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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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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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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