Elementary Geometry


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Elementary Geometry

 

the part of geometry that falls under elementary mathematics.

The boundaries of elementary geometry, like those of elementary mathematics, are somewhat indefinite. Elementary geometry is sometimes said to be the geometry that is studied in secondary school. This definition, however, does not convey effectively the content and nature of elementary geometry. Moreover, many aspects of elementary geometry are not covered by school curricula; examples are axiomatics and spherical geometry.

Since the other branches of geometry developed from it, elementary geometry may be said to be historically—and, accordingly, logically—the first chapter of geometry. Its rudiments were known in ancient Greece, and an exposition of them was given in Euclid’s Elements (third century B.C.). Such a historical definition is not without foundation, but it too fails to give an adequate idea of the general content and nature of elementary geometry, especially since the development of elementary geometry continues today. A fuller, more detailed definition must therefore be provided.

The ancient Greeks investigated not only polygons, the circle, polyhedrons, and other figures studied in school but also conic sections (the ellipse, hyperbola, and parabola) and a number of more complicated curves and figures, such as the quadratrix. In each instance, however, the curve or figure was defined by a specific geometric construction, and only such curves or figures were considered geometric, that is, capable of being studied by geometry; other possible curves and figures were termed mechanical. This point of view was rejected in the 17th century by R. Descartes when he created analytic geometry, and it lost its remaining supporters when analysis was developed and the purview of mathematics was extended to all functions and curves, or at least to all analytic functions and curves.

The logical progression from elementary mathematics—in particular, the progression from elementary geometry—to higher mathematics consists in this historically discernible passage from specially defined curves, such as the circle and ellipse, and specially defined functions, such as a given power of x or the sine, to all curves and functions in a broad class. Elementary geometry does not deal with all possible analytic curves and surfaces, which constitute the subject matter of differential geometry, nor does it deal with all possible members of any other class of geometric objects—for example, the class of convex bodies, which form the subject matter of the geometry of convex bodies. On the other hand, any given curve or convex body that is defined by some construction, such as an ellipse or a cylinder, may be considered by elementary geometry. Thus, with respect to its subject matter elementary geometry is characterized by the investigation not of all possible figures but of sufficiently well-defined individual figures.

More precisely, elementary geometry proceeds from primitive figures—the point, line segment, line, angle, and plane—and from the fundamental concept of the equality of segments and angles (or, in general, the concept of the coincidence of figures upon superposition, whereby equality is determined). Moreover, in the rigorous axiomatic construction of elementary geometry, the concepts of a point lying on a straight line or in a plane and of a point lying between two other points are explicitly singled out.

Elementary geometry deals with the following: (1) Figures defined by a finite number of primitive figures; for example, a polygon can be defined by a finite number of line segments, as can a polyhedron, since it is definable by a finite number of polygons. (2) Figures defined by some property that is formulable in terms of the fundamental concepts; for example, an ellipse with foci A and B is the locus of the points X such that the sum of the segments AX and BX is equal to a given segment. (3) Figures defined by a construction; for example, a cone can be constructed by extending lines from a given point O to all points of some given circle that does not lie in the same plane as O, and a conic section can be defined as the intersection of a cone and a plane. Figures defined in the above ways, no matter how complex, may become subjects of investigation within the framework of elementary geometry.

With regard to the properties of such figures, elementary geometry is confined to the study of properties that can be defined on the basis of the fundamental concepts mentioned. These properties include length, area, volume, the relative positions of figures, and the equality of various elements of a figure. Accordingly, the definitions of the circumference of a circle, the area of an ellipse, the volume of a sphere, and so forth belong to elementary geometry. The general concepts of length, area, and volume, however, fall outside elementary geometry.

Consider for example, the theorem that of all closed curves of a given length, the circle bounds the largest area. Although it deals with a property of the circle, the theorem does not belong to elementary geometry, since it makes use of the concept of the length of any closed curve and of the area bounded thereby. Elementary geometry may study the properties of a tangent to a circle, to an ellipse, to a hyperbola, or to a parabola, but the general concept of tangent is beyond its scope.

The logical difference in generality of concept and in degree of abstraction is in accord with the historical development of geometry: the general concepts of length, area, and volume and the general concept of tangent to a curve were not fully elaborated until the development of analysis, and the theorem about the maximum-area property of a circle was not rigorously proved until the mid-19th century. The geometric constructions and transformations studied in elementary geometry are determined by specific geometric prescriptions on the basis of the primitive concepts of geometry; an example is the transformation known as inversion of a point with respect to a circle.

The methods of elementary geometry are limited in accordance with its subject matter. They, of course, preclude the use of the general concept of any figure, variable, or function, and they avoid references to the general theorems of, for example, limit theory. The fundamental method of elementary geometry is the derivation of theorems by means of clear and concrete reasoning based on initial premises (axioms) or on known theorems of elementary geometry; the derivation may make use of some auxiliary construction that does not employ such general concepts as curve or body. In applying the method we may thus make such statements as “Let us extend the segment AB” or “let us bisect the angle A.” The computational techniques borrowed by elementary geometry from algebra and trigonometry permit of reduction to such constructions.

The concept of limit is not excluded from elementary geometry, since it figures in such theorems as those on the circumference of a circle and the surface of a sphere, which indisputably are included in elementary geometry. In each such case, however, we are dealing with a specific sequence defined by a construction used in elementary geometry, and the approaching of the limit is established directly, without reference to the general theory of limits. An example is the determination of the circumference of a circle through the consideration of a sequence of inscribed and circumscribed regular polygons. In principle this procedure is possible for any given curve, but such a technique cannot be used for an arbitrary curve, since a “curve in general” is not specifically defined. Thus, both elementary geometry (or elementary mathematics in general) and higher geometry use the concept of limit; the difference between the two lies in the degree of generality of the concept.

In accordance with the definition of the method of elementary geometry, a given theory may belong to elementary geometry in formulation but not in proof. An example is Minkowski’s .theorem on the existence of a convex polyhedron with given directions and areas of its faces (seePOLYHEDRON). This theorem is elementary in formulation, but its known proofs are not elementary, since they use general theorems of analysis or even of topology.

In short, elementary geometry may be said to include the problems of geometry that in formulation and solution involve only constructively defined sets (loci) and do not make use of the general concept of an infinite set. Euclidean geometry may be said to be based on Hilbert’s system of axioms or on another system of axioms that is similar in character. The introduction of such general concepts as curve, convex body, and length, however, entails the use of methods of concept formation that are not provided for in the axioms and that rest on the general concept of set, of sequence and limit, or of mapping or functions. The geometry derived from Hilbert’s axioms without such additions constitutes the elementary part of Euclidean geometry. This delimitation can be refined in the terms of mathematical logic. At the same time, such an understanding of elementary geometry permits us to speak of, for example, the elementary geometry of n-dimensional Euclidean space or elementary Lobachevskian geometry. Here, we have in mind those branches, theorems, and derivations of the geometric theories that are characterized by the same features as elementary geometry in the sense defined above.

REFERENCES

Nachala Evklida, books 1–15. Moscow-Leningrad, 1948–50. (Translated from Greek.)
Hadamard, J. Elemenlarnaia geomelriia, part 1, 4th ed. Moscow, 1957. Part 2, 3rd ed.: Moscow, 1958. (Tranlated from French.)
Pogorelov, A. V. Elemenlarnaia geometriia, 2nd ed. Moscow, 1974.
Istoriia matematiki s drevneishikh vremen do nachala XIX stoletiia, vols. 1–3. Moscow, 1970–72.