威理博·斯涅尔Willebrord Snellius

威理博·斯涅尔的名字出现为Snel或Snel van Royen。它也常被写作Willebrordus 斯涅尔，这是斯涅尔的拉丁语版本，他在所有出版物中都使用这个名字。他的父亲是Rudolph Snell（1546-1613），莱顿的数学教授，母亲是来自Oudewater一个显赫家族的Machteld Cornelisdochter。斯涅尔是父母三个孩子中的长子，以他的祖父命名。他的两个弟弟是Jacob（1599年去世，时年16岁）和Hendrik（幼年去世）。让我们对斯涅尔的出生日期作一评论。一些斯涅尔的传记将1591年作为他的出生年份，但这只是从旧传记中抄来的错误。其他传记将年份定为1580或1581，但声称他的出生日期未知。事实上，没有关于他出生的记录，但他的出生日期可以从他父亲在他儿子生日写的一封信中以相当程度的确定性推断出来。更多细节见[3]。

Rudolph Snell（1546-1613）虽然于1581年被任命为莱顿大学的特聘数学教授，但他是一位涉猎广泛的学者，数学技能并不十分高超。他的教学主要基于彼得吕斯·拉米斯的著作，尽管大学当局试图说服他多教欧几里得、少教彼得吕斯·拉米斯。他早年在中学教授希腊语、拉丁语、希伯来语和自由技艺，并研究过医学和亚里士多德的著作。除了大学工作外，鲁道夫斯还经营自己的私立学校，并在大学附近的家中寄宿了大量学生。正是在这座满是学生的房子里，斯涅尔长大。他的教育来自父亲，父亲教他拉丁语、希腊语和哲学。进入大学前他没有接受其他教育，但鉴于父亲经营学校，这并不稀奇。鲁道夫斯本希望儿子在大学学习法律，但斯涅尔热衷于数学，并继续住在家里，成为鲁道夫·范·科伊伦的私人学生。到1600年，斯涅尔在学习法律，并在教授不授课的日子在大学教数学。

从1600年起，他游历了欧洲多个国家，主要是讨论天文学。他访问了维尔茨堡的阿德里安·范·罗门，在那里他是医学教授和“教士团数学家”。在维尔茨堡待了一段时间后，这两位数学家前往布拉格，在那里斯涅尔由阿德里安·范·罗门介绍给了第谷·布拉赫。几年前，斯涅尔的父亲曾安排他在布拉格接受教育，作为交换，第谷·布拉赫的儿子将在莱顿接受教育，但这一安排从未付诸实践。斯涅尔与第谷·布拉赫共度了一段时间，协助他进行观测，并在这次访问中显然学到了很多。然而，1601年10月，当第谷·布拉赫去世时，这段宝贵的经历结束了。在这次访问期间，他结识了当时担任第谷·布拉赫助手的约翰内斯·开普勒。斯涅尔仍在阿德里安·范·罗门的陪伴下，继续拜访德国各城镇的数学家，如阿尔特多夫的Joannes Praetorius、图宾根的米夏埃尔·梅斯特林，以及赫斯费尔德的Wilhelm Hatzfeld和Christophorus Vulteius。1602年春天，他回到莱顿父亲的家，并开始准备手稿以供出版。1603年，他前往巴黎，继续他的法律学习，但也与许多数学家保持联系。这次访问后，他放弃了法律学习，余生大部分时间都在家乡莱顿度过。

尽管此时斯涅尔在莱顿大学没有正式职位，但他开始在那里授课，以帮助健康状况开始恶化的父亲。数年间，父子二人组成了一个在莱顿教授数学的优秀团队。斯涅尔在莱顿的职位慢慢变得更加正式。1609年，他被安排每周六早上8点授课。次年7月，又增加了每天下午的授课，到1612年，他因工作而获得额外报酬。在这个阶段，他得到承诺，当父亲退休时他将获得数学讲席，而斯涅尔利用教学负担较轻的机会，出版了几部著作的译本、评注和版本。这些包括对彼得吕斯·拉米斯著作的评注和版本，以及西蒙·斯蒂文和鲁道夫·范·科伊伦著作的译本。读者不会没有注意到，斯涅尔和他的父亲一样，热衷于彼得吕斯·拉米斯。这无疑是由于他父亲的影响，正如斯涅尔在1607年写给父亲的一封信中所见（例如见[3]）：-

因为您从我幼年起就激励我真正而充分地致力于学术，并在我犹豫时鞭策我前进，所以对我来说，没有什么比借助您的训诫作为规则和规范，更仔细地考察古人证明中的精确性、清晰性和简洁性更重要了，因为您的阿波罗[彼得吕斯·拉米斯]在其《数学导言》中坚决敦促这样做。

他在这段时间做出的其他主要贡献，展现出远比其父高超的数学才能，涉及复原阿波罗尼奥斯关于平面轨迹的著作，这些著作已经失传，但其内容概要由帕普斯保存下来。他以一个希腊文标题出版了两部这样的著作，该标题可译为The Revived Geometry of Cutting off of a Ratio and Cutting off of an Area（1607年）。他还出版了Apollonius BatavusⓉ（阿波罗尼奥斯著作集）（1608年），其中包含对阿波罗尼奥斯第三部著作的重构。斯涅尔此时对阿波罗尼奥斯所做的进一步工作从未出版，已经失传。

斯涅尔于1608年7月12日在莱顿获得文学硕士学位，此前他捍卫了关于自由艺术的论文：语法、修辞、逻辑、算术、几何、分析/代数、物理、光学、天文学、地理学、日晷学、静力学和伦理学。1608年8月，他与Maria de Langhe结婚，她是Janneke Symons和Laurens Adriaens de Langhe的女儿，后者是斯洪霍芬的市长。这对夫妇至少有七个孩子（他的葬礼演说中说他有个18个孩子似乎不太可能），其中只有三个活到成年。1613年2月8日，他接替父亲成为莱顿大学数学教授。安排是他应接管教学职责，因为父亲病重无法继续，但如果父亲康复，他必须辞职。由于Rudolph一个月后去世，斯涅尔被要求继续教学，但他努力争取莱顿大学的适当认可。1614年2月，他获得了更高的薪水，但仍只拿到其他教授薪水的$\large\frac{1}{3}\normalsize$到$\large\frac{1}{2}\normalsize$之间。1615年2月，他被任命为正式数学教授，但薪水没有增加。慢慢地，他获得了加薪，但直到1618年，他才收到他认为与其职位相称的适当金额。

1626年，46岁的斯涅尔死于绞痛，这引起了发烧以及手臂和腿的瘫痪。他的病持续了两周[3]：-

当斯涅尔病倒时，医生们……曾被咨询，但他们未能阻止他病情的恶化。10月30日晚上，[医生们]去拜访斯涅尔，看看一种新药的效果。这完全没有帮助，在给他一个栓剂以缓解症状后，他们离开了。斯涅尔与妻子共进晚餐。由于他无法行走，他的仆人不得不把他抬起来。然后他突然失去知觉并去世，享年46岁。他于11月4日被安葬在莱顿的彼得教堂。二十名学生抬着他的棺材。

现在让我们简要看看他被任命为莱顿教授后所做的贡献。1617年，他出版了Eratosthenes Batavus Ⓣ（埃拉托色尼著作集），其中包含他测量地球的方法。他提出了三角测量法，这项工作是大地的测量学的基础。Bowie写道[6]：-

斯涅尔……通过引入三角学方法测量越野距离，比他的前辈们使用的方法有了很大进步。他确实是三角测量法的创始人，这种方法现在是测量和绘制大片区域的普遍采用的方法。他于1617年在莱顿出版了一本书描述他的工作。在观测他的弧的三角形角度时，他使用了一个半径约两英尺的圆的象限仪。这被刻度到两分钟，读数估计到单分钟。

在这项工作中，斯涅尔试图测量地球的周长，因此需要大量的测量。为了进行这些测量，他不得不在荷兰广泛旅行，但把家人留在莱顿让他不快。他以从家到当地教堂尖顶的距离作为基线，然后建立了一个三角形系统，使他能够确定阿尔克马尔和贝亨奥普佐姆镇之间的距离，大约130公里。他选择了这些城镇，因为它们大致在同一条子午线上（现代数据给出阿尔克马尔东经4°45'0"，贝亨奥普佐姆东经4°18'0"）。他的测量惊人地准确，使他能够推导出地球半径的良好值。他将Eratosthenes Batavus Ⓣ（埃拉托色尼著作集）献给国会，这在财务上是一个好举动，因为作为回报，他们授予他一笔几乎相当于他年薪一半的款项。

在他的整个职业生涯中，斯涅尔对天文学感兴趣，并出版了多部关于该主题的著作，其中一些（但不是全部）包含来自他自己观测的数据。例如，他的Observationes Hassiacae Ⓣ（来自奥托·黑塞的观测）（1618）是一部他使用其他天文学家（包括第谷·布拉赫和约斯特·伯基）的观测数据写成的著作，而Descriptio Cometae Ⓣ（彗星描述）（1619）包含了他自己对1618年11月出现的彗星的观测。在后一部著作中，斯涅尔强烈批评了亚里士多德，并强调继续以如此崇敬的态度对待他的观点对科学的发展是多么有害。当然，这样做时，斯涅尔遵循了彼得吕斯·拉米斯和他自己父亲的教导。尽管他攻击了亚里士多德，斯涅尔并没有接受尼古拉·哥白尼的日心系统，而是坚定地相信地心系统。

斯涅尔还改进了用多边形计算$\pi$近似值的经典方法，他在Cyclometricus Ⓣ（测量圆）（1621）中发表了该方法。使用他的方法，96边多边形给出的$\pi$精确到7位，而经典方法只给出2位。鲁道夫·范·科伊伦的35位可以用$2^{30}$边的多边形而不是$2^{62}$找到。事实上，鲁道夫·范·科伊伦的$\pi$的35位首次出现在斯涅尔的这本书中。

尽管他在1621年发现了折射定律，现在称为“斯涅尔定律”，这是现代几何光学的基础，但他没有发表它，直到1703年克里斯蒂安·惠更斯在Dioptrica中发表了斯涅尔的结果时才为人所知。有一份斯涅尔的手稿，是他打算写的光学论文的提纲。它现在在阿姆斯特丹大学的图书馆里，被称为“阿姆斯特丹手稿”。Klaus Hentschel在[10]中试图构建导致斯涅尔发现斯涅尔定律的路径：-

由于他在大地测量工作中开创了三角测量方法，斯涅尔已经对三角函数有相当多的经验；阿姆斯特丹手稿中两处对测地线的提及表明了这一点，其中一处直接出现在折射定律之前……手稿中的其他评论表明，斯涅尔研究过现有的光学文献，特别是关于折射的文献；手稿中的各个段落，正如他的边注和贯穿手稿的偶尔对其他文本的引用所显示的，在表述和顺序上与其他论著相似。斯涅尔对海什木的experimentum elegans表现出特别的兴趣……他寻求对refractaria的几何描述……这一探索使他以正割形式发现了折射定律……

斯涅尔研究了斜驶线，即球面上与子午线成恒定夹角的路径。这出现在Tiphys batavusⓉ（Tiphys的著作（Typhus是阿尔戈英雄的舵手））中，该书出版于1624年，他在其中研究了航海。该著作分为两部分，一部分是理论数学，另一部分致力于实际应用。

Vollgraff在[16]中将斯涅尔置于其历史背景中：-

斯涅尔……是十七世纪初人物的一个突出例子。他心中没有与过去决裂的压倒性欲望。他既不是作为反对者也不是作为盲目崇拜者，而是作为一个有才智且具有批判性的门徒来思考自己的想法。像他的父亲Rudolph一样，他确实是彼得吕斯·拉米斯的追随者，而彼得吕斯·拉米斯尤其在其青年时期曾强烈反对亚里士多德，或者更确切地说是反对巴黎那些教授亚里士多德逻辑学的教授们；但没有证据表明他自己曾深受逻辑学困扰，因此他从未对亚里士多德表现出任何敌意。与彼得吕斯·拉米斯[和第谷·布拉赫]一样，他认为地球是宇宙的中心。在他关于光学的笔记中，他不加区分地引用柏拉图、亚里士多德、西塞罗、卢克莱修等。在斯涅尔的著作中，我们看到现代物理学从古代物理学中连续地发展出来。

Willebrord Snell's name appears as Snel or Snel van Royen. It is also commonly given as Willebrordus Snellius, the Latin version of Willebrord Snell, which he used for all his publications. His father was Rudolph Snell (1546-1613), the professor of mathematics at Leiden, and his mother was Machteld Cornelisdochter from an leading family from Oudewater. Willebrord, the eldest of his parents' three children, was named after his paternal grandfather. His two younger brothers were Jacob (who died in 1599 aged 16) and Hendrik (who died in childhood). Let us make a comment on Willebrord's date of birth. Some biographies of Snell give 1591 as the year of his birth but this is simply an error copied from an old biography. Others give the year as 1580 or 1581 but claim that his date of birth is unknown. In fact no record of his birth exists but his date of birth can be deduced with a fair degree of certainty from a letter his father wrote on his son's birthday. See [3] for further details.

Rudolph Snell (1546-1613), although appointed as an extraordinary professor of mathematics at the University of Leiden in 1581, was a broad scholar who did not have a great deal of mathematical skill. His teaching was based mainly on the work of Peter Ramus although the university authorities tried to persuade him to teach more Euclid and less Ramus. He had taught Greek, Latin, Hebrew and the liberal arts in a high school earlier in his career and he had studied medicine and Aristotle's works. In addition to his university work, Rudolphus ran his own private school and, in his house near the university, he boarded a large number of students. It is in this house, filled with students, in which Willebrord grew up. His schooling was from his father who taught him Latin, Greek and philosophy. He had no other education before entering university but since his father ran a school this is not remarkable. Rudolphus would have liked his son to study law at university but Willebrord was keen to study mathematics and, continuing to live at home, he became a private pupil of Ludolph Van Ceulen. By 1600 Snell was studying law and teaching mathematics at the university on days when the professor was not teaching.

From 1600 he travelled to various European countries, mostly discussing astronomy. He visited Adriaan van Roomen in Würzburg where he was professor of medicine and "Mathematician to the Chapter". After spending a while in Würzburg, the two mathematicians went to Prague where Snell was introduced to Tycho Brahe by van Roomen. Several years earlier Snell's father had arranged for him to be educated in Prague in an exchange with Brahe's son who would have been educated in Leiden but the arrangement was never put into practice. Snell spent some time with Brahe assisting him in making observations and clearly learnt much during this visit. The valuable experience came to an end in October 1601, however, when Brahe died. During this visit he got to know Johannes Kepler who was Brahe's assistant at the time. Snell, still in the company of van Roomen, continued visits to mathematicians in various German towns such as Joannes Praetorius in Altdorf, Michael Mästlin in Tübingen, and Wilhelm Hatzfeld and Christophorus Vulteius in Hersfeld. He returned to his father's home in Leiden in the spring of 1602 and began preparing manuscripts for publication. In 1603 he went to Paris where his studies of law continued but he also had many contacts with mathematicians. After this visit he gave up the study of law and spent most of the rest of his life in his hometown of Leiden.

Although Snell had no official position at the University of Leiden at this time he began teaching there to assist his father whose health was beginning to fail. For several years the two formed a good team teaching mathematics in Leiden. Slowly Snell's position at Leiden became more official. In 1609 he was assigned teaching at 8 o'clock every Saturday morning. In July of the following year daily afternoon teaching was added and by 1612 he was receiving additional payment for his work. At this stage he was promised the chair of mathematics when his father retired and Snell took the opportunity created by having a low teaching load to publish translations, commentaries and editions of several works. These included commentaries on, and editions of, works by Ramus as well as translations of works by Stevin and Van Ceulen. It will not have escaped the reader's notice that Snell, like his father, was keen on Ramus. This was undoubtedly due to his father's influence as seen by a letter Snell wrote to his father in 1607 (see for example [3]):-

Because you had stimulated me from my youth onwards to apply myself truly and fully to scholarship, and spurred me on when I hesitated, nothing was more important to me than to examine more attentively the exactness, clearness and brevity in the proofs of the ancients by means of your precepts, as rules and norms, because your Apollo [Ramus] resolutely urged to do so in his 'Prooemium Mathematicum'.

Other major contributions he made around this time, showing vastly more mathematical skills than his father, involve the restoration of books by Apollonius on plane loci which had been lost but an outline of their contents had been preserved by Pappus. He published two of these under a Greek title which may be translated as The Revived Geometry of Cutting off of a Ratio and Cutting off of an Area (1607). He also published Apollonius Batavus Ⓣ (1608) containing a reconstruction of a third work by Apollonius. Further work on Apollonius which Snell produced at this time was never published and has been lost.

Snell received the degree for Master of Arts from Leiden on 12 July 1608 after defending theses on artes liberales: grammatica, rhetorica, logica, arithmetica, geometria, analysis/algebra, physica, optica, astronomia, geographia, gnonomica, statica and ethica. He married Maria de Langhe, the daughter of Janneke Symons and Laurens Adriaens de Langhe, a burgomaster of Schoonhoven, in August 1608. The couple had at least seven children (the statement in his funeral oration that he had 18 children seems unlikely), only three of whom survived to adulthood. On 8 February 1613 he succeeded his father as professor of mathematics at the University of Leiden. The arrangement was that he should take over the teaching duties since his father was too ill to continue but, should his father recover, he had to stand down. Since Rudolph died a month later, Snell was required to continue teaching but he struggled to get proper recognition from the University of Leiden. He received a higher salary in February 1614 but was still getting between $\large\frac{1}{3}\normalsize$ and $\large\frac{1}{2}\normalsize$ of the salary of other professors. He was made a full professor of mathematics in February 1615 but his salary was not increased. Slowly he received increases but only in 1618 did he receive what he considered the proper amount for his position.

In 1626, at the age of 46, Snell died from colic which caused a fever and paralysis of his arms and legs. His illness lasted two weeks [3]:-

When Snell had fallen ill, the medical doctors ... had been consulted, but they had not been able to prevent the deterioration of his situation. In the evening of 30 October, [the doctors] went to visit Snell to see the effects of a new medicine. This had not helped at all and after giving him a suppository for some relief, they left. Snell had dinner with his wife. Because he was not able to walk, his servants had to lift him up. He then suddenly lost consciousness and died, 46 years old. He was buried on 4 November in the Pieterskerk in Leiden. Twenty students carried his coffin.

Let us now look briefly at the contributions he made after being appointed as a professor at Leiden. In 1617 he published Eratosthenes Batavus Ⓣ, which contains his methods for measuring the Earth. He proposed the method of triangulation and this work is the foundation of geodesy. Bowie writes [6]:-

Willebrord Snell ... made a great advance over the methods used by his predecessors by introducing trigonometrical methods in the measurement of distances across country. He was really the originator of triangulation, which is now the universally employed method in surveying and mapping large areas. He published a book in Leiden describing his work in 1617. In observing the angles of the triangles of his arc he used a quadrant of a circle of about two feet in radius. This was graduated to two minutes and readings were estimated to single minutes.

In this work Snell attempted to measure the circumference of the earth and so required a considerable number of measurements. To make these he had to travel quite widely in the Netherlands but leaving his family in Leiden caused him unhappiness. He used as a baseline the distance from his house to the local church spire, then built a system of triangles which allowed him to determine the distance between the towns of Alkmaar and Bergen-op-Zoom which is around 130 km. He chose these towns since they were approximately on the same meridian (modern data gives Alkmaar 4° 45' 0" East and Bergen-op-Zoom 4° 18' 0" East). His measurements were surprising accurate allowing him to deduce a good value for the radius of the earth. He dedicated Eratosthenes Batavus Ⓣ to the States General which was a good move financially since in return they awarded him a sum amounting to almost half his annual salary.

Throughout his career Snell was interested in astronomy and published several works on that topic some, but not all, of which contained data from his own observations. For example his Observationes Hassiacae Ⓣ (1618) was a work which he wrote using data from the observations of other astronomers including Tycho Brahe and Joost Bürgi, while Descriptio Cometae Ⓣ (1619) contains his own observations of the comet which appeared in November 1618. In this latter work Snell strongly criticised Aristotle and stressed how harmful to the development of science it was to continue to treat his views with such reverence. Of course, in so doing Snell was following the teaching of Ramus and of his own father. Despite his attack on Aristotle, Snell did not accept Copernicus's heliocentric system but firmly believed in an Earth centred system.

Snell also improved the classical method of calculating approximate values of $\pi$ by polygons which he published in Cyclometricus Ⓣ (1621). Using his method 96 sided polygons gives $\pi$ correct to 7 places while the classical method yields only 2 places. Van Ceulen's 35 places could be found with polygons of $2^{30}$ sides rather than $2^{62}$. In fact Van Ceulen's 35 places of $\pi$ appear in print for the first time in this book by Snell.

Although he discovered the law of refraction now known as "Snell's law" in 1621, a basis of modern geometric optics, he did not publish it and only in 1703 did it become known when Christiaan Huygens published Snell's result in Dioptrica. There is a manuscript by Snell's which is an outline of a treatise he intended to write on optics. It is now in the library of the University of Amsterdam and known as the 'Amsterdam manuscript'. Klaus Hentschel in [10] tries to construct the path which led Snell to discover Snell's law:-

Because of his geodetic work in which he pioneered the method of triangulation, Snell already had considerable experience with trigonometric functions; indicative of this are the two allusions to geodesics in the Amsterdam manuscript, one of them directly before the law of refraction .... Other remarks in the manuscript reveal that Snell had studied the existing literature on optics, particularly on refraction; various passages in the manuscript are analogous in formulation and sequence to other treatises as shown by his marginal notes and occasional references to other texts interspersed throughout the manuscript. Snell showed special interest in Ibn al-Haytham's experimentum elegans .... He sought a geometrical description of the refractaria .... This search led him to find the law of refraction in the secant form ....

Snell studied the loxodrome, the path on the sphere that makes constant angle with the meridians. This appears in Tiphys batavus Ⓣ published in 1624, a work in which he studied navigation. The work was in two parts, one a theoretical piece of mathematics, the other devoted to practical applications.

Vollgraff puts Snell into his historical context in [16]:-

Willebrord Snellius ... is a striking example of the early seventeenth century man. There is in his mind no overwhelming desire to break with the past. He is thinking his own thoughts neither as an oppositionist nor as a blind admirer, but as an intelligent and critical disciple. Like his father Rudolph he is, it is true, a follower of Ramus who was, especially in his youth, in sharp opposition to Aristotle or rather to the professors at Paris who taught Aristotelian logics; but there is no evidence of his having been much troubled with logics himself, so he shows nowhere any animosity against Aristotle. With Ramus [and Brahe] he takes the earth to be the centre of the universe. In his notes on optics he quotes indiscriminately Plato, Aristotle, Cicero, Lucretius, etc. In Snellius's work we see modern physics proceed from ancient physics with continuity.