1
00:00:24,120 --> 00:00:27,200
I'm walking in the
mountains of the moon.

2
00:00:29,280 --> 00:00:33,840
I'm on the trail of the Renaissance
artist, Piero della Francesca,

3
00:00:33,840 --> 00:00:38,280
so I've come to the town in northern
Italy which Piero made his own.

4
00:00:38,280 --> 00:00:41,440
There it is, Urbino.

5
00:00:41,440 --> 00:00:45,240
I've come here to see some
of Piero's finest works,

6
00:00:45,240 --> 00:00:50,520
masterpieces of art, but also
masterpieces of mathematics.

7
00:00:52,280 --> 00:00:55,720
The artists and architects of
the early Renaissance brought back

8
00:00:55,720 --> 00:01:00,400
the use of perspective, a technique
that had been lost for 1,000 years,

9
00:01:00,400 --> 00:01:03,320
but using it properly
turned out to be a lot

10
00:01:03,320 --> 00:01:05,360
more difficult than they'd imagined.

11
00:01:05,360 --> 00:01:10,840
Piero was the first major painter
to fully understand perspective.

12
00:01:10,840 --> 00:01:15,920
That's because he was a
mathematician as well as an artist.

13
00:01:15,920 --> 00:01:18,080
I came here to see his masterpiece,

14
00:01:18,080 --> 00:01:21,640
The Flagellation of Christ,
but there was a problem.

15
00:01:21,640 --> 00:01:24,200
I've just been to see The
Flagellation, and it's an

16
00:01:24,200 --> 00:01:27,520
absolutely stunning picture,
but unfortunately, for various

17
00:01:27,520 --> 00:01:30,880
kind of Italian reasons, we're not
allowed to go and film in there.

18
00:01:30,880 --> 00:01:34,360
But this is a maths programme, after
all, and not an arts programme,

19
00:01:34,360 --> 00:01:38,640
so I've used a bit of mathematics
to bring this picture alive.

20
00:01:38,640 --> 00:01:44,040
We can't go to the picture, but
we can make the picture come to us.

21
00:01:44,040 --> 00:01:46,280
The problem of perspective is how

22
00:01:46,280 --> 00:01:51,360
to represent the three-dimensional
world on a two-dimensional canvas.

23
00:01:51,360 --> 00:01:54,800
To give a sense of depth,
a sense of the third dimension,

24
00:01:54,800 --> 00:01:57,920
Piero used mathematics.

25
00:01:57,920 --> 00:02:00,640
How big is he going to paint Christ,

26
00:02:00,640 --> 00:02:03,600
if this group of men here
were a certain distance away

27
00:02:03,600 --> 00:02:05,720
from these men in the foreground?

28
00:02:07,240 --> 00:02:11,640
Get it wrong and the illusion
of perspective is shattered.

29
00:02:11,640 --> 00:02:14,800
It's far from obvious how
a three-dimensional world

30
00:02:14,800 --> 00:02:19,360
can be accurately represented
on a two-dimensional surface.

31
00:02:19,360 --> 00:02:23,200
Look at how the parallel lines
in the three-dimensional world

32
00:02:23,200 --> 00:02:27,000
are no longer parallel in the
two-dimensional canvas, but meet

33
00:02:27,000 --> 00:02:28,880
at a vanishing point.

34
00:02:30,720 --> 00:02:33,800
And this is what the tiles in
the picture really look like.

35
00:02:39,400 --> 00:02:41,400
What is emerging here is a new

36
00:02:41,400 --> 00:02:45,520
mathematical language which allows
us to map one thing into another.

37
00:02:45,520 --> 00:02:49,320
The power of perspective unleashed
a new way to see the world,

38
00:02:49,320 --> 00:02:53,480
a perspective that would cause
a mathematical revolution.

39
00:02:55,600 --> 00:02:59,840
Piero's work was the beginning of
a new way to understand geometry,

40
00:02:59,840 --> 00:03:02,280
but it would take another 200 years

41
00:03:02,280 --> 00:03:06,000
before other mathematicians
would continue where he left off.

42
00:03:14,040 --> 00:03:16,640
Our journey has come north.

43
00:03:16,640 --> 00:03:19,680
By the 17th century,
Europe had taken over

44
00:03:19,680 --> 00:03:24,160
from the Middle East as the world's
powerhouse of mathematical ideas.

45
00:03:24,160 --> 00:03:26,200
Great strides had been
made in the geometry

46
00:03:26,200 --> 00:03:27,840
of objects fixed in time and space.

47
00:03:27,840 --> 00:03:31,000
In France, Germany,
Holland and Britain,

48
00:03:31,000 --> 00:03:35,520
the race was now on to understand
the mathematics of objects in motion

49
00:03:35,520 --> 00:03:38,800
and the pursuit of this new
mathematics started here in this

50
00:03:38,800 --> 00:03:42,440
village in the centre of France.

51
00:03:42,440 --> 00:03:45,920
Only the French would name a
village after a mathematician.

52
00:03:45,920 --> 00:03:47,600
Imagine in England a town called

53
00:03:47,600 --> 00:03:51,000
Newton or Ball or Cayley.
I don't think so!

54
00:03:51,000 --> 00:03:55,080
But in France, they really
value their mathematicians.

55
00:03:55,080 --> 00:03:57,600
This is the village of
Descartes in the Loire Valley.

56
00:03:57,600 --> 00:04:00,280
It was renamed after
the famous philosopher

57
00:04:00,280 --> 00:04:03,040
and mathematician 200 years ago.

58
00:04:03,040 --> 00:04:08,120
Descartes himself was born here
in 1596, a sickly child who lost

59
00:04:08,120 --> 00:04:11,320
his mother when very young, so
he was allowed to stay in bed every

60
00:04:11,320 --> 00:04:18,200
morning until 11.00am, a practice
he tried to continue all his life.

61
00:04:18,200 --> 00:04:21,160
To do mathematics,
sometimes you just need to remove

62
00:04:21,160 --> 00:04:25,720
all distractions, to float off
into a world of shapes and patterns.

63
00:04:25,720 --> 00:04:28,800
Descartes thought that the bed
was the best place to achieve

64
00:04:28,800 --> 00:04:30,480
this meditative state.

65
00:04:30,480 --> 00:04:32,200
I think I know what he means.

66
00:04:37,000 --> 00:04:39,800
The house where Descartes
undertook his bedtime meditations

67
00:04:39,800 --> 00:04:43,560
is now a museum
dedicated to all things Cartesian.

68
00:04:43,560 --> 00:04:46,960
Come with me.

69
00:04:46,960 --> 00:04:51,120
Its exhibition pieces arranged,
by curator Sylvie Garnier, show how

70
00:04:51,120 --> 00:04:57,240
his philosophical, scientific and
mathematical ideas all fit together.

71
00:04:57,240 --> 00:04:59,000
It also features
less familiar aspects

72
00:04:59,000 --> 00:05:00,720
of Descartes' life and career.

73
00:05:00,720 --> 00:05:03,960
So he decided
to be a soldier...in the army,

74
00:05:05,480 --> 00:05:08,800
in the Protestant Army

75
00:05:08,800 --> 00:05:14,960
and too in the Catholic Army,
not a problem for him

76
00:05:14,960 --> 00:05:17,960
because no patriotism.

77
00:05:17,960 --> 00:05:20,200
Sylvie is putting it very nicely,

78
00:05:20,200 --> 00:05:23,680
but Descartes was
in fact a mercenary.

79
00:05:23,680 --> 00:05:26,600
He fought for the German
Protestants, the French Catholics

80
00:05:26,600 --> 00:05:29,200
and anyone else who would pay him.

81
00:05:29,200 --> 00:05:34,680
Very early one autumn morning
in 1628, he was in the Bavarian Army

82
00:05:34,680 --> 00:05:37,240
camped out on a cold river bank.

83
00:05:37,240 --> 00:05:41,000
Inspiration very often
strikes in very strange places.

84
00:05:41,000 --> 00:05:44,200
The story is told how Descartes
couldn't sleep one night,

85
00:05:44,200 --> 00:05:46,480
maybe because he was getting up
so late

86
00:05:46,480 --> 00:05:49,120
or perhaps
he was celebrating St Martin's Eve

87
00:05:49,120 --> 00:05:50,680
and had just drunk too much.

88
00:05:50,680 --> 00:05:52,960
Problems were tumbling
around in his mind.

89
00:05:52,960 --> 00:05:56,040
He was thinking about his
favourite subject, philosophy.

90
00:05:56,040 --> 00:05:57,960
He was finding it very frustrating.

91
00:05:57,960 --> 00:06:01,840
How can you actually know
anything at all?!

92
00:06:01,840 --> 00:06:04,320
Then he slips into a dream...

93
00:06:06,600 --> 00:06:10,600
and in the dream he understood
that the key was to build philosophy

94
00:06:10,600 --> 00:06:14,320
on the indisputable facts
of mathematics.

95
00:06:14,320 --> 00:06:19,480
Numbers, he realised, could brush
away the cobwebs of uncertainty.

96
00:06:19,480 --> 00:06:23,480
He wanted to publish all his radical
ideas, but he was worried how they'd

97
00:06:23,480 --> 00:06:28,640
be received in Catholic France,
so he packed his bags and left.

98
00:06:31,680 --> 00:06:34,600
Descartes found a
home here in Holland.

99
00:06:34,600 --> 00:06:37,880
He'd been one of the champions
of the new scientific revolution

100
00:06:37,880 --> 00:06:41,520
which rejected the dominant view
that the sun went around the earth,

101
00:06:41,520 --> 00:06:44,760
an opinion that got
scientists like Galileo

102
00:06:44,760 --> 00:06:47,200
into deep trouble with the Vatican.

103
00:06:47,200 --> 00:06:50,760
Descartes reckoned that here
amongst the Protestant Dutch

104
00:06:50,760 --> 00:06:53,240
he would be safe, especially

105
00:06:53,240 --> 00:06:55,800
at the old university town
of Leiden

106
00:06:55,800 --> 00:06:58,560
where they valued maths and
science.

107
00:06:58,560 --> 00:07:00,520
I've come to Leiden too.

108
00:07:00,520 --> 00:07:02,800
Unfortunately, I'm late!

109
00:07:02,800 --> 00:07:05,320
Hello. Yeah, I'm sorry.

110
00:07:05,320 --> 00:07:09,440
I got a puncture. It
took me a bit of time, yeah, yeah.

111
00:07:09,440 --> 00:07:13,400
Henk Bos is one of Europe's
most eminent Cartesian scholars.

112
00:07:13,400 --> 00:07:16,880
He's not surprised the French
scholar ended up in Leiden.

113
00:07:16,880 --> 00:07:21,080
He came to talk with people and
some people were open to his ideas.

114
00:07:21,080 --> 00:07:24,320
This was not only mathematic.
It was also a mechanics specially.

115
00:07:24,320 --> 00:07:26,280
He merged algebra and geometry.

116
00:07:26,280 --> 00:07:31,960
Right. So you could have formulas
and figures and go back and forth.

117
00:07:31,960 --> 00:07:36,240
So a sort of dictionary
between the two? Yeah, yeah.

118
00:07:39,520 --> 00:07:41,560
This dictionary, which was finally
published here in Holland in 1637,

119
00:07:41,560 --> 00:07:43,160
included mainly controversial

120
00:07:43,160 --> 00:07:46,600
philosophical ideas,
but the most radical thoughts

121
00:07:46,600 --> 00:07:51,920
were in the appendix, a proposal
to link algebra and geometry.

122
00:07:53,960 --> 00:07:58,240
Each point in two dimensions
can be described by two numbers,

123
00:07:58,240 --> 00:08:02,480
one giving the horizontal location,
the second number giving the point's

124
00:08:02,480 --> 00:08:04,720
vertical location.

125
00:08:04,720 --> 00:08:08,920
As the point moves around a circle,
these coordinates change,

126
00:08:08,920 --> 00:08:12,800
but we can write down an equation
that identifies the changing value

127
00:08:12,800 --> 00:08:15,360
of these numbers
at any point in the figure.

128
00:08:15,360 --> 00:08:19,040
Suddenly, geometry
has turned into algebra.

129
00:08:19,040 --> 00:08:21,160
Using this transformation

130
00:08:21,160 --> 00:08:24,720
from geometry into numbers,
you could tell, for example,

131
00:08:24,720 --> 00:08:28,040
if the curve on this bridge
was part of a circle or not.

132
00:08:28,040 --> 00:08:29,960
You didn't need to use your eyes.

133
00:08:29,960 --> 00:08:33,880
Instead, the equations of the
curve would reveal its secrets,

134
00:08:33,880 --> 00:08:36,080
but it wouldn't stop there.

135
00:08:36,080 --> 00:08:39,680
Descartes had unlocked the
possibility of navigating geometries

136
00:08:39,680 --> 00:08:44,160
of higher dimensions, worlds our
eyes will never see but are central

137
00:08:44,160 --> 00:08:46,640
to modern technology and physics.

138
00:08:46,640 --> 00:08:51,120
There's no doubt that Descartes was
one of the giants of mathematics.

139
00:08:51,120 --> 00:08:55,320
Unfortunately, though,
he wasn't the nicest of men.

140
00:08:55,320 --> 00:08:59,280
I think he was not
an easy person, so...

141
00:08:59,280 --> 00:09:05,080
And he could be...
he was very much concerned about

142
00:09:05,080 --> 00:09:08,120
his image. He was entirely

143
00:09:08,120 --> 00:09:12,600
self-convinced that he was right,
also when he was wrong and his first

144
00:09:12,600 --> 00:09:16,600
reaction would be that the other one
was stupid that hadn't understood it.

145
00:09:16,600 --> 00:09:20,080
Descartes may not have been
the most congenial person,

146
00:09:20,080 --> 00:09:22,720
but there's no doubt
that his insight into the connection

147
00:09:22,720 --> 00:09:27,640
between algebra and geometry
transformed mathematics forever.

148
00:09:27,640 --> 00:09:31,280
For his mathematical revolution to
work, though, he needed one other

149
00:09:31,280 --> 00:09:32,840
vital ingredient.

150
00:09:32,840 --> 00:09:38,640
To find that, I had to say goodbye
to Henk and Leiden and go to church.

151
00:09:38,640 --> 00:09:40,160
CHORAL SINGING

152
00:09:44,200 --> 00:09:47,160
I'm not a believer myself,
but there's little doubt

153
00:09:47,160 --> 00:09:49,840
that many mathematicians
from the time of Descartes

154
00:09:49,840 --> 00:09:52,240
had strong religious convictions.

155
00:09:56,480 --> 00:09:58,720
Maybe it's just a coincidence,

156
00:09:58,720 --> 00:10:02,520
but perhaps it's because mathematics
and religion are both building ideas

157
00:10:02,520 --> 00:10:08,440
upon an undisputed set of axioms -
one plus one equals two. God exists.

158
00:10:08,440 --> 00:10:11,560
I think I know which set of
axioms I've got my faith in.

159
00:10:14,640 --> 00:10:16,280
In the 17th century,

160
00:10:16,280 --> 00:10:19,880
there was a Parisian monk who went
to the same school as Descartes.

161
00:10:19,880 --> 00:10:23,040
He loved mathematics
as much as he loved God.

162
00:10:23,040 --> 00:10:28,000
Indeed, he saw maths and science as
evidence of the existence of God,

163
00:10:28,000 --> 00:10:31,720
Marin Mersenne
was a first-class mathematician.

164
00:10:31,720 --> 00:10:35,080
One of his discoveries in prime
numbers is still named after him.

165
00:10:36,600 --> 00:10:41,560
But he's also celebrated
for his correspondence.

166
00:10:41,560 --> 00:10:45,000
From his monastery in Paris,
Mersenne acted like some kind of

167
00:10:45,000 --> 00:10:49,880
17th century internet hub, receiving
ideas and then sending them on.

168
00:10:49,880 --> 00:10:51,680
It's not so different now.

169
00:10:51,680 --> 00:10:55,240
We sit like mathematical monks
thinking about our ideas, then

170
00:10:55,240 --> 00:10:59,320
sending a message to a colleague
and hoping for some reply.

171
00:11:01,040 --> 00:11:05,320
There was a spirit of mathematical
communication in 17th century Europe

172
00:11:05,320 --> 00:11:08,520
which had not been seen
since the Greeks.

173
00:11:08,520 --> 00:11:13,280
Mersenne urged people to read
Descartes' new work on geometry.

174
00:11:13,280 --> 00:11:15,480
He also did something
just as important.

175
00:11:15,480 --> 00:11:20,280
He publicised some new findings
on the properties of numbers

176
00:11:20,280 --> 00:11:23,440
by an unknown amateur who would
end up rivalling Descartes as the

177
00:11:23,440 --> 00:11:27,160
greatest mathematician of his time,
Pierre de Fermat.

178
00:11:32,800 --> 00:11:35,280
Here in Beaumont-de-Lomagne

179
00:11:35,280 --> 00:11:37,520
near Toulouse,
residents and visitors have come

180
00:11:37,520 --> 00:11:42,920
out to celebrate the life and work
of the village's most famous son.

181
00:11:42,920 --> 00:11:46,320
But I'm not too sure what
these gladiators are doing here!

182
00:11:46,320 --> 00:11:51,040
And the appearance of this camel
came as a bit of a surprise too.

183
00:11:51,040 --> 00:11:53,360
The man himself
would have hardly approved of

184
00:11:53,360 --> 00:11:57,680
the ideas of using fun and games to
advance an interest in mathematics.

185
00:11:57,680 --> 00:12:01,720
Unlike the aristocratic Descartes,
Fermat wouldn't have considered it

186
00:12:01,720 --> 00:12:05,360
worthless or common
to create a festival of mathematics.

187
00:12:05,360 --> 00:12:08,160
Maths in action, that one.

188
00:12:08,160 --> 00:12:10,440
It's beautiful, really nice, yeah.

189
00:12:10,440 --> 00:12:15,040
Fermat's greatest contribution to
mathematics was to virtually invent

190
00:12:15,040 --> 00:12:16,720
modern number theory.

191
00:12:16,720 --> 00:12:18,680
He devised a wide range
of conjectures

192
00:12:18,680 --> 00:12:22,080
and theorems about numbers
including his famous Last Theorem,

193
00:12:22,080 --> 00:12:27,960
the proof of which would puzzle
mathematicians for over 350 years,

194
00:12:27,960 --> 00:12:29,560
but it's little help to me now.

195
00:12:29,560 --> 00:12:31,440
Getting it apart is the easy bit.

196
00:12:31,440 --> 00:12:34,080
It's putting it together, isn't it,
that's the difficult bit.

197
00:12:34,080 --> 00:12:37,160
How many bits have I got?
I've got six bits.

198
00:12:38,920 --> 00:12:42,240
I think what I need to do is
put some symmetry into this.

199
00:12:42,240 --> 00:12:45,200
I'm afraid he's going to tell me how
to do it and I don't want to see.

200
00:12:45,200 --> 00:12:48,400
I hate being told how to do a
problem. I don't want to look.

201
00:12:48,400 --> 00:12:52,200
And he's laughing at me now
because I can't do it.

202
00:12:52,200 --> 00:12:54,640
That's very unfair!

203
00:12:54,640 --> 00:12:56,040
Here we go.

204
00:12:57,080 --> 00:12:59,760
Can I put them together?

205
00:12:59,760 --> 00:13:01,320
I got it!

206
00:13:01,320 --> 00:13:03,800
Now that's the buzz of
doing mathematics when

207
00:13:03,800 --> 00:13:08,520
the thing clicks together and
suddenly you see the right answer.

208
00:13:08,520 --> 00:13:12,960
Remarkably, Fermat only tackled
mathematics in his spare time.

209
00:13:12,960 --> 00:13:15,400
By day he was a magistrate.

210
00:13:15,400 --> 00:13:20,040
Battling with mathematical problems
was his hobby and his passion.

211
00:13:21,640 --> 00:13:23,240
The wonderful thing
about mathematics is

212
00:13:23,240 --> 00:13:24,800
you can do it anywhere.

213
00:13:24,800 --> 00:13:26,440
You don't have to have a laboratory.

214
00:13:26,440 --> 00:13:28,560
You don't even
really need a library.

215
00:13:28,560 --> 00:13:31,720
Fermat used to do much of his work
while sitting at the kitchen table

216
00:13:31,720 --> 00:13:36,120
or praying in his local church
or up here on his roof.

217
00:13:36,120 --> 00:13:38,480
He may have looked like an amateur,

218
00:13:38,480 --> 00:13:41,240
but he took his mathematics
very seriously indeed.

219
00:13:41,240 --> 00:13:45,040
Fermat managed to find
several new patterns in numbers

220
00:13:45,040 --> 00:13:46,880
that had defeated
mathematicians for centuries.

221
00:13:46,880 --> 00:13:50,200
One of my favourite
theorems of Fermat

222
00:13:50,200 --> 00:13:52,600
is all to do with prime numbers.

223
00:13:52,600 --> 00:13:55,680
If you've got a prime number
which when you divide it by four

224
00:13:55,680 --> 00:13:58,400
leaves remainder one,
then Fermat showed you could

225
00:13:58,400 --> 00:14:02,480
always rewrite this number as
two square numbers added together.

226
00:14:02,480 --> 00:14:05,920
For example,
I've got 13 cloves of garlic here,

227
00:14:05,920 --> 00:14:09,560
a prime number which has remainder
one when I divide it by four.

228
00:14:09,560 --> 00:14:13,960
Fermat proved you can rewrite this
number as two square numbers added

229
00:14:13,960 --> 00:14:18,000
together, so 13 can be rewritten

230
00:14:18,000 --> 00:14:23,080
as three squared plus two squared,
or four plus nine.

231
00:14:23,080 --> 00:14:26,880
The amazing thing is that Fermat
proved this will work however big

232
00:14:26,880 --> 00:14:31,400
the prime number is. Provided it has
remainder one on division by four,

233
00:14:31,400 --> 00:14:33,720
you can always rewrite that number

234
00:14:33,720 --> 00:14:36,600
as two square numbers
added together.

235
00:14:39,800 --> 00:14:42,280
Ah, my God!

236
00:14:44,440 --> 00:14:48,080
What I love about this sort of day
is the playfulness of mathematics

237
00:14:48,080 --> 00:14:51,880
and Fermat certainly enjoyed playing
around with numbers. He loved

238
00:14:51,880 --> 00:14:55,720
looking for patterns in numbers and
then the puzzle side of mathematics,

239
00:14:55,720 --> 00:14:59,040
he wanted to prove that these
patterns would be there forever.

240
00:15:01,000 --> 00:15:04,840
But as well as being the basis for
fun and games in the years to come,

241
00:15:04,840 --> 00:15:09,400
Fermat's mathematics would have
some very serious applications.

242
00:15:09,400 --> 00:15:11,320
One of his theorems,
his Little Theorem, is

243
00:15:11,320 --> 00:15:16,240
the basis of the codes that protect
our credit cards on the internet.

244
00:15:16,240 --> 00:15:20,280
Technology we now rely on today
all comes from the scribblings

245
00:15:20,280 --> 00:15:22,720
of a 17th-century mathematician.

246
00:15:24,720 --> 00:15:28,360
But the usefulness of Fermat's
mathematics is nothing compared to

247
00:15:28,360 --> 00:15:33,400
that of our next great mathematician
and he comes not from France at all,

248
00:15:33,400 --> 00:15:35,080
but from its great rival.

249
00:15:38,480 --> 00:15:43,240
In the 17th century, Britain
was emerging as a world power.

250
00:15:43,240 --> 00:15:46,800
Its expansion and ambitions
required new methods of measurement

251
00:15:46,800 --> 00:15:52,160
and computation and that gave
a great boost to mathematics.

252
00:15:52,160 --> 00:15:54,040
The university towns
of Oxford and Cambridge

253
00:15:54,040 --> 00:15:58,920
were churning out mathematicians
who were in great demand

254
00:15:58,920 --> 00:16:02,880
and the greatest of them
was Isaac Newton.

255
00:16:06,960 --> 00:16:09,560
I'm here in Grantham,
where Isaac Newton grew up,

256
00:16:09,560 --> 00:16:11,440
and they're very proud
of him here.

257
00:16:11,440 --> 00:16:13,320
They have a wonderful statue to him.

258
00:16:13,320 --> 00:16:14,880
They've even got

259
00:16:14,880 --> 00:16:19,120
the Isaac Newton Shopping Centre,
with a nice apple logo up there.

260
00:16:19,120 --> 00:16:22,120
There's a school that he went to
with a nice blue plaque

261
00:16:22,120 --> 00:16:25,680
and there's a museum over here in
the Town Hall, although, actually,

262
00:16:25,680 --> 00:16:28,680
one of the other famous residents
here, Margaret Thatcher,

263
00:16:28,680 --> 00:16:31,120
has got as big a display
as Isaac Newton.

264
00:16:31,120 --> 00:16:32,800
In fact, the Thatcher cups have

265
00:16:32,800 --> 00:16:36,840
sold out and there's loads
of Newton ones still left,

266
00:16:36,840 --> 00:16:41,520
so I thought I would support
mathematics by buying a Newton cup.

267
00:16:41,520 --> 00:16:44,120
And Newton's maths
does need support.

268
00:16:44,120 --> 00:16:49,560
Newton's very famous here. Do you
know what he's famous for? No.

269
00:16:49,560 --> 00:16:54,120
No, I don't. Discovering gravity.
Gravity? Gravity, yes. Gravity?

270
00:16:54,120 --> 00:16:58,560
Apple tree and all that, gravity.
'That pretty much summed it up.

271
00:16:58,560 --> 00:17:01,760
'If people know about Newton's
work at all, it is his physics,

272
00:17:01,760 --> 00:17:05,360
'his laws of gravity in motion,
not his mathematics.'

273
00:17:05,360 --> 00:17:07,560
I'm in a rush! You're in a rush. OK.

274
00:17:07,560 --> 00:17:10,880
Acceleration, you see?
One of Newton's laws!

275
00:17:18,400 --> 00:17:20,280
Eight miles south of Grantham,

276
00:17:20,280 --> 00:17:23,160
in the village of Woolsthorpe,
where Newton was born,

277
00:17:23,160 --> 00:17:26,240
I met up with someone who does share
my passion for his mathematics.

278
00:17:26,240 --> 00:17:28,280
This is the house.

279
00:17:28,280 --> 00:17:32,320
Wow, beautiful. 'Jackie Stedall is
a Newton fan and more than willing

280
00:17:32,320 --> 00:17:35,560
'to show me around the house
where Newton was brought up.'

281
00:17:35,560 --> 00:17:37,560
So here is the...

282
00:17:37,560 --> 00:17:41,040
you might call it the dining room.
I'm sure they didn't call it that,

283
00:17:41,040 --> 00:17:43,760
but the room where they ate,
next to the kitchen.

284
00:17:43,760 --> 00:17:45,720
Of course, there would have
been a huge fire in there.

285
00:17:45,720 --> 00:17:48,320
Yes! Gosh, I wish it was there now!

286
00:17:48,320 --> 00:17:50,840
His father was an illiterate farmer,

287
00:17:50,840 --> 00:17:53,320
but he died shortly
before Newton was born.

288
00:17:53,320 --> 00:17:57,280
Otherwise, the young Isaac's fate
might have been very different.

289
00:17:57,280 --> 00:17:59,280
And here's his room.

290
00:17:59,280 --> 00:18:01,680
Oh, lovely, wow.

291
00:18:01,680 --> 00:18:03,960
They present it really nicely. Yes.

292
00:18:03,960 --> 00:18:07,640
It's got a real feel
of going back in time. It does, yes.

293
00:18:07,640 --> 00:18:10,640
I can see he's as scruffy as I am.
Look at the state of that bed.

294
00:18:10,640 --> 00:18:13,680
That's how,
I think, I left my bed this morning.

295
00:18:13,680 --> 00:18:18,360
Newton hated his stepfather,
but it was this man who ensured

296
00:18:18,360 --> 00:18:21,440
he became a mathematician
rather than a sheep farmer.

297
00:18:21,440 --> 00:18:23,680
I don't think he was
particularly remarkable as a child.

298
00:18:23,680 --> 00:18:27,000
OK. So there's hope for all
those kids out there. Yes, yes.

299
00:18:27,000 --> 00:18:28,600
I think he had a sort
of average school report.

300
00:18:28,600 --> 00:18:32,480
He had very few close friends.
I don't feel he's someone

301
00:18:32,480 --> 00:18:34,080
I particularly would
have wanted to meet,

302
00:18:34,080 --> 00:18:37,960
but I do love his mathematics.
It's wonderful.

303
00:18:37,960 --> 00:18:40,520
Newton came back to
Lincolnshire from Cambridge

304
00:18:40,520 --> 00:18:46,800
during the Great Plague of 1665
when he was just 22 years old.

305
00:18:46,800 --> 00:18:51,080
In two miraculous years here,
he developed a new theory of light,

306
00:18:51,080 --> 00:18:52,600
discovered gravitation

307
00:18:52,600 --> 00:18:58,160
and scribbled out a revolutionary
approach to maths, the calculus.

308
00:18:58,160 --> 00:19:00,080
It works like this.

309
00:19:00,080 --> 00:19:04,120
I'm going to accelerate this car
from 0 to 60 as quickly as I can.

310
00:19:04,120 --> 00:19:07,720
The speedometer is showing me that
the speed's changing all the time,

311
00:19:07,720 --> 00:19:09,560
but this is only an average speed.

312
00:19:09,560 --> 00:19:11,680
How can I tell precisely
what my speed is

313
00:19:11,680 --> 00:19:15,600
at any particular instant?
Well, here's how.

314
00:19:15,600 --> 00:19:20,520
As the car races along the road,
we can draw a graph above the road

315
00:19:20,520 --> 00:19:23,760
where the height above each point in
the road records how long it took

316
00:19:23,760 --> 00:19:26,600
the car to get to that point.

317
00:19:26,600 --> 00:19:29,040
I can calculate the
average speed between

318
00:19:29,040 --> 00:19:33,440
two points, A and B, on my journey
by recording the distance travelled

319
00:19:33,440 --> 00:19:37,960
and dividing by the time it took
to get between these two points,

320
00:19:37,960 --> 00:19:42,200
but what about the precise speed
at the first point, A?

321
00:19:43,720 --> 00:19:48,400
If I move point B closer and closer
to the first point, I take a smaller

322
00:19:48,400 --> 00:19:51,640
and smaller window of time
and the speed gets closer

323
00:19:51,640 --> 00:19:55,440
and closer to the true value,
but eventually, it looks like

324
00:19:55,440 --> 00:19:59,520
I have to calculate 0 divided by 0.

325
00:19:59,520 --> 00:20:04,120
The calculus allows us to
make sense of this calculation.

326
00:20:04,120 --> 00:20:08,520
It enables us to work out the exact
speed and also the precise distance

327
00:20:08,520 --> 00:20:11,480
travelled at any moment in time.

328
00:20:11,480 --> 00:20:15,280
I mean, it does make sense, the
things we take for granted so much,

329
00:20:15,280 --> 00:20:16,920
things like...
if I drop this apple...

330
00:20:16,920 --> 00:20:18,480
Its distance is changing and its

331
00:20:18,480 --> 00:20:21,120
speed is changing and calculus
can deal with all of that.

332
00:20:21,120 --> 00:20:22,680
Which is quite in contrast
to the Greeks.

333
00:20:22,680 --> 00:20:25,320
It was a very static geometry.

334
00:20:25,320 --> 00:20:27,200
Yes, it is. And here we see...

335
00:20:27,200 --> 00:20:30,080
so the calculus is used by

336
00:20:30,080 --> 00:20:33,400
every engineer, physicist, because
it can describe the moving world.

337
00:20:33,400 --> 00:20:36,920
Yes, and it's the only way really
you can deal with the mathematics of

338
00:20:36,920 --> 00:20:38,680
motion or with change.

339
00:20:38,680 --> 00:20:40,280
There's a lot of
mathematics in this apple!

340
00:20:42,560 --> 00:20:46,240
Newton's calculus
enables us to really understand

341
00:20:46,240 --> 00:20:50,800
the changing world, the orbits
of planets, the motions of fluids.

342
00:20:50,800 --> 00:20:54,400
Through the power of the calculus,
we have a way of describing, with

343
00:20:54,400 --> 00:20:59,040
mathematical precision, the complex,
ever-changing natural world.

344
00:21:05,000 --> 00:21:09,280
But it would take 200 years
to realise its full potential.

345
00:21:09,280 --> 00:21:12,840
Newton himself decided not to
publish, but just to circulate

346
00:21:12,840 --> 00:21:15,160
his thoughts among friends.

347
00:21:15,160 --> 00:21:17,440
His reputation, though,
gradually spread.

348
00:21:17,440 --> 00:21:21,680
He became a professor, an MP,
and then Warden of the Royal Mint

349
00:21:21,680 --> 00:21:23,840
here in the City of London.

350
00:21:25,800 --> 00:21:28,960
On his regular trips to the
Royal Society from the Royal Mint,

351
00:21:28,960 --> 00:21:33,320
he preferred to think about theology
and alchemy rather than mathematics.

352
00:21:33,320 --> 00:21:35,640
Developing the calculus
just got crowded out

353
00:21:35,640 --> 00:21:39,920
by all his other interests
until he heard about a rival...

354
00:21:42,000 --> 00:21:46,280
a rival who was also a member of
the Royal Society and who came up

355
00:21:46,280 --> 00:21:49,000
with exactly the same idea as him,

356
00:21:49,000 --> 00:21:51,160
Gottfried Leibniz.

357
00:21:51,160 --> 00:21:54,440
Every word Leibniz wrote
has been preserved and catalogued

358
00:21:54,440 --> 00:21:58,000
in his hometown of Hanover
in northern Germany.

359
00:21:58,000 --> 00:22:01,240
His actual manuscripts
are kept under lock and key,

360
00:22:01,240 --> 00:22:04,560
particularly the manuscript
which shows how Leibniz

361
00:22:04,560 --> 00:22:09,920
also discovered the miracle
of calculus, shortly after Newton.

362
00:22:09,920 --> 00:22:11,720
What age was he when he wrote...

363
00:22:11,720 --> 00:22:16,920
He was 29 years old and that's the
time, within two months, he developed

364
00:22:16,920 --> 00:22:19,840
differential calculus and integral
calculus. In two months?

365
00:22:19,840 --> 00:22:21,800
Yeah. Fast and furious,
when it comes, er...

366
00:22:21,800 --> 00:22:23,440
Yeah.

367
00:22:23,440 --> 00:22:26,640
There is a little scrap of
paper over here. What's that one?

368
00:22:26,640 --> 00:22:30,040
A letter or... That's a small
manuscript of Leibniz's notes.

369
00:22:32,760 --> 00:22:37,480
"Sometimes it happens that in
the morning lying in the bed,

370
00:22:37,480 --> 00:22:41,160
"I have so many ideas that it
takes the whole morning and sometimes

371
00:22:41,160 --> 00:22:45,960
"even longer to note all these ideas
and bring them to paper."

372
00:22:45,960 --> 00:22:47,480
I suppose, that's beautiful.

373
00:22:47,480 --> 00:22:51,680
I suppose that he liked to
lie in the bed in the morning.

374
00:22:51,680 --> 00:22:53,600
A true mathematician. Yeah.

375
00:22:53,600 --> 00:22:55,880
He spends his time thinking in bed.

376
00:22:55,880 --> 00:22:58,840
I see you've got some
paintings down here.

377
00:22:58,840 --> 00:23:00,480
A painting.

378
00:23:00,480 --> 00:23:02,560
This is what he looked like. Right.

379
00:23:04,080 --> 00:23:07,480
Even though he didn't become
quite the 17th century celebrity

380
00:23:07,480 --> 00:23:10,760
that Newton did,
it wasn't such a bad life.

381
00:23:10,760 --> 00:23:12,720
Leibniz worked for the Royal Family

382
00:23:12,720 --> 00:23:16,800
of Hanover and travelled around
Europe representing their interests.

383
00:23:16,800 --> 00:23:19,240
This gave him
plenty of time to indulge in

384
00:23:19,240 --> 00:23:23,600
his favourite intellectual pastimes,
which were wide, even for the time.

385
00:23:23,600 --> 00:23:27,160
He devised a plan for reunifying
the Protestant and Roman Catholic

386
00:23:27,160 --> 00:23:32,200
churches, a proposal for France to
conquer Egypt and contributions to

387
00:23:32,200 --> 00:23:36,480
philosophy and logic
which are still highly rated today.

388
00:23:36,480 --> 00:23:40,080
He wrote all these letters? Yeah.
That's absolutely extraordinary.

389
00:23:40,080 --> 00:23:43,280
He must have cloned himself. I can't
believe there was just one Leibniz!

390
00:23:43,280 --> 00:23:46,240
'But Leibniz was not
just man of words.

391
00:23:46,240 --> 00:23:47,840
'He was also one of the first people

392
00:23:47,840 --> 00:23:49,680
'to invent practical
calculating machines

393
00:23:49,680 --> 00:23:54,720
'that worked on the binary system,
true forerunners of the computer.

394
00:23:54,720 --> 00:23:58,880
'300 years later, the engineering
department at Leibniz University

395
00:23:58,880 --> 00:24:03,080
'in Hanover have put them together
following Leibniz's blueprint.'

396
00:24:03,080 --> 00:24:04,960
I love all the ball bearings,
so these are going to be all

397
00:24:04,960 --> 00:24:06,880
of our zeros and ones.
So a ball bearing is a one.

398
00:24:06,880 --> 00:24:10,920
Only zero and one.
Now we represent a number 127.

399
00:24:10,920 --> 00:24:16,160
In binary, it means that we have
the first seven digits in one. Yeah.

400
00:24:16,160 --> 00:24:19,080
And now I give the number one. OK.

401
00:24:19,080 --> 00:24:24,560
Now we add 127 plus one -
is 128, which is two, power eight.

402
00:24:24,560 --> 00:24:28,200
Oh, OK. So there's going to be lots
of action. Would you show this here?

403
00:24:28,200 --> 00:24:30,680
This is the money shot.

404
00:24:30,680 --> 00:24:33,760
So we're going to add one. Oops.
Here we go. They're all carrying.

405
00:24:33,760 --> 00:24:36,720
So this 128 is two power eight.

406
00:24:36,720 --> 00:24:42,560
Excellent, so 127 in binary is
1, 1, 1, 1, 1, 1, 1, which is

407
00:24:42,560 --> 00:24:44,520
all the ball bearings here.

408
00:24:44,520 --> 00:24:46,520
To add one it all gets

409
00:24:46,520 --> 00:24:51,120
carried, this goes to 0, 0, 0, 0,
and we have a power of two here.

410
00:24:51,120 --> 00:24:53,280
So this mechanism gets rid of
all the ball bearings that you

411
00:24:53,280 --> 00:24:56,880
don't need. It's like pinball,
mathematical pinball. Exactly.

412
00:24:56,880 --> 00:24:58,400
I love this machine!

413
00:25:03,880 --> 00:25:08,320
After a hard day's work,
Leibniz often came here,

414
00:25:08,320 --> 00:25:10,280
the famous gardens of Herrenhausen,

415
00:25:10,280 --> 00:25:15,000
now in the middle of Hanover, but
then on the outskirts of the city.

416
00:25:15,000 --> 00:25:17,600
There's something about
mathematics and walking.

417
00:25:17,600 --> 00:25:21,240
I don't know, you've been working
at your desk all day, all morning

418
00:25:21,240 --> 00:25:22,840
on some problem and your head's all

419
00:25:22,840 --> 00:25:25,240
fuzzy, and you just need
to come and have a walk.

420
00:25:25,240 --> 00:25:27,960
You let your subconscious mind
kind of take over and sometimes

421
00:25:27,960 --> 00:25:32,080
you get your breakthrough just
looking at the trees or whatever.

422
00:25:32,080 --> 00:25:35,360
I've had some of my best ideas
whilst walking in my local park,

423
00:25:35,360 --> 00:25:39,320
so I'm hoping to get a little bit
of inspiration here on Leibniz's

424
00:25:39,320 --> 00:25:40,960
local stomping ground.

425
00:25:44,440 --> 00:25:47,320
I didn't get the chance to purge
my mind of mathematical challenges

426
00:25:47,320 --> 00:25:49,440
because in the years since Leibniz
lived here,

427
00:25:49,440 --> 00:25:50,640
someone has built a maze.

428
00:25:50,640 --> 00:25:53,720
Well, there is a mathematical
formula for getting out of a maze,

429
00:25:53,720 --> 00:25:57,400
which is if you put your left hand
on the side of the maze and just

430
00:25:57,400 --> 00:26:00,960
keep it there, keep on winding
round, you eventually get out.

431
00:26:00,960 --> 00:26:03,960
That's the theory, at least.
Let's see whether it works!

432
00:26:11,280 --> 00:26:13,800
Leibniz had no such distractions.

433
00:26:13,800 --> 00:26:17,520
Within five years, he'd worked
out the details of the calculus,

434
00:26:17,520 --> 00:26:19,360
seemingly independent from Newton,

435
00:26:19,360 --> 00:26:21,880
although he knew
about Newton's work,

436
00:26:21,880 --> 00:26:26,400
but unlike Newton, Leibniz was
quite happy to make his work known

437
00:26:26,400 --> 00:26:29,640
and so mathematicians across Europe
heard about the calculus first

438
00:26:29,640 --> 00:26:35,880
from him and not from Newton, and
that's when all the trouble started.

439
00:26:35,880 --> 00:26:39,400
Throughout mathematical history,
there have been lots of priority

440
00:26:39,400 --> 00:26:41,000
disputes and arguments.

441
00:26:41,000 --> 00:26:44,000
It may seem a little bit
petty and schoolboyish.

442
00:26:44,000 --> 00:26:46,800
We really want
our name to be on that theorem.

443
00:26:46,800 --> 00:26:50,000
This is our one chance for a little
bit of immortality because that

444
00:26:50,000 --> 00:26:54,320
theorem's going to last forever and
that's why we dedicate so much time

445
00:26:54,320 --> 00:26:56,120
to trying to crack these things.

446
00:26:56,120 --> 00:26:58,000
Somehow we can't
believe that somebody else

447
00:26:58,000 --> 00:27:00,200
has got it at the same time as us.

448
00:27:00,200 --> 00:27:03,240
These are our theorems,
our babies, our children and we

449
00:27:03,240 --> 00:27:06,200
don't want to share the credit.

450
00:27:06,200 --> 00:27:08,640
Back in London,
Newton certainly didn't want

451
00:27:08,640 --> 00:27:13,240
to share credit with Leibniz, who he
thought of as a Hanoverian upstart.

452
00:27:13,240 --> 00:27:16,360
After years of acrimony
and accusation, the Royal Society

453
00:27:16,360 --> 00:27:21,320
in London was asked to
adjudicate between the rival claims.

454
00:27:21,320 --> 00:27:23,280
The Royal Society gave Newton credit

455
00:27:23,280 --> 00:27:25,440
for the first discovery
of the calculus

456
00:27:25,440 --> 00:27:29,080
and Leibniz credit
for the first publication,

457
00:27:29,080 --> 00:27:33,600
but in their final judgment,
they accused Leibniz of plagiarism.

458
00:27:33,600 --> 00:27:36,840
However, that might have had
something to do with the fact that

459
00:27:36,840 --> 00:27:42,120
the report was written by their
President, one Sir Isaac Newton.

460
00:27:44,240 --> 00:27:46,640
Leibniz was incredibly hurt.

461
00:27:46,640 --> 00:27:50,600
He admired Newton
and never really recovered.

462
00:27:50,600 --> 00:27:52,640
He died in 1716.

463
00:27:52,640 --> 00:27:56,400
Newton lived on another 11 years
and was buried in the grandeur of

464
00:27:56,400 --> 00:27:58,440
Westminster Abbey.

465
00:27:58,440 --> 00:28:00,560
Leibniz's memorial, by contrast,

466
00:28:00,560 --> 00:28:02,720
is here in this small
church in Hanover.

467
00:28:02,720 --> 00:28:06,240
The irony is that it's
Leibniz's mathematics which

468
00:28:06,240 --> 00:28:09,000
eventually triumphs, not Newton's.

469
00:28:11,240 --> 00:28:13,920
I'm a big Leibniz fan.

470
00:28:13,920 --> 00:28:17,120
Quite often revolutions in
mathematics are about producing the

471
00:28:17,120 --> 00:28:19,880
right language to capture
a new vision and that's what

472
00:28:19,880 --> 00:28:21,720
Leibniz was so good at.

473
00:28:21,720 --> 00:28:25,480
Leibniz's notation,
his way of writing the calculus,

474
00:28:25,480 --> 00:28:27,560
captured its true spirit.

475
00:28:27,560 --> 00:28:30,160
It's still the one
we use in maths today.

476
00:28:30,160 --> 00:28:34,520
Newton's notation was, for many
mathematicians, clumsy and difficult

477
00:28:34,520 --> 00:28:38,800
to use and so while British
mathematics loses its way a little,

478
00:28:38,800 --> 00:28:43,560
the story of maths switches to
the very heart of Europe, Basel.

479
00:28:48,760 --> 00:28:52,480
In its heyday in the 18th century,
the free city of Basel in

480
00:28:52,480 --> 00:28:57,040
Switzerland was the commercial hub
of the entire Western world.

481
00:28:57,040 --> 00:28:59,840
Around this maelstrom of trade,
there developed a tradition of

482
00:28:59,840 --> 00:29:03,720
learning, particularly learning
which connected with commerce

483
00:29:03,720 --> 00:29:06,600
and one family summed all this up.

484
00:29:06,600 --> 00:29:11,360
It's kind of curious - artists
often have children who are artists.

485
00:29:11,360 --> 00:29:15,680
Musicians, their children are often
musicians, but us mathematicians,

486
00:29:15,680 --> 00:29:17,880
our children don't tend
to be mathematicians.

487
00:29:17,880 --> 00:29:19,920
I'm not sure why it is.

488
00:29:19,920 --> 00:29:23,200
At least that's my view,
although others dispute it.

489
00:29:23,200 --> 00:29:25,200
What no-one disagrees with

490
00:29:25,200 --> 00:29:30,280
is there is one great dynasty of
mathematicians, the Bernoullis.

491
00:29:30,280 --> 00:29:33,960
In the 18th and 19th centuries
they produced half a dozen

492
00:29:33,960 --> 00:29:37,240
outstanding mathematicians,
any of which we would have been

493
00:29:37,240 --> 00:29:42,000
proud to have had in Britain,
and they all came from Basel.

494
00:29:42,000 --> 00:29:45,160
You might have great minds
like Newton and Leibniz who make

495
00:29:45,160 --> 00:29:48,640
these fundamental breakthroughs,
but you also need the disciples

496
00:29:48,640 --> 00:29:51,880
who take that message, clarify it,
realise its implications,

497
00:29:51,880 --> 00:29:55,680
then spread it wide. The family
were originally merchants,

498
00:29:55,680 --> 00:29:57,640
and this is one of their houses.

499
00:29:57,640 --> 00:30:00,560
It's now part
of the University of Basel

500
00:30:00,560 --> 00:30:03,640
and it's been completely
refurbished, apart from one room,

501
00:30:03,640 --> 00:30:07,560
which has been kept very much
as the family would have used it.

502
00:30:07,560 --> 00:30:09,920
Dr Fritz Nagel,
keeper of the Bernoulli Archive,

503
00:30:09,920 --> 00:30:12,680
has promised to show it to me.

504
00:30:12,680 --> 00:30:15,320
If we can find it.
No, we're on the wrong floor.

505
00:30:15,320 --> 00:30:17,640
Wrong floor, OK. Right!

506
00:30:17,640 --> 00:30:19,760
Oh, look.

507
00:30:19,760 --> 00:30:21,640
Can we take an apple?

508
00:30:21,640 --> 00:30:24,200
'No, wrong mathematician.

509
00:30:24,200 --> 00:30:26,680
'Eventually, we got there.'

510
00:30:26,680 --> 00:30:29,040
This is where the Bernoullis
would have done

511
00:30:29,040 --> 00:30:30,800
some of their mathematics.

512
00:30:30,800 --> 00:30:33,880
'I was really just being polite.

513
00:30:33,880 --> 00:30:36,600
'The only thing of
interest was an old stove.'

514
00:30:36,600 --> 00:30:40,400
Now, of the Bernoullis,
which is your favourite?

515
00:30:40,400 --> 00:30:44,280
My favourite Bernoulli is Johann I.

516
00:30:44,280 --> 00:30:49,840
He is the most smart mathematician.

517
00:30:49,840 --> 00:30:54,360
Perhaps his brother Jakob
was the mathematician

518
00:30:54,360 --> 00:30:57,360
with the deeper insight
into problems,

519
00:30:57,360 --> 00:31:00,000
but Johann found elegant solutions.

520
00:31:00,000 --> 00:31:04,120
The brothers didn't like each other
much, but both worshipped Leibniz.

521
00:31:04,120 --> 00:31:06,760
They corresponded with him,
stood up for him

522
00:31:06,760 --> 00:31:11,160
against Newton's allies, and spread
his calculus throughout Europe.

523
00:31:11,160 --> 00:31:15,640
Leibnitz was very happy to have found
two gifted mathematicians

524
00:31:15,640 --> 00:31:20,840
outside of his personal circle
of friends who mastered his calculus

525
00:31:20,840 --> 00:31:23,880
and could distribute it
in the scientific community.

526
00:31:23,880 --> 00:31:28,520
That was very important for Leibniz.
And important for maths, too.

527
00:31:28,520 --> 00:31:32,640
Without the Bernoullis, it would
have taken much longer for calculus

528
00:31:32,640 --> 00:31:36,400
to become what it is today,
a cornerstone of mathematics.

529
00:31:36,400 --> 00:31:38,960
At least,
that is Dr Nagel's contention.

530
00:31:38,960 --> 00:31:41,440
And he is a great Bernoulli fan.

531
00:31:41,440 --> 00:31:44,720
He has arranged for me
to meet Professor Daniel Bernoulli,

532
00:31:44,720 --> 00:31:47,160
the latest member of the family,

533
00:31:47,160 --> 00:31:49,880
whose famous name ensures
he gets some odd e-mails.

534
00:31:49,880 --> 00:31:51,520
Another one of which I got was,

535
00:31:51,520 --> 00:31:54,640
"Professor Bernoulli, can you
give me a hand with calculus?"

536
00:31:54,640 --> 00:31:58,760
To find a Bernoulli, you expect
them to be able to do calculus.

537
00:31:58,760 --> 00:32:02,840
'But this Daniel Bernoulli
is a professor of geology.

538
00:32:02,840 --> 00:32:06,080
'The maths gene
seems to have truly died out.

539
00:32:06,080 --> 00:32:08,080
'And during our very hearty dinner,

540
00:32:08,080 --> 00:32:11,400
'I found myself
wandering back to maths.'

541
00:32:11,400 --> 00:32:14,600
It is a bit unfair on the
Bernoullis to describe them simply

542
00:32:14,600 --> 00:32:16,240
as disciples of Leibniz.

543
00:32:16,240 --> 00:32:19,160
One of their many great
contributions to mathematics

544
00:32:19,160 --> 00:32:24,000
was to develop the calculus to
solve a classic problem of the day.

545
00:32:24,000 --> 00:32:26,560
Imagine a ball rolling down a ramp.

546
00:32:26,560 --> 00:32:29,520
The task is to design a ramp
that will get the ball

547
00:32:29,520 --> 00:32:32,640
from the top to the bottom
in the fastest time possible.

548
00:32:32,640 --> 00:32:36,280
You might think that a
straight ramp would be quickest.

549
00:32:36,280 --> 00:32:38,120
Or possibly a curved one like this

550
00:32:38,120 --> 00:32:40,920
that gives the ball
plenty of downward momentum.

551
00:32:40,920 --> 00:32:43,080
In fact, it's neither of these.

552
00:32:43,080 --> 00:32:46,160
Calculus shows
that it is what we call a cycloid,

553
00:32:46,160 --> 00:32:49,840
the path traced by a point on
the rim of a moving bicycle wheel.

554
00:32:49,840 --> 00:32:53,560
This application of the calculus by
the Bernoullis, which became known

555
00:32:53,560 --> 00:32:55,720
as the calculus of variation,

556
00:32:55,720 --> 00:32:58,800
has become one of the most
powerful aspects of the mathematics

557
00:32:58,800 --> 00:33:01,760
of Leibniz and Newton. Investors
use it to maximise profits.

558
00:33:01,760 --> 00:33:05,440
Engineers exploit it
to minimise energy use.

559
00:33:05,440 --> 00:33:08,760
Designers apply it
to optimise construction.

560
00:33:08,760 --> 00:33:10,880
It has now become
one of the linchpins

561
00:33:10,880 --> 00:33:13,040
of our modern technological world.

562
00:33:13,040 --> 00:33:17,360
Meanwhile, things were getting
more interesting in the restaurant.

563
00:33:17,360 --> 00:33:18,960
Here is my second surprise.

564
00:33:18,960 --> 00:33:22,200
Let me introduce Mr Leonhard Euler.

565
00:33:22,200 --> 00:33:23,920
Daniel Bernoulli.

566
00:33:23,920 --> 00:33:28,120
'Leonhard Euler, one of the most
famous names in mathematics.

567
00:33:28,120 --> 00:33:29,800
'This Leonhard is a descendant

568
00:33:29,800 --> 00:33:34,280
'of the original Leonhard Euler,
star pupil of Johann Bernoulli.'

569
00:33:34,280 --> 00:33:36,840
I am the ninth generation,

570
00:33:36,840 --> 00:33:40,040
the fourth Leonhard in our family

571
00:33:40,040 --> 00:33:42,640
after Leonard Euler I,
the mathematician.

572
00:33:42,640 --> 00:33:45,040
OK. And yourself,
are you a mathematician?

573
00:33:45,040 --> 00:33:48,040
Actually, I am a business analyst.

574
00:33:48,040 --> 00:33:52,120
I can't study mathematics
with my name.

575
00:33:52,120 --> 00:33:55,520
Everyone will expect you to prove
that the Riemann hypothesis!

576
00:33:55,520 --> 00:33:58,800
Perhaps it's just as well
that Leonhard decided

577
00:33:58,800 --> 00:34:02,440
not to follow in the footsteps
of his illustrious ancestor.

578
00:34:02,440 --> 00:34:04,800
He'd have had a lot to live up to.

579
00:34:13,200 --> 00:34:15,200
I am going in a boat across
the Rhine,

580
00:34:15,200 --> 00:34:17,760
and I'm feeling a little bit
worse for wear.

581
00:34:17,760 --> 00:34:21,320
Last night's dinner with
Mr Euler and Professor Bernoulli

582
00:34:21,320 --> 00:34:25,680
degenerated into toasts to all the
theorems the Bernoullis and Eulers

583
00:34:25,680 --> 00:34:28,800
have proved, and by God, they
have proved quite a lot of them!

584
00:34:28,800 --> 00:34:31,080
Never again.

585
00:34:31,080 --> 00:34:35,000
I was getting disapproving glances
from my fellow passengers as well.

586
00:34:35,000 --> 00:34:37,560
Luckily, it was only a short trip.

587
00:34:37,560 --> 00:34:42,160
Not like the trip that Euler
took in 1728 to start a new life.

588
00:34:42,160 --> 00:34:45,440
Euler may have been the prodigy
of Johann Bernoulli,

589
00:34:45,440 --> 00:34:48,000
but there was no room for him
in the city.

590
00:34:48,000 --> 00:34:49,720
If your name wasn't Bernoulli,

591
00:34:49,720 --> 00:34:53,440
there was little chance of getting
a job in mathematics here in Basel.

592
00:34:53,440 --> 00:34:55,800
But Daniel,
the son of Johann Bernoulli,

593
00:34:55,800 --> 00:34:57,320
was a great friend of Euler

594
00:34:57,320 --> 00:35:00,560
and managed to get him a job
at his university.

595
00:35:00,560 --> 00:35:03,480
But to get there
would take seven weeks,

596
00:35:03,480 --> 00:35:06,000
because Daniel's university
was in Russia.

597
00:35:08,480 --> 00:35:11,920
It wasn't an intellectual
powerhouse like Berlin or Paris,

598
00:35:11,920 --> 00:35:17,520
but St Petersburg was by no means
unsophisticated in the 18th century.

599
00:35:17,520 --> 00:35:21,640
Peter the Great had created a city
very much in the European style.

600
00:35:21,640 --> 00:35:26,280
And every fashionable city at
the time had a scientific academy.

601
00:35:28,040 --> 00:35:30,240
Peter's Academy is now a museum.

602
00:35:30,240 --> 00:35:34,520
It includes several rooms full of
the kind of grotesque curiosities

603
00:35:34,520 --> 00:35:38,200
that are usually kept out
of the public display in the West.

604
00:35:38,200 --> 00:35:40,160
But in the 1730s,

605
00:35:40,160 --> 00:35:44,600
this building was a centre
for ground-breaking research.

606
00:35:44,600 --> 00:35:47,080
It is where Euler
found his intellectual home.

607
00:35:50,480 --> 00:35:57,200
# I am sure that there could never
be a more contented man than me... #

608
00:35:58,200 --> 00:36:01,040
Many of the ideas that were
bubbling away at the time -

609
00:36:01,040 --> 00:36:02,680
calculus of variation,

610
00:36:02,680 --> 00:36:06,760
Fermat's theory of numbers -
crystallised in Euler's hands.

611
00:36:06,760 --> 00:36:09,760
But he was also creating
incredibly modern mathematics,

612
00:36:09,760 --> 00:36:12,240
topology and analysis.

613
00:36:12,240 --> 00:36:15,440
Much of the notation that
I use today as a mathematician

614
00:36:15,440 --> 00:36:19,440
was created by Euler,
numbers like e and i.

615
00:36:19,440 --> 00:36:23,200
Euler also popularised
the use of the symbol pi.

616
00:36:23,200 --> 00:36:25,400
He even combined
these numbers together

617
00:36:25,400 --> 00:36:28,320
in one of the most beautiful
formulas of mathematics,

618
00:36:28,320 --> 00:36:33,120
e to the power of i times pi
is equal to -1.

619
00:36:33,120 --> 00:36:36,800
An amazing feat
of mathematical alchemy.

620
00:36:36,800 --> 00:36:40,160
His life, in fact,
is full of mathematical magic.

621
00:36:40,160 --> 00:36:43,760
Euler applied his skills
to an immense range of topics,

622
00:36:43,760 --> 00:36:46,640
from prime numbers
to optics to astronomy.

623
00:36:46,640 --> 00:36:50,040
He devised a new system of weights
and measures, wrote a textbook

624
00:36:50,040 --> 00:36:54,720
on mechanics, and even found time
to develop a new theory of music.

625
00:36:59,560 --> 00:37:01,640
I think of him
as the Mozart of maths.

626
00:37:01,640 --> 00:37:05,000
And that view is shared by
the mathematician Nikolai Vavilov,

627
00:37:05,000 --> 00:37:07,560
who met me at the house
that was given to Euler

628
00:37:07,560 --> 00:37:10,240
by Catherine the Great.

629
00:37:10,240 --> 00:37:14,560
Euler lived here from '66 to '83,
which means from the year

630
00:37:14,560 --> 00:37:17,840
he came back to St Petersburg
to the year he died.

631
00:37:17,840 --> 00:37:22,920
And he was a member of
the Russian Academy of Sciences,

632
00:37:22,920 --> 00:37:24,960
and their greatest mathematician.

633
00:37:24,960 --> 00:37:27,560
That is exactly what it says.

634
00:37:27,560 --> 00:37:29,560
What is it now? It is a school.

635
00:37:29,560 --> 00:37:31,120
Shall we go in and see?

636
00:37:31,120 --> 00:37:33,960
OK.

637
00:37:33,960 --> 00:37:39,120
'I'm not sure Nikolai entirely
approved. But nothing ventured...'

638
00:37:39,120 --> 00:37:41,520
Perhaps we should talk
to the head teacher.

639
00:37:46,400 --> 00:37:48,520
The head didn't mind at all.

640
00:37:48,520 --> 00:37:50,880
I rather got the impression
that she was used

641
00:37:50,880 --> 00:37:53,400
to people dropping in
to talk about Euler.

642
00:37:53,400 --> 00:37:57,240
She even had a couple of very able
pupils suspiciously close to hand.

643
00:37:57,240 --> 00:38:02,440
These two young ladies are ready to
tell a few words about the scientist

644
00:38:02,440 --> 00:38:04,600
and about this very building.

645
00:38:04,600 --> 00:38:06,400
They certainly knew their stuff.

646
00:38:06,400 --> 00:38:10,080
They had undertaken an entire
classroom project on Euler,

647
00:38:10,080 --> 00:38:13,360
his long life,
happy marriage and 13 children.

648
00:38:13,360 --> 00:38:16,360
And then his tragedies -
only five of his children

649
00:38:16,360 --> 00:38:17,920
survived to adulthood.

650
00:38:17,920 --> 00:38:21,400
His first wife,
who he adored, died young.

651
00:38:21,400 --> 00:38:23,840
He started losing
most of his eyesight.

652
00:38:26,920 --> 00:38:31,680
So for the last years of his life,
he still continued to work, actually.

653
00:38:31,680 --> 00:38:34,760
He continued
his mathematical research.

654
00:38:34,760 --> 00:38:36,680
I read a quote that said now
with his blindness,

655
00:38:36,680 --> 00:38:38,840
he hasn't got any distractions,

656
00:38:38,840 --> 00:38:42,680
he can finally get on with his
mathematics. A positive attitude.

657
00:38:42,680 --> 00:38:46,400
It was a totally unexpected
and charming visit.

658
00:38:46,400 --> 00:38:49,400
Although I couldn't resist
sneaking back and correcting

659
00:38:49,400 --> 00:38:53,840
one of the equations on the board
when everyone else had left.

660
00:38:55,160 --> 00:39:00,160
To demonstrate one of my favourite
Euler theorems, I needed a drink.

661
00:39:00,160 --> 00:39:03,120
It concerns calculating
infinite sums,

662
00:39:03,120 --> 00:39:06,480
the discovery that shot Euler to
the top of the mathematical pops

663
00:39:06,480 --> 00:39:09,040
when it was announced in 1735.

664
00:39:11,320 --> 00:39:15,880
Take one shot glass full of vodka
and add it to this tall glass here.

665
00:39:18,160 --> 00:39:22,600
Next, take a glass which is a
quarter full, or a half squared,

666
00:39:22,600 --> 00:39:24,320
and add it to the first glass.

667
00:39:26,080 --> 00:39:30,440
Next, take a glass which is
a ninth full, or a third squared,

668
00:39:30,440 --> 00:39:32,120
and add that one.

669
00:39:32,120 --> 00:39:37,080
Now, if I keep on adding infinitely
many glasses where each one

670
00:39:37,080 --> 00:39:43,400
is a fraction squared, how much
will be in this tall glass here?

671
00:39:43,400 --> 00:39:45,280
It was called the Basel problem

672
00:39:45,280 --> 00:39:47,960
after the Bernoullis
tried and failed to solve it.

673
00:39:47,960 --> 00:39:52,800
Daniel Bernoulli knew that you would
not get an infinite amount of vodka.

674
00:39:52,800 --> 00:39:57,480
He estimated that the total would
come to about one and three fifths.

675
00:39:57,480 --> 00:39:59,480
But then Euler came along.

676
00:39:59,480 --> 00:40:03,720
Daniel was close,
but mathematics is about precision.

677
00:40:03,720 --> 00:40:06,840
Euler calculated that
the total height of the vodka

678
00:40:06,840 --> 00:40:11,160
would be exactly pi squared
divided by six.

679
00:40:13,240 --> 00:40:15,360
It was a complete surprise.

680
00:40:15,360 --> 00:40:18,000
What on earth did adding squares
of fractions

681
00:40:18,000 --> 00:40:20,720
have to do with
the special number pi?

682
00:40:20,720 --> 00:40:23,800
But Euler's analysis showed
that they were two sides

683
00:40:23,800 --> 00:40:25,440
of the same equation.

684
00:40:25,440 --> 00:40:29,480
One plus a quarter plus
a ninth plus a sixteenth

685
00:40:29,480 --> 00:40:34,760
and so on to infinity
is equal to pi squared over six.

686
00:40:34,760 --> 00:40:38,280
That's still quite a lot of vodka,
but here goes.

687
00:40:43,480 --> 00:40:46,640
Euler would certainly
be a hard act to follow.

688
00:40:46,640 --> 00:40:49,760
Mathematicians from
two countries would try.

689
00:40:49,760 --> 00:40:53,880
Both France and Germany were caught
up in the age of revolution

690
00:40:53,880 --> 00:40:57,160
that was sweeping Europe
in the late 18th century.

691
00:40:57,160 --> 00:40:59,960
Both desperately
needed mathematicians.

692
00:40:59,960 --> 00:41:04,800
But they went about supporting
mathematics rather differently.

693
00:41:04,800 --> 00:41:06,160
Here in France,

694
00:41:06,160 --> 00:41:09,760
the Revolution emphasised
the usefulness of mathematics.

695
00:41:09,760 --> 00:41:12,480
Napoleon recognised that
if you were going to have

696
00:41:12,480 --> 00:41:15,120
the best military machine,
the best weaponry,

697
00:41:15,120 --> 00:41:17,920
then you needed
the best mathematicians.

698
00:41:17,920 --> 00:41:21,320
Napoleon's reforms
gave mathematics a big boost.

699
00:41:21,320 --> 00:41:24,600
But this was a mathematics
that was going to serve society.

700
00:41:26,120 --> 00:41:30,200
Here in the German states, the great
educationalist Wilhelm von Humboldt

701
00:41:30,200 --> 00:41:34,040
was also committed to mathematics,
but a mathematics that was detached

702
00:41:34,040 --> 00:41:36,560
from the demands
of the State and the military.

703
00:41:36,560 --> 00:41:42,400
Von Humboldt's educational reforms
valued mathematics for its own sake.

704
00:41:42,400 --> 00:41:46,280
In France, they got wonderful
mathematicians, like Joseph Fourier,

705
00:41:46,280 --> 00:41:49,480
whose work on sound waves
we still benefit from today.

706
00:41:49,480 --> 00:41:53,560
MP3 technology
is based on Fourier analysis.

707
00:41:53,560 --> 00:41:56,880
But in Germany,
they got, at least in my opinion,

708
00:41:56,880 --> 00:41:58,880
the greatest mathematician ever.

709
00:42:02,160 --> 00:42:04,120
Quaint and quiet,

710
00:42:04,120 --> 00:42:08,280
the university town of Gottingen
may seem like a bit of a backwater.

711
00:42:08,280 --> 00:42:12,200
But this little town has been home
to some of the giants of maths,

712
00:42:12,200 --> 00:42:14,520
including the man
who's often described

713
00:42:14,520 --> 00:42:19,560
as the Prince of Mathematics,
Carl Friedrich Gauss.

714
00:42:19,560 --> 00:42:23,440
Few non-mathematicians, however,
seem to know anything about him.

715
00:42:23,440 --> 00:42:25,240
Not in Paris.

716
00:42:25,240 --> 00:42:27,200
Qui s'appelle Carl Friedrich Gauss?

717
00:42:27,200 --> 00:42:29,080
Non. Non?

718
00:42:29,080 --> 00:42:30,680
'Not in Oxford.'

719
00:42:30,680 --> 00:42:34,640
I've heard the name but I couldn't
tell you. No idea. No idea? No.

720
00:42:34,640 --> 00:42:37,680
'And I'm afraid to say,
not even in modern Germany.'

721
00:42:37,680 --> 00:42:39,600
Nein. Nein? OK.

722
00:42:39,600 --> 00:42:41,240
I don't know. You don't know?

723
00:42:41,240 --> 00:42:44,800
But in Gottingen,
everyone knows who Gauss is.

724
00:42:44,800 --> 00:42:47,240
He's the local hero.

725
00:42:47,240 --> 00:42:49,640
His father was a stonemason

726
00:42:49,640 --> 00:42:52,760
and it's likely that Gauss
would have become one, too.

727
00:42:52,760 --> 00:42:55,920
But his singular talent
was recognised by his mother,

728
00:42:55,920 --> 00:42:57,760
and she helped ensure

729
00:42:57,760 --> 00:43:01,520
that he received
the best possible education.

730
00:43:01,520 --> 00:43:05,280
Every few years in the news,
you hear about a new prodigy

731
00:43:05,280 --> 00:43:08,440
who's passed their A-levels at ten,
gone to university at 12,

732
00:43:08,440 --> 00:43:10,440
but nobody compares to Gauss.

733
00:43:10,440 --> 00:43:13,880
Already at the age of 12, he
was criticising Euclid's geometry.

734
00:43:13,880 --> 00:43:17,160
At 15, he discovered
a new pattern in prime numbers

735
00:43:17,160 --> 00:43:20,440
which had eluded mathematicians
for 2,000 years.

736
00:43:20,440 --> 00:43:24,200
And at 19, he discovered the
construction of a 17-sided figure

737
00:43:24,200 --> 00:43:27,080
which nobody
had known before this time.

738
00:43:30,400 --> 00:43:34,360
His early successes
encouraged Gauss to keep a diary.

739
00:43:34,360 --> 00:43:36,320
Here at the University of Gottingen,

740
00:43:36,320 --> 00:43:40,200
you can still read it
if you can understand Latin.

741
00:43:40,200 --> 00:43:42,320
Fortunately, I had help.

742
00:43:44,400 --> 00:43:47,160
The first entry is in 1796.

743
00:43:47,160 --> 00:43:49,800
Is it possible to lift it up?
Yes, but be careful.

744
00:43:49,800 --> 00:43:54,360
It's really one of the most valuable
things that this library possesses.

745
00:43:54,360 --> 00:43:56,880
Yes, I can believe that.
He writes beautifully.

746
00:43:56,880 --> 00:43:59,320
It is aesthetically very pleasing,

747
00:43:59,320 --> 00:44:02,760
even if people don't understand
what it is.

748
00:44:02,760 --> 00:44:05,520
I'm going to put this down.
It's very delicate.

749
00:44:05,520 --> 00:44:08,720
The diary proves
that some of Gauss' ideas

750
00:44:08,720 --> 00:44:10,440
were 100 years ahead of their time.

751
00:44:10,440 --> 00:44:15,720
Here are some sines and integrals.
Very different sort of mathematics.

752
00:44:15,720 --> 00:44:20,600
Yes, this was the first
intimations of the theory

753
00:44:20,600 --> 00:44:25,240
of elliptic functions, which was
one of his other great developments.

754
00:44:25,240 --> 00:44:28,800
And here
you see something that is basically

755
00:44:28,800 --> 00:44:30,920
the Riemann zeta function appearing.

756
00:44:30,920 --> 00:44:34,400
Wow, gosh! That's very impressive.

757
00:44:34,400 --> 00:44:39,080
The zeta function has become a vital
element in our present understanding

758
00:44:39,080 --> 00:44:43,800
of the distribution of the building
blocks of all numbers, the primes.

759
00:44:43,800 --> 00:44:47,480
There is somewhere
in the diary here where he says,

760
00:44:47,480 --> 00:44:49,480
"I have made this wonderful discovery

761
00:44:49,480 --> 00:44:52,200
"and incidentally,
a son was born today."

762
00:44:52,200 --> 00:44:53,840
We see his priorities!

763
00:44:53,840 --> 00:44:55,760
Yes, indeed!

764
00:44:55,760 --> 00:44:58,800
I think I know
a few mathematicians like that, too.

765
00:45:00,520 --> 00:45:04,000
My priorities, though, for the
rest of the afternoon were clear.

766
00:45:04,000 --> 00:45:05,760
I needed another walk.

767
00:45:05,760 --> 00:45:09,160
Fortunately, Gottingen is
surrounded by good woodland trails.

768
00:45:09,160 --> 00:45:11,120
It was a perfect setting for me

769
00:45:11,120 --> 00:45:13,640
to think more about
Gauss' discoveries.

770
00:45:22,600 --> 00:45:26,480
Gauss' mathematics has touched many
parts of the mathematical world,

771
00:45:26,480 --> 00:45:31,520
but I'm going to just choose one of
them, a fun one - imaginary numbers.

772
00:45:31,520 --> 00:45:35,120
In the 16th and 17th century,
European mathematicians

773
00:45:35,120 --> 00:45:40,320
imagined the square root of
minus one and gave it the symbol i.

774
00:45:40,320 --> 00:45:42,960
They didn't like it much,
but it solved equations

775
00:45:42,960 --> 00:45:45,440
that couldn't be solved
any other way.

776
00:45:46,520 --> 00:45:49,960
Imaginary numbers have helped
us to understand radio waves,

777
00:45:49,960 --> 00:45:52,200
to build bridges and aeroplanes.

778
00:45:52,200 --> 00:45:54,440
They're even
the key to quantum physics,

779
00:45:54,440 --> 00:45:56,760
the science of the sub-atomic world.

780
00:45:56,760 --> 00:46:01,600
They've provided a map
to see how things really are.

781
00:46:01,600 --> 00:46:05,760
But back in the early 19th century,
they had no map, no picture

782
00:46:05,760 --> 00:46:08,760
of how imaginary numbers
connected with real numbers.

783
00:46:08,760 --> 00:46:10,960
Where is this new number?

784
00:46:10,960 --> 00:46:14,440
There's no room on the number line
for the square root of minus one.

785
00:46:14,440 --> 00:46:16,520
I've got the positive numbers
running out here,

786
00:46:16,520 --> 00:46:18,080
the negative numbers here.

787
00:46:18,080 --> 00:46:21,800
The great step is to create
a new direction of numbers,

788
00:46:21,800 --> 00:46:23,760
perpendicular to the number line,

789
00:46:23,760 --> 00:46:26,920
and that's where
the square root of minus one is.

790
00:46:29,080 --> 00:46:32,800
Gauss was not the first to come up
with this two-dimensional picture

791
00:46:32,800 --> 00:46:36,920
of numbers, but he was the first
person to explain it all clearly.

792
00:46:36,920 --> 00:46:38,960
He gave people
a picture to understand

793
00:46:38,960 --> 00:46:41,120
how imaginary numbers worked.

794
00:46:41,120 --> 00:46:43,280
And once they'd developed
this picture,

795
00:46:43,280 --> 00:46:46,400
their immense potential
could really be unleashed.

796
00:46:46,400 --> 00:46:49,880
Guten Morgen. Ein Kaffee, bitte.

797
00:46:49,880 --> 00:46:53,320
His maths led to a claim and
financial security for Gauss.

798
00:46:53,320 --> 00:46:56,560
He could have gone anywhere,
but he was happy enough

799
00:46:56,560 --> 00:47:01,880
to settle down and spend the rest
of his life in sleepy Gottingen.

800
00:47:01,880 --> 00:47:04,120
Unfortunately,
as his fame developed,

801
00:47:04,120 --> 00:47:06,280
so his character deteriorated.

802
00:47:06,280 --> 00:47:08,640
A naturally conservative, shy man,

803
00:47:08,640 --> 00:47:12,960
he became increasingly
distrustful and grumpy.

804
00:47:12,960 --> 00:47:16,800
Many young mathematicians across
Europe regarded Gauss as a god

805
00:47:16,800 --> 00:47:18,920
and they would send in
their theorems,

806
00:47:18,920 --> 00:47:20,920
their conjectures, even some proofs.

807
00:47:20,920 --> 00:47:23,760
But most of the time, he wouldn't
respond, and even when he did,

808
00:47:23,760 --> 00:47:26,680
it was generally to say
either that they'd got it wrong

809
00:47:26,680 --> 00:47:28,680
or he'd proved it already.

810
00:47:28,680 --> 00:47:32,800
His dismissal or lack of interest
in the work of lesser mortals

811
00:47:32,800 --> 00:47:35,560
sometimes discouraged some
very talented mathematicians

812
00:47:35,560 --> 00:47:38,320
from pursuing their ideas.

813
00:47:38,320 --> 00:47:40,440
But occasionally, Gauss also failed

814
00:47:40,440 --> 00:47:45,240
to follow up on his own insights,
including one very important insight

815
00:47:45,240 --> 00:47:48,440
that might have transformed
the mathematics of his time.

816
00:47:50,600 --> 00:47:53,840
15 kilometres outside Gottingen
stands what is known today

817
00:47:53,840 --> 00:47:55,840
as the Gauss Tower.

818
00:47:55,840 --> 00:47:58,160
Wow, that is stunning.

819
00:47:58,160 --> 00:48:01,840
It is really a fantastic view
here, yes.

820
00:48:01,840 --> 00:48:05,240
Gauss took on many projects
for the Hanoverian government,

821
00:48:05,240 --> 00:48:09,520
including the first proper survey
of all the lands of Hanover.

822
00:48:09,520 --> 00:48:12,760
Was this Gauss' choice
to do this surveying?

823
00:48:12,760 --> 00:48:16,320
For a mathematician, it sounds
like the last thing I'd want to do.

824
00:48:16,320 --> 00:48:17,520
He wanted to do it.

825
00:48:17,520 --> 00:48:23,480
The major point in doing this was
to discover the shape of the earth.

826
00:48:23,480 --> 00:48:25,480
But he also started speculating

827
00:48:25,480 --> 00:48:30,080
about something even more
revolutionary - the shape of space.

828
00:48:30,080 --> 00:48:34,920
So he's thinking there may not
be anything flat in the universe?

829
00:48:34,920 --> 00:48:37,480
Yes. And if we were living
in a curved universe,

830
00:48:37,480 --> 00:48:40,680
there wouldn't be anything flat.

831
00:48:40,680 --> 00:48:44,880
This led Gauss to question one of
the central tenets of mathematics -

832
00:48:44,880 --> 00:48:47,480
Euclid's geometry.

833
00:48:47,480 --> 00:48:50,360
He realised that this geometry,
far from universal,

834
00:48:50,360 --> 00:48:53,160
depended on the idea of space
as flat.

835
00:48:53,160 --> 00:48:56,360
It just didn't apply
to a universe that was curved.

836
00:48:56,360 --> 00:48:59,720
But in the early 19th century,
Euclid's geometry

837
00:48:59,720 --> 00:49:03,520
was seen as God-given
and Gauss didn't want any trouble.

838
00:49:03,520 --> 00:49:05,840
So he never published anything.

839
00:49:05,840 --> 00:49:09,400
Another mathematician, though,
had no such fears.

840
00:49:12,160 --> 00:49:16,200
In mathematics, it's often
helpful to be part of a community

841
00:49:16,200 --> 00:49:19,520
where you can talk to
and bounce ideas off others.

842
00:49:19,520 --> 00:49:22,360
But inside such
a mathematical community,

843
00:49:22,360 --> 00:49:25,600
it can sometimes be difficult
to come up with that one idea

844
00:49:25,600 --> 00:49:28,960
that completely challenges
the status quo,

845
00:49:28,960 --> 00:49:33,760
and then the breakthrough
often comes from somewhere else.

846
00:49:33,760 --> 00:49:37,040
Mathematics can be done
in some pretty weird places.

847
00:49:37,040 --> 00:49:38,640
I'm in Transylvania,

848
00:49:38,640 --> 00:49:42,240
which is fairly appropriate,
cos I'm in search of a lone wolf.

849
00:49:42,240 --> 00:49:45,400
Janos Bolyai spent much of his life

850
00:49:45,400 --> 00:49:49,720
hundreds of miles away from the
mathematical centres of excellence.

851
00:49:49,720 --> 00:49:53,800
This is the only portrait of him
that I was able to find.

852
00:49:53,800 --> 00:49:56,800
Unfortunately,
it isn't actually him.

853
00:49:56,800 --> 00:50:00,240
It's one that the Communist Party
in Romania started circulating

854
00:50:00,240 --> 00:50:04,200
when people got interested
in his theories in the 1960s.

855
00:50:04,200 --> 00:50:06,680
They couldn't find a picture
of Janos.

856
00:50:06,680 --> 00:50:09,720
So they substituted a picture
of somebody else instead.

857
00:50:12,000 --> 00:50:15,720
Born in 1802, Janos
was the son of Farkas Bolyai,

858
00:50:15,720 --> 00:50:17,320
who was a maths teacher.

859
00:50:17,320 --> 00:50:20,600
He realised his son
was a mathematical prodigy,

860
00:50:20,600 --> 00:50:23,920
so he wrote to his old friend
Carl Friedrich Gauss,

861
00:50:23,920 --> 00:50:25,840
asking him to tutor the boy.

862
00:50:25,840 --> 00:50:29,080
Sadly, Gauss declined.

863
00:50:29,080 --> 00:50:31,960
So instead of becoming
a professional mathematician,

864
00:50:31,960 --> 00:50:34,120
Janos joined the Army.

865
00:50:34,120 --> 00:50:37,280
But mathematics
remained his first love.

866
00:50:40,880 --> 00:50:44,520
Maybe there's something about the
air here because Bolyai carried on

867
00:50:44,520 --> 00:50:46,920
doing his mathematics
in his spare time.

868
00:50:46,920 --> 00:50:50,560
He started to explore what
he called imaginary geometries,

869
00:50:50,560 --> 00:50:55,240
where the angles in triangles
add up to less than 180.

870
00:50:55,240 --> 00:50:58,440
The amazing thing
is that these imaginary geometries

871
00:50:58,440 --> 00:51:00,920
make perfect mathematical sense.

872
00:51:04,720 --> 00:51:09,480
Bolyai's new geometry has become
known as hyperbolic geometry.

873
00:51:09,480 --> 00:51:13,000
The best way to imagine it is
a kind of mirror image of a sphere

874
00:51:13,000 --> 00:51:15,640
where lines curve back
on each other.

875
00:51:15,640 --> 00:51:18,520
It's difficult to represent it
since we are so used

876
00:51:18,520 --> 00:51:21,880
to living in space which appears
to be straight and flat.

877
00:51:24,000 --> 00:51:25,680
In his hometown of Targu Mures,

878
00:51:25,680 --> 00:51:29,800
I went looking for more about
Bolyai's revolutionary mathematics.

879
00:51:29,800 --> 00:51:33,240
His memory
is certainly revered here.

880
00:51:33,240 --> 00:51:36,960
The museum contains a collection
of Bolyai-related artefacts,

881
00:51:36,960 --> 00:51:40,720
some of which might be
considered distinctly Transylvanian.

882
00:51:40,720 --> 00:51:42,680
It's still got some hair on it.

883
00:51:42,680 --> 00:51:45,360
It's kind of a little bit gruesome.

884
00:51:45,360 --> 00:51:46,960
But the object I like most here

885
00:51:46,960 --> 00:51:50,600
is a beautiful model
of Bolyai's geometry.

886
00:51:50,600 --> 00:51:54,200
You got the shortest distance
between here and here

887
00:51:54,200 --> 00:51:56,960
if you stick on this surface.
It's not a straight line,

888
00:51:56,960 --> 00:51:59,360
but this curved line which
of bends into the triangle.

889
00:51:59,360 --> 00:52:03,960
Here is a surface where the shortest
distances which define the triangle

890
00:52:03,960 --> 00:52:06,240
add up to less than 180.

891
00:52:06,240 --> 00:52:09,640
Bolyai published his work in 1831.

892
00:52:09,640 --> 00:52:12,560
His father sent
his old friend Gauss a copy.

893
00:52:12,560 --> 00:52:16,480
Gauss wrote back straightaway
giving his approval,

894
00:52:16,480 --> 00:52:19,640
but Gauss refused
to praise the young Bolyai,

895
00:52:19,640 --> 00:52:22,760
because he said the person
he should be praising was himself.

896
00:52:22,760 --> 00:52:26,400
He had worked it all out
a decade or so before.

897
00:52:26,400 --> 00:52:29,960
Actually,
there is a letter from Gauss

898
00:52:29,960 --> 00:52:32,400
to another friend
of his where he says,

899
00:52:32,400 --> 00:52:35,040
"I regard this young geometer boy

900
00:52:35,040 --> 00:52:38,160
"as a genius of the first order."

901
00:52:38,160 --> 00:52:41,760
But Gauss never thought
to tell Bolyai that.

902
00:52:41,760 --> 00:52:44,720
And young Janos
was completely disheartened.

903
00:52:44,720 --> 00:52:47,240
Another body blow soon followed.

904
00:52:47,240 --> 00:52:50,080
Somebody else had developed
exactly the same idea,

905
00:52:50,080 --> 00:52:52,200
but had published two years
before him -

906
00:52:52,200 --> 00:52:55,280
the Russian mathematician
Nicholas Lobachevsky.

907
00:52:57,760 --> 00:53:00,280
It was all downhill
for Bolyai after that.

908
00:53:00,280 --> 00:53:04,280
With no recognition or career,
he didn't publish anything else.

909
00:53:04,280 --> 00:53:07,160
Eventually, he went a little crazy.

910
00:53:08,640 --> 00:53:13,360
In 1860,
Janos Bolyai died in obscurity.

911
00:53:15,480 --> 00:53:19,240
Gauss, by contrast,
was lionised after his death.

912
00:53:19,240 --> 00:53:22,760
A university, the units used
to measure magnetic induction,

913
00:53:22,760 --> 00:53:25,720
even a crater on the moon
would be named after him.

914
00:53:28,960 --> 00:53:31,800
During his lifetime,
Gauss lent his support

915
00:53:31,800 --> 00:53:34,160
to very few mathematicians.

916
00:53:34,160 --> 00:53:39,040
But one exception was another
of Gottingen's mathematical giants -

917
00:53:39,040 --> 00:53:42,040
Bernhard Riemann.

918
00:53:48,480 --> 00:53:50,000
His father was a minister

919
00:53:50,000 --> 00:53:54,280
and he would remain a sincere
Christian all his life.

920
00:53:54,280 --> 00:53:58,480
But Riemann grew up a shy boy
who suffered from consumption.

921
00:53:58,480 --> 00:54:00,840
His family was large and poor
and the only thing

922
00:54:00,840 --> 00:54:04,760
the young boy had going for him
was an excellence at maths.

923
00:54:04,760 --> 00:54:07,920
That was his salvation.

924
00:54:07,920 --> 00:54:11,440
Many mathematicians like Riemann
had very difficult childhoods,

925
00:54:11,440 --> 00:54:15,160
were quite unsociable. Their
lives seemed to be falling apart.

926
00:54:15,160 --> 00:54:19,000
It was mathematics
that gave them a sense of security.

927
00:54:22,120 --> 00:54:25,000
Riemann spent much of his early life
in the town of Luneburg

928
00:54:25,000 --> 00:54:27,040
in northern Germany.

929
00:54:27,040 --> 00:54:30,640
This was his local school,
built as a direct result

930
00:54:30,640 --> 00:54:34,480
of Humboldt's educational
reforms in the early 19th century.

931
00:54:34,480 --> 00:54:37,240
Riemann was one of its first pupils.

932
00:54:37,240 --> 00:54:41,560
The head teacher saw a way
of bringing out the shy boy.

933
00:54:41,560 --> 00:54:44,520
He was given the freedom
of the school's library.

934
00:54:44,520 --> 00:54:47,080
It opened up
a whole new world to him.

935
00:54:47,080 --> 00:54:48,880
One of the books he found in there

936
00:54:48,880 --> 00:54:51,680
was a book by the
French mathematician Legendre,

937
00:54:51,680 --> 00:54:53,200
all about number theory.

938
00:54:53,200 --> 00:54:55,880
His teacher asked him how
he was getting on with it.

939
00:54:55,880 --> 00:55:01,560
He replied, "I have understood all
859 pages of this wonderful book."

940
00:55:01,560 --> 00:55:04,720
It was a strategy
that obviously suited Riemann

941
00:55:04,720 --> 00:55:07,280
because he became
a brilliant mathematician.

942
00:55:07,280 --> 00:55:12,480
One of his most famous contributions
to mathematics was a lecture in 1852

943
00:55:12,480 --> 00:55:16,600
on the foundations of geometry.
In the lecture,

944
00:55:16,600 --> 00:55:20,320
Riemann first described
what geometry actually was

945
00:55:20,320 --> 00:55:22,360
and its relationship with the world.

946
00:55:22,360 --> 00:55:25,440
He then sketched out
what geometry could be -

947
00:55:25,440 --> 00:55:28,440
a mathematics of many different
kinds of space,

948
00:55:28,440 --> 00:55:31,440
only one of which would be
the flat Euclidian space

949
00:55:31,440 --> 00:55:33,080
in which we appear to live.

950
00:55:33,080 --> 00:55:36,280
He was just 26 years old.

951
00:55:36,280 --> 00:55:40,760
Was it received well?
Did people recognise the revolution?

952
00:55:40,760 --> 00:55:43,040
There was no way
that people could actually

953
00:55:43,040 --> 00:55:45,240
make these ideas concrete.

954
00:55:45,240 --> 00:55:50,840
That only occurred 50,
60 years after this, with Einstein.

955
00:55:50,840 --> 00:55:53,600
So this is the beginning,
really, of the revolution

956
00:55:53,600 --> 00:55:57,160
which ends with Einstein's
relativity. Exactly.

957
00:55:57,160 --> 00:56:01,840
Riemann's mathematics
changed how we see the world.

958
00:56:01,840 --> 00:56:04,600
Suddenly, higher dimensional
geometry appeared.

959
00:56:04,600 --> 00:56:06,840
The potential was there
from Descartes,

960
00:56:06,840 --> 00:56:11,320
but it was Riemann's imagination
that made it happen.

961
00:56:11,320 --> 00:56:15,360
He began without putting
any restriction

962
00:56:15,360 --> 00:56:18,880
on the dimensions whatsoever.
This was something quite new,

963
00:56:18,880 --> 00:56:21,520
his way of thinking about things.

964
00:56:21,520 --> 00:56:25,000
Someone like Bolyai was really
thinking about new geometries,

965
00:56:25,000 --> 00:56:27,120
but new two-dimensional geometries.

966
00:56:27,120 --> 00:56:30,360
New two-dimensional geometries.
Riemann then broke away

967
00:56:30,360 --> 00:56:35,440
from all the limitations
of two or three dimensions

968
00:56:35,440 --> 00:56:38,080
and began to think
in in higher dimensions.

969
00:56:38,080 --> 00:56:39,600
And this was quite new.

970
00:56:39,600 --> 00:56:42,160
Multi-dimensional space
is at the heart

971
00:56:42,160 --> 00:56:44,720
of so much mathematics done today.

972
00:56:44,720 --> 00:56:48,280
In geometry, number theory,
and several other branches of maths,

973
00:56:48,280 --> 00:56:52,000
Riemann's ideas
still perplex and amaze.

974
00:56:52,960 --> 00:56:56,120
He died, though, in 1866.

975
00:56:56,120 --> 00:56:59,680
He was only 39 years old.

976
00:56:59,680 --> 00:57:03,160
Today, the results of Riemann's
mathematics are everywhere.

977
00:57:03,160 --> 00:57:07,720
Hyperspace is no longer
science fiction, but science fact.

978
00:57:07,720 --> 00:57:11,480
In Paris, they have even tried
to visualise what shapes

979
00:57:11,480 --> 00:57:14,080
in higher dimensions
might look like.

980
00:57:15,880 --> 00:57:18,840
Just as the Renaissance artist
Piero would have drawn a square

981
00:57:18,840 --> 00:57:23,080
inside a square to represent a
cube on the two-dimensional canvas,

982
00:57:23,080 --> 00:57:27,560
the architect here at La Defense
has built a cube inside a cube

983
00:57:27,560 --> 00:57:31,920
to represent a shadow
of the four-dimensional hypercube.

984
00:57:31,920 --> 00:57:34,840
It is with Riemann's work
that we finally have

985
00:57:34,840 --> 00:57:37,320
the mathematical glasses
to be able to explore

986
00:57:37,320 --> 00:57:39,560
such worlds of the mind.

987
00:57:42,680 --> 00:57:45,120
It's taken a while
to make these glasses fit,

988
00:57:45,120 --> 00:57:47,520
but without this golden age
of mathematics,

989
00:57:47,520 --> 00:57:50,680
from Descartes to Riemann,
there would be no calculus,

990
00:57:50,680 --> 00:57:55,440
no quantum physics, no relativity,
none of the technology we use today.

991
00:57:55,440 --> 00:57:57,640
But even more important than that,

992
00:57:57,640 --> 00:58:01,000
their mathematics blew away
the cobwebs

993
00:58:01,000 --> 00:58:04,720
and allowed us to see the world
as it really is -

994
00:58:04,720 --> 00:58:07,880
a world much stranger
than we ever thought.

995
00:58:11,280 --> 00:58:13,600
You can learn more about
the story of maths

996
00:58:13,600 --> 00:58:16,200
at the Open University at:

997
00:58:26,880 --> 00:58:29,640
Subtitles by Red Bee Media Ltd

998
00:58:29,640 --> 00:58:33,520
Email subtitling@bbc.co.uk