This is the story of a book
that could have changed the history of the World.
Lost for over a thousand years,
it contains a unique record of the world and mind
of one of the greatest men ever,
a mathematical genius
who was centuries ahead of his time:
Archimedes.
When the manuscript first arrived,
you know, shivers ran, ran down my spine.
I have never before in my life handled a book
that is the only material witness
to the mind of someone who died 2.200 years ago.
This is a manuscript of incalculable value
to the history of science.
That feeling, that, that excitement
was, was my, my first reaction.
I think it's fair to say that western science
is a series of footnotes to Archimedes.
People trying to come to terms with the problems of Archimedes,
people trying to produce works that are as great as Archimedes,
that are greater than Archimedes.
This is the goal, this is the goal of western mathematics.
As scientists worked to recover the text from this fragile document,
they are discovering that Archimedes was further ahead of his time
than they had ever believed.
If his secrets had not lain hidden for so long,
the World today could be very different from what we know.
Archimedes' manuscript
is one of the most valuable ever found.
Sold at auction for $2 million.
The buyer refused to reveal his identity,
only that he was a billionaire who'd made his money in IT.
But research institutes all over the world
wanted to work on his precious manuscript.
I did what I think an awful lot of people did
which was to get in touch with the book dealer
who acted on behalf of the anonymous owner of the manuscript.
I sent the e-mail to the book dealer
and three days later I got an e-mail back.
It said:
"Dear Mr. Noel,
I'm sure that you can borrow the Archimedes manuscript
and the owner would be delighted in this idea."
The owner visited the museum, together with the book dealer,
and they left their equipment and kit on this table,
so we went out to lunch...
and I said to him that it was extremely kind of him
to even consider thinking of depositing
the Archimedes manuscript.
He looked at me and he said, "I've already deposited it with you",
and I said, "I'm sorry!"
I was a little alarmed,
and he said, "Yes, it, it was on a duffle bag on my table".
So I had to sit through a three course meal
shaking to get back and thinking sitting on my desk on a duffle bag!...
I can't believe it! And...
But after lunch we came here and opened the duffle bag and,
and, and it was an amazing experience!
This book contains unique works of Archimedes, lost for centuries,
including the most important mathematics he ever wrote.
Because unlike other known writings by the Ancient Greek genius,
this book does much more than list his mathematical achievements.
This book explains how he made his discoveries.
The Archimedes manuscript is,
to all intents and purposes,
the material remains of the thought of the man.
I like to think of it as his brain in a box
and it's for us to dig into that box
and to pull out new thought.
I wake up every day...
knowing that Archimedes is actually dependent upon the team of people
that we've gathered together to, to, to really,
to really allow him to speak for the first time.
Retracing the story of how the Archimedes manuscript
ended its journey at the museum
is a remarkable tale of mystery and intrigue.
The story starts back in Sicily,
in 287 B.C.
when Archimedes was born.
Much about his life remains shrouded in obscurity.
Historians have had to rely on the few surviving records of his work
to try and piece together a picture of Archimedes,
revealing a man
with an extraordinary genius for mathematics.
In antiquity he stands alone.
There is no other mathematician in antiquity,
or for that matter in history, that comes close to Archimedes.
Archimedes has become famous as the man who shouted "Eureka!",
in the bath.
He was trying to solve a problem with a gold wreath
given to the King.
The King suspected the goldsmith who had made it
had slipped in some cheaper silver.
The wreath weighed the right amount,
but silver is lighter than gold.
So the question was:
was it greater in volume than it would have been
if it was made with pure gold?
Archimedes' insight into how to determine a volume
is supposed to have come when he got into a bath,
noticed that the more of him went in,
the more water poured out of the edges of the bath tub,
and realised that this was, in fact,
giving an exact measure of the volume of him going in,
and this would apply to the crown too.
You could find how big the crown was
by immersing it in a vessel of water
and seeing how much water is displaced.
He's supposed to have been so excited by this discovery
that he immediately leaped out of his bath
and without throwing any clothes on
ran naked through the streets of Siracusa shouting the Greek word
"I've discovered it - eureka, eureka!"
It's probably unlikely that the citizens of Sicily
ever saw Archimedes' naked body,
but he did go on to reveal the truth behind the King's wreath.
When the wreath was immersed in the water
then it turned out that in fact its volume was greater than it should have been
if it had been pure gold.
So the smith was clearly not an honest one
and Archimedes had successfully worked out some good detective work.
During his life Archimedes became famous for invention...
And many of his ideas are used in machines today,
but he was best known, and feared,
for his weapons of war.
In a garden in Philadelphia,
an Archimedes enthusiast
has re-created one of his hero's most impressive schemes.
This is a model of the walls of Siracusa,
the Greek city state, in Sicily, in which Archimedes lived.
He was assigned by his King
to be the military adviser
and to design the defences of the city.
and his main defence
were these so-called claws, or iron heads,
that line about a one kilometre long piece of the wall.
The ships would come in close to the wall,
then the, then the claw would be swung around
and the grappling hook dropped.
The ship would be raised a certain amount,
then the grappling hook would be suddenly released.
The ship would come smashing into the ground.
All of these actions just frightened the Romans to death.
But it's through his mathematics
that the true genius of Archimedes is revealed.
He came up with a value for Pi,
probably the most famous mathematical symbol of all.
Vital for calculating the area of a circle
it's one of the most basic building blocks of science,
the mathematical equivalent of the invention of the wheel.
The way he goes about it
is to try to squeeze the circle between polygons.
You can find the perimeter of polygons because they're straight sides
and if he can get polygons
that wrap closer and closer to the perimeter of the circle
then he will have a closer and closer pair of bounds
within which Pi must lie.
He begins by putting a hexagon inside the circle.
Continuing further he next divided the hexagon,
doubled the number of sides to come up with a dodecagon,
a 12 sided figure,
and determining its circumference he has a still better approximation.
No need to stop there.
We can take each of these and put two sides where one was before
and we'll put them all in because you see it's so close to the circle already
that on the drawing it starts to look like the circle.
We now have 24 sides.
He continued this way going from 24 to 48
and finally ending up with 96.
On the outside he does the same thing.
He starts with the hexagon and...
for every side he makes two sides by putting more in like that
so we now have 12 and so on,
until again you have 96 sides outside as well as in.
So in this way he guarantees that the number Pi
is trapped between three and ten seventy-firsts
and three and one-seventh.
An estimate which is accurate to within one part in 2.000,
better than one part in 2.000 !
And indeed this approximation,
three in one-seventh is still used by engineers today
and is more than good enough for all practical purposes.
Obsessed by mathematics...
there was no problem too ambitious for Archimedes!
He even tried to calculate the number of grains of sand
to fill the universe.
The answer: 10...
followed by 62 zeros.
We're told that Archimedes was often so preoccupied
with his mathematical work
that sometimes even just to get him
to go to bathe was difficult.
His, his slaves would have to carry him off forcibly we're told
and even in the bath he would spend his time
drawing little diagrams with the soapsuds presumably on his body.
Ancient historians reported that Archimedes would become ecstatic
as he discovered more and more complex mathematical shapes.
Four triangles and four hexagons constitute
a truncated tetrahedron.
Eight triangles and six squares,
a cubeoctahedron.
Eight triangles and eighteen squares constitute
a rhombic-cubeoctahedron.
Twelve squares, eight hexagons, six octagons -
a truncated cubeoctahedron.
Thirty-two triangles and six squares constitute
a snub cube.
Truncated dodecahedron...
Snub dodecahedron.
Truncated icosa...
Rhombicosi- dodecahedron.
Cut!
But tragically,
Archimedes' genius had brought him to the attention of the Romans
who were eager to capture him.
When they finally managed to invade Siracusa
instructions were issued to take Archimedes prisoner.
Some soldiers had apparently been delegated to find Archimedes
to take him to the Roman General.
But a soldier who wasn't given these instructions...
barged into Archimedes' house,
found him entirely preoccupied with doing mathematics,
making drawings on a dust board.
And...
Archimedes didn't even, hadn't even heard the bustle that was going on!
when he turned to the soldier demanding to not disturb his circles,
and the soldier...
killed him with his sword.
That was the end of Archimedes.
Archimedes' death in 212 B.C.
brought a golden age in Greek mathematics to an end.
There was no-one that could follow him in Europe.
Greek mathematics then gradually declined
and then the Dark Ages,
the Age of Faith entered, where all interest in mathematics was lost.
And as a result, nothing really interested us scientifically.
But Archimedes' writings did survive,
copied by scribes who passed on his precious mathematics
from generation to generation...
until in the 10th-century,
one final copy of his most important work was made.
But interest in mathematics had now died.
Archimedes' name was forgotten.
When one day, in the 12th-century
a monk ran out of parchment...
with devastating results!
The pages were re-used to make a prayer book.
Each of the sheets
that makes a double page in the Archimedes manuscript
was unbound, cut down the binding, turned sideways
and then folded to make a new double page in a prayer book.
The pages were washed, or scraped plain enough
so that it would then be possible to write over them
with the religious texts that are now the...
the obviously visible part of this manuscript.
The ancient works of a mathematical genius
were systematically consigned to oblivion.
Washed clean, re-used and written over,
the manuscript had become what's known as a palimpsest.
It had begun a new life as a book of prayers,
at the Mar Saba monastery,
in the Judean desert.
And there it was used as a prayer book,
the Archimedes text completely unread and unknowing
for many, many centuries.
And so Archimedes' secrets lay hidden
in the library tower of the monastery,
while all around them the rest of the known world moved on.
In the 15th-century
the Renaissance hit Europe.
Now at last, science had advanced enough
for scholars to understand Archimedes' mathematical arguments.
But no-one had the slightest clue
that some of his greatest ideas had been lost.
Renaissance mathematicians had to grapple with concepts and problems
that Archimedes had worked out in his bath
1.500 years before.
If the mathematicians and scientists of the Renaissance
had been aware of these discoveries of Archimedes
this could have had a tremendous impact on the development of mathematics.
The 15th -16th century was a crucial period in mathematics,
It was hundreds of years before the manuscript was heard of again.
No-one knows how, but it turned up
in a library in Constantinople.
The library catalogue listed several lines from the manuscript.
These caught the eye of Johan Ludwig Heiberg,
Danish historian of mathematics and Greek expert.
He realised at once that the words could only come from one source:
Archimedes.
Determined to find out more,
he had arrived in Constantinople, in 1906.
Heiberg must have had his hopes up,
but when he saw the manuscript itself
he must have been flabbergasted.
And...
he of all people knew the significance of what he was reading.
It must have been an extraordinary moment for him.
Heiberg wasn't allowed to remove the manuscript from the library,
so instead he asked a photographer to take pictures of every page
and from these photos he attempted to reconstruct Archimedes' work.
It was an incredibly difficult task.
It is remarkable how much Heiberg was able to get out of this manuscript,
given the condition of the manuscript...
The Archimedes text is really very faint on most of the pages.
He had limited time and, so far as we can tell,
the only help that he had in reading the manuscript was a magnifying glass.
Heiberg's discovery revealed ideas
that had never been seen before.
The discovery of the Archimedes manuscript was,
was so significant that it, that it made the front page of the New York Times.
This was understood at the time
as a major breakthrough in the history of mathematics.
What Heiberg had stumbled across
was like going inside Archimedes' brain.
Here Archimedes didn't just give the answers to his calculations,
but he wrote down the detailed demonstrations.
It was a book he had called, "The Method".
It's a book about discovery
instead of a book about how you get to the result in the first place
before you do the proof.
This is very rare, in fact there's no whole book
that we have from antiquity, aside from The Method,
that addresses that kind of question.
This was a spectacular find for the history of mathematics.
It was very much like getting a glimpse into Archimedes' mind.
If you were a painter, for example,
you would certainly be interested in the finished works of the Masters,
but more than that you want to learn the techniques of the Masters.
What paint did they use, how did they outline their subjects?
Likewise with mathematicians,
they want to know not just what his finished works were,
but how did he arrive at them.
"The Method" revealed that Archimedes had come up with a radical approach
that no mathematician had come close to inventing.
to compare the volumes of curved shapes, he had dreamt up
an entirely imaginary set of scales.
He used this to try and work out the volume of a sphere.
Now prior to Archimedes
the volume of a cone and a cylinder was already known,
and so he tried to use those previously known results
to compute the volume of a sphere.
And so he concocted this very interesting balancing act.
between the sphere and the cone on one side,
with the cylinder on the other.
Using very complex mathematics in his head,
in which he imagined cutting the shapes into an infinite number of slices,
Archimedes was able to work out
how to balance the objects on the scales.
The final result after all the arithmetic was done
was that the volume of a sphere
is precisely two-thirds of the volume of the cylinder that encloses this sphere.
This was a result that he considered most important mathematical discovery
he asked that it be inscribed on his tombstone.
Working out volumes using infinite slicing
suggested that Archimedes was taking the first step
towards a vital branch of mathematics, known as calculus.
1.800 years before it was invented!
The modern world couldn't live without calculus.
It is a form of mathematics essential for scientists and engineers.
21st century technology depends on it.
But back in 1914,
as he was poised to uncover the true genius of Archimedes,
Heiberg's plan to study the manuscript further in Constantinople
was brutally interrupted.
World War 1 turmoil Europe and the Middle East,
and the palimpsest was lost again.
No-one had any idea of the secrets that still lay buried within its covers.
Scholars had little hope that they would ever see the document again.
Then in 1971,
Ancient Greek expert Nigel Wilson
heard about a single page of a manuscript
in a library, in Cambridge.
He decided to take a closer look.
I transcribed a few sentences, almost completely.
They included some rather rare technical terms
and if you go to the Greek lexicon and check where those terms occur
you soon find that you're dealing with essays by Archimedes .
And then I suddenly realised this must be a leaf
detached from the famous palimpsest.
It was a very good moment.
I became rather excited.
But why had a single page, and only a single page
of the Archimedes palimpsest turned up in Cambridge?
A clue lay in a collection of papers that were handed over to the university,
papers that had belonged to a scholar of few scruples,
called Constantine Tischendorf.
Tischendorf travelled a lot in the Near East.
When he got to Constantinople he visited the library,
he said that at the time it had about 30 manuscripts in it,
they weren't of any interest with one exception.
He mentions a palimpsest with a mathematical text in it.
He doesn't say anymore.
When Tischendorf discovered the palimpsest
he was tempted to examine it... in much more detail.
I don't think there's any alternative to the assumption that he stole this page.
He must have waited until the librarian was out of the room.
And I think he wasn't sufficiently interested in Greek science
to know enough to identify the text as Archimedes,
but he probably had a hunch that it was something important.
At the turn of the 20 century
Heiberg only had a magnifying glass with which to read the manuscript.
Now Nigel Wilson had the advantage of modern technology.
When I got the leaf to work on
most of it was legible, not quite all,
but with the ultraviolet lamp,
the corners which one couldn't read, became clear.
I realised that with an ultraviolet lamp
one ought to be able to read most, if not everything,
that had remained a mystery to Heiberg.
This tantalising hi-tech glimpse of a single page
revealed how much more could be gleaned from Archimedes' work
if only they knew what had happened to it
since Heiberg had last held it in his hand.
After World War I, Paris and other European cities
were flooded with works of art from the Middle East.
Yet there had been no sign of any manuscript of Archimedes!
But in 1991,
Felix de Marez Oyens arrived at Christies
to discover that the Archimedes palimpsest
might have been in Paris all along.
At his new office he found a letter from a French family
who claimed they had a palimpsest.
They were talking about this...
amazing palimpsest manuscript
of this... incredibly important scientific classical text.
So you take a little bit of distance
and, you know, you don't immediately get over-excited,
but I did realise immediately that if this thing was authentic
that it would be something incredibly exciting.
Intrigued by the letter, Felix wrote back
and discovered that the owners lived just around the corner from his in Paris.
So he set off to examine the book.
But it was quite, immediately quite clear
that this had to be... that manuscript
that was had been seen, or studied for the first time,
by Heiberg, in 1906.
The owners had an unusual story to tell.
In the 1920s, a member of the Parisian family,
who was a keen amateur collector
had travelled to Turkey,
and somehow had acquired the manuscript in Constantinople.
All the time that it had been assumed lost
the palimpsest had been lying in the family's Paris apartment,
but now they had decided to alert the art world
because they wanted to sell.
Felix had to decide how much the manuscript was worth.
Well the valuation of important manuscripts,
let alone palimpsests, are terribly difficult in general,
But I think I told them, it would have to be worth about £400,000-600,000.
Any valuation of something like that is simply a guess.
Perhaps if you're lucky in it you get an educated guess.
$1,400,000.
$1,800,000.
The manuscript sold for far more than ever Felix had predicted.
$2 million.
An anonymous billionaire paid $2 million!
And so finally the manuscripts arrived
at The Walters Art Museum in Baltimore,
and into the hands of curator Will Noel.
He was in for a terrible shock.
I was horrified, I was aghast.
It's, it's, it's really a disgusting document!
It really looks very, very, very ugly.
It doesn't look like a great object at all !
It looks dreadful, I mean it really looks dreadful !
It's been burnt, it's got modern PVA glue on its spine.
The Archimedes text that we're trying to recover
goes behind that PVA glue.
It's got blue tack on it,
it's got strips of paper that have been stuck on top of it.
It's very hard to describe adequately
the poor condition of the Archimedes palimpsest is in.
Will quickly put a team together from all over the world
to try and rescue the book:
the Greek experts Reviel, Natalie and Nigel,
the imagers Bill, Roger and Keith,
and finally conservationist Abigail.
Detailed examination in the conservation lab
revealed the appalling damage that the book has suffered.
The manuscript was heavily damaged by mould.
This is seen in a lot of these purple spots
all over the surface of the leaves.
In this area of the parchment there's a very intense purple stain.
The parchment is perforated
where the fungi have actually gone through and digested the collagen
and it means that the Archimedes text
is just totally missing in these areas.
It's like Bovine Spongiform Encephalopathy.
What BSE does to the brain mould does to the Archimedes manuscript.
I tend to think of the Archimedes manuscript
as Archimedes' brain in a box,
so this really is
an appalling wasting away of a great mind.
And the team have discovered there is another problem.
On several pages of the palimpsest
there are mysterious illustrations
which completely cover over the text.
When we first saw the manuscript
we were very curious about these paintings.
They could possibly be Medieval,
but the colours were tonally wrong for this period.
The other curious thing was that these leaves
seem to have been intentionally mutilated,
possibly to make them look Medieval
by adding additional knife cuts to the edges of the leaves.
Puzzlingly Heiberg made no mention of these pictures
when he studied the manuscript in 1906.
So the conservation team showed these illustrations
to Byzantine expert John Lowden.
I was convinced I had seen something very similar before.
In 1982, a Gospel book of the 12th-century came up for sale
which I investigated because it had miniatures of the four Evangelists.
John had discovered that the miniatures in the 12th-century Gospel book
were forgeries and had been copied from a French book.
This is the book...
which is Ormont's Manuscrits Grecs de le Bibliotheque Nationale in Paris
which was published in 1929.
When John saw the illustrations in the Archimedes manuscript
he thought they looked very similar to the forgeries in the Gospel book.
He suspected that this same French book
had been used to forge the pictures in the palimpsest.
And indeed I was able, within a few minutes,
to identify... Plate 84 of Ormont
as the source for the four images
of the Evangelists in the Archimedes manuscript.
I am convinced that they're forgeries.
Back in Baltimore, Abigail Quandt has does a test
to see how the forger copied their paintings from the book.
I made this tracing from the painting of John
that appeared in the 1929 publication
and then if you overlay it with the forgery in the Archimedes palimpsest.
It lines up almost exactly.
Abigail did the same experiment
with the other three black-and-white images.
She found each of them was precisely the same size
as those in the palimpsest.
The forger must simply have traced the images.
Much about this mysterious forger remains a mystery,
but John Lowden's investigation has been able to reveal one vital clue.
The earliest the forgeries could have been done is 1929
because that's the date of publication of the book.
But why would someone have spent so long
carefully putting forgeries into this manuscript?
There's only one reason that there are forgeries
put into the Archimedes manuscript
is that it increases the manuscript's value.
Now this is true...
whether you know that the manuscript contains the work of Archimedes or not.
The reason for that
is that it becomes art and it belongs to a completely different clientele.
The question is whether these forgeries were done...
in the knowledge that Archimedes was underneath it.
I rather hope not !
And it's a horrific thought to think that,
that, that, that the illuminations were, were painted over Archimedes text
in full knowledge of the fact.
Back in the lab,
the delicate conservation work on the manuscript has started.
Abigail has the vital job.
To clean and recover the precious Archimedes text.
On some leaves the Archimedes text is almost invisible.
On some leaves though you can see
it is a copper brown colour.
It runs in two columns perpendicular to the Christian text.
I mean the process of removing wax droplets
- the wax is there because, of course,
the manuscript during Medieval times was read by candlelight.
The wax droplets are actually
interfering with the success of the imaging.
To study each page fully,
Abigail has to remove it from its binding in the book.
This is relatively new binding,
probably done in the last hundred years
and removing it has turned out to be far more important
than anyone could have realised.
The way the manuscript is put together...
there are four folded sheets, one nested inside the other.
The Archimedes text goes across the fold, but...
Heiberg's difficulty was that he couldn't actually see those writings.
It had been impossible to read the Archimedes text
that went right into the centre of the binding.
Only now, by taking the book apart,
can the writing be seen.
So we're seeing lines of Archimedes text for the very first time.
And examining the original photos that Heiberg took of the palimpsest
has also shown there was something else that he missed.
Some of the most important pages of the manuscript
were never photographed at all.
People have assumed that Heiberg knew this manuscript extremely well.
He didn't know it that well !
Now that we've got the manuscript, we can fill in
large chunks of Heiberg's transcription, for the first time.
This is really going to surprise the scholarly community.
It's much more than we ever thought that we would,
we would pull out of the Archimedes palimpsest when we began this project.
Recovering the words of Archimedes is a huge technical challenge.
Tackling the problem are teams from Johns Hopkins University
and the Rochester Research Institute.
We are trying to take advantage
of the very slight differences in colour, of the two inks,
of the, the inks from the Archimedes text and the, and the later ink.
So we did that by taking images of a wide variety of wavelengths.
Yeah, that looks good. Alright so UV.
Because the two inks are slightly different colours
they reflect slightly differently in these different bands of wavelengths.
- Now we want to go to... - Flash.
Their latest approach uses a combination of
visible and ultraviolet lights
to try and make the Archimedes text as simple to read as possible.
Here's a piece of the parchment
shown in visible light.
The horizontal text
is the top text that's obscuring the Archimedes text.
The Archimedes text is the vertical lines that you see.
They're very hard to see to the naked eye.
In red light the Archimedes text really is not very visible at all.
As you can see from this image only the top text is showing.
On the other hand, if you look at this piece of parchment
the two sets of text show most clearly
using the blue image from the ultraviolet.
And if you now take the blue image which shows both texts very well
and the red light image and put them together
and do a little extra processing
you can create a false colour image,
which is this image.
The Archimedes text shows up far more clearly than before.
It now appears as the red writing, much easier to read.
I was amazed by the fact that now for the first time
I can look at pages that looks hopeless with the naked eye,
and begin to use them as text from which you just read.
We are able to recover the original text of Archimedes
where it appears to have been lost.
I think we will be able to read everything there for the first time.
One of the most exciting things to emerge already are diagrams.
This is the first time they have ever been examined,
for Heiberg never copied them when he made his great translation.
And Nigel Wilson believes
the diagrams bring us even closer to the mind of Archimedes.
Over the centuries, Archimedes' essays
weren't copied very often,
so the number of intervening copies
between the original and our 10th century manuscript
may be very small.
It might be only four or five.
I would think that these 10th century drawings
reflect very accurately the diagrams that Archimedes himself
intended to be an essential part of his treatise.
The drawings are providing a real insight into the work of Archimedes
revealing the specially important role that diagrams played
in Greek mathematics.
Our mathematics is always based on what you write.
Only what you write down is part of the proof.
In Greek mathematics the proof relies not only on what you write down
as part of the text,
it also relies
on what you write into the diagram,
what you draw in the diagram.
You draw and write simultaneously -
the two work together in Greek mathematics.
If you want to recover the thought of Archimedes
you don't need just the text of Archimedes,
you want also the diagrams of Archimedes.
But the diagrams are only the beginning!
Image after image of the original writing is now emerging.
As Will Noel rapidly e-mails the pictures to Moscow,
Cambridge and London,
the awaiting Greek experts attempt to decipher the text.
Trying to piece together the faint red words is an incredibly painstaking task.
What you do is first of all to stand back from the page
and think what could Archimedes, in principle, have been saying there.
Once you have some sort of guesses
you begin to apply this to the text.
It will look very, very hard
and just to get yourself hypnotised into this text
until somehow certain traces begin to take shape.
Today Reviel, Nigel and Natalie are meeting for the first time
to try and decipher the pages of the manuscript together.
So that means uprights... upright lines.
- And then a row a bit later on. - And then five.
Yeah, there's a row well on yes, towards the end.
Just there is a... (TALKING TOGETHER)
Yeah, you have... (TALKING TOGETHER)
..you have a little hook.
The hook at the bottom is there, yes, that's good!
OK, let's make sure I've got the reading.
(GREEK) - Yes, right.
- Dot. - Yes.
- (GREEK) with an omega. - Yeah.
Could it be actually epsilon! (GREEK)
Look if you, if you look maybe here, and there are no tail...
Ah, there's no tail. Ah, right!
The painfully slow process of unravelling the text
will take years.
But already, Reviel has made one very important new discovery.
He examined a proof in Archimedes revolutionary work, "The Method".
In it Archimedes was trying to work out the volume
of an unusual shape
by dividing it into an infinite number of slices.
Archimedes had drawn a diagram of a triangular prism.
Inside this he drew a circular wedge.
This was the volume that he wanted to calculate.
He then drew a second curve inside the wedge.
Modern mathematicians already understood that Archimedes
had used some very complex ideas
to work out that a slice through the wedge
equals a slice through the curve
times a slice through the prism
divided by a slice through the rectangle.
But what no-one knew was how Archimedes had added up
an infinite number of these slices
to work out the volume of the wedge.
The frustration was that the lines explaining how he had done this
appeared in Heiberg's translation merely as a row of dots.
These vital lines were missing...
but then, with the help of the very latest images of the palimpsest,
Reviel Netz went back to study the manuscript again.
I was looking at those missing lines in the page...
At this point I was stuck...
until I saw a faint trace just above the line.
It didn't appear as if it was part of the writing
because it was just above the line,
but I began to think perhaps it is...
something which belongs to the text of Archimedes,
and this was a breakthrough in the history of Greek mathematics.
Reviel realised that Archimedes had come up
with a set of rules for dealing with infinity.
His system for calculating the value of each slice
and then adding up an infinite number of them.
Well I was there... totally shaken!
I was... ah, ah...
exhilarated and surprised when I saw this argument.
And... I definitely had the sense that
without my knowing yet what the argument is,
what this argument represents
in terms of the mathematical interest of Archimedes
it represents something very important,
something very deep for the history of mathematics.
What was clear
was that Archimedes had made a huge step
towards the understanding of infinity.
Infinity is central to where the history of Western mathematics,
because the history of Western mathematics was determined
by a very Greek problem!
The problem to which Archimedes contributed more than anyone else:
how to calculate the properties of curved objects.
"The theorem of the wedge"
is the first time that we see any Greek mathematician
doing something with infinity.
Actually producing an argument using infinity.
That's something which we simply thought could not happen.
Even today infinity is a concept
that mathematicians can struggle to deal with.
Humans are finite creatures...
and to talk about infinity in any context,
whether it's in a religious context or in a mathematical context,
has always caused us problems.
Possibly the fact that we can even think about infinity,
about the concept, even come up with the concept,
involves that we have some kind of a...
a passport to God.
Now I'm getting very religious here,
but whenever you talk about infinity
you almost have to confront religious issues.
Will we live for infinity, will the Universe last for infinity,
where did the Universe come from?
Is "infinity" something that exists only in our minds
and has no reality in basis?
The new finding in "the wedge theorem"
reveals not only that Archimedes was confident in dealing with infinity,
but also that his use of infinite slices to calculate a volume
was far more sophisticated than anyone had realised.
In fact his technique is similar to the concept
used in modern calculus for the same kind of problem.
Archimedes was even closer to modern science
than had been believed.
It's amazing to think that a branch of mathematics
that has been so crucial to our development
was first begun by a man who died over 2.000 years ago.
We always knew that Archimedes
was making a step in the direction leading to modern calculus.
What we have found right now
is that, in a sense, Archimedes was already there.
He already did develop a special tool with which
you can sum up infinitely many objects in measure of volume.
But perhaps the most interesting question of all
is what might have happened
if this document had not been lost for a millennium.
Supposing it had been available to those mathematicians of the Renaissance.
If the book had been available...
a hundred years before the development of the calculus
then things would have got going sooner.
It would have, of course, changed mathematics,
but also all other sciences.
It serves as the foundation, the language of all of the sciences,
so it's not just mathematicians who need mathematics,
it's all scientists, all physicists, all engineers who need it
and you would basically be raising the tide
by increasing the knowledge of mathematics
several hundred years ago.
It's an extraordinary thought that if scientists
had had access to this document
mathematics might be far more developed now.
Who knows as a result how different the modern world might look,
and all because of something written by a man
in the 3rd century B.C.
If we had been aware of the discoveries of Archimedes
hundreds of years ago
we could have been on Mars today,
we could have a computer as smart as a human being,
and we could have accomplished all of the things that now
people are predicting for a century from now.
Synchronized by varinos.