Principal The Math Book (Big Ideas Simply Explained)
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CONTENTS HOW TO USE THIS EBOOK INTRODUCTION ANCIENT AND CLASSICAL PERIODS 6000 BCE–500 CE Numerals take their places • Positional numbers The square as the highest power • Quadratic equations The accurate reckoning for inquiring into all things • The Rhind papyrus The sum is the same in every direction • Magic squares Number is the cause of gods and daemons • Pythagoras A real number that is not rational • Irrational numbers The quickest runner can never overtake the slowest • Zeno’s paradoxes of motion Their combinations give rise to endless complexities • The Platonic solids Demonstrative knowledge must rest on necessary basic truths • Syllogistic logic The whole is greater than the part • Euclid’s Elements Counting without numbers • The abacus Exploring pi is like exploring the Universe • Calculating pi We separate the numbers as if by some sieve • Eratosthenes’ sieve A geometrical tour de force • Conic sections The art of measuring triangles • Trigonometry Numbers can be less than nothing • Negative numbers The very flower of arithmetic • Diophantine equations An incomparable star in the firmament of wisdom • Hypatia The closest approximation of pi for a millennium • Zu Chongzhi THE MIDDLE AGES 500–1500 A fortune subtracted from zero is a debt • Zero Algebra is a scientific art • Algebra Freeing algebra from the constraints of geometry • The binomial theorem Fourteen forms with all their branches and cases • Cubic equations The ubiquitous music of the spheres • The Fibonacci sequence The power of doubling • Wheat on a chessboard THE RENAISSANCE 1500–1680 The geometry of art and life • The golden ratio Like a large diamond • Mersenne primes Sailing on a rhumb • Rhumb lines A pair of equal-length lines • The equals sign and other symbology Plus of minus times plus of minus makes minus • Imaginary and complex numbers The art of tenths • Decimals Transforming multiplication into addition • Logarithms Nature uses as little as possible of anything • The problem of m; axima The fly on the ceiling • Coordinates A device of marvelous invention • The area under a cycloid Three dimensions made by two • Projective geometry Symmetry is what we see at a glance • Pascal’s triangle Chance is bridled and governed by law • Probability The sum of the distance equals the altitude • Viviani’s triangle theorem The swing of a pendulum • Huygens’s tautochrone curve With calculus I can predict the future • Calculus The perfection of the science of numbers • Binary numbers THE ENLIGHTENMENT 1680–1800 To every action there is an equal and opposite reaction • Newton’s laws of motion Empirical and expected results are the same • The law of large numbers One of those strange numbers that are creatures of their own • Euler’s number Random variation makes a pattern • Normal distribution The seven bridges of Königsberg • Graph theory Every even integer is the sum of two primes • The Goldbach conjecture The most beautiful equation • Euler’s identity No theory is perfect • Bayes’ theorem Simply a question of algebra • The algebraic resolution of equations Let us gather facts • Buffon’s needle experiment Algebra often gives more than is asked of her • The fundamental theorem of algebra THE 19TH CENTURY 1800–1900 Complex numbers are coordinates on a plane • The complex plane Nature is the most fertile source of mathematical discoveries • Fourier analysis The imp that knows the positions of every particle in the Universe • Laplace’s demon What are the chances? • The Poisson distribution An indispensable tool in applied mathematics • Bessel functions It will guide the future course of science • The mechanical computer A new kind of function • Elliptic functions I have created another world out of nothing • Non-Euclidean geometries Algebraic structures have symmetries • Group theory Just like a pocket map • Quaternions Powers of natural numbers are almost never consecutive • Catalan’s conjecture The matrix is everywhere • Matrices An investigation into the laws of thought • Boolean algebra A shape with just one side • The Möbius strip The music of the primes • The Riemann hypothesis Some infinities are bigger than others • Transfinite numbers A diagrammatic representation of reasonings • Venn diagrams The tower will fall and the world will end • The Tower of Hanoi Size and shape do not matter, only connections • Topology Lost in that silent, measured space • The prime number theorem MODERN MATHEMATICS 1900–PRESENT The veil behind which the future lies hidden • 23 problems for the 20th century Statistics is the grammar of science • The birth of modern statistics A freer logic emancipates us • The logic of mathematics The Universe is four-dimensional • Minkowski space Rather a dull number • Taxicab numbers A million monkeys banging on a million typewriters • The infinite monkey theorem She changed the face of algebra • Emmy Noether and abstract algebra Structures are the weapons of the mathematician • The Bourbaki group A single machine to compute any computable sequence • The Turing machine Small things are more numerous than large things • Benford’s law A blueprint for the digital age • Information theory We are all just six steps away from each other • Six degrees of separation A small positive vibration can change the entire cosmos • The butterfly effect Logically things can only partly be true • Fuzzy logic A grand unifying theory of mathematics • The Langlands Program Another roof, another proof • Social mathematics Pentagons are just nice to look at • The Penrose tile Endless variety and unlimited complication • Fractals Four colors but no more • The four-color theorem Securing data with a one-way calculation • Cryptography Jewels strung on an as-yet invisible thread • Finite simple groups A truly marvelous proof • Proving Fermat’s last theorem No other recognition is needed • Proving the Poincaré conjecture DIRECTORY GLOSSARY CONTRIBUTORS QUOTATIONS ACKNOWLEDGMENTS COPYRIGHT How to use this eBook Preferred application settings For the best reading experience, the following application settings are recommended: Color theme: White background Font size: At the smallest point size Orientation: Landscape (for screen sizes over 9”/23cm), Portrait (for screen sizes below 9”/23cm) Scrolling view: [OFF] Text alignment: Auto-justification [OFF] (if the eBook reader has this feature) Auto-hyphenation: [OFF] (if the eBook reader has this feature) Font style: Publisher default setting [ON] (if the eBook reader has this feature) Images: Double tap on the images to see them in full screen and be able to zoom in on them FOREWORD Summarizing all of mathematics in one book is a daunting and indeed impossible task. Humankind has been exploring and discovering mathematics for millennia. Practically, we have relied on math to advance our species, with early arithmetic and geometry providing the foundations for the first cities and civilizations. And philosophically, we have used mathematics as an exercise in pure thought to explore patterns and logic. As a subject, mathematics is surprisingly hard to pin down with one catch-all definition. “Mathematics” is not simply, as many people think, “stuff to do with numbers.” That would exclude a huge range of mathematical topics, including much of the geometry and topology covered in this book. Of course, numbers are still very useful tools to understand even the most esoteric areas of mathematics, but the point is that they are not the most interesting aspect of it. Focusing just on numbers misses the forest for the threes. For the record, my own definition of math as “the sort of things that mathematicians enjoy doing,” while delightfully circular, is largely unhelpful. Big Ideas Simply Explained is actually not a bad definition. Mathematics could be seen as the attempt to find the simplest explanations for the biggest ideas. It is the endeavor of finding and summarizing patterns. Some of those patterns involve the practical triangles required to build pyramids and divide land; other patterns attempt to classify all of the 26 sporadic groups of abstract algebra. These are very different problems in terms of both usefulness and complexity, but both types of pattern have become the obsession of mathematicians throughout the ages. There is no definitive way to organize all of mathematics, but looking at it chronologically is not a bad way to go. This book uses the historical journey of humans discovering math as a way to classify it and wrangle it into a linear progression, which is a valiant but difficult effort. Our current mathematical body of knowledge has been built up by a haphazard and diverse group of people across time and cultures. So something like the short section on magic squares covers thousands of years and the span of the globe. Magic squares—arrangements of numbers where the sum in each row, column, and diagonal is always the same—are one of the oldest areas of recreational mathematics. Starting in the 9th century BCE in China, the story then bounces around via Indian texts from 100 CE, Arab scholars in the Middle Ages, Europe during the Renaissance, and finally modern Sudoku-style puzzles. Across a mere two pages this book has to cover 3,000 years of history ending with geomagic squares in 2001. And even in this small niche of mathematics, there are many magic square developments that there was simply not enough room to include. The whole book should be viewed as a curated tour of mathematical highlights. Studying even just a sample of mathematics is a great reminder of how much humans have achieved. But it also highlights where mathematics could do better; things like the glaring omission of women from the history of mathematics cannot be ignored. A lot of talent has been squandered over the centuries, and a lot of credit has not been appropriately given. But I hope that we are now improving the diversity of mathematicians and encouraging all humans to discover and learn about mathematics. Because going forward, the body of mathematics will continue to grow. Had this book been written a century earlier it would have been much the same up until about page 280. And then it would have ended. No ring theory from Emmy Noether, no computing from Alan Turing, and no six degrees of separation from Kevin Bacon. And no doubt that will be true again 100 years from now. The edition printed a century from now will carry on past page 325, covering patterns totally alien to us. And because anyone can do math, there is no telling who will discover this new math, and where or when. To make the biggest advancement in mathematics during the 21st century, we need to include all people. I hope this book helps inspire everyone to get involved. Matt Parker INTRODUCTION The history of mathematics reaches back to prehistory, when early humans found ways to count and quantify things. In doing so, they began to identify certain patterns and rules in the concepts of numbers, sizes, and shapes. They discovered the basic principles of addition and subtraction—for example, that two things (whether pebbles, berries, or mammoths) when added to another two invariably resulted in four things. While such ideas may seem obvious to us today, they were profound insights for their time. They also demonstrate that the history of mathematics is above all a story of discovery rather than invention. Although it was human curiosity and intuition that recognized the underlying principles of mathematics, and human ingenuity that later provided various means of recording and notating them, those principles themselves are not a human invention. The fact that 2 + 2 = 4 is true, independent of human existence; the rules of mathematics, like the laws of physics, are universal, eternal, and unchanging. When mathematicians first showed that the angles of any triangle in a flat plane when added together come to 180°, a straight line, this was not their invention: they had simply discovered a fact that had always been (and will always be) true. Early applications The process of mathematical discovery began in prehistoric times, with the development of ways of counting things people needed to quantify. At its simplest, this was done by cutting tally marks in a bone or stick, a rudimentary but reliable means of recording numbers of things. In time, words and symbols were assigned to the numbers and the first systems of numerals began to evolve, a means of expressing operations such as acquisition of additional items, or depletion of a stock, the basic operations of arithmetic. As hunter-gatherers turned to trade and farming, and societies became more sophisticated, arithmetical operations and a numeral system became essential tools in all kinds of transactions. To enable trade, stocktaking, and taxes in uncountable goods such as oil, flour, or plots of land, systems of measurement were developed, putting a numerical value on dimensions such as weight and length. Calculations also became more complex, developing the concepts of multiplication and division from addition and subtraction—allowing the area of land to be calculated, for example. In the early civilizations, these new discoveries in mathematics, and specifically the measurement of objects in space, became the foundation of the field of geometry, knowledge that could be used in building and toolmaking. In using these measurements for practical purposes, people found that certain patterns were emerging, which could in turn prove useful. A simple but accurate carpenter’s square can be made from a triangle with sides of three, four, and five units. Without that accurate tool and knowledge, the roads, canals, ziggurats, and pyramids of ancient Mesopotamia and Egypt could not have been built. As new applications for these mathematical discoveries were found—in astronomy, navigation, engineering, bookkeeping, taxation, and so on—further patterns and ideas emerged. The ancient civilizations each established the foundations of mathematics through this interdependent process of application and discovery, but also developed a fascination with mathematics for its own sake, so-called pure mathematics. From the middle of the first millennium BCE, the first pure mathematicians began to appear in Greece, and slightly later in India and China, building on the legacy of the practical pioneers of the subject—the engineers, astronomers, and explorers of earlier civilizations. Although these early mathematicians were not so concerned with the practical applications of their discoveries, they did not restrict their studies to mathematics alone. In their exploration of the properties of numbers, shapes, and processes, they discovered universal rules and patterns that raised metaphysical questions about the nature of the cosmos, and even suggested that these patterns had mystical properties. Often mathematics was therefore seen as a complementary discipline to philosophy—many of the greatest mathematicians through the ages have also been philosophers, and vice versa—and the links between the two subjects have persisted to the present day. It is impossible to be a mathematician without being a poet of the soul. Sofya Kovalevskaya Russian mathematician Arithmetic and algebra So began the history of mathematics as we understand it today—the discoveries, conjectures, and insights of mathematicians that form the bulk of this book. As well as the individual thinkers and their ideas, it is a story of societies and cultures, a continuously developing thread of thought from the ancient civilizations of Mesopotamia and Egypt, through Greece, China, India, and the Islamic empire to Renaissance Europe and into the modern world. As it evolved, mathematics was also seen to comprise several distinct but interconnected fields of study. The first field to emerge, and in many ways the most fundamental, is the study of numbers and quantities, which we now call arithmetic, from the Greek word arithmos (“number”). At its most basic, it is concerned with counting and assigning numerical values to things, but also the operations, such as addition, subtraction, multiplication, and division, that can be applied to numbers. From the simple concept of a system of numbers comes the study of the properties of numbers, and even the study of the very concept itself. Certain numbers—such as the constants π, e, or the prime and irrational numbers—hold a special fascination and have become the subject of considerable study. Another major field in mathematics is algebra, which is the study of structure, the way that mathematics is organized, and therefore has some relevance in every other field. What marks algebra from arithmetic is the use of symbols, such as letters, to represent variables (unknown numbers). In its basic form, algebra is the study of the underlying rules of how those symbols are used in mathematics—in equations, for example. Methods of solving equations, even quite complex quadratic equations, had been discovered as early as the ancient Babylonians, but it was medieval mathematicians of the Islamic Golden Age who pioneered the use of symbols to simplify the process, giving us the word “algebra,” which is derived from the Arabic al-jabr. More recent developments in algebra have extended the idea of abstraction into the study of algebraic structure, known as abstract algebra. Geometry is knowledge of the eternally existent. Pythagoras Ancient Greek mathematician Geometry and calculus A third major field of mathematics, geometry, is concerned with the concept of space, and the relationships of objects in space: the study of the shape, size, and position of figures. It evolved from the very practical business of describing the physical dimensions of things, in engineering and construction projects, measuring and apportioning plots of land, and astronomical observations for navigation and compiling calendars. A particular branch of geometry, trigonometry (the study of the properties of triangles), proved to be especially useful in these pursuits. Perhaps because of its very concrete nature, for many ancient civilizations, geometry was the cornerstone of mathematics, and provided a means of problem-solving and proof in other fields. This was particularly true of ancient Greece, where geometry and mathematics were virtually synonymous. The legacy of great mathematical philosophers such as Pythagoras, Plato, and Aristotle was consolidated by Euclid, whose principles of mathematics based on a combination of geometry and logic were accepted as the subject’s foundation for some 2,000 years. In the 1800s, however, alternatives to classical Euclidean geometry were proposed, opening up new areas of study, including topology, which examines the nature and properties not only of objects in space, but of space itself. Since the Classical period, mathematics had been concerned with static situations, or how things are at any given moment. It failed to offer a means of measuring or calculating continuous change. Calculus, developed independently by Gottfried Leibniz and Isaac Newton in the 1600s, provided an answer to this problem. The two branches of calculus, integral and differential, offered a method of analyzing such things as the slope of curves on a graph and the area beneath them as a way of describing and calculating change. The discovery of calculus opened up a field of analysis that later became particularly relevant to, for example, the theories of quantum mechanics and chaos theory in the 1900s. Revisiting logic The late 19th and early 20th centuries saw the emergence of another field of mathematics—the foundations of mathematics. This revived the link between philosophy and mathematics. Just as Euclid had done in the 3rd century BCE, scholars including Gottlob Frege and Bertrand Russell sought to discover the logical foundations on which mathematical principles are based. Their work inspired a re-examination of the nature of mathematics itself, how it works, and what its limits are. This study of basic mathematical concepts is perhaps the most abstract field, a sort of meta-mathematics, yet an essential adjunct to every other field of modern mathematics. In mathematics, the art of asking questions is more valuable than solving problems. Georg Cantor German mathematician New technology, new ideas The various fields of mathematics—arithmetic, algebra, geometry, calculus, and foundations—are worthy of study for their own sake, and the popular image of academic mathematics is that of an almost incomprehensible abstraction. But applications for mathematical discoveries have usually been found, and advances in science and technology have driven innovations in mathematical thinking. A prime example is the symbiotic relationship between mathematics and computers. Originally developed as a mechanical means of doing the “donkey work” of calculation to provide tables for mathematicians, astronomers and so on, the actual construction of computers required new mathematical thinking. It was mathematicians, as much as engineers, who provided the means of building mechanical, and then electronic computing devices, which in turn could be used as tools in the discovery of new mathematical ideas. No doubt, new applications for mathematical theorems will be found in the future too—and with numerous problems still unsolved, it seems that there is no end to the mathematical discoveries to be made. The story of mathematics is one of exploration of these different fields, and the discovery of new ones. But it is also the story of the explorers, the mathematicians who set out with a definite aim in mind, to find answers to unsolved problems, or to travel into unknown territory in search of new ideas—and those who simply stumbled upon an idea in the course of their mathematical journey, and were inspired to see where it would lead. Sometimes the discovery would come as a game-changing revelation, providing a way into unexplored fields; at other times it was a case of “standing on the shoulders of giants,” developing the ideas of previous thinkers, or finding practical applications for them. This book presents many of the “big ideas” in mathematics, from the earliest discoveries to the present day, explaining them in layperson’s language, where they came from, who discovered them, and what makes them significant. Some may be familiar, others less so. With an understanding of these ideas, and an insight into the people and societies in which they were discovered, we can gain an appreciation of not only the ubiquity and usefulness of mathematics, but also the elegance and beauty that mathematicians find in the subject. Mathematics, rightly viewed, possesses not only truth, but supreme beauty. Bertrand Russell British philosopher and mathematician INTRODUCTION As early as 40,000 years ago, humans were making tally marks on wood and bone as a means of counting. They undoubtedly had a rudimentary sense of number and arithmetic, but the history of mathematics only properly began with the development of numerical systems in early civilizations. The first of these emerged in the sixth millennium BCE, in Mesopotamia, western Asia, home to the world’s earliest agriculture and cities. Here, the Sumerians elaborated on the concept of tally marks, using different symbols to denote different quantities, which the Babylonians then developed into a sophisticated numerical system of cuneiform (wedge-shaped) characters. From about 4000 BCE, the Babylonians used elementary geometry and algebra to solve practical problems—such as building, engineering, and calculating land divisions—alongside the arithmetical skills they used to conduct commerce and levy taxes. A similar story emerges in the slightly later civilization of the ancient Egyptians. Their trade and taxation required a sophisticated numerical system, and their building and engineering works relied on both a means of measurement and some knowledge of geometry and algebra. The Egyptians were also able to use their mathematical skills in conjunction with observations of the heavens to calculate and predict astronomical and seasonal cycles and construct calendars for the religious and agricultural year. They established the study of the principles of arithmetic and geometry as early as 2000 BCE. Greek rigor The 6th century BCE onward saw a rapid rise in the influence of ancient Greece across the eastern Mediterranean. Greek scholars quickly assimilated the mathematical ideas of the Babylonians and Egyptians. The Greeks used a numerical system of base-10 (with ten symbols) derived from the Egyptians. Geometry in particular chimed with Greek culture, which idolized beauty of form and symmetry. Mathematics became a cornerstone of Classical Greek thinking, reflected in its art, architecture, and even philosophy. The almost mystical qualities of geometry and numbers inspired Pythagoras and his followers to establish a cultlike community, dedicated to studying the mathematical principles they believed were the foundations of the Universe and everything in it. Centuries before Pythagoras, the Egyptians had used a triangle with sides of 3, 4, and 5 units as a building tool to ensure corners were square. They had come across this idea by observation, and then applied it as a rule of thumb, whereas the Pythagoreans set about rigorously showing the principle, offering a proof that it is true for all right-angled triangles. It is this notion of proof and rigor that is the Greeks’ greatest contribution to mathematics. Plato’s Academy in Athens was dedicated to the study of philosophy and mathematics, and Plato himself described the five Platonic solids (the tetrahedron, cube, octahedron, dodecahedron, and icosahedron). Other philosophers, notably Zeno of Elea, applied logic to the foundations of mathematics, exposing the problems of infinity and change. They even explored the strange phenomenon of irrational numbers. Plato’s pupil Aristotle, with his methodical analysis of logical forms, identified the difference between inductive reasoning (such as inferring a rule of thumb from observations) and deductive reasoning (using logical steps to reach a certain conclusion from established premises, or axioms). From this basis, Euclid laid out the principles of mathematical proof from axiomatic truths in his Elements, a treatise that was the foundation of mathematics for the next two millennia. With similar rigor, Diophantus pioneered the use of symbols to represent unknown numbers in his equations; this was the first step toward the symbolic notation of algebra. A new dawn in the East Greek dominance was eventually eclipsed by the rise of the Roman Empire. The Romans regarded mathematics as a practical tool rather than worthy of study. At the same time, the ancient civilizations of India and China independently developed their own numerical systems. Chinese mathematics in particular flourished between the 2nd and 5th centuries CE, thanks largely to the work of Liu Hui in revising and expanding the classic texts of Chinese mathematics. IN CONTEXT KEY CIVILIZATION Babylonians FIELD Arithmetic BEFORE 40,000 years ago Stone Age people in Europe and Africa count using tally marks on wood or bone. 6000–5000 BCE Sumerians develop early calculation systems to measure land and to study the night sky. 4000–3000 BCE Babylonians use a small clay cone for 1 and a large cone for 60, along with a clay ball for 10, as their base-60 system evolves. AFTER 2nd century CE The Chinese use an abacus in their base-10 positional number system. 7th century In India, Brahmagupta establishes zero as a number in its own right and not just as a placeholder. It is given to us to calculate, to weigh, to measure, to observe; this is natural philosophy. Voltaire French philosopher The first people known to have used an advanced numeration system were the Sumerians of Mesopotamia, an ancient civilization living between the Tigris and Euphrates rivers in what is present-day Iraq. Sumerian clay tablets from as early as the 6th millennium BCE include symbols denoting different quantities. The Sumerians, followed by the Babylonians, needed efficient mathematical tools in order to administer their empires. What distinguished the Babylonians from neighbors such as Egypt was their use of a positional (place value) number system. In such systems, the value of a number is indicated both by its symbol and its position. Today, for instance, in the decimal system, the position of a digit in a number indicates whether its value is in ones (less than 10), tens, hundreds, or more. Such systems make calculation more efficient because a small set of symbols can represent a huge range of values. By contrast, the ancient Egyptians used separate symbols for ones, tens, hundreds, thousands, and above, and had no place value system. Representing larger numbers could require 50 or more hieroglyphs. Using different bases The Hindu–Arabic numeration that is employed today is a base-10 (decimal) system. It requires only 10 symbols—nine digits (1, 2, 3, 4, 5, 6, 7, 8, 9) and a zero as a placeholder. As in the Babylonian system, the position of a digit indicates its value, and the smallest value digit is always to the right. In a base-10 system, a two-digit number, such as 22, indicates (2 × 101) + 2; the value of the 2 on the left is ten times that of the 2 on the right. Placing digits after the number 22 will create hundreds, thousands, and larger powers of 10. A symbol after a whole number (the standard notation now is a decimal point) can also separate it from its fractional parts, each representing a tenth of the place value of the preceding figure. The Babylonians worked with a more complex sexagesimal (base-60) number system that was probably inherited from the earlier Sumerians and is still used across the world today for measuring time, degrees in a circle (360° = 6 × 60), and geographic coordinates. Why they used 60 as a number base is still not known for sure. It may have been chosen because it can be divided by many other numbers—1, 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30. The Babylonians also based their calendar year on the solar year (365.24 days); the number of days in a year was 360 (6 × 60) with additional days for festivals. In the Babylonian sexagesimal system, a single symbol was used alone and repeated up to nine times to represent symbols for 1 to 9. For 10, a different symbol was used, placed to the left of the one symbol, and repeated two to five times in numbers up to 59. At 60 (60 × 1), the original symbol for one was reused but placed further to the left than the symbol for 1. Because it was a base-60 system, two such symbols signified 61, while three such symbols indicated 3,661, that is, 60 × 60 (602) + 60 + 1. The base-60 system had obvious drawbacks. It necessarily requires many more symbols than a base-10 system. For centuries, the sexagesimal system also had no place value holders, and nothing to separate whole numbers from fractional parts. By around 300 BCE, however, the Babylonians used two wedges to indicate no value, much as we use a placeholder zero today; this was possibly the earliest use of zero. The Babylonian sun-god Shamash awards a rod and a coiled rope, ancient measuring devices, to newly trained surveyors, on a clay tablet dating from around 1000 BCE. Other counting systems In Mesoamerica, on the other side of the world, the Mayan civilization developed its own advanced numeration system in the 1st millennium BCE—apparently in complete isolation. Theirs was a base-20 (vigesimal) number system, which probably evolved from a simple counting method using fingers and toes. In fact, base-20 number systems were used across the world, in Europe, Africa, and Asia. Language often contains remnants of this system. For example, in French, 80 is expressed as quatre-vingt (4 × 20); Welsh and Irish also express some numbers as multiples of 20, while in English a score is 20. In the Bible, for instance, Psalm 90 talks of a human lifespan being “threescore years and ten” or as great as “fourscore years.” From around 500 BCE until the 16th century when Hindu–Arabic numbers were officially adopted in China, the Chinese used rod numerals to represent numbers. This was the first decimal place value system. By alternating quantities of vertical rods with horizontal rods, this system could indicate ones, tens, hundreds, thousands, and more powers of 10, much as the decimal system does today. For example, 45 was written with four horizontal bars representing 4 × 101 (40) and five vertical bars for 5 × 1 (5). However, four vertical rods followed by five vertical rods indicated 405 (4 × 100, or 102) + 5 × 1—the absence of horizontal rods meant there were no tens in the number. Calculations were carried out by manipulating the rods on a counting board. Positive and negative numbers were represented by red and black rods respectively or different cross sections (triangular and rectangular). Rod numerals are still used occasionally in China, just as Roman numerals are sometimes used in Western society. The Chinese place value system is reflected in the Chinese abacus (suanpan). Dating back to at least 200 BCE, it is one of the oldest bead-counting devices, although the Romans used something similar. The Chinese version, which is still used today, has a central bar and a varying number of vertical wires to separate ones from tens, hundreds, or more. In each column, there are two beads above the bar worth five each and five beads below the bar worth one each. The Japanese adopted the Chinese abacus in the 14th century and developed their own abacus, the soroban, which has one bead worth five above the central bar and four beads each worth one below the bar in each column. Japan still uses the soroban today: there are even contests in which young people demonstrate their ability to perform soroban calculations mentally, a skill known as anzan. Cuneiform Cuneiform, a word derived from the Latin cuneus (“wedge”) to describe the shape of the symbols, was inscribed into wet clay, stone, or metal. In the late 1800s, academics deciphered the “cuneiform” (wedge-shaped) markings on clay tablets recovered from Babylonian sites in and around Iraq. Such marks, denoting letters and words as well as an advanced number system, were etched in wet clay with either end of a stylus. Like the Egyptians, the Babylonians needed scribes to administer their complex society, and many of the tablets bearing mathematical records are thought to be from training schools for scribes. A great deal has now been discovered about Babylonian mathematics, which extended to multiplication, division, geometry, fractions, square roots, cube roots, equations, and other forms, because—unlike Egyptian papyrus scrolls—the clay tablets have survived well. Several thousand, mostly dating from between 1800 and 1600 BCE, are housed in museums around the world. The Babylonian base-60 number system was built from two symbols—the single unit symbol, used alone and combined for numbers 1 to 9, and the 10 symbol, repeated for 20, 30, 40, and 50. The Babylonian and Assyrian civilizations have perished…yet Babylonian mathematics is still interesting, and the Babylonian scale of 60 is still used in astronomy. G. H. Hardy British mathematician Modern numeration The Hindu–Arabic decimal system used throughout the world today has its origins in India. In the 1st to 4th centuries CE, the use of nine symbols along with zero was developed to allow any number to be written efficiently, through the use of place value. The system was adopted and refined by Arab mathematicians in the 9th century. They introduced the decimal point, so that the system could also express fractions of whole numbers. Three centuries later, Leonardo of Pisa (Fibonacci) popularized the use of Hindu–Arabic numerals in Europe through his book Liber Abaci (1202). Yet the debate about whether to use the new system rather than Roman numerals and traditional counting methods lasted for several hundred years, before its adoption paved the way for modern mathematical advances. With the advent of electronic computers, other number bases became important—particularly binary, a number system with base 2. Unlike the base-10 system with its 10 symbols, binary has just two: 1 and 0. It is a positional system, but instead of multiplying by 10, each column is multiplied by 2, also expressed as 21, 22, 23 and upward. In binary, the number 111 means 1 × 22 + 1 × 21 + 1 × 20, that is 4 + 2 + 1, or 7 in our decimal number system. In binary, as in all modern number systems whatever their base, the principles of place value are always the same. Place value—the Babylonian legacy—remains a powerful, easily understood, and efficient way to represent large numbers. The fact that we work in 10s as opposed to any other number is purely a consequence of our anatomy. We use our ten fingers to count. Marcus du Sautoy British mathematician Ebisu, the Japanese god of fishermen and one of the seven gods of fortune, uses a soroban to calculate his profits in The Red Snapper’s Dream by Utagawa Toyohiro. Mayan numeral system The Dresden Codex, the oldest surviving Mayan book, dating from the 13th or 14th century, illustrates Mayan number symbols and glyphs. The Mayans, who lived in Central America from around 2000 BCE, used a base-20 (vigesimal) number system from around 1000 BCE to perform astronomical and calendar calculations. Like the Babylonians, they used a calendar of 360 days plus festivals, to make 365.24 days based on the solar year; their calendars helped them work out the growing cycles of crops. The Mayan system employed symbols: a dot representing one and a bar representing five. By using combinations of dots over bars they could generate numerals up to 19. Numbers larger than 19 were written vertically, with the lowest numbers at the bottom, and there is evidence of Mayan calculations up to hundreds of millions. An inscription from 36 BCE shows that they used a shell-shaped symbol to denote zero, which was widely used by the 4th century. The Mayans’ number system was in use in Central America until the Spanish conquests in the 16th century. Its influence, however, never spread further. See also: The Rhind papyrus • The abacus • Negative numbers • Zero • The Fibonacci sequence • Decimals IN CONTEXT KEY CIVILIZATIONS Egyptians (c. 2000 BCE), Babylonians (c. 1600 BCE) FIELD Algebra BEFORE c. 2000 BCE The Berlin papyrus records a quadratic equation solved in ancient Egypt. AFTER 7th century CE The Indian mathematician Brahmagupta solves quadratic equations using only positive integers. 10th century CE Egyptian scholar Abu Kamil Shuja ibn Aslam uses negative and irrational numbers to solve quadratic equations. 1545 Italian mathematician Gerolamo Cardano publishes his Ars Magna, setting out the rules of algebra. Quadratic equations are those involving unknown numbers to the power of 2 but not to a higher power; they contain x2 but not x3, x4, and so on. One of the main rudiments of mathematics is the ability to use equations to work out solutions to real-world problems. Where those problems involve areas or paths of curves such as parabolas, quadratic equations become very useful, and describe physical phenomena, such as the flight of a ball or a rocket. Ancient roots The history of quadratic equations extends across the world. It is likely that these equations first arose from the need to subdivide land for inheritance purposes, or to solve problems involving addition and multiplication. One of the oldest surviving examples of a quadratic equation comes from the ancient Egyptian text known as the Berlin papyrus (c. 2000 BCE). The problem contains the following information: the area of a square of 100 cubits is equal to that of two smaller squares. The side of one of the smaller squares is equal to one half plus a quarter of the side of the other. In modern notation, this translates into two simultaneous equations: x2 + y2 = 100 and x = (1⁄2 + 1⁄4)y = 3⁄4 y. These can be simplified to the quadratic equation (3⁄4 y)2 + y2 = 100 to find the length of a side on each square. The Egyptians used a method called “false position” to determine the solution. In this method, the mathematician selects a convenient number that is usually easy to calculate, then works out what the solution to the equation would be using that number. The result shows how to adjust the number to give the correct solution the equation. For example, in the Berlin papyrus problem, the simplest length to use for the larger of the two small squares is 4, because the problem deals with quarters. For the side of the smallest square, 3 is used because this length is 3⁄4 of the side of the other small square. Two squares created using these false position numbers would have areas of 16 and 9 respectively, which when added together give a total area of 25. This is only 1⁄4 of 100, so the areas must be quadrupled to match the Berlin papyrus equation. The lengths therefore must be doubled from the false positions of 4 and 3 to reach the solutions: 8 and 6. Other early records of quadratic equations are found in Babylonian clay tablets, where the diagonal of a square is given to five decimal places. The Babylonian tablet YBC 7289 (c. 1800–1600 BCE) shows a method of working out the quadratic equation x2 = 2 by drawing rectangles and trimming them down into squares. In the 7th century CE, Indian mathematician Brahmagupta wrote a formula for solving quadratic equations that could be applied to equations in the form ax2 + bx = c. Mathematicians at the time did not use letters or symbols, so he wrote his solution in words, but it was similar to the modern formula shown above. In the 8th century, Persian mathematician al-Khwarizmi employed a geometric solution for quadratic equations known as completing the square. Until the 10th century, geometric methods were were often used, as quadratic equations were used to solve real-world problems involving land rather than abstract algebraic challenges. The Berlin papyrus was copied and published by German Egyptologist Hans Schack-Schackenburg in 1900. It contains two mathematical problems, one of which is a quadratic equation. Negative solutions Indian, Persian, and Arab scholars thus far had used only positive numbers. When solving the equation x2 + 10x = 39, they gave the solution as 3. However, this is one of two correct solutions to the problem; -13 is the other. If x is -13, x2 = 169 and 10x = -130. Adding a negative number gives the same result as subtracting its equivalent positive number, so 169 + -130 = 169 - 130 = 39. In the 10th century, Egyptian scholar Abu Kamil Shuja ibn Aslam made use of negative numbers and algebraic irrational numbers (such as the square root of 2) as both solutions and coefficients (numbers multiplying an unknown quantity). By the 1500s, most mathematicians accepted negative solutions and were comfortable with surds (irrational roots – those that cannot be expressed exactly as a decimal). They had also started using numbers and symbols, rather than writing equations in words. Mathematicians now utilized the plus or minus symbol, ±, in solving quadratic equations. With the equation x2 = 2, the solution is not just x = but x = ±. The plus or minus symbol is included because two negative numbers multiplied together make a positive number. While × = 2, it is also true that - ×- = 2. In 1545, Italian scholar Gerolamo Cardano published his Ars Magna (The Great Art, or the Rules of Algebra) in which he explored the problem: “What pair of numbers have a sum of ten and product of 40?” He found that the problem led to a quadratic equation which, when he completed the square, gave . No numbers available to mathematicians at the time gave a negative number when multiplied by themselves, but Cardano suggested suspending belief and working with the square root of negative 15 to find the equation’s two solutions. Numbers such as would later be known as “imaginary” numbers. The quadratic formula is a way to solve quadratic equations. By modern convention, quadratic equations include a number, a, multiplied by x2; a number, b, multiplied by x; and a number, c, on its own. The illustration above shows how the formula uses a, b, and c to find the value of x. Quadratic equations often equal 0, because this makes them easy to work out on a graph; the x solutions are the points where the curve crosses the x axis. Politics is for the present, but an equation is for eternity. Albert Einstein Structure of equations Modern quadratic equations usually look like ax2 + bx + c = 0. The letters a, b, and c represent known numbers, while x represents the unknown number. Equations contain variables (symbols for numbers that are unknown), coefficients, constants (those that do not multiply variables), and operators (symbols such as the plus and equals sign). Terms are the parts separated by operators; they can be a number or variable, or a combination of both. The modern quadratic equation has four terms: ax2, bx, c, and 0. A graph of the quadratic function y = ax2 + bx + c creates a U-shaped curve called a parabola. This graph plots the points (in black) of the quadratic function where a = 1, b = 3, and c = 2. This expresses the quadratic equation x2 + 3x + 2 = 0. The solutions for x are where y = 0 and the curve crosses the x axis. These are -2 and -1. Parabolas A function is a group of terms that defines a relationship between variables (often x and y). The quadratic function is generally written as y = ax2 + bx + c, which, on a graph, produces a curve called a parabola. When real (not imaginary) solutions to ax2 + bx + c = 0 exist, they will be the roots—the points where the parabola crosses the x axis. Not all parabolas cut the x axis in two places. If the parabola touches the x axis only once, this means that there are coincident roots (the solutions are equal to each other). The simplest equation of this form is y = x2. If the parabola does not touch or cross the x axis, there are no real roots. Parabolas prove useful in the real world because of their reflective. properties. Satellite dishes are parabolic for this reason. Signals received by the dish will reflect off the parabola and be directed to one single point—the receiver. Parabolic objects have special reflective properties. With a parabolic mirror, any ray of light parallel to its line of symmetry will reflect off the surface to the same fixed point (A). Practical applications Quadratic equations are used by military specialists to model the trajectory of projectiles fired by artillery—such as this MIM-104 Patriot surface-to-air missile, commonly used by the US Army. Although they were initially used for working out geometric problems, today quadratic equations are important in many aspects of mathematics, science, and technology. Projectile flight, for example, can be modeled with quadratic equations. An object thrown up into the air will fall back down again as a result of gravity. The quadratic function can predict projectile motion—the height of the object over time. Quadratic equations are used to model the relationship between time, speed, and distance, and in calculations with parabolic objects such as lenses. They can also be used to forecast profits and loss in the world of business. Profit is based on total revenue minus production cost; companies create a quadratic equation known as the profit function with these variables to work out the optimal sale prices to maximize profits. See also: Irrational numbers • Negative numbers • Diophantine equations • Zero • Algebra • The binomial theorem • Cubic equations • Imaginary and complex numbers IN CONTEXT KEY CIVILIZATION Ancient Egyptians (c. 1650 BCE) FIELD Arithmetic BEFORE c. 2480 BCE Stone carvings record flood levels on the River Nile, measured in cubits—about 201⁄2 in (52 cm)—and palms—about 3 in (7.5 cm). c. 1800 BCE The Moscow papyrus provides solutions to 25 mathematical problems, including the calculation of the surface area of a hemisphere and the volume of a pyramid. AFTER c. 1300 BCE The Berlin papyrus is produced. It shows that the ancient Egyptians used quadratic equations. 6th century BCE The Greek scientist Thales travels to Egypt and studies its mathematical theories. The Rhind papyrus in the British Museum in London provides an intriguing account of mathematics in ancient Egypt. Named after Scottish antiquarian Alexander Henry Rhind, who purchased the papyrus in Egypt in 1858, it was copied from earlier documents by a scribe, Ahmose, more than 3,500 years ago. It measures 121⁄2 in (32 cm) by 781⁄2 in (200 cm) and includes 84 problems concerned with arithmetic, algebra, geometry, and measurement. The problems, recorded in this and other ancient Egyptian artifacts such as the earlier Moscow papyrus, illustrated techniques for working out areas, proportions, and volumes. The Eye of Horus, an Egyptian god, was a symbol of power and protection. Parts of it were also used to denote fractions whose denominators were powers of 2. The eyeball, for example, represents 1⁄4, while the eyebrow is 1⁄8. Representing concepts The Egyptian number system was the first decimal system. It used strokes for single digits and a different symbol for each power of 10. The symbols were then repeated to create other numbers. A fraction was shown as a number with a dot above it. The Egyptian concept of a fraction was closest to a unit fraction—that is, 1⁄n, where n is a whole number. When a fraction was doubled, it had to be rewritten as one unit fraction added to another unit fraction; for example, 2⁄3 in modern notation would be 1⁄2 + 1⁄6 in Egyptian notation (not 1⁄3 + 1⁄3 because the Egyptians did not allow repeats of the same fraction). The 84 problems in the Rhind papyrus illustrate the mathematical methods in common use in ancient Egypt. Problem 24, for example, asks what quantity, if added to its seventh part, becomes 19. This translates as x + x⁄7 = 19. The approach applied to problem 24 is known as “false position.” This technique—used well into the Middle Ages—is based on trial and improvement, choosing the simplest, or “false,” value for a variable and adjusting the value using a scaling factor (the required quantity divided by the result). In the workings for problem 24, one-seventh is easiest to find for the number 7, so 7 is used first as a “false” value for the variable. The result of the calculation—7 plus 7⁄7 (or 1)—is 8, not 19, so a scaling factor is needed. To find how far the guess of 7 is from the required quantity, 19 is divided by 8 (the “false” answer). This produces a result of 2 + 1⁄4 + 1⁄8 (not 23⁄8, as Egyptian multiplication was based on doubling and halving fractions), which is the scaling factor that should be applied. So 7 (the original “false” value) is multiplied by 2 + 1⁄4 + 1⁄8 (the scaling factor) to give the quantity 16 + 1⁄2 + 1⁄8 (or 165⁄8). Many problems in the papyrus deal with working out shares of commodities or land. Problem 41 asks for the volume of a cylindrical store with a diameter of 9 cubits and a height of 10 cubits. The method finds the area of a square whose side length is 8⁄9 of the diameter, then multiplies this by the height. The figure of 8⁄9 is used as an approximation for the proportion of the area of a square that would be taken up by a circle if it were drawn within the square. This method is used in problem 50 to find the area of a circle: subtract 1⁄9 from the diameter of the circle, and find the area of the square with the resulting side length. Ancient Egyptians used vertical lines to denote the numbers 1 to 9. Powers of 10, particularly those inscribed on stone, were depicted as hieroglyphs—picture symbols. Level of accuracy Since the Ancient Greeks, the area of a circle has been found by multiplying the square of its radius (r2) with the number pi (π), written as πr2. The ancient Egyptians had no concept of pi, but the calculations in the Rhind papyrus show that they were close to its value. Their circle area calculation—with the circle diameter as twice the radius (2r)—can be expressed as (8⁄9 × 2r)2, which, simplified, is 256⁄81 r2, giving an equivalent for pi of 256⁄81. As a decimal, this is about 0.6 percent greater than the true value of pi. Instruction books The Rhind papyrus scribe used the hieratic system of writing numerals. This cursive style was more compact and practical than drawing complex hieroglyphs. The Rhind and Moscow papyri are the most complete mathematical documents to survive from the height of the ancient Egyptian civilization. They were painstakingly copied by scribes well versed in arithmetic, geometry, and mensuration (the study of measurements) and are likely to have been used for training of other scribes. Although they captured probably the most advanced mathematical knowledge of the time, they were not seen as works of scholarship. Instead, they were instruction manuals for use in trade, accounting, construction, and other activities that involved measurement and calculation. Egyptian engineers, for example, used mathematics in the building of pyramids. The Rhind papyrus includes a calculation for the slope of a pyramid using the seked— a measure for the horizontal distance traveled by a slope for each drop of 1 cubit. The steeper the side of a pyramid, the fewer the sekeds. See also: Positional numbers • Pythagoras • Calculating pi • Algebra • Decimals IN CONTEXT KEY CIVILIZATION Ancient Chinese FIELD Number theory BEFORE 9th century BCE The Chinese I Ching (Book of Changes) lays out trigrams and hexagrams of numbers for use in divination. AFTER 1782 Leonhard Euler writes about Latin squares in his Recherches sur une nouvelle espèce de carrés magiques (Investigations on a new type of magic square). 1979 The first Sudoku-style puzzle is published by Dell Magazines in New York. 2001 British electronics engineer Lee Sallows invents magic squares called “geomagic squares,” which contain geometric shapes rather than numbers. There are thousands of ways in which to arrange the numbers 1 to 9 in a three-by-three grid. Only eight of these produce a magic square, where the sum of the numbers in each row, column, and diagonal—the magic total—is the same. The sum of the numbers 1 to 9 is 45, as is the sum of all three rows or columns. The magic total, therefore, is 1⁄3 of 45, or 15. In fact, there is really just one combination of numbers in a magic square. The other seven are rotations of this combination. Ancient origins Magic squares are probably the earliest example of “recreational mathematics.” Their exact origin is unknown, but the first known reference, in the Chinese legend of Lo Shu (Scroll of the river Lo), dates from 650 BCE. In the legend, a turtle appears to the great King Yu as he faces a devastating flood. The markings on the turtle’s back form a magic square, with numbers from 1 to 9 represented by circular dots. Because of this legend, the arrangement of odd and even numbers (even numbers are always in the corners of the square) were believed to have magical properties and was used as a good luck talisman through the ages. As ideas from China spread along trade routes such as the Silk Road, other cultures became interested in magic squares. Magic squares are discussed in Indian texts dating from 100 CE, and Brihat-Samhita (c. 550 CE), a book of divination, includes the first recorded magic square in India, used to measure out quantities of perfume. Arab scholars, who created a vital link between the learning of ancient civilizations and the European Renaissance, introduced magic squares to Europe in the 14th century. An order-four magic square appears beneath the bell in Melencolia I by the German artist Albrecht Dürer and wittily includes the engraving’s date of 1514. Different-sized squares The number of rows and columns in a magic square is called its order. For example, a three-by-three magic square is said to have an order of three. An order-two magic square does not exist because it would only work if all the numbers were identical. As the orders increase, so do the quantities of magic squares. Order four produces 880 magic squares—with a magic total of 34. There are hundreds of millions of order-five magic squares, while the quantity of order-six magic squares has not yet been calculated. Magic squares have been an enduring source of fascination for mathematicians. The 15th-century Italian mathematician Luca Pacioli, author of De viribus quantitatis (On the Power of Numbers), collected magic squares. In 18th-century Switzerland, Leonhard Euler also became interested in them, and devised a form that he named Latin squares. The rows and columns in a Latin square contain figures or symbols that appear only once in each row and column. One derivation of the Latin square—Sudoku—has become a popular puzzle. Devised in the US in the 1970s (where it was called Number Place), Sudoku took off in Japan in the 1980s, acquiring its now-familiar name, which means “single digits.” A Sudoku puzzle is a nine-by-nine Latin square with the added restriction that subdivisions of the square must also contain all nine numbers. The most magically magical of any magic square ever made by a magician. Benjamin Franklin Talking about a magic square that he discovered Once you have one magic square, you can add the same quantity to every number in the square and still end up with a magic square. Similarly, if you multiply all the numbers by the same quantity, you still have a magic square. See also: Irrational numbers • Eratosthenes’ sieve • Negative numbers • The Fibonacci sequence • The golden ratio • Mersenne primes • Pascal’s triangle IN CONTEXT KEY FIGURE Pythagoras (c. 570 BCE–495 BCE) FIELD Applied geometry BEFORE c. 1800 BCE The columns of cuneiform numbers on the Plimpton 322 clay tablet from Babylon include some numbers related to Pythagorean triples. 6th century BCE Greek philosopher Thales of Miletus proposes a non-mythological explanation of the Universe— pioneering the idea that nature can be interpreted by reason. AFTER c. 380 BCE In the tenth book of his Republic, Plato espouses Pythagoras’s theory of the transmigration of souls. c. 300 BCE Euclid produces a formula to find sets of primitive Pythagorean triples. The 6th-century BCE Greek philosopher Pythagoras is also antiquity’s most famous mathematician. Whether or not he was responsible for all the many achievements attributed to him in math, science, astronomy, music, and medicine, there is no doubt that he founded an exclusive community that lived for the pursuit of mathematics and philosophy, and regarded numbers as the sacred building blocks of the Universe. Thales of Miletus, one of the Seven Sages of ancient Greece, possibly inspired the younger Pythagoras with his geometrical and scientific ideas. They may have met in Egypt. Angles and symmetry The Pythagoreans were masters of geometry and knew that the sum of the three angles of a triangle (180°) is equal to the sum of two right angles (90° + 90°), a fact which two centuries later was described by Euclid as the triangle postulate. Pythagoras’s followers were also aware of some of the regular polyhedra; these are the perfectly symmetrical three-dimensional shapes (such as the cube) that were later known as the Platonic solids. Pythagoras himself is primarily associated with the formula that describes the relationship between the sides of a right-angled triangle. Widely known as Pythagoras’s theorem, it states that a2 + b2 = c2, where c is the longest side of the triangle (the hypotenuse), and a and b represent the other two, shorter sides that are adjacent to the right angle. For example, a right-angled triangle with two shorter sides of lengths 3in and 4in will have a hypotenuse of length 5in. The length of this hypotenuse is found because 32 + 42 = 52 (9 + 16 = 25). Such sets of whole-number solutions to the equation a2 + b2 = c2 are known as Pythagorean triples. Multiplying the triple 3, 4, and 5 by 2 produces another Pythagorean triple: 6, 8, and 10 (36 + 64 = 100). The set 3, 4, 5 is called a “primitive” Pythagorean triple because its components share no common divisor larger than 1. The set 6, 8, 10 is not primitive as its components share the common divisor 2. There is good evidence that the Babylonians and the Chinese were well aware of the mathematical relationship between sides of a right-angled triangle centuries before Pythagoras’s birth. However, Pythagoras is believed to have been the first to prove the truth of the formula that states this relationship, and its validity for all right-angled triangles, which is why the theorem takes his name. Pythagorean triples The smallest, or most primitive, of the Pythagorean triples is a triangle with side lengths 3, 4, and 5. As this graphic shows, 9 plus 16 equals 25. The sets of three integers that solve the equation a2 + b2 = c2 are known as Pythagorean triples, although their existence was known long before Pythagoras. Around 1800 BCE, the Babylonians recorded sets of Pythagorean numbers on the Plimpton 322 clay tablet; these show that triples become more spread out as the number line progresses. The Pythagoreans developed methods for finding sets of triples, and also proved that there are an infinite number of such sets. After many of Pythagoras’s schools were destroyed in a 6th-century BCE political purge, Pythagoreans emigrated to other parts of southern Italy, spreading their knowledge of triples across the ancient world. Two centuries later, Euclid developed a formula to generate triples: a = m2 - n2, b = 2mn, c = m2 + n2. With certain exceptions, m and n can be any two integers, such as 7 and 4, which produce the triple 33, 56, 65 (332 + 562 = 652). The formula dramatically sped up the process of finding new Pythagorean triples. The graphic above demonstrates why the Pythagorean equation (a²+ b²= c²) works. Within a large square there are four right-angled triangles of equal size (sides labeled a, b, and c). They are arranged so that a tilted square is formed in the middle, by the hypotenuses (c sides) of the four triangles. Journeys of discovery Pythagoras was well-traveled, and the ideas he absorbed from other countries undoubtedly fueled his mathematical inspiration. Hailing from Samos, which was not far from Miletus in western Anatolia (present-day Turkey), he may have studied at the school of Thales of Miletus under the philosopher Anaximander. He embarked on his travels at the age of 20, and spent many years away. He is thought to have visited Phoenicia, Persia, Babylon, and Egypt, and may also have reached India. The Egyptians knew that a triangle with sides of 3, 4, and 5 (the first Pythagorean triple) would produce a right angle, so their surveyors used ropes of these lengths to construct perfect right angles for their building projects. Observing this method firsthand may have encouraged Pythagoras to study and prove the underlying mathematical theorem. In Egypt, Pythagoras may also have met Thales of Miletus, a keen geometrician, who calculated the heights of pyramids and applied deductive reasoning to geometry. Reason is immortal, all else is mortal. Pythagoras A Pythagorean community After 20 years of traveling, Pythagoras eventually settled in Croton (now Crotone), southern Italy, a city with a large Greek population. There, he established the Pythagorean brotherhood— a community to whom he could teach both his mathematical and philosophical beliefs. Women were welcome in the brotherhood, and formed a significant part of its 600 members. When they joined, members were obliged to give all their possessions and wealth to the brotherhood, and also swore to keep its mathematical discoveries secret. Under Pythagoras’s leadership, the community gained considerable political influence. As well as his theorem, Pythagoras and his close-knit community made numerous other advances in mathematics, but carefully guarded that knowledge. Among their discoveries were polygonal numbers: these, when represented by dots, can form the shapes of regular polygons. For example, 4 is a polygonal number as 4 dots can form a square, and 10 is a polygonal number as 10 dots can form a triangle with 4 dots at the base, 3 dots on the next row, 2 on the next, and 1 dot at the top of the triangle (4 + 3 + 2 + 1 = 10). Two millennia after Pythagoras, in 1638, Pierre de Fermat enlarged on this idea when he asserted that any number could be written as the sum of up to k k-gonal numbers; in other words, every single number is the sum of up to 3 triangular numbers, up to 4 square numbers, or up to 5 pentagonal numbers, and so on. For example, 19 can be written as the sum of three triangular numbers: 1 + 3 + 15 = 19. Fermat could not prove this conjecture; it was only in 1813 that French mathematician Augustin-Louis Cauchy completed the proof. Strength of mind rests in sobriety; for this keeps your reason unclouded by passion. Pythagoras Fascinated by numbers Another type of number that excited Pythagoras was the perfect number. It was so called because it is the exact sum of all the divisors less than itself. The first perfect number is 6, as its divisors 1, 2, and 3 add up to 6. The second is 28 (1 + 2 + 4 + 7 + 14 = 28), the third 496, and the fourth 8,128. There was no practical value in identifying such numbers, but their quirkiness and the beauty of their patterns fascinated Pythagoras and his brotherhood. By contrast, Pythagoras was said to have an overwhelming fear and disbelief of irrational numbers, numbers that cannot be expressed as fractions of two integers, the most famous example being π. Such numbers had no place among the well-ordered integers and fractions by which Pythagoras claimed the Universe was governed. One story suggests that his fear of irrational numbers drove his followers to drown a fellow Pythagorean—Hippasus— for revealing their existence when attempting to find . Pythagoras’s reputation for ruthlessness is also highlighted in a story about a member of the brotherhood who was executed for publicly disclosing that the Pythagoreans had discovered a new regular polyhedron. The new shape was formed from 12 regular pentagons, and known as the dodecahedron—one of the five Platonic solids. Pythagoreans revered the pentagon, and their symbol was the pentagram, a five-pointed star with a pentagon at its center. Breaking the brotherhood’s rule of secrecy by revealing their knowledge of the dodecahedron would therefore have been an especially heinous crime, punishable by death. The finest type of man gives himself up to discovering the meaning and purpose of life itself… this is the man I call a philosopher. Pythagoras In The School of Athens, painted by Raphael in 1509–11 for the Vatican in Rome, Pythagoras is shown with a book, surrounded by scholars eager to learn from him. I have often admired the mystical way of Pythagoras, and the secret magick of numbers. Sir Thomas Browne English polymath An integrated philosophy In ancient Greece, mathematics and philosophy were considered complementary subjects and were studied together. Pythagoras is credited with coining the term “philosopher,” from the Greek philos (“love”) and sophos (“wisdom”). For Pythagoras and his successors, the duty of a philosopher was the pursuit of wisdom. Pythagoras’s own brand of philosophy integrated spiritual ideas with mathematics, science, and reasoning. Among his beliefs was the idea of metempsychosis, which he may have encountered on his travels in Egypt or elsewhere in the Middle East. This held that souls are immortal and at death transmigrate to occupy a new body. In Athens two centuries later, Plato was entranced by the idea and included it in many of his dialogues. Later, Christianity, too, embraced the idea of a division between body and soul; and Pythagoras’s ideas would become a core part of Western thought. Importantly for mathematics, Pythagoras also believed that everything in the Universe related to numbers and obeyed mathematical rules. Certain numbers were endowed with characteristics and spiritual significance in what amounted to a kind of number worship, and Pythagoras and his followers sought mathematical patterns in everything around them. Numbers in harmony Music was of great importance to Pythagoras. He is said to have considered it a holy science, rather than something simply to be used for entertainment. It was a unifying element in his concept of Harmonia, the joining together of the cosmos and the psyche. This may be why he is credited with discovering the link between mathematical ratios and harmony. It is said that, while walking past a blacksmith’s forge, he noticed that different notes were produced when hammers of unequal weight were struck against equal lengths of metal. If the weights of the hammers were in exact and particular proportions, their resulting notes were harmonic. The hammers in the forge had individual weights of 6, 8, 9, and 12 units. Those weighing 6 and 12 units sounded the same notes at different pitches; in today’s music terminology they would be said to be an octave apart. The frequency of the note produced by the hammer of weight 6 was double that of the hammer weighing 12, which corresponds with the ratio of their weights. The hammers of weights 12 and 9 produced a harmonious sound—a perfect fourth—as their weights were in the ratio 4:3. The notes made by the hammers of weights 12 and 8 were also harmonious—a perfect fifth—as their weights were in the ratio 3:2. In contrast, the hammers of weights 9 and 8 were dissonant, as 9:8 is not a simple mathematical ratio. By noticing that harmonious musical notes were connected to numerical ratios, Pythagoras was the first to uncover the relationship between mathematics and music. Pythagoras was reputedly an excellent lyre player. This drawing of ancient Greek musicians illustrates two members of the lyre family— the trigonon (left) and the cithara. Creating a musical scale Although scholars have questioned the story of the forge, Pythagoras is also widely credited with another musical discovery. He is said to have experimented with notes produced by lyre strings of different lengths. He found that while a vibrating string produces a note with frequency f, halving the length of the string produces a note an octave higher, with frequency 2f. When Pythagoras used the same ratios that produced harmoniously sounding hammers, and applied them to vibrating strings, he similarly produced notes in harmony with one another. Pythagoras then constructed a musical scale, starting with one note and the note an octave above it, filling in the notes between using perfect fifths. This scale was used until the 1500s, when it was replaced by the even-tempered scale, in which the notes between the two octaves are more evenly spaced. Although the Pythagorean scale worked well for music lying within one octave, it was not suited for more modern music, which was written in different keys and extended across several octaves. While there have been many different types of musical scales in use by different cultures, the long tradition of Western music dates back to the Pythagoreans and their quest to understand the relationship between music and mathematical proportions. The numerology of the Divine Comedy by Dante (1265–1331)—pictured here in a fresco from the Duomo in Florence, Italy—reflects the influence of Pythagoras, whom Dante mentions several times in his writings. The legacy of Pythagoras Pythagoras’s status as the most famous mathematician from antiquity is justified by his contributions to geometry, number theory, and music. His ideas were not always original, but the rigor with which he and his followers developed them, using axioms and logic to build a system of mathematics, was a fine legacy for those who succeeded him. There is geometry in the humming of the strings, there is music in the spacing of the spheres. Pythagoras PYTHAGORAS Pythagoras was born around 570 BCE on the Greek island of Samos in the eastern Aegean Sea. His ideas have influenced many of the greatest scholars in history, from Plato to Nicolaus Copernicus, Johannes Kepler, and Isaac Newton. Pythagoras is thought to have traveled widely, assimilating ideas from scholars in Egypt and elsewhere in the Middle East before establishing his community of around 600 people in Croton, southern Italy, around 518 BCE. This ascetic brotherhood required its members to live for intellectual pursuits, while following strict rules of diet and clothing. It is from this time onward that his theorem and other discoveries were probably set down, although no records remain. At the age of 60, Pythagoras is said to have married a young member of the community, Theano, and perhaps had two or three children. Political upheaval in Croton led to a revolt against the Pythagoreans. Pythagoras may have been killed when his school was set on fire, or shortly afterward. He is said to have died around 495 BCE. See also: Irrational numbers • The Platonic solids • Syllogistic logic • Calculating pi • Trigonometry • The golden ratio • Projective geometry IN CONTEXT KEY FIGURE Hippasus (5th century BCE) FIELD Number systems BEFORE 19th century BCE Cuneiform inscriptions show that the Babylonians constructed right-angled triangles and understood their properties. 6th century BCE In Greece, the relationship between the side lengths of a right-angled triangle is discovered, and is later attributed to Pythagoras. AFTER 400 BCE Theodorus of Cyrene proves the irrationality of the square roots of the nonsquare numbers between 3 and 17. 4th century BCE The Greek mathematician Eudoxus of Cnidus establishes a strong mathematical foundation for irrational numbers. Any number that can be expressed as a ratio of two integers—a fraction, a decimal that either ends or repeats in a recurring pattern, or a percentage—is said to be a rational number. All whole numbers are rational as they can be shown as fractions divided by 1. Irrational numbers, however, cannot be expressed as a ratio of two numbers Hippasus, a Greek scholar, is believed to have first identified irrational numbers in the 5th century BCE, as he worked on geometrical problems. He was familiar with Pythagoras’s theorem, which states that the square of the hypotenuse in a right-angled triangle is equal to the sum of the squares of the other two sides. He applied the theorem to a right-angled triangle that has both shorter sides equal to 1. As 12 + 12 = 2, the length of the hypotenuse is the square root of 2. Hippasus realized, however, that the square root of 2 could not be expressed as the ratio of two whole numbers—that is, it could not be written as a fraction, as there is no rational number that can be multiplied by itself to produce precisely 2. This makes the square root of 2 an irrational number, and 2 itself is termed nonsquare or square-free. The numbers 3, 5, 7, and many others are similarly nonsquare and in each case, their square root is irrational. By contrast, numbers such as 4 (22), 9 (32), and 16 (42) are square numbers, with square roots that are also whole numbers and therefore rational. The concept of irrational numbers was not readily accepted, although later Greek and Indian mathematicians explored their properties. In the 9th century, Arab scholars used them in algebra. Hippasus may have encountered irrational numbers while exploring the relationship between the length of the side of a pentagon and one side of a pentagram formed inside it. He found that it was impossible to express it as a ratio between two whole numbers. In decimal terms The positional decimal system of Hindu–Arabic numeration allowed further study of irrational numbers, which can be shown as an infinite series of digits after the decimal point with no recurring pattern. For example, 0.1010010001… with an extra zero between each successive pair of 1s, continuing indefinitely, is an irrational number. Pi (π), which is the ratio of the circumference of a circle to its diameter, is irrational. This was proved in 1761 by Johann Heinrich Lambert—earlier estimations of π had been 3 or 22⁄7. Between any two rational numbers, another rational number can always be found. The average of the two numbers will also be rational, as will the average of that number and either of the original numbers. Irrational numbers can also be found between any two rational numbers. One method is to change a digit in a recurring sequence. For example, an irrational number can be found between the recurring numbers 0.124124… and 0.125125… by changing 1 to 3 in the second cycle of 124, to give 0.124324…, and doing so again at the fifth, then ninth cycle, increasing the gap between the replacement 3s by one cycle each time. One of the great challenges of modern number theory has been establishing whether there are more rational or irrational numbers. Set theory strongly indicates that there are many more irrational numbers than rational numbers, even though there are infinite numbers of each. HIPPASUS Details of Hippasus’s early life are sketchy, but it is thought that he was born in Metapontum, in Magna Graecia (now southern Italy), around 500 BCE. According to the philosopher Iamblichus, who wrote a biography of Pythagoras, Hippasus was a founder of a Pythagorean sect called the Mathematici, which fervently believed that all numbers were rational. Hippasus is usually credited with discovering irrational numbers, an idea that would have been considered heresy by the sect. According to one story, Hippasus drowned when his fellow Pythagoreans threw him over the side of a boat in disgust. Another story suggests that a fellow Pythagorean discovered irrational numbers, but Hippasus was punished for telling the outside world about them. The year of Hippasus’s death is not known but is likely to have been in the 5th century BCE. Key work 5th century BCE Mystic Discourse See also: Positional numbers • Quadratic equations • Pythagoras • Imaginary and complex numbers • Euler’s number IN CONTEXT KEY FIGURE Zeno of Elea (c. 495–430 BCE) FIELD Logic BEFORE Early 5th century BCE The Greek philosopher Parmenides founds the Eleatic school of philosophy in Elea, a Greek colony in southern Italy. AFTER 350 BCE Aristotle produces his treatise Physics, in which he draws on the concept of relative motion to refute Zeno’s paradoxes. 1914 British philosopher Bertrand Russell, who described Zeno’s paradoxes as immeasurably subtle, states that motion is a function of position with respect to time. Zeno of Elea belonged to the Eleatic school of philosophy that flourished in ancient Greece in the 5th century BCE. In contrast to the pluralists, who believed that the Universe could be divided into its constituent atoms, Eleatics believed in the indivisibility of all things. Zeno wrote 40 paradoxes to show the absurdity of the pluralist view. Four of these—the dichotomy paradox, Achilles and the tortoise, the arrow paradox, and the stadium paradox—address motion. The dichotomy paradox shows the absurdity of the pluralist view that motion can be divided. A body moving a certain distance, it says, would have to reach the halfway point before it arrived at the end, and in order to reach that halfway mark, it would first have to reach the quarter-way mark, and so on ad infinitum. Because the body has to pass through an infinite number of points, it would never reach its goal. In the paradox of Achilles and the tortoise, Achilles, who is 100 times faster than the tortoise, gives the creature a head start of 100 meters in a race. At the sound of the starting signal, Achilles runs 100 meters to reach the tortoise’s starting point, while the tortoise runs 1 meter, giving it a 1 meter lead. Undeterred, Achilles runs another meter; however, in the same time, the tortoise runs one-hundredth of a meter, so it is still in the lead. This continues, and Achilles never catches up. The stadium paradox concerns three columns of people, each containing an equal number of people; one group is at rest, while the other two run past each other at the same speed in opposite directions. According to the paradox, a person in one moving group can pass two people in the other moving group in a fixed time, but only one person in the stationary group. The paradoxical conclusion is that half a given time is equivalent to double that time. Over the centuries, many mathematicians have refuted the paradoxes. The development of calculus allowed mathematicians to deal with infinitesimal quantities without resulting in contradiction. The paradox of Achilles and the tortoise maintains that a fast object, such as Achilles, will never catch up with a slow one, such as a tortoise. Achilles will get closer to the tortoise, but never actually overtake it. ZENO OF ELEA Zeno of Elea was born around 495 BCE in the Greek city of Elea (now Velia, in southern Italy). At a young age, he was adopted by the philosopher Parmenides, and was said to have been “beloved” by him. Zeno was inducted into the school of Eleatic thought, founded by Parmenides. At the age of around 40, Zeno traveled to Athens, where he met Socrates. Zeno introduced the Socratic philosophers to Eleatic ideas. Zeno was renowned for his paradoxes, which contributed to the development of mathematical rigor. Aristotle later described him as the inventor of the dialectical method (a method starting from two opposing viewpoints) of logical argument. Zeno collected his arguments in a book, but this did not survive. The paradoxes are known from Aristotle’s treatise Physics, which lists nine of them. Although little is known of Zeno’s life, the ancient Greek biographer Diogenes claimed he was beaten to death for trying to overthrow the tyrant Nearchus. In a clash with Nearchus, Zeno is reported to have bitten off the man’s ear. See also: Pythagoras • Syllogistic logic • Calculus • Transfinite numbers • The logic of mathematics • The infinite monkey theorem IN CONTEXT KEY FIGURE Plato (c. 428–348 BCE) FIELD Geometry BEFORE 6th century BCE Pythagoras identifies the tetrahedron, cube, and dodecahedron. 4th century BCE Theaetetus, an Athenian contemporary of Plato, discusses the octahedron and icosahedron. AFTER c. 300 BCE Euclid’s Elements fully describes the five regular convex polyhedra. 1596 German astronomer Johannes Kepler proposes a model of the Solar System, explaining it geometrically in terms of Platonic solids. 1735 Leonhard Euler devises a formula that links the faces, vertices, and edges of polyhedra. The perfect symmetry of the five Platonic solids was probably known to scholars long before the Greek philosopher Plato popularized the forms in his dialogue Timaeus, written in c. 360 BCE. Each of the five regular convex polyhedra—3-D shapes with flat faces and straight edges—has its own set of identical polygonal faces, the same number of faces meeting at each vertex, as well as equilateral sides, and same-sized angles. Theorizing on the nature of the world, Plato assigned four of the shapes to the classical elements: the cube (also known as a regular hexahedron) was associated with earth; the icosahedron with water; the octahedron with air; and the tetrahedron with fire. The 12-faced dodecahedron was associated with the heavens and its constellations. Composed of polygons Only five regular polyhedra are possible—each one created either from identical equilateral triangles, squares, or regular pentagons, as Euclid explained in Book XIII of his Elements. To create a Platonic solid, a minimum of three identical polygons must meet at a vertex, so the simplest is a tetrahedron— a pyramid made up of four equilateral triangles. Octahedra and icosahedra are also formed with equilateral triangles, while cubes are created from squares, and dodecahedra are constructed with regular pentagons. Platonic solids also display duality: the vertices of one polyhedron correspond to the faces of another. For example, a cube, which has six faces and eight vertices, and an octahedron (eight faces and six vertices) form a dual pair. A dodecahedron (12 faces and 20 vertices), and an icosahedron (20 faces and 12 vertices) form another dual pair. Tetrahedra, which have four faces and four vertices, are said to be self-dual. Shapes in the Universe? Like Plato, later scholars sought Platonic solids in nature and the Universe. In 1596, Johannes Kepler reasoned that the positions of the six planets then known (Mercury, Venus, Earth, Mars, Jupiter, and Saturn) could be explained in terms of the Platonic solids. Kepler later acknowledged he was wrong, but his calculations led him to discover that planets have elliptical orbits. In 1735, Swiss mathematician Leonhard Euler noted a further property of Platonic solids, later shown to be true for all polyhedra. The sum of the vertices (V) minus the number of edges (E) plus the number of faces (F) always equals 2, that is, V ˗ E + F = 2. It is also now known that Platonic solids are indeed found in nature—in certain crystals, viruses, gases, and the clustering of galaxies. PLATO Born around 428 BCE to wealthy Athenian parents, Plato was a student of Socrates, who was also a family friend. Socrates’ execution in 399 BCE deeply affected Plato and he left Greece to travel. During this period his discovery of the work of Pythagoras inspired a love of mathematics. Returning to Athens, in 387 BCE he founded the Academy, inscribing over its entrance the words “Let no one ignorant of geometry enter here.” Teaching mathematics as a branch of philosophy, Plato emphasized the importance of geometry, believing that its forms—especially the five regular convex polyhedra—could explain the properties of the Universe. Plato found perfection in mathematical objects, believing they were the key to understanding the differences between the real and the abstract. He died in Athens around 348 BCE. Key works c. 375 BCE The Republic c. 360 BCE Philebus c. 360 BCE Timaeus See also: Pythagoras • Euclid’s Elements • Conic sections • Trigonometry • Non-Euclidean geometries • Topology • The Penrose tile IN CONTEXT KEY FIGURE Aristotle (384–322 BCE) FIELD Logic BEFORE 6th century BCE Pythagoras and his followers develop a systematic method of proof for geometric theorems. AFTER c. 300 BCE Euclid’s Elements describes geometry in terms of logical deduction from axioms. 1677 Gottfried Leibniz suggests a form of symbolic notation for logic, anticipating the development of mathematical logic. 1854 George Boole publishes The Laws of Thought, his second book on algebraic logic. 1884 The Foundations of Arithmetic by German mathematician Gottlob Frege examines the logical principles underpinning mathematics. In the Square of Opposition, S is a subject, such as “sugar,” and P a predicate, such as “sweet.” A and O are contradictory, as are E and I (if one is true, the other is false, and vice versa). A and E are contrary (both cannot be true but both can be false); I and O are subcontrary: both can be true but both cannot be false. I is a subaltern of A and O is a subaltern of E. In syllogistic logic, this means that if A is true, I must be true, but that if I is false, A must be false as well. In Classical Greece, there was no clear distinction between mathematics and philosophy; the two were considered interdependent. For philosophers, one important principle was the formulation of cogent arguments that followed a logical progression of ideas. The principle was based on Socrates’ dialectal method of questioning assumptions to expose inconsistencies and contradictions. Aristotle, however, did not find this model entirely satisfactory, so he set about determining a systematic structure for logical argument. First, he identified the different kinds of proposition that can be used in logical arguments, and how they can be combined to reach a logical conclusion. In Prior Analytics, he describes the propositions as being of broadly four types, in the form of “all S are P,” “no S are P,” “some S are P,” and “some S are not P,” where S is a subject, such as sugar, and P the predicate—a quality, such as sweet. From just two such propositions an argument can be constructed and a conclusion deduced. This is, in essence, the logical form known as the syllogism: two premises leading to a conclusion. Aristotle identified the structure of syllogisms that are logically valid, those where the conclusion follows from the premises, and those that are not, where the conclusion does not follow from the premises, providing a method for both constructing and analyzing logical arguments. Seeking a rigorous proof Implicit in his discussion of valid syllogistic logic is the process of deduction, working from a general rule in the major premise, such as “All men are mortal,” and a particular case in the minor premise, such as “Aristotle is a man,” to reach a conclusion that necessarily follows—in this case, “Aristotle is mortal.” This form of deductive reasoning is the foundation of mathematical proofs. Aristotle notes in Posterior Analytics that, even in a valid syllogistic argument, a conclusion cannot be true unless it is based on premises accepted as true, such as self-evident truths or axioms. With this idea, he established the principle of axiomatic truths as the basis for a logical progression of ideas—the model for mathematical theorems from Euclid onward. ARISTOTLE The son of a physician at the Macedonian court, Aristotle was born in 384 BCE, in Stagira, Chalkidiki. At the age of about 17, he left to study at Plato’s Academy in Athens, where he excelled. Soon after Plato’s death, anti-Macedonian prejudice forced him to leave Athens. He continued his academic work in Assos (now in Turkey). In 343 BCE, Philip II recalled him to Macedonia to head the school at the court; one of his students was Philip’s son, later known as Alexander the Great. In 335 BCE, Aristotle returned to Athens and founded the Lyceum, a rival institution to the Academy. In 323 BCE, after Alexander’s death, Athens again became fiercely anti-Macedonian, and Aristotle retired to his family estate in Chalcis, on Euboea. He died there in 322 BCE. Key works c. 350 BCE Prior Analytics c. 350 BCE Posterior Analytics c. 350 BCE On Interpretation 335–323 BCE Nichomachean Ethics 335–323 BCE Politics See also: Pythagoras • Zeno’s paradoxes of motion • Euclid’s Elements • Boolean algebra • The logic of mathematics IN CONTEXT KEY FIGURE Euclid (c. 300 BCE) FIELD Geometry BEFORE c. 600 BCE The Greek philosopher, mathematician, and astronomer Thales of Miletus deduces that the angle inscribed inside a semicircle is a right angle. This becomes Proposition 31 of Euclid’s Elements. c. 440 BCE The Greek mathematician Hippocrates of Chios writes the first systematically organized geometry textbook, Elements. AFTER c. 1820 Mathematicians such as Carl Friedrich Gauss, János Bolyai, and Nicolai Ivanovich Lobachevsky begin to move toward hyperbolic non-Euclidean geometry. Euclid’s Elements has a strong claim for being the most influential mathematical work of all time. It dominated human conceptions of space and number for more than 2,000 years and was the standard geometrical textbook until the start of the 1900s. Euclid lived in Alexandria, Egypt, in around 300 BCE, when the city was part of the culturally rich Greek-speaking Hellenistic world that flourished around the Mediterranean Sea. He would have written on papyrus, which is not very durable; all that remains of his work are the copies, translations, and commentaries made by later scholars. There is no royal road to geometry. Euclid Collection of works The Elements is a collection of 13 books that range widely in subject matter. Books I to IV tackle plane geometry—the study of flat surfaces. Book V addresses the idea of ratio and proportion, inspired by the thinking of the Greek mathematician and astronomer Eudoxus of Cnidus. Book VI contains more advanced plane geometry. Books VII to IX are devoted to number theory and discuss the properties and relationships of numbers. The long and difficult Book X deals with incommensurables. Now known as irrational numbers, these numbers cannot be expressed as a ratio of integers. Books XI to XIII examine three-dimensional solid geometry. Book XIII of the Elements is actually attributed to another author—Athenian mathematician and disciple of Plato, Theaetetus, who died in 369 BCE. It covers the five regular convex solids—the tetrahedron, cube, octahedron, dodecahedron, and icosahedron, which are often called the Platonic solids—and is the first recorded example of a classification theorem (one that itemizes all possible figures given certain limitations). Euclid is known to have written an account of conic sections, but this work has not survived. Conic sections are figures formed from the intersection of a plane and a cone and they may be circular, elliptical, or parabolic in shape. EUCLID Details of Euclid’s date and place of birth are unknown and knowledge of his life is scant. It is thought that he studied at the Academy in Athens, which had been founded by Plato. In the 5th century CE, the Greek philosopher Proclus wrote in his history of mathematicians that Euclid taught at Alexandria during the reign of Ptolemy I Soter (323–285 BCE). Euclid’s work covers two areas: elementary geometry and general mathematics. In addition to the Elements, he wrote about perspective, conic sections, spherical geometry, mathematical astronomy, number theory, and the importance of mathematical rigor. Several of the works attributed to Euclid have been lost, but at least five have survived to the 21st century. It is thought that Euclid died between the mid-4th century and the mid-3rd century BCE. Key works Elements Conics Catoptrics Phaenomena Optics World of proof The title of Euclid’s work has a particular meaning that reflects his mathematical approach. In the 1900s, British mathematician John Fauvel maintained that the meaning of the Greek word for “element,” stoicheia, changed over time, from “a constituent of a line,” such as an olive tree in a line of trees, to “a proposition used to prove another,” and eventually evolved to mean “a starting point for many other theorems.” This is the sense in which Euclid used it. In the 5th century CE, the philosopher Proclus talked of an element as “a letter of an alphabet,” with combinations of letters creating words in the same way that combinations of axioms—statements that are self-evidently true—create propositions. This opening page of Euclid’s Elements shows illuminated Latin text with diagrams and comes from the first printed edition, produced in Venice in 1482. Logical deductions Euclid was not writing in a vacuum; he built upon foundations laid by a number of influential Greek mathematicians who came before him. Thales of Miletus, Hippocrates, and Plato (among others) had all begun to move toward the mathematical mindset that Euclid so brilliantly formalized: the world of proof. It is this that makes Euclid unique; his writings are the earliest surviving example of fully axiomatized mathematics. He identified certain basic facts and progressed from there to statements that were sound logical deductions (propositions). Euclid also managed to assemble all the mathematical knowledge of his day, and organize it into a mathematical structure where the logical relationships between the various propositions were carefully explained. Euclid faced a Herculean task when he attempted to systematize the mathematics that lay before him. In devising his axiomatic system, he began with 23 definitions for terms such as point, line, surface, circle, and diameter. He then put forward five postulates: any two points can be joined with a straight line segment; any straight line segment can be extended to infinity; given any straight line segment, a circle can be drawn having the segment as its radius and one endpoint as its center; all right angles are equal to one another; and a postulate about parallel lines (see Euclid’s five postulates). He then went on to add five axioms, or common notions; if A = B and B = C, then A = C; if A = B and C = D, A + C = B + D; if A = B and C = D, then A - C = B - D; if A coincides with B, then A and B are equal; and the whole of A is greater than part of A. To prove Proposition 1, Euclid drew a line with endpoints labeled A and B. Taking each endpoint as a center, he then drew two intersecting circles, so that each had the radius AB. This used his third postulate. Where the circles met, he called that point C, and he could draw two more lines AC and BC, calling on his first postulate. The radius of the two circles is the same, so AC = AB and BC = AB; this means that AC = BC, which is Euclid’s first axiom (things that are equal to the same thing are also equal to one another). It follows that AB = BC = CA, meaning that he had drawn an equilateral triangle on AB. In Latin translations of Elements, deductions end with the letters QEF (quod erat faciendum, meaning “which was to be [and has been] done.” Logical proofs end with QED (quod erat demonstrandum, meaning “which was to be [and has been] demonstrated”). The equilateral triangle construction is a good example of Euclid’s method. Each step has to be justified by reference to the definitions, the postulates, and the axioms. Nothing else can be taken as obvious, and intuition is regarded as potentially suspect. Euclid’s very first proposition was criticized by later writers. They noted, for instance, that Euclid did not justify or explain the existence of C, the point of intersection of the two circles. Although apparent, it is not mentioned in his preliminary assumptions. Postulate 5 talks about a point of intersection, but that is between two lines, and not two circles. Similarly, one of the definitions describes a triangle as a plane figure bounded by three lines, which all lie in that plane. However, it seems that Euclid did not explicitly show that the lines AB, BC, and CA lie in the same plane. Postulate 5 is also known as the “parallel postulate” because it can be used to prove properties of parallel lines. It says that if a straight line crossing two straight lines (A, B) creates interior angles on one side that total less than two right angles (180°), lines A and B will eventually cross on that side, if extended indefinitely. Euclid did not use it until Proposition 29, in which he stated that one condition for a straight line crossing two parallel lines was that the interior angles on the same side were equal to two right angles. The fifth postulate is more elaborate than the other four, and Euclid himself seems to have been wary of it. A vital part of any axiomatic system is to have enough axioms, and postulates in the case of Euclid, to derive every true proposition, but to avoid superfluous axioms that can be derived from others. Some asked whether the parallel postulate could be proved as a proposition using Euclid’s common notions, definitions, and the other four postulates; if it could, the fifth was unnecessary. Euclid’s contemporaries and later scholars made unsuccessful attempts to construct such a proof. Finally, in the 1800s, the fifth postulate was ruled both necessary for Euclid’s geometry and independent of his other four postulates. To construct an equilateral triangle, for Proposition 1, Euclid drew a line and centered a circle on its endpoints, here A and B. By drawing a line from each endpoint to C, where the circles intersect, he created a triangle with sides AB, AC, and BC of equal length. Geometry is knowledge of what always exists. Plato Beyond Euclidean geometry The Elements also examines spherical geometry, an area explored by two of Euclid’s successors, Theodosius of Bithynia and Menelaus of Alexandria. While Euclid’s definition of “a point” addresses a point on the plane, a point can also be understood as a point on a sphere. This raises the question of how Euclid’s five postulates can be applied to the sphere. In spherical geometry, almost all the axioms look different from the postulates set out in Euclid’s Elements. The Elements gave rise to what is called Euclidean geometry; spherical geometry is the first example of a non-Euclidean geometry. The parallel postulate is not true for spherical geometry, where all pairs of lines have points in common, nor for hyperbolic geometry, where they can meet infinite numbers of times. The first 16 propositions in Book 1 Proposition 1 On a given finite straight line, to construct an equilateral triangle. Proposition 2 To place at a given point (as an extremity) a straight line equal to a given straight line. Proposition 3 Given two unequal straight lines, to cut off from the greater a straight line equal to the less. Proposition 4 If two sides of one triangle are equal in length to two sides of another triangle, and if the angles contained by each pair of equal sides are equal, then the base of one triangle will equal the base of the other, the two triangles will be of equal area, and the remaining angles in one triangle will be equal to those in the other triangle. Proposition 5 In an