阿布·瓦法Abu al-Wafa Buzjani

阿布·瓦法'l-Wafa是在一个新王朝建立并统治伊朗的时期长大的。布维希伊斯兰王朝在945年至1055年间统治伊朗西部和伊拉克，处于阿拉伯和土耳其征服之间的时期。该时期始于945年，当时Ahmad Buyeh占领了阿拔斯王朝的首都巴格达。布维希王朝的高峰是在'Adud ad-Dawlah从949年到983年的统治期间。他从巴格达统治整个伊朗南部和现在伊拉克的大部分地区。作为科学和艺术的大赞助人，'Adud ad-Dawlah支持了许多数学家，瓦法'l-Wafa于959年移居到巴格达的'Adud ad-Dawlah宫廷。瓦法'l-Wafa并不是巴格达哈里发宫廷中唯一杰出的科学家，因为像阿布·萨赫勒·库希和阿布·赛义德·西杰兹这样的杰出数学家也在那里工作。

Sharaf ad-Dawlah是'Adud ad-Dawlah的儿子，他于983年成为哈里发。他继续支持数学和天文学，瓦法'l-Wafa和阿布·萨赫勒·库希留在巴格达宫廷为新哈里发工作。Sharaf ad-Dawlah要求建立一座天文台，它建在巴格达宫殿的花园中。天文台于988年6月正式开放，许多著名科学家出席，如阿布·萨赫勒·库希和瓦法'l-Wafa。

天文台中的仪器包括一个超过6米长的象限仪和一个18米的石制六分仪。据说瓦法'l-Wafa是第一个建造墙象限仪来观测恒星的人。然而，哈里发Sharaf ad-Dawlah在次年去世，天文台被关闭。

像他那个时代的许多科学家一样，瓦法'l-Wafa翻译并撰写了关于欧几里得、丢番图和花拉子米著作的注释，这些注释后来丢失了。在961年至976年之间的某个时候，他写了Kitab fi ma yahtaj ilayh al-kuttab wa'l-ummal min 'ilm al-hisab Ⓣ（关于抄写员和商人所需的算术科学）。在这本书的引言中，瓦法'l-Wafa写道，它（[3]或[4]）：-

... 包括所有经验丰富或新手、下属或主管在算术中需要知道的一切，公务员的艺术，土地税的使用以及行政中所需的各种业务，比例，乘法，除法，测量，土地税，分配，交换以及各类人为做生意而使用的所有其他实践，这些实践对他们的日常生活有用。

有趣的是，在此期间，有两种类型的算术书被撰写，一种使用印度符号，另一种是手指计算类型。瓦法'l-Wafa的文本属于第二种类型，没有数字；所有数字都用文字书写，所有计算都是心算。早期历史学家如莫里兹·贝内迪克特·康托尔认为存在对立的作者学派，一个致力于印度方法，另一个致力于希腊方法。然而，这后来被否证了（例如见[9]），现在人们认为数学家为两种不同类型的读者写作。瓦法'l-Wafa本人是使用印度数字的专家，但这些[1]：-

... 在商业圈和东哈里发国的人口中长期没有得到应用。

因此，他使用手指计算算术写了他的文本，因为这是商业界使用的系统。这部作品分为七个部分，每个部分包含七章

第一部分：论比（分数被表示为由“基本”分数$\large\frac{1}{2}\normalsize , \large\frac{1}{3}\normalsize , \large\frac{1}{4}\normalsize , ... ,\large\frac{1}{10}\normalsize$构成）。

第二部分：关于乘法和除法（整数和分数的算术运算）。

第三部分：测量（图形的面积、固体的体积和求距离）。

第四部分：关于税收（不同种类的税和税收计算问题）。

第五部分：论交换与份额（作物种类，以及与其价值和交换有关的问题）。

第六部分：杂项主题（货币单位、士兵的支付、河上船只通行许可的授予与扣留、道路上的商人）。

第七部分：进一步的商业主题。

这部著作在[12]中有详细研究（另见[10]）。特别令人感兴趣的是瓦法'l-Wafa的论著第二部分中对负数的提及，这一特定方面在[11]和[12]中有详细研究（另见[1]）。这似乎是中世纪阿拉伯数学中发现负数的唯一地方。瓦法'l-Wafa给出了一条一般规则，并给出了一个特例，其中从3中减去5得到2的“债务”。然后他将其乘以10得到20的“债务”，当将其加到(10 - 3)(10 - 5) = 35时，得到3和5的乘积，即15。

瓦法'l-Wafa为实际使用而写的另一部文本是A book on those geometric constructions which are necessary for a craftsman。这比他关于算术的文本晚得多，肯定是在990年之后。这本书有十三章，它考虑了绘图工具的设计和测试、直角的构造、近似三等分角、抛物线的构造、正多边形以及将它们内接于给定圆和外切于给定圆的方法、将各种多边形内接于给定多边形、诸如平面多边形等图形的分割，以及将球面分割为正球面多边形。

瓦法'l-Wafa这部特定著作的另一个有趣方面是，他尽可能用尺规作图来解决问题。当这不可能时，他使用近似方法。然而，有一整类问题他是用直尺和固定圆规解决的，即圆规两脚之间的角度是固定的。在[1]中提出：-

对这些作图的兴趣可能是由于在实践中它们比改变圆规开度所能获得的结果更精确。

瓦法'l-Wafa最为人所知的是首次使用tan函数并编制了间隔为15'的正弦和正切表。这项工作是在对月球轨道的调查中完成的，记录在Theories of the Moon中。他还引入了sec和cosec，并研究了与弧相关的六条三角线之间的相互关系。

瓦法'l-Wafa设计了一种计算正弦表的新方法。他的三角函数表精确到8位小数（转换为十进制表示法），而克劳狄乌斯·托勒密的仅精确到3位。

他的其他著作包括Kitab al-Kamil Ⓣ（完整版），这是克劳狄乌斯·托勒密的Almagest Ⓣ的简化版本（主要论题：源自阿拉伯语“al-majisti”——希腊语“Mathematike Syntaxis”的阿拉伯语译本，后来被译为拉丁语“Magna Syntaxis”）。尽管这部著作中似乎没有什么新颖的理论趣味，但其中的观测数据似乎被许多后来的天文学家所使用。

Abu'l-Wafa was brought up during the period that a new dynasty was being established which would rule over Iran. The Buyid Islamic dynasty ruled in western Iran and Iraq from 945 to 1055 in the period between the Arab and Turkish conquests. The period began in 945 when Ahmad Buyeh occupied the 'Abbasid capital of Baghdad. The high point of the Buyid dynasty was during the reign of 'Adud ad-Dawlah from 949 to 983. He ruled from Baghdad over all southern Iran and most of what is now Iraq. A great patron of science and the arts, 'Adud ad-Dawlah supported a number of mathematicians and Abu'l-Wafa moved to 'Adud ad-Dawlah's court in Baghdad in 959. Abu'l-Wafa was not the only distinguished scientist at the Caliph's court in Baghdad, for outstanding mathematicians such as al-Quhi and al-Sijzi also worked there.

Sharaf ad-Dawlah was 'Adud ad-Dawlah's son and he became Caliph in 983. He continued to support mathematics and astronomy and Abu'l-Wafa and al-Quhi remained at the court in Baghdad working for the new Caliph. Sharaf ad-Dawlah required an observatory to be set up, and it was built in the garden of the palace in Baghdad. The observatory was officially opened in June 988 with a number of famous scientists present such as al-Quhi and Abu'l-Wafa.

The instruments in the observatory included a quadrant over 6 metres long and a stone sextant of 18 metres. Abu'l-Wafa is said to have been the first to build a wall quadrant to observe the stars. However, the caliph Sharaf ad-Dawlah died in the following year and the observatory was closed.

Like many scientist of his period, Abu'l-Wafa translated and wrote commentaries, which have since been lost, on the works of Euclid, Diophantus and al-Khwarizmi. Some time between 961 and 976 he wrote Kitab fi ma yahtaj ilayh al-kuttab wa'l-ummal min 'ilm al-hisab Ⓣ. In the introduction to this book Abu'l-Wafa writes that it ([3] or [4]):-

... comprises all that an experienced or novice, subordinate or chief in arithmetic needs to know, the art of civil servants, the employment of land taxes and all kinds of business needed in administrations, proportions, multiplication, division, measurements, land taxes, distribution, exchange and all other practices used by various categories of men for doing business and which are useful to them in their daily life.

It is interesting that during this period there were two types of arithmetic books written, those using Indian symbols and those of finger-reckoning type. Abu'l-Wafa's text is of this second type with no numerals; all the numbers are written in words and all calculations are performed mentally. Early historians such as Moritz Cantor believed that there were opposing schools of authors, one committed to Indian methods, the other to Greek methods. However, this has since been disproved (see for example [9]), and it is now believed that mathematicians wrote for two differing types of readers. Abu'l-Wafa himself was an expert in the use of Indian numerals but these [1]:-

... did not find application in business circles and among the population of the Eastern Caliphate for a long time.

Hence he wrote his text using finger-reckoning arithmetic since this was the system used for by the business community. The work is in seven parts, each part containing seven chapters

Part I: On ratio (fractions are represented as made from the "capital" fractions $\large\frac{1}{2}\normalsize , \large\frac{1}{3}\normalsize , \large\frac{1}{4}\normalsize , ... ,\large\frac{1}{10}\normalsize$).

Part II: On multiplication and division (arithmetical operations with integers and fractions).

Part III: Mensuration (area of figures, volume of solids and finding distances).

Part IV: On taxes (different kinds of taxes and problems of tax calculations).

Part V: On exchange and shares (types of crops, and problems relating to their value and exchange).

Part VI: Miscellaneous topics (units of money, payment of soldiers, the granting and withholding of permits for ships on the river, merchants on the roads).

Part VII: Further business topics.

This work is studied in detail in [12] (see also [10]). Of particular interest is the reference to negative numbers in Part II of Abu'l-Wafa's treatise, and this particular aspect is studied in detail in [11] and [12] (see also [1]). This seems to be the only place that negative numbers have been found in medieval Arabic mathematics. Abu'l-Wafa gives a general rule and gives a special case of this where subtraction of 5 from 3 gives a "debt" of 2. He then multiples this by 10 to obtain a "debt" of 20, which when added to (10 - 3)(10 - 5) = 35 gives the product of 3 and 5, namely 15.

Another text written by Abu'l-Wafa for practical use was A book on those geometric constructions which are necessary for a craftsman. This was written much later than his arithmetic text, certainly after 990. The book is in thirteen chapters and it considered the design and testing of drafting instruments, the construction of right angles, approximate angle trisections, constructions of parabolas, regular polygons and methods of inscribing them in and circumscribing them about given circles, inscribing of various polygons in given polygons, the division of figures such as plane polygons, and the division of spherical surfaces into regular spherical polygons.

Another interesting aspect of this particular work of Abu'l-Wafa's is that he tries where possible to solve his problems with ruler and compass constructions. When this is not possible he uses approximate methods. However, there are a whole collection of problems which he solves using a ruler and fixed compass, that is one where the angle between the legs of the compass is fixed. It is suggested in [1] that:-

Interest in these constructions was probably aroused by the fact that in practice they give more exact results than can be obtained by changing the compass opening.

Abu'l-Wafa is best known for the first use of the tan function and compiling tables of sines and tangents at 15' intervals. This work was done as part of an investigation into the orbit of the Moon, written down in Theories of the Moon. He also introduced the sec and cosec and studied the interrelations between the six trigonometric lines associated with an arc.

Abu'l-Wafa devised a new method of calculating sine tables. His trigonometric tables are accurate to 8 decimal places (converted to decimal notation) while Ptolemy's were only accurate to 3 places.

His other works include Kitab al-Kamil Ⓣ, a simplified version of Ptolemy's Almagest Ⓣ. Although there seems to have been little of novel theoretical interest in this work, the observational data in it seem to have been used by many later astronomers.