数学代写非欧几何代写Non-Euclidean Geometry代考|MATH353 The Infinitude of the Line

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数学代写非欧几何代写Non-Euclidean Geometry代考|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 dee Hypotbesen welche der Geometrie zu Grunde
See Heath, loc. ctt, Vol. I, pp. 224-228.legen, ${ }^{10}$ 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.”

数学代写非欧几何代写Non-Euclidean Geometry代考|THE FIFTH POSTULATE

Even a cursory examination of Book I of Euclid’s Elements will reveal that it comprises three distinct parts, although Euclid did not formally separate them. There is a definite change in the character of the propositions between Proposition 26 and Proposition $27 .$ The first twenty-six propositions deal almost entirely with the elementary theory of triangles. Beginning with Proposition 27, the middle section introduces the important theory of parallels and leads adroitly through Propositions 33 and 34 to the third part. This last section is concerned with the relations of the areas of parallelograms, triangles and squares and culminates in the famous I, 47 and its converse.

In connection with our study of the common notions and postulates we have already had occasion to examine a number of the propositions of the first of the three sections. It is a fact to be noted that the Fifth Postulate was not used by Euclid in the proof of any of these propositions. They would still be valid if the Fifth Postulate were deleted or replaced by another one compatible with the remaining postulates and common notions.

Turning our attention to the second division, consisting of Prop- ositions $27-34$, we shall find it profitable to state the first three and recall their proofs.

数学代写非欧几何代写NON-EUCLIDEAN GEOMETRY代考|THE INFINITUDE OF THE LINE

See Heath, loc。ctt，卷。我，第 224-228.legen，10他在 1854 年读给哥廷根哲学系时指出，无论我们多么肯定空间的无限性，我们都不需要因此推断它的无限性。他说：“在将空间构造扩展到无限大时，我们必须区分无界和无限大；前者属于程度关系，后者属于量度关系。空间是一个无限的三重多样性，这是一个由外部世界的每一个概念发展而来的假设。根据它，每时每刻都完成了真实感知的区域，并构建了所寻找对象的可能位置，并且通过这些应用程序永远证实了自己。以这种方式，空间的无限性比任何外部经验都具有更大的经验确定性。但它的无限范围绝不是由此而来的；另一方面，如果我们假设物体与位置无关，因此将其归因于空间常数曲率，那么它必然是有限的，只要该曲率具有如此小的正值。”

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MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。