The Great Mathematicians – Professor Robin Wilson and Dr Raymond Flood

well good evening everybody and welcome to Gresham College a particular welcome to those who are visiting for the first time and the objects of today as you know is to launch wonderful nubuck the great mathematicians because we’re all familiar with the stories of Isaac Newton and the Apple and we’re familiar with the story of Archimedes running naked along the streets shouting Eureka but which mathematician was killed in a duel which one published books yet did not exist which one was crowned Pope who were dr. Morales and dr. profundus who learned calculus from her nursery wallpaper who was excited by a taxi cab number who measured the chests of 5,730 two Scottish soldiers you’ll find the answers of that later in this hour and what was Geoffrey Chaucer Christopher Wren Napoleon France nightingale and Lewis Carroll got to do with mathematics as his questions indicate mathematics has always been a human endeavor as people have found themselves grappling with a wide range of problems both practical and theoretical for example our book will tell you about time measurers like the Mayans and Christian Huygens arithmetic ins like Pythagoras and al-khwarizmi geometers like Apollonius and Lobachevsky algebraic vs and Galois logicians like Aristotle and Russell number theorists like firma and remark Ramanujan applied mathematicians like Laplace and Maxwell statisticians like Jakob Bernoulli and nightingale textbook writers like include and Bourbaki teachers like Hypatia and Dodgson architects like Brunelleschi and Wren and calculators like Napier and babish what a bargain so I get his hand over to Raymond flood now who’s going to tell you about the structure of the book and you have a handout on the back of your handout you’ll see the contents page with a collection of useful map’s followed by a timeline of mathematicians following these as Robin has just said on the back of your hand Ike you can see that the book is organized into 5 chapters comprising almost a hundred double-page spreads featuring many great mathematicians and illustrated by about 350 pictures the first chapter inch in mathematics covers the period from the Egyptians of Mesopotamians via the Greek period to China India the Mayans and the mathematic mathematics of Islam Islam chapter 2 on early European mathematics / moves with gerbera of aurillac and Fibonacci and precedes five prospective tempters textbook writers and astronomers to the mathematicians of the late 16th century chapter 3 on Awakening and enlightenment covers the 17th and 18th centuries unrighteous from Napier and Briggs in Britain and firma Descartes and Pascal in France to newton and leibniz the bernoulli’s and Euler and in back in France with Lagrangian Laplace the first chapter the image of revolutions deals with the 19th century from guys album and crochet via such varied figures as Babbage bull and Maxwell to riemann cover the sky and sighing finally in Chapter five the modern age recovered the 20th century and beyond starting with Hilbert and conquer a via Russell Einstein and cheering to Mandelbrot wilds Pearlman and the fields medalists so what we’re going to do is to survey these five chapters and time presenting two spreads from each of them which we’ve chosen to try to illustrate the wide range of material that we have featured in the book and Robins going to

start off by presenting the first two spreads thank you very much as we mentioned chapter one years of ancient mathematics and this includes a dozen also of the Greek mathematicians such as Pythagoras so if Ricci spreads I’m going to read if they start with an abstract which I shall read and then I’ll tell you a bit more about the spread as we go on so the double page spread starts the semi-legendary figure right of Pythagoras was born on the island of Samos in the Aegean Sea in his youth he studied mathematics astronomy philosophy and music possibly around 520 BC he left Samos to go to the Greek sea port of Crotona now in southern Italy and formed a philosophical school now known as the pythagorean’s this is a picture of Pythagoras from Raphael’s School of Athens the inner members of the pythagorean’s called the mathematic boy apparently a beta strict regime having no personal possessions and eating only vegetables except beans and the sect is open to both men and women the Pythagorean studied mathematics astronomy and philosophy and they believe that everything is created from whole numbers that anything worthy of study can be quantified and in particular for the pythagorean’s arithmetic meant studying whole numbers which they sometimes represented geometrically for example as you can see on the left square numbers are formed by square patterns of dots or pebbles and using such pictures they could show that square numbers are obtained by adding consecutive odd numbers starting from one so as you can see at the bottom 16 is one plus three plus five plus seven and similarly triangular numbers performed by triangular patterns of dots and as you can see at the bottom of the right-hand bit they could see that the sum of any two consecutive triangular numbers is a square number for example 10 plus 15 is 25 well Pythagorean is also experimented with music and in particular linking certain musical intervals to simple ratios between small numbers they may have discovered these ratios by plucking strings of different lengths and comparing the notes produced for example an octave results from having the length of a string giving a frequency ratio of 2 to 1 while a perfect fifth results from stopping it at 2/3 its length giving a ratio of 3 to 2 this 1492 woodcut features some of Pythagoras is supposed musical experiments finally of course we have missed mentioned the Pythagorean theorem on right-angled triangles although no contemporary historical evidence links it to Pythagoras himself although known to the Mesopotamians a thousand years earlier it was probably the Greeks who were the first to prove it geometrically it says that if we draw squares on the size of a right-angled triangle then the area of the square on the longest side is the sum of the areas of the squares on the other two sides in the picture Z equals x Plus why so for a right angle triangle with sides of length a B a B and C we can write the algebraic equation a squared plus B squared equals C squared although the Greeks would never have written it in that way for example for the 3 4 5 triangle 3 squared plus 4 squared equals 5 squared by way of complete contrast are we then going the chap on in the chapter to talk about the Chinese the Indians and the Mayans so let’s look at the Mayans one of the most interesting counting systems is that of the Mayans of Central America throughout their most productive years from 300 to 1000 AD the Mayans were situated over a large area centered on present-day Guatemala and Belize and extending from the Yucatan Peninsula of Mexico in the north to Honduras in the south most of their calculations involve the construction of calendars for which they developed a place value system a place

value accounting system based mainly on the number 20 well our knowledge of the counting system of their calendars derives mainly from writings on the walls of caves and ruins and from hieroglyphic inscriptions on carved pillars and from a handful of painted manuscripts called codices such as the beautiful Dresden codex from about 1200 ad these codices were intended to guide Mayan priests in ritual ceremonies involving hunting planting and rain making but many of these courtesies were destroyed by the Spanish conquerors who arrived in this area after the Year 1500 you can see a number of patterns on here with lines and dots and this is their their counting system so specifically the Mayan counting system was a place value system with a dot to represent one align to represent five and a special symbol bit like a shell to represent zero and combining these gave the numbers from 0 to 19 so they were one of the earliest cultures to have a zero if they wanted to obtain larger numbers they combine these numbers vertically for example as you can see on the right there’s a bit from our earlier codex and it shows a symbol for 12 above a symbol for 13 and this represents the number 12 20s plus 13 or 253 and in fact they could deal with very very large numbers by having sufficiently tall towers of these numbers so in order to keep track of the passage of time the Mayans used two types of calendar with 260 days and 365 days the 260 day calendar shown here was a ritual one consisting of 13 months of 20 days each day combined a month number from 1 to 13 with one of 20-day pictures shown on the right named after deities such as a mix eke and Iqbal and if you intermesh these two systems as you can see you then get a cycle of 13 times 20 or 260 days for their 365 day calendar they needed to adapt their number system to take account of the number of days in the calendar year so to do this they introduced an 18 into their 20 based system because 1820 a 360 and then we added five extra so-called inauspicious days to make up the full 365 days these two calendars operated independently but they’re also combined to give a calendar round in which the number of days is the least common multiple of 260 and 365 which gives you 18 thousand nine hundred and eighty or 52 calendar years and as I said before they had no problems with dealing with large numbers of this kind they went further in these periods of 52 years were then packaged into even longer time periods the longest of which was their Long Count calendar of 5125 years I don’t want to worry you but the current Long Count comes to an end towards the end of next year we now move on to the chapter two and I’d like to hand over to Raymond who’s going to tell you about about two spreads from there yes the two I’m going to tell you about in Chapter two the European mathematics of the Middle Ages and the Renaissance are Fibonacci and our Fibonacci and then later on about Mercator Leonardo of Pisa known since the 19th century is Fibonacci some son of Bonacci Oh is remembered mainly for his Libra bocce his book of calculations which used to popularize the hindu-arabic numerals and also for a number sequence named after him his work was crucial in bringing Arabic mathematics to wider recognition in Western Europe he was born in Pisa after travelling widely throughout the Mediterranean he returned home and wrote works expanding on what he had learned to help his countrymen deal with calculation and with commerce most of our knowledge about Fibonacci comes from the prologue of his influential book Libra bocce the first edition of which

appeared in 1202 it covers four main areas starting with the use of hindu-arabic numerals in calculation and then using them for the mathematics needed in business the largest part of the book deals with recreational mathematical problems finishing with operations on roofs and little geometry Fibonacci’s Libra bachi contains a wide range of problems including several that we list in the book the famous of these is the problem of the rabbits a farmer has a pair of baby rabbits rabbits take two months to reach maturity and then give birth to another pair each month how many pairs of rabbits are there after a year it can be shown that the number of rabbits in each month follows the so called Fibonacci sequence which has written at the bottom of the screen 1 1 2 3 5 in which each successive number is the sum of the previous two for example 89 equals 34 plus 55 the answer to the problem is therefore the twelfth number in the sequence 144 it’s a fascinating sequence for example the reissues of successive terms of the Fibonacci sequence are 1 over 1 2 over 1 3 over 2 5 over 3 and these 10 to the gold numbers so-called gold number 1/2 into 1 plus root 5 which is about 1.6 1 it and the number has pleasing and remarkable properties for example it squares to point 6 1 8 and it’s reciprocal is 0.618 a rectangle whose sides are in the reissue 1.618 to 1 is often considered to have the most pleasing ship neither too thin nor too fat the picture shows how the Fibonacci numbers can be arranged so as to give rise to a spiral pattern further rectangles can be added to the pattern at will similar spirals occur through nature on a nautilus shell and then the pattern of seats and the sunflower for example the number of seats in any spiral pattern is often 34 45 or 89 all of which are Fibonacci numbers the other spread we’re going to look at from this chapter is a celebrated figure from the 16th century Mercator the Flemish map neighbor map maker and cartographer gerardus mercator is mainly remembered for the Mercator action which proved to be an extremely useful projection and map for navigators this was a projection of the spherical earth onto a flat sheet of paper so the lines of latitude and of longitude as well as the paths of compass constant compass bearing were represented by straight lines a major concern in the sixteenth century an active period for voyages of Treatt and discovery was to develop mathematical methods and maps to eard navigation picture of a ship the basic problem was if you’re on a ship in the middle of the ocean how can you tell where you are and in which direction you should seal to get to your destination you could find your latitude by using astronomical instruments to locate the Sun and stars much more problematic was to find your longitude and the satisfactory method wasn’t available until the end of the 18th century using a magnetic compass Mariners could steer a line of constant compass bearing a rum line such a path crosses all lines of longitude at the same angle however is the 16th century Portuguese multiplication and cosmography teju Nunez discovered a long line spirals towards the pole Mercator obtained his projection by projecting the sphere onto a cylinder as indicated in the left of this diagram which was then unrolled the cylinder was then unrolled and stretched vertically so the rum lines became stripped the amount of stretching increases the further north one goes this is the consequence of exaggerating areas that are far from the equator for example Alaska appears as large as Brazil when Brazil is actually five times bigger I’m Finland appears to have a greater north-south extent than India which it doesn’t the advantages of Mercator projection where the represented lines of latitude and longitude are straight lines meeting at right angle I’m not at all so represented rhumb lines are straight lines on the map if a

navigator knew the latitude and longitude of the ship’s current position and of the destination then the line joining the two places could be find on the map this enabled the appropriate constant compass bearing to be determined but of course didn’t give the shortest distance to the destination but least you’re getting there Makita coined the word atlas for a collection of maps he’s shown here in the title page of his outlets with two bogus Honda’s his publisher surrounded by the tools of the cartographer Mercator didn’t present the mathematical basis for his projection this was first given by Edward Wright in his influential certain errors in navigation of 1599 right also give accurate mathematical tables for its construction but it was Thomas Harriot we eventually discovered the fundamental mathematical formula underlying Mercator projection so Robin night went to Chapter three I was tempted to ask Raymond where the Fibonacci is program of the rabbits could be described as a hair-raising problem but I decided not to so in Chapter 3 awakening an Enlightenment which is on the 17th and 18th centuries double-page spreads begin with Napier and Briggs the inventors of logarithms in 1614 John Napier 8th Laird of Medicine near Edinburgh introduced logarithms as an aid to mathematical calculation they were designed to replace lengthy calculations involving multiplications and divisions by simpler ones using additions and subtractions being somewhat awkward to use they were soon supplanted by simpler ones due to Henry Briggs and their use proves an enormous boon to navigators and astronomers here you can see Napier and you can see the title page of his famous book on logarithms but all the ideas on logarithms had appeared before then around the Year 1500 Nicholas shook a and Michael Stifel listed the first few powers of two alleges that to multiply two of them once simply adds their exponents for example for example to multiply 2 to the power 4 and 2 to the power 7 you get 2 to the foot 2 to the power 11 the idea is not developed until no cure produced his description of the admirable table of logarithms containing extensive tables of logarithms of the signs and tangents of all the angles from 0 to 90 degrees increasing in steps of just one minute at a time quite remarkable how did Napier define his logarithms because he considered two points moving along straight lines the first shown at the top travels a constant speed forever while the second representing its logarithm moves from a fixed point along a straight line in such a way that at each point its speed is proportional to the distance if still has to travel so it’s slowing down of course it follows from Napier’s definition that log of a B is log of a cost log of B minus log of 1 for any numbers a and B so unfortunately when you go to do a calculation you always have to subtract the log of 1 which is very inconvenient shortly after their invention Henry Briggs who of course was the first professor of geometry at Gresham College realized that Napier logarithms were cumbersome and he felt that they could be redefined so as to avoid subtracting the term log one in every calculation and as he described by myself when expounding this doctrine to my auditors aggression college remarked that it would be much more convenient that zero should be kept for the logarithm of the whole sign this picture by the way shows is an early picture of question College and in the quadrangle the top right are the rooms of Henry Briggs used to occupy while he worked on his logarithms Briggs

twice visited Edinburgh Edinburgh displays Napier and sort out the difficulties and on returning to London he devised his logs to base-10 in which the log of one is zero and the log of 10 is 1 and to multiply numbers we simply add their logarithms log of a B is log of a plus log of B and while still at Gresham College he produced his first set of tables of logarithms which you can see here and then in 1624 after he had left London to become the first civilian professor of geometry in Oxford Briggs produced an extensive collection of base 10 logarithms for no fewer than 30,000 integers all calculated by hand to 14 decimal places and to do that he had to take the square root of 10 and then the square root of that and the square root of that and he iterated that no fewer than 54 times meanwhile Napier had also constructed from ivory a set of rods with numbers marked on them now called maples bones or Napier’s rods which could be used to multiply numbers mechanically his invention of logarithms quickly led to the development of further mathematical instruments based on logarithmic scales and most notable about me among these was the slide rule versions of which appeared around 1630 and were widely used for over 300 years until the advent of the pocket calculator in the 1970s I’m now going to hand back to Raymond who’s going to tell us about gravitation but before that just say that I was listening to a lecture by John Barrow who’s Robin’s successor as Grossman professor of geometry and in the course of the lecture he he said he he typed log tables into his search engine and he got an article the only thing he could find was an article and hard to make tables out of logs so write newton successors the appearance of Newton’s Principia and 1687 caused the sensation the scientists were puzzled by the nature of an attractive force of gravity that could apparently act over astronomical distances the Huygens in particular it was an absurd idea that was not capable explaining anything preferable to them with some sort of mechanical theory such as that of Descartes in which the planets were swept along by vortices like leaves in a world pill but there were too many areas in which Newton’s theory caused difficulties the shape of the earth and the motion of the moon those were important for navigation and both of them Newton’s Furies were eventually vindicated and the year after Newton’s death the great French philosopher Voltaire and this is not falter I shall explain in a minute rude about the different world views in France and England in Paris they see the universe as composed of vortices of sudden matter in London they should nothing of the kind for us it is the pressure of the moon that causes the tides of the sea for the English it is to see the gravity it’s towards the moon Voltaire was well-placed to comet to comment as he had the expertise of Emily Marquis du ch√Ętelet shown here to inform him she was the gifted mathematician who translated Newton’s Principia into French Volta continued in Paris you see the Earthship like a lemon in London it is flattened on two sides for Newton the Earth’s rotation causes a flattening of the poles whereas for Descartes there’s an elongation there to decide on the actual ship the Paris Academy said to geodetic expeditions one to Peru and the other to Lapland eventually booth reported and Newton was proved right the earth is indeed flatter at the poles although Newton felt effectively with the motion of two bodies moving under gravity the motion of the moon depends both on the earth and on the Sun it’s an example of a three body problem without the Sun the motion of the moon would be an ellipse Newton simplified the problem by assuming that the Sun causes the moon’s elliptical orbit to revolve slowly his calculation was that it would

take 18 years for the orbit to return to its original position but observations showed that it took only 9 years by the end of the 1740s Newton’s theories were under investigation by those who best understood them such as Clairol that’s our hero Newton and Clairol who taken part in the Lapland expedition tomorrow initially proposed to modify Newton’s inverse square law of gravitation by adding an additional term to it it seems that Newton’s law of gravitation might be wrong but then in 1749 he made a dramatic retraction when he took a new approach to the differential equations that describe the moon’s motion finding that the previous difference between theory and observation was due to the way in which these equations have been approximated so moving to Chapter four then the age of revolutions represent a range of topics and I’m going to present one about Charles Babbage and ADA Lovelace and then Robin will present one on Florence Nightingale the central figure of 19th century computing was Charles Babbage who may be said that pioneered the modern computer age with his different engines and his analytical engine a Louis influence and subsequent generations is hard to assess either kinds of loveliest daughter of Lord Byron and a close friend of Babbage produced a perceptive and clear commentary on the pars and potential of the analytical engine this was essentially an introduction to what we know I call programming Charles Babbage and John Herschel were asked by the Royal Astronomical Society to produce new astronomical tables it was this the cause Babbage to design a calculating machine he wanted to calc II wanted to mechanize the calculation of a polynomial formula by looking at the difference between his successive values this approach can be applied to any polynomial because continuing to take differences eventually use constant values also many functions of interest which are not polynomials like the sine function and the logarithm function can be approximated by polynomials however the construction of the difference engine ran into engineering financial and political difficulties and construction ended near times there three here’s a portion of it as recreated in the 1990s it was to have the feature of being able to print its results as more errors arose in printing and improve reading than an original calculations garbage wondered whether his Difference Engine could act upon the results of his own of its own calculations or as he put it the engine eating its own tail with this in mind he designed a new engine his analytical engine basing its control system on the punched cards used by Jacques hard for his automatic loom the loveliest became interested in the analytical engine describing what it could do and how it could be instructed and giving what is considered to be the first computer program she wrote the distinctive characteristic of the analytical engine is the introduction into it of the principle which jacquard devised for regulating by means of punch cards the most complicated patterns in the fabrication of brocade stuffs it is in this that the distinction between the two engines lies nothing of the sort exists in the difference engine we may say most are claim that the analytical engine moves algebraically patterns just as the J car jacquard loom weaves flowers and leaves and the low the analytical engine was never built modern scholarship is of the view that if it had been constructed it would have worked as Babbage had intended and the name era is now given to a programming language developed for the United States Department of Defense so he now turns in the area of statistics with Florence Nightingale and Adolphe kettle a Florence Nightingale the lady with the lamp who saved lives during the Crimean War was also a fine statistician who collected and analyzed

mortality data from a Crimea and displayed them on her polar diagrams a forerunner of the pie chart her work is strongly influenced by that of the belgian statistician Adolphe kettle a kettle a was supervisor of Statistics for Belgium pioneering techniques for taking the national census his desire to find us these statistical characteristics of an average man led to his compiling the chest measurements of 5,400 5732 Scottish soldiers and observing that the results were arranged around a mean of 40 inches according to the normal or Gaussian distribution and on the right you can see his own diagram his investigations helped to lay the foundations of modern actuarial science Florence Nightingale showed an early interest in mathematics at the age of nine she was already displaying data in tabular form she regarded statistics as the most important science in the world and used statistical methods to support her efforts of the administrative and social reform she was the first woman to be elected a fellow of the Royal statistical Society and an honorary foreign member of the American Statistical Association by 1852 she had established a reputation as an effective administrator and project manager her work on the professionalization of Nursing led to her accepting the position of Superintendent of a female nursing establish in the English general military hospitals in Turkey for the British troops fighting in the Crimean War she arrived in 1854 and was appalled at what she found there in attempting to change attitudes and practices she made use of pictorial diagrams for statistical information developing her polar area graphs as shown here these pictures have 12 sectors one for each month and they reveal changes over the year in the deaths from wounds obtained in battle from diseases from other causes they show dramatically in the extent of the needless deaths among the soldiers during the Crimean War and were used to persuade medical and other professionals that death could be prevented if sanitary and other reforms were made on her return to London in 1858 she continued to use statistics statistics to inform an inference public health policy she argued for the inclusion in the 1861 census of questions on the number of sick people in a household and on the standard of housing because she realized the important relationship between health and housing in another initiative she tried to educate members of the government in the usefulness of statistics and to influence the future by establishing the teaching of the subject in the universities for Nightingale and the collection of data was only the beginning her subsequent analysis and interpretation were crucial and led to medical and social improvements and political reform all with the aim of saving lives so with this we now move to our final chapter the modern age covering essentially the 20th century which opens was an important event in the history of our subject on the 8th of August 1904 the greatest mathematicians of the day gave the most celebrated mathematical lecture of all time for it was on this date at the International Congress of mathematicians in Paris that David Hilbert presented a list of unsolved problems the 20th century mathematicians to tackle trying to solve these problems helped to set the mathematical agenda for the next hundred years Gilbert’s mathematical range was immense from abstract number theory and invariant

theory to more practical topics such as potential theory and the kinetic theory of gases following Cantor’s introduction of set theory and later investigations by various mathematicians into the foundations of arithmetic Hilbert became increasingly involved with the foundations of geometry and in 1899 he produced an influential book in which he developed foolproof axiom systems for Euclidean geometry and for projective geometry and four years later in the second edition of that book he also act sympathised the so-called non-euclidean geometries as he said in his famous Paris lecture who of us would not be glad to lift the veil behind which the future lies hidden to cast a glance at the next advances of our science and at the secrets of its development during future centuries so Hilbert asked this in his Paris lecture in which he presented his list of 23 unsolved problems these include the famous Riemann hypothesis which remains unsolved to this day let’s look at two more of these somewhat simpler ones problem 3 the problem says given two polyhedra with the same volume can we always cut the first into finitely many pieces which can then be reassembled jigsaw like to give the second well to see what the problem involves let’s go down to two dimensions because in 1833 the Hungarian Janos Bolyai had proved that if two polygons have the same area then the first can be cut into pieces that can be arranged to give the second and here you can see an example in which a triangle is reassembled to give a square Helens problem asks whether a similar result holds in three dimensions or in fact in higher dimensions from that but the answer is no within two years of Hilbert supposing the problem the German max Dane proved that a regular tetrahedron cannot be cut in two pieces that can then be reassembled to give a cube with the same volume problem 18 is in three parts all to do with geometry and here is probably the most famous of them what is the most efficient way to stack spheres so the amount of empty space between them is kept as small as possible well this problem is considered by Harriette and Kepler many centuries earlier in two possible ways of stacking spheres are the cubic stacking shown at the top and also hexagonal stacking but it turns out that neither is the most efficient in fact it turns out the way your greengrocer stacks oranges is the best because they’re the proportion of empty space is about 0.36 which is less than the point four eight and point four zero of the other two but to prove this result rigorously was absolutely horrendous in 1998 Thomas Hales gave a computer-aided proof involving no less than three gigabytes of computer power and that hand back to Raymond to do the other topic from Chapter five the last figure the last spread I want to talk about it’s about one of the greatest figures of all time Albert Einstein an iconic figure of the 20th century he was the greatest mathematical physicist since Isaac Newton he revolutionized physics with his theories of special and then general relativity the instru and mathematical ideas not previously used in physics some of which had been developed by Riemann and by Hermann Minkowski Einstein was born in southern Germany slowly learning to speak he showed little promise in his early schooling he was admitted to Zurich Polytechnic at the second attempt and although one of his lectures was makovski he gained little from the formal teaching I’m preferred to read

independently after graduation he worked for the Swiss Patent Office and burn in 1905 Einstein submitted a paper on special relativity to the University of Bern in support of the doctorate but it was rejected however recognition for his work soon arrived as it became more widely known he announced his general theory of relativity in 1915 but was awarded the Nobel Prize in 1921 for his work in quantum theory rather than relativity in 1933 he went to America and eventually to the Institute for studies in Princeton in 1905 his year of Wonders Einstein published four papers of groundbreaking importance one of these on the electrodynamics of moving bodies introduced a new theory linking time distance mass and energy it was consistent with electromagnetism but omitted the force of gravity this became known as the special theory of relativity and assumed that C the speed of light it’s constant irrespective of where you are or how you move another 1905 paper contains one of the most famous equations of all e equals mc-squared asserting the equivalence of mass and energy Hermann Minkowski was born of German parents in Lithuania and 1902 moved to become a colleague of Hilbert and they’re the mathematical foundation of the theory of relativity by developing a new view of space and time no known as space-time this is a four-dimensional non Euclidean geometry that incorporates the three dimensions of space with the one of time and it comes with a way of measuring the distance between two different points of space-time space and time were now no longer separate as Newton had thought but are intermixed Einstein initially thought very little of mid-calf skis approached the space-time but later find it essential when he was trying to extend his theory to include gravity Einstein’s general theory of gravity gravity building also in Raymond’s geometrical ideas for just space-time that was curved as a result of the presence of mass and energy the curvature increased near to massive bodies and it was the curvature of space-time that controlled the motion of bodies the theory predicted that light rays will be bent by the curvature of space-time produced by the Sun an effect that was observed during the 1919 ciliary letter concludes by show you the last double page spread in the book on the fields medalists mathematics is developing as an ever-increasing rate indeed more new mathematics has been discovered since the Second World War then was known up to that time by the time my grandchildren retire more new mathematics will be discovered than is known now one outcome of this tremendous activity has been the regular series of international congresses of mathematicians these are held every four years for the discussion of the latest advances and the award to outstanding young mathematicians of the fields medals shown here on on the left often regarded as the mathematical equivalent of the Nobel Prize so we chose to conclude our book with a double page spread that outlines the history of these congresses and presents a full list of the Fields Medal winners over 70 years and from many countries in this lecture we’ve had the opportunity to show you only eleven of the hundred or so a double-page spreads and of course we want to see all the rest so can I suggest that we now have a break there is a glass of wine or fruit juice in the other room I suggest you help yourself to that and by that time we’ll have the table ready so you can all rush and get your copies of the book thank you very much you