Geometry

VIII

History of Geometry

The derivation of the term geometry—from the Greek words geô, “earth,” and metrein, “to measure”—is an accurate description of the works of the earliest geometers, who were concerned with problems such as measuring the size of fields and laying out accurate right angles for the corners of buildings. In ancient Egypt, for instance, where the Nile River periodically overflowed its banks, geometry was used to reestablish boundary lines on the plots of land affected by the flooding. This type of empirical (based on experience) geometry, which flourished in ancient Egypt, Sumer, and Babylonia, was refined and systematized by the Greeks.

A

Geometry in Ancient Greece

The first important geometer mentioned in history is Thales of Miletus, a Greek who lived about 600 bc. Thales is credited with several simple but important theorems, including the proof that an angle inscribed in a semicircle is a right angle.

He was the first to demonstrate the truth of a geometric relationship by showing that it followed in a logical, orderly fashion from a set of universally accepted statements, called postulates or axioms. These postulates were taken by Thales and his successors to be self-evident truths, but in modern mathematical thinking they are considered to be a group of convenient but arbitrary assumptions. This method of deductive reasoning has dominated all geometry, and in fact all mathematics, to the present day.

One of Thales’ most famous pupils was Pythagoras. Pythagoras and his associates proved many new theorems about triangles, circles, proportions, and certain solids. His most famous proof, a theorem that bears his name, states that the square of the longest side of a right triangle is equal to the sum of the squares of the other two sides.

Typical of the postulates that were developed and accepted by Greek mathematicians is this statement: “A straight line is the shortest distance between two points.” From such postulates a number of theorems about the properties of points, lines, angles, curves, and planes can be logically deduced. However, it was Euclid, who lived about 300 bc, who brought the various unconnected postulates and theorems together into one system in his publication Elements. The 13 'books,' or parchment rolls, of Elements are among the greatest achievements of the human mind. For more than 1,000 years mathematicians could add little of importance to them. Euclid's text served as a basic textbook in geometry almost without alteration into the 20th century. The major importance of Euclid's work lay in his method rather than his results. Most of the theorems he proved had been known for many years. However, it had not been known that they were all closely related or that most of them could be derived from a few basic axioms. By proving this, Euclid established the value of the deductive method.

---

---

The Greeks introduced construction problems, which require a certain line or figure to be constructed by using straightedge and compass alone. (A compass is an instrument used to draw circles; it consists of two pointed arms joined at an adjustable angle.) Simple examples of these problems include the construction of a line that is twice as long as another line or of a line that divides a given angle into two equal angles. Three famous construction problems dating from the time of the ancient Greeks resisted the efforts of many generations of mathematicians to solve them: duplicating the cube (constructing a cube with double the volume of a given cube), squaring the circle (constructing a square equal in area to a given circle), and trisecting the angle (dividing a given angle into three equal parts). None of these constructions is possible with straightedge and compass alone, but the impossibility of squaring the circle was not finally proved until 1882.

Greek mathematician Apollonius of Perga studied the family of curves known as conic sections and discovered many of their fundamental properties about 300 bc. The conic sections are important in many fields of physical science; for example, the orbit of any astronomical object, such as a planet or comet, around any other object, such as the Sun, is always one of the conic sections. Artificial satellites follow elliptical orbits around Earth.

Archimedes, one of the greatest Greek scientists, made a number of important contributions to geometry during the 3rd century bc. He devised ways to measure the areas of a number of curved figures and the surface areas and volumes of solids bounded by curved surfaces, such as cylinders. He also worked out a method for approximating the value of p (the ratio between the diameter and circumference of a circle) and stated that numerically it lay between 3 10/70 and 3 10/71. See Pi.

B

Geometry During the Middle Ages

Geometry, like most other sciences, advanced little from the fall of the Roman Empire in the 5th century ad to the end of the Middle Ages in the 15th century. After the fall of the Greek and Roman civilizations, Europe entered the Dark Ages. Advances in geometry were made largely by Muslims in the Middle East and North Africa and Hindus in India. Most of the works of Greek mathematics were scattered or lost. Some of these, including Elements, were translated and studied by the Muslims and Hindus. Aryabhata, an Indian mathematician living in the 6th century, discovered, or perhaps rediscovered, the formula for the area of an isosceles triangle. He also determined the value of p with remarkable accuracy to four decimal places, setting it equal to 62832/20000, or 3.1416. Between the 4th and 13th centuries other Hindus and Muslims used their geometric knowledge to establish and make advances in the field of trigonometry.

During the 12th and 13th centuries Elements was translated from Greek and Arabic into Latin and the modern European languages, and geometry was added to the curriculum of monastery schools.

C

17th- and 18th-Century Geometry

The next great stride in geometry was taken by French philosopher and mathematician René Descartes, whose influential treatise Discourse on Method was published in 1637. Descartes introduced a method of representing geometric figures within a coordinate system. His work forged a link between geometry and algebra by showing how to apply the methods of one discipline to the other. This link is the basis of analytic geometry, a subject that underlies much modern work in geometry.

Another important 17th-century development was projective geometry, the investigation of the properties of geometrical figures that do not vary when the figures are projected from one plane to another. Gérard Desargues, a French engineer, was led by his study of perspective to develop projective geometry. In the 18th century, Gaspard Monge, a French professor of mathematics, developed still another branch of geometry, called descriptive geometry. Descriptive geometry is the science of making accurate, two-dimensional drawings, or representations, of three-dimensional geometrical forms and of graphically solving problems relating to the size and position in space of such forms. Descriptive geometry is the basis of much of engineering and architectural drafting.

D

Modern Geometry

Analytic, projective, and descriptive geometry came into being within the framework of Euclidean geometry. For many centuries mathematicians believed that Euclid’s fifth postulate of the unique parallel could be proved on the basis of Euclid’s first four postulates, but all efforts to discover such a proof were fruitless. In the 19th century, however, geometries were developed in which Euclid's fifth postulate was replaced by alternative statements. The leaders in developing these non-Euclidean geometries were Carl Friedrich Gauss, János Bolyai, Nikolay I. Lobachevsky, and George Friedrich Bernhard Riemann.

In 1872 German mathematician Felix Klein used a relatively new branch of mathematics called group theory to unify and classify all the geometries of his time. In 1899 David Hilbert, another German mathematician, published his Foundations of Geometry, which provided a rigorous system of axioms for Euclidean geometry and exerted great influence on other branches of mathematics.

In 1916 the theory of relativity showed that many physical phenomena could be deduced from geometric principles. The success of the theory gave impetus to studies in differential geometry and in topology.

Geometry in four or more dimensions (n-dimensional geometry, in mathematical terms) was developed by 19th-century British mathematician Arthur Cayley. Another dimensional concept, that of fractional dimensions, also arose in the 19th century. In the 1970s this concept was developed into a new field of geometry known as fractal geometry.