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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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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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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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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OMAR KHAYYAM (c.1048–1131)

In the service of the Kurdish-Turkish Sultan Salah al-Din ibn Ayyub (Saladin صلاح الدين يوسف بن أيوب‎ ), the nemesis of Richard the Lionheart during the second crusade, there was published a definitive treatise by Khayyām on algebra in which he classified algebraic equations up to the third degree and showed how geometric solutions to the equations could be obtained.

Image depicting OMAR KHAYYAM

Ironically, the source of Khayyām’s most enduring legacy is neither his mathematics nor his science but rather his poetry. The Rubaiyat, a translation, or recomposition, published initially in 1859 by the British poet Edward Fitzgerald, presents his work in a series of melancholic ruminations concerning the irreversibility of fate and the fleeting nature of life.

One explanation of the decline of science in Islāmic civilization, which began in the late fifteenth century, is the general fatalism that pervaded Islāmic culture, as revealed in the melancholia and pathos of Khayyām’s quatrains describing life continuing among the ruins of ancient grandeur.
Another explanation is the emergence in twelfth century Baghdad of an intellectual movement spearheaded by the fundamentalist al-Ghazali, which favoured faith and dogma over reason and direct evidence.

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LEONARDO FIBONACCI (c.1170-c.1250)

Also known as Leonardo Pisano. Published ‘Liber Abaci’ in 1202.

1202 – Italy

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FIBONACCI

Picture of a statue of Leonardo Pisano

FIBONACCI

‘A series of numbers in which each successive term is the sum of the preceding two’

For example:   1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , 89 , 144….

The series is known as the Fibonacci sequence and the numbers themselves as the Fibonacci numbers.

The Fibonacci sequence has other interesting mathematical properties – the ratio of successive terms ( larger to smaller;   1/1, 2/1, 3/2, 5/3, 8/5…. ) approaches the number 1.618
This is known as the golden ratio and is denoted by the Greek letter Phi.

Phi was known to ancient Greeks.
Greek architects used the ratio 1:Phi as part of their design, the most famous example of which is the Parthenon in Athens.

Fibonacci sequence in flower petals. flowers often have a Fibonacci number of petals - link to <http://pinterest.com/mcvjfly/fibonacci/>

Fibonacci sequence in flower petals

Phi also occurs in the natural world.
Flowers often have a Fibonacci number of petals.

      

During his travels in North Africa, Fibonacci learned of the decimal system of numbers that had evolved in India and had been taken up by the Arabs.
In his book Liber Abaci he re-introduced to Europe the Arabic numerals that we use today, adhering roughly to the recipe ‘the value represented must be proportional to the number of straight lines in the symbol’.

Following the Arabs, Fibonacci ( ‘son of the simpleton’ euph. or ‘son of the innocent’ ) introduced the place–value concept, with each position representing a different power of ten and these arranged in ascending order from right to left.

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RENE DESCARTES (1596-1650)

1637 – France

Cogito ergo sum‘ – The result of a thought experiment resolving to cast doubt on any and all of his beliefs, in order to discover which he was logically justified in holding.

Descartes argued that although all his experience could be the product of deception by an evil daemon, the demon could not deceive him if he did not exist.

His theory that all knowledge could be gathered in a single, complete science and his pursuit of a system of thought by which this could be achieved left him to speculate on the source and the truth of all existing knowledge. He rejected much of what was commonly accepted and only recognised facts that could intuitively be taken as being beyond any doubt.

His work ‘Meditations on First Philosophy’ (1641) is centered on his famous maxim. From this he would pursue all ‘certainties’ via a method of systematic, detailed mental analysis. This ultimately led to a detached, mechanistic interpretation of the natural world, reinforced in his metaphysical text ‘Principia Philosophiae‘ (1644) in which he attempted to explain the universe according to the single system of logical, mechanical laws he had earlier envisaged and which, although largely inaccurate, would have an important influence even after Newton. He envisaged the human body as subject to the same mechanical laws as all matter; distinguished only by the mind, which operated as a distinct, separate entity.

Through his belief in the logical certainty of mathematics and his reasoning that the subject could be applied to give a superior interpretation of the universe came his 1637 appendix to the ‘Discourse’, entitled ‘La Geometrie‘, Descartes sought to describe the application of mathematics to the plotting of a single point in space.

This led to the invention of ‘Cartesian Coordinates’ and allowed geometric expressions such as curves to be written for the first time as algebraic equations. He brought the symbolism of analytical geometry to his equations, thus going beyond what could be drawn. This bringing together of geometry and algebra was a significant breakthrough and could in theory predict the future course of any object in space given enough initial knowledge of its physical properties and movement.

Descartes showed that circular motion is in fact accelerated motion, and requires a cause, as opposed to uniform rectilinear motion in a straight line that has the property of inertia – and if there is to be any change in this motion a cause must be invoked.

By the 1660s, there were two rival theories about light. One, espoused by the French physicist Pierre Gassendi (1592-1655) held that it was a stream of tiny particles, traveling at unimaginably high-speed. The other, put forward by Descartes, suggested that instead of anything physically moving from one place to another the universe was filled with some material (dubbed ‘plenum’), which pressed against the eyes. This pressure, or ‘tendency of motion’, was supposed to produce the phenomenon of sight. Some action of a bright object, like the Sun, was supposed to push outwards. This push was transmitted instantaneously, and would be felt by the human eye looking at a bright object.

There were problems with these ideas. If light is a stream of tiny particles, what happens when two people stand face-to-face looking each other in the eye? And if sight is caused by the pressure of the plenum on the eye, then a person running at night should be able to see, because the runner’s motion would make the plenum press against their eyes.

Descartes original theory is only a small step to a theory involving pulses of pressure spreading out from a bright object, like the pulses of pressure that would travel through water if you slap the surface, and exactly equivalent to pressure waves which explain how sound travels outward from its source.

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PIERRE DE FERMAT (1601- 65) ANDREW WILES (b.1953)

1637 – France; 1993 – USA

Portrait of PIERRE DE FERMAT

PIERRE DE FERMAT

Fermat’s theorem proves that there are no whole-number solutions of the equation x n + y n = z n for n greater than 2

The problem is based on Pythagoras’ Theorem; in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares on the other two sides; that is x 2 + y 2 = z 2

If x and y are whole numbers then z can also be a whole number: for example 52+ 122 = 132
If the same equation is taken to a higher power than 2, such as x 3 + y 3 = z 3 then z cannot ever be a whole number.

In about 1637, Fermat wrote an equation in the margin of a book and added ‘I have discovered a truly marvelous proof, which this margin is too small to contain’. The problem now called Fermat’s Last Theorem baffled mathematicians for 356 years.

photo of Andrew Wiles in classroom

ANDREW WILES

In 1993, Wiles, a professor of mathematics at Princeton University, finally proved the theorem.

Wiles, born in England, dreamed of proving the theorem ever since he read it at the age of ten in his local library. It took him years of dedicated work to prove it and the 130-page proof was published in the journal ‘Annals of Mathematics‘ in May 1995.

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BLAISE PASCAL (1623- 62)

1647 – France

Portrait of BLAISE PASCAL

BLAISE PASCAL

‘When pressure is applied anywhere to an enclosed fluid, it is transmitted uniformly in all directions’

EVANGELISTA TORICELLI (1608-47) had argued that air pressure falls at higher altitudes.

Using a mercury barometer, Pascal proved this on the summit of the 1200m high Puy de Dome in 1647. His studies in this area led to the development of PASCAL’S PRINCIPLE, the law that has practical applications in devices such as the car jack and hydraulic brakes. This is because the small force created by moving a lever such as the jacking handle in a sizable sweep equates to a large amount of pressure sufficient to move the jack head a few centimetres.
The unit of pressure is now termed the pascal.

‘The study of the likelihood of an event’

Together with PIERRE DE FERMAT, Pascal developed the theory of probabilities (1654) using the now famous PASCAL’S TRIANGLE.

Chance is something that happens in an unpredictable way. Probability is the mathematical concept that deals with the chances of an event happening.

Probability theory can help you understand everything from your chances of winning a lottery to your chances of being struck by lightning. You can find the probability of an event by simply dividing the number of ways the event can happen by the total number of possible outcomes.
The probability of drawing an ace from a full pack of cards is 4/52 or 0.077.

Probability ranges from 1 (100%) – Absolutely certain, through Very Likely 0.9 (90%) and Quite Likely 0.7 (70%), Evens (Equally Likely) 0.5 (50%), Not Likely 0.3 (30%) and Not Very Likely 0.2 (20%), to Never – Probability 0 (0%).

Picture of the 'Pascaline'. The French mathematician Blaise Pascal invented the a mechanical calculation machine. He called it the Pascaline. The Pascaline was made out of clock gears and levers and could solve basic mathematical problems like addition and subtraction.

 
 

The computer language Pascal is named in recognition of his invention in 1644 of a mechanical calculating machine that could add and subtract.

 
 
 

Like many of his contemporaries, Pascal did not separate philosophy from science; in his book ‘Pensees’ he applies his mathematical probability theory to the problem of the existence of God. In the absence of evidence for or against God’s existence, says Pascal, the wise man will choose to believe, since if he is correct he will gain his reward, and if he is incorrect he stands to lose nothing.

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GOTTFRIED LEIBNIZ (1646-1716)

1684 – Germany

‘A new method for maxima and minima, as well as tangents … and a curious type of calculation’

Newton invented calculus (fluxions) as early as 1665, but did not publish his major work until 1687. The controversy continued for years, but it is now thought that each developed calculus independently.
Terminology and notation of calculus as we know it today is due to Leibniz. He also introduced many other mathematical symbols: the decimal point, the equals sign, the colon (:) for division and ratio, and the dot for multiplication.

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MECHANICSMECHANICS

ISAAC NEWTON (1642-1727)

1687 – England

‘Any two bodies attract each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between them’

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NEWTON

The force is known as gravitation
Expressed as an equation:

F = GmM/r2

where F is Force, m and M the masses of two bodies, r the distance between them and G the gravitational constant
This follows from KEPLER’s laws, Newton’s laws of motion and the laws of conic sections. Gravitation is the same thing as gravity. The word gravity is particularly used for the attraction of the Earth for other objects.

Gravitation
Newton stated that the law of gravitation is universal; it applies to all bodies in the universe. All historical speculation of different mechanical principles for the earth from the rest of the cosmos were cast aside in favour of a single system. He demonstrated that the planets were attracted toward the Sun by a force varying as the inverse square of the distance and generalized that all heavenly bodies mutually attract one another. Simple mathematical laws could explain a huge range of seemingly disconnected physical facts, providing science with the straightforward explanations it had been seeking since the time of the ancients. That the constant of gravitation is in fact constant was proved by careful experiment, that the focus of a body’s centre of gravity appears to be a point at the centre of the object was proved by his calculus.

Calculus
The angle of curve, by definition, is constantly changing, so it is difficult to calculate at any particular point. Similarly, it is difficult to calculate the area under a curve. Using ARCHIMEDES’ method of employing polygons and rectangles to work out the areas of circles and curves, and to show how the tangent or slope of any point of a curve can be analyzed, Newton developed his work on the revolutionary mathematical and scientific ideas of RENE DESCARTES, which were just beginning to filter into England, to create the mathematics of calculus. Calculus studies how fast things change.
The idea of fluxions has become known as differentiation, a means of determining the slope of a line, and integration, of finding the area beneath a curve.

Newton’s ideas on universal gravitation did not emerge until he began a controversial correspondence with ROBERT HOOKE in around 1680. Hooke claimed that he had solved the problem of planetary motion with an inverse square law that governed the way that planets moved. Hooke was right about the inverse square law, but he had no idea how it worked or how to prove it, he lacked the genius that permitted Newton to combine Kepler’s laws of planetary motion with the assumption that an object falling towards Earth was the same kind of motion as the Earth’s falling toward the Sun.
It was not until EDMUND HALLEY challenged Newton in 1684 to show how planets could have the elliptical orbits described by Johannes Kepler, supposing the force of attraction by the Sun to be the reciprocal of their distance from it – and Newton replied that he already knew – that he fully articulated his laws of gravitation.

It amounts to deriving Kepler’s first law by starting with the inverse square hypothesis of gravitation. Here the Sun attracts each of the planets with a force that is inversely proportional to the square of the distance of the planet from the Sun. From Kepler’s second law, the force acting on the planets is centripetal. Newton says this is the same as gravitation.

In the previous half century, Kepler had shown that planets have elliptical orbits and GALILEO had shown that things accelerate at an even pace as they fall towards the ground. Newton realized that his ideas about gravity and the laws of motion, which he had only applied to the Earth, might apply to all physical objects, and work for the heavens too. Any object that has mass will be pulled towards any other object. The larger the mass, the greater the pull. Things were not simply falling but being pulled by an invisible force. Just as this force (of gravity) pulls things towards the Earth, it also keeps the Moon in its orbit round the Earth and the planets moving around the Sun. With mathematical proofs he showed that this force is the same everywhere and that the pull between two things depends on their mass and the square of the distance between them.

Title page of Philosophiae Naturalis Principia Mathematica

Title page of Philosophiae Naturalis Principia Mathematica

Newton published his law of gravitation in his magnum opus Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy) in 1687. In it Newton analyzed the motion of orbiting bodies, projectiles, pendulums and free fall near the Earth.

The first book of Principia states the laws of motion and deals with the general principles of mechanics. The second book is concerned mainly with the motion of fluids. The third book is considered the most spectacular and explains gravitation.

Why do two objects attract each other?
‘I frame no hypotheses’, said Newton

It was Newton’s acceptance of the possibility that there are mysterious forces in the world, his passions for alchemy and the study of the influence of the Divine that led him to the idea of an invisible gravitational force – something that the more rationally minded Galileo had not been able to accept.
Newton’s use of mathematical expression of physical occurrences underlined the standard for modern physics and his laws underpin our basic understanding of how things work on an everyday scale. The universality of the law of gravitation was challenged in 1915 when EINSTEIN published the theory of general relativity.

1670-71 Newton composes ‘Methodis Fluxionum‘, his main work on calculus, which is not published until 1736. His secrecy meant that in the intervening period, the German mathematician LEIBNIZ could publish his own independently discovered version – he gave it the name calculus, which stuck.

LAWS OF MOTION

1687 – England

  • First Law: An object at rest will remain at rest and an object in motion will remain in motion at that velocity until an external force acts on the object

  • Second Law: The sum of all forces (F) that act on an object is equal to the mass (m) of the object multiplied by the acceleration (a), or F = ma

  • Third Law: To every action, there is an equal and opposite reaction

The first law

introduces the concept of inertia, the tendency of a body to resist change in its velocity. The law is completely general, applying to all objects and any force. The inertia of an object is related to its mass. Things keep moving in a straight line until they are acted on by a force. The Moon tries to move in a straight line, but gravity pulls it into an orbit.
Weight is not the same as mass.

The second law

explains the relationship between mass and acceleration, stating that a force can change the motion of an object according to the product of its mass and its acceleration. That is, the rate and direction of any change depends entirely on the strength of the force that causes it and how heavy the object is. If the Moon were closer to the Earth, the pull of gravity between them would be so strong that the Moon would be dragged down to crash into the Earth. If it were further away, gravity would be weaker and the Moon would fly off into space.

The third law

shows that forces always exist in pairs. Every action and reaction is equal and opposite, so that when two things crash together they bounce off one another with equal force.

LIGHT

1672 – New Theory about Light and Colours is his first published work and contains his proof that white light is made up of all colours of the spectrum. By using a prism to split daylight into the colours of the rainbow and then using another to recombine them into white light, he showed that white light is made up of all the colours of the spectrum, each of which is bent to a slightly different extent when it passes through a lens – each type of ray producing a different spectral colour.

Newton also had a practical side. In the 1660s his reflecting telescope bypassed the focusing problems caused by chromatic aberration in the refracting telescope of the type used by Galileo. Newton solved the problem by swapping the lenses for curved mirrors so that the light rays did not have to pass through glass but reflected off it.

At around the same time, the Dutch scientist CHRISTIAAN HUYGENS came up with the convincing but wholly contradictory theory that light travels in waves like ripples on a pond. Newton vigorously challenged anyone who tried to contradict his opinion on the theory of light, as Robert Hooke and Leibniz, who shared similar views to Huygens found out. Given Newton’s standing, science abandoned the wave theory for the best part of two hundred years.

1704 – ‘Optiks’ published. In it he articulates his influential (if partly inaccurate) particle or corpuscle theory of light. Newton suggested that a beam of light is a stream of tiny particles or corpuscles, traveling at huge speed. If so, this would explain why light could travel through a vacuüm, where there is nothing to carry it. It also explained, he argued, why light travels in straight lines and casts sharp shadows – and is reflected from mirrors. His particle theory leads to an inverse square law that says that the intensity of light varies as the square of its distance from the source, just as gravity does. Newton was not dogmatic in Optiks, and shows an awareness of problems with the corpuscular theory.

In the mid-eighteenth century an English optician John Dolland realized that the problem of coloured images could largely be overcome by making two element glass lenses, in which a converging lens made from one kind of glass was sandwiched together with a diverging lens made of another type of glass. In such an ‘achromatic’ lens the spreading of white light into component colours by one element was cancelled out by the other.

During Newton’s time as master of the mint, twenty-seven counterfeiters were executed.

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DANIEL BERNOULLI (1700- 82) JAMES CLERK MAXWELL (1831- 79)

1738 – Switzerland
1859 – England

‘Gases are composed of molecules which are in constant random motion and their properties depend upon this motion’

The volume of a gas is simply the space through which molecules are free to move. Collisions of the molecules with each other and the walls of a container are perfectly elastic, resulting in no decrease in kinetic energy. The average kinetic energy of a gas increases with an increase in temperature and decreases with a decrease in temperature. The theory has been extended to provide a model for two states of matter – liquids and solids.

Bernoulli had a great advantage over DEMOCRITUS. He knew that free atoms were more than simply tiny grains flying though space; they were tiny grains flying through space and obeying NEWTON’s Laws of Motion.
Bernoulli proposed a ‘bombardment theory’, which stated that a gas consisted of tiny particles in rapid, random motion like a swarm of angry bees. He realized that in the case of such a gas visualized as a host of tiny grains in perpetual frenzied motion, the atoms hammering relentlessly on the walls of any containing vessel would produce a force by bombarding the container. The effect of each individual impact would of course be vanishingly small. The effect of billions upon billions of atoms, hammering away incessantly, however, would be to push the walls back. A gas made of atoms would exert a jittery force that we would detect as a ‘pressure’.

Heating a gas would make its particles move faster.
The pressure of a gas such as steam was easy to measure using a piston in a hollow container. This was essentially a moveable wall. To deduce how the pressure of a gas would be affected by different conditions, Bernoulli first made some simplifying assumptions. He assumed the atoms were very small compared to the gulf between them. This allowed Bernoulli to ignore any force – whether of attraction or repulsion – that existed between them, as being unlikely to be ‘long range’. (This is an ‘ideal’ or ‘perfect’ gas. The behaviour of a real gas may differ from the ideal, for example at very high pressure). With the motion of each atom unaffected by its fellows, Newton’s laws dictated that it should fly at a constant speed in a straight line. The exception was when it slammed into a piston or the walls of the container. Bernoulli assumed that in such a collision a gas atom bounced off the walls of the surface without losing any speed, in the process imparting a miniscule force to the wall.

What would happen if the volume of the gas were reduced by applying an outside force to the piston? If the gas were reduced to half its original volume, the atoms would now have to fly only half as far between collisions, in any given time they would collide with the piston twice as many times and would exert twice the pressure. Similarly, if the gas were compressed to a third of its volume, its pressure would triple. This had been observed by ROBERT BOYLE in 1660 and named Boyle’s Law.

What would happen to the pressure of gas in a closed cylinder if the gas were heated while its volume remained unchanged? Exploiting the insight that the temperature of a gas was a measure of how fast on average its atoms were flying about, that when a gas was heated, its atoms speeded up, he deduced that as the atoms would be moving faster they would collide with the piston more often and create a greater force. Consequently the pressure of the gas would rise. This was observed by the French scientist JACQUES ALEXANDRE CESARE CHARLES in 1787, and christened Charles’ law.

After 120 years MAXWELL polished Bernoulli’s ideas into a rigorous mathematical theory. In Germany, LUDWIG  BOLTZMANN championed the atomic hypothesis, but was refuted by the Austrian ERNST MACH, who was convinced that science should not concern itself with any feature of the world that could not be observed directly with the senses.

BERNOULLI’S PRINCIPLE

‘As the velocity of a liquid or gas increases, its pressure decreases; and when the velocity decreases, its pressure increases’

At a narrow constriction in a pipe or tube, the speed of a gas or liquid is increased, but its pressure is decreased, according to Bernoulli’s principle. This effect is named the Venturi effect (and a pipe or tube with a narrow constriction the Venturi tube) after the Italian G.B. Venturi (1746-1822) who first observed it in constrictions in water channels. An atomiser works on the same principle.

 

The principle is expressed as a complex equation, but it can be summed up simply as the faster the flow the lower the pressure.

An aircraft wing’s curved upper surface is longer than the lower one, which ensures that air has to travel further and so faster over the top than it does below the wing. Hence the air pressure underneath is greater than on top of the wing, causing an upward force, called lift.

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