Euclidean Geometry

Geometry is the branch of mathematics that investigates the relations between points, lines, planes, and solids. The system is normally formalized by axioms and derived results.

Overview

Investigating the relationships between points, lines, and planes constitutes the subject of geometry. These relationships are used in engineering and many areas of science. Geometry includes the development of proofs that things can be done and provides methods of construction using only the straightedge and compass. Geometry can be approached from two distinct vantage points. One of these is intuitive, building on the observation and experiences of the investigator.

The second approach, the axiomatic approach, is more formal. The axiomatic approach, first formalized by Euclid of Alexandria (c. 300 B.C.), is based on a fundamental set of elements and their relationships. Demonstrations are developed to justify relationships that are dependent on the axioms and no other information.

The intuitive approach may require the definition of terms such as "congruent," leaving a formalization of the properties to theorems and demonstrations. In a well-formulated axiom system, the word "congruent" would probably be undefined, with its meanings made clear through axioms that characterize its properties. There are arguments for approaching a subject by each of these methods, but in practice both are frequently used. In either approach, the concept of discovering and developing other relationships on the basis of associations that have already been observed is critical. In developing any body of knowledge, this inductive step is important.

These inductive leaps, followed by their justification, provide growth to the field.

Some of the earliest-known observational facts included such statements as "Two lines in a plane intersect in a point" and "Two points determine a line." It was not recognized in the early history of geometry that these and many other observations have a dual in the sense that if one interchanges the words "point" and "line" and other relevant words such as "collinear" and "concurrent," a new and equally valid statement is derived. A characteristic of Euclidean geometry is that most theorems have a dual. In some cases, centuries elapsed before the dual was discovered. Recognizing this dualism is a relatively recent contribution to geometry. Observing properties such as dualism and attempting to organize the thought processes helped formalize the structure of an axiom system that is now known to have four components. These components are undefined terms whose meaning can be made clear only through axioms or postulates. For the purposes of this discussion, "axiom" and "postulate" are synonyms. They describe the relations between the undefined terms. A third element is derived or defined terms. These are introduced primarily to make the subsequent discussion easier and are given in terms of the previously given undefined terms. Finally, there are theorems and propositions, which can be demonstrated to follow logically from the axioms alone. Euclid provided the first axiom system for geometry. It has changed somewhat over the years. Today, there are other formulations, including that of David Hilbert (1862-1943), which seems to be free of inconsistencies. There is always the possibility that someone will replace one axiom system with another, equivalent one.

Among the geometrical relationships that have been recognized is the knowledge of the Pythagorean triple: 3, 4, 5. When these values are used as lengths for the sides of a triangle, a right triangle is formed. This triangle was used by early geometers, or "rope stretchers," to survey the fields in Egypt after the annual floods. These triples are called Pythagorean because they satisfy a theorem credited to Pythagoras (c. 540 B.C.). This theorem states that the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse. For this statement to be complete, one needs to know that a right triangle has a right angle, that the legs are the sides forming the right angle, and that the hypotenuse is the longest side. Finding integral solutions to the Pythagorean theorem, of which there are many, is one of the challenges of geometry.

In organizing geometrical data, it is logical to start with the simplest concepts. Points and lines appear to be the best starting point. Two lines that intersect not only share a common point but also form angles. "Angle" is a derived term.

In building geometry intuitively and seeking the building blocks to develop an axiom system, it can be observed that a line can be drawn through two points. This can be restated as "Two points determine a line." This statement implies an element of uniqueness that leads to the concept of a line being straight, a property that is understood without being stated. If the element connecting the two points is not straight, it is called a curve. There are many possibilities of curves passing through two given points, contrary to the notion of uniqueness. Additional terms are needed for convenience. The "line segment," or the set of points on the line between the two points, is one such. Another is the "ray," a part of the line that starts at a point and extends in one direction from it.

Angles are best described in terms of the figure determined by two rays starting from a common point. The point is called the "vertex" of the angle and the rays, the "sides." It follows that two intersecting lines determine four angles that share the common vertex or point of intersection. When all four of these angles are congruent, the lines are said to be perpendicular and the angles are right angles. An angle divides the plane into two parts, as does a line. This poses the problem of identifying the inside and outside of the angle. Usually the smaller part is considered the inside, but it is possible that the angle selected may be the larger portion. For some purposes, angles can involve several rotations. Intuition is important in developing an understanding of relations such as inside and outside, but it can pose problems when one tries to make precise statements.

Attempts to describe a plane, another fundamental element, are more difficult because of the care needed. One description can be based on an analogy between points and lines. Just as two intersecting lines determine a point, and have angles associated with them, two lines can also determine a plane. In order for the description to be complete, certain conditions must be established, as is illustrated by the following special cases.

First, two lines may intersect in a point (the dual of two points determining a line).

These two lines determine a point, and, on a little reflection, it can be seen that other lines can also pass through this common point, forming a "pencil" of lines. Some of these additional lines lie in the plane determined by the original lines, but some of them do not. This latter case can be illustrated by considering a piece of paper as a model of the plane and observing a line represented by a wire passing through the plane. Lines sharing only one point in a plane are inadequate for determining a plane.

Second, the two lines selected may be parallel. Two such lines lie in a plane, but consider what happens if one takes a family of parallel lines drawn on a piece of paper and then rolls the paper into a cylinder, keeping the lines parallel. The resulting surface obviously is not a plane even though the lines are still parallel.

Third, the two lines may be "skew"--that is, drawn in space so that they never intersect and are not parallel. No plane contains both lines.

To characterize the plane, something more than two intersecting or parallel lines is needed. Starting with two intersecting lines, as in the first case, and taking points other than the point of intersection on each of the two lines, one can construct another line. A plane can be thought of as being described by a line that must either intersect all three lines or intersect two of them and be parallel to the third. This leads to the well-known result that three points determine a plane and provides some insight into how they do so.

The three points used to describe a plane also form the vertices of a triangle, a fundamental plane figure of geometry. The triangle could be known as a "trilateral," since it is formed by the three side segments joining the points.

The properties of triangles provide one of the application areas of geometry. The rigidity of the triangle and the ways in which it can be constructed or shown to be congruent to another triangle prove to be very useful. Generally, three pieces of information are needed. One datum must be a length associated with the triangle. Among the well-known results are that a triangle is uniquely identified if the following are given: three sides, two sides and the included angle, two angles and a side, and two sides and the altitude drawn to one of them. If one is given two sides and the angle opposite one of them, the solution may not be unique; this is known as the ambiguous case.

The Pythagorean theorem is critical to the foundation of trigonometry, which is used in calculating the remaining parts of a triangle and providing identities used in problems of calculus and other applications.

Other geometrical areas of interest and use are found in the constructions. The challenge is to make the drawings with Euclidean tools: the straightedge for drawing lines and the compass for drawing circles. The more fundamental constructions include drawing a line through two points, duplicating a line segment, and constructing a circle with a given center and radius. From these, other constructions can be developed that are apparently more useful. These include duplicating an angle, constructing perpendicular and parallel lines, and bisecting angles and line segments.

In more advanced constructions, one finds problems of constructing triangles when the minimal required information is given, and of locating special points, lines, and circles associated with the triangle. Some of the special points are the centroid, the orthocenter, the incenter of the inscribed circle, and the circumcenter of the circumscribed circle. Useful lines of the triangle are the medians, the angle bisectors, the altitudes, and the perpendicular bisectors of the sides. There are other points and lines that can be described, and new ones are occasionally discovered.

Euclid's fifth postulate involved the property of parallelism. Some believed that this postulate was dependent on the other four. Many equivalent statements have been used in an attempt to demonstrate the expected dependence. One of the statements asserted that through a point not on a line, one and only one line could be drawn parallel to the given line. There are two ways of denying this postulate. One is to assume that there are no such things as parallel lines and the other is to assume that more than one parallel line can be drawn through the point. The initial approach was to try to show that assuming one of these contrary conditions and the other four postulates would lead to a contradiction. No contradictions were found, and ultimately models were created that showed that these assumptions produced self-consistent geometries now known as non-Euclidean geometries.

Applications

Establishing the congruency of triangles has its counterpart in determining how a triangle can be constructed given selected parts. These and other construction problems represent one aspect of applications of geometry.

The center of gravity, or centroid, of a triangle plays an important role in many physical problems. Altitudes are also critical to some applications. Both are standard construction problems. Exploring properties of lines such as medians and altitudes associated with triangles can lead to generalizations that are potentially useful even though they are not necessarily obvious. Ceva's theorem (named for Giovanni Ceva, 1647?-1734) is one of these. It gives the necessary and sufficient condition for the lines drawn from the vertices of a triangle and intersecting the opposite sides to be concurrent. Medians used to find the centroid, as well as altitudes and angle bisectors, satisfy the conditions of this theorem, as do other, little-known lines.

The dual of Ceva's theorem, obtained by interchanging the roles of points and lines in Ceva's theorem and making one other simple change, gives Menelaus' theorem (named for Menelaus of Alexandria, c. A.D. 100), which addresses the points of intersection of a line that cuts the three sides of a triangle. This theorem provides the necessary and sufficient condition for three points on the sides or sides extended of a triangle to be collinear. It would seem that Ceva's theorem should have been discovered before Menelaus' theorem, since the lines involved have long been recognized as important lines of a triangle, but this was not the case.

The problem of finding the center of a circle can be readily solved by geometry. Instead of doing it geometrically, the methods of geometry are used to design a tool to do the job. The center is found by using angle bisectors. If the sides of the angle are tangent to the circle, the angle bisector will pass through the center of the circle. Relocating the angle bisector and finding the point of intersection of the two angle bisectors locates the center.

A slightly more complicated problem is that of finding the center of rotation to carry one object into a congruent image of it in the plane. This construction follows from an understanding of the geometry of rotation. To find the center, one needs two pairs of corresponding points. The point of intersection of the perpendicular bisectors to the line segments connecting the corresponding points will be the center of rotation. Should the congruent figures have corresponding parallel parts, the transformation in getting from one to the other is a translation. The translation, however, can be considered a rotation around a point at "infinity."

A protractor is a device for measuring the number of degrees in an angle. The size of the degree is arbitrary, but today it is accepted that a circle or round angle will be divided into 360 congruent parts called degrees. The construction of a protractor corresponds to inscribing a regular polygon with 360 congruent sides in a circle. The vertices of the polygon will mark off degrees on the circle. Using Euclidean tools, this subdivision cannot be done. It has long been known how to construct regular polygons of three, four, and five sides. Once one of these has been constructed, other polygons with twice the number of sides can be constructed. Carl Friedrich Gauss (1777-1855) proved that regular polygons of 17, 257, and 65,537 sides--and no other regular polygons that have a prime number of sides--can be constructed with Euclidean tools. Others have proved that arbitrary angles cannot be trisected--that is, divided into three congruent parts--with Euclidean tools. A consequence is that a regular polygon with 120 sides can be constructed, but the usual protractor requires a polygon with 360 sides. Fortunately, this curious selection of 360 degrees for a circle admits to a construction that can be resolved by methods that do not use Euclidean tools.

Another problem related to the circle is finding the ratio of the circumference to the diameter. This number, called π (denoted by the Greek letter pi), has an extensive history.

Through a study of similarities, it can be shown that the value does not depend on the size of the circle. Many approximations are known. The early Baby lonians used the approximation (4/3) to the power of 4. Before the advent of the pocket calculator, many elementary mathematics books used 22/7 as a reasonable approximation. Today, the number π is stored in many calculators, and one may find values with many more significant figures, such as 3.141592654. Procedures for determining the value of π are many, but one of the classical methods involved finding the perimeter of regular polygons. Values increasing to π were obtained by starting with a six-sided polygon inscribed in the circle and repeatedly doubling the number of sides. Similarly, a value always too large but approaching π was found by circumscribing a square about the circle and again repeatedly doubling the number of sides. The values obtained through these two approximations came closer and closer together. If the original had a radius of one unit, the diameter would be of length two and the value of π would be one-half the value of the perimeter or circumference.

Geometrical linkages provide another class of applications. The problem of converting rectilinear to circular motion is classical. The solution to this problem was critical in the design of steam engines. Another linkage problem is found in the pantograph, a drawing instrument that is used to trace images that are either larger or smaller than the original. Using computers as a drafting tool has eliminated the need for this application.

Context

The development of geometry as a formal system began with Euclid. Subsequent investigators have contributed to the field, and some observed errors and inconsistencies in Euclid's development. David Hilbert addressed these problems and formulated a system that is generally accepted to be error-free. Though different in organization, it is consistent with Euclid's formulation.

In trying to show that parallelism is a separate idea, elliptic and hyperbolic geometries have been discovered and developed. These geometries are called non-Euclidean. The elliptic geometries, in which there are no parallels, are called Riemannian geometries, named for Georg Friedrich Bernhard Riemann (1826-1866). Hyperbolic geometries can have more than one parallel to a given line through a given point. These geometries were developed by Janos Bolyai (1802-1860) of Hungary and Nikolay Ivanovich Lobachevsky (1793-1856) of Russia. Both geometries have played a role in subjects such as relativity and astronomy, because of the concept of space curvature. In the "small," Euclidean geometry provides a good approximation of these other geometries. By modifying axioms, other geometries have been found, including some with interesting applications.

The work of Rene Descartes (1596-1650) changed the orientation used to study geometry from the synthetic approach to an analytical one. This was accomplished by means of the introduction of coordinate systems, which enhanced many applications and led to other subjects, such as differential geometry. By using coordinates to describe positions, a number of important relationships can be found, all contributing to the quantitative description of observed phenomena, which is so important in the physical sciences.

Euclidean geometry provides the structure from which trigonometry is developed.

From the properties of the angles of right triangles, given in part by the Pythagorean theorem that describes the relationship of the lengths of the sides, and the concept of congruent triangles, the trigonometric functions can be defined. These definitions provide the basis for calculating tables and establishing identities.

Problems involving collinearity of points and concurrency of lines can be addressed through Euclidean geometry and extended to three dimensions. Less obvious are the relationships needed to describe four- and higher-dimensional problems. Although a four-dimensional object cannot be seen in its entirety, one can, through geometry, discover many of its characteristics.

Computer graphics is an application that is gaining in popularity. In its early development, the process was relatively crude. Software has improved and will continue to do so. Some of the improvements take the form of constructions in the traditional mode that minimize the need for analytical solutions. Computer graphics has all but eliminated drafting and plays a major role in engineering design.

Principal terms

AXIOM: a logically self-evident relationship that is assumed to hold between the fundamental elements of geometry

CONGRUENCY: the relationship that exists between two objects if they are alike in all aspects of size and shape

COLLINEARITY: the relationship that exists between points if they lie on the same straight line

CONCURRENCY: the relationship that exists between lines that all pass through the same point

EUCLIDEAN TOOLS: the straightedge and the compass, the classical drawing tools used in making geometrical constructions

POSTULATE: today, synonymous with "axiom"; originally, postulates were empirically self-evident relationships

THEOREM: a property or relationship that can be logically derived from an axiom and/or a previously established theorem

Bibliography

Euclid. THE THIRTEEN BOOKS OF THE ELEMENTS. Translated by Thomas L. Heath. 2d rev. ed. 3 vols. New York: Dover, 1956. Classic reference work on the geometry of Euclid.

Eves, Howard. AN INTRODUCTION TO THE HISTORY OF MATHEMATICS. 5th ed. Philadelphia: W. B. Saunders, 1983. A good overview of history with many illustrations made through exercises.

Eves, Howard. A SURVEY OF GEOMETRY. Rev. ed. Boston: Allyn & Bacon, 1972. A broad perspective of modern geometry, well annotated with historical references.

Krouse, Engene F. TAXICAB GEOMETRY. New York: Dover, 1986. Illustrations of a geometry that does not follow the traditional view of distance.

Seydel, Ken. GEOMETRY: AN EXERCISE IN REASONING. Philadelphia: W. B. Saunders, 1980. A well-illustrated text on elementary geometry.

Coordinate Systems Used in Astronomy