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.

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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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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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GALEN OF PERGAMUM (130-201)

161 – Rome, Italy

Bust of GALEN

GALEN

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

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

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

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

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

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

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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)

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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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