阿尔伯特‧吉拉德Albert Girard

阿尔伯特‧吉拉德是法国人，但作为归正会成员，他作为宗教难民去了荷兰。我们不知道他何时做出这一举动，但我们知道他一生都为被迫生活在祖国之外而悲伤。他就读于莱顿大学，在那里学习数学，22岁进入大学。事实上，他最初的兴趣是音乐，他专业地演奏鲁特琴。Jacob Golius与阿尔伯特‧吉拉德年龄相仿，但早几年就开始在莱顿大学学习数学。 certainly by 1616 the two were engaging in mathematical discussions and there is surviving correspondence from that time in which they are discussing scientific matters. Golius在摩洛哥和叙利亚及阿拉伯地区旅行中度过了几年。他于1629年被任命为莱顿的数学教授（除了他1625年的阿拉伯语教授职位）。当Constantijn 克里斯蒂安·惠更斯（克里斯蒂安·惠更斯的父亲）写信祝贺Golius获得数学任命时，他赞扬了阿尔伯特‧吉拉德的工作，特别是关于折射的工作，这 certainly suggests that the two had continued to exchange ideas. 然而我们知道，当惠更斯写这封信时，阿尔伯特‧吉拉德正在为奥兰治亲王、杰森·约翰·拿骚的弗雷德里克·亨利的军队担任工程师。

阿尔伯特‧吉拉德研究代数、三角学和算术。他通过出版西蒙·斯蒂文的各种著作对数学做出了重大贡献。1625年，他准备了一版西蒙·斯蒂文的ArithmétiqueⓉ（算术）的修订版，但他还加入了丢番图的ArithmeticaⓉ（算术）第5卷和第6卷的希腊文翻译，以及西蒙·斯蒂文的AppendiceⓉ（附录）。西蒙·斯蒂文制作了正弦、正切和正割表，阿尔伯特‧吉拉德对其进行了大幅改进，并于1626年出版了他的版本。在这部关于三角学的著作TrigonométrieⓉ（三角学）中，他首次使用了缩写sin、cos、tan。他还给出了球面三角形面积的公式。在代数方面，他对代数基本定理有一些早期的想法，并在Invention Nouvelle en l'AlgèbreⓉ（代数中的新发明）（1629年）中阐述了这些想法。Gray Funkhouser写道[7]：-

第一个真正在方程根的对称函数史上占有一席之地的人，一个不仅在这一主题上，而且在代数其他方面，对现有材料的清晰性和把握能力，完全可以在一个世纪后仍保持其地位的人，是阿尔伯特‧吉拉德……他的代数著作是一本34页的小册子，名为《代数中的新发明》Ⓣ（代数中的新发明），出版于1629年。阿尔伯特‧吉拉德给出了后来被称为Pascal's triangle的三角形，并以它为基础发展了一个关于对称函数的定理，尽管他并没有把它们本身视为对称函数。

阿尔伯特‧吉拉德称布莱兹·帕斯卡的三角形为“提取三角形”。他把一组数的和称为“第一分数”，把两两乘积的和称为“第二分数”，等等。然后他给出一个定理：如果给定一组数，每个分数的乘积的数量可以用提取三角形中的同一行来表示，与数的数量相同。

他给出了一个方程的例子（我们用现代记号写出）

$x^{4} = 4x^{3} + 7x^{2} - 34x + 24.$

由于未知数的最高次幂是4，阿尔伯特‧吉拉德明确指出有四个根，“不多也不少”。他把偶次幂移到左边，奇次幂移到右边，得到

$x^{4} - 7x^{2} - 24 = 4x^{3} - 34x.$

他接着说，这些系数连同它们各自的符号是 4、-7、-34、-24。于是 4 是第一个分数，即根的和；-7 是第二个分数，即所有两根乘积之和；-34 是第三个分数，即所有三根乘积之和；-24 是第四个分数，即四根之积。接着他给出另一个例子，即 $x^{4} - 4x + 3 = 0$，它有两个虚根，并表明该方法在这种情况下仍给出正确答案。

然后他考察根的和、根的平方和、根的立方和等等。他写道：-

如果第二、第三、第四项等的系数是 $A, B, C$ 等，那么在任何次数的方程中

$A$ 将是根的和；
$A^{2} - 2B$ 将是根的平方和；
$A^{3} -3AB + 3C$ 将是根的立方和；
$A^{4} -4A^{2}B + 4AC + 2B^{2} -4D$ 将是根的四次方和。

查尔斯·赫顿 在 [8] 中详细叙述了 Invention Nouvelle en l'Algèbre Ⓣ（《代数中的新发明》）的内容。他解释说，该书共 63 页，其中 49 页是关于算术和代数的：-

……其余部分是关于球面三角形和多边形的表面度量，由他当时新近发现。

在详细叙述了那49页关于算术与代数的内容之后，查尔斯·赫顿给出了这样的总结：-

1. He was the first person who understood the general doctrine of the formation of the coefficients of the powers, from the sums of their roots, and their products, etc。
2. He was the first who understood the use of negative roots in the solution of geometrical problems。
3. He was the first who spoke of the imaginary roots, and understood that every equation might have as many roots real and imaginary, and no more, as there are units in the index of the highest power. And he was the first who gave the whimsical name of quantities less than nothing to the negative。
4. He was the first who discovered the rules for summing the powers of the roots of any equation。

我们还应当提到他解方程的迭代方法[1]：-

借助三角函数表，阿尔伯特‧吉拉德解出了有三个实根的三次方程。对于只有一个根的方程，除了吉罗拉莫·卡尔达诺的法则之外，他还指出了一种借助三角函数表和迭代进行数值求解的巧妙方法。

他是第一个给出负数几何解释的人，他写道：-

负的解在几何上通过向后移动来解释，当+前进时，负号就后退。

像他那个时代的许多数学家一样，阿尔伯特‧吉拉德对数学在军事上的应用感兴趣，尤其研究了防御工事。他翻译了几部关于防御工事的著作，其中一些是从法语译成弗拉芒语，例如皮埃尔·萨米埃尔 Marolois的Fortification ou architecture militaireⓉ（防御工事或军事建筑），他还为此添加了材料并修订了文本。他对两卷本专著Géométrie contenant la théorie et practique d'icelle. necessaire à la FortificationⓉ（几何学：包含防御工事所必需的理论与实践）也做了同样的事。他还将其他著作从弗拉芒语译成法语，例如Oeuvres de Henry HondiusⓉ（Henry Hondius著作集）（1625年）。

看来阿尔伯特‧吉拉德曾在荷兰军队中担任工程师一段时间，尽管这很可能是在他发表三角学著作之后。皮埃尔·伽桑狄在1629年7月21日写信给他的朋友Nicholas de Peiresc时谈到了阿尔伯特‧吉拉德，并提到了他在荷兰军队中的职位[1]：-

[我在Boisle-Roi前的营地与]阿尔伯特‧吉拉德共进晚餐，他是一位现在在营地的工程师。

在他去世时，他被描述为工程师而非数学家，尽管在他一生中，他自己总是自称数学家。在他编辑的西蒙·斯蒂文的著作中（在他去世后出版），阿尔伯特‧吉拉德说他生活在异国他乡很不快乐，没有人提供经济支持来帮助他抚养他的大家庭[1]：-

他的遗孀在这部作品的献词中说得更具体。她很穷，有十一个孤儿，他们的父亲只留给他们忠实服务并毕生致力于研究数学最崇高秘密的声誉。

阿尔伯特‧吉拉德致力于制作Les Oeuvres mathématiques de Simon Stevin augmentées par Albert GirardⓉ（斯蒂文的数学著作，由阿尔伯特‧吉拉德增补），但在作品出版前的1632年去世；这发生在1634年。由西蒙·斯蒂文的遗孀和子女签名的献词包含了我们上面引用的段落。George Sarton写道[12]：-

阿尔伯特‧吉拉德本人是一位伟大的数学家，他在Stevinian文本中添加了许多自己的观察：这些观察可以很容易地与其余部分区分开来。有些作品由Tuning或斯蒂文翻译，其他作品由他自己翻译并缩写；他自己的添加总是特别标明。因此，《Oeuvres》可以用来研究西蒙·斯蒂文自己的思想，但必须小心不要将阿尔伯特‧吉拉德明显的插入内容归因于西蒙·斯蒂文。

萨顿指出，阿尔伯特‧吉拉德 就 西蒙·斯蒂文 关于“智慧时代”的思想写了一篇“奇怪的评论”。萨顿特别写道：-

阿尔伯特‧吉拉德 在一本法文书中对法语进行攻击，这确实很奇特。

阿尔伯特‧吉拉德 还因首次为 斐波那契数列 表述了（如今众所周知的）归纳定义 $f_{n+2} = f_{n+1} + f_{n}$，并指出 斐波那契 数列各项之比趋于黄金比而闻名，这些出现在他1634年的出版物中。罗伯特·西姆松 在1753年写道［13］：-

阿尔伯特‧吉拉德 给出的第一件事……是一种用 有理的 数表示按中外比分割的线段之比的方法，这些数收敛于真实比值。为此他取数列 0, 1, 1, 2, 3, 5, 8, 13, 21 等，其中每一项都等于它前面两项之和：并且说，这个数列中的任何一个数与其后一个数之比［近似］等于任何另一个数与其后一个数之比。因此 5 与 8 之比近似等于 8 与 13 之比；从而，任何相邻的三个数如 8, 13, 21，近似表示按中外比分割的线段的两段以及整条线段；所以 13, 21, 34 足够近似地构成一个等腰三角形，具有五边形的角……阿尔伯特‧吉拉德 提到的第二件事，是一种给出收敛于任何给定数的平方根的有理分数序列的方法，而且收敛得非常快。他没有告诉我们形成它的方法，只给出了以下两个例子；也就是说，他说 √2 近似等于 $\Large\frac{577}{408}\normalsize$：或者，如果你想要更接近，$\Large\frac{1393}{985}\normalsize$。他的另一个例子是 √10，他说它近似等于 $\Large\frac{1039681}{328776}\normalsize$。而这些……乍看之下，是同一值的连分数。

不能把 连分数 的发明归功于 阿尔伯特‧吉拉德 的那些辉煌观察，但他的天才再次闪耀。事实上，人们不免有些伤感：阿尔伯特‧吉拉德 的名字今天并不广为人知，然而人们觉得，如果他花时间充分解释他显然理解的东西，并且也花些时间把他一些惊人的洞见再推进一步，事情本可能有所不同。Jean Itard 写道［1］：-

［阿尔伯特‧吉拉德］总是时间紧迫，通常又缺少篇幅，他非常吝惜文字，对证明更是如此；因此，他常常提出的多于他证明的。

Albert Girard was French but, being a member of the Reformed church, went as a religious refugee to the Netherlands. We do not know when he made this move, but we do know he was sad throughout his live that he was forced to live outside his native land. He attended the University of Leiden, where he studied mathematics, entering the University at the age of 22. In fact his first interest was music and he played the lute professionally. Jacob Golius was about the same age as Girard but began studying mathematics at the University of Leiden some years earlier. Certainly by 1616 the two were engaging in mathematical discussions and there is surviving correspondence from that time in which they are discussing scientific matters. Golius spent several years in Morocco and on tours of Syria and Arabic lands. He was appointed professor of mathematics at Leiden in 1629 (in addition to his Arabic professorship of 1625). When Constantijn Huygens (Christiaan Huygens' father) wrote a congratulatory note to Golius on his mathematics appointment, he praised the work of Girard, particularly on refraction, which certainly suggests that the two had continued to exchange ideas. We know however that by the time Huygens wrote this letter, Girard was serving as an engineer in the army of the Prince of Orange, Frederick Henry of Nassau.

Girard worked on algebra, trigonometry and arithmetic. He made a major contribution to mathematics by publishing various works by Simon Stevin. In 1625 he prepared a revised edition of Stevin's Arithmétique Ⓣ but he also added to it translations from the Greek of Books 5 and 6 of Diophantus's Arithmetica Ⓣ as well as Stevin's Appendice Ⓣ. Stevin had produced tables of sines, tangents and secants which were greatly improved by Girard who published his version in 1626. In this work Trigonométrie Ⓣ on trigonometry he made the first use of the abbreviations sin, cos, tan. He also gave formulas for the area of a spherical triangle. In algebra he had some early thoughts on the fundamental theorem of algebra which he stated in Invention Nouvelle en l'Algèbre Ⓣ (1629). Gray Funkhouser writes [7]:-

The first man who really has a place in the history of symmetric functions of roots of equations, a man who for clearness and grasp of material at hand in not only this topic but also in other phases of algebra could well hold his place a century later was Albert Girard ... his work on algebra is a little 34-leaf pamphlet called 'Invention Nouvelle en l'Algèbre' Ⓣ, published in 1629. Girard gives the triangle later known as Pascal's triangle and uses it as the basis for developing a theorem on symmetric functions, although he has no idea of them as such.

Girard calls Pascal's triangle the "triangle of extraction". He calls the sum of a group of numbers the "first fraction", the sum of the products of pairs of the numbers the "second fraction", etc. He then gives a theorem: If a group of numbers is given, the multitude of the products of each fraction can be expressed by the same row in the triangle of extraction as the multitude of numbers.

He gives an example of the equation (which we write in modern notation)

$x^{4} = 4x^{3} + 7x^{2} - 34x + 24.$

Since the highest power of the unknown is 4, Girard states clearly that there are four roots "neither more nor less". He takes the even powers to the left, the odd powers to the right giving

$x^{4} - 7x^{2} - 24 = 4x^{3} - 34x.$

He then says that the coefficients, with their proper signs, are 4, -7, -34, -24. Then 4 is the first fraction, namely the sum of the roots, -7 is the second fraction, namely the sum of all products of pairs of roots, -34 is the third fraction, namely the sum of all products of three roots, -24 is the fourth fraction, namely the product of the four roots. Following this he gives another example, namely $x^{4} - 4x + 3 = 0$ which has two imaginary roots, and shows that the method still gives the right answer in this case.

He then looks at the sum of the roots, the sum of the squares of the roots, the sum of the cubes of the roots, etc. He writes:-

If the coefficients of the second, third, fourth terms etc. are $A, B, C$, etc. then in an equation of any degree

$A$ will be the sum of the roots;
$A^{2} - 2B$ will be the sum of the squares of the roots;
$A^{3} -3AB + 3C$will be the sum of the cubes of the roots;
$A^{4} -4A^{2}B + 4AC + 2B^{2} -4D$ will be the sum of the fourth powers of the roots.

Charles Hutton gives a detailed account of the contents of Invention Nouvelle en l'Algèbre Ⓣ in [8]. He explains that of the 63 pages in the book, 49 are on arithmetic and algebra:-

... and the rest on the measure of the superficies of spherical triangles and polygons, by him then lately discovered.

After giving a detailed account of the 49 pages on arithmetic and algebra, Hutton gives this summary:-

1. He was the first person who understood the general doctrine of the formation of the coefficients of the powers, from the sums of their roots, and their products, etc.
2. He was the first who understood the use of negative roots in the solution of geometrical problems.
3. He was the first who spoke of the imaginary roots, and understood that every equation might have as many roots real and imaginary, and no more, as there are units in the index of the highest power. And he was the first who gave the whimsical name of quantities less than nothing to the negative.
4. He was the first who discovered the rules for summing the powers of the roots of any equation.

We should also mention his iterative approach to solving equations [1]:-

With the aid of trigonometric tables Girard solved equations of the third degree having three real roots. For those having only one root he indicated, beside Cardano's rules, an elegant method of numerical solution by means of trigonometric tables and iteration.

He was the first to give a geometric interpretation of negative quantities, writing:-

The negative solution is explained in geometry by moving backward, and the minus sign moves back when the + advances.

Like many mathematicians of his day Albert Girard was interested in military applications of mathematics and in particular studied fortifications. He translated several works on fortifications some from French to Flemish such as Samuel Marolois's Fortification ou architecture militaire Ⓣ to which he also added material and revised the text. He did the same for the two-volume treatise Géométrie contenant la théorie et practique d'icelle. necessaire à la Fortification Ⓣ. Other works he translated from Flemish to French such as Oeuvres de Henry Hondius Ⓣ (1625).

It appears that Girard spent some time as an engineer in the Dutch army although this was probably after he published his work on trigonometry. Pierre Gassendi, writing on 21 July 1629 to his friend Nicholas de Peiresc, talks about Girard and refers to his position in the Dutch army [1]:-

[I dined at the camp before Boisle-Roi with] Albert Girard, an engineer now at the camp.

On his death he was described as an engineer rather than as a mathematician although, throughout his life, he himself always described himself as a mathematician. In the works of Stevin which he edited (published after his death), Girard says he is unhappy living in a foreign country with nobody to provide him with financial support to help him bring up his large family [1]:-

His widow, in the dedication of this work, is more precise. She is poor, with eleven orphans to whom their father has left only his reputation of having faithfully served and having spent all his time on research on the most noble secrets of mathematics.

Girard worked on producing Les Oeuvres mathématiques de Simon Stevin augmentées par Albert Girard Ⓣ but died in 1632 before the work was published; this happened in 1634. The dedication, signed by Stevin's widow and children, contains the passage we quoted above. George Sarton writes [12]:-

Girard was himself a great mathematician, and he added many observations of his own to the Stevinian text: these observations can be easily distinguished from the rest. Some works were translated by Tuning or Stevin, others were translated by himself and abbreviated; his own additions are always specifically mentioned as such. Hence the 'Oeuvres' can be used to study Stevin's own thought, but one must be careful not to ascribe Girard's unmistakable interpolations to Stevin.

Sarton notes that Girard wrote a "strange commentary" on Stevin's ideas on the "age of wisdom". In particular, Sarton writes:-

Girard's attack on the French language in a French book is certainly curious.

Girard is also famed for being the first to formulate the (now well-known) inductive definition $f_{n+2} = f_{n+1} + f_{n}$ for the Fibonacci sequence, and stating that the ratios of terms of the Fibonacci sequence tend to the golden ratio, which appear in this 1634 publication. Robert Simson writes in 1753 [13]:-

The first thing Albert Girard gives ... is a method of expressing the ratio of the segments of a line cut in extreme and mean proportion, by rational numbers, that converge to the true ratio. For this purpose he takes the progression 0, 1, 1, 2, 3, 5, 8, 13, 21, etc. every term of which is equal to the sum of the two terms that precede it: and says, any number in this progression has unto the following the same ratio [nearly] that any other has to that, which follows it. Thus 5 has to 8 nearly the same ratio, that 8 has to 13; consequently, any 3 numbers next to one another as 8, 13, 21, nearly express the segments of a line cut in extreme and mean proportion, and the whole line; so that 13, 21, 34 constitute near enough an isosceles triangle, having the angle of a pentagon ... The second thing which Albert Girard mentions, is a way of exhibiting a series of rational fractions, that converge to the square root of any number proposed, and that very fast. He tells us nothing about the way of forming it, and gives the following two examples; namely he says that √2 is equal nearly to $\Large\frac{577}{408}\normalsize$: or, if you would have it nearer, $\Large\frac{1393}{985}\normalsize$. His other example is of √10, which, he says, is nearly equal to $\Large\frac{1039681}{328776}\normalsize$. And these are ... at first sight, continued fractions of the same value.

Girard cannot be credited with the invention of continued fractions as a result of his brilliant observations but again his genius shines through. In fact one is left with a little sadness that Girard's name is not today well-known yet one feels that things could have been different if he had taken the time to fully explain the things he obviously understood and also taken some time to push a little further some of his amazing insights. Jean Itard writes [1]:-

[Girard was] always pressed for time and generally lacking space, he was very stingy with words and still more so with demonstrations; thus, he very often suggested more than he demonstrated.