The Math Book

by DK

  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 isosceles triangle, the angles at the base are equal to one another, and, if the equal straight lines are extended below the base, the angles under the base will also be equal to one another.

  Proposition 6 If in a triangle two angles are equal to one another, the sides separated from the third side by these angles will also be equal.

  Proposition 7 Given two straight lines constructed on a straight line (from its extremities) and meeting in a point, there cannot be constructed on the same straight line (from its extremities), and on the same side of it, two other straight lines meeting in another point and equal to the former two respectively, namely each to that which starts at the same extremity.

  Proposition 8 If two sides of one triangle are equal in length to two sides of another triangle, and the base of one triangle is equal to the base of the other, the angles of the two triangles will also be equal.

  Proposition 9 To bisect a given rectilineal angle.

  Proposition 10 To bisect a given finite straight line.

  Proposition 11 To draw a straight line at right angles to a given straight line from a given point on it.

  Proposition 12 To a given infinite straight line, from a given point which is not on it, to draw a perpendicular straight line.

  Proposition 13 If a straight line set up on a straight line makes angles, it will make either two right angles or angles equal to two right angles.

  Proposition 14 If with any straight line, and at a point on it, two straight lines not lying on the same side and meeting at the point make adjacent angles equal to two right angles, the two straight lines will be in a straight line with one another.

  Proposition 15 If two straight lines cut one another, they make the vertical angles equal to one another.

  Proposition 16 In any triangle, if one of the sides is extended, the angle between the triangle and the extended side is greater than any of the angles inside the triangle.

  See also: Pythagoras • The Platonic solids • Syllogistic logic • Conic sections • The problem of maxima • Non-Euclidean geometries

  IN CONTEXT

  KEY CIVILIZATION

  Ancient Greeks (c. 300 BCE)

  FIELD

  Number systems

  BEFORE

  c. 18,000 BCE In Central Africa, numbers are recorded on bone as carved marks.

  c. 3000 BCE South American Indians record numbers by tying knots in string.

  c. 2000 BCE The Babylonians develop positional numbers.

  AFTER

  1202 Leonardo of Pisa (Fibonacci) commends the Hindu–Arabic number system in Liber Abaci.

  1621 In England, William Oughtred invents the slide rule, which simplifies the use of logarithms.

  1972 Hewlett Packard invents an electronic scientific calculator for personal use.

  The abacus is a counting device and calculator that has been in use since ancient times. It comes in many forms, but all of them work on the same principles: values of different sizes are represented by “counters” arranged in columns or rows.

  Early abaci

  The word “abacus” may hint at its origins. It is a Latin word derived from the ancient Greek, abax, which means “slab” or “board”— a surface that would have been covered in sand and used as a drawing board. The oldest surviving abacus is the Salamis Tablet, a marble slab made c. 300 BCE that is etched with horizontal lines. Pebbles were placed on these lines to count out values. The bottom line represented 0 to 4; the line above counted 5s, and the lines above that 10s, 50s, and so on. The tablet was discovered on the Greek island of Salamis in 1846.

  Some scholars believe that the Salamis Tablet was actu
ally Babylonian. The Greek abax may have come from the Phoenician or Hebrew word for “dust” (abaq) and may refer to far older counting tables developed in Mesopotamian civilizations, where counters were set out on grids drawn in sand. The Babylonian positional number system, developed c. 2000 BCE, may have been inspired by the abacus.

  The Romans upgraded the Greek counting table into a device that greatly simplified calculations. The horizontal rows of the Greek abacus became vertical columns in the Roman abacus, in which were set small pebbles—or calculi in Latin, from which we get the word “calculation.”

  A type of abacus was also in use in the pre-Columbian civilizations of Central America. Based on a five-digit vigesimal, or base-20, counting system, it used corn kernels threaded on strings to represent numbers. No device has survived, but scholars think that the ancient Olmec people invented it 3,000 years ago. By about 1000 CE, the Aztec people knew it as the nepohualtzintzin—the “personal accounts counter”—and wore it on the wrist as a bracelet.

  The suanpan shown here is set to the number 917,470,346. The suanpan is traditionally a 2:5 abacus—each column has two “heaven” beads, each with a value of 5, and 5 “earth” beads, each with a value of 1, giving a potential value of 15 units. This allows for calculations involving the Chinese base-16 system, which uses 15 units rather than the 9 used in the decimal system. Numbers can be added together by entering the units of one number, starting from the right, then adjusting the beads as further numbers are entered. For subtraction, the units of the first number are entered, then bead values are adjusted downward in each column as further subtracted numbers are entered.