The Math Book

by DK

  Key works

  Mensuram orae ad terram (On the Measurement of the Earth)

  Geographika (Geography)

  See also: Mersenne primes • The Riemann hypothesis • The prime number theorem • Finite simple groups

  IN CONTEXT

  KEY FIGURE

  Apollonius of Perga (c. 262–190 BCE)

  FIELD

  Geometry

  BEFORE

  c. 300 BCE Euclid’s 13-volume Elements sets out the propositions that form the basis of plane geometry.

  c. 250 BCE In On Conoids and Spheroids, Archimedes deals with the solids created by the revolution of conic sections about their axes.

  AFTER

  c. 1079 CE Persian polymath Omar Khayyam uses intersecting conics to solve algebraic equations.

  1639 In France, 16-year-old Blaise Pascal asserts that where a hexagon is inscribed in a circle, the opposite sides of the hexagon meet at three points on a straight line.

  Of the many pioneering mathematicians produced by ancient Greece, Apollonius of Perga was one of the most brilliant. He began studying mathematics after Euclid’s great work Elements had emerged and he employed the Euclidian method of taking “axioms”—statements taken to be true—as starting points for further reasoning and proofs.

  Apollonius wrote on many subjects, including optics (how light rays travel) and astronomy, as well as geometry. Much of his work survives only in fragments, but his most influential, Conics, is relatively intact. It was written in eight volumes, of which seven survive: books 1–4 in Greek, and books 5–7 in Arabic. The work was designed to be read by mathematicians already well versed in geometry.

  I have sent my son… to bring you… the second book of my Conics. Read it carefully and communicate it to such others as are worthy of it.

  Apollonius of Perga

  A new geometry

  Early Greek mathematicians such as Euclid focused on the line and circle as the purest geometric forms. Apollonius viewed these in three-dimensional terms: if a circle is combined with all lines that emanate from it, above or below its plane, and those lines pass through the same fixed point—the vertex—a cone is created. By slicing that cone in different ways, a series of curves, known as conic sections, can be produced.

  In Conics, Apollonius expounded in minute detail this new world of geometric construction, studying and defining the properties of conic sections. He based his workings on the assumption of two cones joined at the same vertex, with the area of their circular bases potentially stretching to infinity. To three of the conic sections he gave the names ellipse, parabola, and hyperbola. An ellipse occurs when a plane intersects a cone on a slant. A parabola emerges if the cut is parallel to the edge of the cone, and a hyperbola results when the plane is vertical. Although he saw the circle as one of the four conic sections, it is really an ellipse with the plane perpendicular to the axis of the cone.

  [Conic sections are] the necessary key with which to attain the knowledge of the most important laws of nature.

  Alfred North Whitehead

  British mathematician

  Paving the way for others

  In his description of these four geometric objects, Apollonius used no algebraic formulae and no numbers. However, his view of a conic curve as a set of ordered parallel lines emanating from an axis looked toward the later creation of coordinate system geometry. He did not achieve the kind of precision that would come 1,800 years later with the work of French mathematicians René Descartes and Pierre de Fermat, but he did get close to coordinate representations of his conic curves. Some things held Apollonius back: he did not use negative numbers, nor did he explicitly work with zero. So while the two-dimensional Cartesian geometry developed by Descartes worked across four quadrants—with both positive and negative coordinates—Apollonius effectively worked in just one.

  Apollonius’s studies inspired many of the advances in geometry seen in the Islamic world during the Middle Ages. His work was then rediscovered in Europe during the Renaissance, leading mathematicians to develop the analytic geometry that helped to fuel the scientific revolution.

  When a plane intersects a cone, it creates a conic section. As well as the sections described by Apollonius, this can be a single point, where the plane cuts across the apex (top vertex), or straight lines cutting through the apex at an angle.

  APOLLONIUS OF PERGA

  Little is known about the life of Apollonius. He was born in c.262 BCE in Perga, a center for the worship of the goddess Artemis, in southern Anatolia (now part of Turkey). After crossing the Mediterranean to Egypt, he was taught by Euclidean scholars in the great cultural city of Alexandria.

  It is thought that all eight volumes of Conics were compiled while Apollonius was in Egypt. The first volumes produced little that was not known to Euclid, but the later works were a significant advance in geometry.

  Beyond his work with conic sections, Apollonius is credited with estimating the value of pi more accurately than his contemporary Archimedes, and with being the first to state that

  Key work

  c. 200 BCE Conics

  See also: Euclid’s Elements • Coordinates • The area under a cycloid • Projective geometry • The complex plane • Non-Euclidean geometries • Proving Fermat’s last theorem

  IN CONTEXT

  KEY FIGURE

  Hipparchus (c. 190–120 BCE)

  FIELD

  Geometry

  BEFORE

  c. 1800 BCE The Babylonian Plimpton 322 tablet contains a list of Pythagorean triples, long before Pythagoras devised his formula a2 + b2 = c2.

  c. 1650 BCE The Egyptian Rhind papyrus includes a method for calculating the slope of a pyramid.

  6th century BCE In ancient Greece, Pythagoras discovers his theorem relating to the geometry of triangles.

  AFTER

  500 CE In India, the first trigonometric tables are used.

  1000 CE In the Islamic world, mathematicians are using all the various ratios between the sides and angles of triangles.

  Trigonometry, a term based on the Greek words for “triangle” and “measure,” is of immense importance in both the historical development of mathematics and in the modern world. Trigonometry is one of the most useful of all the mathematical disciplines, enabling people to navigate the world, to understand electricity, and to measure the height of mountains.

  Since antiquity, civilizations have appreciated the need for right angles in architecture. This led mathematicians to analyze the properties of right-angled triangles: all right-angled triangles contain two shorter sides (which may or may not be of equal length) and a diagonal, or hypotenuse, which is longer than either of the others; all triangles contain three angles; and right-angled triangles have one angle of 90°.

  The Plimpton tablet

  In the early 1900s, an examination of triangles, dating back to around 1800 BCE, was discovered on an ancient Babylonian clay tablet. The tablet, bought by American publisher George Plimpton in 1923 and known as Plimpton 322, is etched with numerical information relating to right-angled triangles. Its exact significance is debated, but the information appears to include Pythagorean triples (three positive numbers representing the lengths of sides of a right-angled triangle), alongside another set of numbers that resemble the ratios of the squares of sides. The tablet’s original purpose is unknown, but it may have been used as a practical manual for measuring dimensions.

  At around the same time as the ancient Babylonians, Egypt’s mathematicians were developing an interest in geometry. This was driven not just by their monumental building program, but also by the annual flooding of the Nile River, which required them to mark out the areas of fields each time the floods subsided. Egyptian interest is evident in the Rhind papyrus, a scroll that contains a set of tables relating to fractions. One of these tables poses the question: “If a pyramid is 250 cubits high and the side of its base is 360 cubits long, what is its seked?” The word seked means slope, so the problem is purely trig
onometrical.

  Even if he did not invent it, Hipparchus is the first person of whose systematic use of trigonometry we have documentary evidence.

  Sir Thomas Heath

  British historian of mathematics

  Hipparchus sets out rules

  Influenced by Babylonian theories on angles, the ancient Greeks developed trigonometry as a branch of mathematics that was governed by definite rules rather than the tables of numbers relied on by the earliest mathematicians. In the 2nd century BCE, the astronomer and mathematician Hipparchus, generally regarded as the founder of trigonometry, was particularly interested in triangles inscribed within circles and spheres, and the relationship between angles and lengths of chords (straight lines drawn between two points on a circle—or on any curve). Hipparchus compiled what was effectively the first true table of trigonometric values.

  In the medieval period, astrolabes applied trigonometric principles to measure the position of celestial bodies. Hipparchus is credited with inventing the device.

  Ptolemy’s contribution

  Around 300 years later, in the Egyptian city of Alexandria, the gifted Greco-Roman polymath Claudius Ptolemaeus, better known as Ptolemy, wrote a mathematical treatise called the Syntaxis Mathematikos (later renamed the Almagest by Islamic scholars). In this work, Ptolemy further developed the ideas of Hipparchus on triangles and chords of circles, building formulae that would allow the prediction of the position of the Sun and other “heavenly bodies” based on the assumption of circular orbits around Earth. Ptolemy, like the mathematicians before him, used the Babylonian system of numbers known as the sexagesimal system, based on the number 60.

  Ptolemy’s work was developed further in India, where the growing discipline of trigonometry was regarded as part of astronomy. The Indian mathematician Aryabhata (474–550 CE) pursued the study of chords to produce the first table of what is now known as the sine function (all the possible values of sine/cosine ratios for determining the unknown length of the side of a triangle when the lengths of the hypotenuse—the triangle’s longest side—and the side opposite the angle are known).

  In the 7th century CE, another great Indian mathematician and astronomer, Brahmagupta, made his own contributions to geometry and trigonometry, including what is now known as Brahmagupta’s formula. This is used to find the area of cyclic quadrilaterals, which are four-sided shapes inscribed within a circle. This area can also be found with a trigonometric method if the quadrilateral is split into two triangles.

  Trigonometry, like other branches of mathematics, was not the work of any one man, or nation.

  Carl Benjamin Boyer

  American historian of mathematics

  Islamic trigonometry

  Brahmagupta had already created a table of sine values, but in the 9th century CE, Persian astronomer and mathematician Habash al-Hasib (“Habash the Calculator”) produced some of the first sine, cosine, and tangent tables to calculate the angles and sides of triangles. Around the same time, al-Battani (Albatenius) developed Ptolemy’s work on the sine function and applied it to astronomical calculations. He recorded highly accurate observations of celestial objects from Raqqah, Syria. The motivation among Arab scholars for developing trigonometry was not just for astronomy, but also for religious purposes, since it was important that Muslims knew the position of the holy city of Mecca from anywhere in the world.

  In the 12th century CE, Indian mathematician and astronomer Bhaskara II invented the study of spherical trigonometry. This explores triangles and other shapes on the surface of a sphere rather than on a plane.

  In later centuries, trigonometry became invaluable in navigation as well as astronomy. Bhaskara II’s work, along with the ideas in Ptolemy’s Almagest, were valued by the Islamic scholars of the medieval world, who had begun studying trigonometry well before Bhaskara II.

  A logarithmic table is a small table by the use of which we can obtain knowledge of all geometrical dimensions and motions in space.

  John Napier

  Aid to astronomy

  Along with the developments in trigonometry, there was a gradual and corresponding shift in the way people viewed the heavens. From passively observing and recording the patterns in the movement of celestial bodies, scholars began to model that movement mathematically so that they could predict future astronomical events with ever greater accuracy. The study of trigonometry purely as an aid to astronomy persisted well into the 1500s, when new developments in Europe began to gain momentum. De Triangulis Omnimodis (On Triangles of all Kinds) was published in 1533. Written by German mathematician Johannes Müller von Königsberg, known as Regiomontanus, it was a compendium of all known theorems for finding sides and angles of both planar (2-D) and spherical triangles (those formed on the surface of a 3-D sphere). The publication of this work marked a turning point for trigonometry. It was no longer merely a branch of astronomy, but a key component of geometry.

  Trigonometry was to develop even further; although geometry was its natural home, it was also increasingly applied to solve algebraic equations. French mathematician François Viète showed how algebraic equations could be solved using trigonometric functions, in conjunction with the new system of imaginary numbers that had been invented by Italian mathematician Rafael Bombelli in 1572.

  At the end of the 1500s, Italian physicist and astronomer Galileo Galilei used trigonometry to model the trajectories of projectiles on which gravity was acting. The same equations are still used to project the motion of rockets and missiles into the atmosphere today. Also in the 1500s, Dutch cartographer and mathematician Gemma Frisius used trigonometry to determine distances, thus enabling accurate maps to be created for the first time.

  To find the unknown angle (θ) in a right-angled triangle, the sine formula is used when the lengths of the opposite (opposite θ) and the hypotenuse are known; the cosine formula is used when the lengths of the adjacent and hypotenuse are known; and the tangent formula is used when the lengths of the opposite and adjacent are known.

  New developments

  Developments in trigonometry gathered pace in the 1600s. Scottish mathematician John Napier’s discovery of logarithms in 1614 enabled the compilation of accurate sine, cosine, and tangent tables. In 1722, Abraham de Moivre, a French mathematician, went a step further than Vieté and showed how trigonometric functions could be used in the analysis of complex numbers. The latter comprised a real part and an imaginary part, and were to be of great significance in the development of mechanical and electrical engineering. Leonhard Euler used de Moivre’s findings to derive the “most elegant equation in mathematics”: eiπ + 1 = 0, also known as Euler’s identity.

  In the 1700s, Joseph Fourier applied trigonometry to his research into different forms of waves and vibrations. The “Fourier trigonometry series” has been used widely in scientific fields such as optics, electromagnetism, and, more recently, quantum mechanics. From its early beginnings, when the Babylonians and ancient Egyptians pondered the lengths of shadows cast by a stick in the ground, through architecture and astronomy to modern applications, trigonometry has formed a part of the language of mathematics in modeling the Universe.

  A network of triangulation stations such as this stone “trig point” in Wales was launched by the Ordnance Survey in 1936 to accurately map the island of Great Britain.

  HIPPARCHUS

  Hipparchus was born in Nicaea (now Iznik in Turkey) in 190 BCE. Although little is known of his life, he achieved fame as an astronomer from the studies he carried out while living on the island of Rhodes. His findings were immortalized in Ptolemy’s Almagest, where he is described as “a lover of truth.”

  The only work of Hipparchus to survive was his commentary on the Phaenomena of the poet Aratus and the mathematician and astronomer Eudoxus, criticizing the inaccuracy of their descriptions of constellations. Hipparchus’s most notable contribution to astronomy was his work Sizes and Distances (now lost, but used by Ptolemy), on the orbits of the Sun and Moon, which enabl
ed him to calculate the dates of the equinoxes and solstices. He also compiled a star catalogue, which may be the one used by Ptolemy in Almagest. Hipparchus died in 120 BCE.

  Key work

  2nd century BCE Sizes and Distances

  See also: The Rhind papyrus • Pythagoras • Euclid’s Elements • Imaginary and complex numbers • Logarithms • Pascal’s triangle • Viviani’s triangle theorem • Fourier analysis

  IN CONTEXT

  KEY CIVILIZATION

  Ancient Chinese (c. 1700 BCE–c. 600 CE)

  FIELD

  Number systems

  BEFORE

  c. 1000 BCE In China, bamboo rods are first used to denote numbers, including negatives.

  AFTER

  628 CE The Indian mathematician Brahmagupta provides rules for arithmetic with negative numbers.

  1631 In Practice of the Art of Analysis, published 10 years after his death, British mathematician Thomas Harriott accepts negative numbers in algebraic notation.

  While practical notions of negative quantities were used from ancient times, particularly in China, negative numbers took far longer to be accepted within mathematics. Ancient Greek thinkers and many later European mathematicians regarded negative numbers—and the concept of something being less than nothing—as absurd. Only in the 1600s did European mathematicians begin to fully accept negative numbers.