约翰·缪勒Regiomontanus

约翰·缪勒出生于缪勒的柯尼斯堡。首先我们要注意，他出生地附近的柯尼斯堡镇并不是东普鲁士那个更有名的柯尼斯堡。柯尼斯堡的拉丁语版本（意为国王之山）是Regio Monte，或者后来变成的缪勒。事实上，在我们开始描述他一生的事件之前，我们应该再多说一点关于他所为人知的各种名字。他上大学注册时用的是Johannes Molitoris de Künigsperg，使用“Molitoris”，这是“缪勒”的拉丁语形式。其他变体包括Johannes Germanus（缪勒德国人）、Johannes Francus（来自弗兰肯的约翰内斯）、缪勒 von Künigsperg（来自柯尼斯堡的缪勒），以及听起来像法语的Joannes de Monte Regio，皮埃尔·伽桑狄在写他的传记时就是这样称呼他的。

他是一个磨坊主的儿子，很小的时候就以数学和天文学神童闻名。他早期在家接受教育，但十一岁就上了大学，从1447年到1450年在莱比锡大学学习辩证法。然后他于1450年4月14日进入维也纳大学，成为格奥尔格·冯·波伊尔巴赫的学生。吸引缪勒到维也纳的主要是这所已有85年历史的大学，尤其是它在数学天文学和宇宙学方面的活动。他于1452年1月16日获得学士学位，但大学规定必须年满21岁才能获得硕士学位。他在1457年达到规定年龄后获得了硕士学位。

1457年11月11日，他被任命到维也纳大学文学院，与他的前老师格奥尔格·冯·波伊尔巴赫合作。从格奥尔格·冯·波伊尔巴赫那里，他了解到《阿方索表》有多么不准确。这两位天文学家对火星进行了观测，显示该行星与预测位置相差2°，他们还观测到一次月食，发生时间比《阿方索表》预测的晚一小时。缪勒在维也纳讲授的课程包括1458年的一门透视学课程、1460年的一门欧几里得课程，以及1461年的一门维吉尔的Bucolics课程。他研究数学、天文学，并制作星盘等仪器。他特别喜欢阅读旧手稿，并为自己抄写了一些，其中一些至今仍存。

1450年，特拉比松的乔治翻译并评论了克劳狄乌斯·托勒密的AlmagestⓉ（主要论题：来自阿拉伯语“al-majisti”——希腊语“Mathematike Syntaxis”的阿拉伯语翻译，后来译为拉丁语“Magna Syntaxis”）。在这部作品中，他攻击了亚历山大的席恩所写的评注，并因此惹恼了红衣主教约翰内斯·贝萨里翁——驻神圣罗马帝国的教皇使节，他是亚历山大的席恩的狂热崇拜者。贝萨里翁红衣主教是一位学者，母语为希腊语，他的使命是在欧洲推广古典希腊著作。1460年5月5日，贝萨里翁和他的兄弟西吉斯蒙德抵达维也纳，进行外交访问，以争取对抵抗土耳其人的支持。贝萨里翁鼓励格奥尔格·冯·波伊尔巴赫制作克劳狄乌斯·托勒密的AlmagestⓉ（主要论题：来自阿拉伯语“al-majisti”——希腊语“Mathematike Syntaxis”的阿拉伯语翻译，后来译为拉丁语“Magna Syntaxis”）的节本。他有两个动机，一是希望有一个更容易理解的克劳狄乌斯·托勒密著作版本；二是支持亚历山大的席恩对抗特拉比松的乔治的攻击。当格奥尔格·冯·波伊尔巴赫在1461年临终时，他恳求缪勒完成Epitome of the Almagest，缪勒热情地继续了这项工作。The Defence of Theon against George of Trebizond是缪勒可能大约在这个时候开始考虑的另一部作品。

贝萨里翁枢机主教现在成为缪勒的赞助人，他随赞助人前往意大利，于1461年11月20日抵达罗马。在[31]中，弗洛伦斯·南丁格尔·大卫金和杰拉德·特纳描述了一组十一件星盘。他们写道：-

该组中的一件星盘具有特殊的历史意义，因为它于1462年在罗马被赠予红衣主教贝萨里翁，并附有奉献铭文。贝萨里翁自1463年起担任君士坦丁堡的拉丁礼名义牧首，也是十五世纪推动文学伟大复兴的杰出希腊学者之一。这件星盘是由Johannes Regiomontanus赠送的，其赞助人是贝萨里翁。当时的欧洲首屈一指的天文学家缪勒受这位红衣主教委托，准备一部克劳狄乌斯·托勒密的《天文学大成》Ⓣ的概要（主要论题：源自阿拉伯语“al-majisti”——即希腊语“Mathematike Syntaxis”的阿拉伯语译本，后来译为拉丁语“Magna Syntaxis”），该书也于1462年献给他。

1461年至1465年间，缪勒作为贝萨里翁大家庭的一员，主要居住在罗马。在此期间，他通过向以希腊语为母语的贝萨里翁学习，提高了语言知识，从而能够阅读其他重要的希腊手稿。他还继续从事Epitome of the Almagest的工作，并于1462年完成。除了在罗马的时间外，他还与贝萨里翁一起在意大利旅行，1462年夏天在维泰博度过，那是贝萨里翁红衣主教最喜欢的夏季住所；同年秋天贝萨里翁前往希腊时，缪勒陪同他一直到了威尼斯。1463年7月5日，贝萨里翁被任命为驻威尼斯共和国的教宗使节，他随即离开罗马。1464年春天，他在帕多瓦大学（位于威尼斯共和国）讲学，并在那里观测了1464年4月21日的月全食。他关于穆斯林科学家al-Farhani的讲座没有留存下来，但他的Introductory discourse on all the mathematical disciplines后来出版了。

他在1464年8月教皇庇护二世去世后返回罗马之前曾到访威尼斯。贝萨里翁此时必须返回参加教皇继任者的选举。匈牙利的皇家天文学家是奥尔库什的马丁·比利卡，他也前往罗马参加新教皇的选举。比利卡与缪勒此时成为朋友。1465年6月19日，缪勒在维泰博再次于夏季住所进行了观测。然而此后，关于他活动的记载有两年空白。在某个阶段，他收到国王马蒂亚斯一世·科尔维努斯的邀请后前往匈牙利，该邀请是在格兰大主教的推荐下发出的，但几乎可以肯定是由于马丁·比利卡的影响。我们确知，到1467年，缪勒已在匈牙利，接受了国王授予的布达皇家图书馆职位。国王刚刚完成了一场对土耳其人的极为成功（从他自己的角度看！）的战役，并带回了许多珍本书籍。这为缪勒提供了理想的职位，因为这使他既能与马丁·比利卡一起从事天文学工作，又能享受他对古书的热情。

缪勒在1462年于威尼斯期间偶然发现的一本古书，是丢番图的Arithmetica的不完整抄本。他在1464年2月11日写信给数学家乔瓦尼·比安基尼，说如果能找到完整抄本，他将翻译希腊文文本。正是在前往费拉拉的一次旅行中，他遇到了比安基尼，后者曾是格奥尔格·冯·波伊尔巴赫的朋友。缪勒从未翻译丢番图的Arithmetica，也从未找到完整版本。事实上，从未有人发现过完整版本，但缪勒的这一重要发现标志着Arithmetica开始为欧洲所知。

缪勒对三角学和天文学做出了重要贡献。我们在上文提到了Epitome of the Almagest，它由格奥尔格·冯·波伊尔巴赫开始，但由缪勒完成。它[1]：-

……对当时的科学研究有所贡献，而不是增进了对过去的理解。此外，《概要》并非仅仅是压缩的翻译……[因为]它增加了后来的观测、修订了计算，并加入了批判性思考——其中一项揭示出，克劳狄乌斯·托勒密的月球理论要求月球的视直径变化幅度远大于其实际变化。这段文字见于在威尼斯印刷的《概要》，引起了当时在博洛尼亚大学就读的学生尼古拉·哥白尼的注意。

在Epitome中，缪勒意识到需要一部系统的三角学论述来支持天文学，于是承诺撰写这样一部论著。他确实做到了，他的书De triangulis omnimodis（1464年）系统论述了解三角形的方法。在引言中他写道：-

你，希望研究伟大而奇妙的事物，对星辰的运动感到惊奇的人，必须阅读这些关于三角形的定理。……因为没有人能够绕过三角形科学而获得对星辰的满意知识。……新学生既不应害怕，也不应绝望。……当某条定理可能带来困难时，他总可以查阅数值例子以寻求帮助。

缪勒 以类似于 欧几里得 的 Elements. De triangulis 的方式组织了他的工作，该书分为五卷，第一卷给出了基本定义：量、比、相等、圆、弧、弦和正弦函数。然后他列出了他将假设的公理，接着是56个关于几何的定理。从第二卷开始，三角学的研究正式开始。正弦定理被陈述（用现代符号表示，缪勒 未使用，即 $\Large \frac {a}{\sin A}\normalsize = \Large \frac {b}{\sin B}\normalsize = \Large \frac {c}{\sin C}$），并用于解三角形。三角形的面积公式以两边和夹角表示出现，但并非完全以人们所期望的形式。第三、四、五卷讨论球面三角学，这在天文学中当然至关重要。

当他在匈牙利时，缪勒 计算了两张正弦表。第一张计算于1467年，是 Tables of directions，基于六十进制数，而次年他在布达计算了十进制基的正弦表。到1471年，他提议搬到纽伦堡，在1471年7月4日给朋友的信中写道：-

最近我在纽伦堡市进行了观测……因为我选择它作为我的永久住所，不仅因为仪器的可用性，特别是整个科学所基于的天文仪器，还因为与各地博学之士的各种交流非常便利，因为由于商人的旅行，这个地方被视为欧洲的中心。

我们知道 缪勒 确实在纽伦堡进行了观测，因为他在1471年6月2日在那里观测了一次月食。到11月29日，他获准在纽伦堡居住，在那里他建造了一个天文台和一个制造仪器的车间。他写了 Scipta，详细介绍了他的仪器，这些仪器包括日晷、象限仪、safea、星盘、浑天星盘、torquetum、视差尺和雅各布杆，在 [53] 中有描述。1472年1月，他使用雅各布杆观测了一颗彗星，这些观测足够准确，使其能够被识别为现在已知的 [55] C/1471 Y1（C = 非周期彗星）。

缪勒对月球运动的兴趣使他做出了重要观测，即月球距离法可用于确定海上经度。然而，许多年过去了，才能以足够的准确度预测月球的位置，使该方法实用，并建造出能以该方法所需的高准确度给出月球位置的仪器。缪勒描述了如何在他在1474-1506年出版的Ephemerides中使用月球位置来确定经度。这是在他于纽伦堡设立的自己的印刷机上印刷的。

欧洲书籍的首次印刷始于1454年，随着缪勒古腾堡发明活字印刷术。缪勒意识到印刷在制作科学文本的相同多份副本方面的潜在价值，这些副本可以经过精心编辑并配有准确的图表。1471-1472年，他在纽伦堡自己的家中设立了一台印刷机，并印刷了一份Prospectus，宣布了他出版许多精心编辑的数学、天文学和地理学文本的详细计划。他因此成为这类科学文献的第一位出版商，其中包括古代、中世纪和现代的作品。他的第一份出版物是他的前老师格奥尔格·冯·波伊尔巴赫的New theory of the planets，接着在1474年，是他自己的历书Kalendarium，以及我们上面提到的他的Ephemerides。这些书被多次重印并产生了巨大影响，例如克里斯托弗·哥伦布和亚美利哥·韦斯普奇都使用缪勒的Ephemerides来测量新世界的经度。

教皇西克斯图斯四世于1475年召缪勒到罗马，为历法改革提供建议，他在7月28日之后某个时间离开纽伦堡，当时他在那里记录了他的最后一次观测。他的富有学生伯恩哈德·瓦尔特，曾资助他的仪器店、天文台和印刷厂，于8月2日开始从缪勒在纽伦堡的天文台进行观测。可以肯定的是，缪勒那时已经离开纽伦堡。他在罗马去世，有些记载说他是被敌人毒死的，其他记载说他死于瘟疫。后者更有可能，但让我们简要看看这两种说法。

缪勒 曾宣布他将出版一部作品，展示 George of Trebizond 的作品毫无价值，其：-

……对“Syntaxis”的评论，他将以最清晰的方式展示其毫无价值，并且他对 克劳狄乌斯·托勒密 作品的翻译并非没有错误。

有人认为这足以构成乔治·特雷比松的两个儿子谋杀他的动机，并有传言如此流传。然而，这种情形似乎不太可能，因为其他人也曾像缪勒那样猛烈批评乔治·特雷比松，却未见有人企图谋害他们的性命。更可能的情况是，1476年1月台伯河泛滥后爆发了瘟疫，缪勒成为了受害者。

Regiomontanus was born Johann Müller of Königsberg. First let us note that the town of Königsberg near which he was born was not the more famous one in East Prussia. The Latin version of Königsberg (meaning King's mountain) is Regio Monte or, as it later became, Regiomontanus. In fact before we begin to describe the events of his life we should say a little more about the variety of names under which he was known. He matriculated at university as Johannes Molitoris de Künigsperg, using 'Molitoris' which is a Latin form of 'Müller'. Other variants included Johannes Germanus (Johann the German), Johannes Francus (Johannes from Franconia), Johann von Künigsperg (Johann from Königsberg), and the French sounding Joannes de Monte Regio which Gassendi called him when he wrote his biography.

He was the son of a miller, and became known as a mathematical and astronomical prodigy at a very early age. His early education was at home, but he entered university at the age of eleven, studying dialectics at the University of Leipzig from 1447 to 1450. He then entered the University of Vienna on 14 April 1450 where he became a pupil of Peurbach. What attracted Regiomontanus to Vienna was principally the 85-year old University and especially its activity in mathematical astronomy and cosmology. He was awarded a baccalaureate on 16 January 1452 but the University regulations required him to be 21 years of age before he could be awarded a Master's Degree. This was awarded once he reached the required age in 1457.

On 11 November 1457 he was appointed to the Arts Faculty of the University of Vienna where he collaborated with his former teacher Peurbach. From Peurbach he learnt how inaccurate the Alphonsine Tables were. The two astronomers made observations of Mars which showed the planet to be 2° from its predicted position, and they also observed an eclipse of the moon which happened one hour later than the Tables predicted. Courses Regiomontanus gave at Vienna included one on perspective in 1458, one on Euclid in 1460, and one on Virgil's Bucolics in 1461. He worked on mathematics, astronomy, and constructed instruments such as astrolabes. He was particularly interested in reading old manuscripts and he made copies for his own use, some of which still survive.

In 1450 George of Trebizond had translated and commented on Ptolemy's Almagest Ⓣ. In this work he had attacked the commentary written by Theon of Alexandria and, in so doing, he upset Cardinal Johannes Bessarion, papal legate to the Holy Roman Empire, who was a great admirer of Theon. Cardinal Bessarion was a scholar and native Greek speaker who had a mission to promote classical Greek works in Europe. On 5 May 1460 Bessarion arrived in Vienna, with his brother Sigismund, on a diplomatic visit to drum up support against the Turks. Bessarion encouraged Peurbach to produce an abridgment of Ptolemy's Almagest Ⓣ. He had two motives, one being a desire to have a more easily understandable version of Ptolemy's work available; the second being to give support to Theon of Alexandria against the attack from George of Trebizond. When Peurbach was on his deathbed in 1461, he begged Regiomontanus to complete the Epitome of the Almagest and Regiomontanus enthusiastically carried on the work. The Defence of Theon against George of Trebizond was another work which Regiomontanus probably began think about around this time.

Cardinal Bessarion now became Regiomontanus's patron and he travelled to Italy with his patron arriving in Rome on 20 November 1461. In [31] David King and Gerard Turner describe a group of eleven astrolabes. They write:-

One astrolabe in the group is of particular historical significance because it was presented at Rome in 1462, with a dedicatory inscription, to Cardinal Bessarion, titular Latin patriarch of Constantinople from 1463, and one of the illustrious Greek scholars who contributed to the great revival of letters in the fifteenth century. The astrolabe was presented by Johannes Regiomontanus, whose patron was Bessarion. Regiomontanus, the foremost European astronomer of the time, was commissioned by the Cardinal to prepare an Epitome of Ptolemy's "Almagest" Ⓣ, also dedicated to him in 1462.

The years 1461 to 1465 Regiomontanus spent as a member of Bessarion's extended household based mostly in Rome. During this time he was able to read other important Greek manuscripts after improving his knowledge of the language with instruction from the native Greek speaker Bessarion. He also continued to work on the Epitome of the Almagest which he completed in 1462. As well as the time spent in Rome, he travelled in Italy with Bessarion spending the summer of 1462 at Viterbo, Cardinal Bessarion's favourite summer residence, and, when Bessarion left for Greece in the autumn of that year, Regiomontanus accompanied him as far as Venice. He left Rome on 5 July 1463 when Bessarion was appointed as papal legate to the Venetian Republic. In the spring of 1464 he lectured at the University of Padua (in the Venetian Republic) and while there he observed the total eclipse of the moon on 21 April 1464. His lectures on the Muslim scientist al-Farhani have not survived, but his Introductory discourse on all the mathematical disciplines was later published.

He visited Venice before returning to Rome in August 1464 after the death of the Pope Pius II. Bessarion had to return at this time to take part in the election of the pope's successor. The astronomer royal for Hungary was Martin Bylica of Olkusz and he had also gone to Rome for the election of the new pope. Bylica and Regiomontanus became friends at this time. On 19 June 1465 Regiomontanus made an observation at Viterbo, again in the summer residence. After this, however, there is a two year gap in our knowledge of his activities. At some stage he travelled to Hungary after receiving an invitation from King Matthias I Corvinus which was sent on the recommendation of the Archbishop of Gran, but was almost certainly due to influence by Martin Bylica. Certainly we know that by 1467 Regiomontanus was in Hungary having accepted an appointment from the King to the Royal Library in Buda. The King had just completed a highly successful (from his point of view!) campaign against the Turks and had returned with many rare books. This provided an ideal position for Regiomontanus since it enabled him to both work with Martin Bylica on astronomy and also to enjoy his passion for old books.

One of the old books which Regiomontanus had come across in 1462 while he was in Venice, was an incomplete copy of Diophantus's Arithmetica. He wrote to the mathematician Giovanni Bianchini on 11 February 1464 saying that if he could find a complete copy he would translate the Greek text. It was while he had been on a trip to Ferrara that he had met Bianchini, who had been a friend of Peurbach. Regiomontanus never translated Diophantus's Arithmetica and he never found a complete version. Indeed nobody has ever discovered a complete version, but this important discovery by Regiomontanus marks the beginning of the Arithmetica becoming known in Europe.

Regiomontanus made important contributions to trigonometry and astronomy. We have mentioned the above the Epitome of the Almagest which was begun by Peurbach but completed by Regiomontanus. It [1]:-

... contributed to current scientific research rather than to improved understanding of the past. Moreover, the "Epitome" was no mere compressed translation... [for] it added later observations, revised computations, and critical reflections - one of which revealed that Ptolemy's lunar theory required the apparent diameter of the moon to vary in length much more than it really does. This passage in the "Epitome", which was printed in Venice, attracted the attention of Copernicus, then a student at the University of Bologna.

In the Epitome Regiomontanus, realising the need for a systematic account of trigonometry to support astronomy, promised to write such a treatise. Indeed he did so and his book De triangulis omnimodis (1464) is a systematic account of methods for solving triangles. In the introduction he writes:-

You, who wish to study great and wondrous things, who wonder about the movement of the stars, must read these theorems about triangles. ... For no one can bypass the science of triangles and reach a satisfying knowledge of the stars. ... A new student should neither be frightened nor despair. ... And where a theorem may present some problem, he may always look down to the numerical examples for help.

Regiomontanus structured his work in a similar way to Euclid's Elements. De triangulis is in five books, the first of which gives the basic definitions: quantity, ratio, equality, circles, arcs, chords, and the sine function. He then gives a list of the axioms he will assume, followed by 56 theorems on geometry. With Book II the study of trigonometry gets under way in earnest. The sine law is stated (in modern notation, not used by Regiomontanus, this is $\Large \frac {a}{\sin A}\normalsize = \Large \frac {b}{\sin B}\normalsize = \Large \frac {c}{\sin C}$) and it is used to solve triangles. The formula for the area of a triangle in terms of two sides and the included angle appears but not in quite the form that one would expect. Books III, IV and V treat spherical trigonometry which, of course, is of major importance in astronomy.

When he was in Hungary, Regiomontanus computed two tables of sines. The first computed in 1467 was Tables of directions which was based on sexagesimal numbers, while in the following year in Buda he computed tables of sines to a decimal base. By 1471 he was proposing to move to Nuremberg, writing in a letter to a friend on 4 July 1471:-

Quite recently I have made observations in the city of Nuremberg ... for I have chosen it as my permanent home not only on account of the availability of instruments, particularly the astronomical instruments on which the entire science is based, but also on account of the great ease of all sorts of communication with learned men living everywhere, since this place is regarded as the centre of Europe because of the journeys of the merchants.

We know that Regiomontanus had indeed been making observations in Nuremberg for he observed a lunar eclipse there on 2 June 1471. By 29 November he had been granted leave to reside in Nuremberg where he built an observatory and a workshop to construct instruments. He wrote Scipta giving details of his instruments and these, including dials, quadrants, safea, astrolabes, armillary astrolabe, torquetum, parallactic ruler, and Jacob's staff are described in [53]. In January 1472 he made observations of a comet, using his Jacob's staff, which were accurate enough to allow it to be identified with what is now known [55] as C/1471 Y1 (C = non periodic comet).

Regiomontanus's interest in the motion of the Moon led him to make the important observation that the method of lunar distances could be used to determine longitude at sea. It was many years, however, before the position of the Moon could be predicted with sufficient accuracy to make the method practical and instruments constructed to give the lunar position with the high degree of accuracy necessary for the method. Regiomontanus describes how the position of the Moon can be used to determine longitude in the Ephemerides for the years 1474-1506 which he published. This was printed on his own printing press which he set up in Nuremberg.

The first European printing of books began in 1454 with the invention of movable type by Johann Gutenberg. Regiomontanus realized the potential value of printing for producing identical multiple copies of scientific texts, which could be carefully edited with accurate diagrams. At Nuremberg in 1471-1472 he set up a printing press in his own house, and printed a Prospectus announcing his detailed plans for publishing many carefully edited mathematical, astronomical and geographical texts. He thus became the first publisher of this type of scientific literature which included ancient, mediaeval and modern works. His first publication was New theory of the planets by his former teacher Peurbach and next, in 1474, his own calendar Kalendarium, and his Ephemerides which we referred to above. These books were reprinted many times and had great influence, for example both Christopher Columbus and Amerigo Vespucci used Regiomontanus's Ephemerides to measure longitudes in the New World.

Pope Sixtus IV summoned Regiomontanus to Rome in 1475 to advise on calendar reform and he left Nuremberg some time after 28 July when he recorded his last observation there. Bernhard Walther, his wealthy pupil who had funded his instrument shop, observatory and printing works, began observations from Regiomontanus's observatory in Nuremberg on 2 August. It is certain that Regiomontanus had left Nuremberg by then. He died in Rome and some accounts say he was poisoned by his enemies, other accounts say he died from the plague. The latter is far more likely but let us look briefly at the two theories.

Regiomontanus had announced that he would publish a work showing the worthlessness of George of Trebizond's whose:-

... commentary on the "Syntaxis" he will show with the utmost clarity to be worthless and his translation of Ptolemy's work not to be free of faults.

This was considered sufficient motive for his murder by the two sons of George of Trebizond and rumours circulated to this effect. It seems a rather unlikely scenario, however, for others criticised George of Trebizond as vigorously as did Regiomontanus yet no attempts appear to have been made on their lives. Much more likely is that, after the Tiber overflowed its banks in January 1476 and there was a resulting outbreak of plague, Regiomontanus became its victim.