约斯特·伯基Jost Bürgi

约斯特·伯基的名字有时写作伯基、Jobst或Justus，而他的第二个名字有时以拉丁形式Byrgius书写。他出生在利希滕施泰格，当时是一个约有400名居民的小村庄。他的祖父Lienhard 伯基是一名锁匠，也是这个因新教改革而分裂的村庄的主要官员。伯基家族在一个新教徒和罗马天主教徒平分秋色的村庄里是新教徒。

据推测，伯基决定离开利希滕施泰格，部分原因是宗教分歧，部分原因是这个小镇缺乏教育机会。在离开之前，他在小学获得了阅读和算术知识，但没有机会超越这一基础阶段。从他在随后几年获得的技能可以清楚地看出，伯基一定当过铁匠、仪器制造者和钟表匠的学徒，但关于他在哪些城镇当学徒没有任何资料。当时钟表制造技能的主要城市是纽伦堡、奥格斯堡和克雷莫纳，因此，鉴于他高超的工艺水平，可以猜测他一定在这些城镇中的一个或多个工作过。我们能够更确定地说，他在1570-74年期间曾在斯特拉斯堡待过一段时间，当时Josia和Isaac Habrecht正在为斯特拉斯堡大教堂建造一座天文钟。瑞士数学家Konrad Dasypodius是斯特拉斯堡大学的数学教授，设计了这座天文钟，伯基很可能从Dasypodius或其学生那里获得了数学专业知识。尽管伯基从未学过拉丁语（当时科学语言），但他在数学和天文学方面知识渊博，其技能与曾融入斯特拉斯堡科学界相符，但肯定不是作为大学课程的一部分。

此时奥托·黑塞-卡塞尔的伯爵是威廉四世，他是一位杰出的数学家和天文学家，曾与斯特拉斯堡保持联系，并在那里接受了科学训练。威廉当然知道伯基是他那个时代最技艺精湛的仪器制造者，因为他从1579年起就雇用了他。1579年7月25日，黑塞-卡塞尔的伯爵威廉四世要求伯基成为卡塞尔宫廷的钟表匠，以研制科学仪器，并协助观测恒星，以证实尼古拉·哥白尼所描述的日心模型。正如我们提到的，这位伯爵受过天文学训练，人们往往没有意识到他是一位非凡的天文学家，其观测，尤其是对恒星的观测，总体上至少与第谷·布拉赫的观测一样精确。在卡塞尔，伯爵建造了一座天文台，这是最早专门为天文观测而建造的建筑之一。作为其职责的一部分，伯基制作了六分仪、天球仪和高度精确的时钟，供这座天文台使用。伯爵还于1584年任命数学家Christoph Rothmann到天文台工作；他在那里工作了六年。伯爵于1586年4月14日写信给第谷·布拉赫，告诉他伯基制造了一台高度精确的时钟，这台时钟首次带有分针、能记录秒，并且在24小时内的误差不到一分钟。Christoph Rothman写到了这台非凡的新时钟[14]：-

一秒的持续时间并不很短，而是类似于一首中等缓慢歌曲中最短音符的长度。摆轮不像当前的模型，而是以这样一种方式发明的：它的每一次摆动代表一秒。

钟及其表盘的图片在THIS LINK。

伯基的一个天球仪的图片见THIS LINK。

伯基的钟有一个创新的交叉擒纵机构，在传统的轮系上增加了一个独立系统，使擒纵机构受到的压力更加恒定，从而提高了精度。这是第一次有钟表足够精确，可用于天文学，通过计时恒星穿过望远镜瞄准器来计算恒星的相对位置。Landgraf对伯基的能力感到高兴，并在给第谷·布拉赫的信中描述他道：-

……具有第二个阿基米德的创新才能。

1590年Rothmann离开卡塞尔天文台后，伯基成为宫廷的官方数学家和天文学家。后来威理博·斯涅尔在[14]中描述伯基道：-

……一个非凡的人物，同时是一位杰出的钟表匠、称职的天文学家和优秀的数学家——这是钟表制造史上独一无二的组合。

1591 年，伯基 在卡塞尔市入籍。他娶了 弗洛伦斯·南丁格尔·大卫 Bramer 的女儿，Bramer 是卡塞尔附近 Felsberg 的牧师，但这段婚姻没有子女。1591 年，大卫 Bramer 去世，伯基 收养了他妻子的弟弟，即 大卫 Bramer 三岁的儿子 Benjamin Bramer。伯基 后来教这个男孩数学和天文学，水平之高使他成为当时最杰出的科学家之一。1591 年对 伯基 来说还有另一层重要意义，因为就在那一年，他完成了基于 尼古拉·哥白尼 日心体系的天文钟；在教会正打压任何持日心说观点之人的时代，这是相当大胆的举动。人们有理由问，不懂拉丁语的 伯基 怎么可能知道哥白尼体系的细节；这里我们可以给出明确答案，因为 Nicolaus Reimers 于 1586—1587 年在卡塞尔天文台工作，并把 尼古拉·哥白尼 的 De revolutionibus orbium coelestium 从拉丁文翻译成德文，供 伯基 研究。这份译本的一个副本，称为“Grazer Handschrift”，一直保存至今。到 伯基 制作他的天文钟时，他已经在使用自己版本的对数，那是他为便于天文计算而自创的。目前尚不清楚他究竟何时开始使用对数，但大多数历史学家认为他大约在 1588 年发明了它们。更早的时候，伯基 就在使用三角公式

$\sin a \sin b = \large\frac{1}{2}\normalsize [\cos(a - b) - \cos(a + b)]$

和

$\cos a \cos b = \large\frac{1}{2}\normalsize [\cos(a - b) + \cos(a + b)]$

为了帮助进行数的乘法运算。为了使用这些公式，伯基计算了称为Canon Sinuum的正弦表，这些表从未出版，如今遗憾的是似乎已经失传。

1592年2月，神圣罗马帝国皇帝鲁道夫二世要求伯基亲自交付一个机械天文地球仪。7月4日，伯基在布拉格觐见了鲁道夫，交付了地球仪。他返回卡塞尔后不久，黑塞-卡塞尔的领主威廉四世于1592年8月去世，由其子莫里茨继位。这对伯基影响不大，因为他从莫里茨那里获得的雇佣条件与其父相似。同样在1592年，伯基为其基于三角测量法的测量仪器获得了专利。1596年以及1604年，伯基再次前往布拉格。第谷·布拉赫已于1599年在布拉格被任命为神圣罗马帝国皇帝鲁道夫二世的皇家数学家，约翰内斯·开普勒为其助手。第谷·布拉赫于1601年去世，约翰内斯·开普勒成为布拉格的皇家数学家。经莫里茨同意，伯基于1604年12月被神圣罗马帝国皇帝鲁道夫二世任命，并迁往布拉格。他在布拉格城堡的赫拉德恰尼获得了一个工作坊，配有两名助手，还与约翰内斯·开普勒合作，后者因伯基引介代数而受其恩惠。有强有力的证据表明，约翰内斯·开普勒关于行星运动第三定律的想法源于对对数的思考，而对数之所以成为赫拉德恰尼的常见话题，必定是通过与伯基的讨论。神圣罗马帝国皇帝鲁道夫二世于1612年去世，伯基继续在布拉格为其继任者马蒂亚斯工作。约翰内斯·开普勒说服伯基写下他关于对数的原创而有趣的工作（手稿大部分是约翰内斯·开普勒的笔迹），并于1620年印刷出版。伯基的方法与约翰·纳皮尔的不同，显然是独立发明的。约翰内斯·开普勒在其Rudolphine Tables （1627年）的引言中写到了伯基的对数：-

……作为计算的辅助手段，尤斯图斯·比尔吉在约翰·纳皮尔的体系出现许多年前就被引向了这些对数；但由于他生性懒散，又很不善交流，他没有为自己的孩子谋取公众利益，而是在其出生时就将其遗弃了。

伯基的第一任妻子去世，他于1611年与卡塔琳娜·布劳恩结婚；他们没有孩子。在伯基于布拉格工作的那些年里，他曾多次回访卡塞尔。1631年他回到卡塞尔，并于次年1月在那里去世。他的坟墓已不复存在，但墓地里立了一块牌匾，以纪念他葬于此地。你可以在THIS LINK看到它。
上面写着：

在这片墓地中安葬着
黑塞的领主和
皇帝的钟表匠与数学家
伯基
1552年2月28日生于瑞士利希滕施泰格
1632年1月31日卒于卡塞尔
测量仪器和天文地球仪的天才设计者，16世纪最精密时钟的制造者，
对数的发明者。

Jost Bürgi's first name is sometime written as Joost, Jobst or Justus while his second name is sometime written in a Latin form Byrgius. He was born in Lichtensteig, a small village of around 400 inhabitants at this time. His grandfather, Lienhard Bürgi, was a locksmith and a leading official of the village which had been divided by the Protestant Reformation. The Bürgi family was Protestant in a village which was equally divided between Protestants and Roman Catholics.

It is conjectured that Jost Bürgi decided to leave Lichtensteig, partly because of the religious divide and partly because of the lack of educational opportunities in the small town. Before leaving he had acquired a knowledge of reading and numeracy at elementary school but had not had the opportunity to progress beyond this basic stage. It is clear from the skills that he acquired over the floowing few years, that Bürgi must have served an apprenticeship to a blacksmith, instrument maker and watchmaker but no knowledge exists of the towns in which he served his apprenticeship. The leading cities for watchmaking skills at this time were Nuremberg, Augsburg and Cremona so, given the high level of his craftsmanship, one can guess that he must have worked in one or more of these towns. We are able to say with more certainty that he was in Strasbourg for some time during 1570-74 when Josia and Isaac Habrecht were building an astronomical clock for Strasbourg cathedral. The Swiss mathematician Konrad Dasypodius, who was the professor of mathematics at the University of Strasbourg, had designed the astronomical clock and is likely that Bürgi acquired his expertise in mathematics from Dasypodius or his pupils. Although Bürgi never learnt Latin (the language of science at this time), he was very knowledgeable in mathematics and astronomy with skills compatible with having been immersed in the scientific circle in Strasbourg, but certainly not as part of a university course.

The Landgraf of Hesse-Kassel at this time was Wilhelm IV, an excellent mathematician and astronomer, who had maintained connections with Strasbourg where he had undergone his scientific training. Certainly Wilhelm was aware that Bürgi was the most skilful instrument maker of his day for he employed him from 1579. On 25 July 1579 the Landgraf of Hesse-Kassel Wilhelm IV asked Bürgi to become a watchmaker to the court in Kassel, to develop scientific instruments, and assist in the observation of stars which would confirm the heliocentric model described by Copernicus. The Landgraf, as we have mentioned, was trained in astronomy and it is often not realised that he was an exceptional astronomer whose observations, particularly those of the fixed stars, were on the whole at least as accurate as those by Tycho Brahe. At Kassel, the Landgraf had built an Observatory, one of the first buildings constructed specifically for astronomical observations. As part of his duties Bürgi made sextants, celestial globes and highly accurate clocks for use in this Observatory. The Landgraf also appointed the mathematician Christoph Rothmann to work in the Observatory in 1584; he worked there for six years. The Landgraf wrote to Tycho Brahe on 14 April 1586 telling him about a highly accurate clock which Bürgi had built which, for the first time, had a minute hand, registered seconds and had an error of less than a minute in 24 hours. Christoph Rothman wrote about this remarkable new clock [14]:-

The duration of a second is not very short but resembles the length of the shortest note in a moderately slow song. The balance is not like the current models, but was invented in such a way that each of its beats represents one second.

A picture of the clock and of its dial is at THIS LINK.

A picture of one of Bürgi's celestial globes is at THIS LINK.

Bürgi's clock had an innovative cross-beat escapement with an independent system added to the traditional wheel-train to give a considerably more constant pressure to the escapement so leading to the greater accuracy. For the first time a clock was sufficiently accurate to be used in astronomy with relative positions of stars being calculated by timing their crossing of the sights of a telescope. The Landgraf was delighted with Bürgi's abilities and described him in a letter to Tycho Brahe as:-

... having the innovative capacity of a second Archimedes.

After Rothmann left the Kassel Observatory in 1590, Bürgi became official mathematician and astronomer to the Court. Later Willebrord Snell described Bürgi as [14]:-

... an extraordinary personality, at the same time a brilliant clockmaker, competent astronomer and excellent mathematician - a unique combination in the history of watchmaking.

In 1591 Bürgi became naturalised in the city of Kassel. He had married the daughter of David Bramer, who was a pastor in Felsberg near Kassel, but the marriage was childless. In 1591 David Bramer died and Bürgi adopted his wife's brother, Benjamin Bramer, David Bramer's three year old son. Bürgi went on to teach the boy mathematics and astronomy to such a high level that he became one of the leading scientists of his day. The year 1591 was significant for Bürgi in yet another way, for in that year he completed his astronomical clock which was based on Copernicus's heliocentric system, quite a bold move at a time when the Church was moving against anyone holding heliocentric views. One could reasonably ask how Bürgi, who knew no Latin, could have known the details of the Copernican system and here we can give a clear answer since Nicolaus Reimers worked at the Observatory in Kassel in 1586-87 and made a translation of Copernicus's De revolutionibus orbium coelestium from Latin to German for Bürgi to study. A copy of this translation, called the "Grazer Handschrift", has survived to this day. By the time Bürgi made his astronomical clock he was using his own version of logarithms, invented for his own use to aid him in his astronomical calculations. It is not clear precisely when he started using logarithms but most historians believe that he invented them around 1588. Even earlier, Bürgi was using the trigonometrical formulas

$\sin a \sin b = \large\frac{1}{2}\normalsize [\cos(a - b) - \cos(a + b)]$

and

$\cos a \cos b = \large\frac{1}{2}\normalsize [\cos(a - b) + \cos(a + b)]$

to aid in multiplying numbers. To use these formulas, Bürgi computed sine tables called the Canon Sinuum which were never published and now sadly appear to be lost.

In February 1592 the Holy Roman Emperor Rudolph II requested a mechanical astronomical globe from Bürgi to be delivered personally. On 4 July Bürgi had an audience with Rudolph in Prague, delivering the globe. Shortly after he returned to Kassel, the Landgraf of Hesse-Kassel Wilhelm IV died in August 1592 and was succeeded by his son Moritz. This had little affect on Bürgi since he had similar terms of employment from Moritz as he had from his father. Also in 1592 Bürgi obtained a patent for his surveying instrument based on the method of triangulation. In 1596, and again in 1604, Bürgi again travelled to Prague. Tycho Brahe had been appointed as Imperial Mathematician to the Holy Roman Emperor, Rudolph II, in Prague in 1599 with Johannes Kepler as his assistant. Tycho Brahe died in 1601 and Kepler became Imperial Mathematician in Prague. With Moritz's consent, Bürgi was appointed by the Holy Roman Emperor Rudolph II in December 1604 and moved to Prague. He was given a workshop at Hradcany, Prague Castle, with two assistants and also worked with Kepler who was indebted to Bürgi for his introduction to algebra. There is strong evidence that Kepler got the idea for his third law of planetary motion from thinking about logarithms, and it must have been through discussions with Bürgi that logarithms were a common topic at Hradcany. The Holy Roman Emperor Rudolph II died in 1612, and Bürgi continued to work for his successor Matthias in Prague. Kepler persuaded Bürgi to write up his original and interesting work on logarithms (the manuscript is largely in Kepler's handwriting), and it was printed in 1620. Bürgi's method is different from that of Napier and was clearly invented independently. Kepler wrote about Bürgi's logarithms in the introduction to his Rudolphine Tables (1627):-

... as aids to calculation Justus Byrgius was led to these very logarithms many years before Napier's system appeared; but being an indolent man, and very uncommunicative, instead of rearing up his child for the public benefit he deserted it at birth.

Bürgi's first wife died and he married Catharina Braun in 1611; they had no children. During the years that Bürgi worked in Prague, he made several visits back to Kassel. In 1631 he returned to Kassel where he died in January of the following year. His grave no longer exists but a plaque has been erected in the cemetery to commemorate his being buried there. You can see it at THIS LINK.
It reads:

In this cemetery lies buried
the Landgrave of Hesse's and
the Emperor's watchmaker and mathematician
Jost Bürgi
born February 28th , 1552 in Lichtensteig, Switzerland
died January 31st , 1632 in Kassel
ingenious designer of measuring instruments
and celestial globes, builder of the
most precise clocks of the 16th century,
inventor of the logarithms.