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.