1
00:00:17,770 --> 00:00:22,540
So welcome. In this talk, I aim to cover the entire history of mathematics in just one hour,
2
00:00:24,010 --> 00:00:29,440
illustrating my story with some of the 1500 mathematical stamps from around the world in my collection.
3
00:00:31,780 --> 00:00:38,770
Now. I usually start by mentioning a set of ten stamps issued by Nicaragua in 1971
4
00:00:38,980 --> 00:00:43,810
and entitled Ten Mathematical Formulas That Changed the face of the Earth.
5
00:00:44,800 --> 00:00:48,190
As you can see, these range from one plus one equals two,
6
00:00:48,580 --> 00:01:00,700
and the Pythagorean theorem via Archimedes Law of moments and in a Newton's inverse square law ending up with Einstein's law equals M.C. squared.
7
00:01:00,760 --> 00:01:06,480
You'll see some of these again later on. So let's go back in time.
8
00:01:06,810 --> 00:01:13,380
My treatment is chronological, going back to the time when people first began to count and measure the objects around them.
9
00:01:14,460 --> 00:01:22,380
Early methods included forming stones into piles, cutting notches in sticks and especially finger counting,
10
00:01:22,770 --> 00:01:26,250
which led eventually to our familiar decimal number system.
11
00:01:29,580 --> 00:01:34,410
Also in several places we find ancient geometrical arrangements of stones,
12
00:01:35,040 --> 00:01:41,640
such as the circular pattern of megaliths at Stonehenge and the linear arrangements at Carnac in Brittany.
13
00:01:42,960 --> 00:01:50,940
We don't know their exact purpose, but they may have celebrated astronomical events such as Sunrise on Midsummer's Day.
14
00:01:55,780 --> 00:02:02,290
So we'll begin our journey with the mathematical achievements of Egypt, Mesopotamia and Greece, all shown on this map.
15
00:02:03,250 --> 00:02:07,270
Here's Mesopotamia. There's Egypt and Greece.
16
00:02:07,450 --> 00:02:14,870
Further up. In Egypt, the main achievements involve the practical scale of measurement.
17
00:02:16,630 --> 00:02:24,550
The magnificent pyramids designed as tombs for their readers, for their rulers, attest to the Egyptians extremely accurate measuring abilities.
18
00:02:25,480 --> 00:02:30,460
And the oldest of these at the top is King Djoser Step Pyramid in Saqqara.
19
00:02:31,240 --> 00:02:35,680
Built in horizontal layers and dating from about 2700 B.C.
20
00:02:37,850 --> 00:02:46,910
Rather better known as the Great Pyramid of Cheops and Giza, constructed from over 2 million large blocks of stone and amazingly,
21
00:02:47,120 --> 00:02:55,130
the size of its square base, which about 230 metres long agree in length to within 100th of 1%.
22
00:02:58,120 --> 00:03:03,100
Our knowledge of Egyptian mathematics comes mainly from two fragile primary sources.
23
00:03:03,550 --> 00:03:08,320
The Moscow and Ryan Papyrus is similar to the papyrus you can see here.
24
00:03:10,980 --> 00:03:16,350
The Moscow and Rhine for passing through tables of fractions and problems in arithmetic and geometry.
25
00:03:16,560 --> 00:03:19,800
Probably probably designed for the teaching of scribes.
26
00:03:21,150 --> 00:03:25,530
On the right you can see the Egyptian God of reckoning self.
27
00:03:30,670 --> 00:03:39,910
Moving to Mesopotamia. Early examples of mathematical writing also appeared there between the rivers, Tigris and Euphrates in present day Iraq.
28
00:03:41,440 --> 00:03:50,110
Top left, you can see an accounting tablet from about 3000 B.C., where the thumbnail indentations at the top represent numbers.
29
00:03:52,910 --> 00:04:00,490
Later the Mesopotamians wrote their mathematics with a wedge shaped styrofoam get on damp clay, which was then baked in the sun.
30
00:04:02,440 --> 00:04:10,680
Many thousands of these cuneiform tablets survive and show basic arithmetic and the calculation of areas and volumes in geometry.
31
00:04:11,840 --> 00:04:14,930
The solving of what we now call linear and quadratic equations.
32
00:04:15,830 --> 00:04:24,580
And an extremely accurate value for the square root of two. The number system was based on 60, which we still use for measuring time.
33
00:04:25,630 --> 00:04:31,150
And they also studied astronomy and observe comets such as the one we now call Halley's Comet.
34
00:04:31,690 --> 00:04:36,970
Although the date of two, three, four, nine B.C. is about 2000 years out.
35
00:04:41,370 --> 00:04:47,050
Let's now move to more familiar territory. From about 600 B.C.
36
00:04:47,080 --> 00:04:50,830
Mathematics flourished in the Greek speaking world of the Eastern Mediterranean.
37
00:04:52,200 --> 00:05:00,840
During this time, the Greeks developed deductive logic, logical reasoning, which became the hallmark of their work, especially in geometry.
38
00:05:02,900 --> 00:05:11,330
Pythagoras, a semi legendary figure, formed a school in Cortona to further the study of mathematics, philosophy and natural science.
39
00:05:12,860 --> 00:05:21,470
His followers supposedly believe that all is number, quantifying many things and emphasising the four mathematical arts of arithmetic,
40
00:05:21,920 --> 00:05:26,210
geometry, astronomy and music, which will meet again later.
41
00:05:28,520 --> 00:05:32,209
We do not know who first proved the Pythagorean theorem that the area of the square
42
00:05:32,210 --> 00:05:35,930
on the hypotenuse is the sum of the areas of the squares on the other two sides.
43
00:05:36,680 --> 00:05:42,710
But it's connection with right angle triangles was already known to the Mesopotamians a thousand years earlier.
44
00:05:44,770 --> 00:05:52,240
The blue stamp was issued very recently in August for the International Congress of Mathematicians in Korea.
45
00:05:57,160 --> 00:06:01,780
The scene then moved to Athens, which became the most important intellectual centre in Greece.
46
00:06:02,290 --> 00:06:05,200
Numbering among its scholars, Plato and Aristotle.
47
00:06:07,340 --> 00:06:17,630
In 387 B.C., Plato founded a school in a suburb of Athens called Academy, and Plato's Academy soon became the focal point for mathematical study.
48
00:06:18,850 --> 00:06:24,370
It said that over the entrance appeared the inscription that no one ignorant of geometry entered here,
49
00:06:24,880 --> 00:06:28,450
like the Greek inscription at the entrance to this very building.
50
00:06:31,220 --> 00:06:38,630
In Raphael's fresco, the school of Athens. On the left, Plato and Aristotle appear together on the steps of the academy.
51
00:06:41,340 --> 00:06:46,560
Plato believed that mathematics and philosophy were essential training for those holding positions of responsibility,
52
00:06:47,490 --> 00:06:55,200
discussing the mathematical arts of arithmetic, geometry, astronomy and music, as I mentioned before, and justifying their importance.
53
00:06:56,750 --> 00:07:01,400
He also described the five regular or platonic solids shown here in the middle.
54
00:07:03,610 --> 00:07:07,180
His pupil, Aristotle, helped to formalise the study of deductive reasoning,
55
00:07:07,570 --> 00:07:16,270
explaining why the square root of two cannot be written as a fraction, and also discussing logical syllogism such as All men are mortal.
56
00:07:16,540 --> 00:07:20,110
Socrates is a man, therefore Socrates is mortal.
57
00:07:23,990 --> 00:07:27,850
Around 300 B.C. following the military successes of Alexander the Great
58
00:07:28,400 --> 00:07:33,710
Mathematical Activity moved to Alexandria in the Egyptian part of the Greek world.
59
00:07:35,750 --> 00:07:39,440
The first important mathematician there was Euclid, shown on the right.
60
00:07:39,920 --> 00:07:41,780
Mainly remembered for his elements.
61
00:07:43,840 --> 00:07:52,840
This most influential compilation of known results consisted of 13 books on plain and solid geometry, arithmetic and number theory.
62
00:07:54,620 --> 00:07:56,120
A model of deductive reasoning.
63
00:07:56,120 --> 00:08:04,730
It started from initial axioms and postulates and used rules of deduction to derive new results in a logical and systematic order.
64
00:08:09,080 --> 00:08:13,400
And we mustn't forget Archimedes from Syracuse in Sicily.
65
00:08:13,700 --> 00:08:22,880
He was one of the greatest mathematicians of all time. Around 250 B.C., he calculated volumes of solids such as spheres and cylinders.
66
00:08:23,990 --> 00:08:27,260
He listed the 13 Archimedes and or Semi-regular Solids.
67
00:08:27,650 --> 00:08:30,950
These are polyhedra with regular faces, but of different types.
68
00:08:32,000 --> 00:08:39,110
And he also obtained a good approximation for PI by considering 96 sided polygons that approximate a circle.
69
00:08:41,610 --> 00:08:48,209
In Mechanics, he found the law of moments for a balance that you can see on the right and is
70
00:08:48,210 --> 00:08:53,070
also credited with inventing the Archimedes screw for raising water from a river.
71
00:08:54,410 --> 00:08:59,660
In statics, he stated Archimedes principle on the weight of an object immersed in water.
72
00:09:00,140 --> 00:09:05,810
As you can see in the middle and he used it to test the purity of King Herod Gold crown.
73
00:09:07,460 --> 00:09:15,770
On discovering the principle he supposedly jumped out of his bath and ran narrate, ran naked through the street shouting, Eureka, I have found it.
74
00:09:16,670 --> 00:09:20,270
But sadly, no stamp has ever been issued featuring that particular episode.
75
00:09:25,490 --> 00:09:32,990
We now move to China, where the mathematics dates back some 3000 years, around 220 B.C. in particular.
76
00:09:33,230 --> 00:09:38,090
The Great Wall was a major triumph of engineering, skill and mathematical calculation.
77
00:09:40,760 --> 00:09:46,160
Most ancient Greek, most ancient Chinese mathematics was written on bamboos bamboo strips.
78
00:09:46,940 --> 00:09:51,710
And you can see an example top left showing a nine by nine multiplication table.
79
00:09:53,510 --> 00:09:57,550
For their calculations. The Chinese used a form of counting board shown.
80
00:09:57,860 --> 00:10:08,660
Bottom left a box with separate compartments for units tens, hundreds, etc., into which small bamboo rods were placed in order to do the calculations.
81
00:10:11,640 --> 00:10:17,430
Chinese mathematicians are fascinated by PI around A.D. 100.
82
00:10:17,610 --> 00:10:22,440
Zhang Heng, who is the first of the three Chinese mathematicians you see up there?
83
00:10:23,100 --> 00:10:26,250
Zhang Heng, inventor of a seismograph for measuring earthquakes.
84
00:10:26,730 --> 00:10:33,600
He gave the value square root of ten, which is about 3.16, which was a popular approximation in many areas.
85
00:10:35,550 --> 00:10:50,530
Later to a. In A.D. 263, showing in the middle of their considered polygons with 192 sighs to deduce that pie lies between 3.1410 and 3.1427.
86
00:10:50,950 --> 00:10:55,929
And here you can see in the middle Liu, who has nine chapter of a mathematical art.
87
00:10:55,930 --> 00:11:06,730
His edition of that. Then around the 500 top, writes Cianci and his son.
88
00:11:07,000 --> 00:11:14,650
Consider Polygon's with about 25,000 sides to obtain pi to six decimal places.
89
00:11:15,770 --> 00:11:21,530
And they also found the value 355 over 113 also correct to six decimal places.
90
00:11:22,400 --> 00:11:27,020
And this approximation wasn't rediscovered in the West until the 16th century.
91
00:11:29,480 --> 00:11:35,600
But a similar situation occurred with pascals triangle of binomial coefficients shown here, bottom right,
92
00:11:36,590 --> 00:11:42,800
which had appeared in a Chinese text of 1303, although in fact it was known earlier than that elsewhere.
93
00:11:43,430 --> 00:11:46,640
But even so, this was 400 years before Pascal.
94
00:11:51,040 --> 00:11:55,810
An ancient Chinese story of perhaps 1100 B.C. concerns Emperor Yu,
95
00:11:56,320 --> 00:12:02,680
who is standing on the banks of the River Luo when a turtle emerged with 1 to 9 on its back,
96
00:12:03,970 --> 00:12:08,440
with the numbers in each row column and diagonal adding to the same some 15.
97
00:12:10,380 --> 00:12:18,240
This arrangement of numbers, the Horseshoe acquired great religious and mystic significance and eventually led to much
98
00:12:18,840 --> 00:12:25,650
to larger such magic squares as pictured on these Macao stamps issued only last month.
99
00:12:27,970 --> 00:12:39,310
Next year, three more stamps will be issued by Michael with the values 816 so that the stamp values themselves will form the new shoe magic square.
100
00:12:39,580 --> 00:12:43,780
492357816.
101
00:12:44,410 --> 00:12:47,320
They obviously have a mathematician in the post office.
102
00:12:51,980 --> 00:13:01,280
India mathematics can be traced back to about 600 B.C. when Vedic manuscripts presented early work on arithmetic permutations and combinations.
103
00:13:01,670 --> 00:13:05,210
Number Theory. And the extraction of square roots.
104
00:13:07,040 --> 00:13:13,760
And around 250 BCE. King Ashoka ruler of most of India, became the first Buddhist monarch.
105
00:13:14,180 --> 00:13:22,130
And in celebration, pillars were constructed. You can see one on the left, and then above you can see the capstone at the top of it,
106
00:13:23,360 --> 00:13:27,140
showing the earliest examples of what became our Hindu Arabic numerals,
107
00:13:28,160 --> 00:13:33,620
a decimal place value system with the position of each digit 0 to 9 indicating
108
00:13:33,620 --> 00:13:38,269
its value and enabling one to carry out calculations one column at a time,
109
00:13:38,270 --> 00:13:49,340
as we do now. The first outstanding Indian mathematician was Aryabhata around the year 500 who showed how to sum arithmetic,
110
00:13:49,520 --> 00:13:53,090
arithmetic progressions and also to some natural numbers.
111
00:13:53,210 --> 00:13:54,740
And this squares and cubes.
112
00:13:55,650 --> 00:14:04,920
There's no stamp featuring Aryabhata, but there is one showing the first Indian space satellite, which was named Aryabhata in his honour.
113
00:14:07,650 --> 00:14:13,290
In later years, Indian mathematicians and astronomers constructed magnificent observatories,
114
00:14:13,290 --> 00:14:17,610
such as you can see on the right, such as the Jantar Mantar in Jaipur.
115
00:14:18,750 --> 00:14:22,140
Comprising 14 massive measuring instruments.
116
00:14:22,710 --> 00:14:27,510
It included the 90 foot samrat Yamcha, the world's largest sundial.
117
00:14:33,600 --> 00:14:39,260
We move now to the mathematics of the Arabs and Persians. United by their new religion.
118
00:14:39,860 --> 00:14:42,710
And with Baghdad lying on the East-West trade routes.
119
00:14:43,310 --> 00:14:52,700
Islamic scholars from 750 to 1200 developed Greek and Roman writings from the West and Hindu writings from India.
120
00:14:54,590 --> 00:14:59,720
And some of our terminology dates from this period. The word algorithm.
121
00:15:00,470 --> 00:15:05,030
A step by step procedure for solving a problem comes from Al-Khwarizmi.
122
00:15:05,660 --> 00:15:13,850
Shown on the left a Persian method mathematicians who lived in Baghdad and wrote influential works on arithmetic and algebra.
123
00:15:15,790 --> 00:15:20,560
His arithmetic introduced the Indian decimal place value system to the Islamic world.
124
00:15:21,340 --> 00:15:28,180
While the title of his algebra book, Kitab Algebra while Kabbalah gives us the word algebra.
125
00:15:28,390 --> 00:15:35,050
Algebra means adding a positive term to both sides of an equation to eliminate negative terms.
126
00:15:38,340 --> 00:15:41,940
Also influential among Islamic works were those of the geometry.
127
00:15:42,180 --> 00:15:47,550
Even a high time in the middle. Known in the West as al-Hassan.
128
00:15:48,300 --> 00:15:49,860
He mainly worked in optics,
129
00:15:50,670 --> 00:16:00,420
and a celebrated problem is al problem where on a spherical mirror is light from a given point source reflected into the eye of a given observer.
130
00:16:03,420 --> 00:16:09,329
A century later, the mathematician and poet Omar Khayyam on the right wrote a celebrated algebra text with
131
00:16:09,330 --> 00:16:14,910
the first systematic classification of cubic equations and a discussion of their solution.
132
00:16:15,870 --> 00:16:20,220
They weren't solved in general until 16th century Italy.
133
00:16:22,220 --> 00:16:28,610
And look, I am also publicly criticised an attempt by Al Hosn to prove Euclid's so-called parallel postulate,
134
00:16:29,060 --> 00:16:32,180
which I'll be explaining later in the West.
135
00:16:32,720 --> 00:16:35,900
He's mainly remembered for his collection of poems, The Rubaiyat.
136
00:16:40,120 --> 00:16:48,579
A later unsuccessful attempt to prove Euclid's parallel postulate was by the Persian mathematician Nasser el-Din Al See shown on
137
00:16:48,580 --> 00:16:57,250
this magnificent stamp there on the left who wrote substantial works on astronomy and constructed the first modern observatory.
138
00:16:58,270 --> 00:17:01,510
He also wrote influential treatises on logic and theology,
139
00:17:02,290 --> 00:17:09,520
and his extensive investigations into plain and spherical trigonometry included the sine rule for triangles.
140
00:17:14,350 --> 00:17:22,480
During the eighth and ninth centuries, the Islamic world spread across the top of Africa and then up into Europe through Spain and Italy.
141
00:17:23,930 --> 00:17:32,210
Cordova became the scientific capital of Europe, while Islamic decorative arts and architecture spread through southern Spain and Portugal,
142
00:17:32,750 --> 00:17:37,860
and included the magnificent geometrical arches in the corner of a mosque, which you can see on the left.
143
00:17:38,870 --> 00:17:42,440
And the tiling patterns in the Alhambra. In Granada.
144
00:17:42,830 --> 00:17:54,330
On the top. Right. Also in Cordova, the John Ritter and astronomer al-Zarqawi, shown in the middle, at the bottom, constructed many astrolabes.
145
00:17:55,940 --> 00:18:00,019
This consisted of a brass disk suspended by a ring that's fixed or held in the
146
00:18:00,020 --> 00:18:06,200
hand with an attached rotating bar and a circular scale on the rim and so on.
147
00:18:06,530 --> 00:18:13,070
To measure the altitude of a planet or star, you view it along the bar and then read the results from the scale.
148
00:18:18,290 --> 00:18:24,140
But in Europe, the period from five 500 to 1000 was known as the Dark Ages.
149
00:18:25,340 --> 00:18:30,680
The legacy of the ancient world was largely forgotten and the level of culture was low.
150
00:18:32,760 --> 00:18:36,300
But revival of interest in mathematics began with Chabot of Horik,
151
00:18:36,840 --> 00:18:41,040
who trained in Catalonia and introduced the Hindu Arabic numerals to Christian Europe.
152
00:18:41,550 --> 00:18:47,190
Using an abacus he designed for the purpose a major figure in the church.
153
00:18:47,490 --> 00:18:50,610
He was crowned pope in the year 999.
154
00:18:51,450 --> 00:18:58,310
That doesn't happen to many mathematicians. To operate.
155
00:18:59,000 --> 00:19:00,920
The Catalan mystic, Ramon Law,
156
00:19:01,190 --> 00:19:10,130
believes that all knowledge can be obtained monetarily by taking mathematical combinations of God's divine attributes such as power,
157
00:19:10,430 --> 00:19:22,130
wisdom, goodness, etc. His commentary, ideas later spread throughout throughout Europe, influencing such mathematicians as Messiaen and Leibniz.
158
00:19:24,440 --> 00:19:32,210
And the surprise appearance here is Geoffrey Chaucer, author of The Canterbury Tales, who also wrote on the Astrolabe.
159
00:19:33,470 --> 00:19:36,350
This is one of the earliest scientific books in English.
160
00:19:41,530 --> 00:19:48,970
The Hindu Arabic numbers were also popularised by Fibonacci or Leonardo of Pisa in his Lieber Obaseki Book of calculation.
161
00:19:50,260 --> 00:19:56,110
This famous work of 1202 contained many problems in arithmetic and algebra,
162
00:19:56,770 --> 00:20:02,050
including his rabbit's problem on the left that leads to the Fibonacci sequence.
163
00:20:02,290 --> 00:20:09,340
One, one, two, three, five, eight, 13 and so on, in which each successive term is the sum of the previous two.
164
00:20:11,550 --> 00:20:17,070
These numbers also arise in the arrangements of seeds and sunflowers and pine cones, for example.
165
00:20:17,460 --> 00:20:24,660
There's a sunflower head there, and if you count round the spirals, you get numbers like 55 and 89 appearing.
166
00:20:27,470 --> 00:20:31,490
The ratios of successive terms of the Fibonacci sequence approach limit.
167
00:20:31,580 --> 00:20:39,500
The golden ratio, which is about 1.618, is the ratio of a diagonal and a side of a regular pentagon.
168
00:20:40,490 --> 00:20:48,410
And also a rectangle with size in this ratio is sometimes considered to have the most pleasing shape not too thin and not too fat.
169
00:20:50,430 --> 00:20:53,860
As shown on the Swiss stamp top lie in the middle.
170
00:20:54,670 --> 00:20:58,000
Removing a square from a golden rectangle leaves another one.
171
00:20:58,750 --> 00:21:04,120
And this stamp also features the logarithmic spiral found on the north of the shell below.
172
00:21:06,420 --> 00:21:08,160
But also in the set for Michael.
173
00:21:09,450 --> 00:21:18,480
It's a stamp featuring a Penrose tiling, which I thought I ought to show you today, consisting of its two stacked two shapes, the kite and the dart.
174
00:21:19,290 --> 00:21:24,960
And if you want to see a Penrose tiling, you don't have to go very far. There's a big one just outside this building, as you know.
175
00:21:29,450 --> 00:21:32,420
The Renaissance in mathematical learning and the mathematic mathematical.
176
00:21:33,440 --> 00:21:40,520
So the Renaissance in mathematical learning in the Middle Ages was mainly due to three things the establishment of universities.
177
00:21:41,460 --> 00:21:46,140
The translation of Arabic text into Latin and the invention of printing.
178
00:21:47,610 --> 00:21:51,930
The first European University was Bologna. Schoen Top Left.
179
00:21:52,380 --> 00:21:58,350
Founded in 1080 1088, with Paris and Oxford following soon after.
180
00:22:00,660 --> 00:22:07,680
What was taught. The curriculum was in two parts. The second of which led to a master's degree and is based on the, quote, Trivium.
181
00:22:08,160 --> 00:22:16,080
You can see the quadrillion subjects on the right. The Greek mathematical arts, again, of arithmetic, geometry, music and astronomy.
182
00:22:17,210 --> 00:22:23,480
And the curriculum also included the study of Ptolemy's Almagest and Euclid's elements.
183
00:22:27,480 --> 00:22:35,130
Gutenberg's Invention of the Printing Press around 1440 enabled mathematical works to be widely available for the first time.
184
00:22:36,330 --> 00:22:38,730
At first, they were printed in Latin for the scholar.
185
00:22:40,370 --> 00:22:49,460
But gradually vernacular works in the language of the people appeared at prices accessible to all, and these included texts in arithmetic,
186
00:22:49,700 --> 00:22:55,520
algebra and geometry, as well as practical works to prepare their readers for a commercial career.
187
00:22:58,100 --> 00:23:06,170
An important, important among these you can see on the top right was Pat Schiller's 600 page Italian Summa,
188
00:23:07,190 --> 00:23:14,150
a compilation of the mathematics of the time, which also included the first published account of double entry bookkeeping.
189
00:23:15,410 --> 00:23:21,670
And then below that, a commercial arithmetic by Adam Rizza, which is so respectable, the phrase no.
190
00:23:21,680 --> 00:23:26,030
Adam Riza, after Adam Riza came to refer to a correct calculation.
191
00:23:28,670 --> 00:23:34,880
The invention of printing led gradually to a standardisation of mathematical notation, and the symbols,
192
00:23:35,030 --> 00:23:42,050
plus and minus first appeared in print in an arithmetic text of 1489,
193
00:23:43,190 --> 00:23:48,170
whereas the symbols four times and divide didn't appear until considerably later.
194
00:23:52,800 --> 00:23:58,440
It was around this time that painters learned how to give visual depth to their works through geometrical perspective.
195
00:24:00,500 --> 00:24:08,840
The first to investigate this were Brunelleschi, who had designed the Dome of Florence Cathedral and his friend Alberti.
196
00:24:09,340 --> 00:24:12,350
They're both shown at the top on the left.
197
00:24:13,370 --> 00:24:20,360
Alberti gave mathematical rules for correct perspective, and he stated that the first duty of a painter is to know geometry.
198
00:24:21,050 --> 00:24:27,810
So if you have any artist friends, you know what to tell them. Bottom left,
199
00:24:27,810 --> 00:24:32,459
Piero della Francesca used a prospective grid for his geometrical studies and his
200
00:24:32,460 --> 00:24:37,770
Madonna and child with Saints shown here is in perfect mathematical perspective.
201
00:24:39,680 --> 00:24:46,530
The Polyhedron woodcuts that appear in Peres books were actually by Peres friend and student Leonardo da Vinci.
202
00:24:46,640 --> 00:24:53,900
Shown on the top right, who investigated perspective more deeply than any other Renaissance painter and warned,
203
00:24:54,350 --> 00:24:57,110
Let no one who is not a mathematician read my work.
204
00:25:00,220 --> 00:25:10,930
Another celebrated artist was Albrecht Dürer on the at the bottom, who learned geometrical perspective from the Italians and introduced it to Germany.
205
00:25:11,470 --> 00:25:16,840
And his engravings such as Saint Jerome and his study show his mastery of it.
206
00:25:21,540 --> 00:25:32,070
We now turn to navigation. The Renaissance period coincided with the great sea sea voyages and explorations of Vasco da Gama, Columbus and others.
207
00:25:33,890 --> 00:25:40,930
The Portuguese explorer sailed south and east. Prince Henry, the navigator on that wonderful stamp at the top,
208
00:25:41,650 --> 00:25:47,170
sent ships down the west coast of Africa, claiming the island groups of Madeira and the Azores.
209
00:25:48,380 --> 00:25:54,530
And Vasco da Gama was the first European to sail around the tip of Africa to reach the west coast of India.
210
00:25:56,550 --> 00:26:02,070
Their rivals, the Spanish headed west, hoping to reach India by sailing around the globe.
211
00:26:02,820 --> 00:26:11,800
And Columbus led four search expeditions, but instead reaching North and Central America, the West Indies and Venezuela.
212
00:26:12,630 --> 00:26:25,400
And you can see Columbus, the bottom right. Such explorations inevitably needed accurate maps and reliable navigational instruments for use at sea.
213
00:26:26,360 --> 00:26:35,030
But trying to represent the Spherical Earth on a flat sheet of paper led to new types of map projection and to improve maps for navigators at sea.
214
00:26:37,490 --> 00:26:45,709
The first modern maps use the Mercator projection obtained by projecting the sphere outwards onto a vertical cylinder and then stretching the
215
00:26:45,710 --> 00:26:53,540
scale vertically so that all the lines of latitude and longitude appear straight while all the angles and compass directions were correct.
216
00:26:54,110 --> 00:27:03,110
And you can see here, Mercator, you can see the Mercator projection and also the Canadian stamp of 1898 at the top also features it.
217
00:27:04,140 --> 00:27:11,490
Incidentally, Mercator was the first person to use the word Atlas for his three volume collection of maps in 1590.
218
00:27:14,370 --> 00:27:22,950
But for navigation at sea you needed instruments. Astrolabes were widely used as were quadrants, sextant occupants and other instruments.
219
00:27:23,550 --> 00:27:29,700
They were used to measure the altitudes of heavenly bodies such as the Sun or Polestar in order to determine latitude.
220
00:27:31,390 --> 00:27:35,830
Quadrants on the top left have the shape of a quarter circle 90 degrees,
221
00:27:36,370 --> 00:27:45,790
while sixteenths in the middle correspond to a sixth of a circle 60 degrees and Orton's bottom left to an eighth of a circle 45 degrees.
222
00:27:46,720 --> 00:27:54,760
To measure an object's altitude, you view it along the top edge of the instrument and the position of immovable rod on the rim gives you the result.
223
00:27:57,270 --> 00:28:06,390
Earlier Lave Ben Gazan had invented the widely used Jacobs staff across staff for measuring angles between celestial bodies,
224
00:28:06,930 --> 00:28:15,060
which you can see on the top. Right. But unfortunately, to measure the angle between the sun and the horizon, you had to look directly at the sun.
225
00:28:16,200 --> 00:28:21,930
The back staff shown below. It was a clever modification where you had your back to the sun.
226
00:28:26,970 --> 00:28:30,330
Well, now move to astronomy, which is completely transformed.
227
00:28:30,330 --> 00:28:36,870
When Copernicus replaced the Greeks earth centred system of planetary motion by a heliocentric one
228
00:28:36,870 --> 00:28:42,330
with the sun at the centre and the earth is just one of the planets in circular orbits around it.
229
00:28:44,350 --> 00:28:49,030
His book On the Revolution of the Heavenly Spheres was published in 1543.
230
00:28:49,450 --> 00:28:52,540
The copy of it was supposedly presented to him on his deathbed.
231
00:28:54,970 --> 00:28:57,580
The Copernican system aroused much controversy,
232
00:28:57,580 --> 00:29:04,030
bringing its supporters into direct conflict with the church who considered Earth to lie at the centre of creation.
233
00:29:04,910 --> 00:29:11,980
And several were arrested by the Inquisition, or even burnt alive for heresy, such as Giordano Bruno.
234
00:29:14,440 --> 00:29:17,680
Indeed, at the famous Inquisition trial in 1633,
235
00:29:18,100 --> 00:29:22,629
Galileo was placed under house arrest after his dialogue concerning the two chief
236
00:29:22,630 --> 00:29:28,480
world systems who presented the Copernican system as superior to the Ptolemaic one.
237
00:29:29,710 --> 00:29:35,680
Galileo wasn't pardoned by the church until 1995, which was a bit late to do him much good.
238
00:29:38,700 --> 00:29:43,320
Galileo was the first to make extensive use of the telescope. He wasn't the first to use it.
239
00:29:43,710 --> 00:29:47,730
I think that claim goes to Harriot in Oxford just a few months earlier.
240
00:29:48,750 --> 00:29:54,340
But anyway, Galileo drew the moon's surface and discovered the moons of Jupiter and Saturn.
241
00:29:54,360 --> 00:29:57,750
And you can see one of his drawings on the left.
242
00:30:00,460 --> 00:30:10,570
In his mechanic's book, Two New Sciences, he discussed uniform and accelerated motion and explained why the path of a projectile must be a parabola.
243
00:30:14,180 --> 00:30:20,460
But the greatest observer of the heavens before the invention of the telescope was the Danish astronomer Tycho Brahe.
244
00:30:21,900 --> 00:30:29,130
In his observatory in Europe. Nyborg on the Island of Heaven, which I had the pleasure of visiting this summer.
245
00:30:29,880 --> 00:30:36,090
He designed astronomical instruments of unequalled accuracy and measured over 700 stars.
246
00:30:37,070 --> 00:30:43,010
And in particular, he discovered a new star or Nova in the constellation of Cassiopeia.
247
00:30:46,200 --> 00:30:53,700
Yohannes Kepler, shown here at the bottom who worked with him, is mainly remembered for his three laws of planetary motion.
248
00:30:54,780 --> 00:31:00,720
Using Tycho Brahe his extensive observations, he was led to consider non circular orbits,
249
00:31:01,260 --> 00:31:08,100
proposing that the planets move in elliptical orbits the sun at one focus and the line from any sun,
250
00:31:08,100 --> 00:31:12,240
from the sun to any planet sweeps out equal areas in equal times.
251
00:31:14,390 --> 00:31:17,150
Kepler was fascinated by ellipses and other clinics,
252
00:31:17,750 --> 00:31:25,280
and he was the one that introduced the word focus into mathematics, rotating such curves around an axis.
253
00:31:25,610 --> 00:31:33,750
He found the volumes of over 90 followers of revolution by some thin disks, thereby anticipating the integral calculus.
254
00:31:33,770 --> 00:31:45,010
A few years later. But a major problem of the time, particularly for astronomers and navigators, were difficulties of calculation.
255
00:31:46,480 --> 00:31:49,990
In 1614, 400 years ago this year,
256
00:31:50,590 --> 00:31:55,329
John Napier of Edinburgh introduces logarithms designed to replace lengthy
257
00:31:55,330 --> 00:32:00,730
multiplications and divisions by easier calculations with additions and subtractions.
258
00:32:03,070 --> 00:32:08,860
Being awkward to use. For example, Napier's log of one was not zero.
259
00:32:09,550 --> 00:32:16,420
They were soon supplanted by Henry Briggs as similar logs to base ten produced while here in Oxford,
260
00:32:17,140 --> 00:32:22,180
whose use whose use proved of enormous benefit to two astronomers and navigators.
261
00:32:24,510 --> 00:32:28,589
The invention of logarithms soon led to practical instruments based on a logarithmic scale,
262
00:32:28,590 --> 00:32:33,330
such as the slide rule shown on the right, dating from about 1630.
263
00:32:33,540 --> 00:32:41,040
They were used for over 300 years until pocket calculators started to appear around the year 1970.
264
00:32:44,680 --> 00:32:48,520
The first mechanical, calculating machines were also introduced around this time.
265
00:32:49,270 --> 00:32:55,180
And in 1623, one was described by William Schick Hart on the Orange Stamp.
266
00:32:55,750 --> 00:33:05,410
In letters to Kepler, while later machines were constructed by Pascal and Leyden, it's Pascal's machine, the Pascaline.
267
00:33:06,550 --> 00:33:13,240
It was operated by contrails and could add and subtract numbers, while leibniz's could also multiply and divide.
268
00:33:13,900 --> 00:33:20,550
But neither instrument worked particularly well in practice. Well.
269
00:33:22,220 --> 00:33:27,350
Will now move to 17th century France, to Descartes, Firma and Pascal.
270
00:33:29,030 --> 00:33:35,750
An ancient problem of purpose. It also the path of a point moving in a specified way relative to some fixed lines.
271
00:33:36,770 --> 00:33:43,819
To solve this, Descartes named to particular length X and Y and calculated every other length in
272
00:33:43,820 --> 00:33:49,610
terms of them obtaining a quadratic expression that is a conic as the required path.
273
00:33:51,420 --> 00:33:54,660
Descartes thus introduced algebraic methods into geometry,
274
00:33:54,840 --> 00:34:00,480
but he didn't initiate the Cartesian coordinates named after him, in which the angles are right angles.
275
00:34:00,870 --> 00:34:09,990
Points appear as pairs of numbers X-Y lines for equations y equals that makes plus C and to find the intersection of two lines,
276
00:34:10,380 --> 00:34:13,140
you solve two simultaneous algebraic equations.
277
00:34:16,170 --> 00:34:25,740
Descartes most celebrated work was his discourse on Method, a philosophical treatise on universal science containing his contributions to geometry.
278
00:34:26,740 --> 00:34:30,880
300 years later, France issued a stamp to commemorate it.
279
00:34:31,240 --> 00:34:35,590
The one on the right at the top. But they got its title wrong.
280
00:34:36,580 --> 00:34:43,690
To score? Sure. No method. The seal was later corrected to die on a later stamp shown below.
281
00:34:49,570 --> 00:34:55,240
Still in France. Fermor, a lawyer in Toulouse, is mainly remembered for his contributions to number theory.
282
00:34:57,480 --> 00:35:01,740
In particular, he famously claimed to have proved that the equation X to the envelopes,
283
00:35:01,740 --> 00:35:08,400
watched their nickels read to the end has no non-zero integer solutions when an is greater than two.
284
00:35:09,000 --> 00:35:17,190
And this result came to be known as Fermat's Last Theorem and was eventually proved by Andrew Wiles in 1995,
285
00:35:17,580 --> 00:35:23,340
as shown by the Red Bar across the Equal Sign, which says Andrew Wiles, 1995.
286
00:35:27,360 --> 00:35:29,999
Pascal showed an early interest in mathematics.
287
00:35:30,000 --> 00:35:36,570
At the age of only 16, he discovered his Hexagon theorem about six points on a conic, joined up in a particular way.
288
00:35:37,900 --> 00:35:44,650
Pascal's Principle in Hydrodynamics was named after him, and he was one of the first to investigate the theory of probability.
289
00:35:45,720 --> 00:35:53,430
He also wrote about the arithmetical triangle of binomial coefficients and Pascal's triangle appears on the pink stamp here.
290
00:35:54,060 --> 00:36:00,180
Another one from the recent International Congress of Mathematicians in Seoul, in Korea.
291
00:36:05,620 --> 00:36:09,370
Isaac Newton was born in 1642 in Cambridge.
292
00:36:09,380 --> 00:36:17,170
He avidly studied such contemporary works as a Latin edition of Descartes Geometry and was appointed Lucasian Professor of Mathematics,
293
00:36:17,650 --> 00:36:25,190
a post later held by Stephen Hawking. The story of Isaac Newton and the apple is well known.
294
00:36:25,610 --> 00:36:31,550
Seeing an apple fall, he proposed that the gravitational force pulling it to Earth is the same as the force
295
00:36:31,670 --> 00:36:35,540
that keeps the moon orbiting around the Earth and the earth orbiting around the sun.
296
00:36:36,980 --> 00:36:41,560
He also claimed that planetary motion is governed by a universal law of gravitation.
297
00:36:41,960 --> 00:36:48,110
The inverse square law, the force of attraction between two objects varies as the product of their masses,
298
00:36:48,500 --> 00:36:51,530
and inversely is a square of the distance between them.
299
00:36:51,980 --> 00:37:02,890
As shown on the stamp here. In his 1687 principle, Mathematica, perhaps the greatest scientific work of all time,
300
00:37:03,280 --> 00:37:08,739
Newton used the inverse square law to justify Kepler's laws of planetary Kepler's
301
00:37:08,740 --> 00:37:12,700
laws of elliptical planetary motion and to account for cometary orbits,
302
00:37:12,850 --> 00:37:16,480
the various variation of tides and much else besides.
303
00:37:20,350 --> 00:37:24,700
But the principle here would never have appeared if Edmund Halley hadn't cajoled
304
00:37:24,700 --> 00:37:29,380
a reluctant Newton into developing his ideas of gravitation and publishing them.
305
00:37:30,160 --> 00:37:36,580
Halley himself paid for this. Halley, of course, later became a civilian professor of geometry here in Oxford.
306
00:37:39,100 --> 00:37:44,830
So Harley or possibly and probably Hawley, but definitely not.
307
00:37:44,860 --> 00:37:52,150
Haley is mainly remembered for the comet named after him observing it in 1682.
308
00:37:52,180 --> 00:37:55,030
He realised it was the same as one seen many times earlier.
309
00:37:55,780 --> 00:38:01,720
He predicted its return in late 1758 and its appearance then several years after his death,
310
00:38:02,020 --> 00:38:08,740
did much to vindicate Newton's theory of gravitation and because of his prediction, it became known as Halley's Comet.
311
00:38:09,220 --> 00:38:11,440
There are quite a few stamps featuring Halley's Comet.
312
00:38:11,850 --> 00:38:20,200
I like the British one top right, which has Halley's head as the head of the comet and his wig as the tail.
313
00:38:24,090 --> 00:38:26,970
But reactions to Newton's approach to gravitation were mixed.
314
00:38:27,750 --> 00:38:32,790
In England, his principle was well received, even though few rude, few readers understood it.
315
00:38:33,750 --> 00:38:36,780
But not everyone was receptive, particularly in France.
316
00:38:37,890 --> 00:38:42,090
The principle was long and difficult and raised awkward questions about the shape of the earth.
317
00:38:43,430 --> 00:38:47,030
Descartes had earlier proposed the vortex theory of the universe,
318
00:38:47,420 --> 00:38:52,550
a consequence of which is that the earth is elongated at the polls that is lemon shaped.
319
00:38:53,810 --> 00:39:00,980
Newton criticised Descartes ideas in the print copier, predicting a flattening of the poles so that the earth is onion shaped.
320
00:39:03,210 --> 00:39:09,270
National pride was at stake and the matter was urgent because inaccurate mapmaking was leading to the loss of lives at sea.
321
00:39:10,680 --> 00:39:13,290
Eventually, two geometric, two g,
322
00:39:13,920 --> 00:39:20,850
two geodetic missions were dispatched to settle the matter by measuring the swing of a pendulum at the equator and near the North Pole.
323
00:39:22,170 --> 00:39:30,390
De la Condamine mission on the left went to Peru and included the Spanish mathematician and Cosmograph Hawk one.
324
00:39:31,740 --> 00:39:37,920
While most of his mission travelled to Lapland, but eventually they confirmed Newton's theory.
325
00:39:38,100 --> 00:39:40,680
The earth is indeed flatter at the poles.
326
00:39:45,510 --> 00:39:52,380
Although Newton could justly claim priority for the calculus, Livni's, who developed it independently, was the first to publish it.
327
00:39:53,280 --> 00:40:01,200
Also, his notation was more versatile than Newton's and amazingly his d y by X for differentiation and the integral
328
00:40:01,200 --> 00:40:09,420
sign were introduced by him within three weeks of each other in the autumn of 1675 and are still used today.
329
00:40:11,450 --> 00:40:18,950
But his calculus is rather different from Newton's being based on geometry and areas rather than on velocity and motion.
330
00:40:23,960 --> 00:40:28,010
The Bernoulli family included several distinguished Swiss mathematicians.
331
00:40:28,490 --> 00:40:32,240
But only one, unfortunately, has been featured on a stamp.
332
00:40:34,440 --> 00:40:40,950
In his book, The Art of Conjecturing, Jacob Bernoulli Top Left proved the law of large numbers.
333
00:40:41,460 --> 00:40:45,990
When an experiment is performed often, it's highly probable the outcome is as we expect.
334
00:40:46,620 --> 00:40:51,630
For example, tossing a fair coin 2000 times gives a number of heads close to a thousand.
335
00:40:53,420 --> 00:40:58,520
With his brother, Johann Jacob Bernoulli was the first to develop Leibniz's calculus,
336
00:40:59,300 --> 00:41:06,590
introducing the word integral and applying calculus to such curves as the logarithmic spiral and the cycle.
337
00:41:06,590 --> 00:41:14,370
Lloyd. Leonard Euler, shown on the rest of the stamps, also grew up in Basel.
338
00:41:15,340 --> 00:41:20,290
But spent most of his working life at the scientific academies of St Petersburg and Berlin.
339
00:41:22,230 --> 00:41:24,470
The most prolific mathematician of all time.
340
00:41:25,350 --> 00:41:35,730
He contributed to almost every branch of mathematics and physics, including number, theory, mechanics, differential equations, astronomy and optics.
341
00:41:37,890 --> 00:41:41,010
Oil reformulated the calculus in terms of functions.
342
00:41:41,640 --> 00:41:50,850
He introduced the notations e for exponential I for the squares for minus one capital sigma for summation and F for a function.
343
00:41:52,300 --> 00:41:56,860
And he also related the exponential and trigonometric functions through the equation.
344
00:41:56,860 --> 00:41:59,770
Each of the IFIs cause phi plus I signed Phi.
345
00:42:00,750 --> 00:42:07,890
And if you're in Switzerland on the way to the post office and you happen to forget that formula, it's on the top right stamp of the side.
346
00:42:12,840 --> 00:42:21,060
In 1750, he discovered his polyhedron formula. Faces plus vertices equals edges plus two shown the two stems on the bottom left.
347
00:42:23,210 --> 00:42:26,960
And in 1735 he solved the famous Königsberg Bridges problem,
348
00:42:26,960 --> 00:42:33,290
which asked whether one can cross all of the seven bridges of the city of Coningsby without visiting any bridge twice.
349
00:42:34,070 --> 00:42:38,780
But he didn't draw the graph shown here on the recent Korean stamp.
350
00:42:38,930 --> 00:42:40,730
Also for the International Congress.
351
00:42:45,090 --> 00:42:52,650
Back now to France, where the turbulent years of the French Revolution and the rise to power of Napoleon led to important developments in mathematics.
352
00:42:54,120 --> 00:43:01,469
Napoleon himself was greatly interested in the subject, and one of his greatest friends was the janitor, Gaspar Moorish,
353
00:43:01,470 --> 00:43:10,920
whom you can see on the left underneath, who had taught at a military school where he studied the properties of lines and planes in three dimensions.
354
00:43:12,920 --> 00:43:19,940
While investigating gun emplacements in the fortress, he greatly improved methods for projecting three dimensional objects onto a plane.
355
00:43:20,660 --> 00:43:23,570
This subject soon became known as descriptive geometry.
356
00:43:26,760 --> 00:43:35,280
An important consequence of the French Revolution was the founding of Paris, founding in Paris of the École Polytechnique showing top prize.
357
00:43:37,130 --> 00:43:42,890
They're the country's finest mathematicians, including Marsh, La Grange, La Plus.
358
00:43:42,890 --> 00:43:48,950
And Koshy taught students designed to serve in both military and civilian capacities.
359
00:43:51,250 --> 00:43:54,640
Lagouranis was always successor at the court of Frederick the Great in Berlin.
360
00:43:55,420 --> 00:44:00,670
He wrote the first three functions and an influential text on mechanics and in number theory,
361
00:44:00,670 --> 00:44:04,540
he proved that every integer can be written as the sum of four perfect squares.
362
00:44:07,010 --> 00:44:09,860
The place was the applied mathematician par excellence.
363
00:44:10,370 --> 00:44:16,250
Remember for the plus transform of a of a function and full of pluses equation in mathematical physics.
364
00:44:17,180 --> 00:44:22,489
He also wrote a celebrated text on probability and his monumental five volume
365
00:44:22,490 --> 00:44:27,470
treatise on Celestial Mechanics earned him the title of the Newton of France.
366
00:44:31,250 --> 00:44:32,930
Shortly after the French Revolution,
367
00:44:32,930 --> 00:44:39,440
a commission was set up to standardise the weights and measures in France and introduce a metrics and metric system.
368
00:44:39,770 --> 00:44:43,190
Here you can see a French stamp celebrating the metric system.
369
00:44:45,820 --> 00:44:51,640
The chairman of this commission was La Roche and its members included the press and more.
370
00:44:53,910 --> 00:44:59,920
Working analysis can continue to Koshy. The calculus would prove to be on shaky grounds.
371
00:44:59,920 --> 00:45:06,880
But because she rescued it by basing it on the limits of on the concepts of limit and continuity.
372
00:45:07,900 --> 00:45:10,900
He also developed complex analysis almost single handedly.
373
00:45:11,470 --> 00:45:19,630
And here on the top of that stamp, you can see Coach's integral formula, which is familiar to all second year undergraduates.
374
00:45:23,720 --> 00:45:28,850
Meanwhile, in Germany, Gauss is working in many areas from complex numbers.
375
00:45:30,080 --> 00:45:35,090
The Gauss in number of planes plane shown on the right factor, rising polynomials.
376
00:45:35,120 --> 00:45:38,329
The fundamental theorem of algebra to statistics.
377
00:45:38,330 --> 00:45:39,680
The Gaussian distribution.
378
00:45:41,330 --> 00:45:48,290
One of the greatest mathematicians of all time, he discovered which regular polygons can be drawn by a ruler and compasses alone.
379
00:45:49,400 --> 00:45:59,060
These include equilateral triangle and regular pentagons, and also the 17 sided polygon shown here, which you thought was a circle.
380
00:46:03,920 --> 00:46:08,090
The early 19th century saw major revolutions in geometry.
381
00:46:09,210 --> 00:46:14,370
UK is elements that can commence with five postures for straight, straightforward,
382
00:46:14,640 --> 00:46:20,430
but one different in style resembling a theorem that ought to be provable from the others.
383
00:46:21,510 --> 00:46:26,410
One version of it is a parallel postulate given any line and any point not on it.
384
00:46:26,430 --> 00:46:29,700
There's a unique line through the point which is parallel to the given line.
385
00:46:33,150 --> 00:46:37,590
For over 2000 years. Mathematicians tried to prove this from the other postures, but failed.
386
00:46:38,790 --> 00:46:43,800
And this is because their non-Euclidean geometries satisfying the first four postulates, but not the fifth.
387
00:46:44,580 --> 00:46:50,220
These geometries have infinitely many lines to the point which is parallel to the given one as shown in the.
388
00:46:50,430 --> 00:46:53,850
In the Hungarian stamp. There. At the bottom.
389
00:46:56,030 --> 00:47:02,360
They were first published around 1830 by Nikolai Lorber Chayefsky from Russia and Polish from Hungary.
390
00:47:05,260 --> 00:47:10,060
Another new geometrical object from the 19th century, actually from 1858,
391
00:47:10,510 --> 00:47:15,700
was the map, a strip shown here, which has only one side and one boundary edge.
392
00:47:19,040 --> 00:47:23,600
All these many developments in geometry, projective geometry, non-Euclidean geometry and so on,
393
00:47:24,140 --> 00:47:28,220
forced mathematicians to ask which geometry corresponds to the world we live in.
394
00:47:29,000 --> 00:47:31,760
Is it Euclidean geometry or is it a non-Euclidean one?
395
00:47:33,030 --> 00:47:39,420
In fact, the geometry that later arose in Einstein's theory of relativity was one designed by the German Pendry man,
396
00:47:39,810 --> 00:47:48,620
who has unfortunately never appeared on a stamp. But here are some pictures of Einstein, including the strange one in the middle of the top row.
397
00:47:49,590 --> 00:47:55,830
It looks as though he's just got up in the morning with his famous equation equals M.C. squared releasing energy in mass.
398
00:47:56,100 --> 00:48:00,390
Shown on the wall paper with a behind him. Obviously, that's where he got the idea.
399
00:48:06,790 --> 00:48:13,360
There's also a major breakthrough in algebra when the Norwegian Niels Abel solved a longstanding problem.
400
00:48:14,250 --> 00:48:18,240
Recall, the quadratic equations have been solved by the ancient Mesopotamians,
401
00:48:18,660 --> 00:48:23,940
while 16th century Italians had shown how to solve equations of degrees three, three and four.
402
00:48:25,400 --> 00:48:30,510
But what about equations of degree? Five or more? All showed that the change stops here.
403
00:48:30,530 --> 00:48:33,680
There is no general formula to solve such equations.
404
00:48:35,200 --> 00:48:42,970
Abel's work was extended by the brilliant young French mathematician Everest Galois, about whom Peter Norman has written extensively,
405
00:48:43,810 --> 00:48:48,970
who explained algebraically which algebraically exactly which equations can be solved.
406
00:48:49,960 --> 00:48:56,410
Galois had a short and turbulent life being sent to jail for political activities and dying tragically in a duel at the age of 20.
407
00:48:56,890 --> 00:49:06,430
Having set out the previous night writing to a colleague about containing his mathematical achievements for post for posterity.
408
00:49:08,780 --> 00:49:13,490
Below is William Rowan Hamilton, a child prodigy who must have several languages of an early age,
409
00:49:13,970 --> 00:49:20,030
discovered an error in Le Plessis, Lester Mechanics as a teenager and became Astronomer Royal of Ireland.
410
00:49:20,030 --> 00:49:28,580
While still a student. He made important advances in mechanics and geometrical optics while attempting to generalise the complex numbers.
411
00:49:28,730 --> 00:49:37,100
Discover the Kryptonians a non-competitive system involving three square roots of minus one called AJ and K,
412
00:49:37,430 --> 00:49:41,720
which you can see here satisfying the equation shown on the Irish stamp.
413
00:49:45,020 --> 00:49:49,700
Unfortunately, not many women mathematicians have featured on stamps, but here are two important ones.
414
00:49:50,540 --> 00:49:54,679
Florence Nightingale is mainly remembered as the lady with the lamp who saved
415
00:49:54,680 --> 00:49:59,360
many lives through her sanitary improvements in in Crimea and war hospitals.
416
00:50:00,800 --> 00:50:09,350
But she is also a fine statistician who collected and analysed mortality data from the Crimea and displayed them using her polar diagrams,
417
00:50:09,710 --> 00:50:11,270
a forerunner of the pie chart.
418
00:50:13,310 --> 00:50:22,070
Meanwhile, Sonya Kovalev Skyler was making valuable contributions to mathematics, physics and astronomy as well as becoming a well-known novelist.
419
00:50:23,150 --> 00:50:32,120
Barred by her gender from studying in Russia, she attended a physics lectures by Kirchhoff and Helmholtz in Germany before settling in Sweden.
420
00:50:32,930 --> 00:50:38,990
She later won a coveted prize from the French Academy of Sciences for a memoir on the rotation of bodies.
421
00:50:42,740 --> 00:50:48,470
So we've at last reached the 20th century when several mathematicians created the subject as we now know it.
422
00:50:49,160 --> 00:50:50,720
So here's a brief selection.
423
00:50:52,200 --> 00:51:02,520
The range of David Hilbert was immense from number theory and so-called Hilbert spaces and analysis to potential theory and the theory of gases.
424
00:51:04,080 --> 00:51:09,659
In 1900, his celebrated lecture at the International Congress of Mathematicians in Paris described
425
00:51:09,660 --> 00:51:14,880
23 problems that set the agenda for mathematical research over the coming century.
426
00:51:16,940 --> 00:51:25,850
Ori Poincaré shown top left wrote on the still unsolved three-body problem determining the simultaneous motion of the sun, earth and moon.
427
00:51:26,930 --> 00:51:28,790
A gifted populariser of mathematics.
428
00:51:28,790 --> 00:51:36,560
He developed many areas including algebraic topology, differential equations, celestial mechanics and much else besides.
429
00:51:39,360 --> 00:51:46,710
Below him. In England, Bertrand Russell made fundamental contributions to mathematical logic, such as Russell's paradox.
430
00:51:47,520 --> 00:51:55,350
With an and WHITEHEAD He wrote Principia Mathematica, a pioneering three volume work on the logical foundation foundations of mathematics,
431
00:51:56,130 --> 00:52:00,300
in which the equation one plus one equals two isn't proved until page 86.
432
00:52:03,490 --> 00:52:08,710
On the right. Ramanujan was one of the most intuitive of all mathematicians, mainly self-taught.
433
00:52:08,740 --> 00:52:15,879
He left India in 1914 to work with G.H. Hardy in Cambridge, producing some spectacular joint work in number,
434
00:52:15,880 --> 00:52:20,530
theory and analysis before his untimely death at the age of 32.
435
00:52:23,370 --> 00:52:23,939
In Poland.
436
00:52:23,940 --> 00:52:33,840
Stefan Brunner, bottom right, helped to create modern functional analysis and to develop links between topology and algebra and Banach spaces.
437
00:52:34,050 --> 00:52:43,840
Then archaeology, eligibles and so on are named after him. Another 20th century development is fractal geometry.
438
00:52:44,620 --> 00:52:50,950
And in 2005, Macao issued a set of stamps featuring featuring fractal patterns.
439
00:52:51,820 --> 00:52:57,370
Such patterns are so similar, reproducing themselves infinitely often when magnified or reduced.
440
00:52:58,180 --> 00:53:07,090
Here, top left, you can see Hong Kong snowflake curve of infinite length, but enclosing a finite area and on top, right.
441
00:53:07,210 --> 00:53:17,470
Hilbert Space Filling Curve. And the series also has a cantor set and a fractal tree and the PINSKY triangle.
442
00:53:20,200 --> 00:53:27,010
When the transformation of squaring and when the transformation of squaring and translating
443
00:53:27,420 --> 00:53:31,660
that said goes to Z squared plus C is applied repeatedly to the complex plane.
444
00:53:31,900 --> 00:53:37,480
Some points shoot off to infinity depending on the value of C, while the rest remain finite.
445
00:53:38,710 --> 00:53:42,010
The boundary between these two sets is a fractal pattern called the Julia set.
446
00:53:42,310 --> 00:53:51,340
After the Frenchman guessed on Julia and the attractive Israelis stamp that you can see there shows part of a Julia set.
447
00:53:53,070 --> 00:54:00,600
The set of all constancy for which the Julia set is in one piece is the famous Mandelbrot set shown here on this miniature sheet.
448
00:54:03,370 --> 00:54:10,600
But you prefer something a little more light-hearted. There's always the Rubik's Cube invented by the Hungarian engineer and a Rubik.
449
00:54:10,690 --> 00:54:18,340
It's a colour cube whose faces can be independently rotated so as to yield over four times 10 to 19 different patterns.
450
00:54:19,670 --> 00:54:29,470
The object is to restore the original colours. In the early 1980s when the Rubik's Cube craze at its height over 1 million.
451
00:54:30,240 --> 00:54:39,020
Over 100 million cubes were sold. Well, we're coming towards the end.
452
00:54:39,590 --> 00:54:42,830
Mathematics continues to advance at an ever increasing rate.
453
00:54:43,310 --> 00:54:50,780
And since 1897, international Congresses of mathematicians have been held around the world at which many thousands
454
00:54:50,780 --> 00:54:54,860
gather for every four years to learn about the most recent developments in their subject.
455
00:54:56,450 --> 00:54:59,870
Several several of these Congresses have been commemorated on stamps.
456
00:55:01,010 --> 00:55:11,050
Here are those moving along the top row from Moscow in 1966, Helsinki in 1978, with a design from differential geometry.
457
00:55:12,110 --> 00:55:15,470
Kyoto in 1990 showing an origami polyhedron.
458
00:55:16,550 --> 00:55:18,860
Berlin in 1998,
459
00:55:19,070 --> 00:55:27,080
showing the division of a rectangle into unequal squares with integer integer sides with the digits of pi swirling around in the background.
460
00:55:28,710 --> 00:55:39,980
In Madrid in 2006. This year, Korea issued no fewer than three stamps, all seen earlier depicting the Pythagorean theorem,
461
00:55:39,980 --> 00:55:42,470
Pascals Triangle and the Königsberg Bridges problem.
462
00:55:45,350 --> 00:55:51,800
And a special feature of the Icmje see awarding of Fields medals to the most outstanding mathematicians under the age of 40.
463
00:55:53,090 --> 00:56:00,800
Starting in 1936, they were financed from the estate of J.C. Fields, who is the top right.
464
00:56:01,910 --> 00:56:05,390
He was president of the 1924 Toronto Congress.
465
00:56:06,440 --> 00:56:14,570
And here in the middle of middle of the bottom row, you can see Simon Donaldson who who was awarded his Fields medal while in Oxford.
466
00:56:15,050 --> 00:56:20,390
Well above him is Alan Baker, the distinguished number theorist from Cambridge.
467
00:56:22,980 --> 00:56:25,770
So to conclude, as we look back over 4000 years,
468
00:56:26,160 --> 00:56:33,630
we might emulate Time magazine's My Man of the Year and ask whether the mathematician of all time has ever appeared on a stamp.
469
00:56:34,230 --> 00:56:39,240
Unfortunately, the answer is yes. Thank you very much. Thank you.