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数学代写非欧几何代写Non-Euclidean Geometry代考|MATH360 The Common Notions

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数学代写非欧几何代写Non-Euclidean Geometry代考|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:

I. Thengs whech are equal to the same theng are also equal to one anotber.

1. If equals be added to equals, the wholes are equal.
2. If equals be subtracted from equals, the remainders are equal.
3. Things which coincide with one another are equal to one another.
4. 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.

数学代写非欧几何代写Non-Euclidean Geometry代考|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 I, 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. ${ }^8$ 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 positzons 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 recognnize 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 approxımate comparıson 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 s1des and the included’ angle of one equal to two s1des and the included angle of the other, the remaining pairs of corresponding angles are equal. $.^9$

数学代写非欧几何代写NON-EUCLIDEAN GEOMETRY代考|THE COMMON NOTIONS

I. 等于同一theng 的Thengs 也等于一个annotber。

1. 如果equals与equals相加，则整体相等。
2. 如果从equals中减去equals，则余数相等。
3. 彼此一致的事物彼此相等。
4. 整体大于部分。
人们在这些假设中认识到曾经经常被描述为“不言而喻”的那种类型的命题。从已经说过的内容来看，应该清楚的是，这根本不是几何假设的特征。事实上，从来没有发现过“不言而喻”的命题。

数学代写非欧几何代写NON-EUCLIDEAN GEOMETRY代考|TACIT ASSUMPTIONS MADE BY EUCLID. SUPERPOSITION

Euclid 对命题 I, 4 使用了与大多数现代基本文本中使用的基本相同的证明。毫无疑问，在证明两个有两条边的三角形和一个等于两条边的夹角和另一个三角形的夹角全等时，他实际上是把一个三角形看作是移动的，以使其与其他。但是在空间图形性质的证明中，有人反对这种诉诸运动而不变形的观念。8看来欧几里得本人并不重视这种方法，很不情愿地使用它。

Matlab代写

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。