The Math Book

by DK

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 Pl
impton 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 perfe
ct 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.