The Logic of Shape First steps in geometry There are two main types of reasoning in mathematics: symbolic and visual. Symbolic reasoning originated in number notation, and we will shortly see how it led to the invention of algebra, in which symbols can represent general numbers ('the unknown') rather than specific ones ('7'). From the Middle Ages onwards, mathematics came to rely increasingly heavily on the use of symbols, as a glance at any modern mathematics text will confirm. The beginnings of geometry As well as symbols, mathematicians use diagrams, opening up various types of visual reasoning. Pictures are less formal than symbols, and their use has sometimes been frowned upon for that reason. There is a widespread feeling that a picture is somehow less rigorous, logically speaking, than a symbolic calculation. It is true that pictures leave more room for differences of interpretation than symbols. Additionally, pictures can contain hidden assumptions — we cannot draw a 'general' triangle; any triangle we draw has a particular size and shape, which may not be representative of an arbitrary triangle. Nonetheless, visual intuition is such a powerful 26 THE LOGIC OF SHAPE It-aii i re of the human brain that pictures play a prominent role in mathematics. In fact, diey introduce a second major concept into the •.nliject, after number. Namely, shape. Mathematicians' fascination with shapes goes back a long way. I here are diagrams on Babylonian tablets. For example, the tablet catalogued as YBC 7289 shows a square and two diagonals. The sides ■ 11 the square are marked with cuneiform numerals for 30. Above inn- diagonal is marked 1 ;24,51,10, and below it 42;25,35, which |l its product by 30 and therefore the length of that diagonal. So I. .'.4,51,10 is the length of the diagonal of a smaller square, with Ides 1 unit. Pythagoras's Theorem tells us that this diagonal is the Mjiiare root of two, which we write as /I.The approximation i1,51,10 to /2 is very good, correct to 6 decimal places. The first systematic use of diagrams, together with a limited use ■ symbols and a heavy dose of logic, occurs in the geometric II ii i rigs of Euclid of Alexandria. Euclid's work followed a tradition ih.u went back at least to the Pythagorean cult, which flourished I.....nd 500 bc, but Euclid insisted that any mathematical statement must be given a logical proof before it could be assumed to be true. Si > Kuclid's writings combine two distinct innovations: die use of I hi lures and the logical structure of proofs. For centuries, the word .....ietry' was closely associated with both. In this chapter we follow the story of geometry from Pythagoras, ill rough Euclid and his forerunner Eudoxus, to the late period of 1| isical Greece and Euclid's successors Archimedes and Apollonius. I" early geometers paved the way for all later work on visual pinking in madiematics.They also set standards of logical proof that wt'ic not surpassed for millennia. Pythagoras »l.i) we almost take it for granted that mathematics provides a key I the underlying laws of nature. The first recorded systematic i Innking along those lines comes from the Pythagoreans, a rather 27 TAMING THE INFINITE mystical cult dating from around 500 bc. Its founder, Pythagoras, was born on Samos around 569 bc. When and where he died is a mystery, but in 460 bc the cult that he founded was attacked and destroyed, its meeting places wrecked and burned. In one, the house of Milo in Croton, more than 50 Pythagoreans were slaughtered. Many survivors fled to Thebes in Upper Egypt. Possibly Pythagoras was one of them, but even this is conjectural, for legends aside, we know virtually nothing about Pythagoras. His name is well known, mainly because of his celebrated theorem about right-angled triangles, but we don't even know whether Pythagoras proved it. We know much more about the Pythagoreans' philosophy and beliefs. They understood that mathematics is about abstract concepts, not reality. However, they also believed that these abstractions were somehow embodied in 'ideal' concepts, existing in some strange realm of the imagination, so that, for instance, a circle drawn in sand with a stick is a flawed attempt to be an ideal circle, perfecdy round and infinitely thin. The most influential aspect of the Pythagorean cult's philosophy is the belief that the universe is founded on numbers. They expressed this belief in mythological symbolism, and supported it with empirical observations. On the mystic side, they considered the number 1 to be the prime source of everything in the universe. The numbers 2 and 3 symbolized the female and male principles. The number 4 symbolized harmony, and also the four elements (earth, air, fire, water) out of which everything is made. The Pythagoreans believed that the 4 9 number 10 had deep mystical significance, because 10=1+2 + 3+4, combining prime unity, the female principle, the male principle and the four elements. Moreover, these numbers formed a triangle, and the whole of Greek geometry hinged upon properties of triangles. T I III I Hull, ill I'.IIAI' • • • • • • • Hu muni nrten forms a lM.lll!|l(! Harmony of the World The main empirical support for the Pythagorean concept of a numerical universe came from music, where they had noticed some remarkable connections between harmonious sounds and simple numerical ratios. Using simple experiments, they discovered that if a plucked string produces a note with a particular pitch, then a string half as long produces an extremely harmonious note, now called the octave. A string two-thirds as long produces the next most harmonious note, and one three-quarters as long also produces a harmonious note. Today these numerical aspects of music are traced to the physics of vibrating strings, which move in patterns of waves. The number of waves that can fit into a given length of string is a whole number, and these whole numbers determine the simple numerical ratios. If the numbers do not form a simple ratio then the corresponding notes interfere with each other, forming discordant 'beats' which are unpleasant to the ear. The full story is more complex, involving what the brain becomes accustomed to, but there is a definite physical rationale behind the Pythagorean discovery. The Pythagoreans recognized the existence of nine heavenly bodies, Sun, Moon, Mercury, Venus, Earth, Mars, Jupiter and Saturn, plus the Central Fire, which differed from the Sun. So i mportant was the number 10 in their view of cosmology that they believed there was a tenth body, Counter-Earth, perpetually hidden from us by the Sun. As we have seen, the whole numbers 1,2,3,..., naturally lead to a second type of number, fractions, which mathematicians call rational numbers. A rational number is a fraction a/b where a, b are whole numbers (and b is non-zero, odierwise the fraction makes no sense). Fractions subdivide whole numbers into arbitrarily fine parts, so that in particular the length of a line in a geometric figure I'M 29 TAMING THE INFINITE THE LOGIC OF SHAPE can be approximated as closely as we wish by a rational number. It seems natural to imagine that enough subdivision would hit the number exactly; if so, all lengths would be rational. If this were true, it would make geometry much simpler, because any two lengths would be whole number multiples of a common (perhaps small) length, and so could be obtained by fitting lots of copies of this common length together.This may not sound very important, but it would make the whole theory of lengths, areas and especially similar figures — figures with the same shape but different sizes — much simpler. Everything could be proved using diagrams formed from lots and lots of copies of one basic shape. Unfortunately, this dream cannot be realized. According to legend, one of the followers of Pythagoras, Hippasus of Metapontum, discovered that this statement was false. Specifically, he proved that the diagonal of a unit square (a square with sides one unit long) is irrational: not an exact fraction. It is said (on dubious grounds, but it's a good story) that he made the mistake of announcing this fact when the Pythagoreans were crossing the Mediterranean by boat, and his fellow cult-members were so incensed that they threw him overboard and he drowned. More likely he was just expelled from the cult. Whatever his punishment, it seems that the Pythagoreans were not pleased by his discovery. The modern interpretation of Hippasus's observation is that v'Z is irrational.To the Pythagoreans, this brutal fact was a body-blow to their almost religious belief that the universe was rooted in numbers — by which they meant whole numbers. Fractions — ratios of whole numbers - fitted neatly enough into this world-view, but mi in hers that were provably not fractions did not. And so, whether drowned or expelled, poor Hippasus became one of the early \ k iiins of the irrationality, so to speak, of religious belief. Taming irrationals ■ iiiually, the Greeks found a way to handle irrationals. It works ■ ■ -i iiso any irrational number can be approximated by a rational ......lher.The better the approximation, the more complicated thai .......lal becomes, and there is always some error. But by making the ■ rror smaller and smaller, there is a prospect of approaching the properties of irrationals by exploiting analogous properties of ■ I 'I iroximating rational numbers. The problem is to set this idea up mi .1 way that is compatible with the Greek approach to geometry i in I proof. This turns out to be feasible, but complicated. The Greek theory of irrationals was invented by Eudoxus around t /() bc. His idea is to represent any magnitude, rational or irrational, ■. i lie ratio of two lengths - that is, in terms of a pair of lengths, i In is Lwo-thirds is represented by two lines, one of length two and Pythagoras's Theorem: ■I II10 triangle has a right .....lie, then the largest i |i i. ire, A, has the same iroa as the other two, B ii i< I C, combined TAMING THE INFINITE THE LOGIC OP SHAPE one of length three (a ratio 2:3). Similarly, v'2 is represented by the pair formed by the diagonal of a unit square, and its side (a ratio V2:1). Note that both pairs of lines can be constructed geometrically. The key point is to define when two such ratios are equal. When is a.b = c:d? Lacking a suitable number system, the Greeks could not do this by dividing one length by the other and comparing a-H with c~M. Instead, Eudoxus found a cumbersome but precise method of comparison that could be performed within the conventions of Greek geometry. The idea is to try to compare a and c by forming integer multiples ma and nc. This can be done by joining m copies of a end to end, and similarly n copies of c. Use the same two multiples m and n to compare mb and nd. If the ratios a:b and c.d are not equal, says Eudoxus, then we can find m and n to exaggerate die difference, to such an extent that ma > nc but mb < nd. Indeed, we can define equality of ratios that way. This definition takes some getting used to. It is tailored very carefully to the limited operations permitted in Greek geometry. Nonetheless, it works; it let the Greek geometers take theorems that could easily be proved for rational ratios, and extend them to irrarional ratios. Often they used a method called 'exhaustion', which let them prove theorems that we would nowadays prove using the idea of a limit and calculus. In this manner they proved that the area of a circle is proportional to the square of its radius. The proof starts from a simpler fact, found in Euclid: the areas of two similar polygons are i n the same proportion as the squares of corresponding sides. The circle poses new problems because it is not a polygon. The Greeks therefore considered two sequences of regular polygons whose vertices are on the circle: one inside the circle, the other outside, llntli sequences get closer and closer to the circle, and Eudoxus's definition implies that the ratio of the areas of the approximating I m ilygons is the same as the ratio of the areas of the circles. luclid Mm best-known Greek geometer, though probably not the most iiia! mathematician, is Euclid of Alexandria. Euclid was a great lynihesizer, and his geometry text, the Elements, became an all-time IwsisHler. Euclid wrote at least ten texts on mathematics, but only live < if them survive - all through later copies, and then only in part. ■ luve no original documents from ancient Greece. The five I in Iilay that particular game must accept the rules; if they don't, they .iir free to play a different game, but it won't be the one determined lay iliose particular rules. hi Euclid's day, and for nearly 2000 years afterwards, mathematicians didn't think that way at all.They generally viewed i In- axioms as self-evident truths, so obvious that no one could leriously question them. So Euclid did his best to make all of his louts obvious — and very nearly succeeded. But one axiom, the 'parallel axiom', is unusually comphcated and unintuitive, and many people tried to deduce it from simpler assumptions. Later, i 'II see the remarkable discoveries that this led to. I'iom these simple beginnings, the Elements proceeded, step by M 35 TAMING THE INFINITE Euclid of Alexandria 325-265BC Euclid is famous for his geometry book the Elements, which was a prominent - indeed, the leading - text in mathematical teaching for two millennia. We know very little about Euclid's life. He taught at Alexandria. Around 45 bc the Greek philosopher Proclus wrote: 'Euclid... lived in the time of the first Ptolemy, for Archimedes, who followed closely upon the first Ptolemy, makes mention of Euclid... Ptolemy once asked [Euclid] if there were a shorter way to study geometry than the Elements, to which he replied that there was no royal road to geometry. He is therefore younger than Plato's circle, but older than Eratosthenes and Archimedes ... he was a Platonist, being in sympathy with this philosophy, whence he made the end of the whole Elements the construction of the so-called Platonic figures [regular solids}.' step, to provide proofs of increasingly sophisticated geometrical theorems. For example, Book I Proposition 5 proves that the angles at the base of an isosceles triangle (one with two equal sides) are equal. This theorem was known to generations of Victorian schoolboys as the pons asinorum or bridge of asses: the diagram looks like a bridge, and it was the first serious stumbling block for students who tried to learn the subject by rote instead of understanding it. Book I Proposition 32 proves that the angles of a triangle add up to 180°. Book I Proposition 47 is Pythagoras's Theorem. Kuclid deduced each theorem from previous theorems and \ .n'iciiis axioms. He built a logical tower, which climbed higher and higher towards the sky, with the axioms as its foundations and il ■ l> ■ Iik iii >n as the mortar that bound the bricks together. I' m lay we are less satisfied with Euclid's logic, because it has many ľ■.. F.uclid takes a lot of things for granted; his list of axioms is 1 here near complete. For example, it may seem obvious that if i luir passes through a point inside a circle then it must cut the circle ininewhere - at least if it is extended far enough. It certainly looks ohvii mis if you draw a picture, but there are examples showing thai ii • li ics not follow from Euclid's axioms. Euclid did pretty well, but In- .issumed that apparently obvious features of diagrams needed m n her proof nor an axiomatic basis. This omission is more serious than it might seem.There are some l.i i n< ms examples of fallacious reasoning arising from subtle errors in pictures. One of them 'proves' that every triangle has two equal lidos. The golden mean li< >nk V of the Elements heads off in a very different, and rather obscure, direction from Books I—IV It doesn't look like conventional geometry. In fact, at first sight it mostly reads like gobbledegook. What, for instance, are we to make of BookV Proposition 1 ? It reads: If certain magnitudes are equimultiples of other magnitudes, then whatever multiple one »f the magnitudes is of one of the others, that multiple also will be of all. The language (which I have simplified a little) doesn't help, but ihe proof makes it clear what Euclid intended. The 19th-century linglish mathematician Augustus De Morgan explained the idea in iniple language in his geometry textbook: 'Ten feet ten inches makes ten times as much as one foot one inch.' What is Euclid up to here? Is it trivialities dressed up as theorems? Mystical nonsense? Not at all. This material may seem obscure, but it leads up to the most profound part of the Elements: Rudoxus's techniques for dealing with irrational ratios. Nowadays iii.ifhematicians prefer to work with numbers, and because these ire more familiar I will often interpret the Greek ideas in that language. ■M TAMING THE INFINITE Euclid could not avoid facing up to the difficulties of irrational numbers, because the climax to the Elements - and, many believe, its main objective — was the proof that there exist precisely five regular solids: the tetrahedron, cube (or hexahedron), octahedron, dodecahedron and icosahedron. Euclid proved two things: there are no other regular solids, and these five actually exist - they can be constructed geometrically, and their faces fit together perfectly, with no tiny errors. Two of the regular solids, the dodecahedron and the icosahedron, involve the regular pentagon: the dodecahedron has pentagonal faces, and the five faces of the icosahedron surrounding any vertex determine a pentagon. Regular pentagons are directly connected with what Euclid called 'extreme and mean ratio'. On a line AB, construct a point C so that the ratio ABAC is equal to AC:BC.That is, the whole line bears the same proportion to the larger segment as the larger segment does to the smaller. If you draw a pentagon and inscribe a five-pointed star, the edges of the star are related to the edges of the pentagon by this particular ratio. ^ +^ Nowadays we call this ratio the golden mean. It is equal to ^ , and this number is irrational. Its numerical value is roughly 1.618. The Greeks could prove it was irrational by exploiting the geometry iln■ i.iiin ill ihe diagonals I'll' Ml I.-MM lOldOII Extreme and mean ratio (now called the golden mean). The ratio of the top line to the middle one is equal to that of the middle one to the bottom one I THE LOGIC OF SHAPE Í the pentagon. So Euclid and his predecessors were aware that, for ■ |in>|>er understanding of the dodecahedron and icosahedron, iln \ must come to grips with irrationals. Iliis, at least, is the conventional view of the Elements. David l' iwler argues in his book The Mathematics of Plato s Academy that there ľ. .in alternative view - essentially, the other way round. Perhaps Ui lu l's main objective was the theory of irrationals, and the regular ■ 'In Is were just a neat application.The evidence can be interpreted ■ Ither way, but one feature of the Elements fits this alternative theory ......e tidily. Much of the material on number theory is not needed Im ilie classification of the regular solids - so why did Euclid in I tide this material? However, the same material is closely related .....rational numbers, which could explain why it was included. Archimedes iln- greatest of the ancient mathematicians was Archimedes. He n inle important contributions to geometry, he was at the forefront ■ 'I applications of mathematics to the natural world, and he was an 11 unplished engineer. But to mathematicians, Archimedes will 11 ways be remembered for his work on circles, spheres and ■ ) li nders, which we now associate with the number n ('pi'), which I ioughly 3.14159. Of course the Greeks did not work with n directly: they viewed it geometrically as the ratio of the m umference of a circle to its diameter, liarlier cultures had realized that the circumference of a circle is II ways the same multiple of its diameter, and knew that this multiple is roughly 3, maybe a bit bigger. The Babylonians used 3%. But Archimedes went much further; his results were accompanied by i Igorous proofs, in the spirit of Eudoxus. As far as the Greeks knew, the ratio of circumference of a circle to its diameter might be n i.itional. We now know that this is indeed die case, but the proof I in I to wait until 1770, when one was devised by Johann Heinrich l nnbert. (The school value of 3'/7 is convenient, but approximate.) 39 TAMING THE INFINITE llll UICIC (II SIIAI'I Be that as it may, since Archimedes could not prove n to be rational, he had to assume that it might not be. Greek geometry worked best with polygons - shapes formed by straight lines. But a circle is curved, so Archimedes sneaked up on it by way of approximating polygons. To estimate %, he compared the circumference of a circle with the perimeters of two series of polygons: one series situated inside the circle, the other surrounding it. The perimeters of polygons inside the circle must be shorter than the circle, whereas those outside the circle must be longer than the circle. To make the calculations easier, Archimedes constructed his polygons by repeatedly bisecting the sides of a regular hexagon (six-sided polygon) getting regular polygons with 12 sides, 24, 48 and so on. He stopped at 96. His calculations proved that 3I0/71 < n < 3'/7; that is, n lies somewhere between 3.1408 and 3.1429 in today's decimal notation. Archimedes's work on the sphere is of special interest, because we know not just his rigorous proof, but how he found it - which was decidedly non-rigorous. The proof is given in his book On the Sphere and Cylinder. He shows that the volume of a sphere is two-thirds that of a circumscribed cylinder, and that the surface areas of Pi to Vast Accuracy The value of n has now been calculated to several billion digits, using more sophisticated methods. Such computations are of interest for their methods, to test computer systems, and for pure curiosity, but the result itself has little significance. Practical applications of it generally require no more than five or six digits. The current record is 1.24 trillion decimal digits, computed by Yasumasa Kanada and a team of nine other people in December 2002. The computation took 600 hours on a Hitachi SR8000 supercomputer. ise 287-212bc Archimedes was born in Syracuse, Greece, son of the astronomer Phidias. He visited Egypt, where supposedly he invented the Archimedean screw, which until very recently was still widely used to raise Nile water for irrigation. He probably visited Euclid in Alexandria; he definitely corresponded with Alexandrian mathematicians. His mathematical skills were unsurpassed and wide-ranging. He turned them to practical use, and constructed gigantic war machines based on his 'law of the lever', able to hurl huge rocks at the enemy. His machines were used to good effect In the Roman siege of Alexandria in 212bc. He even used the geometry of optical reflection to focus the Sun's rays on to an invading Roman fleet, burning the ships. His surviving books {in later copies only) are On Plane Equilibria, Quadrature of the Parabola, On the Sphere and Cylinder, On Spirals, On Conoids and Spheroids, On Floating Bodies, Measurement of a Circle and The Sandreckoner, together with The Method, found in 1906 by Johan Heiberg. 11k >se parts of the sphere and the cylinder that lie between any two l larallel planes are equal. In modern language, Archimedes proved that the volume of a sphere is 4/3jrr3, where r is the radius, and its .iii-face area is 4jtr2.These basic facts are still in use today. The proof is an accomplished use of exhaustion. This method li.is an important limitation: you have to know what the answer is I lefore you have much chance of proving it. For centuries, scholars had no idea how Archimedes guessed the answer. But in 1906 the I lanish scholar Heiberg was studying a 13th-century parchment, with prayers written on it. He noticed faint lines from an earlier hi ( ription, which had been erased to make room for the prayers. 41 TAMING THE INFINITE THE LUUKJ (II SHAPE He discovered that the original document was a copy of several works by Archimedes, some of them previously unknown. Such a document is called a palimpsest — a piece of parchment which has later writing superimposed on erased earlier writing. (Astonishingly, the same manuscript is now known to contain pieces of lost works by two other ancient authors.) One work by Archimedes, the Method of Mechanical Theorems, explains how to guess the volume of a sphere. The idea is to slice the sphere infinitely thinly, and place the slices at one end of a balance; at the other end, similar slices of a cylinder and a cone - whose volumes Archimedes already knew — are hung. The law of the lever produces the required value for the volume. The parchment sold for two million dollars in 1998 to a private buyer. Problems for the Greeks Greek geometry had limitations, some of which it overcame by introducing new methods and concepts. Euclid in effect restricted the permitted geometrical constructions to those that could be performed using an unmarked straight edge (ruler) and a pair of compasses (henceforth 'compass' — the word 'pair' is technically needed, for the same reason that we cut paper with a pair of A sphere and its circumscribed cylinder I!' i ors, but let's not be pedantic.) It is sometimes said that he made this .1 requirement, but it is implicit in his constructions, not .in implicit rule.With extra instruments — idealized in the same way thai i In- rurve drawn by a compass is idealized to a perfect circle — new .....structions are possible. Ii>r example,Archimedes knew that you can trisect an angle using ' 11.light edge with two fixed marks on it.The Greeks called such l>M>redures 'neusis constructions'. We now know (as the Greeks .....si have suspected) drat an exact trisection of the angle with ruler .mi I compass is impossible, so Archimedes's contribution genuinely c iriids what is possible. Two other famous problems from the l»'i iod are duplicating the cube (constructing a cube whose volume |l i wice that of a given cube) and squaring the circle (constructing .1 M|iiare with the same area as a given circle).These are also known to be impossible using ruler and compass. A far-reaching extension of the allowed operations in geometry, \\ Inch bore fruit in Arab work on the cubic equation around ad800 Ind had major applications to mechanics and astronomy, was the introduction of a new class of curves, conic sections. These curves, \\ Inch are extraordinarily important in the history of mathematics, ire obtained by slicing a double-cone with a plane. Today we lliorten the name to conies.They come in three main types: • The ellipse, a closed oval curve obtained when the plane meets only one half of the cone. Circles are special ellipses. • The hyperbola, a curve with two infinite branches, obtained when the plane meets both halves of the cone. • The parabola, a transitional curve lying between ellipses and hyperbolas, in the sense that it is parallel to some line passing through the vertex of the cone and lying on the cone. A parabola has only one branch, but extends to infinity. TAMING THE INFINITE Parabola Circle Ellipse Conic sections Conic sections were studied in detail by Apollonius of Perga, who travelled from Perga in Asia Minor to Alexandria to study under Euclid. His masterwork, the Conic Sections of about 230 bc, contains 487 theorems. Euclid and Archimedes had studied some properties c >l cones, but it would take an entire book to summarize Apollonius s theorems. One important idea deserves mention here. This is the notion of the foci (plural of focus) of an ellipse (or hyperbola). The l< x i .1 re two special points associated with these two types of conic. Among (heir many properties, we single out just one: the distance I'.....one focus of an ellipse, to any point, and back to the other fb< us is constant (equal to the long diameter of the ellipse). The foci of j hyperbola have a similar property, but now we take the difference of the two lengths. AD370-415 Hypatia is the first woman mathematician in the historical record. She was the daughter of Theon of Alexandria, himself a mathematician, and it is probable that she learned her mathematics from him. By 400 she had become the head of the Platonist school in Alexandria, lecturing on philosophy and mathematics. Several historical sources state that she was a brilliant teacher. We do not know whether Hypatia made any original contributions to mathematics, but she helped Theon to write a commentary on Ptolemy's Almagest, and may also have helped him to prepare a new edition of the Elements, upon which all later editions were based. She wrote commentaries on the Arithmetics of Diophantus and the Conies of Apollonius. Among Hypatia's students were several leading figures in the growing religion of Christianity, among them Synesius of Cyrene. Some of his letters to her are recorded, and these praise her abilities. Unfortunately, many early Christians considered Hypatia's philosophy and science to be rooted in paganism, leading some to resent her influence. In 412 the new patriarch of Alexandria, Cyril, engaged in political rivalry with the Roman prefect Orestes. Hypatia was a good friend of Orestes, and her abilities as a teacher and orator were seen as a threat by the Christians. She became a focus for political unrest, and was dismembered by a mob. One source blames a fundamentalist sect, the Nitrian monks, who supported Cyril. Another blames an Alexandrian mob. A third source claims that she was part of a political rebellion, and her death was unavoidable. Her death was brutal, hacked to pieces by a mob wielding sharp files (some say oyster-shells}. Her mangled body was then burned. This punishment may be evidence that Hypatia was condemned for witchcraft - indeed, the first prominent witch to be killed by the early Christians - because the penalty for witchcraft prescribed by Constantius II was for their flesh to be 'torn off their bones with iron hooks'. i I ••I.'. TAMING THE INFINITE THE LOGIC OF SH APE The Greeks knew how to trisect angles and duplicate the cube using conies. With the aid of odier special curves, notably the quadratrix, they could also square the circle. Greek mathematics contributed two crucial ideas to human development. The more obvious was a systematic understanding of geometry. Using geometry as a tool, the Greeks understood the size and shape of our planet, its relation to the Sun and Moon, even the complex motions of the remainder of the solar system. They used geometry to dig long tunnels from both ends, meeting in the middle, which cut construction time in half. They built gigantic and powerful machines, based on simple principles like the law of the lever, for purposes both peaceful and warlike. They exploited geometry in ship-building and in architecture, where buildings like the Parthenon prove to us that mathematics and beauty are not so far apart. The Parthenon's visual elegance derives from a host of clever mathematical tricks, used by the architect to overcome limitations of the human visual system and irregularities in the very ground on which the building rested. The second Greek contribution was the systematic use of logical deduction to make sure that what was being asserted could also be justified. Logical argument emerged from their philosophy, but it found its most developed and explicit form in the geometry of liuclid and his successors. Without solid logical foundations, later mathematics could not have arisen. Roth influences remain vital today. Modern engineering -i :< >i nputer-based design and manufacture, for example - rests heavily on the geometric principles discovered by the Greeks. Every building is designed so that it doesn't fall down of its own accord; many are designed to resist earthquakes. Every tower block, every ■ ■11 ■ ■ In:.i(in bridge, every football stadium is a tribute to the yc> >meters oi ancient Greece. Rational thinking, logical argument, is equally vital. Our world is too complex, potentially too dangerous, for us to base decisions on What geometry did for them -:o Around 250 bc Eratosthenes of Cyrene used geometry to estimate the size of the Earth. He noticed that at midday on the summer solstice, the Sun was almost exactly overhead at Syene (present-day Aswan), because it shone straight down a vertical well. On the same day of the year, the shadow of a tall column indicated that the Sun's position at Alexandria was one fiftieth of a full circle (about 7.2°) away from the vertical. The Greeks knew that the Earth is spherical, and Alexandria was almost due north of Syene, so the geometry of a circular section of the sphere implied that the distance from Alexandria to Syene is one fiftieth of the circumference of the Earth. Eratosthenes knew that camel trains took 50 days to get from Alexandria to Syene, and they travelled a distance of 100 stadia each day. So the distance from Alexandria to Syene is 5000 stadia, making the circumference of the Earth 250,000 stadia. Unfortunately we don't know for sure how long a stadium was, but one estimate is 157 metres, leading to a circumference of 39,250 km (24,500 miles). The modern figure is 39,840 km (24,900 miles). How Eratosthenes measured the size of the Earth Well at Syene TAMING THE INFINITE What geometry does for us Archimedes's expression for the volume of a sphere is still useful today. One application, which requires knowing n to high accuracy, is the standardized unit of mass for the whole of science. For many years, for example, a metre was defined to be the length of a particular metal bar when measured at a particular temperature. Many basic units of measurement are now defined in terms of such things as how long it takes an atom of. some specific element to vibrate some huge number of times. But some are still based on physical objects, and mass is a case in point. The standard unit of mass is the kilogram. One kilogram is currently defined to be the mass of a particular sphere, in III I 111,11. ill Ul/il'l Another modern use of geometry occurs in computer graphics. Movies make extensive use of computer-generated images (CGI), and it is often necessary to generate images that include reflections - in a mirror, a wineglass, anything that catches the light. Without such reflections the image will not appear realistic. An efficient way to do this is by ray-tracing. When we look at a scene from some particular direction, our eye detects a ray of light that has bounded around the objects in the scene and happens to enter the eye from that direction. We can follow the path of this ray by working backwards. At any reflecting surface, the ray bounces so that the original ray and the reflected ray make equal angles at the surface. Translating this geometric fact into numerical calculations allows the computer to trace the ray backwards through however many bounces might be needed before it meets something opaque. (Several bounces may be necessary - if, for example, the wine glass is sitting in front of a mirror.) what we want to believe, rather than on what is actually the case. The ■iientitle method is deliberately constructed to overcome a deep-teated human wish to assume that what we want to be true - what we claim to 'know' - really is true. In science, emphasis is placed on 11 yiag to prove that what you deeply believe to be the case is wrong. Ideas that survive stringent attempts to disprove them are more likely to be correct. 49