乔治·威廉·希尔George Hill

乔治·威廉·希尔的母亲是Catherine Smith，父亲是弗瑞兹·约翰 William Hill，他是一位专攻版画的艺术家。1846年，希尔八岁时，他家搬到纽约西奈阿克附近的一个农场，希尔在那里上学。他在中学接受了有些基础的教育，但他在数学课上表现出色。高中毕业后，他于1855年进入拉特格斯学院。该学院自1766年起就是一所高等教育机构，并在希尔在那里学习之前三十年更名为拉特格斯学院。希尔毕业不久后，1862年的《莫里尔法案》使拉特格斯学院于1864年成为新泽西州的赠地学院。

希尔在拉特格斯学院的本科学习对于这一时期的美国学生来说极不寻常。他在学院由一位非常有能力的数学家Theodore Strong授课，后者是纳撒尼尔·鲍迪奇的朋友。Strong拥有一座经典数学教材的优良藏书室，他很乐意让才华横溢的本科生希尔研读这些书。这些教材包括西尔维斯特·佛朗索瓦·拉克鲁瓦的Traité du calcul différentiel et intégral Ⓣ（《微分与积分学论著》）、约瑟夫·拉格朗日的Mécanique analytique Ⓣ（《分析力学》）、皮埃尔·西蒙·拉普拉斯的Mécanique céleste Ⓣ（《天体力学》）和阿德里安-马里·勒让德的Fonctions elliptiques Ⓣ（《椭圆函数》）。希尔于1859年从拉特格斯学院毕业，获得A.B.学位。次年，他开始研究夏尔-欧仁·德洛奈和Hansen的月球理论。他继续这项研究十二年，之后才发表自己的任何出版物。他于1862年由拉特格斯学院授予A.M.学位。

1861年，希尔加入位于马萨诸塞州剑桥的航海天文历办公室。同年，他凭借论文On the confrontation of the Earth在Runkle's Mathematical Monthly举办的竞赛中获得一等奖。在马萨诸塞州剑桥待了两年后，他回到西奈阿克，在家中工作。这适合喜欢独处的隐士希尔。除了1882年至1892年这十年间他在华盛顿从事木星和土星轨道的理论与表格工作外，这成为他余生的工作模式。

在[4]中，欧内斯特·威廉·布朗写道：-

他基本上属于那种似乎不需要与他人进行个人接触的学者和研究者。虽然少数认识他的人谈到，在频繁穿越华盛顿周边乡间的徒步旅行中与他为伴的乐趣，但无论在工作还是休闲时，他显然都相当乐于独处。

加入航海天文历办公室后不久，他便开始了对金星凌日的研究，并在此早期阶段掌握了夏尔-欧仁·德洛奈在其两卷本专著Théorie du mouvement de la luneⓉ（《月球运动理论》）中阐述的方法。希尔是第一个使用无穷行列式来研究On the part of the motion of the lunar perigee which is a function of the mean motion of the sun and moon中月球轨道的人。他于1877年自费出版了这部著作（1886年在Acta Mathematica中重印）。他的Researches in Lunar Theory于1878年出现在新创办的American Journal of Mathematics上。该出版物包含关于三体问题的重要新思想。他还引入了无穷行列式和其他方法，以提高其结果的精度。欧内斯特·威廉·布朗在1915年写道，希尔的论文Researches in Lunar Theory：-

……虽然只有五十个四开页，却已成为天体力学向三个不同方向发展的基础。……儒勒·昂利·庞加莱的评论说，从中我们可以看出自其发表以来天体力学所取得的一切进展的萌芽，这无疑是完全正确的。

西蒙·纽康说服希尔发展一套木星和土星轨道的理论，而希尔关于这一主题的工作是对数学天文学的又一重大贡献。希尔最重要的工作涉及行星对月球轨道的引力效应，因此在这项工作中他考虑的是四体问题。他在Annals of Mathematics上发表的论文例子包括：On the lunar inequalities produced by the motion of the ecliptic（1884）、Coplanar motion of two planets, one having a zero mass（1887）、On differential equations with periodic integrals（1887）（这些微分方程现在被称为希尔微分方程）、On the interior constitution of the earth as respects density（1888）、The secular perturbations of two planets moving in the same plane; with application to Jupiter and Saturn（1890）、On intermediate orbits（1893）、Literal expression for the motion of the Moon's perigee（1894）和Application of Chebyshev's principle in the projection of maps（1908）。他还在1900年的Transactions of the American Mathematical Society上发表了On the extension of Delaunay's method in the lunar theory to the general problem of planetary motion。

尽管他必须被视为一位数学家，但他的数学完全基于解决其轨道问题所必需的内容。他关于如何求解三体问题的新思想涉及求解限制性问题。在此，他假设三个天体位于同一平面内，两个天体绕其共同质心运行，而第三个天体绕另外两个天体运行，但假设其质量可忽略不计，因此不影响这两个（假定有质量的）天体的轨道。他对数学其他领域中与求解轨道问题无关的任何现代发展都不感兴趣。事实上，希尔研究与约翰·柯西·亚当斯非常相似的问题。约翰·柯西·亚当斯也被引向研究无穷行列式，这并非巧合，而他做这项工作完全独立于希尔。

从1898年到1901年，希尔在哥伦比亚大学授课，但[4]：-

……他颇具特色地退还了薪水，写道他不需要这笔钱，而且照看它让他感到麻烦。

Zund 在12中写道：-

希尔终身未婚，一生大部分时间都在西奈亚克的家族农场度过。他的收入微薄，但需求也同样微薄，而且他的几个兄弟就住在附近。尽管他实际上过着隐居生活，但他在那里的大型藏书中最感快乐，可以自由地进行研究。他在西奈亚克的家中去世。

希尔于1874年当选为国家科学院（美国）院士。他于1902年当选为皇家学会会士，并于1909年获得该学会的科普利奖章。他当选为皇家天文学会院士，该学会于1887年授予他金质奖章；奖章由时任该学会会长的詹姆斯·惠特布雷德·李·格莱舍颁发。希尔于1894年至1896年担任美国数学会会长，并发表了关于Remarks on the progress of celestial mechanics since the middle of the century的会长致辞。他当选为法国科学院院士，该学会于1898年授予他达穆瓦索奖；他还当选为俄罗斯科学院院士，该学会于1905年授予他舒伯特奖。太平洋天文学会于1908年授予他布鲁斯奖章。他还于1908年当选为爱丁堡皇家学会院士，1909年当选为比利时皇家科学院院士，1910年当选为挪威科学院院士，1913年当选为瑞典皇家科学院院士，并于1913年当选为罗马的Accademia dei Lincei院士。

George Hill's mother was Catherine Smith and his father was John William Hill who was an artist specialising in engraving. In 1846, when Hill was eight years old, his family moved to a farm near West Nyack, in New York, where George attended school. He received a somewhat basic education at secondary school, yet he shone in the mathematics classes. After graduating from high school he entered Rutgers College in 1855. It had been an institution of higher education from 1766 and had been renamed Rutgers College thirty years before Hill studied there. Hill had graduated just before the Morrill Act of 1862 led to Rutgers College becoming New Jersey's land-grant college in 1864.

Certainly Hill's undergraduate studies at Rutgers College were highly unusual for American students of this period. He was taught at the College by a very able mathematician named Theodore Strong who was a friend of Bowditch. Strong had a fine library of classic mathematics texts which he was pleased to allow the talented undergraduate Hill to study. These texts included Lacroix' Traité du calcul différentiel et intégral Ⓣ, Lagrange's Mécanique analytique Ⓣ, Laplace's Mécanique céleste Ⓣ and Legendre's Fonctions elliptiques Ⓣ. Hill graduated from Rutgers College in 1859 with his A.B. The following year he began his study of the lunar theory of Delaunay and Hansen. He was to continue this study for twelve years before he produced any publications of his own. He was awarded his A.M. by Rutgers in 1862.

In 1861 Hill joined the Nautical Almanac Office working in Cambridge, Massachusetts. In the same year he won first prize for his essay On the confrontation of the Earth in a competition which was run by Runkle's Mathematical Monthly. After two years in Cambridge, Massachusetts he returned to West Nyack where he worked from his home. This suited the reclusive Hill who prefered being on his own. Except for a period of 10 years from 1882 to 1892 when he worked in Washington on the theory and tables for the orbits of Jupiter and Saturn, this was to be the working pattern for the rest of his life.

Writing in [4], E W Brown says:-

He was essentially of the type of scholar and investigator who seems to feel no need of personal contacts with others. While the few who knew him speak of the pleasure of his companionship in frequent tramps over the country surrounding Washington, he was apparently quite happy alone, whether at work or taking recreation.

He began his studies of the transits of Venus soon after joining the Nautical Almanac Office and at this early stage he mastered the methods set out by Delaunay in his two volume treatise Théorie du mouvement de la lune Ⓣ. Hill was the first to use infinite determinants to study the orbit of the Moon in On the part of the motion of the lunar perigee which is a function of the mean motion of the sun and moon. He published this work at his own expense in 1877 (it was reprinted in Acta Mathematica in 1886). His Researches in Lunar Theory appeared in 1878 in the newly founded American Journal of Mathematics. This publication contains important new ideas on the three-body problem. He also introduced infinite determinants and other methods to give increased accuracy to his results. Brown wrote in 1915 that Hill's memoir Researches in Lunar Theory :-

... of but fifty quarto pages has become fundamental for the development of celestial mechanics in three different directions. ... Poincaré's remark that in it we may perceive the germ of all progress which has been made in celestial mechanics since its publication is doubtless fully justified.

Newcomb persuaded Hill to develop a theory of the orbits of Jupiter and Saturn and Hill's work on this topic is another major contribution to mathematical astronomy. Hill's most important work dealt with the gravitational effects of the planets on the Moon's orbit so in this work he was considering the 4-body problem. Examples of papers he published in the Annals of Mathematics include: On the lunar inequalities produced by the motion of the ecliptic (1884), Coplanar motion of two planets, one having a zero mass (1887), On differential equations with periodic integrals (1887) (these differential equations are now called Hill's differential equation), On the interior constitution of the earth as respects density (1888), The secular perturbations of two planets moving in the same plane; with application to Jupiter and Saturn (1890), On intermediate orbits (1893), Literal expression for the motion of the Moon's perigee (1894) and Application of Chebyshev's principle in the projection of maps (1908). He also published On the extension of Delaunay's method in the lunar theory to the general problem of planetary motion in the Transactions of the American Mathematical Society in 1900.

Although he must be considered a mathematician, his mathematics was entirely based on that necessary to solve his orbits problems. His new idea on how to approach the solution to the three body problem involved solving the restricted problem. In this he assumed that the three bodies lie in the same plane, that two bodies orbited their common centre of mass, and that the third body orbited the other two but it was assumed to be of negligible mass so did not affect the orbits of the two (assumed massive) bodies. He had no interest in any modern developments in other areas of mathematics which did not relate to solving problems about orbits. In fact Hill worked on very similar problems to Adams. It is no coincidence that Adams was also led to investigate infinite determinants and he did this work quite independently of Hill.

From 1898 until 1901 Hill lectured at Columbia University, but [4]:-

... characteristically returned the salary, writing that he did not need the money and that it bothered him to look after it.

Zund writes in [12]:-

Hill never married and spent most of his life on the family farm at West Nyack. His income was small, but so were his needs, and several of his brothers lived nearby. Although a virtual recluse, he was happiest there among his large library, where he was free to pursue his research. He died at his home in West Nyack.

Hill was elected to the National Academy of Sciences (United States) in 1874. He was elected a Fellow of the Royal Society (1902) receiving its Copley Medal in 1909. He was elected to the Royal Astronomical Society and they awarded him their Gold Medal in 1887; it was presented by Glaisher who was president of the Society. Hill was president of the American Mathematical Society from 1894 to 1896 delivering his presidential address on Remarks on the progress of celestial mechanics since the middle of the century. He was elected to the French Academy of Sciences who awarded him their Damoiseau Prize in 1898, and to the Russian Academy of Sciences who presented him with their Schubert Prize (1905). The Astronomical Society of the Pacific awarded him their Bruce Medal in 1908. He was also elected to the Royal Society of Edinburgh in 1908, the Royal Belgium Academy of Science (1909), the Norwegian Academy of Science (1910), the Royal Swedish Academy of Sciences (1913), and the Accademia dei Lincei in Rome (1913).