The direction-theory leads to the completion of a vicious circle. ⁴ These involve the ideas that two parallel lines are lines which have the same direction or which are everywhere equally distant. 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. 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. It is the definition of parallel lines - the best one, viewed from an elementary standpoint, ever devised. Particular attention should be given to the twenty-third, for it will play an important part in what is to follow. The majority of Euclid’s definitions are satisfactory enough. The complete and exact description of these relations follows as a consequence of the axioms of geometry." 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. 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. 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. As a matter of fact, Euclid made no use of them. From the logical viewpoint, such definitions as these are useless. 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. In laying the foundation for his geometry, Euclid ¹ gave twenty-three definitions.² A number of these might very well have been omitted. The other elements and relations are then defined in terms of these fundamental ones. 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 figures of geometry are constructed from various elements such as points, lines, planes, curves, and surfaces. 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. In the next few paragraphs we shall examine briefly the foundation of Euclidean Geometry. 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. But the practical aspects are not to be ignored. We shall, in what follows, wish principally to regard geometry as an abstract science, the postulates as mere assumptions. 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. A geometry carefully built upon such a foundation may be expected to correlate the data of observation very well, perhaps, but certainly not exactly. ![]() ![]() At best these were statements of what seemed from observation to be true or approximately true. In geometry these assumptions originally took the form of postulates suggested by experience and intuition. Any attempt to prove all of the propositions must lead inevitably to the completion of a vicious circle. In building a logical structure, one or more of the propositions must be assumed, the others following by logical deduction. 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.Įuclid and his predecessors recognized what every student of philosophy knows: that not everything can be proved. By the time of Euclid (about 300 B.C) the science of geometry had reached a well-advanced stage. ![]() 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. Geometry, that branch of mathematics in which are treated the properties of figures in space, is of ancient origin. 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.
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