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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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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WILLIAM GILBERT (1540-1603)

1600 – England

‘Gilbert’s principal area of study related to magnetism, however, his method of enquiry is equally significant’

portrait of WILLIAM GILBERT ©

WILLIAM GILBERT

Gilbert rejected the scholastics’ approach to science, preferring the experimental method, which he applied to the Earth’s magnetic properties.
He carried out some of the first systematic studies of the lodestone in Europe and showed that the Earth acts as a bar magnet with magnetic poles.

His celebrated text, ‘De magnete, magnetisque corporibus, et de magno magnete tellure‘ (On the Magnetic, Magnetic Bodies and the Great Magnet Earth – 1600) is considered to be one of the first truly scientific texts.
Gilbert received his medical training in Cambridge and practiced as a physician in London. He became president of the College of Physicians and was physician to Queen Elizabeth I.

In the time of Elizabeth I and Shakespeare, England was still largely a place of superstition and religious fervor. Gilbert concurred with Copernicus, a potentially dangerous sentiment in an era when elsewhere in Europe others such as Giordano Bruno and later GALILEO were being persecuted (and in the case of Bruno, executed) for sharing the same opinion.


Magnetism was to cast its influence in the eighteenth century, displayed through the electric fluid of GALVANI and VOLTA
.

He distinguished the properties of magnetism from the attractive effect produced by friction with amber. In so doing he introduced the term that was to become electricity.
He introduced a number of expressions to the English language including: magnetic pole, electric force and electric attraction.
A term of magneto motive force, the gilbert, is named after him.

Gilbert and others postulated that magnetism is the force holding the planets in their orbits.

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WILLEBRORD SNELLIUS (1580-1626)

1621 – Holland

woodblock print portrait of WILLEBRORD SNELL ©

WILLEBRORD SNELL

‘During refraction of light, the ratio of the sines of the angles of incidence ( i ) and refraction ( r ) is a constant equal to the refractive index of the medium’

In equation form: n1sini = n2 sinr 
where n1 and n2 are the respective refractive indices of the two media.

The refractive index of a substance is a measure of its ability to bend light. The higher the number the better light is refracted. The refractive index of diamond, 2.42, is the highest of all gems.

Refraction is the change in direction of a ray of light when it crosses the boundary between two media. It happens because light has different speeds in different media. A ray of light entering a medium where the speed of light is less (from air to water, for example) bends towards the perpendicular to the boundary of the two media. It bends away from the perpendicular when it crosses from water to air. Refraction was known to ancient Greeks, but Snell, a Dutch mathematician, was the first to study it.

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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’

Portrait of GALILEO GALILEI ©

GALILEO GALILEI

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 Hiero 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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EVANGELISTA TORICELLI (1608- 47)

1640 – Italy

‘Together with VINCENZO VIVIANI (1622-1703) realised that the weight of air pushing on a reservoir of mercury can force the liquid to rise into a tube that contains no air; that is, a vacuüm’

In 1650 OTTO VON GUERICKE (1602-1686) invented an air pump and showed that if you remove the air from the centre of two hemispheres that are resting together, the pressure of the outside air is sufficient to prevent a team of horses from pulling them apart.

1657 – Formed the Accademia del Cimento with eight other Florentines to build their own apparatus and conduct experiments to advance the pursuit of knowledge. Disbanded after ten years as a condition of its patron Leopoldo de Medici’s appointment as cardinal, its dissolution followed Galileo’s trial by the Catholic Church and marked the decline of free scientific research in Italy.

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BLAISE PASCAL (1623- 62)

1647 – France

Portrait of BLAISE PASCAL

BLAISE PASCAL

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

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

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

‘The study of the likelihood of an event’

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

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

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

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

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

 
 

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

 
 
 

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

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