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# 数学代写非欧几何代写Non-Euclidean Geometry代考|MTH431 THE FOUNDATION OF EUCLIDEAN GEOMETRY

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## 数学代写非欧几何代写Non-Euclidean Geometry代考|THE FOUNDATION OF EUCLIDEAN GEOMETRY

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 в.с.) 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.

## 数学代写非欧几何代写Non-Euclidean Geometry代考|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 futule 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 bas no part; a line, according to him, is breadtbless length, while a plane surface is one which lies evenly wath 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 ${ }^1$ In this book, all specific statements pertanning to Eucldd’s text and all quotations from Euclid are based upon $T$. L. Heath’s excellent edition: The Tbrreen Books of Eucldd’s Elements, 2nd edition (Cambridge, 1926). By permission of The Macmillan Company.
2 These definitions are to be found in the Appendix.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.”

## 数学代写非欧几何代写NON-EUCLIDEAN GEOMETRY代考|THE DEFINITIONS.

2 这些定义可以在附录中找到。作为欧几里得几何的逻辑基础的是希尔伯特的定义。3他首先考虑三类事物，点、线和平面。“我们认为这些点、直线和平面，”他解释说，“具有一定的相互关系，我们用诸如位于、在、平行、一致、连续等词来表示。完整而准确的描述这些关系是几何公理的结果。”

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