So welcome. In this talk, I aim to cover the entire history of mathematics in just one hour,
illustrating my story with some of the 1500 mathematical stamps from around the world in my collection.
Now. I usually start by mentioning a set of ten stamps issued by Nicaragua in 1971
and entitled Ten Mathematical Formulas That Changed the face of the Earth.
As you can see, these range from one plus one equals two,
and the Pythagorean theorem via Archimedes Law of moments and in a Newton's inverse square law ending up with Einstein's law equals M.C. squared.
You'll see some of these again later on. So let's go back in time.
My treatment is chronological, going back to the time when people first began to count and measure the objects around them.
Early methods included forming stones into piles, cutting notches in sticks and especially finger counting,
which led eventually to our familiar decimal number system.
Also in several places we find ancient geometrical arrangements of stones,
such as the circular pattern of megaliths at Stonehenge and the linear arrangements at Carnac in Brittany.
We don't know their exact purpose, but they may have celebrated astronomical events such as Sunrise on Midsummer's Day.
So we'll begin our journey with the mathematical achievements of Egypt, Mesopotamia and Greece, all shown on this map.
Here's Mesopotamia. There's Egypt and Greece.
Further up. In Egypt, the main achievements involve the practical scale of measurement.
The magnificent pyramids designed as tombs for their readers, for their rulers, attest to the Egyptians extremely accurate measuring abilities.
And the oldest of these at the top is King Djoser Step Pyramid in Saqqara.
Built in horizontal layers and dating from about 2700 B.C.
Rather better known as the Great Pyramid of Cheops and Giza, constructed from over 2 million large blocks of stone and amazingly,
the size of its square base, which about 230 metres long agree in length to within 100th of 1%.
Our knowledge of Egyptian mathematics comes mainly from two fragile primary sources.
The Moscow and Ryan Papyrus is similar to the papyrus you can see here.
The Moscow and Rhine for passing through tables of fractions and problems in arithmetic and geometry.
Probably probably designed for the teaching of scribes.
On the right you can see the Egyptian God of reckoning self.
Moving to Mesopotamia. Early examples of mathematical writing also appeared there between the rivers, Tigris and Euphrates in present day Iraq.
Top left, you can see an accounting tablet from about 3000 B.C., where the thumbnail indentations at the top represent numbers.
Later the Mesopotamians wrote their mathematics with a wedge shaped styrofoam get on damp clay, which was then baked in the sun.
Many thousands of these cuneiform tablets survive and show basic arithmetic and the calculation of areas and volumes in geometry.
The solving of what we now call linear and quadratic equations.
And an extremely accurate value for the square root of two. The number system was based on 60, which we still use for measuring time.
And they also studied astronomy and observe comets such as the one we now call Halley's Comet.
Although the date of two, three, four, nine B.C. is about 2000 years out.
Let's now move to more familiar territory. From about 600 B.C.
Mathematics flourished in the Greek speaking world of the Eastern Mediterranean.
During this time, the Greeks developed deductive logic, logical reasoning, which became the hallmark of their work, especially in geometry.
Pythagoras, a semi legendary figure, formed a school in Cortona to further the study of mathematics, philosophy and natural science.
His followers supposedly believe that all is number, quantifying many things and emphasising the four mathematical arts of arithmetic,
geometry, astronomy and music, which will meet again later.
We do not know who first proved the Pythagorean theorem that the area of the square
on the hypotenuse is the sum of the areas of the squares on the other two sides.
But it's connection with right angle triangles was already known to the Mesopotamians a thousand years earlier.
The blue stamp was issued very recently in August for the International Congress of Mathematicians in Korea.
The scene then moved to Athens, which became the most important intellectual centre in Greece.
Numbering among its scholars, Plato and Aristotle.
In 387 B.C., Plato founded a school in a suburb of Athens called Academy, and Plato's Academy soon became the focal point for mathematical study.
It said that over the entrance appeared the inscription that no one ignorant of geometry entered here,
like the Greek inscription at the entrance to this very building.
In Raphael's fresco, the school of Athens. On the left, Plato and Aristotle appear together on the steps of the academy.
Plato believed that mathematics and philosophy were essential training for those holding positions of responsibility,
discussing the mathematical arts of arithmetic, geometry, astronomy and music, as I mentioned before, and justifying their importance.
He also described the five regular or platonic solids shown here in the middle.
His pupil, Aristotle, helped to formalise the study of deductive reasoning,
explaining why the square root of two cannot be written as a fraction, and also discussing logical syllogism such as All men are mortal.
Socrates is a man, therefore Socrates is mortal.
Around 300 B.C. following the military successes of Alexander the Great
Mathematical Activity moved to Alexandria in the Egyptian part of the Greek world.
The first important mathematician there was Euclid, shown on the right.
Mainly remembered for his elements.
This most influential compilation of known results consisted of 13 books on plain and solid geometry, arithmetic and number theory.
A model of deductive reasoning.
It started from initial axioms and postulates and used rules of deduction to derive new results in a logical and systematic order.
And we mustn't forget Archimedes from Syracuse in Sicily.
He was one of the greatest mathematicians of all time. Around 250 B.C., he calculated volumes of solids such as spheres and cylinders.
He listed the 13 Archimedes and or Semi-regular Solids.
These are polyhedra with regular faces, but of different types.
And he also obtained a good approximation for PI by considering 96 sided polygons that approximate a circle.
In Mechanics, he found the law of moments for a balance that you can see on the right and is
also credited with inventing the Archimedes screw for raising water from a river.
In statics, he stated Archimedes principle on the weight of an object immersed in water.
As you can see in the middle and he used it to test the purity of King Herod Gold crown.
On discovering the principle he supposedly jumped out of his bath and ran narrate, ran naked through the street shouting, Eureka, I have found it.
But sadly, no stamp has ever been issued featuring that particular episode.
We now move to China, where the mathematics dates back some 3000 years, around 220 B.C. in particular.
The Great Wall was a major triumph of engineering, skill and mathematical calculation.
Most ancient Greek, most ancient Chinese mathematics was written on bamboos bamboo strips.
And you can see an example top left showing a nine by nine multiplication table.
For their calculations. The Chinese used a form of counting board shown.
Bottom left a box with separate compartments for units tens, hundreds, etc., into which small bamboo rods were placed in order to do the calculations.
Chinese mathematicians are fascinated by PI around A.D. 100.
Zhang Heng, who is the first of the three Chinese mathematicians you see up there?
Zhang Heng, inventor of a seismograph for measuring earthquakes.
He gave the value square root of ten, which is about 3.16, which was a popular approximation in many areas.
Later to a. In A.D. 263, showing in the middle of their considered polygons with 192 sighs to deduce that pie lies between 3.1410 and 3.1427.
And here you can see in the middle Liu, who has nine chapter of a mathematical art.
His edition of that. Then around the 500 top, writes Cianci and his son.
Consider Polygon's with about 25,000 sides to obtain pi to six decimal places.
And they also found the value 355 over 113 also correct to six decimal places.
And this approximation wasn't rediscovered in the West until the 16th century.
But a similar situation occurred with pascals triangle of binomial coefficients shown here, bottom right,
which had appeared in a Chinese text of 1303, although in fact it was known earlier than that elsewhere.
But even so, this was 400 years before Pascal.
An ancient Chinese story of perhaps 1100 B.C. concerns Emperor Yu,
who is standing on the banks of the River Luo when a turtle emerged with 1 to 9 on its back,
with the numbers in each row column and diagonal adding to the same some 15.
This arrangement of numbers, the Horseshoe acquired great religious and mystic significance and eventually led to much
to larger such magic squares as pictured on these Macao stamps issued only last month.
Next year, three more stamps will be issued by Michael with the values 816 so that the stamp values themselves will form the new shoe magic square.
492357816.
They obviously have a mathematician in the post office.
India mathematics can be traced back to about 600 B.C. when Vedic manuscripts presented early work on arithmetic permutations and combinations.
Number Theory. And the extraction of square roots.
And around 250 BCE. King Ashoka ruler of most of India, became the first Buddhist monarch.
And in celebration, pillars were constructed. You can see one on the left, and then above you can see the capstone at the top of it,
showing the earliest examples of what became our Hindu Arabic numerals,
a decimal place value system with the position of each digit 0 to 9 indicating
its value and enabling one to carry out calculations one column at a time,
as we do now. The first outstanding Indian mathematician was Aryabhata around the year 500 who showed how to sum arithmetic,
arithmetic progressions and also to some natural numbers.
And this squares and cubes.
There's no stamp featuring Aryabhata, but there is one showing the first Indian space satellite, which was named Aryabhata in his honour.
In later years, Indian mathematicians and astronomers constructed magnificent observatories,
such as you can see on the right, such as the Jantar Mantar in Jaipur.
Comprising 14 massive measuring instruments.
It included the 90 foot samrat Yamcha, the world's largest sundial.
We move now to the mathematics of the Arabs and Persians. United by their new religion.
And with Baghdad lying on the East-West trade routes.
Islamic scholars from 750 to 1200 developed Greek and Roman writings from the West and Hindu writings from India.
And some of our terminology dates from this period. The word algorithm.
A step by step procedure for solving a problem comes from Al-Khwarizmi.
Shown on the left a Persian method mathematicians who lived in Baghdad and wrote influential works on arithmetic and algebra.
His arithmetic introduced the Indian decimal place value system to the Islamic world.
While the title of his algebra book, Kitab Algebra while Kabbalah gives us the word algebra.
Algebra means adding a positive term to both sides of an equation to eliminate negative terms.
Also influential among Islamic works were those of the geometry.
Even a high time in the middle. Known in the West as al-Hassan.
He mainly worked in optics,
and a celebrated problem is al problem where on a spherical mirror is light from a given point source reflected into the eye of a given observer.
A century later, the mathematician and poet Omar Khayyam on the right wrote a celebrated algebra text with
the first systematic classification of cubic equations and a discussion of their solution.
They weren't solved in general until 16th century Italy.
And look, I am also publicly criticised an attempt by Al Hosn to prove Euclid's so-called parallel postulate,
which I'll be explaining later in the West.
He's mainly remembered for his collection of poems, The Rubaiyat.
A later unsuccessful attempt to prove Euclid's parallel postulate was by the Persian mathematician Nasser el-Din Al See shown on
this magnificent stamp there on the left who wrote substantial works on astronomy and constructed the first modern observatory.
He also wrote influential treatises on logic and theology,
and his extensive investigations into plain and spherical trigonometry included the sine rule for triangles.
During the eighth and ninth centuries, the Islamic world spread across the top of Africa and then up into Europe through Spain and Italy.
Cordova became the scientific capital of Europe, while Islamic decorative arts and architecture spread through southern Spain and Portugal,
and included the magnificent geometrical arches in the corner of a mosque, which you can see on the left.
And the tiling patterns in the Alhambra. In Granada.
On the top. Right. Also in Cordova, the John Ritter and astronomer al-Zarqawi, shown in the middle, at the bottom, constructed many astrolabes.
This consisted of a brass disk suspended by a ring that's fixed or held in the
hand with an attached rotating bar and a circular scale on the rim and so on.
To measure the altitude of a planet or star, you view it along the bar and then read the results from the scale.
But in Europe, the period from five 500 to 1000 was known as the Dark Ages.
The legacy of the ancient world was largely forgotten and the level of culture was low.
But revival of interest in mathematics began with Chabot of Horik,
who trained in Catalonia and introduced the Hindu Arabic numerals to Christian Europe.
Using an abacus he designed for the purpose a major figure in the church.
He was crowned pope in the year 999.
That doesn't happen to many mathematicians. To operate.
The Catalan mystic, Ramon Law,
believes that all knowledge can be obtained monetarily by taking mathematical combinations of God's divine attributes such as power,
wisdom, goodness, etc. His commentary, ideas later spread throughout throughout Europe, influencing such mathematicians as Messiaen and Leibniz.
And the surprise appearance here is Geoffrey Chaucer, author of The Canterbury Tales, who also wrote on the Astrolabe.
This is one of the earliest scientific books in English.
The Hindu Arabic numbers were also popularised by Fibonacci or Leonardo of Pisa in his Lieber Obaseki Book of calculation.
This famous work of 1202 contained many problems in arithmetic and algebra,
including his rabbit's problem on the left that leads to the Fibonacci sequence.
One, one, two, three, five, eight, 13 and so on, in which each successive term is the sum of the previous two.
These numbers also arise in the arrangements of seeds and sunflowers and pine cones, for example.
There's a sunflower head there, and if you count round the spirals, you get numbers like 55 and 89 appearing.
The ratios of successive terms of the Fibonacci sequence approach limit.
The golden ratio, which is about 1.618, is the ratio of a diagonal and a side of a regular pentagon.
And also a rectangle with size in this ratio is sometimes considered to have the most pleasing shape not too thin and not too fat.
As shown on the Swiss stamp top lie in the middle.
Removing a square from a golden rectangle leaves another one.
And this stamp also features the logarithmic spiral found on the north of the shell below.
But also in the set for Michael.
It's a stamp featuring a Penrose tiling, which I thought I ought to show you today, consisting of its two stacked two shapes, the kite and the dart.
And if you want to see a Penrose tiling, you don't have to go very far. There's a big one just outside this building, as you know.
The Renaissance in mathematical learning and the mathematic mathematical.
So the Renaissance in mathematical learning in the Middle Ages was mainly due to three things the establishment of universities.
The translation of Arabic text into Latin and the invention of printing.
The first European University was Bologna. Schoen Top Left.
Founded in 1080 1088, with Paris and Oxford following soon after.
What was taught. The curriculum was in two parts. The second of which led to a master's degree and is based on the, quote, Trivium.
You can see the quadrillion subjects on the right. The Greek mathematical arts, again, of arithmetic, geometry, music and astronomy.
And the curriculum also included the study of Ptolemy's Almagest and Euclid's elements.
Gutenberg's Invention of the Printing Press around 1440 enabled mathematical works to be widely available for the first time.
At first, they were printed in Latin for the scholar.
But gradually vernacular works in the language of the people appeared at prices accessible to all, and these included texts in arithmetic,
algebra and geometry, as well as practical works to prepare their readers for a commercial career.
An important, important among these you can see on the top right was Pat Schiller's 600 page Italian Summa,
a compilation of the mathematics of the time, which also included the first published account of double entry bookkeeping.
And then below that, a commercial arithmetic by Adam Rizza, which is so respectable, the phrase no.
Adam Riza, after Adam Riza came to refer to a correct calculation.
The invention of printing led gradually to a standardisation of mathematical notation, and the symbols,
plus and minus first appeared in print in an arithmetic text of 1489,
whereas the symbols four times and divide didn't appear until considerably later.
It was around this time that painters learned how to give visual depth to their works through geometrical perspective.
The first to investigate this were Brunelleschi, who had designed the Dome of Florence Cathedral and his friend Alberti.
They're both shown at the top on the left.
Alberti gave mathematical rules for correct perspective, and he stated that the first duty of a painter is to know geometry.
So if you have any artist friends, you know what to tell them. Bottom left,
Piero della Francesca used a prospective grid for his geometrical studies and his
Madonna and child with Saints shown here is in perfect mathematical perspective.
The Polyhedron woodcuts that appear in Peres books were actually by Peres friend and student Leonardo da Vinci.
Shown on the top right, who investigated perspective more deeply than any other Renaissance painter and warned,
Let no one who is not a mathematician read my work.
Another celebrated artist was Albrecht Dürer on the at the bottom, who learned geometrical perspective from the Italians and introduced it to Germany.
And his engravings such as Saint Jerome and his study show his mastery of it.
We now turn to navigation. The Renaissance period coincided with the great sea sea voyages and explorations of Vasco da Gama, Columbus and others.
The Portuguese explorer sailed south and east. Prince Henry, the navigator on that wonderful stamp at the top,
sent ships down the west coast of Africa, claiming the island groups of Madeira and the Azores.
And Vasco da Gama was the first European to sail around the tip of Africa to reach the west coast of India.
Their rivals, the Spanish headed west, hoping to reach India by sailing around the globe.
And Columbus led four search expeditions, but instead reaching North and Central America, the West Indies and Venezuela.
And you can see Columbus, the bottom right. Such explorations inevitably needed accurate maps and reliable navigational instruments for use at sea.
But trying to represent the Spherical Earth on a flat sheet of paper led to new types of map projection and to improve maps for navigators at sea.
The first modern maps use the Mercator projection obtained by projecting the sphere outwards onto a vertical cylinder and then stretching the
scale vertically so that all the lines of latitude and longitude appear straight while all the angles and compass directions were correct.
And you can see here, Mercator, you can see the Mercator projection and also the Canadian stamp of 1898 at the top also features it.
Incidentally, Mercator was the first person to use the word Atlas for his three volume collection of maps in 1590.
But for navigation at sea you needed instruments. Astrolabes were widely used as were quadrants, sextant occupants and other instruments.
They were used to measure the altitudes of heavenly bodies such as the Sun or Polestar in order to determine latitude.
Quadrants on the top left have the shape of a quarter circle 90 degrees,
while sixteenths in the middle correspond to a sixth of a circle 60 degrees and Orton's bottom left to an eighth of a circle 45 degrees.
To measure an object's altitude, you view it along the top edge of the instrument and the position of immovable rod on the rim gives you the result.
Earlier Lave Ben Gazan had invented the widely used Jacobs staff across staff for measuring angles between celestial bodies,
which you can see on the top. Right. But unfortunately, to measure the angle between the sun and the horizon, you had to look directly at the sun.
The back staff shown below. It was a clever modification where you had your back to the sun.
Well, now move to astronomy, which is completely transformed.
When Copernicus replaced the Greeks earth centred system of planetary motion by a heliocentric one
with the sun at the centre and the earth is just one of the planets in circular orbits around it.
His book On the Revolution of the Heavenly Spheres was published in 1543.
The copy of it was supposedly presented to him on his deathbed.
The Copernican system aroused much controversy,
bringing its supporters into direct conflict with the church who considered Earth to lie at the centre of creation.
And several were arrested by the Inquisition, or even burnt alive for heresy, such as Giordano Bruno.
Indeed, at the famous Inquisition trial in 1633,
Galileo was placed under house arrest after his dialogue concerning the two chief
world systems who presented the Copernican system as superior to the Ptolemaic one.
Galileo wasn't pardoned by the church until 1995, which was a bit late to do him much good.
Galileo was the first to make extensive use of the telescope. He wasn't the first to use it.
I think that claim goes to Harriot in Oxford just a few months earlier.
But anyway, Galileo drew the moon's surface and discovered the moons of Jupiter and Saturn.
And you can see one of his drawings on the left.
In his mechanic's book, Two New Sciences, he discussed uniform and accelerated motion and explained why the path of a projectile must be a parabola.
But the greatest observer of the heavens before the invention of the telescope was the Danish astronomer Tycho Brahe.
In his observatory in Europe. Nyborg on the Island of Heaven, which I had the pleasure of visiting this summer.
He designed astronomical instruments of unequalled accuracy and measured over 700 stars.
And in particular, he discovered a new star or Nova in the constellation of Cassiopeia.
Yohannes Kepler, shown here at the bottom who worked with him, is mainly remembered for his three laws of planetary motion.
Using Tycho Brahe his extensive observations, he was led to consider non circular orbits,
proposing that the planets move in elliptical orbits the sun at one focus and the line from any sun,
from the sun to any planet sweeps out equal areas in equal times.
Kepler was fascinated by ellipses and other clinics,
and he was the one that introduced the word focus into mathematics, rotating such curves around an axis.
He found the volumes of over 90 followers of revolution by some thin disks, thereby anticipating the integral calculus.
A few years later. But a major problem of the time, particularly for astronomers and navigators, were difficulties of calculation.
In 1614, 400 years ago this year,
John Napier of Edinburgh introduces logarithms designed to replace lengthy
multiplications and divisions by easier calculations with additions and subtractions.
Being awkward to use. For example, Napier's log of one was not zero.
They were soon supplanted by Henry Briggs as similar logs to base ten produced while here in Oxford,
whose use whose use proved of enormous benefit to two astronomers and navigators.
The invention of logarithms soon led to practical instruments based on a logarithmic scale,
such as the slide rule shown on the right, dating from about 1630.
They were used for over 300 years until pocket calculators started to appear around the year 1970.
The first mechanical, calculating machines were also introduced around this time.
And in 1623, one was described by William Schick Hart on the Orange Stamp.
In letters to Kepler, while later machines were constructed by Pascal and Leyden, it's Pascal's machine, the Pascaline.
It was operated by contrails and could add and subtract numbers, while leibniz's could also multiply and divide.
But neither instrument worked particularly well in practice. Well.
Will now move to 17th century France, to Descartes, Firma and Pascal.
An ancient problem of purpose. It also the path of a point moving in a specified way relative to some fixed lines.
To solve this, Descartes named to particular length X and Y and calculated every other length in
terms of them obtaining a quadratic expression that is a conic as the required path.
Descartes thus introduced algebraic methods into geometry,
but he didn't initiate the Cartesian coordinates named after him, in which the angles are right angles.
Points appear as pairs of numbers X-Y lines for equations y equals that makes plus C and to find the intersection of two lines,
you solve two simultaneous algebraic equations.
Descartes most celebrated work was his discourse on Method, a philosophical treatise on universal science containing his contributions to geometry.
300 years later, France issued a stamp to commemorate it.
The one on the right at the top. But they got its title wrong.
To score? Sure. No method. The seal was later corrected to die on a later stamp shown below.
Still in France. Fermor, a lawyer in Toulouse, is mainly remembered for his contributions to number theory.
In particular, he famously claimed to have proved that the equation X to the envelopes,
watched their nickels read to the end has no non-zero integer solutions when an is greater than two.
And this result came to be known as Fermat's Last Theorem and was eventually proved by Andrew Wiles in 1995,
as shown by the Red Bar across the Equal Sign, which says Andrew Wiles, 1995.
Pascal showed an early interest in mathematics.
At the age of only 16, he discovered his Hexagon theorem about six points on a conic, joined up in a particular way.
Pascal's Principle in Hydrodynamics was named after him, and he was one of the first to investigate the theory of probability.
He also wrote about the arithmetical triangle of binomial coefficients and Pascal's triangle appears on the pink stamp here.
Another one from the recent International Congress of Mathematicians in Seoul, in Korea.
Isaac Newton was born in 1642 in Cambridge.
He avidly studied such contemporary works as a Latin edition of Descartes Geometry and was appointed Lucasian Professor of Mathematics,
a post later held by Stephen Hawking. The story of Isaac Newton and the apple is well known.
Seeing an apple fall, he proposed that the gravitational force pulling it to Earth is the same as the force
that keeps the moon orbiting around the Earth and the earth orbiting around the sun.
He also claimed that planetary motion is governed by a universal law of gravitation.
The inverse square law, the force of attraction between two objects varies as the product of their masses,
and inversely is a square of the distance between them.
As shown on the stamp here. In his 1687 principle, Mathematica, perhaps the greatest scientific work of all time,
Newton used the inverse square law to justify Kepler's laws of planetary Kepler's
laws of elliptical planetary motion and to account for cometary orbits,
the various variation of tides and much else besides.
But the principle here would never have appeared if Edmund Halley hadn't cajoled
a reluctant Newton into developing his ideas of gravitation and publishing them.
Halley himself paid for this. Halley, of course, later became a civilian professor of geometry here in Oxford.
So Harley or possibly and probably Hawley, but definitely not.
Haley is mainly remembered for the comet named after him observing it in 1682.
He realised it was the same as one seen many times earlier.
He predicted its return in late 1758 and its appearance then several years after his death,
did much to vindicate Newton's theory of gravitation and because of his prediction, it became known as Halley's Comet.
There are quite a few stamps featuring Halley's Comet.
I like the British one top right, which has Halley's head as the head of the comet and his wig as the tail.
But reactions to Newton's approach to gravitation were mixed.
In England, his principle was well received, even though few rude, few readers understood it.
But not everyone was receptive, particularly in France.
The principle was long and difficult and raised awkward questions about the shape of the earth.
Descartes had earlier proposed the vortex theory of the universe,
a consequence of which is that the earth is elongated at the polls that is lemon shaped.
Newton criticised Descartes ideas in the print copier, predicting a flattening of the poles so that the earth is onion shaped.
National pride was at stake and the matter was urgent because inaccurate mapmaking was leading to the loss of lives at sea.
Eventually, two geometric, two g,
two geodetic missions were dispatched to settle the matter by measuring the swing of a pendulum at the equator and near the North Pole.
De la Condamine mission on the left went to Peru and included the Spanish mathematician and Cosmograph Hawk one.
While most of his mission travelled to Lapland, but eventually they confirmed Newton's theory.
The earth is indeed flatter at the poles.
Although Newton could justly claim priority for the calculus, Livni's, who developed it independently, was the first to publish it.
Also, his notation was more versatile than Newton's and amazingly his d y by X for differentiation and the integral
sign were introduced by him within three weeks of each other in the autumn of 1675 and are still used today.
But his calculus is rather different from Newton's being based on geometry and areas rather than on velocity and motion.
The Bernoulli family included several distinguished Swiss mathematicians.
But only one, unfortunately, has been featured on a stamp.
In his book, The Art of Conjecturing, Jacob Bernoulli Top Left proved the law of large numbers.
When an experiment is performed often, it's highly probable the outcome is as we expect.
For example, tossing a fair coin 2000 times gives a number of heads close to a thousand.
With his brother, Johann Jacob Bernoulli was the first to develop Leibniz's calculus,
introducing the word integral and applying calculus to such curves as the logarithmic spiral and the cycle.
Lloyd. Leonard Euler, shown on the rest of the stamps, also grew up in Basel.
But spent most of his working life at the scientific academies of St Petersburg and Berlin.
The most prolific mathematician of all time.
He contributed to almost every branch of mathematics and physics, including number, theory, mechanics, differential equations, astronomy and optics.
Oil reformulated the calculus in terms of functions.
He introduced the notations e for exponential I for the squares for minus one capital sigma for summation and F for a function.
And he also related the exponential and trigonometric functions through the equation.
Each of the IFIs cause phi plus I signed Phi.
And if you're in Switzerland on the way to the post office and you happen to forget that formula, it's on the top right stamp of the side.
In 1750, he discovered his polyhedron formula. Faces plus vertices equals edges plus two shown the two stems on the bottom left.
And in 1735 he solved the famous Königsberg Bridges problem,
which asked whether one can cross all of the seven bridges of the city of Coningsby without visiting any bridge twice.
But he didn't draw the graph shown here on the recent Korean stamp.
Also for the International Congress.
Back now to France, where the turbulent years of the French Revolution and the rise to power of Napoleon led to important developments in mathematics.
Napoleon himself was greatly interested in the subject, and one of his greatest friends was the janitor, Gaspar Moorish,
whom you can see on the left underneath, who had taught at a military school where he studied the properties of lines and planes in three dimensions.
While investigating gun emplacements in the fortress, he greatly improved methods for projecting three dimensional objects onto a plane.
This subject soon became known as descriptive geometry.
An important consequence of the French Revolution was the founding of Paris, founding in Paris of the École Polytechnique showing top prize.
They're the country's finest mathematicians, including Marsh, La Grange, La Plus.
And Koshy taught students designed to serve in both military and civilian capacities.
Lagouranis was always successor at the court of Frederick the Great in Berlin.
He wrote the first three functions and an influential text on mechanics and in number theory,
he proved that every integer can be written as the sum of four perfect squares.
The place was the applied mathematician par excellence.
Remember for the plus transform of a of a function and full of pluses equation in mathematical physics.
He also wrote a celebrated text on probability and his monumental five volume
treatise on Celestial Mechanics earned him the title of the Newton of France.
Shortly after the French Revolution,
a commission was set up to standardise the weights and measures in France and introduce a metrics and metric system.
Here you can see a French stamp celebrating the metric system.
The chairman of this commission was La Roche and its members included the press and more.
Working analysis can continue to Koshy. The calculus would prove to be on shaky grounds.
But because she rescued it by basing it on the limits of on the concepts of limit and continuity.
He also developed complex analysis almost single handedly.
And here on the top of that stamp, you can see Coach's integral formula, which is familiar to all second year undergraduates.
Meanwhile, in Germany, Gauss is working in many areas from complex numbers.
The Gauss in number of planes plane shown on the right factor, rising polynomials.
The fundamental theorem of algebra to statistics.
The Gaussian distribution.
One of the greatest mathematicians of all time, he discovered which regular polygons can be drawn by a ruler and compasses alone.
These include equilateral triangle and regular pentagons, and also the 17 sided polygon shown here, which you thought was a circle.
The early 19th century saw major revolutions in geometry.
UK is elements that can commence with five postures for straight, straightforward,
but one different in style resembling a theorem that ought to be provable from the others.
One version of it is a parallel postulate given any line and any point not on it.
There's a unique line through the point which is parallel to the given line.
For over 2000 years. Mathematicians tried to prove this from the other postures, but failed.
And this is because their non-Euclidean geometries satisfying the first four postulates, but not the fifth.
These geometries have infinitely many lines to the point which is parallel to the given one as shown in the.
In the Hungarian stamp. There. At the bottom.
They were first published around 1830 by Nikolai Lorber Chayefsky from Russia and Polish from Hungary.
Another new geometrical object from the 19th century, actually from 1858,
was the map, a strip shown here, which has only one side and one boundary edge.
All these many developments in geometry, projective geometry, non-Euclidean geometry and so on,
forced mathematicians to ask which geometry corresponds to the world we live in.
Is it Euclidean geometry or is it a non-Euclidean one?
In fact, the geometry that later arose in Einstein's theory of relativity was one designed by the German Pendry man,
who has unfortunately never appeared on a stamp. But here are some pictures of Einstein, including the strange one in the middle of the top row.
It looks as though he's just got up in the morning with his famous equation equals M.C. squared releasing energy in mass.
Shown on the wall paper with a behind him. Obviously, that's where he got the idea.
There's also a major breakthrough in algebra when the Norwegian Niels Abel solved a longstanding problem.
Recall, the quadratic equations have been solved by the ancient Mesopotamians,
while 16th century Italians had shown how to solve equations of degrees three, three and four.
But what about equations of degree? Five or more? All showed that the change stops here.
There is no general formula to solve such equations.
Abel's work was extended by the brilliant young French mathematician Everest Galois, about whom Peter Norman has written extensively,
who explained algebraically which algebraically exactly which equations can be solved.
Galois had a short and turbulent life being sent to jail for political activities and dying tragically in a duel at the age of 20.
Having set out the previous night writing to a colleague about containing his mathematical achievements for post for posterity.
Below is William Rowan Hamilton, a child prodigy who must have several languages of an early age,
discovered an error in Le Plessis, Lester Mechanics as a teenager and became Astronomer Royal of Ireland.
While still a student. He made important advances in mechanics and geometrical optics while attempting to generalise the complex numbers.
Discover the Kryptonians a non-competitive system involving three square roots of minus one called AJ and K,
which you can see here satisfying the equation shown on the Irish stamp.
Unfortunately, not many women mathematicians have featured on stamps, but here are two important ones.
Florence Nightingale is mainly remembered as the lady with the lamp who saved
many lives through her sanitary improvements in in Crimea and war hospitals.
But she is also a fine statistician who collected and analysed mortality data from the Crimea and displayed them using her polar diagrams,
a forerunner of the pie chart.
Meanwhile, Sonya Kovalev Skyler was making valuable contributions to mathematics, physics and astronomy as well as becoming a well-known novelist.
Barred by her gender from studying in Russia, she attended a physics lectures by Kirchhoff and Helmholtz in Germany before settling in Sweden.
She later won a coveted prize from the French Academy of Sciences for a memoir on the rotation of bodies.
So we've at last reached the 20th century when several mathematicians created the subject as we now know it.
So here's a brief selection.
The range of David Hilbert was immense from number theory and so-called Hilbert spaces and analysis to potential theory and the theory of gases.
In 1900, his celebrated lecture at the International Congress of Mathematicians in Paris described
23 problems that set the agenda for mathematical research over the coming century.
Ori Poincaré shown top left wrote on the still unsolved three-body problem determining the simultaneous motion of the sun, earth and moon.
A gifted populariser of mathematics.
He developed many areas including algebraic topology, differential equations, celestial mechanics and much else besides.
Below him. In England, Bertrand Russell made fundamental contributions to mathematical logic, such as Russell's paradox.
With an and WHITEHEAD He wrote Principia Mathematica, a pioneering three volume work on the logical foundation foundations of mathematics,
in which the equation one plus one equals two isn't proved until page 86.
On the right. Ramanujan was one of the most intuitive of all mathematicians, mainly self-taught.
He left India in 1914 to work with G.H. Hardy in Cambridge, producing some spectacular joint work in number,
theory and analysis before his untimely death at the age of 32.
In Poland.
Stefan Brunner, bottom right, helped to create modern functional analysis and to develop links between topology and algebra and Banach spaces.
Then archaeology, eligibles and so on are named after him. Another 20th century development is fractal geometry.
And in 2005, Macao issued a set of stamps featuring featuring fractal patterns.
Such patterns are so similar, reproducing themselves infinitely often when magnified or reduced.
Here, top left, you can see Hong Kong snowflake curve of infinite length, but enclosing a finite area and on top, right.
Hilbert Space Filling Curve. And the series also has a cantor set and a fractal tree and the PINSKY triangle.
When the transformation of squaring and when the transformation of squaring and translating
that said goes to Z squared plus C is applied repeatedly to the complex plane.
Some points shoot off to infinity depending on the value of C, while the rest remain finite.
The boundary between these two sets is a fractal pattern called the Julia set.
After the Frenchman guessed on Julia and the attractive Israelis stamp that you can see there shows part of a Julia set.
The set of all constancy for which the Julia set is in one piece is the famous Mandelbrot set shown here on this miniature sheet.
But you prefer something a little more light-hearted. There's always the Rubik's Cube invented by the Hungarian engineer and a Rubik.
It's a colour cube whose faces can be independently rotated so as to yield over four times 10 to 19 different patterns.
The object is to restore the original colours. In the early 1980s when the Rubik's Cube craze at its height over 1 million.
Over 100 million cubes were sold. Well, we're coming towards the end.
Mathematics continues to advance at an ever increasing rate.
And since 1897, international Congresses of mathematicians have been held around the world at which many thousands
gather for every four years to learn about the most recent developments in their subject.
Several several of these Congresses have been commemorated on stamps.
Here are those moving along the top row from Moscow in 1966, Helsinki in 1978, with a design from differential geometry.
Kyoto in 1990 showing an origami polyhedron.
Berlin in 1998,
showing the division of a rectangle into unequal squares with integer integer sides with the digits of pi swirling around in the background.
In Madrid in 2006. This year, Korea issued no fewer than three stamps, all seen earlier depicting the Pythagorean theorem,
Pascals Triangle and the Königsberg Bridges problem.
And a special feature of the Icmje see awarding of Fields medals to the most outstanding mathematicians under the age of 40.
Starting in 1936, they were financed from the estate of J.C. Fields, who is the top right.
He was president of the 1924 Toronto Congress.
And here in the middle of middle of the bottom row, you can see Simon Donaldson who who was awarded his Fields medal while in Oxford.
Well above him is Alan Baker, the distinguished number theorist from Cambridge.
So to conclude, as we look back over 4000 years,
we might emulate Time magazine's My Man of the Year and ask whether the mathematician of all time has ever appeared on a stamp.
Unfortunately, the answer is yes. Thank you very much. Thank you.