斐波那契Fibonacci

斐波那契 Pisano以其绰号斐波那契更为人所知。他是Guilielmo的儿子，也是Bonacci家族的成员。斐波那契本人有时使用Bigollo这个名字，这可能意味着无用之人或旅行者。正如[1]中所述：-

他的同胞是想用这个绰号来表达对一个关心无实用价值问题之人的蔑视，还是这个词在托斯卡纳方言中意为一个游历甚广的人，而他正是如此？

斐波那契出生于意大利，但在北非接受教育，他的父亲Guilielmo在那里担任外交职务。他父亲的工作是代表在布吉亚（后来称为布吉，现在称为贝贾亚）进行贸易的比萨共和国商人。贝贾亚是阿尔及利亚东北部的一个地中海港口。该镇位于苏马姆河河口，靠近古拉亚山和卡本角。斐波那契在布吉亚学习数学，并随父亲广泛旅行，认识到他们所访问国家使用的数学体系的巨大优势。斐波那契在其著名著作Liber abaciⓉ（算盘书）（1202年）中写道：-

当我的父亲被他的国家任命为布吉亚海关的公证人，为前往那里的比萨商人服务时，他正在任职，在我还是个孩子的时候就把我叫到身边，考虑到实用性和未来的便利，希望我留在那里，在会计学校接受教育。在那里，当我通过出色的教学接触到印度人的九个符号的艺术时，这门艺术的知识很快便令我欣喜不已，胜过一切，我逐渐理解了它，因为无论在埃及、叙利亚、希腊、西西里还是普罗旺斯，这门艺术所研究的一切，无论其形式如何多样。

斐波那契大约在1200年结束了他的旅行，那时他回到了比萨。在那里，他写了许多重要的著作，这些著作在复兴古代数学技能方面发挥了重要作用，他也做出了自己的重大贡献。斐波那契生活在印刷术出现之前，所以他的书都是手写的，要得到他的一本书的副本，唯一的办法就是让人再手抄一份。在他的著作中，我们仍然拥有Liber abaci Ⓣ（《计算之书》）（1202年）、Practica geometriae Ⓣ（《实用几何》）（1220年）、Flos Ⓣ（《花》）（1225年）和Liber quadratorum Ⓣ（《平方数之书》）的副本。鉴于手抄副本的数量相对较少，我们能有幸接触到他在这些著作中的文字。然而，我们知道他还写过一些其他文本，不幸的是，这些文本已经失传。他关于商业算术的书Di minor guisa Ⓣ（《小方法之书》）已经失传，他对欧几里得的Elements第十卷的评注也已失传，该评注包含了对无理数数的数值处理，而欧几里得曾从几何角度对此进行过探讨。

人们可能会认为，在欧洲对学术兴趣不大的时候，斐波那契会在很大程度上被忽视。然而，事实并非如此，对他的作品的广泛兴趣无疑对他的重要性起到了很大的推动作用。斐波那契是Jordanus的同时代人，但他是一位远为精深的数学家，他的成就得到了明确认可，尽管使他同时代人闻名的是实际应用而非抽象定理。

神圣罗马帝国皇帝是腓特烈二世。他于1212年加冕为德国国王，然后于1220年11月在罗马圣彼得教堂被教皇加冕为神圣罗马帝国皇帝。腓特烈二世支持比萨在海上与热那亚的冲突，以及在陆地上与卢卡和佛罗伦萨的冲突，他直到1227年都在巩固他在意大利的权力。国家对贸易和制造实行控制，监督这一垄断的公务员在那不勒斯大学接受培训，腓特烈于1224年为此目的创办了该大学。

腓特烈通过他宫廷中的学者了解到斐波那契的工作，这些学者自斐波那契大约1200年返回比萨以来就与他通信。这些学者包括宫廷占星家斯格特、宫廷哲学家Theodorus Physicus和Dominicus Hispanus，后者向腓特烈建议，当腓特烈的宫廷于1225年左右在比萨集会时，他会见斐波那契。

巴勒莫的约翰内斯，腓特烈二世宫廷的另一位成员，向伟大的数学家斐波那契提出了一些问题作为挑战。其中三个问题由斐波那契解决，他在FlosⓉ（花）中给出了解答，并将其寄给了腓特烈二世。下面我们给出其中一个问题的一些细节。

1228年之后，只有一份已知文件提到了斐波那契。这是比萨共和国在1240年颁布的一项法令，其中授予薪水给：-

……严肃而博学的斐波那契 Bigollo大师……。

这份薪水是给予斐波那契的，以表彰他为城市提供的服务，在会计事务上提供建议并教导市民。

Liber abaciⓉ（算盘书）于1202年斐波那契返回意大利后出版，献给了斯格特。这本书基于斐波那契在旅行期间积累的算术和代数知识。这本书后来被广泛抄写和模仿，将印度-阿拉伯位置值十进制系统和使用阿拉伯数字引入欧洲。事实上，尽管主要是一本关于使用阿拉伯数字的书，后来被称为算法，但在这部著作中也研究了联立线性方程。当然，斐波那契在Liber abaciⓉ（算盘书）中考虑的许多问题与阿拉伯来源中出现的问题相似。

Liber abaciⓉ（算盘书）的第二部分包含大量针对商人的问题。它们涉及商品价格、如何计算交易利润、如何在地中海国家使用的各种货币之间转换，以及起源于中国的问题。

Liber abaci Ⓣ（《计算之书》）第三部分中的一个问题引出了斐波那契数和斐波那契数列，而斐波那契今天最为人所铭记的正是后者：-

某人把一对兔子放在一个四面被墙围住的地方。假设每个月每对兔子都生出一对新兔子，而新兔子从第二个月起就能生育，那么一年内这对兔子能繁殖出多少对兔子？

由此得到的数列是 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...（斐波那契在Liber abaciⓉ（《算盘书》）中省略了第一项）。这个数列中每个数都是前两个数之和，已被证明极为多产，出现在数学和科学的许多不同领域中。Fibonacci Quarterly是一本现代期刊，致力于研究与这个数列相关的数学。

这第三部分还给出了许多其他问题，包括这些类型，以及许许多多其他类型：

一只蜘蛛每天沿墙向上爬若干英尺，每晚又滑回固定的距离，问它需要多少天才能爬上墙顶。
一只速度按算术级数增加的猎犬追赶一只速度也按算术级数增加的野兔，问猎犬追上野兔前它们各跑了多远。
计算两人在若干钱转手后各有多少钱，已知按比例增加和减少的量。

还有涉及完全数的问题、涉及中国剩余定理的问题，以及涉及算术级数和几何级数求和的问题。

斐波那契在第四节中处理了诸如√10这样的数，既使用了有理的近似值，也使用了几何构造。

斐波那契于1228年推出了Liber abaciⓉ（《算盘书》）的第二版，并附有一篇序言，其内容正如许多书籍第二版序言的典型做法，声称：-

……书中增加了新材料，删去了多余的内容……

斐波那契的另一部著作是Practica geometriaeⓉ（《实用几何》），写于1220年，题献给上文提到的Dominicus Hispanus。书中收录了大量几何问题，分为八章，其中的定理以欧几里得的Elements和欧几里得的On Divisions为基础。除了带有严格证明的几何定理外，该书还包含供测量员使用的实用信息，其中有一章讲述如何利用相似三角形计算高大物体的高度。最后一章给出了斐波那契所称的几何精妙之处[1]：——

其中包括由外接的和内接圆的直径计算五边形和十边形的边长；也给出了逆运算，以及由面积计算边长。……为完成关于等边三角形的部分，在这样的三角形内作一个矩形和一个正方形，并用代数方法计算它们的边长……

在FlosⓉ（《花》）中，斐波那契给出了$10x + 2x^{2} + x^{3} = 20$一个根的精确实用的近似值，这是巴勒莫的Johannes向他提出挑战要求解决的问题之一。这个问题并非巴勒莫的Johannes所创，而是取自奥马尔·海亚姆的代数书，在那里是通过一个圆与一条双曲线的交点来求解的。斐波那契证明该方程的根既不是整数也不是分数，也不是分数的平方根。他接着写道：——

由于无法用上述其他任何方法解此方程，我便设法将解法化为一种近似。

斐波那契没有解释他的方法，随后用六十进制记法给出了近似解1.22.7.42.33.4.40（这是以60为基写出的，因此等于$1 + \large\frac{22}{60}\normalsize + \large\frac{7}{60^{2}\normalsize} + \large\frac{42}{60^{3}\normalsize} + ...$）。它换算成十进制为1.3688081075，精确到九位小数，这是一项了不起的成就。

Liber quadratorum写于1225年，是斐波那契最令人印象深刻的作品，尽管并非他最著名的作品。书名意为《平方数之书》，它是一本数论著作，其中除其他内容外，还考察了寻找毕达哥拉斯三元组的方法。斐波那契首先指出，平方数可以构造为奇数之和，本质上描述了使用公式$n^{2} + (2n+1) = (n+1)^{2}$的归纳构造。斐波那契写道：-

我思考了所有平方数的起源，发现它们源于奇数的规则递增。因为1是一个平方数，由它产生第一个平方数，即1；加上3得到第二个平方数，即4，其根为2；若在此和上加上第三个奇数，即5，将产生第三个平方数，即9，其根为3；如此，平方数的序列和级数总是通过奇数的规则相加而上升。

为了构造毕达哥拉斯三元组，斐波那契进行如下操作：-

因此，当我想找到两个平方数，其和产生一个平方数时，我取任意奇平方数作为这两个平方数之一，并通过将从1开始直到但不包括该奇平方数的所有奇数相加来找到另一个平方数。例如，我取9作为所提到的两个平方数之一；剩余的平方数将通过加上9以下的所有奇数，即1、3、5、7来获得，其和为16，是一个平方数，当加到9上时得到25，是一个平方数。

斐波那契还证明了许多有趣的数论结果，例如：

不存在$x, y$使得$x^{2} + y^{2}$和$x^{2} - y^{2}$都是平方数。

而$x^{4} - y^{4}$不可能是平方数。

他定义了congruum的概念，即形如$ab(a + b)(a - b)$的数，如果$a + b$是偶数，以及如果$a + b$是奇数则为此数的4倍。斐波那契证明了同余数必须能被24整除，他还证明了对于满足$x^{2} + c$和$x^{2} - c$均为平方数的$x, c$，则$c$是同余数。他还证明了平方数不可能是同余数。

正如[2]中所述：-

……仅Ⓣ（《平方数之书》）一书就使斐波那契跻身于丢番图与17世纪法国数学家Pierre de 皮埃尔·德·费马之间数论的主要贡献者之列。

斐波那契的影响比人们所期望的更为有限，除了他在传播印度-阿拉伯数字的使用和兔子问题方面的作用外，斐波那契对数学的贡献在很大程度上被忽视了。正如[1]中所解释的：-

直接影响仅由《计算之书》和《实践》中那些用于引入印度-阿拉伯数字和方法并有助于掌握日常生活问题的部分所施加。在这里，斐波那契成为了计算大师和测量员的老师，正如从卢卡·帕西奥利的《大全》Ⓣ（算术、几何、比例和比例性概要）中可以了解到的那样……斐波那契也是“Cossists”的老师，后者得名于'causa'一词，该词在西方首先由斐波那契使用，以代替'res'或'radix'。他对一般数或系数的字母表示法首先由弗朗索瓦·韦达改进……

斐波那契在数论方面的工作在中世纪几乎完全被忽视，实际上无人知晓。三百年后，我们发现同样的结果出现在弗朗西斯科·马若利科的著作中。

上方的肖像来自一幅现代版画，据信并非基于可靠的来源。

Leonardo Pisano is better known by his nickname Fibonacci. He was the son of Guilielmo and a member of the Bonacci family. Fibonacci himself sometimes used the name Bigollo, which may mean good-for-nothing or a traveller. As stated in [1]:-

Did his countrymen wish to express by this epithet their disdain for a man who concerned himself with questions of no practical value, or does the word in the Tuscan dialect mean a much-travelled man, which he was?

Fibonacci was born in Italy but was educated in North Africa where his father, Guilielmo, held a diplomatic post. His father's job was to represent the merchants of the Republic of Pisa who were trading in Bugia, later called Bougie and now called Bejaia. Bejaia is a Mediterranean port in northeastern Algeria. The town lies at the mouth of the Wadi Soummam near Mount Gouraya and Cape Carbon. Fibonacci was taught mathematics in Bugia and travelled widely with his father and recognised the enormous advantages of the mathematical systems used in the countries they visited. Fibonacci writes in his famous book Liber abaci Ⓣ (1202):-

When my father, who had been appointed by his country as public notary in the customs at Bugia acting for the Pisan merchants going there, was in charge, he summoned me to him while I was still a child, and having an eye to usefulness and future convenience, desired me to stay there and receive instruction in the school of accounting. There, when I had been introduced to the art of the Indians' nine symbols through remarkable teaching, knowledge of the art very soon pleased me above all else and I came to understand it, for whatever was studied by the art in Egypt, Syria, Greece, Sicily and Provence, in all its various forms.

Fibonacci ended his travels around the year 1200 and at that time he returned to Pisa. There he wrote a number of important texts which played an important role in reviving ancient mathematical skills and he made significant contributions of his own. Fibonacci lived in the days before printing, so his books were hand written and the only way to have a copy of one of his books was to have another hand-written copy made. Of his books we still have copies of Liber abaci Ⓣ (1202), Practica geometriae Ⓣ (1220), Flos Ⓣ (1225), and Liber quadratorum Ⓣ. Given that relatively few hand-made copies would ever have been produced, we are fortunate to have access to his writing in these works. However, we know that he wrote some other texts which, unfortunately, are lost. His book on commercial arithmetic Di minor guisa Ⓣ is lost, as is his commentary on Book X of Euclid's Elements which contained a numerical treatment of irrational numbers which Euclid had approached from a geometric point of view.

One might have thought that at a time when Europe was little interested in scholarship, Fibonacci would have been largely ignored. This, however, is not so and widespread interest in his work undoubtedly contributed strongly to his importance. Fibonacci was a contemporary of Jordanus but he was a far more sophisticated mathematician and his achievements were clearly recognised, although it was the practical applications rather than the abstract theorems that made him famous to his contemporaries.

The Holy Roman emperor was Frederick II. He had been crowned king of Germany in 1212 and then crowned Holy Roman emperor by the Pope in St Peter's Church in Rome in November 1220. Frederick II supported Pisa in its conflicts with Genoa at sea, and with Lucca and Florence on land, and he spent the years up to 1227 consolidating his power in Italy. State control was introduced on trade and manufacture, and civil servants to oversee this monopoly were trained at the University of Naples which Frederick founded for this purpose in 1224.

Frederick became aware of Fibonacci's work through the scholars at his court who had corresponded with Fibonacci since his return to Pisa around 1200. These scholars included Michael Scotus who was the court astrologer, Theodorus Physicus the court philosopher and Dominicus Hispanus, who suggested to Frederick that he meet Fibonacci when Frederick's court met in Pisa around 1225.

Johannes of Palermo, another member of Frederick II's court, presented a number of problems as challenges to the great mathematician Fibonacci. Three of these problems were solved by Fibonacci and he gives solutions in Flos Ⓣ which he sent to Frederick II. We give some details of one of these problems below.

After 1228 there is only one known document which refers to Fibonacci. This is a decree made by the Republic of Pisa in 1240 in which a salary is awarded to:-

... the serious and learned Master Leonardo Bigollo ... .

This salary was given to Fibonacci in recognition for the services that he had given to the city, advising on matters of accounting and teaching the citizens.

Liber abaci Ⓣ, published in 1202 after Fibonacci's return to Italy, was dedicated to Scotus. The book was based on the arithmetic and algebra that Fibonacci had accumulated during his travels. The book, which went on to be widely copied and imitated, introduced the Hindu-Arabic place-valued decimal system and the use of Arabic numerals into Europe. Indeed, although mainly a book about the use of Arab numerals, which became known as algorism, simultaneous linear equations are also studied in this work. Certainly many of the problems that Fibonacci considers in Liber abaci Ⓣ were similar to those appearing in Arab sources.

The second section of Liber abaci Ⓣ contains a large collection of problems aimed at merchants. They relate to the price of goods, how to calculate profit on transactions, how to convert between the various currencies in use in Mediterranean countries, and problems which had originated in China.

A problem in the third section of Liber abaci Ⓣ led to the introduction of the Fibonacci numbers and the Fibonacci sequence for which Fibonacci is best remembered today:-

A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?

The resulting sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... (Fibonacci omitted the first term in Liber abaci Ⓣ). This sequence, in which each number is the sum of the two preceding numbers, has proved extremely fruitful and appears in many different areas of mathematics and science. The Fibonacci Quarterly is a modern journal devoted to studying mathematics related to this sequence.

Many other problems are given in this third section, including these types, and many many more:

A spider climbs so many feet up a wall each day and slips back a fixed number each night, how many days does it take him to climb the wall.
A hound whose speed increases arithmetically chases a hare whose speed also increases arithmetically, how far do they travel before the hound catches the hare.
Calculate the amount of money two people have after a certain amount changes hands and the proportional increase and decrease are given.

There are also problems involving perfect numbers, problems involving the Chinese remainder theorem and problems involving summing arithmetic and geometric series.

Fibonacci treats numbers such as √10 in the fourth section, both with rational approximations and with geometric constructions.

A second edition of Liber abaci Ⓣ was produced by Fibonacci in 1228 with a preface, typical of so many second editions of books, stating that:-

... new material has been added [to the book] from which superfluous had been removed ...

Another of Fibonacci's books is Practica geometriae Ⓣ written in 1220 which is dedicated to Dominicus Hispanus whom we mentioned above. It contains a large collection of geometry problems arranged into eight chapters with theorems based on Euclid's Elements and Euclid's On Divisions. In addition to geometrical theorems with precise proofs, the book includes practical information for surveyors, including a chapter on how to calculate the height of tall objects using similar triangles. The final chapter presents what Fibonacci called geometrical subtleties [1]:-

Among those included is the calculation of the sides of the pentagon and the decagon from the diameter of circumscribed and inscribed circles; the inverse calculation is also given, as well as that of the sides from the surfaces. ... to complete the section on equilateral triangles, a rectangle and a square are inscribed in such a triangle and their sides are algebraically calculated ...

In Flos Ⓣ Fibonacci gives an accurate approximation to a root of $10x + 2x^{2} + x^{3} = 20$, one of the problems that he was challenged to solve by Johannes of Palermo. This problem was not made up by Johannes of Palermo, rather he took it from Omar Khayyam's algebra book where it is solved by means of the intersection of a circle and a hyperbola. Fibonacci proves that the root of the equation is neither an integer nor a fraction, nor the square root of a fraction. He then continues:-

And because it was not possible to solve this equation in any other of the above ways, I worked to reduce the solution to an approximation.

Without explaining his methods, Fibonacci then gives the approximate solution in sexagesimal notation as 1.22.7.42.33.4.40 (this is written to base 60, so it is $1 + \large\frac{22}{60}\normalsize + \large\frac{7}{60^{2}\normalsize} + \large\frac{42}{60^{3}\normalsize} + ...$). This converts to the decimal 1.3688081075 which is correct to nine decimal places, a remarkable achievement.

Liber quadratorum, written in 1225, is Fibonacci's most impressive piece of work, although not the work for which he is most famous. The book's name means the book of squares and it is a number theory book which, among other things, examines methods to find Pythagorean triples. Fibonacci first notes that square numbers can be constructed as sums of odd numbers, essentially describing an inductive construction using the formula $n^{2} + (2n+1) = (n+1)^{2}$. Fibonacci writes:-

I thought about the origin of all square numbers and discovered that they arose from the regular ascent of odd numbers. For unity is a square and from it is produced the first square, namely 1; adding 3 to this makes the second square, namely 4, whose root is 2; if to this sum is added a third odd number, namely 5, the third square will be produced, namely 9, whose root is 3; and so the sequence and series of square numbers always rise through the regular addition of odd numbers.

To construct the Pythagorean triples, Fibonacci proceeds as follows:-

Thus when I wish to find two square numbers whose addition produces a square number, I take any odd square number as one of the two square numbers and I find the other square number by the addition of all the odd numbers from unity up to but excluding the odd square number. For example, I take 9 as one of the two squares mentioned; the remaining square will be obtained by the addition of all the odd numbers below 9, namely 1, 3, 5, 7, whose sum is 16, a square number, which when added to 9 gives 25, a square number.

Fibonacci also proves many interesting number theory results such as:

there is no $x, y$ such that $x^{2} + y^{2}$ and $x^{2} - y^{2}$ are both squares.

and $x^{4} - y^{4}$ cannot be a square.

He defined the concept of a congruum, a number of the form $ab(a + b)(a - b)$, if $a + b$ is even, and 4 times this if $a + b$ is odd. Fibonacci proved that a congruum must be divisible by 24 and he also showed that for $x, c$ such that $x^{2} + c$ and $x^{2} - c$ are both squares, then $c$ is a congruum. He also proved that a square cannot be a congruum.

As stated in [2]:-

... the Liber quadratorum Ⓣ alone ranks Fibonacci as the major contributor to number theory between Diophantus and the 17th -century French mathematician Pierre de Fermat.

Fibonacci's influence was more limited than one might have hoped and apart from his role in spreading the use of the Hindu-Arabic numerals and his rabbit problem, Fibonacci's contribution to mathematics has been largely overlooked. As explained in [1]:-

Direct influence was exerted only by those portions of the "Liber abaci" and of the "Practica" that served to introduce Indian-Arabic numerals and methods and contributed to the mastering of the problems of daily life. Here Fibonacci became the teacher of the masters of computation and of the surveyors, as one learns from the "Summa" Ⓣ of Luca Pacioli ... Fibonacci was also the teacher of the "Cossists", who took their name from the word 'causa' which was first used in the West by Fibonacci in place of 'res' or 'radix'. His alphabetic designation for the general number or coefficient was first improved by Viète ...

Fibonacci's work in number theory was almost wholly ignored and virtually unknown during the Middle ages. Three hundred years later we find the same results appearing in the work of Maurolico.

The portrait above is from a modern engraving and is believed to not be based on authentic sources.