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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EUDOXUS (c.375 BCE)

‘Pupil of Plato’

Eudoxus flourished around the middle of the 4th century BCE; he was an astronomer initiated into the Egyptian mysteries, obtaining his knowledge of the art from the priests of Isis.

EUDOXUS CRATER Famed for his early contributions to understanding the movement of the planets. His work on proportions shows rigorous treatment of continuous quantities, whole numbers or even rational numbers. Craters on Mars and the Moon are named in his honor.

EUDOXUS CRATER

His work is passed to us through Aristotle.

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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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ERATOSTHENES (c.275 – 194 BCE)

Third Century BCE – Alexandria, Egypt

‘At noon on the day of the summer solstice, the Sun is directly overhead in Syene (now Aswan) and there is no shadow, but at the same time in Alexandria the Sun is at an angle and there is a measurable shadow’

Eratosthenes used this concept to calculate the circumference of the Earth.

In 230 BCE, the Greek philosopher Eratosthenes worked out the circumference of the Earth to be 25,000 miles (40,000 km) by studying shadows cast by the Sun in both Alexandria and Syene on the day of the summer solstice. Eratosthenes knew from his predecessors that at noon on the longest day of the year (the summer solstice), the Sun would be directly overhead at Syene when a vertical post would cast no shadow, whereas a post in Alexandria 800 kilometers to the north would have a measurable shadow

diagram explaining how Eratosthenes was able to calculate the size of the Earth by measuring shadows at different locations a known distance apart

Eratosthenes reasoned that the surface of the Earth was curved, resulting in the Sun’s rays being different in different locations. With the aid of simple geometrical instruments he found that in Alexandria at noon the Sun’s rays were falling at an angle of 7.2 degrees, which is one fiftieth of 360 degrees. Having determined the difference in the angles between the axes of the two posts, these axes, if extrapolated downwards would meet at the centre of a spherical Earth. Knowing the distance between the two places, he calculated that the circumference of the Earth was fifty times that distance.

Drawing of head of ERATOSTHENES © 

As 7 degrees is approximately one-fiftieth of a circle, multiplying the 800 km distance between the posts by 50 gives a circumference for the Earth of 40,000 km and dividing by pi gives a diameter of 12,800 km.

Eratosthenes’ value comes to 39,350 kilometres, compared to a true average length of 40,033 kilometres.

Eratosthenes was a scholar, an astronomer, mathematician, geographer, historian, literary critic and poet. He was nicknamed ‘Beta’ (the second letter of the Greek alphabet) because he was considered the second best at everything.

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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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ASTROLOGY

– throughout the Middle Ages, astrology and astronomy were closely linked in both the Western and the Arabic worlds.
Although astrology was used for prediction, pre-modern astrology required a substantial command of mathematics and an informed astronomical knowledge.

PTOLEMY – ‘ The Almagest ’ how the planets move; ‘ Tetrabiblos ’ what effect the qualities of the planets (Mars – hot & dry, Moon – cold & wet [affect on the tides]) and their relative positions will have.

Belief that the influence of the planets may have an effect on earthly health and other matters (disease and character traits).

Tables of positions of planets became developed from the Babylonian originals in the Islāmic world.

Alphonsine tables produced for King Alphonso X of Castile in 1275.

Prognostication repeatedly condemned by the Church as influence of the planets denies the concept of free will.

Refutation of astrology is difficult owing to its complexity.

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CLAUDIUS PTOLEMY (c.90-168)

(NOT to be confused with the royal dynasty of the Ptolemys)

c.150 – Alexandria, Egypt

‘The Earth is at the centre of all the cosmos’

This erroneous belief dominated astronomy for 14 centuries.

‘The Earth does not rotate; it remains at the centre of things because this is its natural place – it has no tendency to go either one way or the other. Around it and in successively larger spheres revolve the Moon, Mercury, Venus, the Sun, Mars, Jupiter and Saturn, all of them deriving their motion from the immense and outermost spheres of fixed stars’. Ptolemy wrote in the thirteen-volume Almagest (Arabic for ‘The Greatest’), in which he synthesised the work of his predecessors. It provided a definitive compilation of all that was known and accepted in the field of astronomy up to that point.

Almagest’s eminence, importance and influence can only be compared with Euclid’s Elements. A major part of Almagest deals with the mathematics of planetary motion. Ptolemy explained the wandering of the planets by a complicated system of cycles and epicycles. Starting from the Aristotelian notion that the earth was at the centre of the universe, with the stars and the planets rotating in perfect circles around it, the Ptolemaic system argued for a system of ‘deferents’, or large circles, rotating around the earth, and eighty epicycles, or small circles, which circulated within the deferents. He also examined theories of ‘movable eccentrics’. These proposed just one circle of rotation, with its centre slightly offset from the earth, as well as ‘equants’ – imaginary points in space that helped define the focal point of the rotation of the celestial bodies. Ptolemy’s texts were written with such authority that later generations struggled for a thousand years to convincingly challenge his theories and they remained the cornerstone of Western and Arab astronomy until the sixteenth century.

Ptolemy’s theory was challenged by COPERNICUS and demolished by KEPLER. Ptolemy supported Eratosthenes’ view that the Earth is spherical.

Ptolemy’s other major text is his Tetrabiblos, a founding work on the then science of astrology.

Despite that Ptolemy’s ideas of a geocentric universe have been shown to be erroneous by modern researchers it must be remembered that at the time the observable phenomena would support this view of the cosmos. Without a more informed understanding of the mechanisms involved it can appear that heavenly bodies do in fact move according to the Ptolemaeic model and mathematical evidence was available to provide verification and vindication.

 Medieval Astronomy from Melk Abbey Credit: Paul Beck (Univ. Vienna), Georg Zotti (Vienna Inst. Arch. Science) Copyright: Library of Melk Abbey, Frag. 229  Explanation: Discovered by accident, this manuscript page provides graphical insight to astronomy in medieval times, before the Renaissance and the influence of Nicolaus Copernicus, Tycho de Brahe, Johannes Kepler, and Galileo. The intriguing page is from lecture notes on astronomy compiled by the monk Magister Wolfgang de Styria before the year 1490 at Melk Abbey in Austria. The top panels clearly illustrate the necessary geometry for a lunar (left) and solar eclipse in the Earth-centered Ptolemaic system. At lower left is a diagram of the Ptolemaic view of the solar system and at the lower right is a chart to calculate the date of Easter Sunday in the Julian calendar. Text at the upper right explains the movement of the planets according to the Ptolemaic system. The actual manuscript page is on view at historic Melk Abbey as part of a special exhibition during the International Year of Astronomy.

Library of Melk Abbey, Frag. 229

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GALEN OF PERGAMUM (130-201)

161 – Rome, Italy

Bust of GALEN

GALEN

‘A body of work consisting 129 volumes. Some of the deductions were wrong’

Born in Pergamum (now Bergama in Turkey) in the reign of the Emperor Hadrian (76-138AD)
Studied in Corinth and Alexandria
157 – became surgeon to the Pergamum gladiators
161 – became physician to the emperors Marcus Aurielius and Commodus

Famous for the sheer volume of medical thought which he presented. He summarized his observations in books such as ‘On The Usefulness of Parts of The Body’. His works on medical science became accepted as the only authority on the subject for the following 1400 years. One explanation is that Galen not only incorporated the results of his own findings in his texts, but also compiled the best of all other medical knowledge that had gone before him into a single collection, such as that of Hippocrates.
In particular, Galen adopted Hippocrates’ ‘four humors’ approach to the body. This resulted from a desire to see in bodily conditions the attributes of the four Aristotelian elements. Thus earth was reflected in the body as black bile or melancholy; air as yellow bile or choler; fire as blood and water as phlegm.

After the move to Rome in 161 Galen became physician to emperors Marcus Aurelius, Lucius Verus, Commodus and Septimus Severus. This position allowed him the freedom to undertake dissection in the quest for improved knowledge.
Galen was not permitted to scrutinise human cadavers, so he dissected animals and Barbary apes. His most important conclusions concerned the central operation of the human body. Sadly they were only influential in that they limited the search for accurate information for the next millennia and a half.

Many people visited the shrine of Asklepios, the god of healing in Galen’s hometown, to seek cures for ailments and Galen observed first-hand the symptoms and treatment of diseases. After spells in Smyrna (now Izmir), Corinth and Alexandria studying philosophy and medicine and incorporating work on the dissection of animals, he returned to Pergamum in 157, where he took a position as physician to gladiators, giving him further first-hand experience in practical anatomical medicine. He realized that there were two types of blood flow from wounds. In one the blood was bright red and came spurting out, and in the other it was dark blue and flowed out in a steady stream. These observations convinced him these were two different types of blood. He also believed there was a third form of blood that flowed along nerves.

Galen believed that blood was formulated in the liver, the source of ‘natural spirit’. In turn this organ was nourished by the contents of the stomach that was transported to it. Veins from the liver carried blood to the extremes of the body where it was turned into flesh and used up, thus requiring more food on a daily basis to be converted into blood. Some of this blood passed through the heart’s right ventricle, then seeped through to the left ventricle and mixed with air from the lungs, providing ‘vital spirit’ which then passed into the body through the arteries and regulated the body’s heat. A portion of this blood was transported to the brain where it blended with ‘animal spirit’, which was passed through the body by the nerves. This created movement and the senses. The combination of these three spirits managed the body and contributed to the make-up of the soul. It was not until 1628 that WILLIAM HARVEY‘s system of blood circulation conclusively proved the idea of a single, integrated system.

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THE MEDIEVAL ARAB SCIENTISTS

A great deal of what we know about the ancient world and its scientific ideas has come to us from documents which were translated from ancient Greek or other ancient languages into Arabic, and later from Arabic into European languages. The material reached the Arab world in many cases through the Roman empire in the East, Byzantium, which survived until 1453, almost a thousand years after the fall of Rome, during the period known in Europe as the Dark Ages.
During this time the consolidating influence of Islāmic religion saw Arab Muslims begin to build an empire that was to stretch across the Middle East and across North Africa into Spain. At the heart of the Islāmic world the caliphs ruled in Baghdad. Arab scientists sowed the seeds that would later be reaped in the scientific revolution of the seventeenth century, especially under the Abbasid dynasty during the caliphate of Harun al-Rashid and his son al-Mamun, and the Middle East became the intellectual hub of the World.

depiction of early islamic scholars at work at various scientific investigations

In the ninth century, at the House of Wisdom – a mixture of library, research institute and university – scholars worked to translate the great works of the GREEK thinkers. Muslim scholars of this golden age made important and original contributions to mathematics and astronomy, medicine and chemistry. They developed the ASTROLABE, which enabled astronomers to measure the position of the stars with unparalleled accuracy.Astrology & Astronomy in Iran and Ancient Mesopotamia: Astrolabe: An ancient astronomical instrument
In medicine they made the first serious studies of drugs and advanced surgery. A number of mathematicians, including Habash al-Hasib (‘he who calculates’), Abul’l-Wafa al-Buzjani, Abu Nasr al-Iraq and Ibn Yunus formulated trigonometry (including all six trig functions [ sin, cosec, cos, sec, tan, and cot ]) at a level far above that introduced by the Greek astronomer-mathematician HIPPARCHUS in the second century BCE.
It is largely through such efforts that Greek ideas were preserved through the DARK AGES.

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Eight hundred years before COPERNICUS, a model of the solar system was advanced with the Earth as a planet orbiting the Sun along with other planets.

A few centuries later this idea fell into disfavour with the early Christian Church, which placed mankind at the centre of the universe in a geo-centric model. The alternative teaching would be deemed heresy punishable by death and it would not be until the seventeenth century that the work of GALILEO, KEPLER and NEWTON gave credence to the ideas revitalized by Copernicus in 1543.

It is worth noting that even to-day at least half the named stars in the sky bear Arabic names (Aldebaran and Algol amongst others) and many terms used in astronomy, such as Nadir and Azimuth, are originally Arabic words.

 The Ulugh Beg Observatory in Samarkand, Uzbekistan

The elaborate observatory established by the Ulugh Begg in Samarkand in the fifteenth century appeared to function with a dictum meant to challenge PTOLEMY’s geocentric picture of the universe sanctioned by the Church in Europe. Arabic scholars had access to the early teachings of ARISTARCHUS, the astronomer from Samos of the third century BCE. (referred to by Copernicus in the forward of an early draft of De Revolutionibus, although omitted from the final copy)

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J’BIR IHBIN AYAM (722-804)

Around two thousand texts are attributed to this name; the founder of a Shi’ite sect. They were written over a hundred and fifty year period either side of the year 1000.

‘Sulfur and Mercury hypothesis’ (the idea that the glisten of mercury and the yellow of sulphur may somehow be combined in the form of gold).

An Alchemical theory: Accepting the Aristotelian ‘fundamental qualities’ of hot, cold, dry and moist, all metals are composed of two principles. Under the ground two fumes – one dry and smoky (sulfur), one wet and vaporous (mercury) – arising from the centre of the Earth, condense and combine to form metals.

This is said to explain the similarity of all metals; different metals contain different proportions of these two substances. In base metals the combination is impure, in silver and gold they co-exist in a higher state of purity.

The idea underpins the theory of transmutation, as all metals are composed of the same substances in differing proportions, and became the cornerstone of all chemical theory for the next eight hundred years.

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AL-KHWARIZMI (800-847)

820 – Baghdad, Iraq

Portrait of AL-KHWARIZMI

AL-KHWARIZMI

The man often credited with the introduction of ‘Arabic’ numerals was al-Khwarizmi, an Arabian mathematician, geographer and astronomer. Strictly speaking it was neither invented by al-Khwarizmi, nor was it Middle Eastern in origin.

786 – Harun al-Rashid came to power. Around this time al-Khwarizmi born in Khwarizm, now Khiva, in Uzbekistan.

813 – Caliph al-Ma’mun, the patron of al-Khwarizmi, begins his reign in Baghdad.

Arabic notation has its roots in India around 500 AD, thus the current naming as the ‘Hindu-Arabic’ system. al-Khwarizmi, a scholar in the Dar al-ulum (House of Wisdom) in Baghdad in the ninth century, is responsible for introducing the numerals to Europe. The method of using only the digits 0-9, with the value assigned to them determined by their position, as well as introducing a symbol for zero, revolutionised mathematics.

al-Khwarizmi explained how this system worked in his text ‘Calculation with Hindu numerals‘. He was clearly building upon the work of others before him, such as DIOPHANTUS and BRAHMAGUPTA, and on Babylonian sources that he accessed through Hebrew translations. By standardizing units, Arabic numerals made multiplication, division and every other form of mathematical calculation much simpler. His text ‘al-Kitab al-mukhtasar- fi hisab al-jabr w’al-muqabala’ (The Compendious Book on Calculating by Completion and Balancing) gives us the word algebra. In this treatise, al-Khwarizmi provides a practical guide to arithmetic.

In his introduction to the book he says the aim of the work is to introduce ‘what is easiest and most useful in mathematics, such as men constantly require in cases of inheritance, legacies, partition, lawsuits and trade, and in all their dealings with one another, or when measuring lands, digging canals and making geometrical calculations.’ He introduced quadratic equations, although he described them fully in words and did not use symbolic algebra.
It was in his way of handling equations that he created algebra.

The two key concepts were the ideas of completion and balancing of equations. Completion (al-jabr) is the method of expelling negatives from an equation by moving them to the opposite side

4x2 = 54x – 2x2  becomes  6x2 = 54x

Balancing (al-muqabala) meanwhile, is the reduction of common positive terms on both sides of the equation to their simplest forms

x2 + 3x + 22 = 7x + 12  becomes  x2 + 10 = 4x

Thus he was able to reduce every equation to simple, standard forms and then show a method of solving each, showing geometrical proofs for each of his methods – hence preparing the stage for the introduction of analytical geometry and calculus in the seventeenth century.

The name al-Khwarizmi also gives us the word algorithm meaning ‘a rule of calculation’, from the Latin title of the book, Algoritmi de numero Indorum.

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AL-BIRUNI (973-1050)

The Persian scholar al-Biruni lived around the same time as ibn-Sina. He pioneered the idea that light travels faster than sound, promoted the idea that the Earth rotates on its axis and measured the density of 18 precious stones and metals.

portrait of al-biruni

He classified gems according to the properties: colour; powder colour; dispersion (whether white light splits up into the colours of the rainbow when it goes through the gem); hardness; crystal shape; density.
He used crystal shape to help him decide whether a gemstone was quartz or diamond.

He noted that flowers have 3,4,5 or 8 petals, but never 7 or 9.

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IBN SINA (AVICENNA) (980-1037)

‘al Qann fi al-Tibb’ (The Canon of Medicine), also ‘ The Book of the Remedy

Avicenna lived under the Sammarid caliphs in Bukhara. He identified different forms of energy – heat, light and mechanical – and the idea of a force.

drawing of Ibn Sina ©

AVICENNA

Before GALEN, scientists describing nature followed the old Greek traditions of giving the definitions and following them up with the body of logical development. The investigator was then obliged merely to define the various types of ‘nature’ to be found. With Galen this procedure was changed.

Instead of hunting for these natures and defining more and more of them, reproducing ARISTOTLE’s ideas, AVICENNA, a Persian physician, planned inductive and deductive experimental approaches to determine the conditions producing observable results.

His tome surveyed the entire field of medical knowledge from ancient times up to the most up to date Muslim techniques. Avicenna was the first to note that tuberculosis is contagious; that diseases can spread through soil and water and that a person’s emotions can affect their state of physical health. He was the first to describe meningitis and realize that nerves transmit pain. The book also contained a description of 760 drugs. Its comprehensive and systematic approach meant that once it was translated into Latin in the twelfth century it became the standard medical textbook in Europe for the next 600 years.

Arabic Canon of Medicine by Avicenna 1632. Many physicians in the Islamic world were outstanding medical teachers and practitioners. Avicenna (980-1037 CE) was born near Bokhara in Central Asia. Known as the 'Prince of Physicians', his Canon of Medicine (medical encyclopedia) remained the standard text in both the East and West until the 16th century and still forms the basis of Unani theory and practice today. Divided into five books, this opening shows the start of the third book depicting diseases of the brain.

Arabic Canon of Medicine by Avicenna 1632

Avicenna thought of light as being made up of a stream of particles, produced in the Sun and in flames on Earth, which travel in straight lines and bounce off objects that they strike.

A pinhole in a curtain in a darkened room causes an inverted image to be projected, upside-down, onto a wall opposite the curtained window. The key point is that light travels in straight lines. A straight line from the top of a tree some distance away, in a garden that the window of the camera obscura faces onto – passing through the hole in the curtain – will carry on down to a point near the ground on the wall opposite. A straight line from the base of the tree will go upwards through the hole to strike the wall opposite near the ceiling. Straight lines from every other point on the tree will go through the hole to strike the wall in correspondingly determined spots, the result is an upside-down image of the tree (and of everything else in the garden).

He realized that refraction is a result of light traveling at different speeds in water and in air.

He used several logical arguments to support his contention that sight is not a result of some inner light reaching outward from the eye to probe the world around it, but is solely a result of light entering the eye from the world outside – realizing that ‘after-images’ caused by a bright light will persist when the eyes are closed and reasoning that this can only be the result of something from outside affecting the eyes. By effectively reversing the extro-missive theory of Euclid, he formulated the idea of a cone emanating from outside the eye entering and thus forming an image inside the eye – he thus introduced the modern idea of the ray of light.

The idea which was to have the most profound effect on the scientific development of an understanding of the behaviour of light was the thought of the way images are formed on a sunny day by the ‘camera obscura’.

AL HAZEN (c.965-1039)

Born in Basra and working in Egypt under al-Hakim, Abu Ali al-Hassan ibn al-Haytham was one of the three greatest scientists of Islam (along with al-Biruni and ibn-Sina). He explained how vision works in terms of geometric optics and had a huge influence on Western science. He is regarded as one of the earliest advocates of the scientific method.

The mathematical technique of ‘casting out of nines’, used to verify squares and cubes, is attributed to al-Hazen.

Al-Hazen dissented with the J’bir Ayam hypothesis of transmutation, thus providing two different strands for Alchemy in Europe from the Islāmic world.

diagram explaining the working of the eye

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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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