椭圆积分的Maclaurin到Landen的双曲线校正和Lagrange-勒让德变换
The hyperbola rectification from Maclaurin to Landen and the Lagrange-Legendre transformation for the elliptic integrals
摘要：
本文介绍了主要的数学研究进行，在英国和大陆之间的1742年至1827年，对主题的双曲线整流，从而增加了我们的一些贡献。我们开始与马克劳林发明微积分及其显着的作用，在早期中期1700年;接下来，我们重点有点他的评价，1742年，双曲过剩，解释背后的真正动机，他的研究。对于他的几何分析的处理，我们附上我们的，一个纯粹的分析的选择。我们的双曲线调查然后切换到约翰·兰登，一个业余数学家，他可能写更多的是为了确定他的优先事项，而不是解释他的非凡发现。我们跟随他在模糊的证明，他的定理对双曲线整流，解释联系马克劳林等。与链的几何结构，我们重视我们的解释兰登的治疗。我们的现代分析证明他的双曲极限过剩，通过椭圆积分的第一和第二类也提供了，我们证明了为什么所谓的Landen变换的椭圆积分不能归因于他。接下来，这个问题离开英国的大陆：字符拉格朗日介绍，即使我们的兴趣只涉及他1785年的回忆录无理积分，其中算术几何平均，AGM，是由他建立的。然而，我们的目标不是AGM，而是检测所谓的椭圆积分的Landen变换的真正来源。事实上，拉格朗日的文件包含一个微分身份源于AGM：整合它，我们展示了如何可能达到著名的勒让德递归计算的第一类椭圆积分，这出现在他的Traité，1827年，后拉格朗日的文件。
英文摘要：
This article describes the main mathematical researches performed, in England and in the Continent between 1742-1827, on the subject of hyperbola rectification, thereby adding some of our contributions. We start with the Maclaurin inventions on Calculus and their remarkable role in the early mid 1700s; next we focus a bit on his evaluation, 1742, of the hyperbolic excess, explaining the true motivation behind his research. To his geometrical-analytical treatment we attach ours, a purely analytical alternative. Our hyperbola inquiry is then switched to John Landen, an amateur mathematician, who probably was writing more to fix his priorities than to explain his remarkable findings. We follow him in the obscure proofs of his theorem on hyperbola rectification, explaining the links to Maclaurin and so on. With a chain of geometrical constructions, we attach our interpretation to Landen's treatment. Our modern analytical proof to his hyperbolic limit excess, by means of elliptic integrals of the first and second kind is also provided, and we demonstrate why the so called Landen transformation for the elliptic integrals cannot be ascribed to him. Next, the subject leaves England for the Continent: the character of Lagrange is introduced, even if our interest concerns only his 1785 memoir on irrational integrals, where the Arithmetic Geometric Mean, AGM, is established by him. Nevertheless, our objective is not the AGM, but to detect the real source of the so-called Landen transformation for elliptic integrals. In fact, Lagrange's paper encloses a differential identity stemming from the AGM: integrating it, we show how it could be possible to arrive at the well-known Legendre recursive computation of a first kind elliptic integral, which appeared in his Traité, 1827, much after the Lagrange's paper.
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