Mathematics

Article Outline

Introduction; Mathematics: The Language of Science; Branches of Mathematics; History of Mathematics

Number System

The Greek number system was based on the alphabet. The Attic system, in use from 600 bc to 200 bc, used a stroke for 1 and the initial letters of the words for 5, 10, 100, 1,000, and 10,000—namely, the initials of pente, deka, hekaton, khilioi, and myrioi—to represent the respective numbers. A later system assigned number values to the 24 letters of the Greek alphabet and to 3 other letters that were no longer used. The letters could be combined to form numbers through 999. For higher numbers, a stroke preceding the initial letter (1 through 9) indicated a multiple of 1,000 (1,000 through 9,000). For 10,000 and above, the symbol M indicated that the numeral below should be multiplied by 10,000. See also Numerals.

Pythagoras and the Pythagoreans

Pythagoras taught the importance of studying numbers in order to understand the world. We know of his achievements only from his disciples, the Pythagoreans, who made important discoveries about number theory and geometry. The Pythagoreans represented whole numbers by using arrangements of dots or pebbles, and classified these numbers according to the shapes produced. (The English word calculation is derived from the Greek word for stone or pebble.) The numbers 3, 6, 10, and so on were called triangular numbers because the pebbles could be arranged to form triangles. The numbers 4, 9, 16, and so on were called square numbers because the pebbles could be arranged as squares.

From these simple geometrical arrangements some properties of the whole numbers emerged. The Pythagoreans concluded that the sum of two consecutive triangular numbers is always a square number. They called a perfect number one that equaled the sum of its divisors, for example 6 (the sum of 1, 2, and 3), 28 (the sum of 1, 2, 4, 7, and 14), and 496 (the sum of 1, 2, 4, 8, 16, 31, 62, 124, and 248).

To the Pythagoreans a number represented more than a quantity. The number 2 suggested diversity and so was identified with opinion. Four represented justice because it was the first number that was the product of equals (2 x 2). The identification of the square number 4 with justice continues in the phrase square shooter, meaning someone who acts in an impartial and straightforward manner.

The Importance of Geometry

The classical Greeks solved equations involving unknown quantities by using a series of geometrical constructions. Constructions called for adding, subtracting, multiplying, and dividing lines and taking the square root of a line; the method is referred to as geometrical algebra. We still speak of 25 as “the square of 5” and of 27 as “the cube of 3.”

The casting of problems in geometrical form had several major consequences. It separated numbers and geometry, for only geometry could handle incommensurable ratios. Geometry became the basis for almost all rigorous mathematics until at least 1600. As late as the 18th century, when algebra and calculus had already been well developed, rigorous mathematics meant geometry, and geometer meant “mathematician.”

Among great geometers of the 5th century bc were the philosopher Democritus and the mathematician Hippocrates of Chios. Democritus, who is better known for his theory that all matter is made up of atoms, discovered the correct formula for the volume of a pyramid. Hippocrates made discoveries related to the problem of squaring the circle—that is, constructing a square equal in area to a given circle. Two other problems that originated during the 5th century were those of trisecting (dividing into three equal parts) an angle and doubling a cube—that is, constructing a cube that is double in volume to a given cube. These problems were critical to the development of geometry, and the ancient Greeks solved all three in a variety of ways.

Eudoxus and Euclid

In the 4th century bc Eudoxus of Cnidus introduced the notion of magnitude as a ratio rather than an exact number for dealing with entities such as line segments and angles. He also discovered a method for determining the areas and volumes of curved figures, such as the circle, by successive approximations. Through this method of exhausting all possibilities, he closely approximated the area of a circle by inscribing the circle in a polygon. As the number of sides in the polygon increased, the closer its area came to the area of the circle. With his method of exhaustion, Eudoxus took the first step in the creation of calculus.

Eudoxus also provided the first astronomical theory to account for observed planetary motions. His was a purely mathematical theory that showed how combinations of rotating spheres of varying radii and axes of rotation could explain the seemingly irregular motion of the Sun, Moon, and planets.

About 300 bc Euclid, a Greek mathematician who taught in Alexandria, Egypt, organized the work of many Greek mathematicians in a masterful work called the Elements. The work of Eudoxus had established the deductive organization of mathematics on the basis of axioms. From a few well-chosen axioms Euclid deduced some 500 theorems comprising all the important results of Greek mathematics to that time. Euclid began by defining terms such as line, angle, and circle. He then stated ten self-evident truths such as “The whole is greater than any of its parts.” From these ten axioms he was able to deduce all the theorems. Though mathematicians generally regarded the Elements as a model of rigor until well into the 19th century, it had serious defects, including the unconscious use of unstated assumptions.

Greek Mathematics after Euclid

The character of Greek mathematics changed after about 300 bc during the so-called Alexandrian period, the period of Greek civilization following the conquests of Alexander the Great. Alexandrian mathematics was a fusion of classical Greek mathematics and Babylonian and Egyptian mathematics. In general, the mathematicians of the Alexandrian period were more inclined toward engineering than philosophy. The great Alexandrian mathematicians—Eratosthenes, Archimedes, Apollonius of Perga, Ptolemy, Diophantus, and Hipparchus—displayed the Greek genius for theoretical abstraction but were nevertheless willing to apply their talents to practical problems and quantitative (measurable) knowledge.

Eratosthenes found a simple method of accurately calculating the circumference of the earth in the 3rd century bc, and he also devised a calendar that called for an extra day every fourth year. Archimedes, the greatest mathematician of antiquity, produced theorems on complicated areas and volumes and proved them rigorously. He sought exact answers and found upper and lower limits for the values of irrational numbers. Thus, working with a 96-sided polygon, he determined that the value of p lay between and . Archimedes proved some theorems that contained new geometrical algebra. For example, he stated the problem of cutting a given sphere by a plane so that the volumes of the segments correspond to one another in a given ratio. He then solved the problem geometrically by finding the intersection of a parabola and a rectangular hyperbola. Apollonius, a younger colleague of Archimedes, produced an eight-book treatise on conic sections that established the names of the sections: ellipse, parabola, and hyperbola. The work also provided the basic treatment of the geometry of conic sections until the time of French philosopher and scientist René Descartes in the 17th century.

Archimedes was also the greatest mathematical physicist of ancient times. He used geometrical arguments to prove statements in mechanics. His book Floating Bodies is a foundation work in hydrostatics (the study of liquids at rest). Legend has it that while bathing Archimedes discovered the principle that a body immersed in water is buoyed up by a force equal to the weight of the water displaced, and in his elation he ran out naked into the street shouting “Eureka!” (“I have found it”).

During the Alexandrian period arithmetic and algebra were, for the first time, studied independently of geometry. The first sizable book in which arithmetic was treated independently was the Introduction to Arithmetic of Nicomachus, written about ad 100. Its importance for arithmetic was comparable to that of Euclid’s Elements for geometry. It remained the standard text for 1,000 years because it presented a systematic, clear, and comprehensive treatment of the integers and ratios of integers. While repeating many Pythagorean statements, the work went beyond them in seeing, though not proving, general relations. Diophantus, sometimes called the father of algebra, wrote an outstanding work on algebra about ad 250. He introduced symbolism into algebra and studied solutions to algebraic equations with several unknowns, a field now called Diophantine analysis.

The supreme achievement of the Alexandrian mathematicians was the creation of a quantitative astronomy. Hipparchus developed tables of trigonometric ratios in the 2nd century bc, for which he is sometimes credited as having invented trigonometry. His method rested on the geometric theorem that if two triangles are similar, the ratio of the lengths of any two sides of one triangle equals the corresponding ratio of the other triangle. With the tables and a readily measured distance on Earth, he was able to calculate the circumference of the Earth and the distance to the Moon. He found the distance to the Moon equal to between 59 and 67 times the radius of the Earth; the correct figure is 60 times. He found the radius of the Moon to be one-third of the Earth’s radius; the present figure is 27/100.

The development of Greek trigonometry and its application to astronomy culminated in the Almagest of astronomer Ptolemy in the 2nd century ad. The Almagest put trigonometry into the form it retained for more than 1,000 years and presented a theory of the motions of the celestial bodies that prevailed until it was overthrown by the Copernican system in the 16th century.

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Number System

Pythagoras and the Pythagoreans

The Importance of Geometry

Eudoxus and Euclid

Greek Mathematics after Euclid