佩德罗·努内斯Pedro Nunes

佩德罗·努内斯或Nunez以拉丁名Petrus Nonius Salaciensis写作。他出生的城市阿尔卡塞尔杜萨尔被罗马人命名为Salacia，因此Salaciensis只是表明他来自Salacia。关于他的父母一无所知，但可以肯定他们信仰犹太教。努内斯前往西班牙，大约1517年进入萨拉曼卡大学。这所大学在努内斯开始学业时已有300年历史，是欧洲领先的教育中心之一。虽然没有记录证明他入学的日期，但当时男孩通常在十五岁进入大学，从他在萨拉曼卡学习的广泛科目来看，我们知道他一定修读了典型长度的课程。鉴于他获得的技能，他一定学习过哲学、医学、数学和地理学。1523年他毕业，同年与西班牙女孩D Guiomar Areas在萨拉曼卡结婚。努内斯和Guiomar 努内斯有六个孩子：两个男孩Apolónio和努内斯，以及四个女孩Briolanja、Francisca、Isabel和Guiomar。

我们不知道努内斯毕业后在萨拉曼卡待了多久，但到1527年他已回到葡萄牙。在那里他开始教授Luis王子，即Joao o Piedoso的兄弟。Joao于1521年十九岁时成为葡萄牙国王（称为弗瑞兹·约翰三世或虔诚者约翰）。努内斯担任Luis的导师直到1531年7月。从1527年开始，努内斯向Martim Afonso de Sousa教授航海技能，后者于1530年率领第一次殖民远征巴西，还向另一位海军军官Joao de Castro教授。事实上，Joao de Castro除了在葡萄牙海军于北非和印度的非常成功的职业生涯外，本人也是一位杰出的科学家，于1538年、1538-39年和1541年撰写了三部关于航海科学的重要著作。1529年11月16日，国王Joao o Piedoso任命努内斯为“葡萄牙王国宇宙志学家”。作为宇宙志学家，他的官方职责之一是为海员教授航海课程。几周后，12月4日，他开始在里斯本大学教授道德哲学。这一任命只是作为教授的替代，但1530年1月15日他被任命为逻辑学讲席。他教授逻辑学至少一年，但似乎他的课程对学生不太成功，因此他于1532年4月4日转到形而上学讲席。然而在此期间，努内斯正在攻读医学博士学位，并在1532年2月16日成功通过考试后，于3月3日获得医学博士学位。

关于1532年至1544年期间努内斯身在何处，并不完全清楚。部分时间他在里斯本，但也有证据表明他在西班牙待过。当然，在此期间他正式受雇于里斯本大学，并在这些年里指导了一些博士生的学习。他也在这些年里撰写了一些书籍。他于1534年开始撰写Libro de Algebra，尽管直到三十三年后才出版完整著作。该书分为三部分，第一部分论述一次和二次方程，第三部分论述三次方程。文本的中间部分研究比例理论和代数运算。努内斯出版的第一本书是1537年的Tratado da Sphera。这本书包含努内斯编辑的约翰尼斯·德·萨克罗博斯科的Tractatus de Sphaera、格奥尔格·冯·波伊尔巴赫的Theoricae nouae planetarum以及克劳狄乌斯·托勒密的Geographia第一卷。此外，努内斯还加入了自己两部关于航海的著作。他下一部出版的作品是De Crepusculis（1542年），其中他解答了一位学生向他提出的问题，这位学生是曼努埃尔一世的儿子红衣主教恩里克王子，他于1534年成为布拉加大主教，并在教会职业生涯后于1578年成为葡萄牙国王。努内斯从1531年10月起教授恩里克，直到他在布拉加的任命。他向努内斯提出的问题是“哪一天的暮光最短？”而努内斯在De Crepusculis中讨论的正是暮光持续时间的问题。Gonzalo de Reparaz Ruiz写道[10]：-

[努内斯]是D Manuel之子、Infant Henry的数学老师，他向他演示了通过太阳高度确定某点纬度的方法，以及如何在图表上标示这一点。他在1533年国王面前重复了这一演示。在他为⟦E1⟧所写并献给其前学生、当时担任国家摄政的红衣主教亨利的序言中，努内斯声明，这本书的出版因其他“非常多样”的著作而延迟，他声称这些著作占用了他数年的时间。

努内斯于1544年10月16日转到科英布拉大学担任数学讲席，他担任此职直到1562年，尽管在1557-61年期间他离开大学在里斯本生活了四年。数学讲席是科英布拉大学的一个新职位，设立它是为了提供航海技术要求方面的教学，这显然是当时葡萄牙极为重要的主题，当时控制海上贸易是葡萄牙财富的主要来源。在科英布拉期间，努内斯教授克里斯托佛·克拉乌。他已于1529年被任命为皇家宇宙志学家，但于1547年12月22日晋升为首席皇家宇宙志学家。他担任此职直到去世。1548年，国王若昂三世授予努内斯Cavaleiro do Hábito de Cristo（基督骑士团骑士）的荣誉。

搬到科英布拉后不久，努内斯于1546年出版了De erratis Orontii Finaei。这本书旨在表明奥文斯·菲内解决化圆为方、三等分任意角和倍立方这三个古典问题的尝试是错误的。大约在1550年，他做出了今天最为人所知的发现，即他对斜驶线的研究，或者用努内斯发明的名称，即恒向线。在许多方面，等角航线对于对航海感兴趣的人来说是一条自然的曲线，因为它是船只始终保持同一罗盘方位时所遵循的曲线。另一种理解这条曲线的方式是，它是通过以恒定角度切割球面上的子午线而得到的。事实上，等角航线会螺旋朝向极点，但它对地图投影的重要性在于，在球面的墨卡托投影中，等角航线变成一条直线。正是威理博·斯涅尔将努内斯的恒向线命名为等角航线。

1566年，努内斯出版了自己的文集Petri Nonii Salaciensis Opera，其中包含他早期作品的改进版本以及附加材料[8]：-

本书是一部关于航海的努内斯著作的大幅扩充汇编，在当时代表了航海科学的最先进水平。书中还包含了对亚里士多德关于由桨推动的船只运动的力学问题的评注。

这本书的第二改进版于1573年以不同的标题出版，即De arte atque ratione navigandi。1567年，他出版了Libro de Algebra，正如我们上面指出的，大部分是多年前写的。我们注意到，约翰 Martyn在[4]中发表了据称是努内斯代数教材早期版本的内容。然而，Giovanna Cifoletti [5]对出版的手稿是由努内斯所写，还是由努内斯的学生所做的笔记集，提出了疑问。1571年，他将之前出版的两部文本De Crepusculis和De erratis Orontii Finaei合并成一卷出版。在数学之外，他从事地理学、物理学、宇宙论的工作，并写诗。他的诗歌例子发表在[4]中。

努内斯设计了一种可以测量度的分数部分的系统。他这样描述它：-

……在用于测量角度的象限仪面上画45条同心弧，其中一条被分成90等份或度，其余依次分成89、88、87、86等份，最后一条被分成46等份。当指标没有恰好切到度弧的某一分划时，它会经过或靠近其他某条弧的某一分划；通过记下该分划的位置，就可以计算出一度的分数部分。

1572年，努内斯被召入宫廷主持度量衡改革。他在里斯本待了两年，最后才回到科英布拉。

最后，让我们注意一下约翰·迪伊与努内斯之间的友谊。约翰·迪伊在1558年写道：-

……如果我的著作在我生前无法完成或出版，我已将它遗赠给我们当中数学技艺唯一的遗存、装饰和支柱，那位最博学、最严肃的人，萨拉西亚的 D D 努内斯，并且不久前曾恳切地请求他，如果我这部著作在我死后被送到他那里，希望他能仁慈而人道地将其置于自己的保护之下，并像对待自己的著作一样以各种方式加以利用：希望他最终能屈尊完成它、校正它，并像完全属于他自己的一样，为哲学家的公共使用而润色它。我毫不怀疑，只要他的生命和健康不受损害，他本人也会赞同我的愿望，因为他忠实地爱我，而且勤勉地培育对基督教国家最为必要的技艺，这在他身上出于天性，并由意志、勤勉和习惯所强化。

Pedro Nunes or Nunez wrote under the Latin name of Petrus Nonius Salaciensis. The city of his birth, Alcácer do Sal, was named Salacia by the Romans so Salaciensis simply indicates that he came from Salacia. Nothing is known of his parents but it is certain that they were of the Jewish religion. Nunes travelled to Spain and entered the University of Salamanca around 1517. This university, already 300 years old when Nunes began his studies, was one of the leading centres for education in Europe. Although no records exist to prove the date of his admission, nevertheless at this time it was typical for boys to enter university at age fifteen and from the wide range of topics that he studied at Salamanca we know that he must have taken a typical length course. Given the skills he acquired he must have studied philosophy, medicine, mathematics and geography. In 1523 he graduated and, in the same year, married the Spanish girl D Guiomar Areas in Salamanca. Pedro and Guiomar Nunes had six children: two boys Apolónio and Pedro, and four girls Briolanja, Francisca, Isabel and Guiomar.

We do not know exactly how long Nunes remained at Salamanca after he graduated but by 1527 he had returned to Portugal. There he began teaching Prince Luis, the brother of Joao o Piedoso. Joao had become king of Portugal (known as John III or John the pious) in 1521 at the age of nineteen. Nunes acted as tutor to Luis until July 1531. Starting in 1527, Nunes taught navigation skills to Martim Afonso de Sousa, who led the first colonizing expedition to Brazil in 1530, and to another naval officer Joao de Castro. In fact Joao de Castro, in addition to a highly successful career in the Portuguese navy in North Africa and India, was himself an excellent scientist writing three important works on the science of navigation in 1538, 1538-39 and 1541. On 16 November 1529 king Joao o Piedoso appointed Nunes as "Cosmographer of the Kingdom of Portugal". One of his official duties as Cosmographer was to give classes in navigation to seafarers. A few weeks later, on 4 December, he began teaching moral philosophy at the University of Lisbon. This appointment was only as a substitute for the professor, but on 15 January 1530 he was appointed to the chair of Logic. He taught logic for at least a year but it appears that his classes were not very successful with students so he moved to the chair of metaphysics on 4 April 1532. During this period, however, Nunes was working towards a doctorate in medicine and after successfully taking examinations on 16 February 1532 be was awarded his medical doctorate on 3 March.

It is not entirely clear where Nunes was over the period between 1532 and 1544. Part of the time he was in Lisbon but there is also evidence that he spent time in Spain. Certainly during this period he was officially employed by the University of Lisbon and he supervised the studies of a number of doctoral students over these years. He was also working during these years on writing a number of books. He began writing Libro de Algebra in 1534 although only published the completed work thirty-three years later. The book is in three parts, the first part dealing with equations of the first and second degree and the third part dealing with equations of the third degree. The middle part of the text studies the theory of proportion and algebraic manipulation. The first of Nunes books to be published was Tratado da Sphera in 1537. This book contained Nunes' edition of Sacrobosco's Tractatus de Sphaera, of Peurbach's Theoricae nouae planetarum, and of Book I of Ptolemy's Geographia. In addition Nunes added two works of his own on navigation. His next published work was De Crepusculis (1542) in which he addressed a problem posed to him by one of his pupils, Prince Cardinal Henrique the son of Manuel I who became archbishop of Braga in 1534 and, after a career in the church, king of Portugal in 1578. Nunes taught Henrique from October 1531 until his appointment in Braga. The question he had posed to Nunes was "Which day has the shortest twilight?" and it was the question of the duration of twilight that Nunes addressed in De Crepusculis. Gonzalo de Reparaz Ruiz writes [10]:-

[Nunes] was teacher of mathematics of the son of D Manuel, the Infant Henry, to whom he demonstrated the method of establishing the latitude of a point by the height of the sun and how to indicate this on a chart. This demonstration he repeated before the King in 1533. In the preface which he had written to his Libro de Algebra and dedicated to his former pupil the Cardinal Henry, who was then Regent of the country, Nunes states that the publication of this book was delayed by other "very varied" works which, he asserts, had taken his time for several years.

Nunes moved to the University of Coimbra to take up the Chair of Mathematics on 16 October 1544, a post he held until 1562 although he spent four years away from the University living in Lisbon during the period 1557-61. The Chair of Mathematics was a new post in the University of Coimbra and it was set up to provide instruction in the technical requirements for navigation, clearly a topic of great importance in Portugal at this period when control of sea trade was the chief source of Portuguese wealth. While at Coimbra Nunes taught Christopher Clavius. He had been appointed Royal Cosmographer in 1529 but was promoted to Chief Royal Cosmographer on 22 December 1547. He held this post until his death. In 1548 the king, Joao o Piedoso, honoured Nunes by naming him Cavaleiro do Hábito de Cristo (Knight of the Order of Christ).

Soon after moving to Coimbra Nunes published De erratis Orontii Finaei in 1546. This book was written to show that Oronce Fine's attempts to solve the three classical problems of squaring the circle, trisecting an arbitrary angle and to doubling the cube were incorrect. Around 1550 he made the discovery for which today he is best known, namely his investigation of the loxodrome or, to use the name that Nunes invented, the rhumb line. In many ways the loxodrome is a natural curve for someone interested in navigation to examine for it is the curve that a ship will follow if it always follows the same compass setting. Another way to think of the curve is that it is obtained by cutting the meridians on the sphere at a constant angle. A loxodrome will, in fact, spiral towards the pole but its importance to map projections is that the loxodrome become a straight line in the Mercator projection of the sphere. It was Willebrord Snell who named Nunes' rhumb line the loxodrome.

In 1566 Nunes published his own collected works as Petri Nonii Salaciensis Opera which contains improved versions of his earlier works with additional material [8]:-

This book is a very much extended compilation of Nunes works about navigation and was, at the time, the state of the art of nautical science. It also included a commentary to Aristotle's mechanical problem on the motion of a boat propelled by oars.

A second improved edition of this book was published in 1573 under a different title, namely De arte atque ratione navigandi. In 1567 he published Libro de Algebra mostly written many years earlier as we noted above. We note that John Martyn published in [4] what is claimed to be an early version of Nunes' algebra text. However, Giovanna Cifoletti [5] casts doubt on whether the published manuscript was written by Nunes or whether it was a set of notes taken by one of Nunes' students. In 1571 he put together two of his previously published texts, De Crepusculis and De erratis Orontii Finaei, bring them out in a single volume. Outside mathematics he worked in geography, physics, cosmology and he wrote poetry. Examples of his poetry are published in [4].

Nunes devised a system to allow fractional parts of a degree to be measured. He describes it as:-

... drawing on the face of a quadrant for measuring angles 45 concentric arcs, one of which was divided into 90 equal parts or degrees, and the remainder into 89, 88, 87, 86, etc., successively, the last being divided into 46 equal parts. When the index did not exactly cut one of the divisions of the arc of degrees, it passed through or near to one of the divisions of one or other of the other arcs; and by noting the place of that division the fractional parts of a degree were calculated.

In 1572 Nunes was called to court to preside over the reform of weights and measures. He stayed in Lisbon for two years before finally returning to Coimbra.

Finally, let us note the friendship between John Dee and Nunes. Dee wrote in 1558:-

... if my work cannot be finished or published while I remain alive, I have bequeathed it to that most learned and grave man who is the sole relic and ornament and prop of the mathematical arts among us, D D Pedro Nunes, of Salacia, and not long since prayed him strenuously that, if this work of mine should be brought to him after my death, he would kindly and humanely take it under his protection and use it in every way as if it were his own: that he would deign to complete it, finally, correct it, and polish it for the public use of philosophers as if it were entirely his. And I do not doubt that he will himself be a party to my wish if his life and health remain unimpaired, since he loves me faithfully and it is inborn in him by nature, and reinforced by will, industry, and habit, to cultivate diligently the arts most necessary to a Christian state.