《算数之钥》中球形穹顶的测量法

The Measurement Method of Spherical Domes in The Key to Arithmetic

中世纪中亚数学家阿尔·卡西在《算术之钥》中不但给出了5种伊斯兰建筑风格拱形门和球形穹顶的设计方案,而且用初等方法计算了其中拱形部分的表面积和体积,遗憾的是,他没有给出球形穹顶部分的表面积和体积的具体计算过程,只用文字叙述了用初等数学方法来计算其中第四类球形穹顶表面积和体积的基本思路,并直接给出了球形穹顶表面积和体积分别与其直径平方和直径立方之比的近似值.讨论了阿尔·卡西的初等算法,并用微积分方法来验证其算法的正确性,同时指出阿尔·卡西得到的数据与实际数据之间的误差.

In his book The Key to Arithmetic ,the central Asian mathematician al-Kashi,who belongs to middle ages,not only designed five types of Islamic arches and spherical domes,but also computed the areas and volumes of their arched parts by using elementary method.Unfortunately,he did not provide concrete algorithm about the computation of area and volume of the spherical dome parts,and only literally stated the basic idea of the computation of area and volume of the fourth type spherical dome part.He also provided the approximate values of the ratio of its surface to the square of diameter and the ratio of its volume to the third power of the diameter,respectively.In this paper,we discussed the elementary algo-rithm of al-Kashi and checked its exact value by using calculus,and at the same time,get the error between the two values.