卡拉吉Al-Karaji

我们必须做的第一个评论是关于卡拉吉的名字。它既以卡拉吉出现，也以al-Karkhi出现，但这并不是同一个阿拉伯名字的两种不同音译的简单问题。其意义在于，Karaj是伊朗的一座城市，如果这位数学家的名字是卡拉吉，那么他的家族肯定来自那座城市。另一方面，Karkh是巴格达最初的郊区之一，是在原城市南门外发展起来的。al-Karkhi这个名字将表明这位数学家来自巴格达郊区。

历史学家对于哪种解释是正确的似乎存在分歧。al-Karkhi的版本是由Woepcke提出的（见[7]或[8]），但卡拉吉，即今天文本中最常使用的版本，是由della Vida在1933年提出为最可能的。Rashed评论道（见[1]或[5]）：-

在我们目前的知识状态下，delle Vida的论点是合理的，但并非决定性的。根据所查阅的手稿，要决定支持哪一个名字远非易事。

当然，我们知道卡拉吉一生大部分时间生活在巴格达，他的主要数学著作都是在那座城市生活期间写成的。他关于代数的重要著作Al-Fakhri是献给巴格达统治者的，也是在巴格达写成的。然而，在他职业生涯的某个后期阶段，卡拉吉离开了巴格达，去往被描述为“山地国家”的地方生活。他似乎在那时放弃了数学，专注于诸如钻井之类的工程课题。

不同作者对卡拉吉在数学发展中的重要性看法相当不同。其原因，与对花拉子米的不同看法颇为相似，取决于人们赋予其数学风格的意义。一些人认为他的工作仅仅是对早期数学家思想的重新加工，而另一些人则认为他是第一个将代数从几何运算中完全解放出来，并用算术类型的运算取而代之的人，而这些运算正是今天代数的核心。

Crossley [3]对卡拉吉的贡献显得相对不以为然（尽管他准确地描述了内容）：-

[卡拉吉]给出了算术运算的规则，包括（本质上）多项式的乘法。……卡拉吉通常为他的规则给出一个数值例子，但除了给出几何图形之外，没有给出任何形式的证明。他常常明确地说，他给出的解法是丢番图风格的。他不处理高于二次的方程，除非是那些可以容易地化为至多二次方程、然后开根的方程。quadratics中的解法明确基于欧几里得定理……

Woepcke在[7]中（另见重印本[8]）是第一位认识到卡拉吉工作重要性的历史学家，后来的历史学家大多同意他的解读。他将其描述为以下内容的首次出现：-

……代数演算理论……。

Rashed（见[5]，其中包含Rashed来自[1]的文章以及Rashed关于卡拉吉的其他著作）同意Woepcke的解读，并且在强调卡拉吉的重要性方面也许走得更远。他写道：-

……卡拉吉阐述的一个多少有些明确的目标，是找到实现代数的自主性和特异性的手段，从而能够尤其拒斥代数运算的几何表示。

再引用一段Rashed对卡拉吉贡献的描述：——

卡拉吉的工作在数学史上占有特别重要的地位。……在花拉子米和其他阿拉伯代数家的代数概念和方法的启发下，对丢番图算术著作的发现和解读，使卡拉吉在代数中实现了一种新的出发……

那么，这种代数中的新出发是什么呢？也许最好由al-Samawal来描述，他是卡拉吉的继承者之一，他将其描述为[5]：——

……用所有算术工具对未知量进行运算，就像算术家对已知量进行运算一样。

卡拉吉在Al-Fakhri中取得的成就是首先定义了单项式$x, x^{2}, x^{3}, ...$和$\Large \frac 1 {x}\normalsize, \Large \frac 1 {x^2}\normalsize, \Large \frac 1 {x^3}\normalsize, ...$，并给出了其中任意两者乘积的法则。因此，他在这里取得的成就是定义了这些项的乘积，而完全不涉及几何。事实上，他几乎给出了公式

$x^{n}x^{m} = x^{m+n}$ 对所有整数 $n$ 和 $m$

但他未能给出定义 $x^{0} = 1$，因此只差了一点点。

在给出了单项式乘法和除法的规则之后，卡拉吉 接着考察了“复合量”或单项式之和。对于这些，他给出了加法、减法和乘法的规则，但没有给出一般情况下的除法规则，只给出了复合量除以单项式的规则。他能够给出求复合量平方根的规则，这并非完全一般，因为它要求系数为正，但这仍然是一项了不起的成就。

卡拉吉 在他的论证中也使用了某种形式的数学归纳法，尽管他当然没有给出该原理的严格阐述。基本上，卡拉吉 所做的是为 $n = 1$ 演示一个论证，然后基于他对 $n = 1$ 的结果证明 $n = 2$ 的情形，再基于他对 $n = 2$ 的结果证明 $n = 3$ 的情形，并继续到大约 $n = 5$，然后指出人们可以无限地继续这个过程。虽然这不是真正的归纳法，但它是理解归纳证明的重要一步。

卡拉吉 使用这种归纳法形式所依据的结果之一，来自他关于 二项式定理、二项式系数 和 帕斯卡三角形 的工作。在 Al-Fakhri 中，卡拉吉 计算了 $(a+b)^{3}$，在 Al-Badi 中，他计算了 $(a-b)^{3}$ 和 $(a+b)^{4}$。帕斯卡三角形的一般构造由 卡拉吉 给出，其工作记载于 al-Samawal 后来的著作中。在 Rashed 和 Ahmad 的译本中（例如见 [5]），al-Samawal 写道：-

现在让我们回顾一个原理，用于知道这些次数彼此相乘的必要次数，对于任意分成两部分的数。卡拉吉 说，为了成功，我们必须把“一”放在一张桌子上，把“一”放在第一个“一”下面，把第一个“一”移到第二列，把第一个“一”加到它下面的“一”上。这样我们得到“二”，我们把它放在移过来的“一”下面，并把第二个“一”放在“二”下面。因此我们有“一”、“二”和“一”。

为了看出 1,2,1 的第二列如何对应于 a+b 的平方，al-Samawal 继续描述 卡拉吉 的工作，写道：-

这表明，对于每个由两个数组成的数，如果我们把每个数自身相乘一次——因为两个端项是“一”和“一”——并且如果我们把每个数与另一个数相乘两次——因为中间项是“二”——我们就得到这个数的平方。

这是用帕斯卡三角形对二项式定理的精彩描述。描述一直延续到给出$(a+b)^{5}$的二项式系数，但我们只引用卡拉吉如何从第二列构造第三列：-

如果我们把第二列中的“一”移到第三列，然后把第二列中的“一”加到它下面的“二”上，我们得到“三”，写在第三列的“一”下面。如果我们再把第二列中的“二”加到它下面的“一”上，我们得到“三”，写在“三”下面，然后我们在这个“三”下面写“一”；这样我们就得到了第三列，其数字为“一”、“三”、“三”和“一”。

插图：Karaji.gif ↗
卡拉吉构造的表格看起来像侧放的帕斯卡三角形。

卡拉吉得到的其他结果包括前$n$个自然数之和、前$n$个自然数的平方和以及这些数的立方和。他证明了前$n$个自然数之和为${1\over2}n(n + 1)$。他还给出（在Rashed和Ahmad的译本中，例如见[5]）：-

从一按自然顺序相继的各个数的平方和等于这些数之和加上每个数与其前一个数乘积之和。

用现代记号表示，这个结果是

$\sum i ^{2} = \sum i + \sum i (i - 1)$。

卡拉吉还考虑了前$n$个自然数的立方和，写道（在Rashed和Ahmad的译本中，例如见[5]）：-

如果我们想将按自然顺序相继的数的立方相加，我们将它们的和自乘。

用现代记号表示

$\sum i ^{3} = (\sum i)^{2}$。

卡拉吉证明了$(1 + 2 + 3 + ... + 10)^{2}$等于$1^{3} + 2^{3} + 3^{3} + ... + 10^{3}$。他首先证明$(1 + 2 + 3 + ... + 10)^{2} = (1 + 2 + 3 + ... + 9)^{2} + 10^{3}$来做到这一点。他现在可以对$(1 + 2 + 3 + ... + 9)^{2}$使用同样的规则，然后对$(1 + 2 + 3 + ... + 8)^{2}$等等，得到

$( 1 + 2 + ... + 10)^{2}$
$= (1 + 2 + 3 + ... + 8)^{2} + 9^{3} + 10^{3}$
$= (1 + 2 + 3 + ... + 7)^{2} + 8^{3} + 9^{3} + 10^{3}$
= . . .
$= 1^{3} + 2^{3} + 3^{3} + ... + 10^{3}.$

最后我们应该提到丢番图对卡拉吉的影响。丢番图的Arithmetica的前五卷在约870年由ibn Liqa翻译成阿拉伯文，这些被卡拉吉研究过。Woepcke在Al-Fakhri（[7]或[8]）的导言中写道，他发现：-

……丢番图第一卷中超过三分之一的问题、第二卷从第八个问题开始的问题，以及第三卷几乎所有的问题，都被卡拉吉收录在他的文集中。

卡拉吉还自己发明了许多新问题，但即使是丢番图的那些问题，也肯定不是未经进一步发展就被直接采用的。他总是试图推广丢番图的结果，并寻找更普遍适用的方法。

卡拉吉不仅对代数有贡献。论文[9]讨论了他的一些几何工作。这出现在题为On measurement and balances for measuring of buildings and structures的一章中。卡拉吉定义了点、线、面、体和角。他还给出了测量平面和立体图形的规则，经常使用拱作为例子。他还给出了称量不同物质的方法。

The first comment that we must make regards al-Karaji's name. It appears both as al-Karaji and as al-Karkhi but this is not a simple matter of two different transliterations of the same Arabic name. The significance is that Karaj is a city in Iran and if the mathematician's name is al-Karaji then certainly his family were from that city. On the other hand Karkh is one of the original suburbs of Baghdad which grew up outside the southern gate of the original city. The name al-Karkhi would indicate that the mathematician came from the suburb of Baghdad.

Historians seem divided as to which of these interpretations is correct. The version al-Karkhi was proposed by Woepcke (see [7] or [8]) but al-Karaji, the version which is most often used in texts today, was suggested as most likely by della Vida in 1933. Rashed comments (see [1] or [5]):-

In the present state of our knowledge delle Vida's argument is plausible but not decisive. On the basis of the manuscripts consulted it is far from easy to decide in favour of either name.

Certainly we know that al-Karaji lived in Baghdad for most of his life and that his chief mathematical works were written during the time when he lived in that city. His important treatise on algebra Al-Fakhri was dedicated to the ruler of Baghdad and was written in the city. However, at some later point in his career, al-Karaji left Baghdad to live in what are described as the "mountain countries". He seems to have given up mathematics at this time and concentrated on engineering topics such as the drilling of wells.

The importance of al-Karaji in the development of mathematics is viewed rather differently by different authors. The reason for this, rather in the same spirit as the different views on al-Khwarizmi, depends on the significance one attaches to the style of his mathematics. Some consider that his work is merely reworking ideas from earlier mathematicians while others see him as the first person to completely free algebra from geometrical operations and replace them with the arithmetical type of operations which are at the core of algebra today.

Crossley [3] sounds relatively unimpressed by al-Karaji's contributions (although he describes the content accurately):-

[Al-Karaji] gives rules for the arithmetic operations including (essentially) the multiplication of polynomials. ... al-Karaji usually gives a numerical example for his rules but does not give any sort of proof beyond giving geometrical pictures. Often he explicitely says that he is giving a solution in the style of Diophantus. He does not treat equations above the second degree except for ones which can easily be reduced to at most second degree equations followed by the extraction of roots. The solutions of quadratics are based explicitly on the Euclidean theorems ...

Woepcke in [7] (see also the reprint [8]) was the first historian to realise the importance of al-Karaji's work and later historians mostly agree with his interpretation. He describes it as the first appearance of a:-

... theory of algebraic calculus ... .

Rashed (see [5] which contains Rashed's article from [1] and other writings by Rashed on al-Karaji) agrees with Woepcke's interpretation and perhaps goes even further in stressing al-Karaji's importance. He writes:-

... the more-or-less explicit aim of [al-Karaji's] exposition was to find the means of realising the autonomy and specificity of algebra, so as to be in a position to reject, in particular, the geometric representation of algebraic operations.

To give another quote from Rashed's description of al-Karaji's contribution:-

Al-Karaji's work holds an especially important place in the history of mathematics. ... the discovery and reading of the arithmetical work of Diophantus, in the light of the algebraic conceptions and methods of al-Khwarizmi and other Arab algebraists, made possible a new departure in algebra by Al-Karaji ...

So what was this new departure in algebra? Perhaps it is best described by al-Samawal, one of al-Karaji's successors, who described it as [5]:-

... operating on unknowns using all the arithmetical tools, in the same way as the arithmetician operates on the known.

What al-Karaji achieved in Al-Fakhri was first to define the monomials $x, x^{2}, x^{3}, ...$ and $\Large \frac 1 {x}\normalsize, \Large \frac 1 {x^2}\normalsize, \Large \frac 1 {x^3}\normalsize, ...$ and to give rules for products of any two of these. So what he achieved here was defining the product of these terms without any reference to geometry. In fact he almost gave the formula

$x^{n}x^{m} = x^{m+n}$ for all integers $n$ and $m$

but he failed to make the definition $x^{0} = 1$ so he fell just a little short.

Having given rules for multiplication and division of monomials al-Karaji then looked at "composite quantities" or sums of monomials. For these he gave rules for addition, subtraction and multiplication but not for division in the general case, only giving rules for the division of a composite quantity by a monomial. He was able to give a rule for finding the square root of a composite quantity which is not completely general since it required the coefficients to be positive, but it is still a remarkable achievement.

Al-Karaji also uses a form of mathematical induction in his arguments, although he certainly does not give a rigorous exposition of the principle. Basically what al-Karaji does is to demonstrate an argument for $n = 1$, then prove the case $n = 2$ based on his result for $n = 1$, then prove the case $n = 3$ based on his result for $n = 2$, and carry on to around $n = 5$ before remarking that one can continue the process indefinitely. Although this is not induction proper, it is a major step towards understanding inductive proofs.

One of the results on which al-Karaji uses this form of induction comes from his work on the binomial theorem, the binomial coefficients and the Pascal triangle. In Al-Fakhri al-Karaji computed $(a+b)^{3}$ and in Al-Badi he computed $(a-b)^{3}$ and $(a+b)^{4}$. The general construction of the Pascal triangle was given by al-Karaji in work described in the later writings of al-Samawal. In the translation by Rashed and Ahmad (see for example [5]) al-Samawal writes:-

Let us now recall a principle for knowing the necessary number of multiplications of these degrees by each other, for any number divided into two parts. Al-Karaji said that in order to succeed we must place 'one' on a table and 'one' below the first 'one', move the first 'one' into a second column, add the first 'one' to the 'one' below it. Thus we obtain 'two', which we put below the transferred 'one' and we place the second 'one' below the 'two'. We have therefore 'one', 'two', and 'one'.

To see how the second column of 1,2,1 corresponds to squaring a+b al-Samawal continues to describe Al-Karaji's work writing:-

This shows that for every number composed of two numbers, if we multiple each of them by itself once - since the two extremes are 'one' and 'one' - and if we multiply each one by the other twice - since the intermediate term is 'two' - we obtain the square of this number.

This is a beautiful description of the binomial theorem using the Pascal triangle. The description continues up to the binomial coefficients which give $(a+b)^{5}$ but we shall only quote how al-Karaji constructs the third column from the second:-

If we transfer the 'one' in the second column into a third column, then add 'one' from the second column to 'two' below it, we obtain 'three' to be written under the 'one' in the third column. If we then add 'two' from the second column to ''one' below it we have 'three' which is written under the 'three', then we write 'one' under this 'three'; we thus obtain a third column whose numbers are 'one', 'three', 'three', and 'one'.

插图：Karaji.gif ↗
The table al-Karaji constructed looks like the Pascal triangle on its side.

Other results obtained by al-Karaji include summing the first $n$ natural numbers, the squares of the first $n$ natural numbers and the cubes of these numbers. He proved that the sum of the first $n$ natural numbers was ${1\over2}n(n + 1)$. He also gave (in Rashed and Ahmad's translation, see for example [5]):-

The sum of the squares of the numbers that follow one another in natural order from one is equal to the sum of these numbers and the product of each of them by its predecessor.

In modern notation this result is

$\sum i ^{2} = \sum i + \sum i (i - 1)$.

Al-Karaji also considered sums of the cubes of the first $n$ natural numbers writing (in Rashed and Ahmad's translation, see for example [5]):-

If we want to add the cubes of the numbers that follow one another in their natural order we multiply their sum by itself.

In modern notation

$\sum i ^{3} = (\sum i)^{2}$.

Al-Karaji showed that $(1 + 2 + 3 + ... + 10)^{2}$ was equal to $1^{3} + 2^{3} + 3^{3} + ... + 10^{3}$. He did this by first showing that $(1 + 2 + 3 + ... + 10)^{2} = (1 + 2 + 3 + ... + 9)^{2} + 10^{3}$. He could now use the same rule on $(1 + 2 + 3 + ... + 9)^{2}$, then on $(1 + 2 + 3 + ... + 8)^{2}$ etc. to get

$( 1 + 2 + ... + 10)^{2}$
$= (1 + 2 + 3 + ... + 8)^{2} + 9^{3} + 10^{3}$
$= (1 + 2 + 3 + ... + 7)^{2} + 8^{3} + 9^{3} + 10^{3}$
= . . .
$= 1^{3} + 2^{3} + 3^{3} + ... + 10^{3}.$

Finally we should mention the influence of Diophantus on al-Karaji. The first five books of Diophantus's Arithmetica had been translated into Arabic by ibn Liqa around 870 and these were studied by al-Karaji. Woepcke in his introduction to Al-Fakhri ([7] or [8]) writes that he found:-

... more than a third of the problems of the first book of Diophantus, the problems of the second book starting with the eighth, and virtually all the problems of the third book were included by al-Karaji in his collection.

Al-Karaji also invented many new problem of his own but even those of Diophantus were certainly not just taken without further development. He always tried to generalise Diophantus's results and to find methods which were more generally applicable.

It was not only to algebra that al-Karaji contributed. The paper [9] discusses some of his geometrical work. This occurs in a chapter entitled On measurement and balances for measuring of buildings and structures. al-Karaji defines points, lines, surfaces, solids and angles. He also gives rules for measuring both plane and solid figures, often using arches as examples. He also gives methods of weighing different substances.