One of the first college-level texts for elementary courses in non-Euclidean geometry, this concise, readable volume is geared toward students familiar with calculus. A full treatment of the historical background explores the centuries-long efforts to prove Euclid's parallel postulate and their triumphant conclusion. Numerous original exercises form an integral part of the book.
Topics include hyperbolic plane geometry and hyperbolic plane trigonometry, applications of calculus to the solutions of some problems in hyperbolic geometry, elliptic plane geometry and trigonometry, and the consistency of the non-Euclidean geometries. Extensive appendixes offer background information on the foundation of Euclidean geometry, circular and hyperbolic functions, the theory of orthogonal circles and allied topics, and the elements of inversion.
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Introduction to Non-Euclidean Geometry
By Harold E. Wolfe
Dover Publications, Inc.Copyright © 2012 Dover Publications, Inc.
All rights reserved.
THE FOUNDATION OF EUCLIDEAN GEOMETRY
"This book has been for nearly twenty-two centuries the encouragement and guide of that scientific thought which is one thing with the progress of man from a worse to a better state." – Clifford
Geometry, that branch of mathematics in which are treated the properties of figures in space, is of ancient origin. Much of its development has been the result of efforts made throughout many centuries to construct a body of logical doctrine for correlating the geometrical data obtained from observation and measurement. By the time of Euclid (about 300 B.C) the science of geometry had reached a well-advanced stage. From the accumulated material Euclid compiled his Elements, the most remarkable textbook ever written, one which, despite a number of grave imperfections, has served as a model for scientific treatises for over two thousand years.
Euclid and his predecessors recognized what every student of philosophy knows: that not everything can be proved. In building a logical structure, one or more of the propositions must be assumed, the others following by logical deduction. Any attempt to prove all of the propositions must lead inevitably to the completion of a vicious circle. In geometry these assumptions originally took the form of postulates suggested by experience and intuition. At best these were statements of what seemed from observation to be true or approximately true. A geometry carefully built upon such a foundation may be expected to correlate the data of observation very well, perhaps, but certainly not exactly. Indeed, it should be clear that the mere change of some more-or-less doubtful postulate of one geometry may lead to another geometry which, although radically different from the first, relates the same data quite as well.
We shall, in what follows, wish principally to regard geometry as an abstract science, the postulates as mere assumptions. But the practical aspects are not to be ignored. They have played no small role in the evolution of abstract geometry and a consideration of them will frequently throw light on the significance of our results and help us to determine whether these results are important or trivial.
In the next few paragraphs we shall examine briefly the foundation of Euclidean Geometry. These investigations will serve the double purpose of introducing the Non-Euclidean Geometries and of furnishing the background for a good understanding of their nature and significance.
2. The Definitions.
The figures of geometry are constructed from various elements such as points, lines, planes, curves, and surfaces. Some of these elements, as well as their relations to each other, must be left undefined, for it is futile to attempt to define all of the elements of geometry, just as it is to prove all of the propositions. The other elements and relations are then defined in terms of these fundamental ones. In laying the foundation for his geometry, Euclid gave twenty-three definitions. A number of these might very well have been omitted. For example, he defined a point as that which has no part; a line, according to him, is breadthless length, while a plane surface is one which lies evenly with the straight lines on itself. From the logical viewpoint, such definitions as these are useless. As a matter of fact, Euclid made no use of them. In modern geometries, point, line, and plane are not defined directly; they are described by being restricted to satisfy certain relations, defined or undefined, and certain postulates. One of the best of the systems constructed to serve as a logical basis for Euclidean Geometry is that of Hilbert. He begins by considering three classes of things, points, lines, and planes. "We think of these points, straight lines, and planes," he explains, "as having certain mutual relations, which we indicate by such words as are situated, between, parallel, congruent, continuous, etc. The complete and exact description of these relations follows as a consequence of the axioms of geometry."
The majority of Euclid's definitions are satisfactory enough. Particular attention should be given to the twenty-third, for it will play an important part in what is to follow. It is the definition of parallel lines — the best one, viewed from an elementary standpoint, ever devised.
Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.
In contrast with this definition, which is based on the concept of parallel lines not meeting, it seems important to call attention to two other concepts which have been used extensively since ancient times. These involve the ideas that two parallel lines are lines which have the same direction or which are everywhere equally distant. Neither is satisfactory.
The direction-theory leads to the completion of a vicious circle. If the idea of direction is left undefined, there can be no test to apply to determine whether two given lines are parallel. On the other hand, any attempt to define direction must depend upon some knowledge of the behavior of parallels and their properties.
The equidistant-theory is equally unsatisfactory. It depends upon the assumption that, for the particular geometry under consideration, the locus of points equidistant from a straight line is a straight line. But this must be proved, or at least shown to be compatible with the other assumptions. Strange as it may seem, we shall shortly encounter geometries in which this is not true.
Finally, it is worth emphasizing that, according to Euclid, two lines in a plane either meet or are parallel. There is no other possible relation.
3. The Common Notions.
The ten assumptions of Euclid are divided into two sets: five are classified as common notions, the others as postulates. The distinction between them is not thoroughly clear. We do not care to go further than to remark that the common notions seem to have been regarded as assumptions acceptable to all sciences or to all intelligent people, while the postulates were considered as assumptions peculiar to the science of geometry. The five common notions are:
1. Things which are equal to the same thing are also equal to one another.
2. If equals be added to equals, the wholes are equal.
3. If equals be subtracted from equals, the remainders are equal.
4. Things which coincide with one another are equal to one another.
5. The whole is greater than the part.
One recognizes in these assumptions propositions of the type which at one time were so frequently described as "self-evident." From what has already been said, it should be clear that this is not the character of the assumptions of geometry at all. As a matter of fact, no "self-evident" proposition has ever been found.
4. The Postulates.
Euclid postulated the following:
1. To draw a straight line from any point to any point.
2. To produce a finite straight line continuously in a straight line.
3. To describe a circle with any center and distance.
4. That all right angles are equal to one another.
5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
Although Euclid does not specifically say so, it seems clear that the First Postulate carries with it the idea that the line joining two points is unique and that two lines cannot therefore enclose a space. For example, Euclid tacitly assumed this in his proof of 1, 4. Likewise it must be inferred from the Second Postulate that the finite straight line can be produced at each extremity in only one way, so that two different straight lines cannot have a common segment. Explicit evidence of this implication first appears in the proof of XI, I, although critical examination shows that it is needed from the very beginning of Book I. In regard to the Third Postulate, we merely remark that the word distance is used in place of radius, implying that each point of the circumference is at this distance from the center. The Fourth Postulate provides a standard or unit angle in terms of which other angles can be measured. Immediate use of this unit is made in Postulate 5.
The Fifth Postulate plays a major role in what follows. In fact it is the starting point in the study of Non-Euclidean Geometry. One can hardly overestimate the effect which this postulate, together with the controversies which surrounded it, has had upon geometry, mathematics in general, and logic. It has been described as "perhaps the most famous single utterance in the history of science." On account of its importance, we shall return to it soon and treat it at length.
5. Tacit Assumptions Made by Euclid. Superposition.
In this and the remaining sections of the chapter we wish to call attention to certain other assumptions made by Euclid. With the exception of the one concerned with superposition, they were probably made unconsciously; at any rate they were not stated and included among the common notions and postulates. These omissions constitute what is regarded by geometers as one of the gravest defects of Euclid's geometry.
Euclid uses essentially the same proof for Proposition 1, 4 that is used in most modern elementary texts. There is little doubt that, in proving the congruence of two triangles having two sides and the included angle of one equal to two sides and the included angle of the other, he actually regarded one triangle as being moved in order to make it coincide with the other. But there are objections to such recourse to the idea of motion without deformation in the proofs of properties of figures in space. It appears that Euclid himself had no high regard for the method and used it reluctantly.
Objections arise, for example, from the standpoint that points are positions and are thus incapable of motion. On the other hand, if one regards geometry from the viewpoint of its application to physical space and chooses to consider the figures as capable of displacement, he must recognize that the material bodies which are encountered are always more-or-less subject to distortion and change. Nor, in this connection, may there be ignored the modern physical concept that the dimensions of bodies in motion are not the same as when they are at rest. However, in practice, it is of course possible to make an approximate comparison of certain material bodies by methods which resemble superposition. This may suggest the formulation in geometry of a postulate rendering superposition legitimate. But Euclid did not do this, although there is evidence that he may have intended Common Notion 4 to authorize the method. In answer to the objections, it also may be pointed out that what has been regarded as motion in superposition is, strictly speaking, merely a transference of attention from one figure to another.
The use of superposition can be avoided. Some modern geometers do this, for example, by assuming that, if two triangles have two sides and the included angle of one equal to two sides and the included angle of the other, the remaining pairs of corresponding angles are equal.
6. The Infinitude of the Line.
Postulate 2, which asserts that a straight line can be produced continuously, does not necessarily imply that straight lines are infinite. However, as we shall discover directly, Euclid unconsciously assumed the infinitude of the line.
It was Riemann who first suggested the substitution of the more general postulate that the straight line is unbounded. In his remarkable dissertation, Über die Hypothesen welche der Geometrie zu Grunde liegen, read in 1854 to the Philosophical Faculty at Göttingen, he pointed out that, however certain we may be of the unboundedness of space, we need not as a consequence infer its infinitude. He said, "In the extension of space-construction to the infinitely great, we must distinguish between unboundedness and infinite extent; the former belongs to the extent relations, the latter to the measure relations. That space is an unbounded threefold manifoldness is an assumption which is developed by every conception of the outer world; according to which every instant the region of real perception is completed and the possible positions of a sought object are constructed, and which by these applications is forever confirming itself. The unboundedness of space possesses in this way a greater empirical certainty than any external experience. But its infinite extent by no means follows from this; on the other hand if we assume independence of bodies from position, and therefore ascribe to space constant curvature, it must necessarily be finite provided this curvature has ever so small a positive value."
We shall learn later that geometries, logically as sound as Euclid's, can be constructed upon the hypothesis that straight lines are boundless, being closed, but not infinite. In attempting to conceive straight lines of this character, the reader may find it helpful, provided he does not carry the analogy too far, to consider the great circles of a sphere. It is well known that in spherical geometry the great circles are geodesies, i.e., they are the "lines" of shortest distance between points. It will not be difficult to discover that they have many other properties analogous to those of straight lines in Euclidean Plane Geometry. On the other hand there are many striking differences. We note, for example, that these "lines," while endless, are not infinite; that, while in general two points determine a "line," two points may be so situated that an infinite number of "lines" can be drawn through them; that two "lines" always intersect in two points and enclose a space.
Even a cursory consideration of the consequences of attributing to straight lines the character of being boundless, but not infinite, convinces one that Euclid tacitly assumed the infinitude of the line. One critical point at which this was done is found in the proof of 1, 16. This proposition is of such importance in what follows and its consequences are so far-reaching that we present the proof here.
Proposition 1, 16. In any triangle, if one of the sides be produced, the exterior angle is greater than either of the interior and opposite angles.
Let ABC (Fig. 1) be the given triangle, with BC produced to D. We shall prove that
[angle] ACD [angle] BAC.
Let E be the midpoint of AC. Draw BE and produce it to F, making EF equal to BE. Draw CF. Then triangles BEA and FEC are congruent and consequently angles FCE and BAC are equal.
[angle] ACD [angle] FCE.
[angle] ACD [angle] BAC.
It should be clear now that this proof may fail, if the straight line is not infinite. As a matter of fact, the same proof can be used in spherical geometry for a spherical triangle but is valid only so long as BF is less than a semicircle. If F lies on CD, angle ACD is equal to angle ECF and consequently to angle BAC. If BF is greater than a semicircle, angle ACD will be less than angle BAC. Even if one conceives a geometry in which any two of the closed lines intersect in only one point, BF may be so long that F will coincide with B or lie on segment BE. In either case the proof fails.
The proofs of a number of important propositions in Euclidean Geometry depend upon 1, 16. Such propositions as 1, 17, 18, 19, 20 and 21 will not be valid without restrictions when 1, 16 does not hold.
Prove Propositions 1, 17, 18, 19, 20, 21.
7. Pasch's Axiom.
Another important assumption made by Euclid, without explicit statement, has been formulated by Pasch as follows.
Let A, B, C be three points not lying in the same straight line and let a be a straight line lying in the plane of ABC and not passing through any of points A, B, C. Then, if the line a passes through a point of the segment AB, it will also pass through a point of the segment BC or a point of the segment AC.
It readily follows from this that, if a line enters a triangle at a vertex, it must cut the opposite side. Euclid tacitly assumed this frequently as, for example, in the proof of 1, 21.
We shall make use of Pasch's Axiom many times in what follows at points where intuition cannot be depended upon to guide us as safely as it did in Euclid. In order to emphasize the importance of an explicit statement of this axiom as a characteristic of Euclidean Geometry, we remark that there are geometries in which it holds only with restrictions. It will be recognized that it is true for spherical triangles only if they are limited in size.
Pasch's Axiom is one of those assumptions classified by modern geometers as axioms of order. These important axioms bring out the idea expressed by the word between and make possible an order of sequence of the points on a straight line.
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Table of Contents
Chapter I The Foundation of Euclidean Geometry
1 Introduction 1
2 The Definitions 2
3 The Common Notions 4
4 The Postulates 4
5 Tacit Assumptions Made by Euclid. Superposition 5
6 The Infinitude of the Line 6
7 Pasch's Axiom 9
8 The Principle of Continuity 10
9 The Postulate System of Hilbert 12
Chapter II The Fifth Postulate
10 Introduction 17
11 Substitutes for the Fifth Postulate 20
12 Playfair's Axiom 20
13 The Angle-Sum of a Triangle 21
14 The Existence of Similar Figures 23
15 Equidistant Straight Lines 25
16 Other Substitutes 25
17 Attempts to Prove the Fifth Postulate 26
18 Ptolemy 26
19 Proclus 27
20 Nasiraddin 28
21 Wallis 29
22 Saccheri 30
23 Lambert 33
24 Legendre 34
25 Some Fallacies in Attempts to Prove the Postulate 39
26 The Rotation Proof 40
27 Comparison of Infinite Areas 41
Chapter III The Discovery of Non-Euclidean Geometry
28 Introduction 44
29 Gauss 45
30 Bolyai 48
31 Lobachewsky 53
32 Wachter, Schweikart and Taurinus 56
33 Riemann 60
34 Further Developments 63
35 Conclusion 63
Chapter IV Hyperbolic Plane Geometry
36 Introduction 65
37 The Characteristic Postulate of Hyperbolic Geometry 66
38 Elementary Properties of Parallels 68
39 Ideal Points 71
40 Some Properties of an Important Figure 72
41 The Angle of Parallelism 76
42 The Saccheri Quadrilateral 77
43 The Lambert Quadrilateral 79
44 The Sum of the Angles of a Triangle 81
45 The Common Perpendicular of Two Non-Intersecting Lines 84
46 Ultra-Ideal Points 85
47 The Variation in the Distance between Two Lines 86
48 The Perpendicular Bisectors of the Sides of a Triangle 90
49 The Construction of the Parallels to a Line through a Point 93
50 The Construction of a Common Parallel to Two Intersecting Lines 97
51 The Construction of a Line Perpendicular to One of Two Intersecting Lines and Parallel to the Other 99
52 Units of Length and Angle 100
53 Associated Right Triangles 101
54 The Construction of a Triangle when Its Angles Are Given 105
55 The Absolute 107
56 Circles 109
57 Corresponding Points 110
58 Limiting Curves and Their Properties 113
59 Equidistant Curves and Their Properties 117
60 The Limiting Curve as Related to Circles and Equidistant Curves 119
61 Area 120
62 Equivalence of Polygons and Triangles 122
63 Measure of Area 127
64 The Triangle with Maximum Area 128
Chapter V Hyperbolic Plane Trigonometry
65 Introduction 131
66 The Ratio of Corresponding Arcs of Concentric Limiting Curves 131
67 Relations between the Parts of an Important Figure 136
68 A Coördinate System and another Important Figure 138
69 The Relations between Complementary Segments 140
70 Relations among the Parts of a Right Triangle 142
71 Relations among the Parts of the General Triangle 145
72 The Relation between a Segment and Its Angle of Parallelism 148
73 Simplified Formulas for the Right Triangle and the General Triangle 152
74 The Parameter 153
Chapter VI Applications of Calculus to the Solutions of Some Problems in Hyperbolic Geometry
75 Introduction 157
76 The Differential of Arc in Cartesian Coördinates 158
77 The Differential of Arc in Polar Coördinates 160
78 The Circumference of a Circle and the Lengths of Arcs of Limiting Curve and Equidistant Curve 161
79 The Area of a Fundamental Figure 163
80 Limiting Curve Coördinates 165
81 The Element of Area 166
82 The Area of a Circle 169
83 The Area of a Lambert Quadrilateral 169
84 The Area of a Triangle 171
Chapter VII Elliptic Plane Geometry and Trigonometry
85 Introduction 173
86 The Characteristic Postulate of Elliptic Geometry and Its Immediate Consequences 174
87 The Relation between Geometry on a Sphere and Elliptic Geometry 177
88 The Two Elliptic Geometries 179
89 Properties of Certain Quadrilaterals 180
90 The Sum of the Angles of a Triangle 182
91 The Trigonometry of the Elliptic Plane 185
92 The Trigonometric Functions of an Angle 185
93 Properties of a Variable Lambert Quadrilateral 190
94 The Continuity of the Function φ(x) 192
95 An Important Functional Equation 193
96 The Function φ(x) 194
97 The Relation among the Parts of a Right Triangle 196
Chapter VIII The Consistency of the Non-Euclidean Geometries
98 Introduction 201
99 The Geometry of the Circles Orthogonal to a Fixed Circle 204
100 The Nominal Length of a Segment of Nominal Line 206
101 Displacement by Reflection 208
102 Displacement in the Geometry of the Nominal Lines 209
103 The Counterparts of Circles, Limiting Curves and Equidistant Curves 213
104 The Relation between a Nominal Distance and Its Angle of Parallelism 214
105 Conclusion 217
I The Foundation of Euclidean Geometry
1 The Definitions of Book I 218
2 The Postulates 219
3 The Common Notions 220
4 The Forty-Eight Propositions of Book I 220
II Circular and Hyperbolic Functions
5 The Trigonometric Functions 223
6 The Hyperbolic Functions 225
7 The Inverse Hyperbolic Functions 228
8 Geometric Interpretation of Circular and Hyperbolic Functions 229
III The Theory of Orthogonal Circles and Allied Topics
9 The Power of a Point with Regard to a Circle 232
10 The Radical Axis of Two Circles 232
11 Orthogonal Circles 234
12 Systems of Coaxal Circles 235
IV The Elements of Inversion
13 Inversion 237
14 The Inverse of a Circle and the Inverse of a Line 238
15 The Effect of Inversion on Angles 240
16 The Peaucellier Inversor 240