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	668年; 瞿折罗-普罗蒂诃罗
职业	印度数学家搭天文学家

婆罗摩笈多（梵文：ब्रह्मगुप्त}}，IAST: Brahmagupta，598年—668年），是一位印度数学家搭天文学家，出生垃印度拉贾斯坦邦宾马尔^[1]，佢个一生作兴大多数时间侪垃出生地度过。当时辰光上述地区属于哈尔沙帝国。婆罗摩笈多为乌贾因天文台台长，在佢任职期间，书写仔两部关于数学搭天文学个书籍，当中包括于628年写成功个《婆罗摩历算书》。

婆罗摩笈多是第一个提出有关0个计算规则个数学家。婆罗摩笈多搭当时交关个印度数学家一样，会担文字编排成椭圆形个句子，而且末脚会得有一个环状排列个诗。由于弗曾提出证明，弗知其中个数学推导过程^[2]。

生平搭著作

在《婆罗摩历算书》第十四篇个第7句及第8句提及婆罗摩笈多是垃佢三十岁伊年著作此书个，也是628年，因此可以推得婆罗摩笈多是在598年出生^[3] ^[1]。婆罗摩笈多写仔四本有关数学及天文学个书，分别为624年个《Cadamekela》、628年个《婆罗摩历算书》、665年个《Khandakhadyaka》及672年个《Durkeamynarda》，其中顶顶著名个是《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个哈里发马蒙曾派大使到印度，并担一本“算书”带到巴格达翻译成功阿拉伯文，一般认为昰本算书就是《婆罗摩历算书》。

数学

《婆罗摩历算书》中有四章半讲个是纯数学，第12章讲个是演算系列搭少许几何学。第18章是关于代数，婆罗摩笈多在昰𡍲引入仔一个解二次丢番图方程像nx² + 1 = y²个方法。

婆罗摩笈多还提供了计算任何四边已知个圆内接四边形个面积个公式。海伦公式是婆罗摩笈多给出个公式个一个特殊形式（一边为零）。婆罗摩笈多公式与海伦公式之间个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》第十八章畀了线性方程个解：

色之间个数交换后个差除以未知数个差，就是方程个解。^[4]

当中方程 $bx+c=dx+e$ 个解是 $x={\tfrac {e-c}{b-d}}$ ，而色是指常数项c搭e。佢然后进一步给了二次方程两只解：

18.44：色和二次项和4相乘个积加一次项个二次方个数，担昰只数开方后减一次项，再担整个数除一次项个2倍，就是方程个解。^{[注 1]}
18.45：色和二次项个积加一次项一半个二次方个数，担昰只数开方后减一次项个一半，再担整个数除一次项就是方程个解。^{[注 2]}^[4]

其实渠场分别说明方程 $ax^{2}+bx=c$ 恒等于

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

搭

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 $n$ 个平方和及立方和个算法：

12.20. 平方和是[头几个整数直接和]乘以两倍[项数]与1个和后再除以3个结果。立方和是这直接和个平方。^{[注 3]}^[5]

婆罗摩笈多个方法搭现代个形式比较接近。

昰𡍲婆罗摩笈多所畀个头 $n$ 个自然数个平方搭立方个算法，分别为 ${\tfrac {n(n+1)(2n+1)}{6}}$ 搭 $[{\tfrac {n(n+1)}{2}}]^{2}$

零

婆罗摩笈多普及了数学里向一个邪气重要个概念：0。《婆罗摩历算书》是至今为止已知个第一部担0当作一个普通个数字来使用个著作。除此之外昰部书还阐述了负数搭0个运算规则。昰些规则搭乃朝个规则非常接近。

婆罗摩笈多在《婆罗摩历算书》第十八章中昰然提到：

18.30：正数加正数为正数，负数加负数为负数。正数加负数为渠场彼此个差，假使渠场相等，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，正数减负数为渠场彼此个和。^{[注 5]}^[4]

渠昰介描述乘法：

18.33：正负得负，负负得负，正正得正，正数乘零﹑负数乘零搭零乘零侪是零。^{[注 6]}^[4]

顶大个区别在于婆罗摩笈多试图定义除以零，在现代数学里向昰只运算是弗确定个。

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]}^[4]
18.35：正数或负数除零有零作为该数个除数，零除正数或负数有正数或负数作为该数个除数。正数或负数个平方为正数，零个平方为零。^{[注 8]}^[4]

婆罗摩笈多个定义弗实用，比方佢认为 ${\tfrac {0}{0}}=0$ 。而佢并呒没保证 ${\tfrac {a}{0}}$ 且 $a\neq 0$ 个说法是对个。

几何

婆罗摩笈多公式

主文章：婆罗摩笈多公式

婆罗摩笈多在《婆罗摩历算书》第十二章中昰然提到个

12.21：一个四边形或三角形个大约面积是边搭对边个和个一半。四边形个准确面积是每一个边分别畀另外三条边减个和个一半个开方。^{[注 9]}^[5]

设一个圆内接四边形个四条边为p﹑q﹑r﹑s，大约面积为 $({\tfrac {p+r}{2}})({\tfrac {q+s}{2}})$ ，设 $t={\tfrac {p+q+r+s}{2}}$ ，准确面积嚜为 ${\sqrt {(t-p)(t-q)(t-r)(t-s)}}.$ 。

虽然婆罗摩笈多并呒没话四边形为圆内接四边形，但实质丄昰个是明显个。^[6]

圆周率

婆罗摩笈多还提供了一个化圆为方个几何方法：

12.40：直径搭半径个二次方每个乘3分别地为圆形大约个周界搭面积。而准确值咾为直径搭半径个二次方乘开方10。^{[注 10]}^[5]

昰个方法弗是老精确，按照渠个计算得出个圆周率为 $\pi ={\sqrt {10}}\approx 3.162$ 。

天文学

婆罗摩笈多是最早使用代数解决天文问题个人。一般认为阿拉伯人是通过《婆罗摩历算书》了解到印度天文学个^[7]。770年阿拔斯王朝第二代哈里发曼苏尔邀请乌贾因个学者赴巴格达使用《婆罗摩历算书》介绍印度代数天文学。佢还请人担婆罗摩笈多个著作译成功阿拉伯语。

婆罗摩笈多其它重要个天文成就垃拉：计算星历表、天体出生搭下降个时间、合相、日食搭月食个方法。婆罗摩笈多批评往世书中大地是平个或者像碗一样中空个理论。相反地佢个观察认为大地搭天空是圆个，不过佢错误个认为大地弗运动。

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（2003）．Brāhmasphuṭasiddhānta (CH. 21) of Brahmagupta with Commentary of Pṛthūdhaka, critically edited with English translation and notes．INSA．
↑ Brahmagupta biography. School of Mathematics and Statistics University of St Andrews, Scotland. 访问日脚2013-07-15.
↑ David Pingree．Census of the Exact Sciences in Sanskrit (CESS)．American Philosophical Society，p254．
↑ ^4.0 ^4.1 ^4.2 ^4.3 ^4.4 ^4.5 Template:Harv
↑ ^5.0 ^5.1 ^5.2 Template:Harv
↑ Template:Harv 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). 访问日脚2013-07-15.

[5] 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].”

[6] 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.”

[7] 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]”

[9] 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. [...]”

[10] 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 [...]”

[11] 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.”

[12] 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.”

[13] 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.”

[14] 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.”

[16] 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（2003）．Brāhmasphuṭasiddhānta (CH. 21) of Brahmagupta with Commentary of Pṛthūdhaka, critically edited with English translation and notes．INSA．

[2] Brahmagupta biography. School of Mathematics and Statistics University of St Andrews, Scotland. 访问日脚2013-07-15.

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

[Plofker_Chapter_18_Brahmasphutasiddhanta-4] 4.0 ^4.1 ^4.2 ^4.3 ^4.4 ^4.5 Template:Harv

[Plofker_Brahmagupta_quote_Chapter_12-8] 5.0 ^5.1 ^5.2 Template:Harv

[15] Template:Harv 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.

[17] Brahmagupta, and the influence on Arabia. School of Mathematical and Computational Sciences University of St Andrews (2002-05). 访问日脚2013-07-15.

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[注 1]

[注 2]

[注 3]

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[注 4]

[注 5]

[注 6]

[注 7]

[注 8]

[注 9]

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[注 10]

[7]

生平搭著作

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