GALILEO GALILEI (1564-1642)

1632 – Italy

‘Discounting air resistance, all bodies fall with the same motion; started together, they fall together. The motion is one with constant acceleration; the body gains speed at a steady rate’

From this idea we get the equations of accelerated motion:
v = at and s = 1/2at2
where v is the velocity, a is the acceleration and s is the distance traveled in time t

The Greek philosopher ARISTOTLE (384-322 BCE) was the first to speculate on the motion of bodies. He said that the heavier the body, the faster it fell.
It was not until 18 centuries later that this notion was challenged by Galileo.

The philosophers of ancient Greece had known about statics but were ignorant of the science of dynamics.
They could see that a cart moves because a horse pulls it, they could see that an arrow flies because of the power of the bow, but they had no explanation for why an arrow goes on flying through the air when there is nothing to pull it like the horse pulls the cart. Aristotle made the assumption that there must be a force to keep things moving. Galileo contradicted. He believed that something will keep moving at the same speed unless a force slows it down.

He contended that an arrow or a thrown stone had two forces acting upon it at the same time – ‘momentum’ pushes it horizontally and it only falls to the ground because the resistance of the air (a force) slows it down enough for it to be pulled to the ground by another force pushing downwards upon it; that which we now know as ‘gravity’.
This is the principle of inertia and led him to correctly predict that the path of a projectile is a parabola.

His insights were similar to the first two of the three laws of motion that Newton described 46 years later in ‘Principia’. Although he did not formulate laws with the clarity and mathematical certainty of Newton, he did lay the foundations of the modern understanding of how things move.

Galileo resisted the notion of gravity because he felt the idea of what seemed to be a mystical force seemed unconvincing, but he appreciated the concept of inertia and realized that there is no real difference between something that is moving at a steady speed and something that is not moving at all – both are unaffected by forces. To make an object go faster or slower, or begin to move, a force is needed.

Galileo would take a problem, break it down into a series of simple parts, experiment on those parts and then analyse the results until he could describe them in a series of mathematical expressions. His meticulous experiments (‘cimento‘) on inclined planes provided a study of the motion of falling bodies.

He correctly assumed that gravity would act on a ball rolling down a sloping wooden board that had a polished, parchment lined groove cut into it to act as a guide, in proportion to the angle of the slope. He discovered that whatever the angle of the slope, the time for the ball to travel along the first quarter of the track was the same as that required to complete the remaining three-quarters. The ball was constantly accelerating. He repeated his experiments hundreds of times, getting the same results. From these experiments he formulated his laws of falling bodies.
Mathematics provided the clue to the pattern – double the distance traveled and the ball will be traveling four times faster, treble it and the ball will be moving nine times faster. The speed increases as a square of the distance.
He found that the size of the ball made no difference to the timing and surmised that, neglecting friction, if the surface was horizontal – once a ball was pushed it would neither speed up nor slow down.

His findings were published in his book, ‘Dialogue Concerning the Two Chief World Systems’, which summarised his work on motion, acceleration and gravity.

His theory of uniform acceleration for falling bodies contended that in a vacuum all objects would accelerate at exactly the same rate towards the earth.

Legend has it that Galileo gave a demonstration, dropping a light object and a heavy one from the top of the leaning Tower of Pisa. Dropping two cannonballs of different sizes and weights he showed that they landed at the same time. The demonstration probably never happened, but in 1991 Apollo 15 astronauts re-performed Galileo’s experiment on the moon. Astronaut David Scott dropped a feather and a hammer from the same height. Both reached the surface at the same time, proving that Galileo was right.

Another myth has it that whilst sitting in Pisa cathedral he was distracted by a lantern that was swinging gently on the end of a chain. It seemed to swing with remarkable regularity and experimenting with pendulums, he discovered that a pendulum takes the same amount of time to swing from side to side – whether it is given a small push and it swings with a small amplitude, or it is given a large push. If something moves faster, he realised, then the rate at which it accelerates depends on the strength of the force that is moving it faster, and how heavy the object is. A large force accelerates a light object rapidly, while a small force accelerates a heavy object slowly. The way to vary the rate of swing is to either change the weight on the end of the arm or to alter the length of the supporting rope.
The practical outcome of these observations was the creation of a timing device that he called a ‘pulsilogium’.

Drawing by GALILEO of the surface of the moon

Galileo confirmed and advanced COPERNICUS’ sun centered system by observing the skies through his refracting telescope, which he constructed in 1609. Galileo is mistakenly credited with the invention of the telescope. He did, however, produce an instrument from a description of the Dutch spectacle maker Hans Lippershey’s earlier invention (patent 1608).

He discovered that Venus goes through phases, much like the phases of the Moon. From this he concluded that Venus must be orbiting the Sun. His findings, published in the ‘Sidereal Messenger‘ (1610) provided evidence to back his interpretation of the universe. He discovered that Jupiter has four moons, which rotate around it, directly contradicting the view that all celestial bodies orbited earth, ‘the centre of the universe’.

‘The Earth and the planets not only spin on their axes; they also revolve about the Sun in circular orbits. Dark ‘spots’ on the surface of the Sun appear to move; therefore, the Sun must also rotate’

1610 – Galileo appointed chief mathematician to Cosmo II, the Grand Duke of Tuscany, a move that took him out of Papal jurisdiction.

1613 – writes to Father Castelli, suggesting that biblical interpretation be reconciled with the new findings of science.

1615 – a copy of the letter is handed to the inquisition in Rome.

1616 – Galileo warned by the Pope to stop his heretical teachings or face imprisonment.

1632 – when Galileo published his masterpiece, ‘Dialogue Concerning the Two Chief World Systems’ – (Ptolemaic and Copernican) – which eloquently defended and extended the Copernican system, he was struggling against a society dominated by religious dogma, bent on suppressing his radical ideas – his theories were thought to contravene the teachings of the Catholic Church. He again attracted the attention of the Catholic Inquisition.
His book took the form of a discussion between three characters; the clever Sagredo (who argues for Copernicus), the dullard Simplicio (who argues hopelessly for Aristotle) and Salviati (who takes the apparently neutral line but is clearly for Sagredo).

In 1633 he was tried for heresy.

‘That thou heldest as true the false doctrine taught by many that the Sun was the centre of the universe and immoveable, and that the Earth moved, and had also a diurnal motion. That on this same matter thou didst hold a correspondence with certain German mathematicians.’
‘…a proposition absurd and false in philosophy and considered in theology ad minus erroneous in faith…’.

Threatened with torture, Galileo was forced to renounce his theories and deny that the Earth moves around the Sun. He was put under house arrest for the rest of his life.

After Galileo’s death in 1642 scientific thought gradually accepted the idea of the Sun-centered solar system. In 1992, after more than three and a half centuries, the Vatican officially reversed the verdict of Galileo’s trial.

Galileo’s thermoscope operated on the principle that liquids expand when their temperature increases. A thermoscope with a scale on it is basically a thermometer and in its construction Galileo was probably following directions given by Heron of Alexandria 1500 years earlier in ‘Pneumatics’. As with the telescope, Galileo is often incorrectly given credit for the invention of the thermometer.

Wikipedia-logo © (link to wikipedia)

NEXT buttonTIMELINE

NEXT button - THE STARSTHE STARS

Related sites

link to http://www.museogalileo.it/en/index.html

<< top of page

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.

Wikipedia-logo © (link to wikipedia)

NEXT buttonNEXT

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).”

Wikipedia-logo © (link to wikipedia)

NEXT buttonMAIN INDEX

Related articles

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.

Wikipedia-logo © (link to wikipedia)

NEXT buttonNEXT

<< top of page

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.

Wikipedia-logo © (link to wikipedia)

NEXT buttonNEXT

NEXT buttonTHE STARS

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.

Wikipedia-logo © (link to wikipedia)

NEXT buttonTIMELINE

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.

Wikipedia-logo © (link to wikipedia)

NEXT buttonTIMELINE

Related sites
  • Pi (math.com)

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.

Wikipedia-logo © (link to wikipedia)

NEXT buttonNEXT

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.

Wikipedia-logo © (link to wikipedia)

NEXT buttonNEXT

Related sites

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

Wikipedia-logo © (link to wikipedia)

NEXT buttonNEXT

NEXT buttonTHE STARS

Related sites

GALEN OF PERGAMUM (130-201)

161 – Rome, Italy

Bust of GALEN

GALEN

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

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

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

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

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

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

Wikipedia-logo © (link to wikipedia)

NEXT buttonNEXT

NEXT buttonMEDICINE

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.

more

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)

NEXT buttonNEXT

J’BIR IHBIN AYAM (722-804)

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

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

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

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

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

Wikipedia-logo © (link to wikipedia)

NEXT buttonALCHEMY

NEXT buttonNEXT

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.

Wikipedia-logo © (link to wikipedia)

NEXT buttonNEXT

AL-BIRUNI (973-1050)

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

portrait of al-biruni

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

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

Wikipedia-logo © (link to wikipedia)

NEXT buttonNEXT

4ADiamond

IBN SINA (AVICENNA) (980-1037)

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

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

drawing of Ibn Sina ©

AVICENNA

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

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

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

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

Arabic Canon of Medicine by Avicenna 1632

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

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

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

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

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

AL HAZEN (c.965-1039)

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

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

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

diagram explaining the working of the eye

Wikipedia-logo © (link to wikipedia)

NEXT buttonNEXT

 

NEXT buttonMEDICINE

<< top of page

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.

Wikipedia-logo © (link to wikipedia)

NEXT buttonTHE DARK AGES

CHRISTIAN THEOLOGY & WESTERN SCIENCE

bust said to depict a likeness of Socrates

The speculative Greek philosophers, considering the great overarching principles that controlled the Cosmos, were handicapped by a reluctance to test their speculations by experimentation.
At the other end of the spectrum were the craftsmen who fired and glazed pottery, who forged weapons out of bronze and iron. They in turn were hindered by their reluctance to speculate about the principles that governed their craft.

WESTERN SCIENCE is often credited with discoveries and inventions that have been observed in other cultures in earlier centuries.
This can be due to a lack of reliable records, difficulty in discerning fact from legend, problems in pinning down a finding to an individual or group or simple ignorance.

The Romans were technologists and made little contribution to pure science and then from the fall of Rome to the Renaissance science regressed. Through this time, science and technology evolved independently and to a large extent one could have science without technology and technology without science.

Later, there developed a movement to ‘Christianise Platonism’ (Thierry of Chartres).

Platonism at its simplest is the study and debate of the various arguments put forward by the Greek philosopher PLATO (428/7-348/7 BCE).
The philosopher Plotinus is attributed with having founded neo-Platonism, linking Christian and Gnostic beliefs to debate various arguments within their doctrines. One strand of thought linked together three intellectual states of being: the Good (or the One), the Intelligence and the Soul. The neo-Platonic Academy in Greece was closed by the Emperor Justinian (CE 483-565) in CE 529.
During the early years of the Renaissance, texts on neo-platonism began to be reconsidered, translated and discoursed.

Aristotle’s four causes from the ‘Timaeus’ were attributed to the Christian God, who works through secondary causes (such as angels).

Efficient Cause – Creator – God the Father

Formal Cause – Secondary agent – God the Son

Material Cause – The four elements: earth, air, fire & water.
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. ( ALCHEMY )

Final Cause – Holy Spirit

All other is ‘natural’ – underwritten by God in maintaining the laws of nature without recourse to the supernatural.
Science was the method for investigating the world. It involved carrying out careful experiments, with nature as the ultimate arbiter of which theories were right and which were wrong.

Robert Grosseteste (1168-1253) Bishop of Lincoln (Robert ‘Bighead’) – neo-platonic reading of Genesis – emanation of God’s goodness, like light, begins creation. Light is thus a vehicle of creation and likewise knowledge (hence ‘illumination’), a dimensionless point of matter with a dimensionless point of light superimposed upon it (dimensions are created by God). Spherical radiation of light carries matter with it until it is dissipated. Led to studies of optical phenomena (rainbow, refraction, reflection).

stained glass window depicting Robert Grosseteste (created 1896)

Wikipedia-logo © (link to wikipedia)

NEXT buttonNEXT

JOHANNES GUTENBERG (1395-1468)

Typographic resetting of Gutenberg's 42-line bible of 1452-55, using modern Fraktur and decorative initial in METAFONT by Yannis Haralambous. (Beginning of St. John's Gospel) from a LaTex advertising flyer.

1450 – Mainz, Germany

‘Movable type’

  

Hand-held block printing – a laborious process of carving whole pages of fixed text out of wooden slabs and reproducing copies using dies – had been used for many decades before the German inventor appeared. What Gutenberg mastered was the idea of placing individual metal letters – (his family background was in minting and metalworking, an ideal foundation for his training as an engraver and goldsmith. His skills enabled him to craft the first individual metal letter moulds) – into temporary mounts, which could then be dismantled or ‘moved’ once a page of text had been completed and reused to produce other pages.

In comparison to engraving and the single use of wooden blocks, the theoretically infinite number of sides which could be made out of a set of metal characters, together with the speed at which a template could be created, revolutionised printing and the spread of the printed word.

engraving of Johannes Gutenberg

Gutenberg printing press. Johannes Gutenberg (c. 13951468) invented the printing press sometime in the mid-fifteenth century. The moveable printing blocks it employed made it far simpler to operate than the complicated machinery of the Far East

Some sources credit the Chinese with inventing moveable type printing, using characters made of wood. What is notable is the quality of Gutenberg’s metal casts and press – they are almost as important as the idea of moveable type itself.

By the end of the fifteenth century tens of thousands of books and pamphlets were already in existence, giving academics the opportunity to share scientific knowledge widely and cheaply.

Wikipedia-logo © (link to wikipedia)

NEXT buttonNEXT

LEONARDO DA VINCI (1452-1519)

1502 – Florence, Italy

‘In the Renaissance science was reinvented’

Image of the VITRUVIAN MAN

VITRUVIAN MAN

Leonardo is celebrated as the Renaissance artist who created the masterpieces ‘The Last Supper’ (1495-97) and ‘The Mona Lisa’ (1503-06). Much of his time was spent in scientific enquiry, although most of his work remained unpublished and largely forgotten centuries after his death. The genius of his designs so far outstripped contemporary technology that they were rendered literally inconceivable.

The range of his studies included astronomy, geography, palaeontology, geology, botany, zoölogy, hydrodynamics, optics, aerodynamics and anatomy. In the latter field he undertook a number of human dissections, largely on stolen corpses, to make detailed sketches of the body. He also dissected bears, cows, frogs, monkeys and birds to compare their anatomy with that of humans.

It is perhaps in his study of muscles where Leonardo’s blend of artistry and scientific analysis is best seen. In order to display the layers of the body, he developed the drawing technique of cross-sections and illustrated three-dimensional arrays of muscles and organs from different perspectives.

Leonardo’s superlative skill in illustration and his obsession with accuracy made his anatomical drawings the finest the world had ever seen. One of Leonardo’s special interests was the eye and he was fascinated by how the eye and brain worked together. He was probably the first anatomist to see how the optic nerve leaves the back of the eye and connects to the brain. He was probably the first, too, to realise how nerves link the brain to muscles. There had been no such idea in GALEN’s anatomy.

Possibly the most important contribution Leonardo made to science was the method of his enquiry, introducing a rational, systematic approach to the study of nature after a thousand years of superstition. He would begin by setting himself straightforward scientific queries such as ‘how does a bird fly?’ He would observe his subject in its natural environment, make notes on its behaviour, then repeat the observation over and over to ensure accuracy, before making sketches and ultimately drawing conclusions. In many instances he would directly apply the results of his enquiries into nature to designs for inventions for human use.

Self portrait of LEONARDO DA VINCI

LEONARDO DA VINCI

He wrote ‘Things of the mind left untested by the senses are useless’. This methodical approach to science marks a significant stepping-stone from the DARK AGES to the modern era.

1469 Leonardo apprenticed to the studio of Andrea Verrocchio in Florence

1482 -1499 Leonardo’s work for Ludovico Siorza, the Duke of Milan, included designs for weaponry such as catapults and missiles.
Pictor et iggeniarius ducalis ( painter and engineer of the Duke )’.
Work on architecture, military and hydraulic engineering, flying machines and anatomy.

1502 Returns to Florence to work for Pope Alexander VI’s son, Cesare Borgia, as his military engineer and architect.

1503 Begins to paint the ‘Mona Lisa’.

1505-07 Wrote about the flight of birds and filled his notebooks with ideas for flying machines, including a helicopter and a parachute. In drawing machines he was keen to show how individual components worked.

1508 Studies anatomy in Milan.

1509 Draws maps and geological surveys of Lombardy and Lake Isea.

1516 Journeys to France on invitation of Francis I.

1519 April 23 – Dies in Clos-Luce, near Amboise, France.

Wikipedia-logo © (link to wikipedia)

NEXT buttonTIMELINE

NEXT buttonNEXT

NEXT buttonMECHANICS