数学代写非欧几何代写Non-Euclidean Geometry代考|EDCI8118 Saccheri

如果你也在 怎样代写非欧几何Non-Euclidean Geometry EDCI8118 这个学科遇到相关的难题,请随时右上角联系我们的24/7代写客服。非欧几何Non-Euclidean Geometry实际上是任何与欧几里得几何不同的几何。 尽管该术语经常仅指双曲几何,但常见用法包括与欧几里得几何不同但非常接近的少数几何(双曲和球面)

非欧几何Non-Euclidean Geometry在数学中,非欧几里得几何由两个几何组成,它们基于与欧几里得几何密切相关的公理。由于欧几里得几何位于度量几何和仿射几何的交点,非欧几里得几何的产生要么是用另一种方法替换平行公设,要么是放宽度量要求。在前一种情况下,人们得到双曲几何和椭圆几何,传统的非欧几里德几何。当度量要求放宽时,就会有与平面代数相关联的仿射平面,这就产生了运动学几何,也被称为非欧几里得几何。

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数学代写非欧几何代写Non-Euclidean Geometry代考|MATH-3355 Ptolemy

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

In the next chapter we shall learn of the discovery of Non-Euclidean Geometry by Bolyai and Lobachewsky early in the nineteenth century. However, this discovery had all but been made by an Italian Jesuit priest almost one hundred years earlier. In 1889 there was brought to light a little book which had been published in Milan in 1733 and long since forgotten. The title of the book was Euclides ab omn naevo vindicatus ${ }^{12}$ (Euclid Freed of Every Flaw), and the author was Gerolamo Saccheri $(1667-1733)$, Professor of Mathematics at the University of Pavia.

While teaching grammar and studying philosophy at Milan, Saccheri had read Euclid’s Elements and apparently had been particularly impressed by his use of the method of reductio ad absurdum. This method consists of assuming, by way of hypothesis, that a proposition to be proved is false; if an absurdity results, the conclusion is reached that the original proposition is true. Later, before going to Pavia in 1697 , Saccheri taught philosophy for three years at Turin. The result of these experiences was the publication of an earlier volume, a treatise on logic. In this, his Logica demonstrativa, the innovation was the application of the ancient, powerful method described above to the treatment of formal logic.

It was only natural that, in casting about for material to which his favorite method might be applied, Saccheri should eventually try it out on that famous and baffling problem, the proof of the Fifth Postulate. So far as we know, this was the first time anyone had thought of denying the Postulate, of substituting for it a contradictory statement in order to observe the consequences.

Saccheri was well prepared to undertake the task. In his Logica demonstrativa he had dealt ably and at length with such topics as definitions and postulates. He was acquainted with the work of others who had attempted to prove the Postulate, and had pointed out the flaws in the proofs of Nasiraddin and Wallis. As a matter of fact, it was essentially Saccheri’s proof which we used above to show that the assumption of Wallis is equivalent to the Postulate.

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

In Germany, a little later, Johann Heınrich Lambert (1728-1777) also came close to the discovery of Non-Euclidean Geometry. His investigations on the theory of parallels were stimulated by a dissertation by Georgius Simon Klügel which appeared in 1763 . It appears that Klugel was the first to express some doubt about the possibility of proving the Fifth Postulate.

There is a striking resemblance between Saccheri’s Eucledes Vindzcatus and Lambert’s Theorie der Parallellineen, ${ }^{13}$ which was written in 1766 , but appeared posthumously. Lambert chose for his fundamental figure a quadrilateral with three right angles, that is, one-half the isosceles quadrilateral used by Saccheri. He proposed three hypotheses in which the fourth angle of this quadrilateral was in turn right, obtuse and acute. In deducing propositions under the second and third hypotheses, he was able to go much further than Saccher1. He actually proved that the area of a triangle is proportional to the difference between the sum of its angles and two right angles, to the excess in the case of the second hypothesis and to the deficit in the case of the third. He noted the resemblance of the geometry based on the second hypothesis to spherical geometry in which the area of a triangle is proportional to its spherical excess, and was bold enough to lean toward the conclusion that in a like manner the geometry based on the third hypothesis could be verified on a sphere with imaginary radius. He even remarked that in the third case there is an absolute unit of length.

He, like Saccheri, was able to rule out the geometry of the second hypothesis, but he made the same tacit assumptions without which no contradictions would have been reached. His final conclusions for the third geometry were indefinite and unsatisfactory. $\mathrm{He}$ seemed to realize that the arguments against it were largely the results of tradition and sentiment. They were, as he said, argumenta $a b$ amore et invidia ducta, arguments of a kind which must be banished altogether from geometry, as from all science.

One cannot fail to note that, while geometers at this time were still attempting to prove the Postulate, nevertheless they were attacking the problem with more open minds. The change had been slow, but there is no doubt that old prejudices were beginning to disappear. The time was almost ripe for far-reaching discoveries to be made.

数学代写非欧几何代写Non-Euclidean Geometry代考|MATH-3355 Ptolemy

非欧几何代写

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

在下一章中,我们将了解 19 世纪早期 Bolyai 和 Lobachewsky 对非欧几何的发现。然而,这一发现几乎是一百年前由一位意大利耶稣会神父做出的。1889 年,一本于 1733 年在米兰出版但早已被人遗忘的小书被曝光。书名是 Euclides ab omn naevo vindicatus12(Euclid Freed of Every Flaw),作者是 Gerolamo Saccheri(1667−1733),帕维亚大学数学教授。

在米兰教授语法和学习哲学时,萨凯里读过欧几里得的《几何原本》,显然对他使用归谬法的方法印象深刻。该方法包括通过假设假设要证明的命题是错误的;如果结果是荒谬的,则得出原命题为真的结论。后来,在 1697 年去帕维亚之前,萨凯里在都灵教了三年哲学。这些经验的结果是出版了较早的一本关于逻辑的论文。在这本书中,他的 Logica demostrativa,创新之处在于将上述古老而强大的方法应用于形式逻辑的处理。

很自然,在寻找可能应用他最喜欢的方法的材料时,萨凯里最终应该尝试解决那个著名且令人费解的问题,即第五公设的证明。据我们所知,这是第一次有人想否认公设,用一个自相矛盾的陈述代替它以观察其后果。

萨凯里已经做好了承担这项任务的准备。在他的逻辑论证中,他干练而详尽地处理了诸如定义和假设之类的主题。他熟悉其他试图证明公设的人的工作,并指出了纳西拉丁和瓦利斯证明中的缺陷。事实上,它本质上是萨凯里的证明,我们在上面使用它来证明瓦利斯的假设等价于公设。

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

在德国,不久之后,约翰·海恩里希·兰伯特 (1728-1777) 也接近发现了非欧几里得几何。1763 年 Georgius Simon Klügel 的一篇论文激发了他对平行理论的研究。看来克鲁格尔是第一个对证明第五公设的可能性表示怀疑的人。

Saccheri 的 Eucledes Vindzcatus 和 Lambert 的 Theorie der Parallellineen 之间有着惊人的相似之处,13它写于 1766 年,但在死后出现。兰伯特为他的基本图形选择了一个具有三个直角的四边形,即萨凯里使用的等腰四边形的一半。他提出了三个假设,其中这个四边形的第四个角依次是右、钝和锐角。在推导第二和第三个假设下的命题时,他能够比 Saccher 走得更远。他实际上证明了三角形的面积与它的角与两个直角之和之间的差成正比,在第二个假设的情况下与过剩成正比,在第三个假设的情况下与赤字成正比。他注意到基于第二个假设的几何与球面几何的相似之处,其中三角形的面积与其球面过剩成正比,并且大胆地倾向于得出结论,以类似的方式,基于第三个假设的几何可以在具有假想半径的球体上得到验证。他甚至指出,在第三种情况下,有一个绝对长度单位。

他和萨凯里一样,能够排除第二个假设的几何学,但他做了同样的默认假设,没有这些假设就不会产生任何矛盾。他对第三几何的最终结论是不确定的和不能令人满意的。H和似乎意识到反对它的论点主要是传统和情感的结果。正如他所说,它们是论据一个bamore et invidia ducta,一种必须从几何学中完全摒弃的论点,就像从所有科学中一样。

不能不注意到,虽然此时的几何学家仍在试图证明公设,但他们正在以更开放的心态来解决这个问题。变化一直很缓慢,但毫无疑问,旧的偏见开始消失。进行具有深远意义的发现的时机几乎已经成熟

数学代写非欧几何代写Non-Euclidean Geometry代考

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