亚伯拉罕·哥特黑尔弗·凯斯特纳Abraham Gotthelf Kästner

亚伯拉罕·哥特黑尔弗·凯斯特纳的父亲是一位大学法理学教授。他希望儿子能继承自己的事业，凯斯特纳起初上大学时打算学习法哲学，但很快发现其他学科更合自己的心意，于是开始更多地专注于哲学、物理学和数学。

凯斯特纳在莱比锡大学撰写了教授资格论文（Habilitation）学位论文，并于1739年获得了在该校任教的资格。他在莱比锡大学担任Privatdozent，直到1746年被任命为编外教授。十年后的1756年，他被任命为哥廷根大学的数学和物理学教授，接替了约翰·安德烈亚斯·泽格纳的讲席。他是一位出色的数学讲解者，尽管据报告说卡尔·弗里德里希·高斯觉得他的讲座太基础而不愿去听。然而他确实影响了卡尔·弗里德里希·高斯，尤其是通过他对欧几里得的平行公设的兴趣。

论文[5]考察了凯斯特纳的贡献，Sinaceur将其描述为18世纪中后期一位重要的德国数学家。凯斯特纳并非以原创研究闻名，而是参与编纂百科全书和撰写教科书。他关注数学以及其他领域如逻辑学中的哲学问题。然而凯斯特纳对逻辑学相当不热心，但这对于一位对几何学感兴趣的这一时期的数学家来说并不奇怪。尽管如此，他对数学哲学感兴趣，并广泛地以长篇卷册撰写关于数学在光学、动力学和天文学中应用的著作。

也许他最著名的两部作品，均为四卷本，是Mathematische AnfangsgründeⓉ（数学基础）和Geschichte der MathematikⓉ（数学史）（1796-1800）。后一部作品旨在为一部数学史奠定基础。例如，1787年出版的第2卷被认为是数学光学史的标准参考资料之一。

也许 凯斯特纳 贡献的最重要特点是他对平行公设的兴趣，这也间接影响了 鲍耶 和 罗巴切夫斯基。凯斯特纳 教过 鲍耶 的父亲，而 凯斯特纳 的一名学生 马丁·巴特尔斯 教过 罗巴切夫斯基。Folta 在 [3] 中写到 凯斯特纳 关于几何学的工作：-

凯斯特纳 [是] 18 世纪的数学家之一，其广泛的兴趣迫使他关注几何学的主要问题。[他的] 成果包含一些新特点，更精确地表述了初等几何的传统解释。事实上，[他] 开始了有意识地为基本概念建立精确公理化的尝试。凯斯特纳 尽管对 欧几里得 的《几何原本》有相当大的偏好，却在《纲要》中把他的几何公理学版本建立在其他原则（例如运动）之上，并试图同时抓住其他基本性质（连续性、顺序性），并确定把平行公理选作基础。

文章 [1] 给了我们更多关于 凯斯特纳 生平的细节，这些细节在 [2] 中有详细描述，而 [2] 是一部自传。Goe 在 [1] 中写道：-

凯斯特纳 在德国文学中也很有名，尤其是因为他的警句。他是一位虔诚的路德宗信徒。凯斯特纳 结过两次婚，与第二任妻子有一个女儿。

Abraham Gotthelf Kästner's father was a university professor of jurisprudence. He hoped that his son would follow in his footsteps and Kästner started out on a university course with the intention of studying the philosophy of law, but he soon found other topics more to his liking and began to concentrate more on philosophy, physics and mathematics.

Kästner wrote his habilitation thesis at the University of Leipzig, and was awarded the qualification which allowed him to teach there in 1739. He taught at the University of Leipzig as a Privatdozent until 1746 when he was appointed as an extraordinary professor. Ten years later, in 1756, he was appointed as professor of mathematics and physics at Göttingen where he succeeded to Segner's chair. He was an excellent expositor of mathematics although it is reported that Gauss did not bother to go to his lectures as he found them too elementary. However he did influence Gauss, in particular with his interest in Euclid's parallel postulate.

The paper [5] examines the contribution of Kästner who Sinaceur describes as an important German mathematician of the mid- and late 18th century. Kästner is not famed for original research but rather he was involved in compiling encyclopedias and in writing textbooks. He was concerned with philosophical questions in mathematics and other areas such as logic. However Kästner was quite unenthusiastic about logic, but this is not surprising for a mathematician of this period who was interested in geometry. Despite this he was interested in the philosophy of mathematics and he wrote widely, in long volumes, about the applications of mathematics to optics, dynamics and astronomy.

Perhaps his two most famous works, both in four volumes, were Mathematische Anfangsgründe Ⓣ and Geschichte der Mathematik Ⓣ (1796-1800). This latter work was intended to form the basis for a history of mathematics. For example Volume 2, published in 1787, is considered one of the standard sources on the history of mathematical optics.

Perhaps the most important feature of Kästner's contributions was his interest in the parallel postulate which indirectly influenced Bolyai and Lobachevsky too. Kästner taught Bolyai's father and J M C Bartels, one of Kästner's students, taught Lobachevsky. Folta writes in [3] about Kästner's work on geometry:-

Kästner [is] among the mathematicians of the 18th century whose broad interests compelled [him] to concern [himself] with the principal problems of geometry. [His] results included new features that more precisely formulated the traditional interpretation of elementary geometry. In fact, [he] began the conscious attempt to make a precise axiomatisation of the fundamental concepts. Kästner, in spite of his rather great inclination for Euclid's Elements, based his version of the axiomatics of geometry in his Kompendium on other principles (e.g., on motions) and attempted both to seize on other fundamental properties (continuity, ordering) and to determine the selection of the parallel axiom as a foundation.

The article [1] gives us a few more details of Kästner's life, which is described in detail in [2] which is an autobiography. Goe writes in [1]:-

Kästner is also known in German literature, notably for his epigrams. He was a devout Lutheran. Kästner married twice and had a daughter by his second wife.