Geometry I INTRODUCTION Geometry, branch of mathematics that deals with shapes and sizes. Geometry may be thought of as the science of space. Just as arithmetic deals with experiences that involve counting, so geometry describes and relates experiences that involve space. Basic geometry allows us to determine properties such as the areas and perimeters of two-dimensional shapes and the surface areas and volumes of three-dimensional shapes. People use formulas derived from geometry in everyday life for tasks such as figuring how much paint they will need to cover the walls of a house or calculating the amount of water a fish tank holds. II METHODOLOGY Geometry combines simple conceptual building blocks to construct complex logical structures. These building blocks include undefined terms, defined terms, and postulates. Combining these components creates chains of reasoning that support conclusions called theorems. A Undefined terms Some concepts central to geometry are not defined in terms of simpler concepts. The most familiar of these undefined terms are point, line, and plane. These fundamental concepts arose from everyday experiences. Thus, the experience of where an object is leads to the idea of an exact, fixed location. This is the intuitive idea to which the term point refers. Many physical objects suggest the idea of a point. Examples include the corner of a block, the tip of a pencil, or a dot on a sheet of paper. Such things are called models or representations or pictures of points, although they show only approximately the idea in mind. Similarly, a row of points suggested by a tightly stretched string, the edge of a desk, or a flagpole, extended infinitely in both directions, is called a line. The word plane describes a flat surface--such as a floor, desktop, or chalkboard--but it is imagined as extending infinitely in all directions. This means that a plane has no edges just as a line has no ends. Other undefined terms describe relations among points, lines, and planes, such as the relation described by the phrase "a point that lies on a line." B Defined Terms Undefined terms can be combined to define other terms. Noncollinear points, for example, are points that do not lie on the same line. A line segment is the portion of a line that includes two particular points and all points that lie between them, while a ray is the portion of a line that includes a particular point, called the end point, and all points extending infinitely to one side of the end point. Defined terms can be combined with each other and with undefined terms to define still more terms. An angle, for example, is a combination of two different rays or line segments that share a single end point. Similarly, a triangle is composed of three noncollinear points and the line segments that lie between them. C Postulates Postulates, or axioms, are unproven but universally accepted assumptions, such as "there is one and only one line that passes through two distinct points." A system consisting of a set of noncontradictory postulates concerning the undefined terms point, line, and plane, together with the theorems deduced from these postulates, is called a geometry. Different sets of postulates determine whole different systems of geometry. If the postulates selected are suggested by experience with physical space, then it is reasonable to expect that the conclusions will also correspond closely to experiences related to space. However, since any set of postulates must be selected on the basis of incomplete and approximate observation, they quite possibly apply only approximately to actual space. Thus, it is no surprise if any particular geometry should turn out to be inapplicable, or only approximately applicable, to problems in actual space. D Theorems Theorems are logically deduced from postulates. This process of deduction is called a proof. Each step of a proof must be justified by one of the postulates or by a theorem that has already been proved. One simple theorem, for example, asserts that a line that is parallel to one of a pair of parallel lines is parallel to both lines. Parallel lines are lines that are equally far apart from each other along their entire lengths. In proving a theorem in geometry, we deduce a conclusion from a set of assumptions. III EUCLIDEAN GEOMETRY Perhaps the most familiar and intuitive geometry is called Euclidean geometry. Euclidean geometry describes most aspects of the everyday world and was named after Euclid, the ancient Greek mathematician who developed it. While the postulates of Euclidean geometry do seem plausible when applied to physical space in our universe, there is evidence that Euclidean geometry is not the perfect system for describing space. Two-dimensional Euclidean geometry is often called plane geometry; three-dimensional Euclidean geometry is frequently referred to as solid geometry. Plane geometry deals with figures that lie wholly in one plane. A plane may be measured in terms of two dimensions: length and width. Solid geometry deals with figures that have three dimensions: length, width, and height. Conic sections, a commonly studied topic of geometry, are two-dimensional curves created by slicing a plane through a three-dimensional hollow cone. A Euclid's Postulates Euclid, who lived about 300 BC, realized that only a small number of postulates underlay the various geometric theorems known at the time. He determined that these theorems could be deduced from just five postulates. 1. A straight line may be drawn through any two given points. 2. A straight line may be drawn infinitely or be limited at any point. 3. A circle may be drawn using any given point as the center, and with any given radius (the distance from the center to any point on the circle). 4. All right angles are congruent. (A right angle is an angle that measures 90°. Two geometric figures are congruent if they can be moved or rotated so that they exactly overlap.) 5. Given a straight line and a point that does not lie on the line, one and only one straight line may be drawn that is parallel to the first line and passes through the point. These five postulates can be used in combination with various defined terms to prove the properties of two- and three-dimensional figures, such as areas and circumferences. These properties can in turn be used to prove more complex geometric theorems. B Two-Dimensional Euclidean Figures Figures commonly encountered in two-dimensional geometry include circles, polygons, triangles, and quadrilaterals. Triangles are actually three-sided polygons; quadrilaterals are polygons with four sides. B1 Circles A circle is a plane curve where all points are equidistant from a point in the plane called the center. Only one circle may be drawn that passes through three noncollinear points. The word circle is sometimes used to mean the entire portion of the plane enclosed by the curve rather than just the points that lie on the curve. Concentric circles are circles that have a common center. An angle is called a central angle of a circle if its vertex (point where the two arms of the angle meet) is at the center and its sides are radii of the circle. The circumference of a circle is divided into 360 equal degrees, and the number of degrees in a central angle is equal to the number of degrees in the intercepted arc on the circle. The area of a circle is equal to the product of the circumference and the diameter divided by 4, or A = Cd/4. The ratio of the circumference to the diameter is approximately 3.14159265. This constant number, called pi (p), has an infinite number of nonrepeating digits. The area of a circle may also be written A = pr2, where r is the radius. Similarly, the circumference is equal to the product of the diameter and the constant pi: C = pd B2 Polygons Any plane figure bounded by straight lines is a polygon. If all of a polygon's sides are of equal length and the angles are also equal, the figure is a regular polygon. The apothem of a regular polygon is the distance from the center of the polygon to a side. The area of a regular polygon is equal to the product of one half the apothem and the perimeter, or A = ½ap: B3 Triangles A triangle is a plane figure bounded by three straight lines. A scalene triangle has three sides of unequal lengths, an isosceles triangle has two equal sides, and an equilateral triangle has three equal sides: In the isosceles triangle the angles opposite the equal sides are equal, and in an equilateral triangle all three angles are equal. A right triangle is a triangle in which one angle is a right angle. The side opposite the right angle is called the hypotenuse; the two adjacent sides, the legs. The famous Pythagorean theorem states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the legs, or c2 = a2 + b 2. The angles inside a triangle are called the interior angles; those formed by extending a side of the triangle, exterior angles. The sum of the interior angles of any triangle equals 180°. Also, an exterior angle is equal to the sum of the remote interior angles (the two interior angles that do not share a side with the exterior angle): ÐD = ÐA+ÐB. A line drawn from a vertex of a triangle to the midpoint of the opposite side is called a median. The three medians of a triangle meet at a point two-thirds of the distance from the vertex to the midpoint of the opposite side. An altitude of a triangle is the length of the line connecting a vertex and the side opposite that vertex that is also perpendicular to the opposite side. (Two lines are perpendicular if they meet in a right angle.) Two triangles are congruent if they satisfy any of the three following sets of conditions: (1) two angles and a side of one triangle are equal to the corresponding side and two angles of the other triangle; (2) two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle; or (3) three sides of one triangle are equal to three sides of the other triangle. If the triangles can be perfectly overlapped without removing either from their common plane, they are directly congruent; if one must be flipped over, they are inversely congruent. If two triangles have equal angles, the triangles are said to be similar and the corresponding sides are in proportion to one another. The area of any triangle is equal to the product of one-half of a base and the altitude perpendicular to that base: A = ybh. (Any side can be considered the base of a triangle, but usually the side on the bottom is so designated.) If the triangle is equilateral, the area is given by , where a is the length of the side. If the sides of any triangle are a, b, and c, the area is given by a relation credited to the ancient Greek mathematician Archimedes: B4 where s is half the perimeter (s = ya + yb + yc). Quadrilaterals A quadrilateral is a plane figure bounded by four straight lines. There are several familiar types of quadrilaterals. Trapezoids are quadrilaterals that have two parallel sides of unequal lengths. Parallelograms are quadrilaterals that have opposite sides of equal length. A rhombus is a parallelogram (and therefore also a quadrilateral) whose sides are equal, a rectangle is a parallelogram whose angles are all right angles, and a square is a parallelogram whose angles are right angles and whose sides are of equal length. The diagonals of a parallelogram bisect each other; if the parallelogram is a rectangle, the diagonals are also equal. Irregular quadrilaterals have four unequal and nonparallel sides: The area of a trapezoid is half the sum of the bases times the altitude, or A = [(b1 + b2)/2]h. For a parallelogram, area equals base times height: A = bh. For irregular quadrilaterals, a good method for determining the area is to divide the figure into two triangles by means of a diagonal, then find the individual areas of the triangles and add them together. C Three-Dimensional Euclidean Figures Figures commonly encountered in three-dimensional geometry include spheres, polyhedrons, prisms, pyramids, cylinders, and cones. Cylinders are actually special cases of prisms; cones are special cases of pyramids. C1 Spheres A sphere is a surface where all points are equidistant from one point, called the center. If a plane cuts a sphere, the points where they intersect form a circle. The largest circle (called a great circle) is produced when the plane passes through the center of the sphere. The equator on Earth is a great circle. The surface area of a sphere is given by A = 4pr2, its volume by V = 4/3pr3. C2 Polyhedrons A polyhedron is a figure bounded by plane surfaces. If the faces of the polyhedron are all congruent regular polygons, the polyhedron is said to be regular. It has been proven that the five regular polyhedrons--the tetrahedron (four sides), cube (six sides), octahedron (eight sides), dodecahedron (12 sides), and icosahedron (20 sides)--are the only ones possible. These five polyhedrons were known to the ancient Greek geometers. All polyhedrons (regular or not) have the remarkable property that the number of faces (the flat sides) plus the number of vertices (the angles where edges intersect) equals the number of edges plus 2. Up to relatively recent times, polyhedrons were believed to have mystic associations with natural phenomena. C3 Prisms A prism is a polyhedron that has parallel and congruent polygons, called bases, for two faces and parallelograms for all other faces. A parallelepiped is a variety of prism whose bases are parallelograms. A right prism has rectangles for sides (but not necessarily for bases). The volume of any prism is equal to the area of one of its bases times its height: V = bh. C4 Pyramids A pyramid is a polyhedron that has a polygon as its base and sides that consist of triangles having a common vertex, called the apex. A pyramid is a regular right pyramid if its base is a regular polygon and if a line joining the center of its base to its apex is perpendicular to its base. The volume of any pyramid is equal to one-third the area of its base times its height: V = ? bh. C5 Cylinders and Cones A cylinder is a prism with circular bases. The formula for the volume of a cylinder is therefore the same as for a prism: A = bh. If the line connecting the centers of the two bases is perpendicular to those bases, the cylinder is a right cylinder; otherwise, it is oblique. A cone is a pyramid with a circular base. A cone is a right cone if a line joining the center of its base to its apex is perpendicular to its base. The formula for the volume of a cone is the same as for a pyramid: V = ? bh. D Conic Sections Conic sections are curves formed by the intersection of a plane with the surface of a cone. (When discussing conic sections, cone means two right circular cones placed apex to apex.) The surface of the cone on either side of the apex is called a nappe of the cone. If A is the angle between the axis of the cone and its surface and the cone is cut by a plane that makes an angle with the axis that is greater than A, the intersection is a closed curve called an ellipse. If the plane and the axis are perpendicular, the intersection is a circle, which is considered a special case of the ellipse. If the plane intersects the axis at an angle equal to A, so that the plane is parallel to the surface of the cone, the intersection is an open curve of infinite extent called a parabola. If the cone is intersected by a plane that is either parallel to the axis or makes an angle with it that is smaller than A, and if the plane does not contain the apex of the cone, the intersection is called a hyperbola. In this case the cone is necessarily intersected in both nappes, and it follows that the hyperbola has two branches, each of which is infinite in extent. Conic sections are two-dimensional or plane curves, and therefore a desirable definition of conic sections avoids the notion of a cone, which is three-dimensional. A conic section may be two-dimensionally defined as the set of points of which the distances from some fixed point are in a constant ratio to the distances of the points from a fixed line that does not pass through the fixed point. The fixed point is called the focus, and the fixed line is called the directrix. The constant ratio is called the eccentricity of the conic section and is usually denoted by the letter e. If P is a point and Q is the foot of a line from P perpendicular to the directrix, the point P is on the conic section if and only if [FP] = e[QP], in which [FP] and [QP] are the distances between the respective points. When e = 1, the conic section is a parabola; when e > 1, it is a hyperbola; and when e < 1, it is an ellipse. The conic sections have numerous mathematical properties that give them important applications in mathematical physics. For example, light reflected by mirrors molded to the curve of a conic section has particular characteristics: Rays emanating in any direction from the center of a circle are reflected back to the center; rays emanating in any direction from one of the two foci (geometrical centers) of an ellipse are reflected to the other focus. Parabolic mirrors are often used in spotlights because the rays emanating from the focus of a parabola are reflected out in parallel lines, minimizing spread: Rays emanating from one focus of a hyperbola are reflected in such a direction that they appear to emanate from the other focus. IV ANALYTIC GEOMETRY Analytic geometry arose from the recognition that certain numerical and algebraic equations (see Algebra) correspond to points, lines, and geometric figures. Graphing the equations using a set of axes and coordinates draws the points, lines, or figures. For example, any point in a plane can be located with respect to a pair of perpendicular axes by specifying the distance of the point from each of these axes. Positive x numbers are located at the right side of the y-axis and negative numbers to the left; positive y numbers are located above the x-axis and negative y numbers below. Point E is 1 unit from the vertical y-axis and 4 units from the horizontal x-axis. The coordinates of point e are therefore 1 and 4, and the point is located by the equations x = 1, y = 4. Similarly, a straight line always corresponds to an equation of the form ax + by + c = 0. For example, the collection of points that lie on the straight line passing through points E and F satisfies the equation x + y = 5 (in this simple equation, a and b are both equal to one and c is equal to zero). Every combination of x and y values that satisfies the equation locates a point on the line. Other more complex equations correspond to circles, ellipses, conic sections, and other figures. The problems dealt with in analytic geometry are of two classic kinds. The first kind of problem: Given a geometric description of a set of points, determine the algebraic equation that is satisfied by these points. The second kind of problem: Given an algebraic statement, describe the locations of the points that satisfy the statement in geometric terms. For example, a circle of radius 3 with its center at the point of intersection of the x-axis and the y-axis (the origin) is the collection of points that satisfy the equation x2 + y2 = 9. From such equations as these it is possible to solve geometrical construction problems such as bisecting (dividing exactly in half) a given line segment or angle, constructing a perpendicular to a given line at a given point, or drawing a circle that will pass through three given points that are not on the same straight line. Points, lines, and figures in three-dimensional space can be similarly located with respect to three axes, of which the third, usually called the z-axis, is perpendicular to the other two at their point of intersection, which is also called the origin. Analytic geometry was of great value in the development of mathematics because it unified the concepts of analysis (number relationships) and geometry (space relationships). The techniques of analytic geometry, which made possible the representation of numbers and of algebraic expressions in geometric terms, have cast new light on calculus, the theory of functions, and other problems in higher mathematics. The study of non-Euclidean geometry and the geometries of spaces that have more than three dimensions would not have been possible without the analytic approach. V NON-EUCLIDEAN GEOMETRY Euclid's fifth postulate states that through a point outside a given line it is possible to draw only one line parallel to that line--that is, one that will never meet the given line no matter how far the lines are extended in either direction. In the first part of the 19th century German mathematician Carl Friedrich Gauss, Russian mathematician Nikolay Ivanovich Lobachevsky, and Hungarian mathematician János Bolyai independently demonstrated the possibility of constructing a consistent system of geometry in which Euclid's postulate of the unique parallel was replaced by a postulate stating that through any point not on a given straight line an infinite number of parallels to the given line could be drawn. Later, about 1860, German mathematician Georg Friedrich Bernhard Riemann showed that another geometry in which no parallel lines occurred was equally possible. The details of these two types of non-Euclidean geometry are complex, but both systems can be demonstrated by means of simple models. The Bolyai-Lobachevsky geometry, often called hyperbolic non-Euclidean geometry, describes the geometry of a plane consisting only of the points on the inside of a circle in which all possible straight lines are chords of the circle. (A chord is any straight-line segment that is bounded at each end by a circle.) Since the definition of parallel requires only that two parallel lines never meet no matter how far they are extended, and lines in hyperbolic geometry cannot be extended beyond the edge of a circle, an infinite number of segments parallel to line L can be drawn through the point P that will never meet line L because in this geometry the "universe" ends at the edge of the circle: Similarly Riemannian, or elliptic non-Euclidean geometry, is the geometry of the surface of a sphere in which all straight lines are great circles. It is impossible to draw any pair of parallel lines on this surface: For comparatively small distances, such as those experienced in everyday life, Euclidean geometry and the non-Euclidean geometries are essentially equivalent. However, in dealing with astronomical distances and problems of modern physics such as relativity, non-Euclidean geometries give a more precise description of the observed phenomena than does Euclidean geometry. For example, the theory of relativity developed primarily by Albert Einstein is based on a Riemannian geometry of curved space. VI PROJECTIVE GEOMETRY The effect of various projections upon the properties of geometric figures spawned another branch of geometry during the 17th century. Conic sections, for instance, may be transformed into one another by suitable projections. For example, a flashlight projects a circular patch of light on a wall when held perpendicular to the wall, but projects an ellipse when held at an angle: Some properties of figures do not change when the figure is projected into a new form. A simple example of a theorem describing such a projective property is illustrated in the following figure. If points A, B, C and a, b, c are placed anywhere on a conic section, such as a circle, and these points are connected A to b and c, B to c and a, and C to b and a, the three points at which the corresponding lines intersect will lie in a straight line: The points will still intersect in a straight line even if the circle is transformed via projection into an ellipse. Similarly, if any six tangents (lines that touch the curve but do not cross it) are drawn to a conic section, and lines are drawn connecting the opposite intersections of these tangents, the connecting lines will meet at a single point: This theorem is also projective, since it is equally true for all the conic sections. VII GEOMETRY IN FOUR OR MORE DIMENSIONS The development of projective and analytical geometry led mathematicians to the possibility of studying the geometry of spaces with more than three dimensions. Understanding such spaces does not require extraordinary feats of visualization or other mental gymnastics. Any point in the physical universe can be located by reference to three given axes (usually designated x, y, and z); the physical universe is said, therefore, to be three-dimensional. The same three-dimensional space, however, becomes four-dimensional if each point in it is replaced by a sphere, because then four references must be given to determine or locate each individual sphere: the three coordinates of the sphere's center point and the length of the sphere's radius. Similarly a "three-dimensional" space can be simulated in two dimensions by replacing every possible point with a circle. In this case the three dimensions consist of the coordinates of the center of a circle and the length of its radius: The use of geometrical concepts involving more than three dimensions has had a number of important applications in the physical sciences, particularly in the development of the theory of relativity. Analytical methods may also be used to investigate regular geometrical figures in four or more dimensions and to compare them with similar figures in three or fewer dimensions. Such geometry is called structural geometry. A simple example of this approach to geometry is the definition of the geometrical figure with the smallest number of vertices, edges, and sides that can be drawn in spaces of zero, one, two, three, four, or more dimensions. In the first four of these spaces--zero, one, two, and three dimensions--the figures are the familiar point, line, triangle, and tetrahedron. In a space of four dimensions the simplest figure can be shown to have five vertices, ten edges connecting its vertices, and ten faces. Here are the simplest figures, represented in two dimensions: 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. Microsoft ® Encarta ® 2009. © 1993-2008 Microsoft Corporation. All rights reserved.
