阿布·卡米勒Abū Kāmil Shujāʿ ibn Aslam

阿布·卡米勒有时被称为al-Hasib al-Misri，意为来自埃及的计算者。关于卡米勒卡米勒的生平，我们所知甚少——也许这样说都算夸张，更诚实的说法是我们根本没有他的传记细节，只知道他来自埃及，并且我们相当确定地知道他的生卒年份。

Fihrist（索引）是书商Ibn an-Nadim在988年左右编纂的一部著作。它全面记述了10世纪可获得的阿拉伯文献，并简要描述了其中一些作者。Fihrist中提到了卡米勒卡米勒，其中列出的他的著作包括：(i)Book of fortune，(ii)Book of the key to fortune，(iii)Book on algebra，(vi)Book on surveying and geometry，(v)Book of the adequate，(vi)Book on omens，(vii)Book of the kernel，(viii)Book of the two errors，以及(ix)Book on augmentation and diminution。卡米勒卡米勒留存下来并将在下文讨论的著作包括Book on algebra, Book of rare things in the art of calculation和Book on surveying and geometry。

虽然我们对卡米勒卡米勒的生平一无所知，但我们确实了解他在代数发展中所扮演的角色。在花拉子米之前，我们没有关于代数在阿拉伯国家如何发展的信息，但许多数学史家相对较新的工作给出了自花拉子米之后该学科如何发展的合理图景。卡米勒卡米勒的作用在这里很重要，因为他是花拉子米的直接继承者之一。事实上，卡米勒卡米勒本人强调花拉子米作为“代数发明者”的作用。他将花拉子米描述为（例如见[4]或[5]）：-

……他是第一个在代数著作中取得成功的人，并开创和发明了其中的所有原理。

卡米勒卡米勒再次写道：-

我在我的第二本书中，确立了Muhammad ibn Musa花拉子米在代数中的权威和先例的证明，并且我回答了那个冲动的人Ibn Barza关于他归因于Abd al-Hamid的说法，他称其为他的祖父。

毫无疑问，卡米勒卡米勒认为他是在花拉子米所建立的代数基础上进行构建的，事实上，他在花拉子米和卡拉吉之间的代数发展中构成了一个重要环节。然而，卡米勒卡米勒的重要性还有另一个原因，即他的工作是斐波那契著作的基础。因此，卡米勒卡米勒不仅在阿拉伯代数的发展中重要，而且通过斐波那契，他在将代数引入欧洲方面也具有根本重要性。[12]的作者列出了卡米勒卡米勒的代数著作与斐波那契著作之间的平行之处，并且他还讨论了卡米勒卡米勒对卡拉吉的两部代数文本的影响。

此处一段未译出，以下为英文原文The Book on algebra by Abu Kamil is in three parts: (i) On the solution of 二次方程, (ii) On applications of algebra to the regular pentagon and decagon, and (iii) On 丢番图方程 and problems of recreational mathematics. The part on the regular pentagon and decagon is studied in detail in [7], while the remainder of the work is described in [10]. The content of the work is the application of algebra to geometrical problems. It is the combination of the geometric methods developed by the Greeks together with the practical methods developed by al-Khwarizmi mixed with Babylonian methods.

卡米勒卡米勒的代数向前迈出的重要一步是他能够处理比$x^{2}$更高次的未知数幂。这些幂不是用符号给出，而是用文字写出，但幂的命名告诉我们卡米勒卡米勒已经开始理解我们会用符号写成$x^{n}x^{m} = x^{n+m}$的东西。例如，他用“平方平方根”表示$x^{5}$（即$x^{2} \times x^{2} \times x$），“立方立方”表示$x^{6}$（即$x^{3} \times x^{3}$），“平方平方平方平方”表示$x^{8}$（即$x^{2} \times x^{2} \times x^{2} \times x^{2}$）。事实上，卡米勒卡米勒能轻松处理文本中出现的直到$x^{8}$的幂。该代数著作包含69个问题，其中包括花拉子米考虑过的40个问题中的许多问题，但处理它们的方法相当不同。

Book on surveying and geometry在[9]中有详细研究。它由卡米勒卡米勒撰写，不是为数学家，而是为政府土地测量员。由于它所针对的人群，该著作不包含证明。相反，它给出了一系列规则，其中一些远非容易，每条规则都用于几何问题的数值解法。每条规则都用一个已解出的数值例子加以说明。这些规则主要用于计算正方形、矩形和各种不同类型三角形等图形的面积、周长、对角线等。卡米勒卡米勒还给出了计算各种立体（如长方体、正圆柱、正四棱锥和圆锥）的体积和表面积的规则。

这部著作还讨论了圆，这里卡米勒 卡米勒 取 $\pi = \large\frac{22}{7}\normalsize$。整整一节专门讨论计算圆弓形的面积。著作的最后部分给出了计算3、4、5、6、8和10边的正多边形的边长的法则，这些正多边形要么内接给定直径的圆内，要么外接的给定直径的圆外。对于五边形和十边形，卡米勒 卡米勒 给出的法则，尽管在这部著作中没有证明，但在他的代数书中得到了充分证明。

Book of rare things in the art of calculation涉及不定方程的解。Sesiano在[11]中讨论了卡米勒卡米勒关于不定方程的工作，他认为他的方法非常有趣，原因有三。首先，卡米勒卡米勒是我们所知的第一位解决了丢番图著作中那种类型不定问题的阿拉伯数学家。其次，据我们所知，卡米勒卡米勒的写作时间早于阿拉伯人深入研究丢番图的Arithmetica。第三，卡米勒卡米勒解释了某些在Arithmetica已知书籍中找不到的方法。

Abu Kamil Shuja is sometimes known as al-Hasib al-Misri, meaning the calculator from Egypt. Very little is known about Abu Kamil's life - perhaps even this is an exaggeration and it would be more honest to say that we have no biographical details at all except that he came from Egypt and we know his dates with a fair degree of certainty.

The Fihrist (Index) was a work compiled by the bookseller Ibn an-Nadim around 988. It gives a full account of the Arabic literature which was available in the 10th century and it describes briefly some of the authors of this literature. The Fihrist includes a reference to Abu Kamil and among his works listed there are: (i) Book of fortune, (ii) Book of the key to fortune, (iii) Book on algebra, (vi) Book on surveying and geometry, (v) Book of the adequate, (vi) Book on omens, (vii) Book of the kernel, (viii) Book of the two errors, and (ix) Book on augmentation and diminution. Works by Abu Kamil which have survived, and will be discussed below, include Book on algebra, Book of rare things in the art of calculation, and Book on surveying and geometry.

Although we know nothing of Abu Kamil's life we do understand something of the role he plays in the development of algebra. Before al-Khwarizmi we have no information of how algebra developed in Arabic countries, but relatively recent work by a number of historians of mathematics as given a reasonable picture of how the subject developed after al-Khwarizmi. The role of Abu Kamil is important here as he was one of al-Khwarizmi's immediate successors. In fact Abu Kamil himself stresses al-Khwarizmi's role as the "inventor of algebra". He described al-Khwarizmi as (see for example [4] or [5]):-

... the one who was first to succeed in a book of algebra and who pioneered and invented all the principles in it.

Again Abu Kamil wrote:-

I have established, in my second book, proof of the authority and precedent in algebra of Muhammad ibn Musa al-Khwarizmi, and I have answered that impetuous man Ibn Barza on his attribution to Abd al-Hamid, whom he said was his grandfather.

There is certainly no doubt that Abu Kamil considered that he was building on the foundations of algebra as set up by al-Khwarizmi and indeed he forms an important link in the development of algebra between al-Khwarizmi and al-Karaji. There is another reason for Abu Kamil's importance, however, which is that his work was the basis of Fibonacci's books. So not only is Abu Kamil important in the development of Arabic algebra, but, through Fibonacci, he is also of fundamental importance in the introduction of algebra into Europe. The author of [12] presents a list of parallels between Abu Kamil's works on algebra and the works of Fibonacci, and he also discusses the influence of Abu Kamil on two algebra texts of al-Karaji.

The Book on algebra by Abu Kamil is in three parts: (i) On the solution of quadratic equations, (ii) On applications of algebra to the regular pentagon and decagon, and (iii) On Diophantine equations and problems of recreational mathematics. The part on the regular pentagon and decagon is studied in detail in [7], while the remainder of the work is described in [10]. The content of the work is the application of algebra to geometrical problems. It is the combination of the geometric methods developed by the Greeks together with the practical methods developed by al-Khwarizmi mixed with Babylonian methods.

An important step forward in Abu Kamil's algebra is his ability to work with higher powers of the unknown than $x^{2}$. These powers are not given in symbols but are written in words, yet the naming of the powers tell us that Abu Kamil had begun to understand what we would write in symbols as $x^{n}x^{m} = x^{n+m}$. For example he uses the expression "square square root" for $x^{5}$ (i.e. $x^{2} \times x^{2} \times x$), "cube cube" for $x^{6}$ (i.e. $x^{3} \times x^{3}$), "square square square square" for $x^{8}$ (i.e. $x^{2} \times x^{2} \times x^{2} \times x^{2}$). In fact Abu Kamil works easily with the powers up to $x^{8}$ which appear in the text. The algebra contains 69 problems which include many of the 40 problems considered by al-Khwarizmi, but with a rather different approach to them.

The Book on surveying and geometry is studied in detail in [9]. It was written by Abu Kamil, not for mathematicians, but rather for government land surveyors. Because of the people that it was aimed at, the work contains no proofs. Rather it presents a number of rules, some of which are far from easy, each given for the numerical solution of a geometric problem. Each rule is illustrated with a worked numerical example. Mainly the rules are for calculating the area, perimeter, diagonals etc. of figures such as squares, rectangles, and various different types of triangle. Abu Kamil also gives rules to calculate the volume and surface area of various solids such as rectangular parallelepipeds, right circular prisms, square pyramids, and circular cones.

The work also deals with circles and here Abu Kamil takes $\pi = \large\frac{22}{7}\normalsize$. A whole section is devoted to calculating the area of the segment of a circle. The final part of the work gives rules for calculating the side of regular polygons of 3, 4, 5, 6, 8, and 10 sides either inscribed in, or circumscribed about, a circle of given diameter. For the pentagon and decagon the rules which Abu Kamil gives, although without proof in this work, were fully proved in his algebra book.

The Book of rare things in the art of calculation is concerned with solutions to indeterminate equations. Sesiano in [11] discusses Abu Kamil's work on indeterminate equations and he argues that his methods are very interesting for three reasons. Firstly Abu Kamil is the first Arabic mathematician whom we know solved indeterminate problems of the type found in Diophantus's work. Secondly, as far as we know, Abu Kamil wrote before Diophantus's Arithmetica had been studied in depth by the Arabs. Thirdly, Abu Kamil explains certain methods which are not found in the known books of the Arithmetica.