ALCHEMY

Alchemical symbols

Understanding of the alchemists is hampered by their predilection for making their writings incomprehensible ( instant knowledge was not to be available to the uninitiated ) and the popular view that their quest was simply to isolate the Philosophers’ Stone and to use it to transform base metals into gold.
There was in fact a genuine search for mental and spiritual advance.

Using a world-view totally unlike that recognised today, the alchemists’ ideas of ‘spirit’ and ‘matter’ were intermingled – the ability to use ‘spirit’ in their experiments was the difficult part.

To transform copper to gold: – copper could be heated with sulphur to reduce it to its ‘basic form’ (a black mass which is in fact copper sulphide) – its ‘metallic form’ being ousted by the treatment. The idea of introducing the ‘form of gold’ to this mass by manipulating and mixing suitable quantities of spirit stymied alchemists for over fifteen centuries.

Whilst this transmutation of metals was the mainstream concern of alchemy, there emerged in the sixteenth century a school that brought the techniques and philosophies of alchemy to bear on the preparation of medicines, two of the main figures involved being PARACELSUS and JOHANN VAN HELMONT

THE EIGHTEENTH CENTURY

COMBUSTION and PHLOGISTON

 

Noticing that burning a candle in an upturned container, the open end of which is submerged in water, causes the water to rise into the container, Philon of Byzantium inferred correctly that some of the air in the container had been used up in the combustion. However, he proposed that this is because this portion of the air had been converted into ‘fire particles’, which were smaller than ‘air particles’.

In 1700 the German physician Georg Ernst Stahl (1660-1734) invoked ‘phlogiston’ to explain what happens when things burn. He suggested that a burning substance was losing an undetectable elementary principle, analogous to the ‘sulfur’ of J’BIR IHBIN AYAM, which he re-named ‘phlogiston’. This could explain why a log (rich in phlogiston) could seem to be heavier than its ashes (deficient in phlogiston). The air that is required for burning served to transport the phlogiston away.

The English chemist JOSEPH PRIESTLY (1733-1804), although a supporter of the phlogiston theory, ironically contributed to its downfall. He heated mercury in air to form red mercuric oxide and then applied concentrated heat to the oxide and noticed that it decomposed again to form mercury whilst giving off a strange gas in which things burnt brightly and vigorously. He concluded that this gas must be ‘phlogiston poor’.

Priestly combined this result with the work of the Scottish physician Daniel Rutherford (1749-1819), who had found that keeping a mouse in an enclosed airtight space resulted in its death (by suffocation) and that nothing could be burnt in the enclosed atmosphere; he formed the idea that the trapped air was so rich in phlogiston that it could accept no more. Rutherford regarded this as ‘phlogisticated air’ and so Priestly called his own gas ‘dephlogisticated air’.

In 1774 Priestley visited the French chemist ANTOINE LAVOISIER (1743-1794).
Using chemical reactions, Lavoisier had combined a portion of normal air with other substances.
Lavoisier repeated Priestly’s experiments with careful measurements.
Reasoning that air is made up of a combination of two gases – one that will support combustion and life, another that will not; what was important about Lavoisier’s experiments was not the observation – others had reached a similar conclusion – but the interpretation.

Lavoisier called Priestley’s ‘dephlogisticated air’, ‘oxygene’, meaning ‘acidifying principle’, believing at the time that the active principle was present in all acids (it is not). Classifying oxygen as an element, he called the remaining portion of normal air ‘phlogisticated air’, ‘azote’, meaning ‘without life’.

Oxygen is the mirror image of phlogiston. In burning and rusting (the two processes being essentially the same) a substance picks up one of the gases from the air. Oxygen is consumed, there is no expulsion of ‘phlogiston’.

Lavoisier had been left with almost pure nitrogen, which makes up about four fifths of the air we breath. We now know azote as nitrogen. Rutherford’s ‘mephitic air’ was carbon dioxide.

CALORIC

Like phlogiston, caloric was a weightless fluid, rather like elemental fire; a quality that could be transmitted from one substance to another, so that the first warmed the second up.

It was believed that all substances contained caloric and that when a kettle was being heated over a fire, the fuel gave up its caloric to the flame, which passed it into the metal, which passed it on to the water. Similarly, two pieces of wood rubbed together would give heat because abrasion was releasing caloric trapped within.

What is being transmitted is heat energy. It was the crucial distinction between the physical and the chemical nature of substances that confused the Ancients and led to their minimal elemental schemes.

ASTROLOGY

 

HEAT

 

CHRISTIAN THEOLOGY & WESTERN SCIENCE

WILLIAM GILBERT (1540-1603)

1600 – England

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

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.

WILLIAM GILBERT

His celebrated work,’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 animal magnetism and 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.
Gilbert 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.

TIMELINE

JOHANNES KEPLER (1571-1630)

1609-19 – Germany

1600 – Kepler works in Prague with TYCHO BRAHE the imperial mathematician, under the patronage of Rudolph II
1601 – On Brahe’s death, Kepler inherits his position (and crucially, his astronomical notes)

KEPLER

• First Law: The planets move in elliptical orbits with the Sun at one focus

• Second Law: The straight line joining the Sun and any planet sweeps out equal areas in equal periods of time

• Third Law: The squares of orbital periods of the planets are proportional to the cube of their mean distances from the Sun

Modern measurements of the planets show that they do not precisely follow these laws; however, their development is considered a major landmark in science.

Kepler’s ardent faith in the Copernican system – ‘The Sun not only stands at the centre of the universe, but is its moving spirit’, he asserted – brought him the disfavour of religious leaders. With his realisation that the planets do not rotate in perfect circles but in fact orbit in an ellipse, he provided the mathematical explanation for planetary motion, which had eluded Copernicus and Ptolemy.

The first two laws were published in 1609 ( Astronomia Nova – New Astronomy ) and the third in 1619 ( Harmonicses Mundi – Harmonics of the World ). Their publication put an end to PTOLEMY’s cycles & epicycles. His work provided the observational and arithmetical proof to support COPERNICUS‘ theories.

His second law states that an imaginary line between the Sun and the planets sweeps out an equal area in equal periods of time.

Stating that the planets ‘sweep’ or cover equal areas in equal amounts of time regardless of which location of their orbit they are in means that, as the Sun is only one of two centres of rotation in a planet’s orbit, a planet is nearer to the Sun at some times than at others. Thus the planet must speed up when it is nearer the Sun and slow down when it is further away.

His third law finds that the period (the time for one complete orbit – a year for the Earth, for instance) of a planet squared is the same as the distance from the planet to the Sun cubed (in astronomical units). This allows distances of planets to be worked out from observing their cycles alone.

Kepler was a versatile genius who, besides discovering these three laws, compiled tables of star positions ( Tabulae Rudolphinae – 1627 ) and developed the astronomical telescope.

Kepler also studied the anatomy of the human eye and founded the science of geometrical optics ( ‘Dioptrics’ – 1611 ), proposing the ray theory of light after ALHAZEN’s discussion in Opticae Thesaurus ; he described the eye in the same terms – as a pinhole camera, with light entering through the pupil and forming an image of the outside world on the retina at the back of the eye.

His credible solution to predicting planetary motion would act as the stimulus for questions that would lead to ISAAC NEWTON‘s theory of gravity.

TIMELINE

 

THE STARS

FRANCIS BACON (1561-1626)

1620 – England

Scientific laws must be based on observations and experiments

Bacon rejected ARISTOTLE‘s deductive or a priori, approach to reasoning and suggested his own, inductive, or a posteriori, approach. Bacon developed the scientific method – but he did not make any significant scientific discovery.

‘I shall content myself to awake better spirits like a bell-ringer, who is first up to call others to church’

FRANCIS BACON

Bacon, a philosopher, advocated a new method of enquiry, completely different from the philosophical methods of the ancient Greeks, in his book Novum Organum – which has influenced scientists since its publication in 1620.

The text proposed the sentiment of ‘The Advancement of Learning’ (1605) signaling dissatisfaction with the limits of, and approaches to, knowledge to date and foresaw a future where the ancient masters would be far surpassed – Aristotle had written a text called Organum or ‘Logical Works’ and Bacon’s ‘new’ approach suggested an alternative direction for scientific study.

Bacon strongly criticised Aristotle’s deductive method of science, which involved formulating abstract ideas and ‘logically’ building upon them step-by-step to find ‘truths’, without thorough consideration of whether the theoretical foundation in itself was ever valid.

Rather than rely on superstition or accept unquestioningly the flawed solutions of the ancient academics as had largely been the case for two thousand years, Bacon’s alternative was to argue for ‘inductive’ reason, where the only ‘certain’ statements that should ever be made were based on observation and proof collected from the natural world. The essence of his method is to collect masses of data by observations and experiments, analyse facts by drawing up tables of negative, affirmative and variable instances of the phenomenon ( ‘Tables of Comparative Instances’ ), draw (induce) hypotheses from the evidence, then to collect further evidence to proceed towards a more general theory. The most important aspect of this method was the idea of drawing up tentative hypotheses from available data and then verifying them by further investigations.

‘A true and fruitful natural philosophy has a double scale or ladder ascendant or descendant, ascending from experiments to axioms and descending from axioms to the invention of new experiments’, he wrote in Novum Organum.

Bacon cautioned those trying to practice his new method, urging them to repudiate four kinds of intellectual idol

• Perceptual Illusions – ‘idols of the tribe’
• Personal biases – ‘idols of the cave’
• Linguistic confusions – ‘idols of the market place’
• Dogmatic philosophical systems – ‘idols of the theatre’

TIMELINE

WILLEBRORD SNELLIUS (1580-1626)

1621 – Holland

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: n₁ sin_i = n₂ sin_r 
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 Snellius, a Dutch mathematician, was the first to study it.

TIMELINE

 

LIGHT

JOHANN VAN HELMONT (1579-1644)

1621 – Brussels, Belgium

JOHANN VAN HELMONT

There are gases other than air

Van Helmont coined the term ‘gas’.

He hypothesised that the proof that matter is made entirely of water was provided by his experiment of growing a tree-shoot in a weighed quantity of soil and finding that the weight of the tree increased by over a thousand fold whilst that of the soil decreased only slightly. He failed to consider the contribution of the air.

TIMELINE

 

CHEMISTRY

WILLIAM HARVEY (1578-1657)

1628 – London, England

Circulation of the blood

WILLIAM HARVEY

As WILLIAM GILBERT had begun in physics, and FRANCIS BACON had subsequently implored, Harvey was the first to take a rational, modern, scientific approach to his observations in biology.
Rather than taking the approach of the philosophers, which placed great emphasis upon thinking about what might be the case, Harvey cast aside prejudices and only ‘induced’ conclusions based on the results of experiments and dissections, which could be repeated identically again and again.

After what GALEN had begun and VESALIUS had challenged, Harvey credibly launched perhaps the most significant theory in his field of biology. He postulated and convincingly proved that blood circulated in the body via the heart – itself little more than a biological pump.

Galen had concluded that blood was made in the liver from food, which acted as a fuel, which the body used up, thereby requiring more food to keep a constant supply. Vesalius added little to this theory. Harvey, physician to Kings James I and later Charles I proved his theory of circulation through rigorous and repeated experimentation. He correctly concluded that blood was not used up, but is recycled around the body.

His dissections proved that the arteries took blood from the heart to the extremities of the body, able to do so because of the heart’s pump-like action. He could see that the pulses in arteries came immediately after the heart contracted, and became certain that the pulse was due to blood flowing into the vessels.
By careful observation he found that blood entered the right side of the heart and was forced into the lungs, before returning to the left side of the heart. From there it was pumped via the aorta into the arteries around the body.

Harvey realized that the amount of blood flowing around the system was too much for the liver to produce. The blood had to be circulating back to the veins; which, with their series of one-way valves, brought blood back to the heart.
Without a microscope it was impossible to see the minute capillaries that linked the arteries to the veins.

Harvey published his findings in the 720 page ‘Exercitatio Anatomica de Motu Cordis et Sanguinis in Animalibus‘ ( Anatomical Exercise on the Motion of The Heart and Blood in Animals ) at the Frankfurt Book Fair in 1628.

Initially supported by some academics, an equal number rejected his ideas. One area of weakness was that he was unable to offer a proven explanation for how the blood moved from the arteries to the veins. He speculated that the exchange took place through vessels too small for the human eye to see, which was confirmed shortly after his death with the discovery of capillaries by Marcello Malphigi with the recently invented microscope.

Even then, nobody knew what blood was doing. It would take another hundred years before ANTOINE LAVOISIER discovered oxygen and worked out what it did in the body.

In 1651, Harvey published ‘Exercitationes de Generatione Animalium‘ ( Essays on the Generation of Animals ), a work in the area of reproduction which included conjecture that rejected the ‘spontaneous generation’ theory of reproduction which had hitherto persisted. His belief that the egg was at the root of life gained acceptance long before the observational proof some two centuries later.

TIMELINE

Related sites

• The Old Operating Theatre (www.thegarret.org.uk)

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

GALILEO

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 BC) 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’.

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.

TIMELINE

 

THE STARS

Related sites

• The Galileo Project (galileo.rice.edu)
• Museo Galileo (www.imss.fi.it)
• Dialogue Concerning the Two Chief World Systems (calstatela.edu)

‘The Starry Messenger’ – Galileo (download pdf)

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RENE DESCARTES (1596-1650)

1637 – France

RENE DESCARTES

Cogito ergo sum

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

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

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

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

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

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

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

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

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

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

TIMELINE

Related material

• Descartes — Dioptrique
• download pdf (science.larouchepac.com)

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

1637 – France; 1993 – USA

FERMAT

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

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

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

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

ANDREW WILES

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

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

TIMELINE

 

MATHEMATICS

Related sites

• Another proof of Fermat’s Last Theorem (plus.maths.org)