Ibn al-Banna' al-Marrakushi

伊本al-Banna也被称为阿布·阿巴斯·艾哈迈德·伊本·穆罕默德·伊本·奥斯曼·阿兹迪。关于al-Banna是出生在马拉喀什城，还是欧洲人将马拉喀什地区命名为摩洛哥，这一点尚不完全清楚。有一种说法是al-Banna出生在西班牙的格拉纳达，并前往北非接受教育。可以确定的是，他一生大部分时间都在摩洛哥度过。

马林部落是科尔多瓦伍麦叶哈里发的盟友。该部落当时居住在摩洛哥东部，在他们的统治者阿布·叶海亚的领导下，开始征服该地区。马林人于1248年占领非斯，并将其定为首都。他们于1269年从执政的阿尔摩哈德部落手中夺取马拉喀什，从而控制了整个摩洛哥。征服摩洛哥后，马林人试图帮助格拉纳达阻止基督徒穿越他们的国家。格拉纳达与摩洛哥之间的紧密联系可能解释了关于al-Banna是哪个国家原住民的混淆。

摩洛哥无疑是al-Banna接受教育的国家，他学习了当时领先的数学技能。他学习了一般几何学，特别是欧几里得的Elements。他还学习了分数，并了解到阿拉伯人在过去400年中对数学做出的许多令人印象深刻的贡献。马林人有着浓厚的学习文化，非斯成为他们的学习中心。在非斯的大学里，al-Banna教授所有数学分支，当时包括算术、代数、几何和天文学。非斯是一个繁荣的城市，正在建造一个包含王宫和毗邻大清真寺的新区。许多学生在这个繁荣的学术社区中师从⟦N3⟧。

显然，al-Banna写了大量著作，事实上al-Banna列出了82部（例如见[9]）。并非所有都是数学著作，但数学文本包括欧几里得的Elements导论、一本代数教科书以及各种天文学著作。数学著作的一个困难在于，要知道al-Banna所呈现的材料中有多少是原创的，有多少只是他对已丢失的早期阿拉伯数学家著作的版本。我们当然应该说，⟦N4⟧没有声称任何原创性，事实上，他的写作风格表明他是在汇集从其他数学家那里学到的思想。

al-Banna有两个“第一”：他似乎是最早把分数视为两数之比的人（详见[12]），并且他是在一部包含天文与气象数据的著作中最早使用almanakc（阿拉伯语al-manakh，意为天气）这一表达的人。

也许al-Banna最著名的著作是Talkhis amal al-hisab（《算术运算概要》）以及Raf al-Hijab，后者是al-Banna对Talkhis amal al-hisab的评注。正是在这部著作中，al-Banna引入了一些数学记号，这使某些作者相信代数符号体系最早是在伊斯兰世界由ibn al-Banna和卡尔·卡拉迪发展出来的（例如见[6]）。我们请读者参阅卡尔·卡拉迪的传记，其中我们提出论据表明，al-Banna和卡尔·卡拉迪都不是数学记号的发明者。

然而，在Raf al-Hijab中出现了许多有趣的数学思想和结果。例如，它包含连分数，它们被用来计算近似平方根。关于级数求和的其它有趣结果是

$1^{3} + 3^{3} + 5^{3} + ... + (2n-1)^{3} = n^{2}(2n^{2} - 1)$和

$1^{2} + 3^{2} + 5^{2} + ... + (2n-1)^{2} = \large\frac{1}{6}\normalsize (2n + 1)2n(2n - 1)$。

也许其中最有趣的是关于二项式系数的工作，在[2]和[3]中有详细描述。如果我们用$_{p}C_{k}$表示二项式系数$p$选$k$，那么al-Banna表明

$_{p}C_{2} = \large\frac{1}{2}\normalsize p(p-1)$

然后

$_{p}C_{3} = \large\frac{1}{3}\normalsize _{p}C_{2}(p-2)$。

他写道（例如参见[2]或[3]）：-

……三元组合是这样得到的：用给定数前面的第三项的第三个数相乘；因此我们总是用所求组合的前一个组合乘以给定数前面的数，该数与给定数的距离等于所求组合的数目。从乘积中，我们取命名组合数目的部分。

虽然这有点难以解释，但al-Banna在这里陈述的是

$_{p}C_{k} = _{p}C_{k-1}\large\frac{1}{k}\normalsize (p - (k - 1) )$。

然后他继续给出（对我们来说）熟悉的结果

$_{p}C_{k} = \large\frac{1}{k!}\normalsize p(p - 1)(p - 2)...(p - k + 1)$

正如al-Banna在2中指出的，这与三百年前由卡拉吉给出的帕斯卡三角形结果相比只是一小步，而那时距离al-Banna的al-Samawal还有一百年。然而al-Banna写道：-

……在我们看来，有比[帕斯卡三角形]结果更基本的东西；确切地说，是al-Banna论述的组合外观，以及他部分建立的多边形数与组合之间的关系。这首先涉及三角数和$p$个对象两两组合，然后是4阶多边形数和$p$个对象三三组合。

Ibn al-Banna is also known as Abu'l-Abbas Ahmad ibn Muhammad ibn Uthman al-Azdi. It is a little unclear whether al-Banna was born in the city of Marrakesh or whether it was the region of Marrakesh which was named Morocco by Europeans. There is a claim that al-Banna was born in Granada in Spain and moved to North Africa for his education. What is certain is that he spent most of his life in Morocco.

The Marinids tribe were allies of the Umayyad caliphs of Córdoba. The tribe lived in eastern Morocco then, under their ruler Abu Yahya, they began to conquer the region. The Marinids captured Fez in 1248 and made it their capital. They captured Marrakesh from the ruling Almohads tribe in 1269, thus taking control of the whole of Morocco. Having conquered Morocco, the Marinids tried to help Granada to prevent the Christian advance through their country. The strong link between Granada and Morocco may account for the confusion as to which country al-Banna was a native.

Morocco was certainly the country that al-Banna was educated in, learning the leading mathematical skills of the period. He studied geometry in general, and Euclid's Elements in particular. He also studied fractional numbers and learnt much of the impressive contributions that the Arabs had made to mathematics over the preceding 400 years. The Marinids had a strong culture for learning and Fez became their centre of learning. At the university in Fez Al-Banna taught all branches of mathematics, which at this time included arithmetic, algebra, geometry and astronomy. Fez was a thriving city with a new quarter being built housing the Royal Palace and the adjoining Great Mosque. Many students studied under al-Banna in this thriving academic community.

It is clear that al-Banna wrote a large number of works, in fact 82 are listed by Renaud (see for example [9]). Not all are on mathematics, but the mathematical texts included an introduction to Euclid's Elements, an algebra text and various works on astronomy. One difficulty with the works on mathematics is knowing how much of the material which al-Banna presents is original and how much is simply his version of work by earlier Arab mathematicians which has been lost. We should certainly say that al-Banna does not claim any originality and, indeed, the style of his writing would suggest that he is collecting together ideas that he has learnt from other mathematicians.

Two "firsts" for al-Banna are that he seems to have been the first to consider a fraction as a ratio between two numbers (see [12] for more details) and he is the first to use the expression almanakc (in Arabic al-manakh meaning weather) in a work containing astronomical and meteorological data.

Perhaps al-Banna's most famous work is Talkhis amal al-hisab (Summary of arithmetical operations) and the Raf al-Hijab which is al-Banna's own commentary on the Talkhis amal al-hisab. It is in this work that al-Banna introduces some mathematical notation which has led certain authors to believe that algebraic symbolism was first developed in Islam by ibn al-Banna and al-Qalasadi (see for example [6]). We refer the reader to the biography of al-Qalasadi where we present arguments to show that neither al-Banna nor al-Qalasadi were the inventors of mathematical notation.

There are, however, many interesting mathematical ideas and results which appear in the Raf al-Hijab. For example it contains continued fractions and they are used to compute approximate square roots. Other interesting results on summing series are the results

$1^{3} + 3^{3} + 5^{3} + ... + (2n-1)^{3} = n^{2}(2n^{2} - 1)$ and

$1^{2} + 3^{2} + 5^{2} + ... + (2n-1)^{2} = \large\frac{1}{6}\normalsize (2n + 1)2n(2n - 1)$.

Perhaps the most interesting of all is the work on binomial coefficients which is described in detail in [2] and [3]. If we denote the binomial coefficient $p$ choose $k$ by $_{p}C_{k}$ then al-Banna shows that

$_{p}C_{2} = \large\frac{1}{2}\normalsize p(p-1)$

and then that

$_{p}C_{3} = \large\frac{1}{3}\normalsize _{p}C_{2}(p-2)$.

He writes (see for example [2] or [3]):-

... the ternary combination is thus obtained by multiplying the third of the third term preceding the given number; and so we always multiply the combination that precedes the combination sought by the number that precedes the given number, and whose distance to it is equal to the number of combinations sought. From the product, we take the part that names the number of combinations.

Although this is a little difficult to interpret, what al-Banna is stating here is that

$_{p}C_{k} = _{p}C_{k-1}\large\frac{1}{k}\normalsize (p - (k - 1) )$.

He then goes on to give the familiar (to us) result

$_{p}C_{k} = \large\frac{1}{k!}\normalsize p(p - 1)(p - 2)...(p - k + 1)$

As Rashed points out in [2], this is only a small step from the Pascal triangle results given three hundred years earlier by al-Karaji, then still one hundred years before al-Banna by al-Samawal. However Rashed writes:-

... in our opinion, there is something more fundamental than [the Pascal triangle] results; it is precisely the combinatorial appearance of ibn al-Banna's exposition, together with the relation he partially establishes between polygonal numbers and combinations. It concern, in the first place, triangular numbers and combinations of $p$ objects in twos, and then polygonal numbers of order 4 and combinations of $p$ objects in threes.