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[Music]
[Music]
our world is made up of patterns and
sequences
they're all around us day becomes night
animals travel across the earth in
ever-changing formations
landscapes are constantly altering
one of the reasons mathematics began was
because we needed to find a way
of making sense of these natural
patterns
[Music]

00:38

the most basic concepts of maths space
and quantity
are hardwired into our brains
even animals have a sense of distance
and number
assessing when their pack is outnumbered
whether to fight
or fly calculating whether their preys
within striking distance
understanding maths is the difference
between life
and death
but it was man who took these basic
concepts and started to build upon these

01:10

foundations at some point
humans started to spot patterns to make
connections
to count and to order the world around
them
and with this a whole new mathematical
universe began to emerge
[Music]
this is the river nile it's been the
lifeline of egypt for millennia
i've come here because it's where some
of the first signs of mathematics as we
know it today
emerged
people abandoned nomadic life and began

01:46

settling here as early as 6000 bc
the conditions were perfect for farming
the most important event for egyptian
agriculture each year
was the flooding of the nile so this was
used as
a marker to start each new year
egyptians did
record what was going on over periods of
time so in order to establish a calendar
like this you need to count
how many days for example happened
in between lunar phases

02:22

or how many days happened in between two
floodings of the nile
recording the patterns of the seasons
was essential not only to their
management of the land
but also their religion the ancient
egyptians who settled on the nile banks
believed it was the river god happy who
flooded the river
each year and in return for the
life-giving water
the citizens offered a portion of the
yield as a thanksgiving
as settlements grew larger it became
necessary to find ways to administer

02:57

them
areas of land needed to be calculated
crop yields predicted
taxes charged and collated in short
people needed account and measure
the egyptians use their bodies to
measure the world and it's how their
units of measurements evolved
a palm was the width of a hand a cubit
an
arm length from elbow to fingertips
land qubits strips of land measuring a
qubit by a hundred
were used by the pharaoh's surveyors to

03:28

calculate areas
there's a very strong link between
bureaucracy
and the development of mathematics in
ancient egypt and we can see this link
right from the beginning from the
invention of the number system
throughout egyptian history really for
the old kingdom
the only evidence we have are
metrological systems
that is measurements for areas for
length
this points to a bureaucratic need to
develop such things

04:00

it was vital to know the error of a
farmer's land so he could be taxed
accordingly
or the nile robbed him of part of his
land so he could request a rebate
it meant that the pharaoh's surveyors
were often calculating the area
of irregular parcels of land and it was
the need to solve such practical
problems
that made them the earliest mathematical
innovators
the egyptians needed some way to record
the results of their calculations
amongst all the hieroglyphs that cover

04:36

the tourist souvenirs scattered around
the city of cairo
i was on the hunt for those that
recorded some of the first numbers in
history
they were difficult to track down
but i did find them in the end
the egyptians were using a decimal
system motivated by the ten fingers on
our hands
the sign for one was a stroke ten a heel
bone
a hundred a coil of rope and a thousand
and lotus plants yeah how much is this
t-shirt
25. so that'll be two

05:17

knee bends and five strokes so you're
not gonna charge me anything up here
here one million one million oh my god
no no no is this one million one billion
yeah that's
pretty big the hieroglyphs are beautiful
but the egyptian number system was
fundamentally flawed
they had no concept of a place value so
one stroke could only represent
one unit not a hundred or a thousand
although you can write a million with
just one character rather than the seven

05:48

that we use
if you want to write a million minus one
then the poor old egyptian scribe
has got to write nine strokes nine heel
bones
nine coils of rope and so on a total of
54
characters
despite the drawback of this number
system the egyptians were brilliant
problem solvers
we know this because of the few records
that have survived
the egyptian scribes use sheets of
papyrus to record their mathematical
discoveries
this delicate material made from reeds

06:23

decayed over time
and many secrets perished with it
but there's one revealing document that
has survived
the right mathematical papyrus is the
most important document we have today
for egyptian mathematics
we get a good overview of what types of
problems
the egyptians would have dealt with in
their mathematics
we also get explicitly stated how
multiplications and divisions were
carried out

06:54

the papyri show how to multiply two
large numbers together
[Music]
but to illustrate the method let's take
two smaller numbers
let's do three times six the scribe
would take the first number three
and put it in one column
in the second column you would place the
number one
then he would double the numbers in each
column so 3 becomes 6
and 6 will become 12.
and then in the second column one would
become two

07:34

and two becomes four
[Music]
now here's the really clever bit the
scribe wants to multiply three
by six so he takes the powers of two
in the second column which add up to six
so that's two
plus four then he moves back to the
first column
and just takes those rows corresponding
to the two and the four
so that's 6 and the 12 he adds those
together
to get the answer of 18.
but for me the most striking thing about

08:06

this method is the scribe has
effectively written that second number
in binary six is one lot of four
one lot of two and no units which is
one one zero the egyptians have
understood the power of binary
over three thousand years before the
mathematician and philosopher leibniz
would reveal their potential
today the whole technological world
depends on the same principles
that were used in ancient egypt
the rhine papyrus was recorded by a
scribe called ahmez

08:39

around 1650 bc and its problems are
concerned with finding solutions
to everyday situations several of the
problems mentioned bread and beer
which isn't surprising as egyptian
workers were paid in food and drink
one is concerned with how to divide nine
loaves equally between
ten people without a fight breaking out
i've got nine loaves of bread here i'm
going to take five of them
and cut them into halves of course
nine people could shave a tenth of their

09:10

loaf and give the pile of crumbs to the
tenth person
but the egyptians developed a far more
elegant solution
take the next four and divide those into
thirds
but two of the thirds i'm now going to
cut into fifths so each piece will be
one fifteenth
each person then gets one half
and one third and one fifties
it's through such seemingly practical
problems that we start to see a more
abstract mathematics developing

09:48

suddenly new numbers are on the scene
fractions and it isn't too long before
the egyptians are exploring the
mathematics of these numbers
fractions are clearly of practical
importance to anyone dividing quantities
for trade in the market
to log these transactions the egyptians
developed notation which recorded these
new numbers
one of the earliest representations of
these fractions
came from a hieroglyph which had great
mystical significance
it's called the eye of horus horus was

10:23

an old kingdom god
depicted as half man half falcon
according to legend horace's father was
killed by his other son seth
horus was determined to avenge the
murder
during one particularly fierce battle
seth ripped out
horace's eye tore it up and scattered it
over egypt
but the gods were looking favorably on
horus they gathered up the scattered
pieces
and reassembled the eye
[Music]
each part of the eye represented a

10:58

different fraction
each one half the fraction before
although the original eye represented a
whole unit
the reassembled eye is 1 64 short
although the egyptians stopped at 1 over
64.
implicit in this picture is the
possibility of adding more fractions
halving them each time the sum getting
closer
and closer to one but never quite
reaching it
this is the first hint of something

11:28

called a geometric series
and it appears at a number of points in
the rhine papyrus
but the concept of infinite series would
remain hidden until the mathematicians
of asia
discovered it centuries later
having worked out a system of numbers
including these new fractions
it was time for the egyptians to apply
their knowledge to understanding shapes
that they encountered day to day
these shapes were rarely regular squares
or rectangles
and in the rhine papyrus we find the

12:00

area of a more organic form
the circle what is astounding
in the calculation of the area of a
circle is its exactness
really how they would have found their
method
is open to speculation because the texts
we have do not
show us the methods how they were found
this calculation is particularly
striking because it depends on seeing
how the shape of the circle
can be approximated by shapes the
egyptians already understood

12:31

the rhine papyrus states that a circular
field with a diameter of 9 units
is close an area to a square with sides
of eight
but how would this relationship have
been discovered
my favorite theory sees the answer in
the ancient game
of mancala mancala boards were found
carved on the roofs of temples
each player starts with an equal number
of stones and the object of the game
is to move them around the board
capturing your opponent's counters on

13:01

the way
as the players sat around waiting to
make their next move
perhaps one of them realized that
sometimes the balls fill the circular
holes of the mancala board in a rather
nice way
he might have gone on to experiment with
trying to make larger circles
perhaps he noticed that 64 stones the
square of eight
can be used to make a circle with
diameter nine stones
by rearranging the stones the circle has
been approximated
by a square and because the area of a

13:33

circle is pi
times the radius squared the egyptian
calculation
gives us the first accurate value for pi
the area of the circle is 64. divide
this
by the radius squared in this case 4.5
squared
and you get a value for pi so 64 divided
by 4.5 squared
is 3.16 just a little under two
hundredths away from its true value
but the really brilliant thing is the
egyptians are using these smaller shapes

14:02

to capture the larger shape
but there's one imposing and majestic
symbol of egyptian mathematics
we haven't attempted to unravel yet the
pyramid
i've seen so many pictures that i
couldn't believe i'd be impressed by
them
but meeting them face to face you
understand why they're called one of the
seven
wonders of the ancient world they're
simply breathtaking
and how much more impressive they must
have been in their day
when the sides were as smooth as glass
reflecting the desert sun

14:37

[Music]
to me it looks like there might be
mirror pyramids hiding underneath the
desert
which will complete these shapes to make
perfectly symmetrical octahedrons
sometimes in the shimmer of the desert
heat you can almost see these shapes
it's the hint of symmetry hidden inside
these shapes that makes them so
impressive for a mathematician
the pyramids are just a little short to
create these perfect shapes
but some have suggested that another
important mathematical concept

15:10

might be hidden inside the proportions
of the great pyramid
the golden ratio two lengths are in the
golden ratio
if the relationship of the longest to
the shortest
is the same as the sum of the two to the
longest side
such a ratio has been associated with
the perfect proportions
one finds all over the natural world as
well as in the work of artists
architects and designers for millennium

15:41

whether the architects of the pyramids
were conscious of this important
mathematical idea
or were instinctively drawn to it
because of its satisfying aesthetic
properties
we'll never know for me the most
impressive thing about
the pyramids is the mathematical
brilliance that went into making them
including the first inkling of one of
the great theorems of the ancient world
pythagoras theorem in order to get
perfect right angled corners on their
buildings and pyramids
the egyptians would have used a rope

16:11

with knots tied in it
at some point the egyptians realized
that if they took a triangle
with sides marked with three knots four
knots and five knots
it guaranteed them a perfect right angle
this is because
three squared plus four squared is equal
to five squared
so we've got a perfect pythagorean
triangle
in fact any triangle whose side
satisfied this relationship
will give me a 90 degree angle but i'm
pretty sure that the egyptians hadn't
got this sweeping generalization

16:44

of their 345 triangle
we would not expect to find a general
proof
because this is not the style of
egyptian mathematics
every problem was solved using concrete
numbers and then if a verification would
be carried out
at the end it would use the result and
these concrete given numbers there's no
general proof
within the egyptian mathematical texts
it would be some two thousand years
before the greeks and pythagoras
were proved that all right angled

17:14

triangles share certain properties
this wasn't the only mathematical idea
that the egyptians would anticipate
in a 4 000 year old document called the
moscow papyrus
we find a formula for the volume of a
pyramid with its peak sliced off
which shows the first hint of calculus
at work
for a culture like egypt that is famous
for its pyramids
you would expect problems like this to
have
been a regular feature within the
mathematical texts

17:46

the calculation of the volume of a
truncated pyramid is one of the most
advanced bits
according to our modern standards of
mathematics that we have
from ancient egypt the architects and
engineers could certainly have wanted
such a formula
to calculate the amount of materials
required to build it
but it's a mark of the sophistication of
egyptian mathematics
that they were able to produce such a
beautiful method
to understand how they derive their

18:20

formula start with a pyramid build such
that the highest point
sits directly over one corner
three of these can be put together to
make a rectangular box
so the volume of the skewed pyramid is a
third the volume of the box
that is the height times the length
times the width
divided by three
now comes an argument which shows the
very first hints of the calculus at work
thousands of years before godfred
leibniz and isaac newton would come up
with a theory

18:54

suppose you could cut the pyramid into
slices
you could then slide the layers across
to make the more symmetrical pyramids
you see in giza
however the volume of the pyramid has
not changed despite the rearrangement of
the layers
so the same formula works
[Music]
the egyptians were amazing innovators
and their ability to generate new
mathematics was staggering
for me they revealed the power of
geometry and numbers

19:25

and made the first moves towards some of
the exciting mathematical discoveries to
come
but there was another civilization that
had mathematics to rival that of egypt
and we know much more about their
achievements
this is damascus over 5000 years old
and still vibrant and bustling today it
used to be the most important point on
the trade routes
linking old mesopotamia with egypt
the babylonians controlled much of
modern-day iraq iran

20:00

and syria from 1800 bc
in order to expand and run their empire
they became masters of managing
and manipulating numbers we have law
codes for instance that tell us about
the way society is ordered now the
people we know most about are the
scribes
the professionally literate and numeric
people who kept the records
for the wealthy families and for the
temples and palaces
scribe schools existed from around 2500
bc
aspiring scribes were sent there as

20:31

children and learned how to read
write and work with numbers
scribe records were kept on clay tablets
which allowed the babylonians to manage
and advance their empire however many of
the tablets we have today
aren't official documents but children's
exercises
it's these unlikely relics that give us
a rare insight
into how the babylonians approached
mathematics
so this is a geometrical textbook from
about the 18th century bc
and i hope you can see that there are

21:02

lots of pictures on it and underneath
each picture is a text
that sets a problem about the picture so
for instance this one here says i drew a
square
60 units long and inside it i drew four
circles what are their areas
this little tablet here was written a
thousand years at least
later than the tablet here but has a
very interesting relationship to it
it also has four circles on in a square
roughly drawn

21:32

but this isn't a textbook it's a school
exercise
so that the adult squad who's teaching
the student is being given this as an
example
of completed homework or something like
that
like the egyptians the babylonians
appeared interested in solving practical
problems to do with measuring and
weighing
the babylonian solutions to these
problems are written like mathematical
recipes
a scribe would simply follow and record
a set of instructions
to get a result here's an example of a

22:04

kind of problem they'd solve
now i've got a bundle of cinnamon sticks
here but i'm not going to weigh them
instead i'm going to take four times
their weight and add them to the scales
now i'm going to add 20 gin gin was the
ancient babylonian measure of weight
i'm going to take half of everything
here and add it again
two bundles and tension now everything
on this side
is equal to one mana one manner was 60
gin
and here we have one of the first

22:37

mathematical equations in history
everything on this side is equal to one
manner
but how much does the bundle of cinnamon
sixth weigh without any algebraic
language
they were able to manipulate the
quantities to be able to prove
that the cinnamon sticks weighed five
gin
in my mind it's this kind of problem
which gives mathematics a bit of a bad
name
you can blame those ancient babylonians
for all those tortuous problems you had
at school
but the ancient babylonian scribes
excelled at this kind of problem

23:10

intriguingly they weren't using powers
of 10 like the egyptians
they were using powers of 60.
the babylonians invented their number
system like the egyptians by using their
fingers
but instead of counting through the ten
fingers on their hands babylonians found
a much more intriguing way to count body
parts
they used the twelve knuckles on one
hand and the five fingers on the other
to be able to count 12 x 5 ie 60
different numbers
so for example this number would have
been 2 lots of 12

23:43

24 and then 1 2 3 4 5
to make 29.
but the number 60 had another powerful
property
it can be perfectly divided in a
multitude of ways
here are 60 beans i can arrange them
in two rows of thirty
[Music]
three rows of twenty
four rows of fifteen five rows of twelve

24:15

or six rows of ten the divisibility of
sixty makes it a perfect base in which
to do arithmetic
the base60 system was so successful we
still use
elements of it today every time we want
to tell the time
we recognize units of 60 60 seconds in a
minute
60 minutes in an hour but the most
important feature of the babylonians
number system
was that it recognized place value just
as our decimal numbers count
how many lots of tens hundreds and

24:47

thousands you're recording
the position of each babylonian number
records the power of 60 you're counting
instead of inventing new symbols for
bigger and bigger numbers
they would write one one one so this
number would be 3
61.
[Music]
the catalyst for this discovery was the
babylonian's desire
to chart the course of the night sky

25:18

[Music]
the babylonians calendar was based on
the cycles of the moon
and so they needed a way of recording
astronomically large numbers
month by month year by year these cycles
were recorded
from about 800 bc there were complete
lists of lunar eclipses
the babylonian system of measurement was
quite sophisticated
at that time they had a system of
angular measurement
360 degrees in the full circle each

25:54

degree was divided into 60 minutes
a minute was further divided into 60
seconds
so they had a regular system for
measurement
and it was in perfect harmony with their
number system so it was
well suited not only for observation but
also for calculation
but in order to calculate and cope with
these large numbers
the babylonians needed to invent a new
symbol
and in so doing they prepared the ground
for one of the great breakthroughs in
the history of mathematics

26:25

zero in the early days the babylonians
in order to
mark an empty place in the middle of a
number would simply leave a blank space
so they needed a way of representing
nothing in the middle of a number
so they used a sign as a sort of
breathing mark or a punctuation mark and
it
it comes to mean zero in the middle of a
number
this was the first time zero in any form
had appeared in the mathematical
universe
but it'd be over a thousand years before
this little placeholder

26:57

will become a number in its own right
having established such a sophisticated
system of numbers
they harnessed it to tame the arid and
inhospitable land that ran through
mesopotamia
babylonian engineers and surveyors found
ingenious ways of
accessing water and channeling it to the
crop fields
yet again they use mathematics to come
up with solutions

27:35

the urantis valley in syria is still an
agricultural hub
and the old methods of irrigation are
being exploited today
just as they were thousands of years ago
many of the problems in babylonian
mathematics are concerned with measuring
land
and it's here we see for the first time
the use of quadratic equations
one of the greatest legacies of
babylonian mathematics
quadratic equations involve things where
the unknown quantity you're trying to
identify
is multiplied by itself we call this

28:06

squaring because it gives the area of a
square
and it's in the context of calculating
the error of land that these quadratic
equations naturally arise
here's a typical problem if a field has
an area of 55
units and one side is six units longer
than the other
how long is the shorter side
the babylonian solution was to
reconfigure the field as a square
cut three units off the end and move
this round

28:44

now there's a three by three piece
missing so let's add this in
the area of the field has increased by
nine units
this makes the new area 64.
so the size of the square are eight
units
the problem solver knows that they've
added three to this side
so the original length must be five
it may not look like it but this is one
of the first quadratic equations in

29:13

history
in modern mathematics i would use the
symbolic language of algebra to solve
this problem
the amazing feat of the babylonians is
that they were using these geometric
games to find the value
without any recourse to symbols or
formulas the babylonians were enjoying
problem solving for its own sake
they were falling in love with
mathematics
[Music]
[Applause]
the babylonians fascination with numbers

29:50

soon found a place in their leisure time
too
they were avid game players the
babylonians and their descendants have
been playing a version of backgammon for
over 5 000
years so the babylonians played
board games from very posh board games
in royal tombs
to little bits of board games found in
schools
to board games scratched on the
entrances
of palaces so the guardsmen must have
played at times when they were bored
and they used dice to move their

30:21

counters round
people who played games were using
numbers in their leisure time
to try and outwit their opponent doing
mental arithmetic
very fast and so they were
calculating in their leisure time
without even thinking about it as being
mathematical hard work
there's my chance i hadn't played
backgammon for ages
but i reckon my maths would be good
enough to give me a fighting chance so
it's
six and i need to move something so but
it wasn't as easy as i thought

30:57

what the hell is that yeah yeah this is
one this
is two now you are in trouble so i can't
move anything
can i i can't move these anymore you
cannot move these i just gotta
oh gosh there you go you seem to be all
right
three and four so just like the ancient
babylonians
my opponents were masters of tactical
mathematics
yeah put it there
all right again
the babylonians are recognized as one of

31:28

the first cultures to use symmetrical
mathematical shapes to make dice
but there is more heated debate about
whether they might also have been the
first to discover the secrets
of another important shape the right
angle
triangle we've already seen how the
egyptians use a 3 4
5 right angled triangle but what the
babylonians knew about this shape
and others like it is much more
sophisticated
this is the most famous and
controversial ancient tablet we have

32:01

it's called plympton322
many mathematicians are convinced it
shows the babylonians
could well have known the principle
regarding right angle triangles
that the square on the diagonal is the
sum of the squares on the sides
and known it centuries before the greeks
claimed it
this is a copy of the arguably most
famous babylonian tablet which is
plympton 322 and these numbers here
reflect the width or height of a
triangle
this being the diagonal the other side

32:34

would be over here and
the square of this column
plus the square of the number in this
column equals the square of the
diagonal they are arranged in an order
of
uh steadily decreasing angle on a very
uniform basis showing
that somebody had a lot of understanding
of
how the numbers fit together
here were 15 perfect pythagorean
triangles
all of whose sides had whole number
lengths it's tempting to think that the

33:11

babylonians were the first custodians of
pythagoras theory
and it's a conclusion that generations
of historians have been seduced by
but there could be a much simpler
explanation for the sets of three
numbers which fulfill pythagoras theorem
it's not a systematic explanation of
pythagorean triples it's simply
a mathematics teacher doing some quite
complicated calculations
but in order to produce some very simple
numbers
in order to set his students problems

33:42

about right angle triangles
and in that sense it's about pythagorean
triples only incidentally
the most valuable clues to what they
understood could lie
elsewhere this small school exercise
tablet
is nearly 4 000 years old and reveals
just what the babylonians did know about
right angle triangles
it uses the principle of pythagoras
theorem to find the value
of an astounding new number

34:13

[Music]
drawn along the diagonal is
a really very good approximation to the
square root of two
and so that shows us that it was known
and used
in school environments now why is this
important
is because the square root of two is
what we now call
an irrational number that is if we write
it out in decimals or even in sex
decimal places then it doesn't
doesn't end the numbers go on forever

34:46

after the decimal point
the implications of this calculation are
far reaching
firstly it means the babylonians knew
something of pythagoras's theorem
a thousand years before pythagoras
secondly
the fact that they can calculate this
number to an accuracy of four decimal
places
shows an amazing arithmetic facility as
well as a passion for mathematical
detail
the babylonian's mathematical dexterity
was astounding
and for nearly 2 000 years they
spearheaded intellectual progress in the

35:19

ancient world
but when their imperial power began to
wane so did their intellectual vigor
by 330 bc the greeks had advanced their
imperial reach
into old mesopotamia
this is palmyra in central syria a once
great city built by the greeks
the mathematical expertise needed to
build structures with such geometric
perfection
is impressive

35:59

[Music]
just like the babylonians before them
the greeks were passionate about
mathematics
the greeks were clever colonists they
took the best from the civilizations
they invaded
to advance their own power and influence
but they were soon making contributions
themselves
in my opinion their greatest innovation
was to do with a shift in the mind
what they initiated would influence
humanity for centuries
they gave us the power of proof

36:33

somehow they decided that they had to
have a deductive system
for their mathematics and the typical
deductive system was to begin with
certain axioms
which you assume are true it's as if you
assume a certain theorem is true
without proving it and then using
logical methods
and very careful uh steps
from these axioms you prove theorems and
then from those theorems
you prove more theorems and it just
snowballs
that way proof is what gives mathematics

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its strength
it's the power of proof which means that
the discoveries of the greeks
are as true today as they were 2 000
years ago
i needed to head west into the heart of
the old greek empire to learn more
for me greek mathematics has always been
heroic
and romantic i'm on my way to samos
less than a mile from the turkish coast
this place has become synonymous with

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the birth of greek mathematics
and it's down to the legend of one man
his name is pythagoras the legends that
surround his life and work
have contributed to the celebrity status
he has gained over the last 2
000 years he's credited rightly or
wrongly
with beginning the transformation from
mathematics as a tool for accounting
to the analytic subject we recognize
today
[Music]
pythagoras is a controversial figure

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because he left no mathematical writings
many have questioned whether he
indeed solved any of the theorems
attributed to him
he founded a school in samos in the 6th
century bc
but his teachings were considered
suspect and the pythagoreans a bizarre
sect
there is good evidence that there were
schools of pythagoreans
they may have looked more like sex than
what we associate with philosophical
schools because they didn't just
share knowledge they also shared

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a way of life they may have been a
communal living
and they all seem to have been involved
in the politics of their cities
one feature that makes them unusual in
the ancient world
is that they included women
but pythagoras is synonymous with
understanding something that eluded the
egyptians and the babylonians
the properties of right angle triangles
what's known as pythagoras theorem
states that if you take any right angle
triangle

39:20

build squares on all the sides then the
area of the largest square
is equal to the sum of the squares on
the two smaller sides
it's at this point for me that
mathematics is born and a gulf opens up
between the other sciences
and the proof is as simple as it is
devastating in its implications
place four copies of the right angled
triangle on top of this surface
the square that you now see has sides
equal to the hypotenuse of the triangle

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by sliding these triangles around we see
how we can break the area of the large
square up
into the sum of two smaller squares
whose sides
are given by the two short sides of the
triangle
in other words the square on the
hypotenuse is equal
to the sum of the squares on the other
sides
pythagoras theorem
[Music]
it illustrates one of the characteristic
themes of greek mathematics
the appeal to beautiful arguments in
geometry

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rather than a reliance on number
pythagoras may have fallen out of favor
and many of the discoveries accredited
to him have been contested recently
but there's one mathematical theory that
i'm loath to take away from him
it's to do with music and the discovery
of the harmonics series
[Music]
the story goes that walking past a
blacksmith's one day
pythagoras heard anvils being struck and
notice how the notes being produced
sounded in perfect harmony he believed
that there must be some rational

41:00

explanation
to make sense of why the notes sounded
so appealing
the answer was mathematics
[Music]
experimenting with a stringed instrument
pythagoras discovered
that the intervals between harmonious
musical notes were always represented
as whole number ratios
and here's how he might have constructed
his theory
first play a note on the open string
next take half the length

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the note almost sounds the same as the
first note in fact it's an octave higher
but the relationship is so strong we
give these notes the same name
now take a third the length
[Music]
we get another note which sounds
harmonious next to the first two
but take a length of string which is not
in a whole number ratio
and all we get is dissonance
according to legend pythagoras was so
excited by this discovery
that he concluded the whole universe was

42:11

built from numbers
but he and his followers were in for a
rather unsettling challenge to their
world view
and it came about as a result of the
theorem which bears pythagoras's name
legend has it one of his followers a
mathematician called hepassus
set out to find the length of the
diagonal for a right angled triangle
with two sides measuring one unit
pythagoras theorem implied that the
length of the diagonal
was a number whose square was two the

42:45

pythagoreans assumed that the answer
would be a fraction
but when hapassas tried to express it in
this way no matter how he tried
he couldn't capture it eventually he
realized his mistake
it was the assumption that the value was
a fraction at all which was wrong
the value of the square root of two was
the number that the babylonians etched
into the yale tablet
however they didn't recognize the
special character of this number
but who passes did it was an irrational

43:15

number
the discovery of this new number and
others like it
is akin to an explorer discovering a new
continent
or a naturalist finding a new species
but these irrational numbers didn't fit
the pythagorean world view
later greek commentators tell the story
of how pythagoras swore his sect to
secrecy
but hipaas let slip the discovery and
was promptly drowned for his attempts to
broadcast their research
but these mathematical discoveries could
not be easily suppressed

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schools of philosophy and science
started to flourish all over greece
building on these foundations the most
famous of these
was the academy plato founded this
school in athens
in 387 bc
although we think of him today as a
philosopher he was one of mathematics
most important patrons
plato was enraptured by the pythagorean
worldview
and considered mathematics the bedrock
of knowledge
some people would say that plato is

44:23

possibly the most influential figure
for our perception of greek mathematics
he
argued that mathematics is an important
form of knowledge
and does have a connection with reality
so by knowing mathematics we know more
about reality
in his dialogue tomas plato proposes the
thesis
that geometry is the key to unlocking
the secrets of the universe
a view still held by scientists today
indeed the importance plato attached to
geometry

44:55

is encapsulated in the sign that was
mounted above the academy
let no one ignorant of geometry enter
here
plato proposed that the universe could
be crystallized
into five regular symmetrical shapes
these shapes which we now call the
platonic solids were composed of regular
polygons
assembled to create three-dimensional
symmetrical objects
the tetrahedron represented fire the
icosahedron made from 20 triangles

45:27

represented water the stable cube was
earth
the eight-faced octahedron was air
and the fifth platonic solid the
dodecahedron made out of twelve
pentagons
was reserved for the shape which
captured plato's view of the universe
plato's theory would have a seismic
influence and continue to inspire
mathematicians and astronomers
for over 1500 years
in addition to the breakthroughs made in
the academy mathematical triumphs were

46:02

also emerging
from the edge of the greek empire and
owed as much to the mathematical
heritage of the egyptians
as the greeks alexandria became a hub of
academic excellence under the rule of
the ptolemies in the 3rd century
bc and its famous library soon gained a
reputation
to rival plato's academy
the kings of alexandria were prepared to
invest
in the arts in culture in technology

46:32

mathematics
grammar because patronage
for cultural pursuits was
one way of showing that you
were a more prestigious ruler and a
better entitlement
to greatness
the old library and its precious
contents were destroyed when the muslims
conquered egypt
in the 7th century but its spirit is
alive
in a new building
today the library remains a place of

47:05

discovery and scholarship
mathematicians and philosophers flocked
to alexandria
driven by their thirsts for knowledge in
the pursuit of excellence
the patrons of the library were the
first professional scientists
individuals who are paid for their
devotion to research
but of all those early pioneers my hero
is the enigmatic greek mathematician
euclid
[Music]
we know very little about euclid's life
but his greatest achievements were as a

47:39

chronicler of mathematics
around 300 bc he wrote the most
important textbook
of all time the elements
in the elements we find the culmination
of the mathematical revolution which had
taken place in greece
it's built on a series of mathematical
assumptions called axioms
for example a line can be drawn between
any two points
from these axioms logical deductions are
made
and mathematical theorems established

48:14

the elements contains formulas for
calculating the volumes
of cones and cylinders proofs about
geometric series
perfect numbers and primes
the climax of the elements is a proof
that there are only
five platonic solids
for me this last theorem captures the
power of mathematics
it's one thing to build five symmetrical
solids quite another to come up with a
watertight
logical argument for why there can't be
a sixth
the elements unfolds like a wonderful

48:47

logical mystery novel
but this is a story which transcends
time
scientific theories get knocked down
from one generation to the next
but the theorems and the elements are as
true today as they were
two thousand years ago
when you stop and think about it's
really amazing that it's the same
theorems that we teach we may teach them
in a slightly different way
how we may organize them differently but
it's euclidean geometry that is still
valid and even in in higher mathematics

49:18

when you go to higher dimensional spaces
you're still using euclidean geometry
alexandria must have been an inspiring
place for the ancient scholars
and euclid spain would have attracted
even more eager young intellectuals to
the egyptian port
one mathematician who particularly
enjoyed the intellectual environment in
alexandria
was archimedes he would become
a mathematical visionary the best
greek mathematicians they were always
pushing the limits

49:50

pushing the envelope so akimidis
did what he could with the polygons with
solids it then moved on to centers of
gravity or if they moved on to
the spiral this
instinct to try and mathematize
everything is something that
i see as a legacy one of archimedes
specialities
was weapons of mass destruction they

50:23

were used against the romans when they
invaded his home of syracuse
in 212 bc he also designed mirrors
which harnessed the power of the sun to
set the roman ships on fire
but to archimedes these endeavors were
mere amusements in geometry
he had loftier ambitions
archimedes was enraptured by pure
mathematics
and believed in studying mathematics for
its own sake
not for the ignoble trade of engineering
or the sordid quest for profit

50:57

one of his finest investigations into
pure mathematics
was to produce formulas to calculate the
areas of regular shapes
archimedes method was to capture new
shapes by using shapes he already
understood
so for example to calculate the area of
a circle
we would enclose it inside a triangle
then by doubling the number of sides on
the triangle
the enclosing shape would get closer and
closer to the circle
indeed we sometimes call a circle a
polygon with an infinite number of sides

51:30

but by estimating the error of the
circle archimedes is in fact getting a
value for pi
the most important number in mathematics
however it was calculating the volumes
of solid objects
where archimedes excelled
he found a way to calculate the volume
of a sphere by
slicing it up and approximating each
slice as a cylinder
he then added up the volumes of the
slices to get an approximate value for
the sphere
but his act of genius was to see what

52:01

happens if you make the slices
thinner and thinner in the limit the
approximation
becomes an exact calculation
but it was archimedes commitment to
mathematics it will be his undoing
archimedes was contemplating a problem
about circles
traced in the sand when a roman soldier
accosted him
archimedes was so engrossed in his
problem that he insisted he'd be allowed
to finish his theorem
but the roman soldier was not interested
in archimedes problem

52:37

and killed him on the spot even in death
archimedes devotion to mathematics was
unwavering
[Music]
by the middle of the first century bc

53:09

the romans had tightened their grip
on the old greek empire they were less
smitten with the beauty of mathematics
were more concerned with its practical
applications
this pragmatic attitude signaled the
beginning of the end for the great
library of alexandria
but one mathematician was determined to
keep the legacy of the greeks alive
hypatia was exceptional a female
mathematician
and a pagan in the piously christian
roman empire

53:39

i think she was a very prestigious and
very influential
in her time she was a teacher with a lot
of students a lot of followers she was
a politically influential in alexandria
so it's this combination of a high
knowledge
and high prestige that may have made
her a figure of hatred
for the christian mob

54:14

one morning during lent hypatia was
dragged off her chariot
by a zealous christian mob and taken to
a church
there she was tortured and brutally
murdered
the dramatic circumstances of her life
and death fascinated later generations
sadly her cult status eclipsed her
mathematical achievements
she was in fact a brilliant teacher and
theorist
and her death dealt a final blow to the
greek mathematical heritage of

54:50

alexandria
my travels have taken me on a
fascinating journey to uncover the
passion and innovation of the world's
earliest mathematicians
it's the breakthroughs made by those
early pioneers of egypt
babylon and greece that are the
foundations on which my subject is built
today
but this is just the beginning of my
mathematical odyssey
the next leg of my journey lies east in
the depths of asia
where mathematicians scaled even greater
heights in pursuit of knowledge

55:27

with this new era came a new language of
algebra and numbers
better suited to telling the next
chapter in the story
of maths you can learn more about the
story of maths with the open university
at open2.net and the story of maths
continues here on bbc4 next monday at
the same time
next tonight british novelists become
agents of provocation as they describe a
time when nothing was sacred

55:57

in their own words
[Music]
you

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