卢卡·帕西奥利Luca Pacioli

卢卡·帕西奥利的父亲是Bartolomeo 帕西奥利，但帕西奥利似乎并不是在父母的家中长大的。他小时候住在圣塞波尔克罗的Befolci家，那是他出生的城镇。这个城镇位于意大利相当中心的位置，在佩鲁贾市以北约60公里处。就帕西奥利而言，也许这个小型商业城镇最重要的特点是，Piero della 皮耶罗·德拉·弗朗西斯卡在那里有一个工作室和作坊，而della 皮耶罗·德拉·弗朗西斯卡尽管经常在其他城镇有委托，仍在那里度过了相当长的时间。

尽管我们对帕西奥利的早年生活知之甚少，但关于他可能至少在皮耶罗·德拉·弗朗西斯卡于圣塞波尔克罗的工作室中接受了部分教育的猜测，至少很有可能正确。这似乎很可能为真的一个原因是，帕西奥利对皮耶罗·德拉·弗朗西斯卡的工作有广泛的了解，而帕西奥利的著作受到皮耶罗·德拉·弗朗西斯卡著作的强烈影响。

帕西奥利还是个年轻小伙子时就离开了圣塞波尔克罗。他搬到威尼斯，为富商Antonio Rompiasi服务，后者位于该城非常令人向往的Giudecca区。人们必须假定，帕西奥利在圣塞波尔克罗的学习中已经很好地掌握了基础数学，而且他肯定受过良好的通识教育，才会被选为Rompiasi三个儿子的家庭教师。然而，帕西奥利在威尼斯期间抓住机会继续更高层次的数学学习，在Domenico Bragadino指导下学习数学。在此期间，帕西奥利既从家庭教师的角色中获得了教学经验，也从帮助处理Rompiasi事务的角色中获得了商业经验。

正是在威尼斯期间，帕西奥利写了他的第一部著作，一本关于算术的书，献给他的雇主的三位儿子。此书完成于1470年，很可能就在Rompiasi去世的那一年。帕西奥利显然认识所有该认识的人，因为他离开威尼斯前往罗马，在那里住了几个月，住在莱昂·巴蒂斯塔·阿尔伯蒂家中，莱昂·巴蒂斯塔·阿尔伯蒂是教廷秘书处的秘书。莱昂·巴蒂斯塔·阿尔伯蒂不仅是一位杰出的学者和数学家，还能为帕西奥利提供良好的宗教人脉。此时帕西奥利转而研习神学，并在随后几年中的某个时候成为方济各会的修士。

1477年，帕西奥利开始了游历生涯，在各所大学讲授数学，尤其是算术。他从1477年到1480年在佩鲁贾大学任教，在那里期间，他为所教的班级写了第二部算术著作。他在扎拉（今克罗地亚的扎达尔或亚德拉，当时属威尼斯帝国）任教，并在那里写了第三部算术书。这三部算术著作均未出版，只有为佩鲁贾学生写的那一部留存下来。扎拉之后，帕西奥利再次在佩鲁贾大学任教，随后在那不勒斯大学，再后在罗马大学。可以肯定的是，帕西奥利在此期间某个时候结识了乌尔比诺公爵。教皇西克斯图斯四世于1474年封费德里科·达·蒙特费尔特罗为乌尔比诺公爵，帕西奥利似乎曾一度担任费德里科之子圭多巴尔多（Guidobaldo）的导师，圭多巴尔多在其父1482年去世后成为最后一位执政的蒙特费尔特罗。乌尔比诺宫廷是著名的文化中心，帕西奥利必定在多年间与其有密切接触。

1489年，在罗马待了两年后，帕西奥利回到了家乡圣塞波尔克罗。然而，在家乡，帕西奥利并非一切顺利。他曾获得教皇授予的一些特权，圣塞波尔克罗的修会人士中颇有几分嫉妒。事实上，帕西奥利在1491年被禁止在当地教书，但这种嫉妒似乎夹杂着对其学识的尊重，因为1493年他受邀宣讲四旬期布道。

在Sansepolcro的这段时间里，帕西奥利撰写了他最著名的著作之一Summa de arithmetica, geometria, proportioni et proportionalitaⓉ（算术、几何、比与比例概要），并将其献给了乌尔比诺公爵Guidobaldo。帕西奥利于1494年前往威尼斯出版SummaⓉ（算术、几何、比与比例概要）。这部著作总结了当时已知的数学，尽管它几乎没有表现出什么原创思想。该著作研究算术、代数、几何和三角学，尽管缺乏原创性，却为此后不久在欧洲发生的数学重大进展提供了基础。正如在1中所说，SummaⓉ（算术、几何、比与比例概要）是：-

……并非针对社会的某一特定群体。这是一部用意大利语写成的百科全书式著作（对开本，600页密排印刷），包含一篇关于理论算术和实践算术的一般论著；代数学的基础；一张意大利各邦使用的货币、重量和度量表；一篇关于复式记账的论著；以及欧几里得几何学的概要。他承认曾自由地借用欧几里得、波爱修斯、约翰尼斯·德·萨克罗博斯科、斐波那契……

帕西奥利的SummaⓉ（算术、几何、比与比例概要）中的几何部分在6中得到了详细讨论。作者们写道：-

L 帕西奥利的Summa[威尼斯，1494]的意大利语几何部分是最早印刷的数学书籍之一。帕西奥利广泛使用了欧几里得的Elements，复述了其中一些部分。他还提到了比萨的Leonardo（斐波那契）。

Summa Ⓣ（算术、几何、比例和比例性概要）的另一个有趣方面是它研究了机会游戏。帕西奥利研究了点数问题，见[9]，尽管他给出的解法是不正确的。

卢多维科·斯福尔扎是弗朗切斯科·斯福尔扎的次子，后者自立为米兰公爵。1466年弗朗切斯科去世后，卢多维科的兄长加莱亚佐·斯福尔扎成为米兰公爵。然而，加莱亚佐于1476年被谋杀，他七岁的儿子成为米兰公爵。经过一番政治运作，卢多维科于1480年成为这位年轻人的摄政。卢多维科·斯福尔扎对艺术家和学者极为慷慨地赞助，着手将他在米兰的宫廷打造成全欧洲最出色的宫廷。1482年，列奥纳多·达·芬奇进入卢多维科麾下，担任宫廷画家和工程师。1494年，卢多维科成为米兰公爵，大约在1496年，帕西奥利应卢多维科之邀前往米兰，在卢多维科·斯福尔扎的宫廷教授数学。这一邀请可能是由列奥纳多·达·芬奇促成的，后者对数学怀有热切的兴趣。

帕西奥利和列奥纳多·达·芬奇很快成为亲密的朋友。数学和艺术是他们长时间讨论的话题，双方都从中获益匪浅。此时，帕西奥利开始撰写他两部著名作品中的第二部，Divina proportione Ⓣ（神圣比例），书中的插图由列奥纳多·达·芬奇绘制。很少有数学家能拥有如此才华横溢的插画师为自己的书作画！帕西奥利在1497年从事的这本书最终成为他于1509年以Divina proportione Ⓣ（神圣比例）为标题出版的三本书中的第一本（例如参见[3]）。这是最终构成这篇论著的三本书中的第一本，它研究了“神圣比例”或“黄金比例”，即比率$a : b = b : (a + b)$。它包含了与这个比率相关的欧几里得的定理，还研究了正多边形和半正多边形（特别参见[4]中关于帕西奥利在正多边形方面工作的讨论）。显然，列奥纳多·达·芬奇对这个在数学和艺术角度上都令人满意的比率的兴趣对这项工作产生了重要影响。黄金比例在建筑设计中也很重要，这个主题后来构成了帕西奥利撰写的论著的第二部分。论著中的第三本书是皮耶罗·德拉·弗朗西斯卡一部作品的意大利文译本。

路易十二于1498年成为法国国王，作为米兰第一公爵的后裔，他声称拥有公国。威尼斯支持路易对抗米兰，1499年法国军队进入米兰。次年，卢多维科·斯福尔扎在试图夺回该城时被俘。帕西奥利和列奥纳多·达·芬奇于1499年12月一起逃离，即法国人占领米兰三个月后。他们首先在曼图亚停留，在那里他们是伊莎贝拉·德·埃斯特侯爵夫人的客人，然后于1500年3月继续前往威尼斯。从威尼斯他们返回佛罗伦萨，在那里帕西奥利和列奥纳多·达·芬奇合住一所房子。

比萨大学在1494年遭遇了一场叛乱，并迁至佛罗伦萨。帕西奥利于1500年被任命在佛罗伦萨的比萨大学教授几何学。他留在佛罗伦萨，在大学教几何，直到1506年。列奥纳多·达·芬奇虽然离开十个月为切萨雷·博尔贾工作，也留在佛罗伦萨直到1506年。帕西奥利像列奥纳多·达·芬奇一样，在1501-02年间在博洛尼亚大学任教时也离开佛罗伦萨一段时间。在此期间，帕西奥利与希皮奥内·德尔·费罗合作，关于两人是否讨论过三次方程的代数解有很多猜测。当然，帕西奥利在Summa Ⓣ（算术、几何、比例和比例性概要）中讨论了这个主题，而在帕西奥利访问博洛尼亚之后一段时间，希皮奥内·德尔·费罗解决了这个经典问题两种情况中的一种。

在Florence Pacioli期间，他既参与教会事务也参与数学。他被选为他在罗马涅的修会会长，然后于1506年进入佛罗伦萨的圣十字修道院。离开佛罗伦萨后，帕西奥利去了威尼斯，在那里他获得了在接下来的十五年内独家出版其作品的权利。1509年，他出版了三卷本著作Divina proportione Ⓣ（神圣比例）以及欧几里得的Elements的拉丁文译本。欧几里得的Elements的第一个印刷版是诺瓦拉的坎帕努斯的十三世纪译本，该译本于1482年在威尼斯以印刷形式出版。帕西奥利的版本基于诺瓦拉的坎帕努斯的版本，但其中包含大量帕西奥利本人的注释。

1510年，帕西奥利回到佩鲁贾再次讲学。他还在1514年在罗马再次讲学，但此时帕西奥利已70岁，接近其学术和教学生涯的尾声。他回到圣塞波尔克罗，于1517年在那里去世，留下未出版的主要著作De Viribus Quantitatis Ⓣ（论数的幂），涉及娱乐问题、几何问题和谚语。这部作品频繁提到与他合作该项目的列奥纳多·达·芬奇，这篇论著中的许多问题也出现在列奥纳多·达·芬奇的笔记本中。这又是一部帕西奥利声称没有原创性的作品，将其描述为汇编。

尽管帕西奥利的作品缺乏原创性，但他对数学的贡献很重要，特别是因为他的书在很长一段时间内产生了影响。在[10]中讨论了帕西奥利工作的重要性，特别是他计算平方根的近似值（使用艾萨克·牛顿方法的一个特例），他对某些机会游戏的不正确分析（类似于布莱兹·帕斯卡研究的那些，引发了theory of probability），他涉及数论的问题（类似问题出现在克劳德-加斯帕·巴歇·德·梅齐里亚克的汇编中），以及他收集的许多magic squares。

1550年出现了一部由乔治·瓦萨里撰写的皮耶罗·德拉·弗朗西斯卡传记。这部传记指控帕西奥利剽窃，并声称他窃取了皮耶罗·德拉·弗朗西斯卡关于透视、算术和几何的著作。这是一个不公平的指控，因为尽管帕西奥利确实严重依赖他人的工作，尤其是皮耶罗·德拉·弗朗西斯卡的工作，但他从未试图将这些工作据为己有，而是承认了他所使用的来源。

Luca Pacioli's father was Bartolomeo Pacioli, but Pacioli does not appear to have been brought up in his parents house. He lived as a child with the Befolci family in Sansepolcro which was the town of his birth. This town is very much in the centre of Italy about 60 km north of the city of Perugia. As far as Pacioli was concerned, perhaps the most important feature of this small commercial town was the fact that Piero della Francesca had a studio and workshop in there and della Francesca spent quite some time there despite frequent commissions in other towns

Although we know little of Pacioli's early life, the conjecture that he may have received at least a part of his education in the studio of della Francesca in Sansepolcro must at least have a strong chance of being correct. One reason that this seems likely to be true is the extensive knowledge that Pacioli had of the work of Piero della Francesca and Pacioli's writings were very strongly influenced by those of Piero.

Pacioli moved away from Sansepolcro while he was still a young lad. He moved to Venice to enter the service of the wealthy merchant Antonio Rompiasi whose house was in the highly desirable Giudecca district of that city. One has to assume that Pacioli was already well educated in basic mathematics from studies in Sansepolcro and he certainly must have been well educated generally to have been chosen as a tutor to Rompiasi's three sons. However, Pacioli took the opportunity to continue his mathematical studies at a higher level while in Venice, studying mathematics under Domenico Bragadino. During this time Pacioli gained experience both in teaching, from his role as tutor, and also in business from his role helping with Rompiasi's affairs.

It was during his time in Venice that Pacioli wrote his first work, a book on arithmetic which he dedicated to his employer's three sons. This was completed in 1470 probably in the year that Rompiasi died. Pacioli certainly seemed to know all the right people for he left Venice and travelled to Rome where he spent several months living in the house of Leone Battista Alberti who was secretary in the Papal Chancery. As well as being an excellent scholar and mathematician, Alberti was able to provide Pacioli with good religious connections. At this time Pacioli then studied theology and, at some time during the next few years, he became a friar in the Franciscan Order.

In 1477 Pacioli began a life of travelling, spending time at various universities teaching mathematics, particularly arithmetic. He taught at the University of Perugia from 1477 to 1480 and while there he wrote a second work on arithmetic designed for the classes that he was teaching. He taught at Zara (now called Zadar or Jadera in Croatia but at that time in the Venetian Empire) and there wrote a third book on arithmetic. None of the three arithmetic texts were published, and only the one written for the students in Perugia has survived. After Zara, Pacioli taught again at the University of Perugia, then at the University of Naples, then at the University of Rome. Certainly Pacioli become acquainted with the duke of Urbino at some time during this period. Pope Sixtus IV had made Federico da Montefeltro the duke of Urbino in 1474 and Pacioli seems to have spent some time as a tutor to Federico's son Guidobaldo who was to become the last ruling Montefeltro when his father died in 1482. The court at Urbino was a notable centre of culture and Pacioli must have had close contact with it over a number of years.

In 1489, after two years in Rome, Pacioli returned to his home town of Sansepolcro. Not all went smoothly for Pacioli in his home town, however. He had been granted some privileges by the Pope and there was a degree of jealousy among the men from the religious orders in Sansepolcro. In fact Pacioli was banned from teaching there in 1491 but the jealousy seemed to be mixed with a respect for his learning and scholarship for in 1493 he was invited to preach the Lent sermons.

During this time in Sansepolcro, Pacioli worked on one of his most famous books the Summa de arithmetica, geometria, proportioni et proportionalita Ⓣ which he dedicated to Guidobaldo, the duke of Urbino. Pacioli travelled to Venice in 1494 to publish the Summa Ⓣ. The work gives a summary of the mathematics known at that time although it shows little in the way of original ideas. The work studies arithmetic, algebra, geometry and trigonometry and, despite the lack of originality, was to provide a basis for the major progress in mathematics which took place in Europe shortly after this time. As stated in [1] the Summa Ⓣ was:-

... not addressed to a particular section of the community. An encyclopaedic work (600 pages of close print, in folio) written in Italian, it contains a general treatise on theoretical and practical arithmetic; the elements of algebra; a table of moneys, weights and measures used in the various Italian states; a treatise on double-entry bookkeeping; and a summary of Euclid's geometry. He admitted to having borrowed freely from Euclid, Boethius, Sacrobosco, Fibonacci, ...

The geometrical part of Pacioli's Summa Ⓣ is discussed in detail in [6]. The authors write:-

The geometrical part of L Pacioli's Summa [Venice, 1494] in Italian is one of the earliest printed mathematical books. Pacioli broadly used Euclid's Elements, retelling some parts of it. He referred also to Leonardo of Pisa (Fibonacci).

Another interesting aspect of the Summa Ⓣ was the fact that it studied games of chance. Pacioli studied the problem of points, see [9], although the solution he gave is incorrect.

Ludovico Sforza was the second son of Francesco Sforza, who had made himself duke of Milan. When Francesco died in 1466, Ludovico's elder brother Galeazzo Sforza became duke of Milan. However, Galeazzo was murdered in 1476 and his seven year old son became duke of Milan. Ludovico, after some political intrigue, became regent to the young man in 1480. With very generous patronage of artists and scholars, Ludovico Sforza set about making his court in Milan the finest in the whole of Europe. In 1482 Leonardo da Vinci entered Ludovico's service as a court painter and engineer. In 1494 Ludovico became the duke of Milan and, around 1496, Pacioli was invited by Ludovico to go to Milan to teach mathematics at Ludovico Sforza's court. This invitation may have been made at the prompting of Leonardo da Vinci who had an enthusiastic interest in mathematics.

At Milan Pacioli and Leonardo quickly became close friends. Mathematics and art were topics which they discussed at length, both gaining greatly from the other. At this time Pacioli began work on the second of his two famous works, Divina proportione Ⓣ and the figures for the text were drawn by Leonardo. Few mathematicians can have had a more talented illustrator for their book! The book which Pacioli worked on during 1497 would eventually form the first of three books which he published in 1509 under the title Divina proportione Ⓣ (see for example [3]). This was the first of the three books which finally made up this treatise, and it studied the 'Divine Proportion' or 'golden ratio' which is the ratio $a : b = b : (a + b)$. It contains the theorems of Euclid which relate to this ratio, and it also studies regular and semiregular polygons (see in particular [4] for a discussion of Pacioli's work on regular polygons). Clearly the interest of Leonardo in this aesthetically satisfying ratio both from a mathematical and artistic point of view was an important influence on the work. The golden ratio was also of importance in architectural design and this topic was to form the second part of the treatise which Pacioli wrote later. The third book in the treatise was a translation into Italian of one of della Francesca's works.

Louis XII became king of France in 1498 and, being a descendant of the first duke of Milan, he claimed the duchy. Venice supported Louis against Milan and in 1499 the French armies entered Milan In the following year Ludovico Sforza was captured when he attempted to retake the city. Pacioli and Leonardo fled together in December 1499, three months after the French captured Milan. They stopped first at Mantua, where they were the guests of Marchioness Isabella d'Este, and then in March 1500 they continued to Venice. From Venice they returned to Florence, where Pacioli and Leonardo shared a house.

The University of Pisa had suffered a revolt in 1494 and had moved to Florence. Pacioli was appointed to teach geometry at the University of Pisa in Florence in 1500. He remained in Florence, teaching geometry at the university, until 1506. Leonardo, although spending ten months away working for Cesare Borgia, also remained in Florence until 1506. Pacioli, like Leonardo, had a spell away from Florence when he taught at the University of Bologna during 1501-02. During this time Pacioli worked with Scipione del Ferro and there has been much conjecture as to whether the two discussed the algebraic solution of cubic equations. Certainly Pacioli discussed this topic in the Summa Ⓣ and some time after Pacioli's visit to Bologna, del Ferro solved one of the two cases of this classic problem.

During his time in Florence Pacioli was involved with Church affairs as well as with mathematics. He was elected the superior of his Order in Romagna and then, in 1506, he entered the monastery of Santa Croce in Florence. After leaving Florence, Pacioli went to Venice where he was given the sole rights to publish his works there for the following fifteen years. In 1509 he published the three volume work Divina proportione Ⓣ and also a Latin translation of Euclid's Elements. The first printed edition of Euclid's Elements was the thirteenth century translation by Campanus which had been published in printed form in Venice in 1482. Pacioli's edition was based on that of Campanus but it contained much in the way of annotation by Pacioli himself.

In 1510 Pacioli returned to Perugia to lecture there again. He also lectured again in Rome in 1514 but by this time Pacioli was 70 years of age and nearing the end of his active life of scholarship and teaching. He returned to Sansepolcro where he died in 1517 leaving unpublished a major work De Viribus Quantitatis Ⓣ on recreational problems, geometrical problems and proverbs. This work makes frequent reference to Leonardo da Vinci who worked with him on the project, and many of the problems in this treatise are also in Leonardo's notebooks. Again it is a work for which Pacioli claimed no originality, describing it as a compendium.

Despite the lack of originality in Pacioli's work, his contributions to mathematics are important, particularly because of the influence which his book were to have over a long period. In [10] the importance of Pacioli's work is discussed, in particular his computation of approximate values of a square root (using a special case of Newton's method), his incorrect analysis of certain games of chance (similar to those studied by Pascal which gave rise to the theory of probability), his problems involving number theory (similar problems appeared in Bachet's compilation), and his collection of many magic squares.

In 1550 there appeared a biography of Piero della Francesca written by Giorgio Vasari. This biography accused Pacioli of plagiarism and claimed that he stole della Francesca's work on perspective, on arithmetic and on geometry. This is an unfair accusation, for although there is truth that Pacioli relied heavily on the work of others, and certainly on that of della Francesca in particular, he never attempted to claim the work as his own but acknowledged the sources which he used.