Their most significant trade was slaves, and according to his niece, he lost around £20,000. Newton died in his sleep in London on 20 March 1727 and was buried in Westminster Abbey.

The mathematician Joseph-Louis Lagrange often said that Newton was the greatest genius who ever lived, and once added that Newton was also “the most fortunate, for we cannot find more than once a system of the world to establish.

” I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me” Sir I. Newton

Newton’s Law of Gravity and Motion

Newton derived the law of gravitation between two masses. Since the Sun was the most massive object in the planetary system, all of the planets would naturally be attracted to it and revolve around it, in the same manner as the Moon revolves around the Earth.

Based on Galileo’s and Kepler’s works, Newton published “Principia” in 1687. In this book, Newton posed the theory of Gravity, in which the force that makes planets to move around the Sun is the same force that makes object to fall in the Earth: force of gravity.

In his theory, Newton deduced gravity is a force of mutual interaction of body with mass and this force is inversely proportional to the square of the distance between objects.

The heliocentric model was established by Newton but there were some question about the gravity, for example, its action at a distance and immediately action. Even Newton had doubts about the gravity action at a distance. How can massive objects attract each other at distance without mediation of anything? And how can attraction force between them be immediately without a time to action?

Newton eventually wrote about gravitation and the heliocentric theory in Principia Mathematica in 1687, at the prompting of another famous astronomer, Edmund Halley (1656-1742). Halley used Newton’s equations to predict that a comet seen in 1682 would return in 1758. The return of Halley’s comet gave final proof to the heliocentric theory and is now known as “Halley’s Comet”.

Final “proof”, according to the heliocentric theory for the solar system came in 1838, when F.W. Bessel (1784-1846) determined the first firm trigonometric parallax for the two stars of 61 Cygni (Gliese 820). Their parallax (difference in apparent direction of an object as seen from two different points) ellipses were consistent with orbital motion of Earth around the Sun.

Bessel was a German astronomer, mathematician. He was the first astronomer who determined reliable values for the distance from the sun to another star by the method of parallax. Although he left school at the age of 14, he was appointed in January 1810 as director of the Königsberg Observatory by King Frederick William III of Prussia. Bessel won the Gold Medal of the Royal Astronomical Society in 1829 and 1841.

in 1915, Albert Einstein (1879 – 1955) published the general theory of relativity, in which gravity is not a force but it is a consequence of the curvature of space-time. Thus, massive body creates a curve in the space-time then inertial trajectory that was straight lines became curved. These inertial trajectories are called geodesics.

An object can inertial follow a geodesic without an interaction of forces. As consequence, heavy objects create a “big” curvature on space-time that makes other object fall towards them by a geodesic. If an object has extremely big mass even the light will suffer a noticeable deflection. This object is called of black hole.

Einstein’s master insight was that the constant, familiar pull of the Earth’s gravitational field is fundamentally the same as these fictitious forces.

The apparent magnitude of the fictitious forces always appears to be proportional to the mass of any object on which they act – for instance, the driver’s seat exerts just enough force to accelerate the driver at the same rate as the car. By analogy, Einstein proposed that an object in a gravitational field should feel a gravitational force proportional to its mass, as embodied in Newton’s law of gravitation.

President Dwight D. Eisenhower established the National Aeronautics and Space Administration (NASA) in 1958. Former SS Nazi, Wernher Magnus Maximilian, Freiherr Von Braun (1912-1977) led NASA’s rocketeers.

Von Braun was already the central figure in the Nazis’ rocket development program, responsible for the design and realization of the V-2 rocket during World War II. His was said to be directly responsible for the killing of prisoners by hanging at the Peenemeunde Rocket Facility he commanded during WWII as well as the death of tens of thousands in England and France from Nazi V-1 and V-2 rockets. (Source)

After the war, he and selected members of his rocket team were taken to the United States as part of the secret Operation Paperclip.

Braun worked on the United States Army’s intermediate-range ballistic missile (IRBM) program before his group was assimilated by NASA. Under NASA, he served as director of the newly formed Marshall Space Flight Center and as the chief architect of the Saturn V launch vehicle, the superbooster that propelled the Apollo spacecraft to the Moon.

According to one NASA source, he is “without doubt, the greatest rocket scientist in history”. In 1975 he received the National Medal of Science. (Source).

Additionally, C. Fred Kleinknecht, head of NASA at the time of the Apollo Space Program, is now the Sovereign Grand Commander of the Council of the 33rd Degree of the Ancient and Accepted Scottish Rite of Freemasonry of the Southern Jurisdiction.

In 1992, the Roman Catholic Church finally repealed the ruling of the Inquisition against Galileo. The Church gave a pardon to Galileo and admitted that the heliocentric theory was correct. This pardon came 350 years after Galileo’s death.

Aristarchus- The First mind to depict The Heliocentric Model

If you browse “who proposed the heliocentric model first” the most presumable answer will be Nicolaus Copernicus, who published a book on heliocentrism from his death bed in 1543 afraid of the mass opposition from people who believed in Geocentrism. But was he the first?

Even before the mathematician, Claudius Ptolemy supported the idea of geocentrism, that survived for 1500 years, there lived a Greek astronomer and mathematician named Aristarchus of Samos (c.310 – c.230 BC).

Aristarchus of Samos is credited as the first person to present the Heliocentric model. The book where he depicted his idea was among the vast marvel collection of the Great Library of Alexandria but it didn’t survive its destruction. The only main reference of Aristarchus’ work is Archimedes’ book named ‘The Sand Reckoner’. But, the true essence of his first-hand understandings is lost in the obscured pages of history.

When Plato and Aristotle emphasized on the geocentric model a generation ago, Aristarchus wasn’t appealing to the thought. Probably inspired by the views of Philolaus of Croton, he believed in a Sun-centered Universe and went on to determine the distance from the Earth to the Moon and Sun with the primitive instruments of the time and geometry.

He knew that at the first and last quarter moons, a right triangle would be created between the three celestial bodies and he observed the angle between the Sun and Moon to be 87°(the actual angle was about 89°50′). Further, he calculated the Sun to be 18-20 times (400 times in actual) farther than the Moon and about 6.3-7.2 times (109 times in actual) larger than the Earth.

Even though highly inaccurate, his predictions were truly an astounding achievement for an astronomer living 2300 years ago. Aristarchus’ model went unnoticed as geocentrism ascended to its peak during the time and had to wait another 2000 years until Copernicus. Also, Aristarchus was the first person to place the planets in order from the Sun and suspected all the other stars to be far-away Suns.

Aristarchus was a right man at the right place in the wrong time and a pure genius the world wasn’t ready for.

Historical Astronomy: Ancient Greeks: Aristarchus

  • Relative distance to moon and sun.
  • Relative sizes of earth, moon and sun.
  • Heliocentric theory.

Born in Samos, not a lot is known about Aristarchus. Most of his work is lost, and we only know about him because other ancient Greek people talked about him.

Only one book of Aristarchus survives, "On the Sizes and Distances of the Sun and Moon." In it he proves:

  • The distance to the sun is greater than 18, but less than 20, times the distance to the moon.
  • The radius of the sun is greater than 18, but less than 20, times greater than the radius of the moon.
  • The radius of the sun is greater than 19/3 (6.3), but less than 43/6 (7.2), times the radius of the earth.

While the results are off, his basic geometry and methods are sound. (Actually, the sun is about 400 times farther than the moon, and about 109 times bigger than the earth.)

Aristarchus' method for determining the relative distances to the moon and the sun is pretty easy to understand. Imagine drawing a triangle by connecting the centers of earth, moon and sun, as in the diagram below.

When the moon is "exactly" 1/2 full, and looks like a semicircle, then the angle earth-moon-sun is 90, so that the distance between the earth and the sun is the hypoteneuse of the right triangle. One just has to measure the angle theta in the diagram, and we can say that the ratio of the distance to the moon to the distance to the sun is equal to the cosine of theta. Aristarchus said that the angle theta was 87, which is too small. It turns out that the angle would be just under 90. In practice, it is also difficult to accurately decide when the moon is exactly half full, and so difficult to accurately measure the angle, so while the method is correct, it turns out to be difficult to do.

Aristarchus notes that the angular size of the moon and sun are the same, which is basically true. Because of this, if the sun is about 19 times farther away than the moon, then it must be about 19 times larger.

Having calculated how much farther away the sun is than the moon, Aristarchus is then able to calculate how much bigger the sun is than the earth. To do this, he notes that during a lunar eclipse, when the moon enters the shadow of the earth, the size of the shadow is about twice the size of the moon. (Again, his data is a little off: it is closer to 3 times the size of the moon.) The image below show the moon, earth and sun during a lunar eclipse.

The image below is the one above, with some triangles highlighted.

Lastly, the image below is the triangles from above, but drawn larger and labeled.

Ds = Distance to the sun Dm = Distance to the moon and D = Distance from earth to apex of its shadow.
Rs = Radius of sun Re = Radius of the earth and R = Radius of the earth's shadow at the moon's position.

Using the above diagram, we can make a couple approximations, and then use some geometry and algebra to find the relative sizes of the earth and sun. First, notice that the two triangles with the dotted-line bases are similar, so that we can say:

Since we know that Dm/Ds = Rm/Rs, we can rewrite the equation above as:

Factoring out Re/Rs from the right side:

Finally rearranging we get the ratio of the radii of the sun to the earth:

So we find that the ratio of the sun's radius to the earth's radius depends on two other ratios: the size of the sun to the moon and the size of the eclipse shadow to the moon. From earlier, Aristarchus had already found that the sun was about 19 times farther away than the moon. Because the moon and sun are the same angular size in the sky, the sun must therefor be about 19 times bigger than the moon. Aristarchus had also said that the size of earth's shadow was twice the size of the moon during a lunar eclipse. So we plug in 19 and 2 for those ratios to get:

So we end up with the sun being about 7 times bigger than the earth. If we use "correct" values for those ratios, the sun is 400 times the size of the moon, and the the average eclipse shadow is about 3 times the size of the moon, which makes Rs/Re about 100 which isn't too far from the actual value of 109. In addition, since Aristarchus knew the angular size of the sun, and now the size of the sun, he can calculate how far away the sun is.

Aristarchus can then find the size and distance for the moon. Since the moon will also be 19 times smaller and closer than the sun, we know that the moon is therefor about 7/19 times the size of the earth, or about 1/3 the size of the the earth. (The correct value is about 1/4.) And lastly, knowing the angular size of the moon and the actual size of the moon, we can calculate the actual distance to the moon. Using the actual angular size of 1/2, and calling "R" the radius of the moon and "D" the distance to the moon, we can use a little trig to see that the moon is almost 240 moon radii away, which means that it is almost 60 earth radii away.

It should be noted that Aristarchus didn't use degrees or trigonometry as neither had been invented yet. His basic geometry and methods are valid, but for some reason his claims on some of the measurements are way off. Archimedes also states that Aristarchus had actually measured the angular sizes of the sun and moon to be 1/2, which is correct. Historians tend to think Aristarchus wrote "On the Sizes and Distances of the Sun and Moon" early on in his career, before he made more accurate measurements.

Aristarchus is also the first person to propose a heliocentric theory, though none of the actual details survive. In "The Sand Reckoner," Archimedes says:

He is basically saying that the universe is a lot bigger than everyone else is proposing at the time, and that the stars are infinitely (or at least immeasurably) far away. This way there wouldn't be any measurable stellar parallax. No one really buys into his theory, though.

Watch the video: Aristarchus: The Greek Copernicus. Ancient Greece Revisited


  1. Akil

    It will not go to him in vain.

  2. Gauvain

    Quick answer, a sign of intelligence :)

  3. Orson

    I think this is a brilliant thought.

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