There is no place and no moment in history
where I could stand and say, "Arithmetic begins here now."
People have been counting as they've been talking in every culture.
Arithmetic, like language, begins in legend.
But mathematics, in our sense, that is, reasoning with numbers,
that is another matter.
And it's to look for the origin of that,
at the hinge of legend and history,
that I have come sailing to the island of Samos.
In legendary times,
Samos was a centre of the Greek worship of Hera, the queen of heaven,
the lawful and jealous wife of Zeus.
What remains of her temple, the Heraion, dates from the sixth century before Christ.
At that time, there was born on Samos, about 580 BC,
the first genius and the founder of Greek mathematics, Pythagoras.
During his lifetime, the island was taken over by the tyrant Polycrates.
There is a tradition that before Pythagoras fled,
he taught for a while in hiding, in this small white cave in the mountains.
Samos is a magical island.
The air is full of sea and trees and music.
Perhaps Pythagoras was a kind of magician to his followers,
because he taught them that nature is commanded by numbers.
"There is a harmony in nature," he said, "a unity in her variety.
And it has a language. Numbers are the language of nature."
Pythagoras found a basic relation between musical harmony and mathematics.
The story of his discovery survives only in a garbled form, like a folk tale,
- but what he discovered was precise. (Bouzouki music)
A single stretched string vibrating as a whole
produces a ground note
The notes that sound harmonious with it
are produced by dividing the string into an exact number of parts.
- Into exactly two parts. (Lighter note)
- Into exactly three parts. (High note)
- Into exactly four equal parts. (Light hum)
And so on.
If the still point in the string, the node,
does not come at one of these exact points, the sound is discordant.
(Shrill hum)
(Low note) - This is the ground note.
(Lighter note) - This is the octave above it.
(High note) - This is the fourth above that.
(Light hum) - This is the fifth, another octave above.
(Shrill hum) - And this, which Pythagoras did not reach,
is the major third above that.
(Low note)
(Lighter note)
(High note)
Pythagoras had discovered that the chords that sound pleasing to the ear,
the Western ear,
correspond to exact divisions of the string by whole numbers.
To the Pythagoreans, that discovery had a mystic force.
They felt that all the regularities in nature are musical.
The movements of the heavens were, for them, the music of the spheres.
(Horn and bouzouki music)
Early Greek music must have sounded like this,
which has been composed on Pythagorean harmonies.
Pythagoras had proved that the world of sound is governed by exact numbers,
and he went on to prove that the same thing is true of the world of vision.
That's an extraordinary achievement.
Here I am in this marvelous colored landscape of Greece.
The wild, natural forms,
the Orphic dells, the sea.
Where under this can there lie a simple numerical structure?
Well, it's clear that it must begin
from two experiences on which our visual world is based.
That gravity is vertical and that the horizon stands at right angles to it.
And it's that which fixes the nature of the right angle.
So, if I were to turn this four times,
back I'd come to the cross of gravity and horizon.
In the world of vision.
But also in the horizontal world of experience in which in fact we live.
Consider that world.
Here I am looking across the straits from Samos to Asia Minor due south.
Pointing there.
If I turn that right angle through one right angle,
it points due west.
If I now turn it through a second right angle,
it points due north.
And if I now point that through a third right angle,
it points due east.
And the last turn would take it due south.
Not only the world as we experience it,
but the world as we construct it, is built on that relation.
Since the time the Babylonians built the Hanging Gardens,
since the time that the Egyptians built the Pyramids,
they knew...
...that there is a build, a set square,
in which the numerical relations make the right angle.
The Babylonians knew hundreds of formulae for this, oh, 2000 BC.
The Indians, the Egyptians knew them.
The Egyptians always used a set square made of three, four and five units.
It was not until 550 BC or thereabouts
that Pythagoras moved this knowledge
out of the world of empirical fact into the world of what we should now call proof.
That is, that he asked the question:
"How do such numbers follow
from the fact that a right angle is what you turn four times to point the same way?"
His proof, we think, ran something like this.
It's not the proof that stands in the schoolbooks.
Put that there.
Move this into this position.
Move that into that position.
And now we have constructed a square
on the long side of the right-angled triangle, the hypotenuse.
Just so that we should know...
...what is part of the area and what is not,
I will fill in the area with that tile.
I use tiles because all tile patterns - in Rome, in the Orient -
from now on, derive from this kind of wedding of mathematical relation to thought about nature.
Now we have a square on the hypotenuse
And we can, of course, relate that by calculation to the squares on the two shorter sides.
But we don 't need any calculation.
A small game such as children and mathematicians play
will transpose that triangle there
and this triangle here.
And now we've constructed an L-shaped figure
with the same area, of course, because it's made with the same pieces,
whose size we can see at once in terms of the smaller sides of the triangle.
Let me put that divider down.
Then it's clear to you.
That there is now a square here on the shorter side of the triangle
and a square here on the longer of the two sides enclosing the right angle.
Pythagoras had proved
that, not just for the three-four-five triangle or any Babylonian triangle,
but for every triangle,
the square on that hypotenuse is equal to the square on here and the square on here,
if, and only if, that angle is a right angle.
To this day, that remains the most important single theorem in the whole of mathematics.
That seems an extraordinary thing to say.
But it's because it's the first time
that the structure of nature is translated into numbers.
And the exact fit of the numbers describes the exact laws...
...that bind the universe.
When Pythagoras had proved the great theorem,
he offered 100 oxen to the Muses in thanks for the inspiration.
It's a... gesture of pride and humility together
such as every scientist feels to this day
when the numbers dovetail and say,
"This is part of the key to the structure of nature herself."
Pythagoras was a philosopher
and something of a religious figure to his followers as well.
The fact is that there was in him something of that Asiatic influence
which flows all through Greek culture.
We tend to think of Greece as part of the West.
But here, Samos, the edge of Classical Greece,
stands one mile from the coast of Asia Minor.
From there, the thought that inspired Greece flowed.
And it flowed back to Asia in the centuries after,
before ever it reached Western Europe.
Knowledge makes prodigious journeys.
And what seems to us a leap in time
often turns out to be a long progression from place to place,
from one city to another.
The caravans carry with their merchandise the methods of trade of their countries.
The weights and measures, the methods of reckoning.
The mathematics of Pythagoras has not come to us directly.
It fired the imagination of the Greeks,
but the place where it was formed into an orderly system was the Nile city of Alexandria.
The man who made the system and made it famous was Euclid,
who probably took it to Alexandria around 300 BC.
The impact of Euclid as a model of mathematical reasoning
was immense and lasting.
His book was translated and copied
more than any other book except the Bible, right into modern times.
I was first taught mathematics
by a man who still quoted the theorems by the numbers that Euclid gave them.
And that was not uncommon, even 50 years ago
The other science practiced in Alexandria in the centuries around the birth of Christ
was astronomy.
When the Bible says that three wise men followed a star to Bethlehem,
there sounds in the story the echo of an age when wise men are star-gazers.
The secret of the heavens that wise men looked for in antiquity
was read by a Greek called Ptolemy
working in Alexandria about 150 AD.
His work came to Europe in Arabic texts.
The Moon revolved around the Earth, obviously.
And it seemed just as obvious to Ptolemy that the Sun and the planets do the same.
The ancients thought of the Moon and the Sun as planets.
The Greeks had believed that the perfect form of motion is a circle.
And so, Ptolemy made the planets run on circles
or on circles running in their turn on other circles.
To us, that seems both simple-minded and artificial.
Yet, in fact, the system was a beautiful and a workable invention
and an article of faith for Arabs and Christians right through the Middle Ages.
Every so often, the spread of ideas demands a new impulse.
The coming of Islam, 600 years after Christ,
was the new powerful impulse.
It started as a local event, uncertain in its outcome.
But once Mohammed conquered Mecca in 630 AD,
it took the Southern world by storm.
In 100 years, Islam captured Alexandria,
established a fabulous city of learning in Baghdad,
and thrust its frontier to the East, beyond Isfahan in Persia.
In this proselytising religion, the first domed mosques were built
with no more sophisticated apparatus than the ancient builder set square that is still used.
By 730 AD,
the Muslim empire reached from Spain and Southern France
to the borders of China and India -
an empire of spectacular strength and grace, while Europe lapsed in the Dark Ages.
(Call to prayer)
This is the Masjid-i-Jami in Isfahan, the Friday Mosque,
one of the statuesque monuments of early Islam.
In centres like these, the knowledge of Greece and of the East
was treasured, absorbed and diversified.
Mohammed had been firm that Islam was not to be a religion of miracles.
It became, in intellectual content, a pattern of contemplation and analysis.
" Allah is the light of the heavens and the earth
His light may be compared to a niche that enshrines a lamp
the lamp within a crystal of star-like brilliance
His light is found in temples
which Allah has sanctioned to be built for the remembrance of his name
in them morning and evening
his praise is sung by men
whom neither trade nor profit can divert from remembering him "
One of the Greek inventions that Islam elaborated and spread was the astrolabe.
As an observational device, it is primitive.
It only measures the elevation of the sun or star, and that crudely.
But by coupling that single observation with one or more star maps,
the astrolabe also carried an elaborate scheme of computations
that could determine latitude, sunrise and sunset, the time for prayer,
the direction of Mecca for the traveler.
And over the star map, the astrolabe was embellished
with astrological and religious details, of course, for mystic comfort.
For a long time, the astrolabe was the pocket watch and the slide rule of the world.
When the poet Geoffrey Chaucer in 1391
wrote a primer to teach his son how to use the astrolabe,
he copied it from an Arab astronomer of the 8th century.
Calculation was an endless delight to Moorish scholars.
They loved problems.
This is a more elaborate ready reckoner than the astrolabe.
It's an astrological or astronomical computer, something like an automatic calendar,
made in the Caliphate of Baghdad in the 13th century.
The calculations it makes are not deep.
Yet the dials and cogs, the working machinery,
are a testimony to the mechanical skill of those who made it 700 years ago
and to their passion for playing with numbers.
The most important single innovation
that the eager, inquisitive and tolerant Arab scholars brought from afar
was in writing numbers.
The European notation for numbers was the clumsy Roman style by simple addition.
Islam replaced that by the modern decimal notation that we still call Arabic.
In this marginal note in an Arab manuscript,
the numbers in the top row are 18 and 25.
We recognize one and two at once as our own symbols,
though the two is stood on end.
The Arabic notation requires the invention of a zero.
The symbol for zero occurs twice on this page
and several more times on the next, looking just like our own.
The words zero and cipher are Arab words
So are the words algebra, almanac and a dozen others in mathematics and astronomy.
The Arabs brought the decimal system from India about 750 AD.
But it did not take hold in Europe for another 500 years after that.
It may be the size of the Moorish empire that made it a kind of bazaar of knowledge.
It may be a quality in Islam as a religion,
which, though it strove to convert people, did not despise their knowledge.
In the East, the Persian city of Isfahan is its monument.
In the West, there survives an equally remarkable outpost,
the Alhambra in Southern Spain.
Seen from the outside, it's a square, brutal fortress
that does not hint at Arab forms.
Inside, it's not a fortress, but a palace.
And a palace designed deliberately to prefigure on earth the bliss of heaven.
? Mwashah
The Alhambra is a late construction.
It has the lassitude of an empire past its peak, unadventurous and, it thought, safe.
The religion of meditation has become sensuous and self-satisfied.
It sounds with the music of water,
whose sinuous line runs through all Arab melodies,
though it's based fair and square on the Pythagorean scale.
(Fountain splashes in distance)
Each court in turn is the echo and the memory of a dream
through which the Sultan floated.
He didn�t walk, he was carried.
The Alhambra is most nearly the description of Paradise from the Koran.
"Blessed is the reward of those who labor patiently and put their trust in Allah
Those that embrace the true faith and do good work
shall be forever lodged in the mansions of Paradise
where rivers will roll at their feet
On that day there shall be radiant faces
of men well pleased with their labors in a lofty garden
A gushing fountain shall be there
and raised soft couches with goblets placed before them
silken cushions ranged in order and carpets richly spread "
The Alhambra is the last and most exquisite monument
of Arab civilization in Europe.
The last Moorish king reigned here until 1492.
And this is the most secret place in the palace.
This is where the girls of the harem came after the bath and reclined naked.
Blind musicians played in the gallery.
The eunuchs padded about.
And the Sultan watched from above, and sent an apple down
to signal to the girl of his choice that she would spend the night with him.
In a western civilization,
this room would be filled with marvelous drawings of the female form, erotic pictures.
Not so here.
The representation of the human body was forbidden to Mohammedans.
Indeed... even the study of anatomy at all was forbidden,
and that was a major handicap to Muslim science.
So, here we find colored,
but extraordinarily simple geometric designs.
The artist and the mathematician in Arab civilization have become one.
And I mean that quite literally.
These patterns represent a high point
of the Arab exploration of the subtleties and symmetries of space itself.
Begin with the very straightforward one.
Here, obviously, the translations, the reflections as symmetries are straightforward.
But note one more delicate point.
The Arabs were fond of designs
in which the dark unit of the pattern
and the light unit of the pattern
are identical.
And so, if for one moment you ignore the colors,
then you can see that you can turn this dark unit
once this way through a right angle into this position.
Then, always round this point, into this position.
And again round this point into this, and back on itself,
exactly like the Pythagorean square.
A much more subtle pattern.
These windswept triangles...
...form only one very straightforward kind of symmetry.
You could move the pattern this way,
or up into a new position, if it went there.
But suppose you neglect
the difference between the green, the yellow, the black and the royal blue.
Think of the distinction as simply between dark triangles and light triangles.
Then there is also a symmetry of rotation.
Fix your attention on this point.
This triangle can be rotated then into that position.
Then into that position.
And back here.
A threefold symmetry.
And indeed, if you forget about the colors at all,
then you could move this triangle into the white space -
because it's identical in shape -
into the dark, into the white, into the dark, into the white,
back six folds symmetry of space
Which in fact is the one that we know best.
Because it's the symmetry of the snow crystal.
So what?
Is that what mathematics is about?
Did Arab professors, do modern mathematicians,
spend their time with that kind of elegant game?
Well, it's not a game.
It brings us face to face with something which is hard to remember.
And that is that we live in a special kind of space.
Three-dimensional.
Flat.
And the properties of that space are unbreakable.
There are only certain kinds of symmetries,
not only in man -made patterns,
but in the regularities which nature herself imposes
on her fundamental atomic structures.
Here they are.
The beautiful constructs that, not man, but nature makes.
And when you look at one -
Iceland Spar -
a crystal untouched, until this moment, by human hand -
there is a shock of surprise in realizing
that flat plane is the way in which the atoms had to come together.
And that one. And that one.
And that that has been forced by space on matter
with the same finality as space made that pattern have those symmetries.
Look at the beautiful... cube of pyrites.
Or this, to me, the most exquisite crystal of all.
This is fluorite.
An octahedron.
It's also the natural shape of the diamond crystal.
Because these symmetries are imposed on them
by the nature of the space we live in.
The three dimensions, the flatness, within which we live.
And no assembly of atoms can break that crucial law of nature.
They could not have anything but the symmetries we show here.
That is, rotation through twice,
four times, three times,
or six times - but not more.
And not five.
You cannot make an assembly of atoms
to make triangles which fit space five at a time.
Thinking about these forms,
exhausting the possibilities of the symmetries of space,
was the great achievement of Arab mathematics.
And it has a wonderful finality, 1,000 years old.
The king, the naked women, the eunuchs and the blind musicians,
made a marvelous formal pattern in which the exploration of what exists...
...was perfect, but which was also not looking for any change.
There's nothing new in mathematics because there's nothing new in human thought,
until the ascent of man...
...moves forward to a different dynamic.
(Bell tolls)
Christianity began to surge back in Northern Spain about 1000 AD,
from footholds like the village of Santillana here, which the Moors never conquered.
(Men sing hymn)
This is a religion of the earth, expressed in the simple images of the village.
The ox and the ass, the lamb of God.
And not only the animal form is allowed.
The Son of God is a child.
His mother is a woman and is the object of personal worship.
When the Virgin is carried in procession,
we are in a different universe of vision,
not of abstract patterns but of abounding and irrepressible life.
When Christianity came to win back Spain,
the excitement of the struggle was on the frontier.
(Bell tolls)
Here, Moors and Christians and Jews, too,
mingled and made an extraordinary culture of different faiths.
In 1085, the centre of this mixed culture was fixed for a time in the city of Toledo.
Toledo was the intellectual port of entry into Christian Europe
of all the classics that the Arabs had brought together,
from Greece, from the Middle East, from Asia.
We think of Italy as the birthplace of the Renaissance.
But the conception was in Spain in the 12th century.
And it's symbolized and expressed by the famous school of translation at Toledo,
where the ancient texts were turned from Greek, which Europe had forgotten,
through Arabic and Hebrew into Latin.
In Toledo, an early set of astronomical tables was drawn up
as an encyclopedia of star positions.
The tables are Christian, but the numerals are Arabic
and are now recognizably modern.
The most famous of the translators and the most brilliant was Gerard of Cremona,
who had come from Italy specifically to find a copy of Ptolemy's book of astronomy,
and who stayed on in Toledo
to translate Archimedes, Hippocrates, Galen, Euclid - the classics of Greek science.
And yet, to me personally,
the most remarkable and, in the long run, the most influential man who was translated,
was not a Greek.
That's because... I am interested in the perception of objects in space.
And that was a subject about which the Greeks were totally wrong.
It was understood for the first time, about the year 1000,
by an eccentric mathematician whom we call Alhazan,
who was the one really original scientific mind that Arab culture produced.
Alhazan first recognized that we see an object
because each point of it directs and reflects a ray into the eye.
So that the cone of rays
that comes from the outline and shape of my hand
grows smaller as I move my hand away.
As I move my hand towards you,
the cone of rays that enters your eye
becomes larger and subtends a larger angle.
And that, and only that, accounts for the difference in size.
It's so simple a notion
that it's astonishing that scientists paid almost no attention to it for 600 years.
But artists attended to it almost at once.
The concept of the cone of rays from object to the eye
becomes the foundation of perspective.
And perspective is the new idea which now revivifies mathematics.
? THOMAS SIMPSON: Intrada
The excitement of perspective passed into art in North Italy,
in Florence and Venice, in the 15th century.
This is Carpaccio's painting of St Ursula leaving a vaguely Venetian port,
painted in 1495.
The obvious effect is to give to visual space a third dimension,
just as the ear about this time hears another depth and dimension
in the new harmonies in European music.
Contrast this fresco of Florence painted 100 years earlier, about 1350 AD.
There is no attempt at perspective because the painter thought of himself
as recording things, not as they look, but as they are.
A God's-eye view.
A map of eternal truth.
The perspective painter makes a step away from this absolute and abstract view.
Not so much a place as a moment is fixed for us.
All this was achieved by exact and mathematical means.
The apparatus has been recorded with care by the German artist Albrecht Durer,
who traveled to Italy in 1506 to learn the secret art of perspective.
Durer, of course, has himself fixed a moment in time.
And if we recreate his scene, we see the artist choosing the dramatic moment.
He could have stopped here.
He could have moved and frozen the vision here.
But he chose to open his eye like a camera shutter
understandably at the strong moment here.
In early perspective, it was customary to use a sight and a grid
to hold the instant of vision.
The sighting device comes from astronomy.
And the squared paper on which the picture was drawn
is now the standby of mathematics.
All the natural details in which Durer delights are expressions of the dynamic of time.
The ox and the ass, the blush of youth on the cheek of the Virgin.
The picture is The Adoration Of The Magi.
The Three Wise Men from the East have found their star
and what it announces is the birth of time.
The chalice at the centre of the painting was a test piece in teaching perspective.
This is Uccello's analysis of the way the chalice looks.
We can turn it on the computer, as the perspective artist did.
His eye worked like this to follow and explore its shifting shape,
the elongation of the circles into ellipses,
and to catch the moment of time as a trace in space.
Analyzing the changing movement of an object, as I'm doing on the computer,
was quite foreign to Greek and to Islamic minds.
They looked always for what is unchanging and static.
A timeless world of perfect order.
The most perfect shape to them was the circle.
Motion must run smoothly and uniformly in circles -
that was the music of the spheres.
That's why the Ptolemaic system was built up of circles,
along which time ran uniformly and imperturbably.
But movements in the real world are not uniform.
And they cannot be analyzed with the mathematics of antiquity.
That's a theoretical problem in the heavens,
but it's practical and immediate here on Earth.
In the flight of a projectile.
In the spurting growth of a plant.
In the single splash of a drop of liquid
that goes through abrupt changes of shape and direction.
The Renaissance did not have the technical equipment to stop the picture frame
instant by instant.
But the Renaissance had the intellectual equipment,
the inner eye of the painter
and the logic of the mathematician.
That's how Kepler, after the year 1600, became convinced
that the motion of a planet is not circular and not uniform.
It's an ellipse along which the planet runs at varying speeds.
That means that you need a new mathematics to define that instantaneous motion.
And that was invented by those two superb minds of the late 17th century,
Isaac Newton and Leibnitz.
It's now so familiar to us that we think of time as a natural element in a description of nature.
But not always so.
It was they who brought in the idea of tangent, the idea of acceleration,
the idea of slope, the idea of infinitesimal, of differential.
And the word that has been forgotten
but is really the best word for that flux of time that Newton stopped like a shutter...
Newton called it "fluxions".
The laws of nature had always been made of numbers
since Pythagoras said that was the language of nature.
But now, the language of nature had to include numbers which described time.
The laws of nature become laws of motion.
Nature herself becomes not a series of static frames,
but a moving process.