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# 数学代写非欧几何代写Non-Euclidean Geometry代考|MATH353 The Rotation Proof

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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:

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