婆罗摩笈多

婆罗摩笈多
婆罗摩笈多
出生	598年; 哈尔沙帝国拉贾斯坦邦宾马尔
逝世	670年; 瞿折罗-普罗蒂诃罗
职业	印度数学家和天文学家

婆罗摩笈多（梵语：ब्रह्मगुप्त，IAST: Brahmagupta，598年—668年），印度数学家和天文学家，生于印度拉贾斯坦邦宾马尔^[1]，一生可能大多数时间都在生地度过。当时，该地隶属于哈尔沙帝国。婆罗摩笈多为乌贾因天文台台长，于任职期间著书二部，乃关于数学和天文学，当中包括于628年写成的《婆罗摩历算书（英语：Brāhmasphuṭasiddhānta）》。

婆罗摩笈多系首位提出0计算规则的数学家。和当时许多的印度数学家一样，会将文字编排成椭圆形的句子，而且最后会有一个环状排列的诗。由于未提出证明，不知其中的数学推导过程^[2]。

生平和著作

在《婆罗摩历算书》第十四篇的第7句及第8句，提及婆罗摩笈多于三十岁著作此书，也是628年，便可推得婆罗摩笈多是在598年出生^[3] ^[1]。婆罗摩笈多写了四本有关数学及天文学的书，分别为624年的《Cadamekela》、628年的《婆罗摩历算书（英语：Brāhmasphuṭasiddhānta）》、665年的《Khandakhadyaka》及672年的《Durkeamynarda》，其中最著名的是《婆罗摩历算书》。波斯历史学家比鲁尼在其著作《Tariq al-Hind》提到，阿拉伯帝国阿拔斯王朝的哈里发马蒙曾派大使到印度，并将一本“算书”带到巴格达，翻译为阿拉伯文，一般认为这本算书就是《婆罗摩历算书》^[4]。

数学

《婆罗摩历算书》中，有四章半讲述纯数学，第12章讲述演算系列和少许几何学。第18章是关于代数，婆罗摩笈多在这里引入了一个解二次丢番图方程如nx² + 1 = y²的方法。

婆罗摩笈多还提供了计算任何四边已知的圆内接四边形的面积的公式。海伦公式是婆罗摩笈多给出的公式的一个特殊形式（一边为零）。婆罗摩笈多公式与海伦公式之间的关系，类似馀弦定理扩展了勾股定理。

代数

婆罗摩笈多在《婆罗摩历算书》第十八章给了线性方程的解：

色之间的数交换后的差除以未知数的差，就是方程的解。^[5]

当中方程 $bx+c=dx+e$ 的解是 $x={\tfrac {e-c}{b-d}}$ ，而色是指常数项c和e。他然后进一步给了二次方程两个解：

18.44：色和二次项和4相乘的积加一次项的二次方的数，把这个数开方后减一次项，再把整个数除一次项的2倍，就是方程的解。^{[注 1]}
18.45：色和二次项的积加一次项一半的二次方的数，把这个数开方后减一次项的一半，再把整个数除一次项就是方程的解。^{[注 2]}^[5]

其实它们分别说了方程 $ax^{2}+bx=c$ 的解恒等于

x={\frac {{\sqrt {4ac+b^{2}}}-b}{2a}}

和

x={\frac {{\sqrt {ac+{\tfrac {b^{2}}{4}}}}-{\tfrac {b}{2}}}{a}}

。

运算

级数

婆罗摩笈多提供了头 $n$ 个平方和及立方和的算法：

12.20. 平方和是[头几个整数直接和]乘以两倍[项数]与1的和后再除以3的结果。立方和是这直接和的平方。^{[注 3]}^[6]

婆罗摩笈多的方法较近似于现代形式。

这里，婆罗摩笈多所给的头 $n$ 个自然数的平方和立方的算法，分别为 ${\tfrac {n(n+1)(2n+1)}{6}}$ 和 $[{\tfrac {n(n+1)}{2}}]^{2}$

零

婆罗摩笈多普及了数学中的重要概念：0。《婆罗摩历算书》是至今为止，已知首部将0视为普通数字使用之著作。除此之外，这部书还阐述了负数和0的运算规则。这些规则与现代规则非常接近。

婆罗摩笈多在《婆罗摩历算书》第十八章中这样提到：

18.30：正数加正数为正数，负数加负数为负数。正数加负数为他们彼此的差，如果它们相等，结果就是零。负数加零为负数，正数加零为正数，零加零为零^{[注 4]}
18.32：负数减零为负数，正数减零为正数，零减零为零，正数减负数为他们彼此的和。^{[注 5]}^[5]

他这样描述乘法：

18.33：正负得负，负负得正，正正得正，正数乘零﹑负数乘零和零乘零都是零。^{[注 6]}^[5]

最大的区别在于，婆罗摩笈多试图定义除以零，在现代数学中这个运算无解。

18.34：正数除正数或负数除负数为正数，正数除负数或负数除正数为负数，零除零为零^{[注 7]}^[5]
18.35：正数或负数除以零有零作为该数的除数，零除以正数或负数有正数或负数作为该数的除数。正数或负数的平方为正数，零的平方为零。^{[注 8]}^[5]

婆罗摩笈多的定义不实用，比如他认为 ${\tfrac {0}{0}}=0$ 。而他并没有保证 ${\tfrac {a}{0}}$ 且 $a\neq 0$ 的说法是对的。^[7]

几何

婆罗摩笈多公式

婆罗摩笈多在《婆罗摩历算书》第十二章中这样提到

12.21：一个四边形或三角形的大约面积是边和对边的和的一半。四边形的准确面积是每一个边分别地被另外三条边减的和的一半的开方。^{[注 9]}^[6]

设一个圆内接四边形的四条边为p﹑q﹑r﹑s，大约面积为 $({\tfrac {p+r}{2}})({\tfrac {q+s}{2}})$ ，设 $t={\tfrac {p+q+r+s}{2}}$ ，准确面积则为 ${\sqrt {(t-p)(t-q)(t-r)(t-s)}}.$ 。

虽然婆罗摩笈多并没有说四边形为圆内接四边形，但其实这是明显的。^[8]

圆周率

婆罗摩笈多还提供了一个化圆为方的几何方法：

12.40：直径和半径的二次方每个乘3分别地为圆形大约的周界和面积。而准确值则为直径和半径的二次方乘开方10。^{[注 10]}^[6]

这个方法不十分精确，按照它的计算得出的圆周率为 $\pi ={\sqrt {10}}\approx 3.162$ 。

天文学

婆罗摩笈多是最早使用代数解决天文问题的人。一般认为，阿拉伯人是通过《婆罗摩历算书》了解到印度天文学的^[9]。770年，阿拔斯王朝第二代哈里发曼苏尔邀请乌贾因的学者赴巴格达，使用《婆罗摩历算书》介绍印度代数天文学。他还请人将婆罗摩笈多的著作译成阿拉伯语。

婆罗摩笈多其它重要的天文成就在于：计算星历表、天体出生和下降的时间、合相、日食和月食的方法。婆罗摩笈多批评往世书中大地平坦或者像碗一样中空的理论。相反地，他的观察认为大地和天空是圆的，不过他误认为大地不运动。

原文引注

^ 英文原文是：“18.44. Diminish by the middle [number] the square-root of the rupas multiplied by four times the square and increased by the square of the middle [number]; divide the remainder by twice the square. [The result is] the middle [number].”
^ 英文原文是：“18.45. Whatever is the square-root of the rupas multiplied by the square [and] increased by the square of half the unknown, diminish that by half the unknown [and] divide [the remainder] by its square. [The result is] the unknown.”
^ 英文原文是：“12.20. The sum of the squares is that [sum] multiplied by twice the [number of] step[s] increased by one [and] divided by three. The sum of the cubes is the square of that [sum] Piles of these with identical balls [can also be computed]”
^ 英文原文是：“18.30. [The sum] of two positives is positives, of two negatives negative; of a positive and a negative [the sum] is their difference; if they are equal it is zero. The sum of a negative and zero is negative, [that] of a positive and zero positive, [and that] of two zeros zero. [...]”
^ 英文原文是：“18.32. A negative minus zero is negative, a positive [minus zero] positive; zero [minus zero] is zero. When a positive is to be subtracted from a negative or a negative from a positive, then it is to be added [...]”
^ 英文原文是：“18.33. The product of a negative and a positive is negative, of two negatives positive, and of positives positive; the product of zero and a negative, of zero and a positive, or of two zeros is zero.”
^ 英文原文是：“18.34. A positive divided by a positive or a negative divided by a negative is positive; a zero divided by a zero is zero; a positive divided by a negative is negative; a negative divided by a positive is [also] negative.”
^ 英文原文是：“18.35. A negative or a positive divided by zero has that [zero] as its divisor, or zero divided by a negative or a positive [has that negative or positive as its divisor]. The square of a negative or of a positive is positive; [the square] of zero is zero. That of which [the square] is the square is [its] square-root.”
^ 英文原文是：“12.21. The approximate area is the product of the halves of the sums of the sides and opposite sides of a triangle and a quadrilateral. The accurate [area] is the square root from the product of the halves of the sums of the sides diminished by [each] side of the quadrilateral.”
^ 英文原文是：“12.40. The diameter and the square of the radius [each] multiplied by 3 are [respectively] the practical circumference and the area [of a circle]. The accurate [values] are the square-roots from the squares of those two multiplied by ten.”

参考资料

^ ^1.0 ^1.1 Seturo Ikeyama. Brāhmasphuṭasiddhānta (CH. 21) of Brahmagupta with Commentary of Pṛthūdhaka, critically edited with English translation and notes. INSA. 2003.
^ Brahmagupta biography. School of Mathematics and Statistics University of St Andrews, Scotland. [2013-07-15]. （原始内容存档于2013-09-15）.
^ David Pingree. Census of the Exact Sciences in Sanskrit (CESS). American Philosophical Society. : p254.
^ Boyer. The Arabic Hegemony. 1991: 226. By 766 we learn that an astronomical-mathematical work, known to the Arabs as the Sindhind, was brought to Baghdad from India. It is generally thought that this was the Brahmasphuta Siddhanta, although it may have been the Surya Siddhanata. A few years later, perhaps about 775, this Siddhanata was translated into Arabic, and it was not long afterwards (ca. 780) that Ptolemy's astrological Tetrabiblos was translated into Arabic from the Greek. 缺少或|title=为空 (帮助)
^ ^5.0 ^5.1 ^5.2 ^5.3 ^5.4 ^5.5 （Plofker 2007，pp.428–434）
^ ^6.0 ^6.1 ^6.2 （Plofker 2007，pp.421–427）
^ Boyer. China and India. 1991: 220. However, here again Brahmagupta spoiled matters somewhat by asserting that $0\div 0=0$ , and on the touchy matter of $a\div 0$ , he did not commit himself: 缺少或|title=为空 (帮助)
^ （Plofker 2007，p.424） Brahmagupta does not explicitly state that he is discussing only figures inscribed in circles, but it is implied by these rules for computing their circumradius.
^ Brahmagupta, and the influence on Arabia. School of Mathematical and Computational Sciences University of St Andrews. 2002-05 [2013-07-15]. （原始内容存档于2013-09-15）.

[6] 英文原文是：“18.44. Diminish by the middle [number] the square-root of the rupas multiplied by four times the square and increased by the square of the middle [number]; divide the remainder by twice the square. [The result is] the middle [number].”

[7] 英文原文是：“18.45. Whatever is the square-root of the rupas multiplied by the square [and] increased by the square of half the unknown, diminish that by half the unknown [and] divide [the remainder] by its square. [The result is] the unknown.”

[8] 英文原文是：“12.20. The sum of the squares is that [sum] multiplied by twice the [number of] step[s] increased by one [and] divided by three. The sum of the cubes is the square of that [sum] Piles of these with identical balls [can also be computed]”

[10] 英文原文是：“18.30. [The sum] of two positives is positives, of two negatives negative; of a positive and a negative [the sum] is their difference; if they are equal it is zero. The sum of a negative and zero is negative, [that] of a positive and zero positive, [and that] of two zeros zero. [...]”

[11] 英文原文是：“18.32. A negative minus zero is negative, a positive [minus zero] positive; zero [minus zero] is zero. When a positive is to be subtracted from a negative or a negative from a positive, then it is to be added [...]”

[12] 英文原文是：“18.33. The product of a negative and a positive is negative, of two negatives positive, and of positives positive; the product of zero and a negative, of zero and a positive, or of two zeros is zero.”

[13] 英文原文是：“18.34. A positive divided by a positive or a negative divided by a negative is positive; a zero divided by a zero is zero; a positive divided by a negative is negative; a negative divided by a positive is [also] negative.”

[14] 英文原文是：“18.35. A negative or a positive divided by zero has that [zero] as its divisor, or zero divided by a negative or a positive [has that negative or positive as its divisor]. The square of a negative or of a positive is positive; [the square] of zero is zero. That of which [the square] is the square is [its] square-root.”

[16] 英文原文是：“12.21. The approximate area is the product of the halves of the sums of the sides and opposite sides of a triangle and a quadrilateral. The accurate [area] is the square root from the product of the halves of the sums of the sides diminished by [each] side of the quadrilateral.”

[18] 英文原文是：“12.40. The diameter and the square of the radius [each] multiplied by 3 are [respectively] the practical circumference and the area [of a circle]. The accurate [values] are the square-roots from the squares of those two multiplied by ten.”

[Seturo_2003-1] 1.0 ^1.1 Seturo Ikeyama. Brāhmasphuṭasiddhānta (CH. 21) of Brahmagupta with Commentary of Pṛthūdhaka, critically edited with English translation and notes. INSA. 2003.

[2] Brahmagupta biography. School of Mathematics and Statistics University of St Andrews, Scotland. [2013-07-15]. （原始内容存档于2013-09-15）.

[3] David Pingree. Census of the Exact Sciences in Sanskrit (CESS). American Philosophical Society. : p254.

[Boyer_Siddhanta-4] Boyer. The Arabic Hegemony. 1991: 226. By 766 we learn that an astronomical-mathematical work, known to the Arabs as the Sindhind, was brought to Baghdad from India. It is generally thought that this was the Brahmasphuta Siddhanta, although it may have been the Surya Siddhanata. A few years later, perhaps about 775, this Siddhanata was translated into Arabic, and it was not long afterwards (ca. 780) that Ptolemy's astrological Tetrabiblos was translated into Arabic from the Greek. 缺少或|title=为空 (帮助)

[Plofker_Chapter_18_Brahmasphutasiddhanta-5] 5.0 ^5.1 ^5.2 ^5.3 ^5.4 ^5.5 （Plofker 2007，pp.428–434）

[Plofker_Brahmagupta_quote_Chapter_12-9] 6.0 ^6.1 ^6.2 （Plofker 2007，pp.421–427）

[Boyer_Brahmagupta_p220-15] Boyer. China and India. 1991: 220. However, here again Brahmagupta spoiled matters somewhat by asserting that $0\div 0=0$ , and on the touchy matter of $a\div 0$ , he did not commit himself: 缺少或|title=为空 (帮助)

[17] （Plofker 2007，p.424） Brahmagupta does not explicitly state that he is discussing only figures inscribed in circles, but it is implied by these rules for computing their circumradius.

[19] Brahmagupta, and the influence on Arabia. School of Mathematical and Computational Sciences University of St Andrews. 2002-05 [2013-07-15]. （原始内容存档于2013-09-15）.

[1]

[2]

[3]

[4]

[5]

[注 1]

[注 2]

[注 3]

[6]

[注 4]

[注 5]

[注 6]

[注 7]

[注 8]

[7]

[注 9]

[8]

[注 10]

[9]

生平和著作

数学

代数

运算

级数

零

几何

婆罗摩笈多公式

圆周率

天文学

相关条目

原文引注

参考资料