婆羅摩笈多

婆羅摩笈多
婆羅摩笈多
出生	598年; 哈尔沙帝国拉贾斯坦邦宾马尔
逝世	668年; 瞿折羅-普羅蒂訶羅
职业	印度数学家搭天文学家

婆羅摩笈多（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]，佢个一生作興大多数时间儕垃出生地度过。当时辰光上述地区属於哈尔沙帝国。婆羅摩笈多爲乌贾因天文臺臺长，在佢任职期间，書写仔两部关於数学搭天文学个书籍，當中包括於628年寫成功个《婆罗摩历算书》。

婆羅摩笈多是第一個提出有關0个計算規則个數學家。婆羅摩笈多搭當時交關个印度數學家一樣，會擔文字編排成橢圓形个句子，而且末腳會得有一個環狀排列个詩。由於弗曾提出證明，弗知其中个數學推導過程^[2]。

生平搭著作

在《婆罗摩历算书》第十四篇个第7句及第8句提及婆羅摩笈多是垃佢三十歲伊年著作此書个，也是628年，因此可以推得婆羅摩笈多是在598年出生^[3] ^[1]。婆羅摩笈多寫仔四本有關數學及天文學个書，分別爲624年个《Cadamekela》、628年个《婆罗摩历算书》、665年个《Khandakhadyaka》及672年个《Durkeamynarda》，其中頂頂著名个是《婆罗摩历算书》。波斯歷史學家比魯尼在其著作《Tariq al-Hind》提到阿拉伯帝國阿拔斯王朝个哈里發馬蒙曾派大使到印度，並擔一本「算書」帶到巴格達翻譯成功阿拉伯文，一般認爲昰本算書就是《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章是关於代数，婆羅摩笈多在昰𡍲引入仔一個解二次丟番圖方程像nx² + 1 = y²个方法。

婆羅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。海伦公式是婆羅摩笈多给出个公式个一個特殊形式（一边为零）。婆羅摩笈多公式与海伦公式之间个关系类似餘弦定理扩展了勾股定理。

代數

婆羅摩笈多在《婆羅摩曆算書》第十八章畀了線性方程个解：

色之間个數交換後个差除以未知數个差，就是方程个解。^[4]

當中方程 $bx+c=dx+e$ 个解是 $x={\tfrac {e-c}{b-d}}$ ，而色是指常數項c搭e。佢然後進一步給了二次方程兩隻解：

18.44：色和二次項和4相乘个積加一次項个二次方个數，擔昰隻數開方後減一次項，再擔整個數除一次項个2倍，就是方程个解。^{[注 1]}
18.45：色和二次項个積加一次項一半个二次方个數，擔昰隻數開方後減一次項个一半，再擔整個數除一次項就是方程个解。^{[注 2]}^[4]

其實渠場分別說明方程 $ax^{2}+bx=c$ 恆等於

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

搭

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

。

運算

級數

婆羅摩笈多提供了頭 $n$ 個平方和及立方和个算法：

12.20. 平方和是[头几個整数直接和]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]

婆羅摩笈多个方法搭現代个形式比較接近。

昰𡍲婆羅摩笈多所畀个頭 $n$ 個自然數个平方搭立方个算法，分別爲 ${\tfrac {n(n+1)(2n+1)}{6}}$ 搭 $[{\tfrac {n(n+1)}{2}}]^{2}$

零

婆羅摩笈多普及了數學裏向一個邪氣重要个概念：0。《婆羅摩曆算書》是至今爲止已知个第一部擔0當作一個普通个數字來使用个著作。除此之外昰部書還闡述了負數搭0个運算規則。昰些規則搭乃朝个規則非常接近。

婆羅摩笈多在《婆羅摩曆算書》第十八章中昰然提到：

18.30：正數加正數爲正數，負數加負數爲負數。正數加負數爲渠場彼此个差，假使渠場相等，結果就是零。負數加零爲負數，正數加零爲正數，零加零爲零^{[注 4]}
18.32：負數減零爲負數，正數減零爲正數，零減零爲零，正數減負數爲渠場彼此个和。^{[注 5]}^[4]

渠昰介描述乘法：

18.33：正負得負，負負得負，正正得正，正數乘零﹑負數乘零搭零乘零儕是零。^{[注 6]}^[4]

頂大个區別在於婆羅摩笈多試圖定義除以零，在現代數學裏向昰隻運算是弗確定个。

18.34：正數除正數或負數除負數爲正數，正數除負數或負數除正數爲負數，零除零爲零^{[注 7]}^[4]
18.35：正數或負數除零有零作爲該數个除數，零除正數或負數有正數或負數作爲該數个除數。正數或負數个平方爲正數，零个平方爲零。^{[注 8]}^[4]

婆羅摩笈多个定義弗實用，比方佢認爲 ${\tfrac {0}{0}}=0$ 。而佢並嘸沒保證 ${\tfrac {a}{0}}$ 且 $a\neq 0$ 个說法是對个。

幾何

婆羅摩笈多公式

主文章：婆羅摩笈多公式

婆羅摩笈多在《婆羅摩曆算書》第十二章中昰然提到个

12.21：一個四邊形或三角形个大約面積是邊搭對邊个和个一半。四邊形个準確面積是每一個邊分別畀另外三條邊減个和个一半个開方。^{[注 9]}^[5]

設一個圓內接四邊形个四條邊爲p﹑q﹑r﹑s，大約面積爲 $({\tfrac {p+r}{2}})({\tfrac {q+s}{2}})$ ，設 $t={\tfrac {p+q+r+s}{2}}$ ，準確面積嚜爲 ${\sqrt {(t-p)(t-q)(t-r)(t-s)}}.$ 。

雖然婆羅摩笈多並嘸沒話四邊形爲圓內接四邊形，但實質丄昰個是明顯个。^[6]

圆周率

婆羅摩笈多還提供了一個化圓爲方个幾何方法：

12.40：直徑搭半徑个二次方每個乘3分別地爲圓形大約个周界搭面積。而準確值咾爲直徑搭半徑个二次方乘開方10。^{[注 10]}^[5]

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

天文学

婆羅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。佢还请人擔婆羅摩笈多个著作译成功阿拉伯语。

婆羅摩笈多其它重要个天文成就垃拉：计算星曆表、天体出生搭下降个时间、合相、日食搭月食个方法。婆羅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.”

參考資料

↑ ^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]

[6]

[注 10]

[7]

生平搭著作

数学

代數

運算

級數

零

幾何

婆羅摩笈多公式

圆周率

天文学

相關條目

原文引注

參考資料