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Fundamental Concepts of Geometry
By Bruce E. Meserve
Dover Publications, Inc.Copyright © 1983 Bruce E. Meserve
All rights reserved.
FOUNDATIONS OF GEOMETRY
The word "geometry" is derived from the Greek words for "earth measure." Since the earth was assumed to be flat, early geometers considered measurements of line segments, angles, and other figures on a plane. Gradually, the meaning of "geometry" was extended to include the study of lines and planes in the ordinary space of solids, and the study of spaces based upon systems of coordinates, as in analytic plane geometry, where points are represented by sets of numbers (coordinates) and lines by sets of points whose coordinates satisfy linear equations. During the last century, geometry has been still further extended to include the study of abstract spaces in which points, lines, and planes may be represented in many ways. We shall be primarily concerned with the fundamental concepts of the ordinary high-school geometry—euclidean plane geometry. Our discussion of these concepts is divided into three parts: the study of the foundations of mathematics (Chapter 1), the development of euclidean plane geometry from the assumption of a few fundamental properties of points and lines (Chapters 2 through 6), and a comparison of euclidean plane geometry with some other plane geometries (Chapters 7 through 9). The treatment of the second part—the development of euclidean plane geometry—forms the core of this text and emphasizes the significance of the assumptions underlying euclidean geometry. Together, the three parts of our study enable us to develop an understanding of and appreciation for many fundamental concepts of geometry.
1–1 Logical systems.
We shall consider geometries as logical systems. That is, we shall start with certain elements (points, lines, ...) and relations (two points determine a line, ...) and try to deduce the properties of the geometry. In other words, we shall assume certain properties and try to deduce other properties that are implied by these assumptions.
In a geometry or any other logical system, some undefined elements or terms are necessary in order to avoid a "circle" of definitions. For example, one makes little progress by defining a point to be the intersection of two distinct lines and defining a line to be the join of two distinct points. A nonmathematical example would be obtained by defining a child to be a young adult and an adult to be a full-grown child.
Thus, in any geometry, some of the elements must be accepted without formal definition; all other elements may be defined in terms of these undefined elements. Similarly, in any geometry some of the relations among the elements must be accepted without formal proof. These assumed relations are often called assumptions, axioms, or postulates. Other relations, which may be proved or deduced, are called theorems.
Definitions enable us to associate names with elements and relations that may be expressed in terms of the undefined elements, postulates, previously defined elements, and previously proved relations. The postulates and definitions are combined according to the rules of logic (Sections 1–2 and 1–3) to obtain statements of properties of the geometry. Necessary and desirable properties of postulates are considered in Sections 1–4 through 1–6. Definitions should be concisely stated, should give the distinguishing characteristics of the element or relation being defined with reference to the element or relation most similar to it, should be reversible, and should not contain any new elements or relations.
A reversible statement may be expressed in the form of "if and only if." For example, the statement: A triangle is equilateral if and only if its three sides are equal is reversible. This statement is commonly expressed as a definition: An equilateral triangle is a triangle having three equal sides. That is, we interpret the definition to mean that all equilateral triangles have three equal sides and that all triangles having three equal sides are equilateral. In other words, we usually assume that definitions are reversible.
A definition of a new term is not acceptable if it involves terms that have not been previously defined. A proof of a theorem is not acceptable if it involves relationships that have not been previously proved, postulated, or stated as assumptions in the hypothesis of the theorem.
Such faulty definitions and proofs often involve "circular reasoning." If a statement such as: A zig is a zag and its converse statement (Section 1–2): A zag is a zig are both taken as definitions, we have a circle of definitions and have not improved our understanding of either zigs or zags. In general, whenever the definitions of two elements, say A and B, are related such that the definition of A depends upon the element B and the definition of B depends upon the element A, we have an example of reasoning in a circle in setting up our definitions. Whenever the conclusion of a theorem is used as a basis for a step in the proof of the theorem, we have an example of reasoning in a circle in the proof of a theorem. To avoid such reasoning in a circle we must have undefined elements and unproved relations (postulates) among these elements. Throughout this text we shall be concerned with definitions and theorems based upon sets of undefined elements and unproved postulates.
1. Discuss the reasoning in the following story:
Long ago some Christian monks heard that in a certain medieval village there lived a holy man who talked with angels. In order to verify this report the monks traveled to the village and talked with some of the local people who knew the holy man. These people repeated the story of the holy man and, when asked how they knew that he talked with angels, said that he had told them of his experiences. The monks then asked the people how they knew that the holy man was telling the truth. The local people were astounded at the question and replied "What! A man lie who talks with angels?"
2. Look through current newspapers and magazines for 20th century examples of reasoning similar to that illustrated in Exercise 1.
3. Give a nonmathematical example of reasoning in a circle.
4. Give a mathematical example of reasoning in a circle.
5. In Euclid's Elements, a point is that which has no parts, or which has no magnitude; a line is length without breadth. Discuss the effectiveness of these two definitions.
6. Identify the elements upon which each of the following definitions of Euclid is based: a straight line is that which lies evenly between its extreme points; a surface is that which has only length and breadth; a plane surface is that in which any two points being taken, the straight line between them lies wholly in that surface.
7. Indicate which of the following statements are reversible.
(a) A duck is a bird.
(b) A line is on a plane if and only if at least two points of the line are on the plane.
(c) If a circle is on a plane, then at least two points of the circle are on the plane.
(d) The boy is a McCoy if he has red hair.
(e) The equality of the lengths of two sides of a quadrilateral is necessary and sufficient for the quadrilateral to be a square.
(f) Rat is tar if and only if r is t.
(g) A circle is a square if and only if two radii are equal.
(h) A necessary and sufficient condition for a rectangle to be a square is that the diagonals be equal.
1–2 Logical notation.
In any system, logical thinking may be hampered by the language used in stating the propositions under consideration. Sometimes the statements become so complex that they are difficult to comprehend. At other times lack of preciseness in the language (English is a good example) is a handicap. These difficulties may be minimized by using special symbols. We shall consider a few such symbols and their applications to the concepts that have just been introduced. Further information regarding these symbols may be found in [11; 1–119] and introductory treatments of symbolic logic.
All formal proofs are based upon implications of the form "p ->q," that is, "the statement p implies the statement q." The logical processes underlying geometric proofs may be most easily represented in terms of the symbols
<-> is equivalent to,
For any given statements p,q we use the symbols as follows:
p p is valid,
~ p p is not valid,
p ^ q both p and q are valid,
p v q at least one of the statements p, q is valid,
where a statement is valid if it holds in the logical system under consideration.
Two statements p,q are equivalent if p ->q and q ->p, that is, p<->>q. A reversible definition asserts that two elements or statements are equivalent. For example, the definition of an equilateral triangle may be considered in the form p<->q, where the statements are
p: a triangle has three equal sides,
q: a triangle is equilateral.
Given any statement of the form p ->q, the converse statement is q ->p, the inverse (or opposite) statement is (~ p) -> (~ q), and the contrapositive (opposite of converse) statement is (~ q ) -> (~ p). Briefly then, we have
statement p ->q,
inverse (~ p ) -> (~ q),
converse q ->p,
contrapositive (~ q ) -> (~ p).
Two statements p,q are contradictory if
[p -> (~ q )] ^ [(~ p) ->q]
For example, the statement
p: the apple is all red,
and the statement
q: the apple is not all red
are contradictory statements. If the statement, "The apple is all red," is valid, then the statement, "The apple is not all red," must be invalid. In other words, if the first statement holds, then the second statement does not hold; i.e., if p, then ~ q. Similarly, if the first statement does not hold, then the second statement does hold; i.e., if ~ p, then q.
The word "contradictory" should not be confused with the word "contrary." Two statements p and q are contrary if they cannot both hold; i.e., if ~ (p ^ q). The following statements are contrary: The apple is all red. The apple is all green. Notice that two contrary statements may both be false. In the above example, the apple may be yellow. All contradictory statements are contrary, but many contrary statements are not contradictory. The above distinction between contrary and contradictory statements is often overlooked in popular discussions.
We shall not attempt to give a complete discussion of the logical processes that may be used to derive or deduce theorems from a given set of undefined elements and unproved postulates. Instead we shall consider very briefly the following laws of logic.
ARISTOTLE'S LAWS OF LOGIC
(i) p<->p, law of identity,
(ii) ~ [p ^ (~ p)], law of noncontradiction—a statement and its contradictory cannot both be valid,
(iii) p v (~ p), law of the excluded middle—at least one of any two contradictory statements must be valid.
Throughout this text we shall assume that Aristotle's Laws of Logic apply to the development of any geometry from its postulates. In particular, these laws provide a basis for the indirect method of proof (Exercise 4) and for the inverse and contrapositive statements of any given statement.
1. Indicate which of the following pairs of statements are contrary.
a. That is Bill. That is Jim.
b. It is a citrus fruit. It is an orange.
c. The car is a Ford. The car is not a Chevrolet.
d. x is positive. x is negative.
e. x< 0. x ≥ 0.
f. x< 4. x = 2.
g. x< 4. x = 5.
h. x2 = 4. x ≠ 2.
2. Indicate which of the pairs of statements in Exercise 1 are contradictory.
3. Bring in current examples of the confusion of contrary and contradictory statements.
4. Discuss the dependence of the method of indirect proof upon Aristotle's Laws of Logic.
5. State and discuss the validity of the converse, inverse, and contrapositive of each of the following statements in euclidean geometry:
(a) If two lines are parallel, they do not intersect.
(b) If the angles A and B of a triangle ABC are equal, the triangle is isosceles.
(c) If a triangle is on a euclidean plane, its angle sum is 180°.
6. Describe three advertisements in current periodicals or newspapers in which the advertiser hopes that the reader will assume the converse or inverse of the fact or situation presented.
7. Compare the following statements:
(b) p if and only if q.
(c) p is a necessary and sufficient condition for q.
(d) p is equivalent to q.
*8. Discuss the basis for and the validity of each of the following statements. Give examples of each statement.
(a) A statement does not necessarily imply its converse.
(b) A statement does not necessarily imply its inverse.
(c) Any statement implies its contrapositive.
(d) The contrapositive of the contrapositive of a statement is the statement.
(e) The contrapositive of a statement implies the statement.
(f) Any statement is equivalent to its contrapositive.
(g) A direct proof of the contrapositive of a statement is an indirect proof of the statement.
(h) The converse of any statement is equivalent to the inverse of the statement.
9. Compare the following statements:
a. The statement p ->q is reversible.
b. The statement p ->q implies its inverse.
c. The statement p ->q implies its converse.
10. Give an example of at least one statement in each of the following forms:
b. (~ p ) ->q.
c. (p ^ q ) ->r.
d. p -> (r v q).
e. (p ^ q ) -> (r V 8 ).
f. p v (~ q ).
11. Is <-> an equivalence relation? In other words, is it reflexive (p<->p) symmetric (p<->q implies q<->p), and transitive (p<->q and q<->r imply p<->r )? Explain.
12. Consult an appropriate reference and discuss briefly the use of Venn diagrams in any two-valued logical system.
1–3 Inductive and deductive reasoning.
Before endeavoring to derive properties of a particular geometry, let us consider two types of reasoning that are often used in the development of a geometry. We have seen that no terms or elements can be defined until at least one element is accepted and allowed to remain undefined. In plane geometry it is customary to accept points and lines as undefined elements. In any science, the elements to be left undefined are generally chosen for their simplicity and fundamental relation to other elements that are to be considered.
Similarly, not all statements or theorems can be proved. Some statements must be accepted as postulates or "rules of the game." These statements often arise as generalizations based upon observations of specific cases. It is not known how specific cases suggest general rules or laws in the minds of men, but most scientific progress originates in this manner. Specific cases and happenings lead men to wonder and finally to devise theories, principles, or laws. This process of formulating a generalization based upon the observation of specific cases is called inductive reasoning or induction.
The statements that are used as postulates are frequently devised by induction. Each postulate must be accepted without proof. If there are several postulates, it is desirable (Section 1–4) that no two postulates be contrary and, furthermore, that no two theorems obtainable from them be contrary. The logical process that we use to obtain a proposition or theorem from a set of undefined terms and unproved postulates is called deductive reasoning or deduction.
The proof or deduction of a theorem of the form p ->q from a given set of undefined elements, unproved statements of relations (postulates), defined elements, and previously proved relations (theorems) may be direct or indirect (Exercise 8g, Section 1–2). A direct proof consists of a finite sequence of statements having the form
p -> p1, p1 ->p2, ..., pn-1 -> ITLpn, pn ->q
and such that each statement is valid in the logical system under consideration. When all the undefined elements and postulates of a science have been accepted and the proved theorems arranged in an order such that each theorem may be proved without using any succeeding (later) theorems, the science is said to be well established as a deductive science.
Excerpted from Fundamental Concepts of Geometry by Bruce E. Meserve. Copyright © 1983 Bruce E. Meserve. Excerpted by permission of Dover Publications, Inc..
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