Introduction to Non-Euclidean Geometry

Harold E. Wolfe

• Publisher: Dover Publications
• Year: 2013
• ISBN:
  9780486320373
• Topic: Mathematik
• Subtopic: Geometrie

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.

I

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

1. Introduction.

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, Euclid¹ gave twenty-three definitions.² 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.³ 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.⁴ 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, 4.⁵ Likewise it must be inferred from the Second Postulate that the finite ...