According to Plutarch in his work On exile, Anaxagoras is the first Greek to attempt the problem of squaring the circle, a problem he worked on while in prison.He is considered to be both the geographical and theoretical successor to the earliest Ionian philosophers, particularly Anaximenes. Eventually, Anaxagoras made his way to Athens and he is often credited with making her the home of Western philosophical and physical speculation.

Anaxagoras (/ˌænækˈsæɡərəs/; Greek: Ἀναξαγόρας, Anaxagóras, “lord of the assembly”; c. 500 – c. 428 BC) was a Pre-Socratic Greek philosopher. Born in Clazomenae at a time when parts of IONIA(= “Asia Minor”) came under the control of the Persian Empire, untill when later Alexander the Great liberated and became Greek again Anaxagoras came to Athens.Although Anaxagoras lived in Athens when Socrates was a youth and young adult, there are no reports that Anaxagoras and Socrates ever met.

About 480 Anaxagoras moved to Athens, then becoming the centre of Greek culture, and brought from Ionia the new practice of philosophy and the spirit of scientific inquiry. He was a philosopher of nature remembered for his cosmology and for his discovery of the true cause of eclipses. He was associated with the Athenian statesman Pericles. Although Anaxagoras proposed theories on a variety of subjects, he is most noted for two theories. First, he speculated that in the physical world everything contains a portion of everything else. His observation of how nutrition works in animals led him to conclude that in order for the food an animal eats to turn into bone, hair, flesh, and so forth, it must already contain all of those constituents within it. The second theory of significance is Anaxagoras’ postulation of Mind (Nous) as the initiating and governing principle of the cosmos.

After 30 years’ residence in Athens, he was prosecuted on a charge of impiety for asserting that the Sun is an incandescent stone somewhat larger than the region of the Peloponnese. The attack on him was intended as an indirect blow at Pericles, and, although Pericles managed to save him, Anaxagoras was compelled to leave Athens. He spent his last years in retirement at Lampsacus.

Thales of Miletus was a Greek mathematician who lived 2600 years ago. He formulated a theorem with the following meaning: Every triangle inscribed in a circle has a right angle if one of the sides of the triangle is the diameter of the circle. One easy way to construct a right-angled triangle, is to use Thales’ theorem. Thales theorem states that if one of the sides of a triangle is along the diameter of a circle, and if the third vertex also lies on the circle, then the angle at the third vertex is a right angle.The triangle A B C is inscribed in a circle in such a way.

The Pythagorean theorem

For a right-angled triangle with shorter sides a and b, and the hypotenuse c, following holds:

c2=a2+b2�2=�2+�2

Conversely, if three positive numbers a, b, c satisfy c2=a2+b2�2=�2+�2; then the numbers can be the lengths of the sides of a right-angled triangle.

There are many proofs of the Pythagorean theorem that are based on interpreting the square of a number as the area of a square. You then prove that the area of the two smaller squares in the image below, have the same total area as the large square.

One of the easiest proofs is shown in the worksheet above. A more difficult proof where the smaller squares are cut and then put together as a larger square is shown below.

Exercise 1

Use the picture above to prove Thales’ theorem!

Dynamic Pythagoras’ tree

By using Thales theorem, it is possible to make a dynamic variant of a fractal called Pythagoras tree.

Exercise 2

Input two points A� and B� and a slider α� representing an angle. Use the tool Regular Polygon to make a square as in the picture above.

Use the tool Midpoint or Centre to make the midpoint E� between C� and D�. Usethe tool Rotate, to rotate C� around E� by the angle α�. Move the slider to see the point C′�′ move. Make a triangle as in the picture above. The triangle will be a right-angled triangle due to Thales’ theorem.

Pick Tools -> Create New Tool. Make a tool that has the objects C�, C′�′, D�, and the two polygons as output objects. The tool will have A�, B�, and α� as input objects.

Build a tree by using the tool. Start by clicking two points, then enter α in the input box that shows up.

Much of our modern science, and astronomy in particular, has roots in the ancient world. In particular, the Greek philosophers studied the cosmos and tried to use the language of mathematics to explain everything. The Greek philosopher Thales was one such man. He was born around 624 BCE,from (Miletus was a Greek island in Ionia Asia Minor, now modern Turkey) and he came from a distinguished family.

It is difficult to write about Thales since none of his own writing survives. He was known to be a prolific writer, but as with so many documents from the ancient world, his vanished through the ages. He is mentioned in other people’s works and seems to have been quite well-known for his time among fellow philosophers and writers. Thales was an engineer, scientist, mathematician, and philosopher interested in nature. He may have been the teacher of Anaximander (611 BC – 545 BCE), another philosopher.

Some researchers think Thales wrote a book on navigation, but there is little evidence of such a tome. In fact, if he wrote any works at all, they did not even survive until the time of Aristotle (384 BCE- 322 BCE). Even though the existence of his book is debatable, it turns out that Thales probably did define the constellation Ursa Minor.

Despite the fact that much of what is known about Thales is mostly hearsay, he was definitely well-respected in ancient Greece. He was the only philosopher before Socrates to be counted among the Seven Sages. These were philosophers in the 6th century BCE who were statesmen and law-givers, and in Thales’s case, a natural philosopher (scientist).

There are reports that Thales predicted an eclipse of the Sun in 585 BCE. While the 19-year cycle for lunar eclipses was well known by this time, solar eclipses were harder to predict, since they were visible from different locations on Earth and people were not aware of the orbital motions of the Sun, Moon, and Earth that contributed to solar eclipses. Most likely, if he did make such a prediction, it was a lucky guess based on experience saying that another eclipse was due.

After the eclipse on 28 May, 585 BCE, Herodotus wrote, “Day was all of a sudden changed into night. This event had been foretold by Thales, the Milesian, who forewarned the Ionians of it, fixing for it the very year in which it took place. The Medes and Lydians, when they observed the change, ceased fighting, and were alike anxious to have terms of peace agreed on.”

Impressive but Human

Thales is often credited with some impressive work with geometry. It is said he determined the heights of pyramids by measuring their shadows and could deduce the distances of ships from a vantage point onshore.

How much of our knowledge of Thales is accurate is anyone’s guess. Most of what we know is due to Aristotle who wrote in his Metaphysics: “Thales of Miletus taught that ‘all things are water’.” Apparently Thales believed the Earth floated in water and everything came from water.

Like the absent-minded professor stereotype still popular today, Thales has been described in both glowing and derogatory tales. One story, told by Aristotle, says Thales used his skills to predict that the next season’s olive crop would be bountiful. He then purchased all the olive presses and made a fortune when the prediction came true. Plato, on the other hand, told a story of how one night Thales was gazing at the sky as he walked and fell into a ditch. There was a pretty servant girl nearby who came to his rescue, who then said to him “How do you expect to understand what is going on up in the sky if you do not even see what is at your feet?”

Era Pre-Socratic philosophy Region Western philosophy

School Atomism

Main interests Nature

Notable ideas Atoms and the void as the fundamental constituents of the physical world

Democritus (/dɪˈmɒkrɪtəs/, dim-OCK-rit-əs; Greek: Δημόκριτος, Dēmókritos, meaning “chosen of the people”; c. 460 – c. 370 BC) was an Ancient Greek pre-Socratic philosopher from Abdera, primarily remembered today for his formulation of an atomic theory of the universe.

FEW of Democritus’ original work has survived, MOST references. Many of these references come from ARISTOTELES, who viewed him as an important rival in the field of natural philosophy

Atomic hypothesis See also: Atomism We have various quotes from Democritus on atoms, one of them being:

Δημόκριτος, DEMOKRITOS, meaning “chosen of the people”; c. 460 – c. 370 BC) was an ANCIENT-GREEK PRE-SOCRATIC philosopher from THRACE/ABDERA, primarily remembered today for HIS FORMULATION of an atomic theory of the universe. Democritus knew that if a stone was divided in half, the two halves would have essentially the same properties as the whole.Therefore, he reasoned that if the stone were to be continually cut into smaller and smaller pieces then; at some point, there would be a piece which would be so small as to be indivisible. He called these small pieces of matter “atomos,” the Greek word for indivisible. Democritus, theorized that atoms were specific to the material which they composed. In addition, Democritus believed that the atoms differed in size and shape, were in constant motion in a void, collided with each other; and during these collisions, could rebound or stick together. Therefore, changes in matter were a result of dissociations or combinations of the atoms as they moved throughout the void. Although Democritus’ theory was remarkable, it was rejected by Aristotle, one of the most influential philosophers of Ancient Greece; and the atomic theory was ignored for nearly 2,000 years.

None of Democritus’ original work has survived, except through second-hand references. Many of these references come from Aristotle, who viewed him as an important rival in the field of natural philosophy.

Along with Leucippus the most important representative of the ancient atomic theory (compare atomic theory). According to Democritus and his teacher Leucippus – whose ideas cannot be separated from Democritus – reality consists of an infinite number of indivisible bodies, atoms,

LIFE Although many anecdotes about Democritus’ life survive, their authenticity cannot be verified and modern scholars doubt their accuracy. Democritus was said to be born in the city of Abdera in Thrace, an Ionian colony of Teos. Ancient accounts of his life have claimed that he lived to a very old age, with some writers claiming that he was over a hundred years old at the time of his death. Ancient accounts of his life have claimed that he lived to a very old age, with some writers claiming that he was over a hundred years old at the time of his death.

PHILOSOPHY AND SCIENCE Christopher Charles Whiston Taylor [de] states that the relation between Democritus and his predecessor Leucippus is not clear; while earlier ancient sources such as Aristotle and Theophrastus credit Leucippus with the invention of atomism and credit its doctrines to both philosophers, later sources credit only Democritus, making definitive identification of specific doctrines difficult.

Atomic hypothesis See also: Atomism We have various quotes from Democritus on atoms, one of them being:

δοκεῖ δὲ αὐτῶι τάδε· ἀρχὰς εἶναι τῶν ὅλων ἀτόμους καὶ κενόν, τὰ δ’ἀλλα πάντα νενομίσθαι [δοξάζεσθαι]. (Diogenes Laërtius, Democritus, Vol. IX, 44) Now his principal doctrines were these. That atoms and the vacuum were the beginning of the universe; and that everything else existed only in opinion. (trans. Yonge 1853)

The theory of Democritus held that everything is composed of “atoms,” which are physically, but not geometrically, indivisible; that between atoms, there lies empty space; that atoms are indestructible, and have always been and always will be in motion; that there is an infinite number of atoms and of kinds of atoms, which differ in shape and size. Of the mass of atoms, Democritus said, “The more any indivisible exceeds, the heavier it is.” However, his exact position on atomic weight is disputed. His exact contributions are difficult to disentangle from those of his mentor Leucippus, as they are often mentioned together in texts. Their speculation on atoms, taken from Leucippus, bears a passing and partial resemblance to the 19th-century understanding of atomic structure that has led some to regard Democritus as more of a scientist than other Greek philosophers; however, their ideas rested on very different bases.

Democritus, along with Leucippus and Epicurus, proposed the earliest views on the shapes and connectivity of atoms. They reasoned that the solidness of the material corresponded to the shape of the atoms involved.Using analogies from humans’ sense experiences, he gave a picture or an image of an atom that distinguished them from each other by their shape, their size, and the arrangement of their parts. Moreover, connections were explained by material links in which single atoms were supplied with attachments: some with hooks and eyes, others with balls and sockets.

The Democritean atom is an inert solid (merely excluding other bodies from its volume) that interacts with other atoms mechanically. In contrast, modern, quantum-mechanical atoms interact via electric and magnetic forces and are dynamic.

Correlation with modern science The theory of the atomists appears to be more nearly aligned with that of modern science than any other theory of antiquity. However, the similarity with modern concepts of science can be confusing when trying to understand where the hypothesis came from. Classical atomists could not have had an empirical basis for modern concepts of atoms and molecules.

The atomistic void hypothesis was a response to the paradoxes of Parmenides and Zeno, the founders of metaphysical logic, who put forth difficult-to-answer arguments in favor of the idea that there can be no movement. They held that any movement would require a void—which is nothing—but a nothing cannot exist. The Parmenidean position was “You say there is a void; therefore the void is not nothing; therefore there is not the void. The position of Parmenides appeared validated by the observation that where there seems to be nothing there is air, and indeed even where there is not matter there is something, for instance light waves.

The atomists agreed that motion required a void, but simply rejected the argument of Parmenides on the grounds that motion was an observable fact. Therefore, they asserted, there must be a void.

Democritus held that originally the universe was composed of nothing but tiny atoms churning in chaos, until they collided together to form larger units—including the earth and everything on it. He surmised that there are many worlds, some growing, some decaying; some with no sun or moon, some with several. He held that every world has a beginning and an end and that a world could be destroyed by collision with another world.

MATHEMATICS

Democritus argued that the circular cross-section of a cone would need step-like sides, rather than being shaped like a cylinder. Democritus was also a pioneer of mathematics and geometry in particular. According to Archimedes, Democritus was among the first to observe that a cone and pyramid with the same base area and height has one-third the volume of a cylinder or prism respectively, a result which Archimedes states was later proved by Eudoxus of Cnidus. Plutarch also reports that Democritus worked on a problem involving the cross-section of a cone that Thomas Heath suggests may be an early version of infinitesimal calculus.

ANTHROPOLOGY Democritus thought that the first humans lived an anarchic and animal sort of life, going out to forage individually and living off the most palatable herbs and the fruit which grew wild on the trees. They were driven together into societies for fear of wild animals, he said. He believed that these early people had no language, but that they gradually began to articulate their expressions, establishing symbols for every sort of object, and in this manner came to understand each other. He says that the earliest men lived laboriously, having none of the utilities of life; clothing, houses, fire, domestication, and farming were unknown to them. Democritus presents the early period of mankind as one of learning by trial and error, and says that each step slowly led to more discoveries; they took refuge in the caves in winter, stored fruits that could be preserved, and through reason and keenness of mind came to build upon each new idea.

ETHICS AND POLITICS The ethics and politics of Democritus come to us mostly in the form of maxims. As such, the Stanford Encyclopedia of Philosophy has gone as far as to say that: “despite the large number of ethical sayings, it is difficult to construct a coherent account of Democritus’s ethical views,” noting that there is a “difficulty of deciding which fragments are genuinely Democritean.”

AESTHETICS Later Greek historians consider Democritus to have established aesthetics as a subject of investigation and study,[8] as he wrote theoretically on poetry and fine art long before authors such as Aristotle. Specifically, Thrasyllus identified six works in the philosopher’s oeuvre which had belonged to aesthetics as a discipline, but only fragments of the relevant works are extant; hence of all Democritus’s writings on these matters, only a small percentage of his thoughts and ideas can be known.

WORKS Diogenes Laertius attributes several works to Democritus, but none of them have survived in a complete form. ,,That Organisms first came from moist soil.

Everything that exists in man is made up of atoms.

The SOUL=PSYCHE=ΨΥΧΗ consists of very small, smooth, round atoms, like fire.

Spirit, soul, vital heat, vital principle are all one and the same thing.

They are not limited to humans or animals, but are scattered throughout the world.

And in man and other animals, the intellectual atoms, with which we think, are distributed throughout the body.

However, these subtle atoms, which make up the soul, are the noblest and most wonderful part of the body.

The wise man cultivates thought, frees himself from passion, superstition, and fear, and seeks in contemplation and understanding the humble happiness that exists in human life.

Happiness does not come from external goods.

Man must get used to finding within himself the sources of pleasure!!

Cultivation is better than riches.

No power and no treasure can compare with the expansion of our knowledge.

Happiness is fickle and carnal pleasure provides only short-term satisfaction.

We achieve lasting satisfaction by gaining calmness, cheerfulness, mediocrity and some order and symmetry in our lives.

We can learn a lot from animals – spinning from the spider, thrift from the swallow, singing from the nightingale and the swan.

But strength of body is kindness only to beasts of burden, but strength of character is kindness to man.

Like the heretics of Victorian England, Democritus raises above his scandalous metaphysics a very presentable morality.

Good deeds must be done not out of compulsion, but out of conviction, not out of hope of reward, but for their own sake.

Man should feel more shame towards himself when he does evil, than towards the whole world.

Eventually Democritus proved his principles and perhaps vindicated his advice by living to the age of 109.

Diogenes Laertius narrates that when Democritus publicly read his greatest work, the <<Great Diacosmos>>, the city of Abdira gave him a hundred talents!!

But perhaps Abdera had undervalued their currency.

When someone asked him what was the secret of his longevity, he replied that he ate honey daily and bathed his body in olive oil.

Finally, when he lived long enough, he daily reduced his food, determined to die of starvation.

He was very old, Diogenes tells us, and he looked like he was dying.

His sister mourned him because she would die on the feast of Thesmophoria, which would prevent her from fulfilling her duties to the goddess.

He then reassured her and ordered her to bring him daily warm breads or some honey and by bringing these into his nostrils he was kept alive during the holy festival.

But when three days had passed since the feast, he breathed his last without pain, as Hipparchus assures us, having lived 109 years!!!

His city issued him at public expense, and Timon the Athenian praised him.

Democritus did not found a school, but formulated the most famous scientific hypotheses and gave philosophy a system, which, despite being denounced by many, survived everyone and reappears in every generation!!!!

Hipparchus is considered the greatest ancient astronomical observer and, by some, the greatest overall astronomer of antiquity. He was the first whose quantitative and accurate models for the motion of the Sun and Moon survive. For this he certainly made use of the observations and perhaps the mathematical techniques accumulated over centuries by Meton of Athens (fifth century BC), Timocharis, Aristyllus, Aristarchus of Samos, and Eratosthenes, among others.

He developed trigonometry and constructed trigonometric tables, and he solved several problems of spherical trigonometry. With his solar and lunar theories and his trigonometry, he may have been the first to develop a reliable method to predict solar eclipses.

His other reputed achievements include the discovery and measurement of Earth’s precession, the compilation of the first known comprehensive star catalog from the western world, and possibly the invention of the astrolabe, as well as of the armillary sphere that he may have used in creating the star catalogue. Hipparchus is sometimes called the “father of astronomy”, a title conferred on him by Jean Baptiste Joseph Delambre in 1817^{]}

Life and work

Hipparchus was born in Nicaea (Greek: Νίκαια), in Bithynia. The exact dates of his life are not known, but Ptolemy attributes astronomical observations to him in the period from 147 to 127 BC, and some of these are stated as made in Rhodes; earlier observations since 162 BC might also have been made by him. His birth date (c. 190 BC) was calculated by Delambre based on clues in his work. Hipparchus must have lived some time after 127 BC because he analyzed and published his observations from that year.

In the second and third centuries, coins were made in his honour in Bithynia that bear his name and show him with a globe.

Relatively little of Hipparchus’s direct work survives into modern times. Although he wrote at least fourteen books, only his commentary on the popular astronomical poem by Aratos was preserved by later copyists. Most of what is known about Hipparchus comes from Strabo‘s Geography and Pliny‘s Natural History in the first century; Ptolemy’s second-century Almagest; and additional references to him in the fourth century by Pappus and Theon of Alexandria in their commentaries on the Almagest.

Hipparchus’s only preserved work is Commentary on the Phaenomena of Eudoxus and Aratus (Greek: Τῶν Ἀράτου καὶ Εὐδόξου φαινομένων ἐξήγησις). This is a highly critical commentary in the form of two books on a popular poem by Aratus based on the work by Eudoxos.^{[}Hipparchus also made a list of his major works that apparently mentioned about fourteen books, but which is only known from references by later authors. His famous star catalog was incorporated into the one by Ptolemy and may be almost perfectly reconstructed by subtraction of two and two-thirds degrees from the longitudes of Ptolemy’s stars. The first trigonometric table was apparently compiled by Hipparchus, who is consequently now known as “the father of Trigonometry

Eudoxus in the 4th century BC and Timocharis and Aristillus in the 3rd century BC already divided the ecliptic in 360 parts (our degrees, Greek: moira) of 60 arcminutes and Hipparchus continued this tradition. It was only in Hipparchus’s time (2nd century BC) when this division was introduced (probably by Hipparchus’s contemporary Hypsikles) for all circles in mathematics. Eratosthenes (3rd century BC), in contrast, used a simpler sexagesimal system dividing a circle into 60 parts. Hipparchus also adopted the Aristarhus astronomical Greek πῆχυς pēchys) that was equivalent to 2° or 2.5° (‘large cubit’).

Hipparchus’s long draconitic lunar period (5,458 months = 5,923 is post-Hipparchus so the direction of transmission is not settled by the tablets.

Geometry, trigonometry and other mathematical techniques

Hipparchus was recognized as the first mathematician known to have possessed a trigonometric table, which he needed when computing the eccentricity of the orbits of the Moon and Sun. He tabulated values for the chord function, which for a central angle in a circle gives the length of the straight line segment between the points where the angle intersects the circle. He may have computed this for a circle with a circumference of 21,600 units and a radius (rounded) of 3,438 units; this circle has a unit length for each arcminute along its perimeter. (This was “proven” by Toomer, but he later “cast doubt“ upon his earlier affirmation^{.} Other authors have argued that a circle of radius 3,600 units may instead have been used by Hipparchus.) He tabulated the chords for angles with increments of 7.5°. In modern terms, the chord subtended by a central angle in a circle of given radius R equals Rtimes twice the sine of half of the angle, i.e.:chord�=2�⋅sin12�The now-lost work in which Hipparchus is said to have developed his chord table, is called Tōn en kuklōi eutheiōn (Of Lines Inside a Circle) in Theon of Alexandria‘s fourth-century commentary on section I.10 Trigonometry was a significant innovation, because it allowed Greek astronomers to solve any triangle, and made it possible to make quantitative astronomical models and predictions using their preferred geometric techniques.

Hipparchus must have used a better approximation for π than the one given by Archimedes of between 3+^{10}⁄_{71} (≈ 3.1408) and 3+^{1}⁄_{7} (≈ 3.1429). Perhaps he had the approximation later used by Ptolemy, sexagesimal 3;08,30 (≈ 3.1417) (Almagest VI.7).

Hipparchus could have constructed his chord table using the Pythagorean theorem and a theorem known to Archimedes.

The stereographic projection was ambiguously attributed to Hipparchus by Synesios (c. 400 AD), and on that basis Hipparchus is often credited with inventing it or at least knowing of it. However, some scholars believe this conclusion to be unjustified by available evidence.The oldest extant description of the stereographic projection is found in Ptolemy‘s Planisphere (2nd century AD)

There are several indications that Hipparchus knew spherical trigonometry, but the first surviving text discussing it is by Menelaus of Alexandria in the first century, who now, on that basis, commonly is credited with its discovery. (Previous to the finding of the proofs of Menelaus a century ago, Ptolemy was credited with the invention of spherical trigonometry.) Ptolemy later used spherical trigonometry to compute things such as the rising and setting points of the ecliptic, or to take account of the lunar parallax. If he did not use spherical trigonometry, Hipparchus may have used a globe for these tasks, reading values off coordinate grids drawn on it, or he may have made approximations from planar geometry, or perhaps used arithmetical approximations developed by Pythagoras.

The now lost work in which Hipparchus is said to have developed his chord table, is called Tōn en kuklōi eutheiōn (Of Lines Inside a Circle) in Theon of Alexandria’s 4th-century commentary on section Trigonometry was a significant innovation, because it allowed Greek astronomers to solve any triangle, and made it possible to make quantitative astronomical models and predictions using their preferred geometric techniques.

Hipparchus must have used a better approximation for π than the one from Archimedes of between 3 10⁄71 (3.14085) and 3 1⁄7 (3.14286). Perhaps he had the one later used by Ptolemy: 3;8,30 (sexagesimal)(3.1417) (Almagest VI.7), but it is not known whether he computed an improved value himself.