Posted on Categories:Non-Euclidean Geometry, 数学代写, 非欧几何

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For our next example we pass to the thirteenth century and consider the contributions of Nasiraddin (1201-1274), Persian astronomer and mathematician, who compiled an Arabic version of Euclid and wrote a treatise on the Euclidean postulates. He seems to have been the first to direct attention to the importance, in the study of the Fifth Postulate, of the theorem on the sum of the angles of a triangle. In his attempt to prove the Postulate one finds the germs of important ideas which were to be developed later.
Nasiraddin first asserted, without proof, the following:

cessive perpendiculars such as $E F, G H, I J$, etc., drawn to $C D$ from points $E, G, I$, etc. of $A B$, always make unequal angles with $A B$, which are always acute on the side toward $B$, and consequently always obtuse on the side towards $A$, then the lines $A B$ and $C D$ continually diverge in the direction of $A$ and $C$ and, so long as they do not meet, continually converge in the direction of $B$ and $D$, the perpendiculars continually growing longer in the first direction and shorter in the second. Conversely, if the perpendiculars continually become longer in the direction of $A$ and $C$ and shorter in the direction of $B$ and $D$, the lines diverge in the first direction and converge in the other, and the perpendiculars will make with $A B$ unequal angles, the obtuse angles all lying on the side toward $A$ and $C$ and the acute angles on the side towards $B$ and $D$.

## 数学代写非欧几何代写Non-Euclidean Geometry代考|Wallis

John Wallis (1616-1703) became interested in the work of Nasiraddin and described his demonstrations in a lecture at Oxford in 165 I. In 1663 he offered a proof of his own. We describe it here because it is typical of those proofs which make use of an assumption equivalent to the Fifth Postulate.

Wallis suggested the assumption that, given a triangle, it is possible to construct another triangle similar to it and of any size. Then he argues essentially as follows:

Given lines $A B$ and $C D$ (Fig. I2), cut by the transversal $E F$ in points $G$ and $H$, respectively, and with the sum of angles $B G H$ and $D H G$ less than two right angles. It is to be proved that $A B$ and $C D$ will meet if sufficiently produced.
It is easy to show that
$$\angle E G B>\angle G H D \text {. }$$
Then, if segment $H G$ is moved along $E F$, with $H D$ rigidly attached to it, untıl $H$ coincides with the initial position of $G, H D$ takes the position $G I$, lying entirely above $G B$. Hence, during its motion, $H D$ must at some time cut $G B$ as, for example, when it coincides with $J K$, cutting $G B$ at $L$. Now if one constructs a triangle on base $G H$ sımilar to triangle $G J L$ – and this has been assumed to be possible – it is evident that $H D$ must cut $G B$.

## 数学代写非欧几何代写Non-Euclidean Geometry代考|Wallis

$\angle E G B>\angle G H D$

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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。