克里斯托佛·克拉乌Christopher Clavius

克里斯托佛·克拉乌出生在德国地区，在采用拉丁名'克拉乌'之前一定有一个德国名字。有人猜测如'Clau'或'Klau'，甚至'Schlüssel'，意思是'钥匙'，因此可能导致他采用拉丁名'克拉乌'，这也意味着'钥匙'，但从未有任何证据证实。尽管出生在新教革命在德国蔓延的时期，他出生的弗兰肯地区几乎未受影响，并保持坚定的罗马天主教。他于1555年加入耶稣会（耶稣会），在罗马被接纳，并在会内接受教育。他于1556年被送到葡萄牙的科英布拉大学，在耶稣会学院学习，该学院已在那里成立。

数学一直是克拉乌感兴趣的话题，他在作为普通学位一部分所修的数学课程中表现出色。对他产生重大影响的一件事是1560年8月21日发生的日食。以下是他自己在其1593年出版的著作In sphaeram Ioannis de Sacro Bosco CommentariusⓉ（对约翰尼斯·德·萨克罗博斯科《天球》的评注）中对这次日食的描述：-

我将引用两次非凡的日食，它们发生在我自己的时代，因此不久之前。其中一次是我于1559年[原文如此]在卢西塔尼亚[葡萄牙]的科英布拉附近正午时分观测到的，当时月亮位于我的视线与太阳之间，结果它在相当长的时间内遮住了整个太阳。黑暗在某种程度上比夜晚更甚；人们甚至看不见自己踩在哪里。星星出现在天空中，而且（令人惊叹）鸟儿因如此可怕的黑暗而惊恐地从天空坠落到地面。

我们要注意，尽管克拉乌给出的年份是1559年，但他犯了一个错误，因为我们知道，唯一符合他描述的日食发生在1560年8月21日。这一事件对克拉乌很重要，因为它使他决心将一生献给数学和天文学研究。在这次日食观测之后，他于1560年晚些时候前往意大利，在罗马的耶稣会罗马学院学习神学。他于1564年被任命为神父，但留在罗马学院，并在被任命那年开始了数学教学。事实上，除了大约1596年在拿不勒斯的一段时间和1597年访问西班牙之外，克拉乌余生一直担任罗马学院的数学教授。当然，当时学习神学是一个漫长的过程，尽管他在1564年成为教师，但他继续学习，直到1575年才成为耶稣会的正式成员。

我们上面引用的来自In sphaeram Ioannis de Sacro Bosco CommentariusⓉ（对德·萨克罗博斯科《论球》的评注）的引文开头提到了 'two remarkable eclipses of the Sun'。他在罗马学院观察到的第二次日食发生在1567年4月9日。以下是克拉乌对那次事件的描述：-

另一次我于1567年在罗马看到，也大约在正午，尽管月亮位于我的视线和太阳之间，但它并没有像之前那样遮住整个太阳，而是（这也许是以前从未在任何其他时候发生过的事情）在太阳上留下了一个狭窄的圆圈，从四面八方环绕着整个月亮。

他报告的“大约正午”的时间相当准确，因为最大值发生在正午过后10分钟。该报告表明这是一次环食，月亮的视角度略小于太阳。然而，[18]的作者们写道：-

克拉乌对1567年日食的描述，从表面上看，也许似乎他目睹了一次环食的环阶段。然而，计算表明这次日食不是环食。从罗马看，以观测者为中心的月球半径（955".92）略大于太阳半径（953".56），比值为1.0025...。因此，如果假设月球边缘是精确的圆形，1567年的日食将只是边缘全食，月亮在中心线附近仅完全遮住太阳14秒。然而，当考虑到月球边缘的真实轮廓时，即使在日食中心处，也会看到几个光球层的光点。因此，这次日食既不是完全全食，也不是环食。

另一种说法是，克拉乌看到的“某个狭窄的圆圈”可能是内冕。

也许克拉乌最为人所知的是他在历法改革方面的工作。儒略·凯撒于公元前46年引入的儒略闰年规则，在每385年中多出了3个闰年。结果，二分点和solstices的实际发生时间逐渐偏离了它们的历法日期。春分的日期决定了复活节的日期，因此教会开始敦促改革。1511年的会议，以及随后1563年的特伦托会议，都敦促教皇采取行动。教皇格里高利十三世咨询了他的数学专家，其中克拉乌最为资深。克拉乌提议，1582年10月4日星期三（儒略历）之后应为1582年10月15日星期四（格里高利历）。他提议闰年应出现在能被四整除的年份，但以00结尾的年份必须能被400整除才是闰年。这一规则至今仍在使用，并且如此精确，以至于许多世纪内都不需要进一步改革历法。事实上，需要3500年才会达到一天的误差。

弗朗索瓦·韦达不喜欢克拉乌的历法，法兰克福的民众暴动，反对教皇和数学家，他们认为这些人合谋夺走了他们的11天。克拉乌写了Novi calendarii romani apologia Ⓣ（为新罗马历法辩护）（1595年），为新的历法改革辩护，以抵御这些攻击。

克拉乌 本人原创的数学成果不多，但他在推动数学知识传播方面所做的贡献超过16世纪任何其他德国学者。不过，他是第一个使用小数点的人。他是一位有天赋的教师和教科书作者，1574年出版了一部欧几里得的Elements的版本，其中包含了他自己的想法。另一部写得很好的书是Algebra（1608）。他的算术书被许多数学家使用，包括哥特弗里德·威廉·莱布尼茨和勒内·笛卡儿。克拉乌还制作了一些仪器，其中最有趣的也许是一种测量角度分数的仪器。值得一提的是，他对将测量刻度细分为更小单位作了极好的论述，这是直到皮埃尔·韦尼耶的工作之前最为精密的方法。他还设计了日晷，并开发了一种用于测量的象限仪。

也许对他工作的最好总结，是简要地看一下五卷本的Christophori Clavii e Scoietate Jesu opera mathematica, quinque tomis distributaⓉ（耶稣会士克拉乌数学著作五卷），这部著作是在他晚年编成的，收录了他的文集。第一卷包含上文提到的他的欧几里得以及比提尼亚的狄奥多西的《球面学》（Sphaericorum Libri III）。第二卷包含他在几何学和代数学方面的著作，而第三卷包含他对约翰尼斯·德·萨克罗博斯科（亦称圣木的弗瑞兹·约翰）的《球面学》（Sphaera）的评注——我们曾从中引用过关于两次日食的内容——以及他关于星盘的论著。第四卷包含一篇关于日晷制作的引人入胜的叙述，而最后一卷包含他关于历法改革的著作。

最后让我们看看克拉乌对伽利略所作天文发现的反应。这两位科学家是在伽利略1587年访问罗马时相识的，从那时起他们偶尔就数学问题通信。当克拉乌出版一本书时，他总是会寄一本给他的朋友伽利略。当伽利略在1610年出版Sidereus Nuncius时，克拉乌已是一位老人，无论从科学还是宗教的角度来看，要领会这些新发现都必定极其困难。作为罗马学院的资深科学家，他需要对伽利略的工作作出评判，然而有一段时间他没有质量足够好的望远镜来进行自己的观测。不过，在In sphaeram Ioannis de Sacro Bosco CommentariusⓉ（对德·萨克罗博斯科《论球》的评注）的最后一版中，他讨论了这些问题：-

我不想向读者隐瞒，不久前从比利时带来了一件仪器。它形如一根长管，两端装有两块玻璃，或者说是透镜，借此远离我们的物体看起来比物体本身近得多……这件仪器在天穹中显示出的恒星比不用它所能看到的要多得多，尤其是在昴星团、巨蟹座和猎户座星云周围、银河以及其他地方……而当月亮是新月或半满时，它显得如此明显地破裂和粗糙，以至于我对月球天体竟有如此不平整惊叹不已。请参阅伽利略于1610年在威尼斯印行、名为Sidereus Nuncius的那本可靠的小书，其中描述了他最初对恒星所作的各种观测。

用这件仪器看到的事物中，绝非最不重要的是，金星像月亮一样从太阳获得光线，因此根据它与太阳的距离，有时它看起来更像新月，有时则不那么像。在罗马，我在其他人面前不止一次观察到这一点。土星旁边有两颗较小的星星，一颗在东，一颗在西。最后，木星有四颗游动的星星，它们在彼此之间以及相对于木星的位置上以显著的方式变化——正如伽利略仔细而准确地描述的那样。

既然情况如此，天文学家就应当考虑如何安排天体轨道，以拯救这些现象。

如果读者对土星东西两侧各有一颗较小恒星的描述感到困惑，那么我们应当注意，直到1656年望远镜才改进到足以让克里斯蒂安·惠更斯发现土星环的真实形状。当然，对金星的观测最令人不安，因为克劳狄乌斯·托勒密的太阳和行星模型无法解释所观测到的相位。因此克拉乌呼吁天文学家提出一种理论来'save these phenomena'。

上图是根据一幅写生肖像制作的版画。

Christopher Clavius was born in a German region and must have had a German name before adopting the Latin 'Clavius'. Several guesses such as 'Clau' or 'Klau', or even 'Schlüssel' which means 'key' so might have led to him taking the Latin 'Clavius' which also means 'key', have been made but none have ever been substantiated with any evidence. Despite being born at a time when the Protestant revolution was spreading in Germany, the region of Franconia where he was born was virtually unaffected and remained solidly Roman Catholic. He entered the Jesuit Order (the Society of Jesus) in 1555, being admitted in Rome, and received his education within the Order. He was sent to the University of Coimbra in Portugal in 1556 to study at the Jesuit College that had been founded there.

Mathematics had always been a topic which interested Clavius, and he excelled in the mathematical courses which he took as part of a general degree. An event which had a large influence on him was an eclipse of the sun which occurred on 21 August 1560. Here is his own description of it taken from his work In sphaeram Ioannis de Sacro Bosco Commentarius Ⓣ published in 1593:-

I shall cite two remarkable eclipses of the Sun, which happened in my own time and thus not long ago. One of these I observed about midday at Coimbra in Lusitania [Portugal] in the year 1559 [sic], in which the Moon was placed between my sight and the Sun with the result that it covered the whole Sun for a considerable length of time. There was darkness in some manner greater than night; neither could one see where one stepped. Stars appeared in the sky and (marvellous to behold) the birds fell down from the sky to the ground in terror of such horrid darkness.

Let us note that although Clavius gives the year as 1559, he has made an error since we know that the only possible eclipse that fits his description occurred on 21 August 1560. This event was important for Clavius since it convinced him to devote his life to mathematical and astronomical study. Following this eclipse observation, he went to Italy later in 1560 and studied theology at the Jesuit Collegio Romano in Rome. He was ordained in 1564 but remained at the Collegio Romano were he began teaching mathematics in the year of his ordination. In fact, except for a period in Naples around 1596 and a visit to Spain in 1597, Clavius was to remain Professor of Mathematics at the Collegio Romano for the rest of his life. Of course, the study of theology was a lengthy process at this time and, despite becoming a teacher in 1564, he continued with his studies and did not become a full member of the Jesuit Order until 1575.

The quotation from In sphaeram Ioannis de Sacro Bosco Commentarius Ⓣ which we gave above begins by referring to 'two remarkable eclipses of the Sun'. The second one, which he observed while at the Collegio Romano occurred on 9 April 1567. Here is Clavius's description of that event:-

The other I saw at Rome in the year 1567 also about midday in which although the Moon was placed between my sight and the Sun it did not obscure the whole Sun as previously but (a thing which perhaps never before occurred at any other time) a certain narrow circle was left on the Sun, surrounding the whole of the Moon on all sides.

He is quite accurate with his reported time of 'about midday' since the maximum occurred at 10 minutes after midday. The report suggests an annular eclipse with the moon subtending a slightly smaller angle than the sun. However the authors of [18] write:-

Taking Clavius's account of the eclipse of 1567 at face value, it might perhaps seem that he witnessed the ring phase of an annular eclipse. However, computation shows that the eclipse was not annular. As seen from Rome, the topocentric lunar semi-diameter (955".92) was slightly greater than the solar semi- diameter (953".56) by a factor of 1.0025.... Hence if the lunar limb was assumed to be accurately circular, the eclipse of 1567 would be only marginally total, the Moon covering the Sun completely for just 14 sec near the central line. However, when the true profile of the lunar limb is taken into account, several beads of photospheric light would be visible even where the eclipse was central. In consequence, the eclipse was neither fully total nor annular.

Another suggestion is that the 'certain narrow circle' seen by Clavius might be the inner corona.

It is perhaps for his work on the reform of the calendar that Clavius is most widely known. The Julian leap-year rule, introduced by Julius Caesar in 46 BC, created 3 leap years too many in every period of 385 years. As a result, the actual occurrence of the equinoxes and solstices slowly moved away from their calendar dates. The date of the spring equinox determines the date of Easter so the church began to press for reform. The Council of 1511, then again the Trent Council of 1563, urged the Pope to act. Pope Gregory XIII consulted his mathematical experts, of whom Clavius was the most senior. Clavius proposed that Wednesday, 4 October 1582 (Julian) should be followed by Thursday, 15 October, 1582 (Gregorian). He proposed that leap years occur in years exactly divisible by four, except that years ending in 00 must be divisible by 400 to be leap years. This rule is still used today and is so accurate that no further reform of the calendar will be necessary for many centuries. In fact it requires 3500 years before an error of one day is reached.

Viète did not like Clavius's calendar and the people of Frankfurt rioted against the Pope and mathematicians who, they believed, had conspired together to rob them of 11 days. Clavius wrote Novi calendarii romani apologia Ⓣ (1595) which justified the new calendar reforms defending them against these attacks.

Although Clavius produced little original mathematics of his own, he did more than any other German scholar of the 16th Century to promote a knowledge of mathematics. He was the first, however, to use the decimal point. He was a gifted teacher and writer of textbooks, producing a version of Euclid's Elements in 1574 which contains ideas of his own. Another well written book was Algebra (1608). His arithmetic books were used by many mathematicians including Leibniz and Descartes. Clavius also produced a number of instruments, perhaps the most interesting being an instrument to measure fractions of angles. It is worth mentioning that he gave an excellent account of dividing a measuring scale into subdivisions which is by far the most sophisticated to be produced until the work of Vernier. He also designed sundials and developed a quadrant for use in surveying.

Perhaps the best summary of his work would be to look briefly at the five volumes Christophori Clavii e Scoietate Jesu opera mathematica, quinque tomis distributa Ⓣ which was produced near the end of his life and contains his collected works. The first volume contains his Euclid referred to above and the "Spheric" of Theodosius (Sphaericorum Libri III). The second volume contains his works on geometry and algebra, while the third volume contains his commentary on the "Sphaera" of Johannes de Sacrobosco (also known as John of Holywood) from which we have quoted regarding the two solar eclipses, and his treatise on the astrolabe. The fourth volume contains a fascinating account of the construction of sundials, while the final volume contains his written works on calendar reform.

Finally let us look at Clavius's reaction to the astronomical discoveries made by Galileo. The two scientists met when Galileo visited Rome in 1587 and from that time on they corresponded occasionally concerning mathematical questions. When Clavius published a book he would always send a copy to his friend Galileo. When Galileo published Sidereus Nuncius in 1610, Clavius was an old man and it must have been extremely difficult to grasp these new discoveries both from a scientific and religious point of view. As the senior scientist in the Collegio Romano he was required to pass judgement on Galileo's work yet for some time he did not have a telescope of sufficient quality to make his own observations. However in the final edition of In sphaeram Ioannis de Sacro Bosco Commentarius Ⓣ he addressed the issues:-

I do not want to hide from the reader that not long ago a certain instrument was brought from Belgium. It has the form of a long tube in the bases of which are set two glasses, or rather lenses, by which objects far away from us appear very much closer ... than the things themselves are. This instrument shows many more stars in the firmament than can be seen in any way without it, especially in the Pleiades, around the nebulas of Cancer and Orion, in the Milky Way, and other places ... and when the Moon is a crescent or half full, it appears so remarkably fractured and rough that I cannot marvel enough that there is such unevenness in the lunar body. Consult the reliable little book by Galileo Galilei, printed at Venice in 1610 and called Sidereus Nuncius, which describes various observations of the stars first made by him.

Far from the least important of the things seen with this instrument is that Venus receives its light from the Sun as does the Moon, so that sometimes it appears to be more like a crescent, sometimes less, according to its distance from the Sun. At Rome I have observed this, in the presence of others, more than once. Saturn has joined to it two smaller stars, one on the east, the other on the west. Finally Jupiter has four roving stars, which vary their places in a remarkable way both among themselves and with respect to Jupiter - as Galileo Galilei carefully and accurately describes.

Since things are thus, astronomers ought to consider how the celestial orbs may be arranged in order to save these phenomena.

If the reader is puzzled about the description of Saturn with two smaller stars to the east and west, then we should note that it was only in 1656 that telescopes improved sufficiently for Huygens to discover the true shape of the rings of Saturn. Of course the observations of Venus were the most disturbing since Ptolemy's version of the sun and planets would not account for the observed phases. Hence Clavius's plea to astronomers to come up with a theory to 'save these phenomena'.

The picture above is from a engraving made from a portrait from life.