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## 数学代写非欧几何代写Non-Euclidean Geometry代考|The Rotation Proof

This ostensible proof, due to Bernhard Friedrich Thibaut ${ }^{14}$ $\left(1775^{-1} 832\right)$ is worthy of note because it has from time to time appeared in elementary texts and has otherwise been indorsed. The substance of the proof is as follows:

In triangle $A B C$ (Fig. i 8 ), allow side $A B$ to rotate ahout $A$, clock- wise, until it coincides with $C A$ produced to $L$. Let $C L$ rotate clockwise about $C$ until it coincides with $B C$ produced to $M$. Finally, when $B M$ has been rotated clockwise about $B$, untıl it coincides with $A B$ produced to $N$, it appears that $A B$ has undergone a complete rotation through four right angles. But the three angles of rotation are the three exterior angles of the triangle, and since their sum is equal to four right angles, the sum of the interior angles must be equal to two right angles.This proof is typical of those which depend upon the idea of direction. The circumspect reader will observe that the rotations take place about different points on the rotating line, so that not only rotation, but translation, is involved. In fact, one sees that the segment $A B$, after the rotations described, has finally been translated along $A B$ through a distance equal to the perımeter of the triangle. Thus it is assumed in the proof that the translations and rotations are independent, and that the translations may be ignored. But this is only truc in Euclidean Geometry and its assumption amounts to taking for granted the Fifth Postulate. The very same argument can be used for a spherical triangle, with the same conclusion, although the sum of the angles of any such trıangle is always greater than two right angles.

## 数学代写非欧几何代写Non-Euclidean Geometry代考|Comparison of Infinite Areas

Another proof, which has from time to time captured the favor of the unwary, is due to the Swiss mathematician, Louis Bertrand ${ }^{16}$
${ }^{16}$ See his correspondence with Schumacher, Engel and Stackel, loc ctt, pp. 227-230.
${ }^{16}$ Développement nouveau de la partie élémentaire des Mathématiques, Vol. II,
p. 19 (Geneva, 1778 )

$(1731-1812)$. He attempted to prove the Fifth Postulate directly, using in essence the following argument:

Given two lines $A P_1$ and $A_1 B_1$ (Fig. 19) cut by the transversal $A A_1$ in such a way that the sum of angles $P_1 A A_1$ and $A A_1 B_1$ is less than two right angles, it is to be proved that $A P_1$ and $A_1 B_1$ meet if sufficiently produced.

Construct $A B$ so that angle $B A A_1$ is equal to angle $B_1 A_1 A_2$, where $A_2$ is a point on $A A_1$ produced through $A_1$. Then $A P_1$ will lie within angle $B A A_1$, since angle $P_1 A A_1$ is less than angle $B_1 A_1 A_2$. Construct $A P_2, A P_3, \ldots, A P_n$ so that angles $P_1 A P_2, P_2 A P_3, \ldots, P_{n-1} A P_n$ are all equal to angle $B A P_1$. Since an integral multiple of angle $B A P_1$ can be found which exceeds angle $B A A_1, n$ can be chosen so large that $A P_n$ will fall below $A A_1$ and angle $B A P_n$ be greater than angle $B A A_1$. Since the infinite sectors $B A P_1, P_1 A P_2, \ldots, P_{n-1} A P_n$ can be superposed, they have equal areas and each has an area equal to that of the infinite sector $B A P_n$ divided by $n$.

## 数学代写非欧几何代写Non-Euclidean Geometry代考|Comparison of InfiniteAreas

16 㕕见他与 Schumacher、Engel 和 Stackel 的通信， loc ctt，第 227-230页。
${ }^{16}$ 数学葚础部分的新发展，卷。二，

(1731-1812). 他试图直接证明第五公设, 本质上使用以下论证:

## 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。

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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

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

## 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。

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

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## 数学代写非欧几何代写Non-Euclidean Geometry代考|The Existence of Similar Figures

The following statement is also equivalent to the Fifth Postulate and may be substituted for it, leading to the same consequences.
There exists a pair of similar triangles, i.e., triangles which are not congruent, but bave the three angles of one equal, respectively, to the three angles of the otber.

To show that this is equivalent to the Fifth Postulate, we need only show how to deduce the latter from it, since every student of Euclid knows that the use of the Postulate leads to a geometry in which similar figures exist.

Given two triangles $A B C$ and $A^{\prime} B^{\prime} C^{\prime}$ (Fig. 7) with angles $A, B$ and $C$ equal, respectively, to angles $A^{\prime}, B^{\prime}$ and $C^{\prime}$. Let $A B$ be greater than $A^{\prime} B^{\prime}$. On $A B$ construct $A D$ equal to $A^{\prime} B^{\prime}$ and on $A C$ construct $A E$ equal to $A^{\prime} C^{\prime}$. Draw $D E$. Then triangles $A D E$ and $A^{\prime} B^{\prime} C^{\prime}$ are congruent. The reader can easily show that $A E$ is less than $A C$, for the assumption that $A E$ is greater than or equal to $A C$ leads to a contradiction. It will not be difficult now to prove that the quadrilateral $B C E D$ has the sum of its four angles equal to four right angles.

Very shortly we shall prove, ${ }^8$ without the use of the Fifth Postulate or its equivalent, that (a) the sum of the angles of a triangle can never be greater than two right angles, provided the straight line is assumed to be infinite, and (b) if one triangle has the sum of its angles equal to two right angles, then the sum of the angles of every triangle is equal to two right angles. By the use of these facts, our proof is easily completed.

By drawing $B E$, two triangles, $B D E$ and $B C E$, are formed. The angle-sum for nelther is greater than two right angles; if the anglesum for either were less than two right angles, that for the other would have to be greater. We conclude that the sum of the angles for each triangle is equal to two right angles and that the same is then true for every triangle.

## 数学代写非欧几何代写Non-Euclidean Geometry代考|Equidistant Straight Lines

Another noteworthy substitute is the following:
There exists a pair of straight lines everywbere equally distant from one anotber.

Once the Fifth Postulate is adopted, this statement follows, for then all parallels have this property of being everywhere equally distant. If the above statement is postulated, we can easily deduce the Fifth Postulate by first proving that there exists a triangle with the sum of its angles equal to two right angles.

Let $A B$ and $C D$ (Fig. 8) be the two lines everywhere equally distant. From any two points $O$ and $Q$ on $C D$ draw $O P$ and $Q R$ perpendicular to $A B$, and from any point $S$ on $A B$ draw $S T$ perpendicular to $C D$. By hypothesis $O P, Q R$ and $S T$ are equal. Since right triangles $O P S$ and $O T S$ are congruent,
Similarly
$\angle P S O=\angle T O S$.
$\angle R S Q=\angle T Q S$.
It follows that the sum of the angles of triangle $O S Q$ is equal to two right angles.

∠磷小号○=∠吨○小号.
∠R小号问=∠吨问小号.

## 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。