PYTHAGORAS (c.560 – c.480 BCE)

diagrammatic proof of Pythagoras' theoremSixth Century BCE – Greece

‘In a right-angled triangle, the square on the hypotenuse is the sum of the squares on the other two sides’

The Theorem may also be written as a general law:  a2 + b2 = c2  where c is the length of the hypotenuse of a right-angled triangle, and a and b the lengths of the other two sides. Pythagoras’ theorem is a starting point for trigonometry, which has many practical applications such as calculating the height of mountains and measuring distances.

c.525 BCE – Pythagoras taken prisoner by the Babylonians

c.518 BCE – establishes his own academy at Croton (now Crotone) in southern Italy

c.500 BCE – Pythagoras moves to Metapontum

Pythagoras was the first to prove the relationship between the sides of a right-angled triangle, but he did not discover it – it was known to Babylonians for nearly 1000 years before him.

His disciples, members of the semi-religious, philosophical school he founded, may have actually found many of the mathematical discoveries credited to Pythagoras. The inner circle of followers were known as mathematikoi and, unusually for the time, included women among its membership. An outer circle, the akousmatics, lived in their own homes and came in to the school by day.

Of the five key beliefs the Pythagoreans held, the idea that ‘all is number’ was dominant; the belief that reality at its fundamental level is mathematical and that all physical things like musical scales, or the spherical earth and its companions the stars and the universe, are mathematically related. Pythagoras was responsible for the widely held Greek belief that real knowledge had to be like mathematics – universal, permanent, obtained by pure thought and uncontaminated by the senses.

Because of the reverence with which the originator of the Pythagoreans was treated by his followers and biographers, it is difficult to discern legend from fact, such as the notion that he was the first to offer a three-part argument that the shape of the Earth is spherical:
The field of stars changes with the latitude of the observer; the mast of a ship comes into view before its hull as the ship approaches the shore from a distance; and the shadow of the Earth cast on the moon during a lunar eclipse is always round.

After Pythagoras, the idea of a ‘perfect’ mathematical interrelation between a globe moving in circles and the stars behaving similarly in a spherical universe inspired later Greek scholars, including ARISTOTLE, to seek and ultimately find physical and mathematical evidence to reinforce the theory of the world as an orb.

Attributed to the Pythagoreans is the discovery that simple whole number ratios of string lengths produce harmonious tones when plucked, probably the first time a physical law had been mathematically expressed.

Numerous other discoveries such as ‘the sum of a triangle’s angles is the equal to two right angles’ and ‘the sum of the interior angles in a polygon of n-sides is equal to 2n-4 right angles’ were made. They also discovered irrational numbers, from the realisation that the square root of two cannot be expressed as a perfect fraction. This was a major blow to the Pythagorean idea of perfection and according to some, attempts were made to try to conceal the discovery.

PLATONIC SOLIDS

To the Pythagoreans, the fifth polyhedron had monumental significance. Outnumbering by one the number of recognized elements, the dodecahedron was considered to represent the shape of the universe. 
A omerta, or code of silence, was imposed regarding the dodecahedron and divulging this secret to outsiders could mean a death penalty.

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PROTAGORAS of ABDERA (c.480 – c.411 BCE)

Drawing of PROTAGORAS

PROTAGORAS

‘Of all things the measure is Man, of the things that are, that they are, and of the things that are not, that they are not’
‘About the gods, I am not able to know whether they exist or do not exist, nor what they are like in form; for the factors preventing knowledge are many: the obscurity of the subject, and the shortness of human life’

A contemporary of Socrates
In 415 BCE he was forced to flee Athens because his works were condemned for impiety

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THE PLATONIC SOLIDS

The properties of solid figures have kept mathematicians occupied for centuries. Regular polyhedra are formed from regular polygons such as squares or triangles and mathematicians have failed to find any more than five of them.

the five Platonic solids - tetrahedron, cube, octahedron, dodecahedron, icosahedron

Although they were defined by Pythagoras two hundred years before Plato was born, they are known collectively as the platonic solids, named in honour of PLATO by the geometer Euclid.

“THE PLATONIC SOLIDS – The regular polyhedron is defined as a three-dimensional solid comprising regular polygons for its surfaces – and with all its surfaces, edges and vertices identical. The five regular polyhedra are the tetrahedron (four triangular faces), the cube (six square faces), the octahedron (eight triangular faces), the dodecahedron (twelve pentagonal faces) and the icosahedron (twenty triangular faces).”

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PLATO (c.427 – c.347 BCE)

387 BCE – Athens

‘The safest general characterisation of the European philosophical tradition is that it consists of a series of footnotes to Plato’

So said the English mathematician and philosopher Alfred North Whitehead (1861-1947)

A pupil of Socrates, Plato was introduced to the notion of ‘reality’ being distorted by human perceptions, which became important in his approach to science and to metaphysics. Socrates taught a method of thinking that elucidated truth though a series of questions and answers. The Socratic method was to ask for definitions of familiar concepts like ‘justice’ and ‘courage’, and then probe the definition by asking a series of questions. His intention was to lead people to start to contradict themselves and in so doing uncover any weaknesses in the initial definition. Socratic dialogue is thus often better at revealing ignorance than producing answers. Socrates was convinced that the wisest people are those who are aware of how little they know.
Socrates fell foul of a newly elected democratic government and was put on trial for allegedly corrupting the youth of Athens with his rebellious ideas. He was sentenced to death and elected to drink hemlock rather than argue in favour of a fine or accepting the offer of help to escape. His rational for his obstinacy was that he believed that doing harm damaged one’s soul. As the soul survives death then it would be better to die.

PHILOSOPHY

Plato determined to reveal a world of certainty that existed beyond the world of change and decay. The physical world we see is merely the world of ‘becoming’ – a poor copy of the ‘real’ world of the Forms which can only ever be grasped through thought.
After PYTHAGORAS and HERACLITUS, most Greek philosophers believed that knowledge had to be as stable and fixed as the certainties of mathematics, kept safe from Heraclitan change and from sceptical relativism.
Knowledge could only come through thought and although observation was useful, it was an inferior and misleading way of understanding the world and the place of human beings within it. Such a view helps to explain why it is that the ancient Greeks invented extremely sophisticated mathematics, astronomy and philosophy but little in the way of technology.

Plato produced nearly all the central questions for philosophy in epistemology, metaphysics, ethics, politics and aesthetics.
Central to Plato’s thinking is that people should seek virtue studying what he called the Good, a non-physical absolute concept that never changes. If you know Good, you will live well because your thoughts and desires will automatically be shaped by that knowledge.

During the decade of his travels after the execution of Socrates, Plato wrote his first group of ‘dialogues’ – which include the ‘Euthyphro‘, ‘Apology’, the ‘Crito’, ‘Phaedo‘ – concerning the trial and death of Socrates.
Further accounts of Socrates debates with friends on various subjects are found in other works; ‘Charmides’ (temperance); ‘Laches’ (courage); ‘Lysis’ (friendship); ‘Hippas Minor’, ‘Hippas Major’, ‘Gorgias’, ‘Ion‘, and ‘Protagoras’ (ethics and education). In his plays, Plato used Socrates as a character, bringing his mentor back from the grave and throwing light on his concepts. In Gorgias, Plato portrays Socrates confronting Polus, or the sophist Callicles, who holds that immoral acts can bring the greatest amount of pleasure, measuring actions in terms of their immediate material outcome. Socrates disagrees. Whatever the immediate pleasure, he says, immorality will damage the soul.

Opposing DEMOCRITUS, Plato believed that all substances are composed of one kind of matter, possessing the qualities of form and spirit. He accepted the Greek notion, first suggested by EMPEDOCLES in the fifth century BCE, that matter was made up of mixtures of the four elements – earth, water, air or fire. Because these four are only fundamental forms of the single type of matter, they cannot be related to any idea of ‘elements’ as understood by modern science – they could be transmuted into each other. Different substances, although composed of matter would have different properties due to the differing amounts of the qualities of form and spirit. Thus a lump of lead is made of the same type of matter (fundamental form) as a lump of gold, but has a different aggregation of constituents. Neither lead nor gold would contain much spirit – not as much as air, say, and certainly not as much as God, who is purely spiritual.

399 BCE on the execution of SOCRATES, Plato leaves Athens in disgust.

387 BCE Returns to Athens. Plato founds his academy (‘ Let no one enter here who is ignorant of geometry ‘) – a bastion of intellectual achievement until its closure on the orders of the emperor Justinian in CE 529.

Plato’s ‘Theory of Forms’ consisted of the argument that Nature, as seen through human eyes, is merely a flawed version of true ‘reality’, or ‘forms’.

Plato argued that everything we see and call beautiful in some way resembles the form of Beauty. Two people independently come to the conclusion that a person or an object is beautiful because they both recognise the form of Beauty. In a similar way, everything that we see as ‘Just’ resembles the form of ‘Justice’. Disputes about the rightness of actions then depend on how well the outcome will conform to the form of Good. For a person to act justly requires that while they seek the form of Good, they keep the three parts of their personality in balance. The person needs wisdom, which comes from reason; courage, which comes from the spirited part of man; and self-control, which rules the passions.

In ‘The Republic’ Plato expands the idea that if you educate a person so that he can see that a particular action is not good for them, then they will not perform that action. This knowledge will enable them to make good decisions and to rule wisely, hence the idea of a philosopher king who has mastered the discipline of ‘dialectic’ and studied the hierarchy of Forms. The society is organised into a rigid hierarchy of workers, soldiers and rulers who all know their relative positions and there is a communism of property and family. The rulers have totalitarian powers and a harmonious communal life can only be achieved at the expense of individual freedoms.
Plato’s educational syllabus in ‘The Republic’ is based on Spartan methods – selfless dedication to the welfare of the State is essential.

SCIENCE

Plato encountered the Pythagoreans in Croton, who became a major influence. For Plato, there had always existed an eternal, underlying mathematical form and order to the universe, and what humans saw were merely imperfect glimpses of it, usually corrupted by their own irrational perceptions and prejudices about the way things ‘are’. Consequently, for Plato, the only valid approach to science was a rational mathematical one, which sought to establish universal truths irrespective of the human condition. This has strongly impacted on modern science; for example, arithmetic calculations suggesting that future discoveries would have particular properties has led to the naming of unknown elements in DMITRI MENDELEEV‘s first periodic table.

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ARISTOTLE (c.384 – c.322 BCE)

335 BCE – Athens

Bust of ARISTOTLE

ARISTOTLE

384 BCE – Born in the Greek colony of Stagira. The son of Nicomachus, court physician to the king of Macedonia
367 BCE – Enters Plato’s Academy in Athens
347 BCE – On Plato’s death Speusippus succeeds Plato as head of the Academy. Aristotle leaves the Academy for Lesbos
342 BCE – Becomes tutor to the young Alexander (the Great), son of Phillip of Macedon
335 BCE – Returns to Athens and founds the Lyceum
321 BCE – Accused of impiety, returns to Chalcis where he dies a year later

Aristotle reinforced the view espoused by PYTHAGORAS that the earth is spherical. The arc shaped shadow of the earth cast upon the moon during a lunar eclipse is consistent with this view. He also noted that when traveling north or south, stars ‘move’ on the horizon until some gradually disappear from view.

Proposing that there was no infinity and no void he accepted the notion of the earth at the centre of the universe, with the moon, planets, sun and stars all orbiting around it in perfect circles.
The universe existed as beautiful spheres surrounding the Earth, placed at the centre of the cosmos. This system was later refined by the Alexandrian astronomer Ptolemy and become the dominant philosophy in the Western world.

Explaining why the heavens rotate in perfect, uniform order, with none of the disturbances associated with earthly elements; he described the fifth element added to the traditional four, ‘Aether‘, as having a naturally circular motion. Everything beyond the moon was regulated by aether, explaining both its perfect movement and stability, while everything below it was subject to the laws of the four other elements.

Aristotle rejected the ideas of zero and infinity, hence he had explained away Zeno’s paradoxes – Achilles runs smoothly past the tortoise because the infinite points are simply a figment of Zeno’s imagination; infinity was just a construct of the human mind.
By rejecting zero and infinity, Aristotle denied the atomists’ idea of matter existing in an infinite vacuum, infinity and zero wrapped into one.
In contrast to the theory of atoms, like Plato, Aristotle believed that matter is composed of four elements ( Ignis, Aqua, Aer and Terra ) with differing qualities ( hot, wet, cold, dry )

[ Fire – hot + dry ; Water – cold + wet ; Air – hot + wet ; Earth – cold + dry ]

He believed that the qualities of heat, cold, wetness and dryness were the keys to transformation, each element being converted into another by changing one of these two qualities to its opposite.

Agreeing that things were composed of a single, primal substance (prote hyle) that was too remote and unknowable, he accepted EMPEDOCLES elements as intermediaries between the imponderable and the tangible world, concealing the complications behind a philosophy of matter.

The four elements always sought to return to their ‘natural place’. Thus a rock, for example, would drop to the earth as soon as any obstacles preventing it from doing so were removed – because ‘earth’ elements, being denser and heavier, would naturally seek to move downwards towards the centre of the planet. Water elements would float around the surface, air would rise above that and fire would seek to rise above them all, explaining the leaping, upward direction of flames.

Although the Aristotelian view of matter has been undermined as experiments proved that neither air nor water are indivisible; today, scientists define matter as existing in four phases, solid, liquid, gas and plasma.

MATTER (hyle); FORM (morphe); CAUSE; PURPOSE;

The place where his ideas converge with Plato’s is that for Aristotle, the pinnacle of the tower of superiority is the Good. According to Aristotle, all aims eventually lead to the Good, not necessarily of the individual but of humankind. Humans by nature are social and moral and everyone is part of a group, a family, village, town or city-state. There is no place for individualism or freethinkers, as without the happiness of the group then the individual cannot be happy.
The consequence of this emphasis on the community as opposed to the individual is hierarchy and subordination and as a result slavery was a very normal part of a well-ordered society.

  • Matter is itself only one component of the world – others being form and spirit. There are different sorts of living being in the world.
    Human beings possess immortal souls.
    He believed that there is in living creatures a fundamental vital principle, a ‘life force’, which distinguishes them from non-living material. The gods breathed this vital principle into living things, and thereby gave them their life – ( nous – spontaneous generation ).

The soul is governed by reason, spirit and appetite.
‘All human actions have one or more of these seven causes: chance, nature, compulsion, habit, reason, passion, and desire’ ( source )

  • Forms are incorporated in individual particulars as potentiality.
    All particular acorns possess the form of the potential oak tree.

Although Aristotle was a pupil at Plato’s Academy for almost twenty years, the two great thinkers were diametrically opposed on a number of subjects; he criticised Platonic forms for being impossibly transcendent and mystical.

Aristotle pursued his ideas unrestricted by Socratic theories that non-physical forms such as Truth and Beauty were the keys to understanding.

  • Four Causes – efficient, formal, material, final – (agent, form, matter, goal). – The ‘Timaeus’ – ( Plato’s work in which the chief speaker is encouraged to provide his account of the origins of the universe.)

  • ‘Action exists not in the agent but in the patient’
    To study a situation, or an action, Aristotle would categorise it into a series of subordinate and superior aims.

 

MOTION

  • Motion of Place – A to B

  • Motion of Quantity – change in amount

  • Motion of Quality – green apples turning red or from sour to sweet

 

Aristotle could explain why a rock, when thrown, would travel upwards through the air first before heading downwards, rather than straight down towards the earth. This was because the air, seeking to close the gap made by the invasion of the rock, would propel it along until it lost its horizontal speed and it tumbled to the ground.

Such notions made a lasting impact for the next two thousand years, if only by slowing down progress due to their unchallenged acceptance.

Some of Aristotle’s biology was faulty, such as defining the heart, not the brain as the seat of the mind.

 

Aristotle’s model of ‘the hydrologic cycle’ is uncannily close to the ideas we have today. The Sun’s heat changes water into air ( as defined as ‘elements’ by EMPEDOCLES ). Heat rises, so the heat in this air pulls the air up to the skies ( modern explanations of the nature of heat give a fuller understanding of the mechanisms involved ). The heat then leaves the vapour, which thus becomes progressively more watery again, and this process is marked by the formation of a cloud. The positive feedback of the increased ‘wateriness’ of the mixture in the cloud driving away its opposite ( the ‘heat’ ) and causing the cloud to become colder and shrink results in restoration of the true wateriness of the water, which falls as rain or, if the cloud is now cold enough, as hail or snow.

Aristotle was one of the first to attempt a methodical classification of animals; in ‘Generation of Animals’ he used means of reproduction to differentiate between those animals which give birth to live young and those which lay eggs, a system which is the forerunner of modern taxonomy. He noted that dolphins give birth to live young who were attached to their mothers by umbilical cords and so he classified dolphins as mammals.

Based on the Pythagorean universe, the Aristotelian cosmos had the planets moving in crystalline orbs.
Since there is no infinity, there cannot be an endless number of spheres; there must be a last one. There was no such thing as ‘beyond’ the final sphere and the universe ended with the outermost layer.
With no infinite and no void, the universe was contained within the sphere of fixed stars. The cosmos was finite in extent and entirely filled with matter.
The consequence of this line of reasoning, accounting for Aristotle’s philosophy enduring for two millennia was that this system proved the existence of God.

The heavenly spheres are slowly spinning in their places, making a divine music that suffuses the cosmos. The stationary earth cannot be the cause of that motion, so the innermost sphere must be moved by the next sphere out, which, in its turn must be moved by by its larger neighbour, and on and on. With a finite number of spheres, something must be the ultimate cause of motion of the final sphere of fixed stars. This is the Prime Mover.
Christianity came to rely on Aristotle’s view of the universe and this proof of God’s existence.
Atomism became associated with atheism.

The ideas of Aristotle were picked up by the twelfth century Andalusian philosopher Abu al-Walid Muhammad ibn Ahmed ibn Rushd (AVERROES) and were later adopted by the medieval philosopher THOMAS AQUINAS in the thirteenth century; whose concept of Natural Law is the basis of much thinking in the Christian world.

Aristotle had greater influence on medieval scholastic thought than Plato, whose rediscovery in the Italian renaissance influenced Petrarch, Erasmus, Thomas More and other scholars to question the dogmas of scholasticism.

Aristotle’s work in physics and cosmology dominated Western thought until the time of GALILEO and NEWTON, when much of it was subsequently refuted, though his work still underpins both Christian and Islāmic philosophy. His importance lies as much in his analytical method as in the conclusions he reached.

 

Aristotle expanded Plato’s concept of ‘virtue’ by dividing virtues into two groups, the 12 ‘moral’ and 9 ‘intellectual’ virtues, believing that each lay between the non-virtuous extremes of excess and deficiency.

Deficiency Virtue Excess
Cowardice Courage Rashness
Licentiousness (disregarding convention, unrestrained) Temperance (restraint or moderation) Insensibility (indifference)
Illiberality (meanness) Liberality (generosity) Prodigality (wasteful, extravagant)
Pettiness Magnificence Vulgarity
Humble-mindedness High-mindedness Vanity
Lack of ambition Proper ambition Over ambition
Irascibility (easily angered) Patience Lack of spirit
Understatement Truthfulness Boastfulness
Boorishness Wittiness Buffoonery
Cantankerousness Friendliness Obsequiousness
Shamelessness Modesty Shyness
Malicious enjoyment Righteous indignation Envy/spitefulness

His intellectual virtues consisted of :

art    scientific knowledge    prudence    intelligence    wisdom    resourcefulness    understanding    judgment    cleverness

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EUCLID (c.330 – c.260 BCE)

Fourth century BCE – Alexandria, Egypt

Euclid

EUCLID

  1. A straight line can be drawn between any two points

  2. A straight line can be extended indefinitely in either direction

  3. A circle can be drawn with any given centre and radius

  4. All right angles are equal

  5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines will eventually meet (or, parallel lines never meet)

These five postulates form the basis of Euclidean geometry. Many mathematicians do not consider the fifth postulate (or parallel postulate) as a true postulate, but rather as a theorem that can be derived from the first four postulates. This ‘parallel’ axiom means that if a point lies outside a straight line, then only one straight line can be drawn through the point that never meets the other line in that plane.

The ideas of earlier Greek mathematicians, such as EUDOXUS, THEAETETUS and PYTHAGORAS are all evident, though much of the systematic proof of theories, as well as other original contributions, was Euclid’s.

The first six of his thirteen volumes were concerned with plane geometry; for example laying out the basic principles of triangles, squares, rectangles and circles; as well as outlining other mathematical cornerstones, including Eudoxus’ theory of proportion. The next four books looked at number theory, including the proof that there is an infinite number of prime numbers. The final three works focused on solid geometry.

Virtually nothing is known about Euclid’s life. He studied in Athens and then worked in Alexandria during the reign of Ptolemy I

Euclid’s approach to his writings was systematic, laying out a set of axioms (truths) at the beginning and constructing each proof of theorem that followed on the basis of proven truths that had gone before.

Elements begins with 23 definitions (such as point, line, circle and right angle), the five postulates and five ‘common notions’. From these foundations Euclid proved 465 theorems.

A postulate (or axiom) claims something is true or is the basis for an argument. A theorem is a proven position, which is a statement with logical constraints.

Euclid’s common notions are not about geometry; they are elegant assertions of logic:

  • Two things that are both equal to a third thing are also equal to each other

  • If equals are added to equals, the wholes are equal

  • If equals are subtracted from equals, the remainders are equal

  • Things that coincide with one and other are equal to one and other

  • The whole is greater than the part

One of the dilemmas that he presented was how to deal with a cone. It was known that the volume of a cone was one-third of the volume of a cylinder that had the same height and base diameter. He asked if you cut through a cone parallel to its base, would the circle formed on the top section be the same size as that on the bottom of the new, smaller cone?

If it were, then the cone would in fact be a cylinder and clearly that was not true. If they were not equal, then the surface of a cone must consist of a series of steps or indentations.

NON-EUCLIDEAN MATHEMATICS

Statue of Janus Bolyai

Janus Bolyai

The essential weakness in Euclidean mathematics lay in its treatment of two- and three- dimensional figures. This was examined in the nineteenth century by the Romanian mathematician Janus Bolyai. He attempted to prove Euclid’s parallel postulate, only to discover that it is in fact unprovable. The postulate means that only one line can be drawn parallel to another through a given point, but if space is curved and multidimensional, many other parallel lines can be drawn. Similarly the angles of a triangle drawn on the surface of a ball add up to more than 180 degrees.
CARL FRIEDRICH GAUSS was perhaps the first to ‘doubt the truth of geometry’ and began to develop a new geometry for curved and multidimensional space. The final and conclusive push came from BERNHARD RIEMANN, who developed Gauss’s ideas on the intrinsic curvature of surfaces.

Riemann argued that we should ignore Euclidean geometry and treat each surface by itself. This had a profound effect on mathematics, removing a priori reasoning and ensuring that any future investigation of the geometric nature of the universe would have to be at least in part, empirical. This provides a mechanism for examinations of multidimensional space using an adaptation of the calculus.

However, the discoveries of the last two hundred years that have shown time and space to be other than Euclidean under certain circumstances should not be seen to undermine Euclid’s achievements.

Moreover, Euclid’s method of establishing basic truths by logic, deductive reasoning, evidence and proof is so powerful that it is regarded as common sense.

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ARCHIMEDES (c.287 – c.212 BCE)

Third Century BCE – Syracuse (a Greek city in Sicily)

‘Archimedes’ Screw – a device used to pump water out of ships and to irrigate fields’

Archimedes investigated the principles of static mechanics and pycnometry (the measurement of the volume or density of an object). He was responsible for the science of hydrostatics, the study of the displacement of bodies in water.

Archimedes’ Principle

Buoyancy – ‘A body fully or partially immersed in a fluid is buoyed up by a force equal to the weight of the fluid displaced by the body’
The upthrust (upward force) on a floating object such as a ship is the same as the weight of water it displaces. The volume of the displaced liquid is the same as the volume of the immersed object. This is why an object will float. When an object is immersed in water, its weight pulls it down, but the water, as Archimedes realised, pushes back up with a force that is equal to the weight of water the object pushes out-of-the-way. The object sinks until its weight is equal to the upthrust of the water, at which point it floats.
Objects that weigh less than the water displaced will float and objects that weigh more will sink. Archimedes showed this to be a precise and easily calculated mathematical principle.

 
 

Syracuse’s King Hiero, suspecting that the goldsmith had not made his crown of pure gold as instructed, asked Archimedes to find out the truth without damaging the crown.

Archimedes first immersed in water a piece of gold that weighed the same as the crown and pointed out the subsequent rise in water level. He then immersed the crown and showed that the water level was higher than before. This meant that the crown must have a greater volume than the gold, even though it was the same weight. Therefore it could not be pure gold and Archimedes thus concluded that the goldsmith had substituted some gold with a metal of lesser density such as silver. The fraudulent goldsmith was executed.

Archimedes came to understand and explain the principles behind the compound pulley, windlass, wedge and screw, as well as finding ways to determine the centre of gravity of objects.
He showed that the ratio of weights to one another on each end of a balance goes down in exact mathematical proportion to the distance from the pivot of the balance.

Perhaps the most important inventions to his peers were the devices created during the Roman siege of Syracuse in the second Punic war.

He was killed by a Roman soldier during the sack of the city.

 
 
 
 

(image source)

Π The Greek symbol pi (enclosed in a picture of an apple) - Pi is a name given to the ratio of the circumference of a circle to the diameterPi

‘All circles are similar and the ratio of the circumference to the diameter of a circle is always the same number, known as the constant, Pi’

Pi-unrolled-720.gif

 
 

The Greek tradition disdained the practical.  Following PLATO the Greeks believed pure mathematics was the key to the perfect truth that lay behind the imperfect real world, so that anything that could not be completely worked out with a ruler and compass and elegant calculations was not true.

In the eighteenth century CE the Swiss mathematician LEONHARD EULER was the first person to use the letter  Π , the initial letter of the Greek word for perimeter, to represent this ratio.

The earliest reference to the ratio of the circumference of a circle to the diameter is an Egyptian papyrus written in 1650 BCE, but Archimedes first calculated the most accurate value.

He calculated Pi to be 22/7, a figure which was widely used for the next 1500 years. His value lies between 3 1/2 and 3 10/71, or between 3.142 and 3.141 accurate to two decimal places.

 

‘The Method of Exhaustion – an integral-like limiting process used to compute the area and volume of two-dimensional lamina and three-dimensional solids’

Archimedes realised how much could be achieved through practical approximations, or, as the Greeks called them, mechanics. He was able to calculate the approximate area of a circle by first working out the area of the biggest hexagon that would fit inside it and then the area of the smallest that would fit around it, with the idea in mind that the area of the circle must lie approximately halfway between.

By going from hexagons to polygons with 96 sides, he could narrow the margin for error considerably. In the same way he worked out the approximate area contained by all kinds of different curves from the area of rectangles fitted into the curve. The smaller and more numerous the rectangles, the closer to the right figure the approximation became.

This is the basis of what thousands of years later came to be called integral calculus.
Archimedes’ reckonings were later used by Kepler, Fermat, Leibniz and Newton.

In his treatise ‘On the Sphere and the Cylinder’, Archimedes was the first to deduce that the volume of a sphere is 4/3 Pi r3  where r  is the radius.
He also deduced that a sphere’s surface area can be worked out by multiplying that of its greatest circle by four; or, similarly, a sphere’s volume is two-thirds that of its circumscribing cylinder.

Like the square and cube roots of 2, Pi is an irrational number; it takes a never-ending string of digits to express Pi as a number. It is impossible to find the exact value of Pi – however, the value can be calculated to any required degree of accuracy.
In 2002 Yasumasa Kanada (b.1949) of Tokyo University used a supercomputer with a memory of 1024GB to compute the value to 124,100,000,000 decimal places. It took 602 hours to perform the calculation.

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EPICURUS (341 – 270 BCE)

Third Century BCE

“Epicurus’s philosophy combines a physics based on an atomistic materialism with a rational hedonistic ethics that emphasizes moderation of desires and cultivation of friendships.”

Summarized by the Roman author Lucretius, who wrote ‘On the Nature of the Universe’ in 55 BCE – “The light and heat of the Sun; these are composed of minute atoms which, when they are shoved off, lose no time in shooting right across the interspace of air in the direction imparted by the shove”. This may be considered as accurate for the time, when most people thought that sight was associated with something reaching out from the eye (EMPEDOCLES) .

Plato wrote of a marriage between the inner light and the outer light.

Euclid worried about the speed with which sight worked. He pointed out that if you close your eyes, then open them again, even the distant stars reappear immediately in your sight, although the influence of sight has had to travel all the way from your eyes to the stars and back again before you could see them.

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HIPPARCHUS (c.190 – c.125 BCE)

134 BCE – Nicaea, Turkey

‘Observation of a new star in the constellation Scorpio’

The ‘Precession of the Equinoxes’

Image of Hipparchus surveying the sky

HIPPARCHUS

By the time Hipparchus was born, astronomy was already an ancient art.

Hipparchus plotted a catalogue of the stars – despite warnings that he was thus guilty of impiety. Comparing his observations with earlier recordings from Babylonia he noted that the celestial pole changed over time.
He speculated that the stars are not fixed as had previously been thought and recorded the positions of 850 stars.

Hipparchus‘ astronomical calculations enabled him to plot the ecliptic, which is the path of the Sun through the sky. The ecliptic is at an angle to the Earth‘s equator, and crosses it at two points, the equinoxes (the astronomical event when the Sun is at zenith over the equator, marking the two occasions during the year when both hemispheres are at right angles to the Sun and day and night are of equal length).

The extreme positions of summer and winter mark the times in the Earth’s orbit where one of the hemispheres is directed towards or away from the Sun.

Solstice
The Sun is furthest away at the solstices.

From his observations, he was able to make calculations on the length of the year.
There are several ways of measuring a year astronomically and Hipparchus measured the ‘tropical year’, the time between equinoxes.

Schematic presentation of a seasonal cycle. Note the importance of the fixed direction in space of the rotation axis on these short time scales (today towards Polaris): if the axis were not tilted relative to the plane of orbit, then there would be no seasons.

Schematic presentation of a seasonal cycle

Hipparchus puzzled that even though the Sun apparently traveled a circular path, the seasons – the time between the solstices and equinoxes – were not of equal length. Intrigued, he worked out a method of calculating the Sun’s path that would show its exact location on any date.

To facilitate his celestial observations he developed an early version of trigonometry.
With no notion of sine, he developed a table of chords which calculated the relationship between the length of a line joining two points on a circle and the corresponding angle at the centre.
By comparing his observations with those noted by Timocharis of Alexandria a century and a half previously, Hipparchus noted that the points at which the equinox occurred seemed to move slowly but consistently from east to west against the backdrop of fixed stars.

We now know that this phenomenon is not caused by a shift in the stars.
Because of gravitational effects, over time the axis through the geographic North and South poles of the Earth points towards different parts of space and of the night sky.
The Earth’s rotation experiences movement caused by a slow change in the direction of the planet’s tilt; the axis of the Earth ‘wobbles’, or traces out a cone, changing the Earth’s orientation as it orbits the Sun.
The shift in the orbital position of the equinoxes relative to the Sun is now known as ‘the precession of the equinoxes’, but Hipparchus was basically right.

Hipparchus‘ only large error was to assume, like all those of his time except ARISTARCHUS that the Earth is stationary and that the Sun, moon, planets and stars revolve around it. The fact that the stars are fixed and the Earth is moving makes such a tiny difference to the way the Sun, moon and stars appear to move that Hipparchus was still able to make highly accurate calculations.

These explanations may show how many people become confused by claims that the Earth remains stationary as was believed by the ancients – from our point-of-view on Earth that IS how things could appear.

a) demonstration of precession.


youtube=https://www.youtube.com/watch?v=qlVgEoZDjok

b) demonstration of the equinoxes, but not of the precession, which takes place slowly over a cycle of 26,000 years.


youtube=http://www.youtube.com/watch?v=q4_-R1vnJyw&w=420&h=315

Because the Babylonians kept records dating back millennia, the Greeks were able to formulate their ideas of the truth.

Hipparchus gave a value for the annual precession of around 46 seconds of arc (compared to a modern figure of 50.26 seconds). He concluded that the whole star pattern was moving slowly eastwards and that it would revolve once every 26,000 years.

Hipparchus also made observations and calculations to determine the orbit of the moon, the dates of eclipses and devised the scale of magnitude or brightness that, considerably amended, is still in use.

PTOLEMY cited Hipparchus as his most important predecessor.

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EPICURUS (341 – 270 BCE)

Third Century BCE

Bust of EPICURUS

EPICURUS

Summarized by the Roman author Lucretius, who wrote ‘On the Nature of the Universe’ in 55 BCE – “The light and heat of the sun; these are composed of minute atoms which, when they are shoved off, lose no time in shooting right across the interspace of air in the direction imparted by the shove”. Accurate for the time, when most people thought that sight was associated with something reaching out from the eye ( EMPEDOCLES ).

Plato wrote of a marriage between the inner light and the outer light.

Euclid worried about the speed with which sight worked. He pointed out that if you close your eyes, then open them again, even the distant stars reappear immediately in your sight, although the influence of sight has had to travel all the way from your eyes to the stars and back again before you could see them.

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ZENO of ELEA (c.490 – after 445 BCE)

Fifth Century BCE – Greece

‘A number of paradoxes; some of which seemed to prove the impossibility of motion’

zeno

ZENO

Zeno was a member of the Eleatic school of thought founded by Parmenides. The School held the belief that the underlying nature of the Universe was unvarying and immobile.

In support of Parmenides’ argument, Zeno’s puzzles appeared to show that change and motion were paradoxical and that everything is one – and changeless.

Zeno’s paradoxes were based on the false assumption that space & time are infinitely divisible: that is, the sum of an infinite number of numbers is always infinite. Though they were based on fallacies, the paradoxes remained unsolved for two millennia.

Without the concepts of zero, infinity and the idea of limits, Greek philosophy and mathematics were not equipped to solve the puzzles.

The race between Achilles and the tortoise is not made up of a number of tiny distances but it is continuous until the end.

In the seventeenth century CE the Scottish mathematician James Gregory showed that it is sometimes possible to add infinite terms together to get a finite result – but to do so the terms being added together must approach zero. (This is a necessary, but not sufficient, condition – if the terms go to zero too slowly, then the sum of the terms does not converge to a finite number).
Such a series of numbers is called the convergence series, which occurs when the difference between each number and the one following it becomes smaller throughout the sequence.

The numbers 1, 1/2, 1/4, 1/8, 1/16… are approaching zero as their limit.

When you add up the distance that Achilles runs, with the terms becoming smaller and smaller, each term becomes closer to zero; each term is a step along a journey where the destination is zero.

Since the Greeks rejected the number zero they could not understand that this journey could ever have an end. To them, the terms aren’t approaching anything; the destination does not exist. Instead the Greeks saw the terms as simply getting smaller and smaller.

Using modern definitions, we know that the terms have a limit. The journey has a destination. We now may ask how far away is that destination and how long it will take to get there?

We sum up the distances that Achilles runs:
1 + 1/2 + 1/4 + 1/8 + 1/16… + 1/2n + …
As each step that Achilles takes gets smaller and smaller, it gets closer and closer to zero and the sum of these steps gets closer and closer to 2. How do we know this?
Starting with 2, we subtract the terms of the sum, one by one:

2-1=1 ; 1-1/2=1/2 ; 1/2-1/4=1/4 ; 1/4-1/8=1/8 ; 1/8-1/16=1/16…

We already know that series has a limit of zero, thus as we subtract the terms from 2 we have nothing left and the limit of the sum 1 + 1/2 + 1/4 + 1/8 + … is 2

The source of the trouble is infinity – Zeno had taken continuous motion and divided it into an infinite number of steps, assuming that the race would continue for ever as the steps get smaller and smaller. The race would never finish in finite time, but with the concept of zero we find a key to solving the puzzle.

Of course, Achilles wins the race.
Achilles takes two steps in catching up with the tortoise, even though he does it in an infinite number of increments.

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DEMOCRITUS of ABDERA (c.460 – c.370 BCE)

Democritus of Abdera

DEMOCRITUS

Fifth century BCE – Greece

‘Matter is made up of empty-space and an infinite number of tiny invisible particles called atomos or atoms’

Democritus’ atomic theory was probably based on previous ideas of other Greek philosophers. It was the first scientific attempt to explain the nature of matter; however, many of Democritus’ assumptions have now been proved wrong.

Democritus left no written record of his work and little is known about his life, but we know about his atomic theory from the second century CE Greek cynic and biographer Diogenes Laertius’ book ‘Lives of Eminent Philosophers’.

Democritus reasoned that, if he were to attempt to cut an object in half over and over, he would eventually reach a tiny grain of matter that could not be cut in half. Democritus called these hypothetical building blocks of matter “atoms”, after the Greek atomos, ‘uncuttable’ – suggesting that atoms could not be divided indefinitely into smaller parts and that it is impossible to create new matter. All that you can see is a product of packing together a myriad of miniscule atoms.
He said that atoms were always in motion and as they moved about they collided with other atoms; sometimes they interlocked and held together, sometimes they rebounded from collisions. The Roman poet Lucretius (c.94 – c.55 BCE) imagined Democritus’ atoms with hooks that fastened them together.

His remains one of the earliest attempts to explain the universe with a few simple physical and mathematical laws. For Democritus, there were only two things, space and atoms. Both had always existed and always would exist because de nihilo nihil ‘nothing could come from nothing’. Balancing the idea that the atom is the only unit of being, the Void is the single kind of not being. The higher the atom-to-void ratio, the denser the material. Atoms simply combined with other atoms in the Void; solid, impenetrable, indivisible blocks which never change; they combine to form different things, from rocks to plants to animals. When these things died or fell apart the structure disintegrated and the atoms were free to form new things by combining again in a different shape with other atoms.

Democritus reasoned that the method by which the atoms could combine was through their different shapes. While all the atoms were the same in substance, liquids were thought to have smooth round edges so they could fall over each other, while those that made up solids had toothed rough edges, which could hook on to each other.
Atoms of fire had tetragonal shape; atoms of earth, cubic; atoms of air, octahedral; and those of water were icosahedral.

 

Democritus’ thesis completely rejects the notion of the spiritual or the religious. The soul was explainable through a fast-moving group of atoms brought together by encasement in the body. The motion produced sensations that interacted with the mind (itself a collection of atoms) to produce thoughts and feelings. Once dead, the object that held the fast-moving atoms disintegrated and thus released they could separate and interact with other atoms to form new things. Leaving no place for abstract notions of the supernatural or an afterlife, this marked the arrival of materialism.

THE PLATONIC SOLIDS

As with physical form, Democritus argued that other perceived differences in things, such as their taste, could be explained by the edges of the atoms. Likewise the colour of things was explained by the position of the atoms within a compound, which would result in darker or lighter shades.

Democritus had envisaged free atoms as flying about ceaselessly through empty space. What was needed therefore, was a precise picture of how atoms moved through space. Such a picture required knowledge of the laws that governed all motion.

Picture of the moon crater named after Democritus

DEMOCRITUS CRATER

The Greek philosopher ARISTOTLE rejected Democritus’ idea of the atom and said that matter was completely uniform and continuous. The influence of Aristotle was extraordinary. His concept of matter was at variance with modern thinking, but it was accepted for around 20 centuries until it was replaced by DALTON‘s atomic theory in 1808.

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HIPPOCRATES of COS (c.470 – c.410 BCE)

Fifth/Fourth Century BCE – Greece

‘About 60 medical writings, known collectively as the Hippocratic corpus’

Before the time of Hippocrates there had been little science in medicine. Disease was believed to be the punishment of the gods. Divine intervention came not from the natural, but from the supernatural.

‘Treatment’, therefore, came also from the supernatural, through magic, witchcraft, superstition or religious ritual. Hippocrates’ approach was remarkable given the age in which he lived. ‘There are in fact two things,’ said Hippocrates, ‘science and opinion’; the former begets knowledge, the latter ignorance.’

Although little is known about him, Hippocrates (Bukarat in the Muslim world) is now remembered as the Father of Medicine. He was a contemporary of Socrates and lived on the island of Cos. It is widely acknowledged that Hippocrates himself could not have written many of the texts attributed to him. Written over a century and varying widely in style and argument, it is thought they came from the medical school library of Cos, possibly put together in the first instance by the author to whom they later became attributed.

The corpus is the oldest surviving Western scientific text. It laid the foundations of the western medical tradition. Although its remedies are now considered ‘imaginative’, the corpus speaks the language of science; it does not speak of spells, daemons or gods.

Hippocrates cast aside superstition and focused on the natural, in particular observing, recording and analysing the symptoms and passages of disease. The prognosis of an illness was central to this approach to medicine, partly with a view to being able to avoid in the future the circumstances that were perceived to have initiated the problems in the first place. The development of far-fetched cures or drugs was not important. What came from nature should be cured by nature; therefore rest, healthy diet, exercise, hygiene and air were prescribed for the treatment and prevention of illness. [Vis medicatrix naturae – the healing power of nature]. The fact that Hippocrates was prescribing such a natural solution at all was a major advancement.

Bust said to be of HIPPOCRATES ©

HIPPOCRATES

Hippocratic medicine was based on the balance of four elements – water (cold and moist), air (moist and hot), fire (hot and dry) and earth (cold and dry) – and four humors (bodily fluids) – phlegm, blood, yellow bile and black bile.

Whereas homeopathy attempts to treat disease by inducing in the patient symptoms similar to those displayed, according to the theory that symptoms are a manifestation of the body’s way of coping with the disease, allopathy describes a therapeutic technique that attempts a cure by inducing in the patient different symptoms. Basing treatment on symptoms, not causes, this enantiopathic technique can be fatal when applied to a system based on the humors. A person suffering from influenza is hot and thirsty (dry), so treatment with a mixture of cold and wet could result in a patient sitting a bath of cold water in a draughty room, awaiting a cure.

Personality or temperament was viewed as having four major types – sanguine, choleric, melancholic and phlegmatic.

Hippocrates regarded the body as a single entity, or whole, and the key lay in preserving the natural balance within this entity. Sickness was the sign of imbalance. When each of the factors was present in equal quantities, a healthy body would result. If one element became too dominant, then illness or disease would take over. If, for example, the sickness consisted of an excess of cold, moist humors, then the task of the physician was to restore the balance. The way to treat the problem would be by trying to undertake activities or eat foods which would stimulate the other humors, while at the same time trying to restrain the dominant one, in order to restore the balance and consequently, health.

The concept and treatment of humors endured for the next two thousand years, at least as far as the seventeenth century. The answers he prescribed for healthy living such as diet and exercise are still ‘good medicine’ and moreover, language introduced by Hippocrates still endures; an excess of black bile in Greek was ‘melancholic’, whilst someone with a too dominant phlegm humor became ‘phlegmatic’.

Physicians no longer practice Hippocratic medicine, but his name survives in the Hippocratic Oath that medical students take today. The Oath, probably penned by one of his followers, is a short passage constituting a code of conduct to which henceforth all physicians were obliged to pledge themselves. It outlines, amongst other things, the ethical responsibilities of the doctor to his patients and a commitment to patient confidentiality – an attempt to set physicians in the Hippocratic tradition apart from the spiritual and superstitious healers of their day.

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EMPEDOCLES of AGRIGENTUM (Sicily) (c.494 – c.434 BCE)

‘The four roots of all things are: AIR, WATER, FIRE and EARTH’

Two forces exist – LOVE and STRIFE.

The view of Empedocles developed the monists’ ideas that all substances are derived from a single source, into the concept of objects consisting of different compositions of these four basic elements.
The materials of the natural world being wrought from different blends of the four elemental principles, brought about through the eternal conflict between Love and Strife; their waxing and waning applied to cause mixing when Love is dominant, or separation by Strife.
Empedocles argued that this was the cause of transformation not just of the elements but also of the lives of people and cultures.

Popular mythology described how Aphrodite fashioned the human eye out of the four elements, held together by Love. She kindled the fire of the eye at the hearth fire of the universe, so that it would act like a lantern, transmitting the fire of the eye out into the world and making sight possible.
Empedocles realised that there must be more to sight than this, and that the darkness of night is caused by the body of the Earth getting in the way of light from the sun.

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ANAXAGORAS (c.500 – c.428 BCE)

‘The notion of the indivisible particle’

Anaxagoras came from Ionia but settled in Athens, where he remained for thirty years and taught both Pericles and Euripides. Charged with impiety because of his theory that the Sun is a red-hot stone (such an explanation, denying the role of Helios the sun-god, was enough to warrant prosecution) he fled Athens before the trial and settled in Asia Minor.
What we know about Anaxagoras is based on references to him by later writers.

In the cosmology of Anaxagoras, the Universe began as a homogenous sea of identical basic particles. Nous gave this sea a stir, in the knowledge that in time the particles would so combine to arrange themselves such that everything would be as it is today.

Bust said to be of ANAXAGORAS ©

ANAXAGORAS

Picture of a document showing a seal with a likeness of Anaxagoras ©

Nous was a vital principle akin to the life force of vitalism – the nearest English words being ‘mind’ or ‘intellect’.
The range of the word ‘nous’ is vastly greater, however, as it refers to the combination of insight and intuition which permits the apprehension of the fundamental principles of the cosmos – the concept is closer to the oriental idea of ‘seeing’ than the occidental notion of intelligence founded upon EUCLIDEAN LOGIC.

At the same time, Nous could be the creative, motive intelligence behind the cosmos, almost indistinguishable from the Christian concept of the will of God.

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THALES of MILETUS (624 – 545 BCE)

bust of Thales

THALES

‘WATER is the basis of all matter’

Thales is remembered because he disregarded the classical, mythical understanding of life and the universe in favour of a more physical understanding. Thales and his followers worked with the conviction that there exist natural laws governing the behaviour of natural processes and that future events could be predicted by understanding these laws. One did not have to examine chicken entrails in order to understand the vagaries and whims of the gods who ultimately determined these events.
The basic principle that you can understand nature by studying these natural laws and what things are made of forms the foundation of future philosophy and science.

Born in Miletus on the Aegean coast, Thales engaged in disciplines as varied as engineering and statesmanship.
He had a belief that there must be a fundamental substance or building block from which everything else is made. His conclusion was that this was water. Water was essential for life, drink it and you will grow, take it away and everything dies. Water can exist in a solid form as ice or snow; a bowl of water left exposed will vanish into the air. Water, he thought, could readily be made very fine, in which case it becomes air; alternatively it could be compacted into a slime that becomes earth. Not only was everything made of this elemental substance, but also all life was supported on it, literally. The Earth, said Thales, floated on water.
There were two compelling strands of evidence. First, it couldn’t be supported by air because air was incapable of supporting anything, but water could hold up large objects like logs or ships. Secondly, you could observe the effects of the Earth floating on water in that on occasions it rocked suddenly – an earthquake. Quite obviously this was caused by the water’s movement.

Legend has it that Thales, while traveling in Egypt observed that the ratio of the length of a shadow and the height of an object is the same for all objects if measured at the same time of day, hence when his shadow was the same length as his height, then at the same moment the height of the pyramid would be equal to the length of its shadow.

Nothing remains of Thales’ original work, and all knowledge of him is derived from later philosophers. His way of thinking marks the beginning of a way of investigating the hidden nature of natural things by investigation and systematisation.

The THALES RECTANGLE

THALES RECTANGLE

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ANAXIMANDER of MILETUS (c.611 – c.547 BCE)

‘Apeiron is the basis of all matter’

mosaic of Anaximander

Anaximander

A pupil of THALES of MILETUS. As with Thales, little is known about Anaximander’s life and most of what we know comes from later Greeks, notably Aristotle and Theophrastus.
Anaximander conceptualised the Earth as suspended completely unsupported at the centre of the universe. It had been assumed by other thinkers that the Earth was a flat disc held in place by water, pillars or some other physical structure. Although without a notion of gravity, Anaximander supported his argument by supposing that the Earth, being at the centre of the universe, at equal distances from the extremes,

‘has no inclination to move up rather than down or sideways; and since it is impossible to move in opposite directions at the same time, it necessarily stays where it is’ – ( ARISTOTLE explaining Anaximander’s theory).

Moreover, because the Earth was suspended freely, it allowed Anaximander to propose the idea that the sun, moon and stars orbited in full circles around the Earth.

Anaximander proposed the idea of space or a universe with depth. Rather than the view of the Earth caged in a planetarium style ‘celestial vault’, he argued the celestial bodies (the sun, moon and stars) were different distances away from the Earth, with space or air between them.

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ANAXIMENES

ANAXIMENES of MILETUS (c.585 – c.528 BCE)

‘AIR is the basis of all matter’

Bust of HERACLEITOS ©

HERACLEITUS


HERACLEITUS of EPHESUS (c.535 – c.475 BCE)

‘FIRE is the basis of all matter’

Thales’ pupil Anaximander avoided the issue of ‘prote hyle‘, or first matter by contending that all materials were composed of apeiron, the indefinite and unknowable first substance. Wanting to understand the transformations observed daily in the world, Anaximander believed that change came about through the agency of contending opposite qualities; hot and cold, and dry and moist.

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IMHOTEP (c.2650 BC)

Photograph of King Huni pyramid at Maydoum the 3rd dynasty Old Kingdom at Elfayom (427 x 500) ©

‘Architect of civilisation’

Medical sage, astronomer, mathematician, architect.

Architect of the first pyramid built during the reign of the second pharaoh of the Third Dynasty, with the unification of Upper and Lower Egypt.
Came to be revered as a god of healing and identified by the Greeks with their own Aesculapius.

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ARISTARCHUS of SAMOS (c.320 – c.250 BCE)

 

‘The first Greek astronomer to suggest the Sun was the centre of the solar system was Aristarchus of Samos, around 290 BCE’

No one took him seriously, and most of his writings were lost. We know of him today primarily because Archimedes (whose writings do exist) referred to Aristarchus as holding this apparently nonsensical notion.

Aristarchus believed that the stars must be infinitely far away because they seemed motionless and the motions of the heavenly bodies could easily be understood if it were assumed that all of the planets, including Earth, revolved around the Sun. Copernicus knew of Aristarchus’ views and mentioned them in a passage in De Revolutionibus Orbium Coelestium that he later eliminated, as though not wishing to compromise his own originality.

Aristarchus reasoned that when the moon is illuminated by the Sun as a half-disc, the angle made by the line connecting the Earth and the moon and the line connecting the moon and the Sun would be exactly 90 degrees.

Accordingly, he constructed a pair of similar triangles, and found the relative distances for those two lines to have a ratio of 1:19. His method was correct but his crude instruments gave a false reading.

10th century CE Greek copy of Aristarchus of Samos's 2nd century BCE calculations of the relative sizes of the Sun, Moon and the Earth

The correct ratio is also virtually the same ratio as the diameter of the moon to that of the Sun, which explains why during a total solar eclipse, we view the disc of the moon to fit almost precisely over the disc of the Sun.

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