鲁道夫·范·科伊伦Ludolph van Ceulen

鲁道夫·范·科伊伦的名字意为来自科隆的范·科伊伦，但实际上，他出生在希尔德斯海姆。他的母亲是Hester de Roode，而他的父亲Johannes 范·科伊伦是一名商人，但财力有限。范·科伊伦只能接受初等教育，当然，他没有接受过大学教育，因为他的父母没有足够的财富供他上大学。这使得他的数学学习更加困难，因为他不会读拉丁语或希腊语，所以不得不依靠朋友为他翻译重要文本。他担任过许多职位，不仅是数学教师，还是击剑教师。首先，让我们给出范·科伊伦在1578年之前的生平的一些细节。这类信息的唯一来源是Meursius [2]和范·科伊伦自己为其著作Vanden circkel Ⓣ（论圆）（1596年）所写的序言。根据Meursius [2]，范·科伊伦来自一个大家庭。在他的父亲Johannes去世后，范·科伊伦前往利沃尼亚，这是现今拉脱维亚和爱沙尼亚的一个地区。然后他前往安特卫普拜访他的兄弟Gert。之后他搬到了代尔夫特。第一份给出确切日期的档案材料是他女儿的洗礼证书，她于1578年5月4日出生在代尔夫特。范·科伊伦在序言中告诉我们，他于1569年去德国旅行，访问了科隆并在那里买了一本书。他还告诉我们，他在安特卫普学习数学，师从Iohan Pouwelsz。序言中的信息还告诉我们，范·科伊伦大约从1566年开始以教授数学为生。

荷兰对其西班牙统治者的强烈反抗，是在1567年前后西班牙在南方开始恐怖统治之后发生的。阿尔巴公爵，一位西班牙军人和政治家，在1567年至1573年担任荷兰总督期间因其暴政而臭名昭著。在民众起义之后，腓力二世于1567年8月派遣阿尔巴率领一支庞大的军队前往荷兰惩罚叛乱者。他的任务包括根除异端，并重建腓力二世的权威。范·科伊伦是荷兰加尔文教会的成员，因此像成千上万的其他人一样，遭受了西班牙试图镇压荷兰叛乱的影响。阿尔巴设立了一个新法庭，即“动乱委员会”（后来被称为“血腥委员会”），它无视国家法律，判处约12000人因支持叛乱而有罪。其中许多人没有等待被捕就逃离了该国。很可能此时范·科伊伦仍在安特卫普，那里已成为新教活动的中心，风险最大。当时许多安特卫普的新教居民逃往荷兰北部，而另一次大规模逃亡发生在1576年西班牙军队抢劫之后。很可能范·科伊伦是1576年逃往代尔夫特的大量人群中的一员。1579年1月23日的乌得勒支联盟旨在在低地国家的更大联盟内形成一个集团（称为三级会议），以抵抗西班牙统治。它在荷兰北部产生了一个联盟，名义上仍处于西班牙国王的统治之下，但与南方不同。北方主要是加尔文教徒，实际上由奥兰治亲王威廉统治。

范·科伊伦结过两次婚。他的第一任妻子是Mariken Jansen；他们有五个孩子，包括1578年5月出生的女儿。Mariken于1590年去世，范·科伊伦于同年6月17日再婚，娶了Adriana Simondochter，她是Bartholomew Cloot的遗孀。Cloot一家像许多其他人一样从安特卫普搬到了代尔夫特，由于Bartholomew Cloot是一名会计师和数学教师，可以合理推测这两家人在安特卫普时就已相识。一个表明两家关系密切的迹象是，Bartholomew Cloot在1578年5月见证了范·科伊伦女儿出生证明的签署。像Mariken 范·科伊伦一样，Bartholomew Cloot于1590年去世。Adriana与Bartholomew的婚姻中有八个孩子，所以当她嫁给范·科伊伦时，这对夫妇共有十三个孩子。

到达代尔夫特后，范·科伊伦在那里教授数学，然后于1580年5月13日向代尔夫特镇议会提交请求，允许他在镇上开设一所击剑学校。议会同意了他的请求，并告知他圣阿加塔修道院的教堂可供使用。要理解为什么范·科伊伦被提供一座教堂来举办他的击剑学校，必须明白代尔夫特主要是一个加尔文主义城市，许多以前属于罗马天主教会的建筑已被接管并改作其他用途。议会显然非常高兴范·科伊伦在该市开设击剑学校，因为除了提供建筑外，他们还授予他每年25荷兰盾的津贴。在代尔夫特期间，范·科伊伦卷入了一些数学争议。第一个是与来自哈勒姆的数学教师William Goudaan的争议。Goudaan提出了一个几何问题，范·科伊伦解决了，但他的解决方案未被Goudaan接受。当Goudaan发表自己的解决方案时，范·科伊伦意识到它是错误的。1584年，范·科伊伦发表了Solutie ende werckinghe op twee geometrische vraghen by Willem Goudaen inde jaeren 1580 ende 83 binnen Haerlem aenden kerckdeure ghestelt: mitsgaders propositie van twee andere geometrische vraghen Ⓣ（解决William Goudaen于1580年和83年在哈勒姆提出的两个几何问题及更正：连同另外两个几何问题），阐述了他对争议的看法。这一时期的第二个争议是与Simon van der Eycke的争议，他于1584年发表了圆求积的错误证明。范·科伊伦在两篇出版物中指出了van der Eycke的错误：Kort claar bewijs dat die nieuwe ghevonden proportie eens circkels iegens zyn diameter te groot is ende ouerzulcx de quadratura circuli des zeluen vinders onrecht zy Ⓣ（简短清晰的证明，表明新发现的圆与其直径的比例太大，因此这位发现者的圆求积不正确。）（1585年）和Proefsteen ende claerder wederleggingh dat het claarder bewijs (so dat ghenaempt is) op de gheroemde ervindingh vande quadrature des circkels een onrecht te kennen gheven, ende gheen waerachtich bewijs is: hier by gevoeght Een corte verclaringh aengaende het onverstant ende misbruyck inde reductie op simpel interest. Den ghemeenen volcke tot nut Ⓣ（试金石和清晰的反驳，表明所谓的著名圆求积发现的更清晰证明揭示了一个错误，并非真正的证明：在此添加关于简单论证中不明智和滥用的简短陈述。为了（普通）人民的利益。）（1586年）。直到此时，范·科伊伦还没有读过阿基米德关于使用96边的正多边形来表明$\large\frac{223}{71}\normalsize < \pi < \large\frac{22}{7}\normalsize$的工作。范·科伊伦有一个问题，因为他不懂希腊语，但代尔夫特市长、法学家、学者、政治家和外交家Hugo Grotius的父亲Jan Cornets de Groot为范·科伊伦翻译了阿基米德对π的近似值。这被证明是范·科伊伦一生中的一个重要点，因为他余生都在使用阿基米德的方法和许多边的正多边形来获得更好的π近似值。

范·科伊伦在1594年之前一直以代尔夫特为基地，他定期前往其他城镇，例如他在Vanden circkel的序言中告诉我们，他于1587年去了不来梅，1589年他在阿纳姆的Gelderse法院。1594年，他与家人搬到莱顿，在那里他再次教授数学和击剑。1594年6月9日，他向莱顿议会请求允许他在Catharina医院开设一所击剑学校。议会接受了他开设击剑学校的请求，但没有接受关于其地点的请求。他们向范·科伊伦提供了Faliedenbegijnkerk，因为像在代尔夫特一样，前罗马天主教教堂被改作不同用途。然而，许可信中包括一项条款，使范·科伊伦对因他或其学生造成的建筑损坏负责。没有其他人被允许在莱顿经营击剑学校，1602年，当他意识到他的助手Pieter Bailly在经营自己的学校时，范·科伊伦向议会投诉，议会强制关闭了Bailly的击剑学校。

莱顿的著名教授约瑟夫·斯卡利格于1594年出版了Cyclometrica elementa duoⓉ（《论圆测量的两个要素》）。在这部著作中，他声称π等于√10。范·科伊伦知道这是错误的——他已经有了π的精确值，而√10远远超出了他的界限。事实上，他知道√10甚至不在阿基米德的界限之内。然而，他现在遇到了一个问题，因为他觉得自己不能公开批评斯卡利格，鉴于后者的地位，而且他还面临一个问题：斯卡利格的书是用拉丁文写的，他看不懂（必定是朋友翻译了相关部分）。他联系了其他学者，向他们解释斯卡利格的错误，并希望他们能让斯卡利格看到自己错在哪里。然而，斯卡利格不会让一个击剑教练来教他如何做数学，于是挑战范·科伊伦将他的反对意见写下来。范·科伊伦从未这样做，也许是因为他觉得无法与大学里的著名教授进行公开争论，也许还因为他不会写拉丁文，这意味着他无法按通常的条件参与争论。

除了教授数学和击剑，范·科伊伦还在撰写他最著名的作品，即1596年出版的Vanden circkelⓉ（论圆）。在这本书中，他使用$15 \times 2^{31}$边的正多边形将π计算到20位小数。迪尔克·扬·斯特勒伊克写道[1]：-

《论圆》Ⓣ（Vanden circkel）由四个部分组成。第一部分包含π的计算。第二部分展示了如何计算任意边数的正多边形的边，用现代术语来说，这相当于用$\sin A$表示$\sin nA$（n为整数）。第三部分包含半径达$10^{7}$的正弦表（并非原创成就），第四部分有利息表。第一和第二部分最具原创性；它们不仅包含了当时达到的π的最佳近似值，还表明范·科伊伦在三角学方面与同时代的弗朗索瓦·韦达一样精通。1595年，两人竞争解决阿德里安·范·罗门在其《数学理念》（1593年）中提出的四十五次方程，并认识到它与用$\sin A$表示$\sin 45A$的关系。

范·科伊伦被三级会议任命到多个委员会。第一个是在1598年3月13日，他被任命到一个委员会，审议用于海上仪器的专利。同在委员会里的还有Joseph Scaliger、Rudolph Snell和西蒙·斯蒂文。他还曾在1598年6月26日成立的一个类似委员会中任职。然后在1599年，莱顿市要求他加入他们为研究税收和利息而设立的一个委员会。Joseph Scaliger是这个委员会的主席。

奥兰治的威廉于1584年7月10日在代尔夫特被一名罗马天主教徒暗杀，该教徒认为暗杀威廉将阻止对天主教西班牙的反叛。威廉的长子菲利普·威廉忠于西班牙，因此，奥兰治的威廉的次子毛里茨亲王于1584年被任命为荷兰和泽兰，即荷兰联合省的总督。毛里茨理解军事战略、战术和工程在军事成功中的重要性。1600年，他要求他的亲密顾问西蒙·斯蒂文在莱顿大学内建立一所工程学校。坚持在那里用荷兰语授课是一个很好的政治举措。1600年1月10日，范·科伊伦被任命到工程学校。学校设在Faliedenbegijnkerk，因此范·科伊伦可以在同一栋楼里教授击剑和数学。在他生命的最后十年里，他在工程学校教授算术、测量和防御工事。学校采用的教学方法是课程包括半小时的讲座，然后是半小时的辅导，学生可以向讲师提问。范·科伊伦在当时数学家中结交了几位朋友。特别是他与西蒙·斯蒂文和阿德里安·范·罗门的友谊对他的职业生涯很重要。他最著名的学生，在莱顿跟随他学习的是威理博·斯涅尔。威理博·斯涅尔将范·科伊伦的两部作品翻译成拉丁文，使它们更容易被全世界的数学界所接受。

范·科伊伦以用$2^{62}$边的多边形将π计算到35位小数而闻名。他在1596年的书中发表了π的20位小数，更精确的结果只在他去世后才发表。1615年，他的遗孀Adriana Simondochter出版了范·科伊伦的一部遗著，题为De arithmetische en geometrische fondamenten Ⓣ（论算术与基础几何）。其中包含了他对π的33位小数的计算。完整的35位小数近似值直到1621年才在威理博·斯涅尔的Cyclometricus Ⓣ（测量圆）中发表。他一生的大部分时间都用于计算这个近似值，因此π的35位小数被刻在范·科伊伦的墓碑上是很合适的。事实上，范·科伊伦于1602年11月11日在彼得教堂购买了一座坟墓，但在范·科伊伦于1610年12月31日去世后，他的遗孀Adriana将这座坟墓换成了另一座，仍在彼得教堂，而范·科伊伦于1611年1月2日被埋葬在这第二座坟墓中。墓碑上给出了范·科伊伦的下界3.14159265358979323846264338327950288和上界3.14159265358979323846264338327950289。然而，原来的墓碑在1800年左右消失了，两百年后被复制品取代。墓碑上的原始文字是已知的，因为它被记录在1712年的一本指南中，之后又在许多文章中被转载。Vajta写道[11]：-

2000年7月5日，在荷兰莱顿的圣彼得教堂举行了一场非常特别的仪式。由于原墓碑已消失，范·科伊伦原墓碑的复制品被安放在教堂中。……因此，当2000年7月5日星期三，威廉-亚历山大王子（王位继承人）在莱顿圣彼得教堂为纪念墓碑揭幕时，这是对范·科伊伦的纪念。

在德国，π长期以来被称为“鲁道夫数”。

Ludolph Van Ceulen's name means Ludolph from Cologne but, in fact, he was born in Hildesheim. His mother was Hester de Roode while his father, Johannes Van Ceulen, was a merchant but of limited financial means. Van Ceulen could receive no more than an elementary education and, certainly, he did not have a university education as his parents were not sufficiently wealthy to pay for one. This made his mathematical studies much harder since he could not read Latin or Greek so had to rely on friends to make translations of important texts for him. He held a number of posts not only as a teacher of mathematics but also as a fencing teacher. First let us give the few details of Van Ceulen's life prior to 1578. The only sources of such information are Meursius [2] and Van Ceulen's own Preface to his book Vanden circkel Ⓣ (1596). According Meursius [2], Van Ceulen came from a large family. After the death of his father Johannes, Van Ceulen travelled to Livonia, a region in present day Latvia and Estonia. Then he travelled to visit his brother Gert who lived in Antwerp. After this he moved to Delft. The first archival material giving us a definite date is the baptism certificate of his daughter who was born in Delft on 4 May 1578. Van Ceulen tells us in his Preface that he made a trip to Germany in 1569 when he visited Cologne and purchased a book there. He also tells us that he learnt mathematics in Antwerp, where he was taught by Iohan Pouwelsz. The information from the Preface also tells us that Van Ceulen earned his living teaching mathematics beginning around 1566.

The strong reaction in the Netherlands against their Spanish rulers followed the start of a reign of terror by the Spanish occupation in the south beginning around 1567. The Duke of Alba, a Spanish soldier and statesman, became notorious for his tyranny as governor-general of the Netherlands from 1567 to 1573. Following the popular uprisings, Philip II sent Alba to the Netherlands in August 1567 with a large army to punish the rebels. His tasks included rooting out heresy, and re-establishing Philip II's authority. Van Ceulen was a member of the Calvinist church of the Netherlands, so like thousands of others, suffered from the attempts of the Spanish to subdue rebellions in the Netherlands. Alba set up a new court, the Council of Troubles (later known as the Council of Blood) which ignored the law of the land and condemned around 12,000 people for their support of the rebellion. Many of these had not waited to be arrested but had fled the country. It is likely that at this time Van Ceulen was still in Antwerp which had became a centre of Protestant activity and was most at risk. Many of the Protestant inhabitants of Antwerp fled to the northern Netherlands at this time, while another major exodus occurred after looting by the Spanish armies in 1576. It is likely that Van Ceulen was among the large numbers who fled to Delft in 1576. The Union of Utrecht on 23 January 1579 was designed to form a block (known as the States-General) within the larger union of the Low Countries which would resist Spanish rule. It produced a union in the north Netherlands, still officially under the rule of the King of Spain, but distinct from the south. The north was predominantly Calvinist and effectively ruled by William, Prince of Orange.

Van Ceulen was married twice. His first wife was Mariken Jansen; they had five children including the daughter born in May 1578. Mariken died in 1590 and Van Ceulen remarried on 17 June of that year to Adriana Simondochter who was the widow of Bartholomew Cloot. The Cloots had, like many others, moved from Antwerp to Delft and since Bartholomew Cloot was an accountant and mathematics teacher, it is quite reasonable to guess that they the families had known each other since both lived in Antwerp. An indication of what close friends the families were is seen from the fact that Bartholomew Cloot was a witness to the signing of Van Ceulen's daughter's birth certificate in May 1578. Like Mariken Van Ceulen, Bartholomew Cloot died in 1590. Adriana had eight children from her marriage to Bartholomew so when she married Van Ceulen the couple had thirteen children.

After arriving in Delft, Van Ceulen taught mathematics there then, on 13 May 1580, he submitted a request to the Delft town council to be allowed to open a fencing school in the town. The Council agreed to his request and informed him that the Church of the St Aghata monastery was available. To understand why Van Ceulen was being offered a church in which to hold his fencing school, one has to understand that Delft was largely a Calvinist city and many of the buildings previously owned by the Roman Catholic Church had been taken over and put to other uses. The Council were obviously very pleased to have Van Ceulen open a fencing school in the city for, in addition to offering him the building, they awarded him an annual allowance of 25 guilders. During his time in Delft, Van Ceulen was involved in a number of mathematical disputes. The first was with William Goudaan, a mathematics teacher from Haarlem. Goudaan had posed a geometric problem which Van Ceulen solved but his solution was not accepted by Goudaan. When Goudaan published his own solution to the problem, Van Ceulen realised that it was incorrect. In 1584 Van Ceulen published Solutie ende werckinghe op twee geometrische vraghen by Willem Goudaen inde jaeren 1580 ende 83 binnen Haerlem aenden kerckdeure ghestelt: mitsgaders propositie van twee andere geometrische vraghen Ⓣ putting his side of the dispute. A second dispute from this period was with Simon van der Eycke who had published an incorrect proof of the quadrature of the circle in 1584. Van Ceulen showed van der Eycke's error in two publications: Kort claar bewijs dat die nieuwe ghevonden proportie eens circkels iegens zyn diameter te groot is ende ouerzulcx de quadratura circuli des zeluen vinders onrecht zy Ⓣ (1585) and Proefsteen ende claerder wederleggingh dat het claarder bewijs (so dat ghenaempt is) op de gheroemde ervindingh vande quadrature des circkels een onrecht te kennen gheven, ende gheen waerachtich bewijs is: hier by gevoeght Een corte verclaringh aengaende het onverstant ende misbruyck inde reductie op simpel interest. Den ghemeenen volcke tot nut Ⓣ (1586). Up to this time Van Ceulen had not read Archimedes' work on in which he had used a regular polygon of 96 sides to show that $\large\frac{223}{71}\normalsize < \pi < \large\frac{22}{7}\normalsize$. Van Ceulen had a problem since he could not read Greek, but Jan Cornets de Groot, the burgomaster of Delft and father of the jurist, scholar, statesman and diplomat, Hugo Grotius, translated Archimedes' approximation to π for Van Ceulen. This proved a significant point in Van Ceulen's life for he spent the rest of his life obtaining better approximations to π using Archimedes' method with regular polygons with many sides.

Although Van Ceulen was based in Delft until 1594, he made regular trips to other towns, for example he tells us in the Preface to Vanden circkel that he made a trip to Bremen in 1587 and in 1589 he was in Arnhem at the Gelderse court. In 1594 he moved with his family to Leiden where again he taught mathematics and fencing. On 9 June 1594 he made a request of the Leiden Council that he be given permission to open a fencing school in the Catharina Hospital. The Council accepted his request to open a fencing school, but did not accept his request regarding its location. They offered Van Ceulen the Faliedenbegijnkerk for, as in Delft, former Roman Catholic churches were being put to different uses. However, included in the letter of permission was a clause with made Van Ceulen responsible for any damage caused to the building by either him or his students. Nobody else was allowed to run a fencing school in Leiden and in 1602, when he realised that his assistant Pieter Bailly was running his own school, Van Ceulen complained to the Council who forced the closure of Bailly's fencing school.

The leading Leiden professor Joseph Scaliger published Cyclometrica elementa duo Ⓣ in 1594. In this work he claimed that π was equal to √10. Van Ceulen knew that was incorrect - he already had an accurate value of π with √10 well outside his bounds. In fact he knew that √10 did not even fall within Archimedes' bounds. However, he now had a problem since he felt he could not openly criticise Scaliger, given his position, and he also he had the problem that Scaliger's book was in Latin which he could not read (a friend must have translated the relevant parts). He approached other academics, explaining Scaliger's error to them, and hoped that they would let Scaliger see where he had gone wrong. However Scaliger was not going to be told how to do mathematics by a fencing instructor, and challenged Van Ceulen to put his objections in writing. Van Ceulen never did so, perhaps because he felt he could not enter a public dispute with a leading professor at the university, perhaps also because his inability to write Latin would have meant that he could not take part in the dispute on the usual terms.

In addition to teaching mathematics and fencing, Van Ceulen was writing his most famous work, the book Vanden circkel Ⓣ which he published in 1596. In this book he gave π correct to 20 decimal places using a regular polygon of $15 \times 2^{31}$ sides. Struik writes [1]:-

The 'Vanden circkel' Ⓣ consists of four sections. The first contains the computation of π. The second shows how to compute the sides of regular polygons of any number of sides, which in modern terms amounts to the expression of $\sin nA$ in terms of $\sin A$ (n an integer). The third section contains tables of sines up to a radius of $10^{7}$ (not an original achievement), and the fourth has tables of interest. The first and second sections are the most original; they contain not only the best approximation of π reached at that time but also shows Van Ceulen to be as expert in trigonometry as his contemporary Viète. In 1595 the two men competed in the solution of a forty-fifth degree equation proposed by van Roomen in his 'Ideae mathematicae' (1593) and recognised its relation to the expression of $\sin 45A$ in terms of $\sin A$.

Van Ceulen was appointed to a number of committees by the States-General. The first was on 13 March 1598 when he was appointed to a committee to consider applications for patents for instruments to be used at sea. Also on the committee were Joseph Scaliger, Rudolph Snell and Simon Stevin. He served on a similar committee set up on 26 June 1598. Then in 1599 the city of Leiden asked him to serve on a committee they had set up to study tax and interest. Joseph Scaliger was the chairman of this committee.

William of Orange was assassinated at Delft on 10 July 1584 by a Roman Catholic who believed that William's assassination would prevent a rebellion against Catholic Spain. William's eldest son Philip William was loyal to Spain so it was Prince Maurits, William of Orange's second son, who was appointed stadholder of Holland and Zeeland, or the United Provinces of the Netherlands, in 1584. Maurits understood the importance of military strategy, tactics, and engineering in military success. In 1600 he asked his close advisor, Simon Stevin, to set up an engineering school within the University of Leiden. It was a good political move to insist that the courses were taught there in the Dutch language. On 10 January 1600, Van Ceulen was appointed to the Engineering School. The School was set up in the Faliedenbegijnkerk so Van Ceulen could teach fencing and mathematics in the same building. For the last ten years of his life he taught arithmetic, surveying and fortification in the Engineering School. The method of teaching adopted by the School was for lessons consisting of a half-hour lecture followed by a half-hour tutorial during which students could ask the lecturer questions. Van Ceulen had several friends among the mathematicians of the time. In particular his friendships with Simon Stevin and Adriaan van Roomen were important for his career. His most famous student, who studied under him at Leiden, was Willebrord Snell. Snell translated two of Van Ceulen's works into Latin to make them more accessible to the world-wide mathematical community.

Van Ceulen is famed for his calculation of π to 35 places which he did using polygons with $2^{62}$ sides. Having published 20 places of π in his book of 1596, the more accurate results were only published after his death. In 1615 his widow Adriana Simondochter published a posthumous work by Van Ceulen entitled De arithmetische en geometrische fondamenten Ⓣ. This contained his computation of 33 decimal places for π. The complete 35 decimal place approximation was only published in 1621 in Snell's Cyclometricus Ⓣ. Having spent most of his life computing this approximation, it is fitting that the 35 places of π were engraved on Van Ceulen's tombstone. In fact Van Ceulen had purchased a grave in the Pieterskerk on 11 November 1602 but, after Van Ceulen's death on 31 December 1610, his widow Adriana exchanged this grave for another, still in the Pieterskerk, and it was in this second grave that Van Ceulen was buried on 2 January 1611. The tombstone gave both Van Ceulen's lower bound of 3.14159265358979323846264338327950288 and his upper bound of 3.14159265358979323846264338327950289. However, the original tombstone disappeared around 1800 to be replaced by a replica two hundred years later. The original text on the tombstone was known since it had been recorded in a guidebook of 1712 and after that reprinted in many articles. Vajta writes [11]:-

On July 5, 2000 a very special ceremony took place in the St Pieterskerk (St Peter's Church) at Leiden, the Netherlands. A replica of the original tombstone of Ludolph Van Ceulen was placed into the Church since the original disappeared. ... It was therefore a tribute to the memory of Ludolph Van Ceulen, when on Wednesday 5 July, 2000 prince Willem-Alexander (heir to the throne), unveiled the memorial tombstone in the St Peter's Church, in Leiden.

In Germany π was called the "Ludolphine number" for a long time.