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There is no place and no moment in history
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where I could stand and say,
"Arithmetic begins here now."
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People have been counting
as they've been talking in every culture.
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Arithmetic, like language, begins in legend.
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But mathematics, in our sense,
that is, reasoning with numbers,
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that is another matter.
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And it's to look for the origin of that,
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at the hinge of legend and history,
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that I have come sailing to the island of Samos.
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In legendary times,
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Samos was a centre of the Greek worship
of Hera, the queen of heaven,
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the lawful and jealous wife of Zeus.
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What remains of her temple, the Heraion,
dates from the sixth century before Christ.
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At that time, there was born on Samos,
about 580 BC,
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the first genius and the founder
of Greek mathematics, Pythagoras.
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During his lifetime, the island was taken over
by the tyrant Polycrates.
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There is a tradition that before Pythagoras fled,
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he taught for a while in hiding,
in this small white cave in the mountains.
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Samos is a magical island.
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The air is full of sea and trees and music.
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Perhaps Pythagoras was a kind of magician
to his followers,
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because he taught them
that nature is commanded by numbers.
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"There is a harmony in nature," he said,
"a unity in her variety.
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And it has a language.
Numbers are the language of nature."
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Pythagoras found a basic relation
between musical harmony and mathematics.
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The story of his discovery survives
only in a garbled form, like a folk tale,
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- but what he discovered was precise.
(Bouzouki music)
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A single stretched string vibrating as a whole
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produces a ground note
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The notes that sound harmonious with it
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are produced by dividing the string
into an exact number of parts.
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- Into exactly two parts.
(Lighter note)
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- Into exactly three parts.
(High note)
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- Into exactly four equal parts.
(Light hum)
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And so on.
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If the still point in the string, the node,
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does not come at one of these exact points,
the sound is discordant.
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(Shrill hum)
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(Low note)
- This is the ground note.
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(Lighter note)
- This is the octave above it.
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(High note)
- This is the fourth above that.
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(Light hum)
- This is the fifth, another octave above.
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(Shrill hum)
- And this, which Pythagoras did not reach,
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is the major third above that.
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(Low note)
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(Lighter note)
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(High note)
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Pythagoras had discovered
that the chords that sound pleasing to the ear,
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the Western ear,
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correspond to exact divisions of the string
by whole numbers.
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To the Pythagoreans,
that discovery had a mystic force.
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They felt that all the regularities in nature
are musical.
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The movements of the heavens were, for them,
the music of the spheres.
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(Horn and bouzouki music)
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Early Greek music must have sounded like this,
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which has been composed
on Pythagorean harmonies.
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Pythagoras had proved that the world of sound
is governed by exact numbers,
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and he went on to prove
that the same thing is true of the world of vision.
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That's an extraordinary achievement.
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Here I am in this marvelous colored landscape
of Greece.
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The wild, natural forms,
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the Orphic dells, the sea.
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Where under this can there lie
a simple numerical structure?
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Well, it's clear that it must begin
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from two experiences
on which our visual world is based.
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That gravity is vertical
and that the horizon stands at right angles to it.
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And it's that which fixes
the nature of the right angle.
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So, if I were to turn this four times,
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back I'd come to the cross
of gravity and horizon.
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In the world of vision.
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But also in the horizontal world of experience
in which in fact we live.
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Consider that world.
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Here I am looking across the straits
from Samos to Asia Minor due south.
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Pointing there.
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If I turn that right angle through one right angle,
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it points due west.
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If I now turn it through a second right angle,
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it points due north.
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And if I now point that through a third right angle,
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it points due east.
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And the last turn would take it due south.
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Not only the world as we experience it,
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but the world as we construct it,
is built on that relation.
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Since the time the Babylonians built
the Hanging Gardens,
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since the time
that the Egyptians built the Pyramids,
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they knew...
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...that there is a build, a set square,
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in which the numerical relations
make the right angle.
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The Babylonians knew hundreds of formulae
for this, oh, 2000 BC.
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The Indians, the Egyptians knew them.
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The Egyptians always used a set square
made of three, four and five units.
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It was not until 550 BC or thereabouts
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that Pythagoras moved this knowledge
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out of the world of empirical fact
into the world of what we should now call proof.
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That is, that he asked the question:
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"How do such numbers follow
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from the fact that a right angle is what you turn
four times to point the same way?"
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His proof, we think, ran something like this.
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It's not the proof that stands in the schoolbooks.
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Put that there.
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Move this into this position.
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Move that into that position.
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And now we have constructed a square
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on the long side of the right-angled triangle,
the hypotenuse.
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Just so that we should know...
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...what is part of the area and what is not,
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I will fill in the area with that tile.
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I use tiles because all tile patterns -
in Rome, in the Orient -
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from now on, derive from this kind of wedding
of mathematical relation to thought about nature.
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Now we have a square on the hypotenuse
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And we can, of course, relate that by calculation
to the squares on the two shorter sides.
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But we don 't need any calculation.
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A small game
such as children and mathematicians play
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will transpose that triangle there
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and this triangle here.
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And now we've constructed an L-shaped figure
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with the same area, of course,
because it's made with the same pieces,
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whose size we can see at once
in terms of the smaller sides of the triangle.
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Let me put that divider down.
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Then it's clear to you.
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That there is now a square here
on the shorter side of the triangle
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and a square here on the longer of the two sides
enclosing the right angle.
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Pythagoras had proved
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that, not just for the three-four-five triangle
or any Babylonian triangle,
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but for every triangle,
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the square on that hypotenuse is equal
to the square on here and the square on here,
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if, and only if, that angle is a right angle.
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To this day, that remains the most important
single theorem in the whole of mathematics.
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That seems an extraordinary thing to say.
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But it's because it's the first time
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that the structure of nature
is translated into numbers.
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And the exact fit of the numbers
describes the exact laws...
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...that bind the universe.
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When Pythagoras had proved
the great theorem,
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he offered 100 oxen to the Muses
in thanks for the inspiration.
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It's a... gesture of pride and humility together
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such as every scientist feels to this day
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when the numbers dovetail and say,
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"This is part of the key
to the structure of nature herself."
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Pythagoras was a philosopher
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and something of a religious figure
to his followers as well.
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The fact is that there was in him
something of that Asiatic influence
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which flows all through Greek culture.
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We tend to think of Greece as part of the West.
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But here, Samos, the edge of Classical Greece,
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stands one mile from the coast of Asia Minor.
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From there,
the thought that inspired Greece flowed.
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And it flowed back to Asia in the centuries after,
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before ever it reached Western Europe.
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Knowledge makes prodigious journeys.
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And what seems to us a leap in time
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often turns out to be a long progression
from place to place,
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from one city to another.
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The caravans carry with their merchandise
the methods of trade of their countries.
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The weights and measures,
the methods of reckoning.
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The mathematics of Pythagoras
has not come to us directly.
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It fired the imagination of the Greeks,
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but the place where it was formed into an orderly
system was the Nile city of Alexandria.
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The man who made the system
and made it famous was Euclid,
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who probably took it to Alexandria
around 300 BC.
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The impact of Euclid
as a model of mathematical reasoning
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was immense and lasting.
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His book was translated and copied
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more than any other book except the Bible,
right into modern times.
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I was first taught mathematics
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by a man who still quoted the theorems
by the numbers that Euclid gave them.
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And that was not uncommon, even 50 years ago
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The other science practiced in Alexandria
in the centuries around the birth of Christ
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was astronomy.
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When the Bible says that three wise men
followed a star to Bethlehem,
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there sounds in the story the echo of an age
when wise men are star-gazers.
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The secret of the heavens
that wise men looked for in antiquity
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was read by a Greek called Ptolemy
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working in Alexandria about 150 AD.
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His work came to Europe in Arabic texts.
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The Moon revolved around the Earth, obviously.
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And it seemed just as obvious to Ptolemy
that the Sun and the planets do the same.
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The ancients thought of the Moon and the Sun
as planets.
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00:17:04,160 --> 00:17:08,073
The Greeks had believed
that the perfect form of motion is a circle.
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And so, Ptolemy made the planets run on circles
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or on circles running in their turn
on other circles.
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To us, that seems
both simple-minded and artificial.
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Yet, in fact, the system was
a beautiful and a workable invention
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and an article of faith for Arabs and Christians
right through the Middle Ages.
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Every so often, the spread of ideas
demands a new impulse.
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The coming of Islam, 600 years after Christ,
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was the new powerful impulse.
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It started as a local event,
uncertain in its outcome.
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But once Mohammed conquered Mecca
in 630 AD,
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it took the Southern world by storm.
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In 100 years, Islam captured Alexandria,
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00:18:07,440 --> 00:18:11,115
established a fabulous city of learning
in Baghdad,
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and thrust its frontier to the East,
beyond Isfahan in Persia.
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In this proselytising religion,
the first domed mosques were built
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with no more sophisticated apparatus than
the ancient builder set square that is still used.
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By 730 AD,
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the Muslim empire reached
from Spain and Southern France
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to the borders of China and India -
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an empire of spectacular strength and grace,
while Europe lapsed in the Dark Ages.
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(Call to prayer)
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This is the Masjid-i-Jami in Isfahan,
the Friday Mosque,
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00:19:03,760 --> 00:19:06,433
one of the statuesque monuments of early Islam.
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In centres like these,
the knowledge of Greece and of the East
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was treasured, absorbed and diversified.
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Mohammed had been firm
that Islam was not to be a religion of miracles.
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00:19:20,680 --> 00:19:26,949
It became, in intellectual content,
a pattern of contemplation and analysis.
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00:19:28,280 --> 00:19:30,919
" Allah is the light of the heavens and the earth
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00:19:31,000 --> 00:19:35,391
His light may be compared to a niche
that enshrines a lamp
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the lamp within a crystal of star-like brilliance
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His light is found in temples
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00:19:42,520 --> 00:19:45,796
which Allah has sanctioned to be built
for the remembrance of his name
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00:19:46,800 --> 00:19:48,756
in them morning and evening
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00:19:48,840 --> 00:19:51,149
his praise is sung by men
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whom neither trade nor profit can divert
from remembering him "
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00:20:05,760 --> 00:20:11,437
One of the Greek inventions that Islam
elaborated and spread was the astrolabe.
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00:20:13,440 --> 00:20:16,910
As an observational device, it is primitive.
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It only measures the elevation of the sun or star,
and that crudely.
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00:20:24,520 --> 00:20:29,719
But by coupling that single observation
with one or more star maps,
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00:20:29,800 --> 00:20:34,430
the astrolabe also carried
an elaborate scheme of computations
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00:20:34,520 --> 00:20:39,719
that could determine latitude,
sunrise and sunset, the time for prayer,
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00:20:39,800 --> 00:20:42,234
the direction of Mecca for the traveler.
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And over the star map,
the astrolabe was embellished
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00:20:46,400 --> 00:20:51,110
with astrological and religious details, of course,
for mystic comfort.
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00:20:53,040 --> 00:20:57,750
For a long time, the astrolabe was
the pocket watch and the slide rule of the world.
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00:20:57,840 --> 00:21:01,674
When the poet Geoffrey Chaucer in 1391
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wrote a primer to teach his son
how to use the astrolabe,
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00:21:06,400 --> 00:21:10,075
he copied it from an Arab astronomer
of the 8th century.
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00:21:13,880 --> 00:21:17,759
Calculation was an endless delight
to Moorish scholars.
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00:21:17,840 --> 00:21:19,796
They loved problems.
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This is a more elaborate ready reckoner
than the astrolabe.
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00:21:24,200 --> 00:21:30,389
It's an astrological or astronomical computer,
something like an automatic calendar,
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00:21:30,480 --> 00:21:34,268
made in the Caliphate of Baghdad
in the 13th century.
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The calculations it makes are not deep.
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00:21:45,760 --> 00:21:49,389
Yet the dials and cogs, the working machinery,
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00:21:49,480 --> 00:21:54,508
are a testimony to the mechanical skill
of those who made it 700 years ago
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00:21:54,600 --> 00:21:58,388
and to their passion for playing with numbers.
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The most important single innovation
238
00:22:02,600 --> 00:22:08,118
that the eager, inquisitive and tolerant
Arab scholars brought from afar
239
00:22:08,200 --> 00:22:10,156
was in writing numbers.
240
00:22:11,000 --> 00:22:15,915
The European notation for numbers was
the clumsy Roman style by simple addition.
241
00:22:17,400 --> 00:22:23,077
Islam replaced that by the modern
decimal notation that we still call Arabic.
242
00:22:23,960 --> 00:22:26,872
In this marginal note in an Arab manuscript,
243
00:22:26,960 --> 00:22:30,873
the numbers in the top row are 18 and 25.
244
00:22:30,960 --> 00:22:35,238
We recognize one and two at once
as our own symbols,
245
00:22:35,320 --> 00:22:37,311
though the two is stood on end.
246
00:22:38,200 --> 00:22:41,431
The Arabic notation
requires the invention of a zero.
247
00:22:42,440 --> 00:22:45,637
The symbol for zero occurs twice on this page
248
00:22:45,720 --> 00:22:49,713
and several more times on the next,
looking just like our own.
249
00:22:49,800 --> 00:22:53,793
The words zero and cipher are Arab words
250
00:22:53,880 --> 00:22:59,557
So are the words algebra, almanac and
a dozen others in mathematics and astronomy.
251
00:23:04,720 --> 00:23:09,919
The Arabs brought the decimal system
from India about 750 AD.
252
00:23:10,000 --> 00:23:14,994
But it did not take hold in Europe
for another 500 years after that.
253
00:23:33,400 --> 00:23:39,509
It may be the size of the Moorish empire
that made it a kind of bazaar of knowledge.
254
00:23:40,360 --> 00:23:42,874
It may be a quality in Islam as a religion,
255
00:23:42,960 --> 00:23:48,239
which, though it strove to convert people,
did not despise their knowledge.
256
00:23:50,120 --> 00:23:54,716
In the East, the Persian city of Isfahan
is its monument.
257
00:23:57,720 --> 00:24:02,316
In the West,
there survives an equally remarkable outpost,
258
00:24:02,360 --> 00:24:04,715
the Alhambra in Southern Spain.
259
00:24:04,800 --> 00:24:10,352
Seen from the outside,
it's a square, brutal fortress
260
00:24:10,440 --> 00:24:13,750
that does not hint at Arab forms.
261
00:24:19,520 --> 00:24:22,796
Inside, it's not a fortress, but a palace.
262
00:24:22,880 --> 00:24:28,989
And a palace designed deliberately
to prefigure on earth the bliss of heaven.
263
00:24:30,760 --> 00:24:32,716
? Mwashah
264
00:24:39,320 --> 00:24:41,356
The Alhambra is a late construction.
265
00:24:41,440 --> 00:24:49,279
It has the lassitude of an empire past its peak,
unadventurous and, it thought, safe.
266
00:24:49,360 --> 00:24:54,832
The religion of meditation
has become sensuous and self-satisfied.
267
00:24:54,920 --> 00:24:57,434
It sounds with the music of water,
268
00:24:57,520 --> 00:25:01,274
whose sinuous line
runs through all Arab melodies,
269
00:25:01,360 --> 00:25:05,512
though it's based fair and square
on the Pythagorean scale.
270
00:25:07,640 --> 00:25:09,596
(Fountain splashes in distance)
271
00:25:21,520 --> 00:25:26,310
Each court in turn is the echo
and the memory of a dream
272
00:25:26,400 --> 00:25:28,914
through which the Sultan floated.
273
00:25:29,000 --> 00:25:30,956
He didn’t walk, he was carried.
274
00:25:35,240 --> 00:25:40,792
The Alhambra is most nearly the description
of Paradise from the Koran.
275
00:25:43,160 --> 00:25:47,915
"Blessed is the reward of those
who labor patiently and put their trust in Allah
276
00:25:48,000 --> 00:25:51,515
Those that embrace the true faith
and do good work
277
00:25:51,600 --> 00:25:54,637
shall be forever lodged
in the mansions of Paradise
278
00:25:54,720 --> 00:25:57,314
where rivers will roll at their feet
279
00:25:58,800 --> 00:26:01,439
On that day there shall be radiant faces
280
00:26:01,520 --> 00:26:06,355
of men well pleased with their labors
in a lofty garden
281
00:26:06,440 --> 00:26:08,874
A gushing fountain shall be there
282
00:26:08,960 --> 00:26:12,350
and raised soft couches
with goblets placed before them
283
00:26:12,440 --> 00:26:18,072
silken cushions ranged in order
and carpets richly spread "
284
00:26:34,280 --> 00:26:40,435
The Alhambra is the last
and most exquisite monument
285
00:26:40,520 --> 00:26:43,080
of Arab civilization in Europe.
286
00:26:43,160 --> 00:26:48,598
The last Moorish king reigned here until 1492.
287
00:26:48,680 --> 00:26:53,674
And this is the most secret place in the palace.
288
00:26:53,760 --> 00:27:00,552
This is where the girls of the harem came
after the bath and reclined naked.
289
00:27:00,640 --> 00:27:03,438
Blind musicians played in the gallery.
290
00:27:03,520 --> 00:27:05,476
The eunuchs padded about.
291
00:27:05,560 --> 00:27:10,634
And the Sultan watched from above,
and sent an apple down
292
00:27:11,800 --> 00:27:16,794
to signal to the girl of his choice
that she would spend the night with him.
293
00:27:16,880 --> 00:27:18,836
In a western civilization,
294
00:27:18,920 --> 00:27:25,268
this room would be filled with marvelous
drawings of the female form, erotic pictures.
295
00:27:25,360 --> 00:27:27,794
Not so here.
296
00:27:27,880 --> 00:27:33,034
The representation of the human body
was forbidden to Mohammedans.
297
00:27:33,120 --> 00:27:39,559
Indeed... even the study of anatomy at all
was forbidden,
298
00:27:39,640 --> 00:27:42,996
and that was a major handicap
to Muslim science.
299
00:27:44,040 --> 00:27:48,909
So, here we find colored,
300
00:27:49,000 --> 00:27:54,233
but extraordinarily simple geometric designs.
301
00:27:54,320 --> 00:28:01,510
The artist and the mathematician
in Arab civilization have become one.
302
00:28:01,600 --> 00:28:04,034
And I mean that quite literally.
303
00:28:05,120 --> 00:28:08,271
These patterns represent a high point
304
00:28:09,840 --> 00:28:16,791
of the Arab exploration of the subtleties
and symmetries of space itself.
305
00:28:18,160 --> 00:28:21,277
Begin with the very straightforward one.
306
00:28:22,920 --> 00:28:29,871
Here, obviously, the translations, the reflections
as symmetries are straightforward.
307
00:28:30,720 --> 00:28:35,191
But note one more delicate point.
308
00:28:35,280 --> 00:28:37,748
The Arabs were fond of designs
309
00:28:37,840 --> 00:28:41,355
in which the dark unit of the pattern
310
00:28:43,360 --> 00:28:45,316
and the light unit of the pattern
311
00:28:45,400 --> 00:28:47,436
are identical.
312
00:28:47,520 --> 00:28:50,751
And so, if for one moment
you ignore the colors,
313
00:28:50,840 --> 00:28:54,435
then you can see
that you can turn this dark unit
314
00:28:54,520 --> 00:28:58,877
once this way through a right angle
into this position.
315
00:28:58,960 --> 00:29:02,350
Then, always round this point, into this position.
316
00:29:02,440 --> 00:29:06,638
And again round this point into this,
and back on itself,
317
00:29:06,720 --> 00:29:09,712
exactly like the Pythagorean square.
318
00:29:13,280 --> 00:29:15,236
A much more subtle pattern.
319
00:29:17,040 --> 00:29:20,316
These windswept triangles...
320
00:29:22,320 --> 00:29:27,713
...form only one very straightforward
kind of symmetry.
321
00:29:27,800 --> 00:29:29,756
You could move the pattern this way,
322
00:29:30,640 --> 00:29:32,915
or up into a new position, if it went there.
323
00:29:36,920 --> 00:29:38,876
But suppose you neglect
324
00:29:38,960 --> 00:29:44,478
the difference between the green,
the yellow, the black and the royal blue.
325
00:29:44,560 --> 00:29:51,193
Think of the distinction as simply between
dark triangles and light triangles.
326
00:29:52,280 --> 00:29:54,874
Then there is also a symmetry of rotation.
327
00:29:54,960 --> 00:29:56,951
Fix your attention on this point.
328
00:29:57,960 --> 00:30:02,078
This triangle can be rotated then
into that position.
329
00:30:04,040 --> 00:30:05,996
Then into that position.
330
00:30:06,080 --> 00:30:07,798
And back here.
331
00:30:07,880 --> 00:30:10,155
A threefold symmetry.
332
00:30:12,400 --> 00:30:17,793
And indeed, if you forget about the colors at all,
333
00:30:19,440 --> 00:30:22,830
then you could move this triangle
into the white space -
334
00:30:22,920 --> 00:30:24,717
because it's identical in shape -
335
00:30:24,800 --> 00:30:28,076
into the dark, into the white,
into the dark, into the white,
336
00:30:28,160 --> 00:30:31,277
back six folds symmetry of space
337
00:30:33,320 --> 00:30:36,073
Which in fact is the one that we know best.
338
00:30:36,160 --> 00:30:39,709
Because it's the symmetry of the snow crystal.
339
00:30:41,920 --> 00:30:43,876
So what?
340
00:30:43,960 --> 00:30:46,190
Is that what mathematics is about?
341
00:30:47,040 --> 00:30:51,113
Did Arab professors,
do modern mathematicians,
342
00:30:51,200 --> 00:30:57,116
spend their time with that kind of elegant game?
343
00:30:59,840 --> 00:31:01,796
Well, it's not a game.
344
00:31:06,760 --> 00:31:11,072
It brings us face to face
with something which is hard to remember.
345
00:31:11,160 --> 00:31:15,836
And that is that
we live in a special kind of space.
346
00:31:15,920 --> 00:31:17,876
Three-dimensional.
347
00:31:17,960 --> 00:31:19,916
Flat.
348
00:31:20,000 --> 00:31:24,915
And the properties of that space
are unbreakable.
349
00:31:26,400 --> 00:31:28,834
There are only certain kinds of symmetries,
350
00:31:28,920 --> 00:31:31,514
not only in man -made patterns,
351
00:31:32,640 --> 00:31:38,192
but in the regularities
which nature herself imposes
352
00:31:38,280 --> 00:31:42,239
on her fundamental atomic structures.
353
00:31:47,440 --> 00:31:49,396
Here they are.
354
00:31:49,480 --> 00:31:54,998
The beautiful constructs
that, not man, but nature makes.
355
00:31:56,200 --> 00:31:58,156
And when you look at one -
356
00:31:58,240 --> 00:32:00,310
Iceland Spar -
357
00:32:00,400 --> 00:32:05,599
a crystal untouched, until this moment,
by human hand -
358
00:32:07,320 --> 00:32:09,959
there is a shock of surprise in realizing
359
00:32:10,040 --> 00:32:16,559
that flat plane is the way in which
the atoms had to come together.
360
00:32:17,560 --> 00:32:20,074
And that one. And that one.
361
00:32:20,880 --> 00:32:27,831
And that that has been forced
by space on matter
362
00:32:27,920 --> 00:32:35,838
with the same finality as space made
that pattern have those symmetries.
363
00:32:38,320 --> 00:32:43,838
Look at the beautiful... cube of pyrites.
364
00:32:46,120 --> 00:32:50,716
Or this, to me, the most exquisite crystal of all.
365
00:32:50,800 --> 00:32:53,234
This is fluorite.
366
00:32:54,440 --> 00:32:56,032
An octahedron.
367
00:32:56,120 --> 00:32:59,510
It's also the natural shape
of the diamond crystal.
368
00:33:01,360 --> 00:33:05,353
Because these symmetries
are imposed on them
369
00:33:05,440 --> 00:33:08,238
by the nature of the space we live in.
370
00:33:08,320 --> 00:33:12,472
The three dimensions, the flatness,
within which we live.
371
00:33:12,560 --> 00:33:19,477
And no assembly of atoms
can break that crucial law of nature.
372
00:33:19,560 --> 00:33:25,430
They could not have
anything but the symmetries we show here.
373
00:33:26,360 --> 00:33:29,636
That is, rotation through twice,
374
00:33:29,720 --> 00:33:31,950
four times, three times,
375
00:33:32,040 --> 00:33:33,996
or six times - but not more.
376
00:33:34,080 --> 00:33:36,036
And not five.
377
00:33:36,120 --> 00:33:40,955
You cannot make an assembly of atoms
378
00:33:41,040 --> 00:33:45,716
to make triangles which fit space five at a time.
379
00:33:47,840 --> 00:33:49,796
Thinking about these forms,
380
00:33:49,880 --> 00:33:54,351
exhausting the possibilities
of the symmetries of space,
381
00:33:55,560 --> 00:33:59,712
was the great achievement
of Arab mathematics.
382
00:34:00,720 --> 00:34:06,829
And it has a wonderful finality, 1,000 years old.
383
00:34:07,680 --> 00:34:13,198
The king, the naked women,
the eunuchs and the blind musicians,
384
00:34:13,960 --> 00:34:21,992
made a marvelous formal pattern
in which the exploration of what exists...
385
00:34:23,840 --> 00:34:31,269
...was perfect,
but which was also not looking for any change.
386
00:34:32,360 --> 00:34:37,195
There's nothing new in mathematics
because there's nothing new in human thought,
387
00:34:39,280 --> 00:34:41,874
until the ascent of man...
388
00:34:45,880 --> 00:34:50,795
...moves forward to a different dynamic.
389
00:34:51,280 --> 00:34:52,838
(Bell tolls)
390
00:35:06,240 --> 00:35:12,554
Christianity began to surge back
in Northern Spain about 1000 AD,
391
00:35:12,640 --> 00:35:17,998
from footholds like the village of Santillana here,
which the Moors never conquered.
392
00:35:18,080 --> 00:35:19,957
(Men sing hymn)
393
00:35:20,040 --> 00:35:25,273
This is a religion of the earth,
expressed in the simple images of the village.
394
00:35:25,360 --> 00:35:29,353
The ox and the ass, the lamb of God.
395
00:35:29,440 --> 00:35:31,908
And not only the animal form is allowed.
396
00:35:32,000 --> 00:35:34,468
The Son of God is a child.
397
00:35:34,560 --> 00:35:39,156
His mother is a woman
and is the object of personal worship.
398
00:35:39,960 --> 00:35:42,474
When the Virgin is carried in procession,
399
00:35:42,560 --> 00:35:45,996
we are in a different universe of vision,
400
00:35:46,080 --> 00:35:51,757
not of abstract patterns
but of abounding and irrepressible life.
401
00:35:51,840 --> 00:35:53,876
402
00:36:46,520 --> 00:36:49,398
When Christianity came to win back Spain,
403
00:36:49,480 --> 00:36:53,155
the excitement of the struggle
was on the frontier.
404
00:36:53,240 --> 00:36:55,196
(Bell tolls)
405
00:37:00,280 --> 00:37:02,794
Here, Moors and Christians and Jews, too,
406
00:37:02,880 --> 00:37:08,352
mingled and made an extraordinary culture
of different faiths.
407
00:37:08,440 --> 00:37:15,949
In 1085, the centre of this mixed culture
was fixed for a time in the city of Toledo.
408
00:37:31,200 --> 00:37:35,478
Toledo was the intellectual port of entry
into Christian Europe
409
00:37:35,560 --> 00:37:38,836
of all the classics
that the Arabs had brought together,
410
00:37:38,920 --> 00:37:41,753
from Greece, from the Middle East, from Asia.
411
00:37:42,800 --> 00:37:46,190
We think of Italy as the birthplace
of the Renaissance.
412
00:37:46,280 --> 00:37:50,592
But the conception was in Spain
in the 12th century.
413
00:37:50,680 --> 00:37:56,357
And it's symbolized and expressed
by the famous school of translation at Toledo,
414
00:37:56,440 --> 00:38:00,877
where the ancient texts were turned from Greek,
which Europe had forgotten,
415
00:38:00,960 --> 00:38:03,952
through Arabic and Hebrew into Latin.
416
00:38:06,280 --> 00:38:09,909
In Toledo, an early set of astronomical tables
was drawn up
417
00:38:10,000 --> 00:38:13,310
as an encyclopedia of star positions.
418
00:38:13,400 --> 00:38:17,473
The tables are Christian,
but the numerals are Arabic
419
00:38:17,560 --> 00:38:20,472
and are now recognizably modern.
420
00:38:24,200 --> 00:38:29,513
The most famous of the translators
and the most brilliant was Gerard of Cremona,
421
00:38:29,600 --> 00:38:35,630
who had come from Italy specifically
to find a copy of Ptolemy's book of astronomy,
422
00:38:36,680 --> 00:38:38,636
and who stayed on in Toledo
423
00:38:38,720 --> 00:38:46,308
to translate Archimedes, Hippocrates,
Galen, Euclid - the classics of Greek science.
424
00:38:50,600 --> 00:38:52,556
And yet, to me personally,
425
00:38:52,640 --> 00:38:58,192
the most remarkable and, in the long run,
the most influential man who was translated,
426
00:38:58,280 --> 00:39:00,236
was not a Greek.
427
00:39:02,720 --> 00:39:09,876
That's because... I am interested
in the perception of objects in space.
428
00:39:10,920 --> 00:39:15,038
And that was a subject about which
the Greeks were totally wrong.
429
00:39:16,880 --> 00:39:21,396
It was understood for the first time,
about the year 1000,
430
00:39:21,480 --> 00:39:25,268
by an eccentric mathematician
whom we call Alhazan,
431
00:39:27,120 --> 00:39:32,513
who was the one really original scientific mind
that Arab culture produced.
432
00:39:34,840 --> 00:39:41,109
Alhazan first recognized
that we see an object
433
00:39:42,160 --> 00:39:49,794
because each point of it
directs and reflects a ray into the eye.
434
00:39:50,800 --> 00:39:55,351
So that the cone of rays
435
00:39:55,440 --> 00:40:00,514
that comes from the outline and shape
of my hand
436
00:40:00,600 --> 00:40:05,549
grows smaller as I move my hand away.
437
00:40:09,520 --> 00:40:11,476
As I move my hand towards you,
438
00:40:11,560 --> 00:40:15,394
the cone of rays that enters your eye
439
00:40:16,560 --> 00:40:20,075
becomes larger and subtends a larger angle.
440
00:40:21,400 --> 00:40:25,473
And that, and only that,
accounts for the difference in size.
441
00:40:26,880 --> 00:40:28,836
It's so simple a notion
442
00:40:28,920 --> 00:40:34,313
that it's astonishing that scientists paid
almost no attention to it for 600 years.
443
00:40:36,480 --> 00:40:38,869
But artists attended to it almost at once.
444
00:40:41,040 --> 00:40:44,112
The concept of the cone of rays
from object to the eye
445
00:40:44,200 --> 00:40:48,273
becomes the foundation of perspective.
446
00:40:49,560 --> 00:40:56,910
And perspective is the new idea
which now revivifies mathematics.
447
00:40:58,000 --> 00:40:59,718
? THOMAS SIMPSON: Intrada
448
00:41:06,400 --> 00:41:10,473
The excitement of perspective passed into art
in North Italy,
449
00:41:10,560 --> 00:41:13,358
in Florence and Venice, in the 15th century.
450
00:41:13,440 --> 00:41:20,152
This is Carpaccio's painting
of St Ursula leaving a vaguely Venetian port,
451
00:41:20,240 --> 00:41:22,595
painted in 1495.
452
00:41:29,520 --> 00:41:34,548
The obvious effect is to give to visual space
a third dimension,
453
00:41:34,640 --> 00:41:38,713
just as the ear about this time hears
another depth and dimension
454
00:41:38,800 --> 00:41:42,475
in the new harmonies in European music.
455
00:41:50,920 --> 00:41:57,598
Contrast this fresco of Florence
painted 100 years earlier, about 1350 AD.
456
00:41:57,680 --> 00:42:02,435
There is no attempt at perspective
because the painter thought of himself
457
00:42:02,520 --> 00:42:07,548
as recording things,
not as they look, but as they are.
458
00:42:07,640 --> 00:42:09,596
A God's-eye view.
459
00:42:09,680 --> 00:42:11,636
A map of eternal truth.
460
00:42:12,560 --> 00:42:18,590
The perspective painter makes a step away
from this absolute and abstract view.
461
00:42:25,280 --> 00:42:29,319
Not so much a place as a moment is fixed for us.
462
00:42:31,120 --> 00:42:35,750
All this was achieved
by exact and mathematical means.
463
00:42:35,840 --> 00:42:41,233
The apparatus has been recorded with care
by the German artist Albrecht Durer,
464
00:42:41,320 --> 00:42:48,556
who traveled to Italy in 1506
to learn the secret art of perspective.
465
00:42:49,600 --> 00:42:52,956
Durer, of course,
has himself fixed a moment in time.
466
00:42:53,040 --> 00:42:59,434
And if we recreate his scene,
we see the artist choosing the dramatic moment.
467
00:43:03,440 --> 00:43:05,396
He could have stopped here.
468
00:43:07,080 --> 00:43:09,719
He could have moved
and frozen the vision here.
469
00:43:11,080 --> 00:43:15,039
But he chose to open his eye
like a camera shutter
470
00:43:15,120 --> 00:43:19,193
understandably at the strong moment here.
471
00:43:21,680 --> 00:43:26,959
In early perspective, it was customary
to use a sight and a grid
472
00:43:27,040 --> 00:43:28,996
to hold the instant of vision.
473
00:43:29,080 --> 00:43:32,038
The sighting device comes from astronomy.
474
00:43:32,120 --> 00:43:35,157
And the squared paper
on which the picture was drawn
475
00:43:35,240 --> 00:43:37,993
is now the standby of mathematics.
476
00:43:45,680 --> 00:43:51,357
All the natural details in which Durer delights
are expressions of the dynamic of time.
477
00:43:51,440 --> 00:43:56,434
The ox and the ass,
the blush of youth on the cheek of the Virgin.
478
00:43:59,960 --> 00:44:03,077
The picture is The Adoration Of The Magi.
479
00:44:03,160 --> 00:44:07,597
The Three Wise Men from the East
have found their star
480
00:44:07,680 --> 00:44:11,275
and what it announces is the birth of time.
481
00:44:19,400 --> 00:44:24,030
The chalice at the centre of the painting
was a test piece in teaching perspective.
482
00:44:25,440 --> 00:44:29,149
This is Uccello's analysis
of the way the chalice looks.
483
00:44:30,520 --> 00:44:34,274
We can turn it on the computer,
as the perspective artist did.
484
00:44:34,360 --> 00:44:39,718
His eye worked like this
to follow and explore its shifting shape,
485
00:44:39,800 --> 00:44:42,678
the elongation of the circles into ellipses,
486
00:44:43,480 --> 00:44:47,234
and to catch the moment of time
as a trace in space.
487
00:44:51,840 --> 00:44:55,833
Analyzing the changing movement of an object,
as I'm doing on the computer,
488
00:44:55,920 --> 00:45:00,436
was quite foreign to Greek and to Islamic minds.
489
00:45:00,520 --> 00:45:05,355
They looked always
for what is unchanging and static.
490
00:45:05,440 --> 00:45:08,671
A timeless world of perfect order.
491
00:45:15,400 --> 00:45:18,358
The most perfect shape to them was the circle.
492
00:45:18,440 --> 00:45:23,116
Motion must run smoothly and uniformly
in circles -
493
00:45:23,200 --> 00:45:25,236
that was the music of the spheres.
494
00:45:34,480 --> 00:45:37,278
That's why the Ptolemaic system
was built up of circles,
495
00:45:37,360 --> 00:45:41,148
along which time ran uniformly
and imperturbably.
496
00:45:41,240 --> 00:45:44,471
But movements in the real world are not uniform.
497
00:45:44,560 --> 00:45:49,236
And they cannot be analyzed
with the mathematics of antiquity.
498
00:45:49,320 --> 00:45:52,153
That's a theoretical problem in the heavens,
499
00:45:52,240 --> 00:45:55,038
but it's practical and immediate here on Earth.
500
00:45:56,400 --> 00:45:58,356
In the flight of a projectile.
501
00:46:00,000 --> 00:46:01,956
In the spurting growth of a plant.
502
00:46:10,080 --> 00:46:12,674
In the single splash of a drop of liquid
503
00:46:12,760 --> 00:46:16,719
that goes through abrupt changes
of shape and direction.
504
00:46:18,920 --> 00:46:23,152
The Renaissance did not have the technical
equipment to stop the picture frame
505
00:46:23,240 --> 00:46:25,196
instant by instant.
506
00:46:27,840 --> 00:46:30,434
But the Renaissance had
the intellectual equipment,
507
00:46:30,520 --> 00:46:32,476
the inner eye of the painter
508
00:46:32,560 --> 00:46:34,516
and the logic of the mathematician.
509
00:46:37,160 --> 00:46:41,278
That's how Kepler, after the year 1600,
became convinced
510
00:46:41,360 --> 00:46:45,319
that the motion of a planet is not circular
and not uniform.
511
00:46:45,400 --> 00:46:51,191
It's an ellipse along which the planet runs
at varying speeds.
512
00:46:54,880 --> 00:47:01,069
That means that you need a new mathematics
to define that instantaneous motion.
513
00:47:01,160 --> 00:47:06,871
And that was invented by those
two superb minds of the late 17th century,
514
00:47:06,960 --> 00:47:10,316
Isaac Newton and Leibnitz.
515
00:47:11,800 --> 00:47:17,557
It's now so familiar to us that we think of time
as a natural element in a description of nature.
516
00:47:19,160 --> 00:47:21,116
But not always so.
517
00:47:22,280 --> 00:47:27,035
It was they who brought in the idea of tangent,
the idea of acceleration,
518
00:47:27,120 --> 00:47:32,717
the idea of slope, the idea of infinitesimal,
of differential.
519
00:47:32,800 --> 00:47:35,189
And the word that has been forgotten
520
00:47:35,280 --> 00:47:40,673
but is really the best word for that flux of time
that Newton stopped like a shutter...
521
00:47:42,040 --> 00:47:43,996
Newton called it "fluxions".
522
00:47:46,600 --> 00:47:50,832
The laws of nature
had always been made of numbers
523
00:47:50,920 --> 00:47:53,992
since Pythagoras said
that was the language of nature.
524
00:47:54,080 --> 00:47:59,029
But now, the language of nature had to include
numbers which described time.
525
00:48:00,160 --> 00:48:02,993
The laws of nature become laws of motion.
526
00:48:03,080 --> 00:48:07,278
Nature herself becomes
not a series of static frames,
527
00:48:07,360 --> 00:48:09,954
but a moving process.