婆羅摩笈多

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婆羅摩笈多
Brahmagupta.jpg
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 印度数学家天文学家

婆羅摩笈多梵文ब्रह्मगुप्तIAST: Brahmagupta,598年-668年),是一位印度数学家天文学家,出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。当时上述地区属于哈尔沙帝国。婆羅摩笈多為乌贾因天文台台长,在他任职期间,書写了两部关于数学天文学的书籍,當中包括於628年寫成的《婆罗摩历算书英语Brāhmasphuṭasiddhānta》。

婆羅摩笈多是第一個提出有關0的計算規則的數學家。婆羅摩笈多和當時許多的印度數學家一樣,會將文字編排成橢圓形的句子,而且最後會有一個環狀排列的詩。由於沒有提出證明,不知其中的數學推導過程[2]

生平和著作[编辑]

在《婆罗摩历算书》第十四篇的第7句及第8句提及婆羅摩笈多是在三十歲那年著作此書的,也是628年,因此可以推得婆羅摩笈多是在598年出生[3] [1]。婆羅摩笈多寫了四本有關數學及天文學的書,分別為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哈里發馬蒙曾派大使到印度,並將一本「算書」帶到巴格達翻譯為阿拉伯文,一般認為這本算書就是《婆罗摩历算书》[4]

数学[编辑]

《婆罗摩历算书》中有四章半讲的是纯数学,第12章讲的是演算系列和少许几何学。第18章是关于代数,婆羅摩笈多在这里引入了一个解二次丟番圖方程nx² + 1 = y²的方法。

婆羅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。海伦公式是婆羅摩笈多给出的公式的一个特殊形式(一边为零)。婆羅摩笈多公式与海伦公式之间的关系类似餘弦定理扩展了勾股定理

代數[编辑]

婆羅摩笈多在《婆羅摩曆算書》第十八章給了線性方程的解:

之間的數交換後的差除以未知數的差,就是方程的解。[5]

當中方程 b x + c = d x + e的解是x = \tfrac{e-c}{b-d},而色是指常數項c和e。他然後進一步給了二次方程兩個解:

18.44:色和二次項和4相乘的積加一次項的二次方的數,把這個數開方後減一次項,再把整個數除一次項的2倍,就是方程的解。[注 1]
18.45:色和二次項的積加一次項一半的二次方的數,把這個數開方後減一次項的一半,再把整個數除一次項就是方程的解。[注 2][5]

其實它們分別說了方程a x^2 + b x = c恆等於

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

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

運算[编辑]

級數[编辑]

婆羅摩笈多提供了頭n個平方或立方和的算法:

那些平方的和是平方的項數乘項數加1乘項數加1的兩倍除6。立方的和是本身數的和的二次。[注 3][6]

這個婆羅摩笈多的方法比較接近現代的形式。

這裏婆羅摩笈多給了頭n個自然數的平方和立方的算法,分別為\tfrac{n(n+1)(2n+1)}{6}(\tfrac{n(n+1)}{2})^2

[编辑]

婆羅摩笈多普及了數學中一個非常重要的概念:0。《婆羅摩曆算書》是至今為止已知的第一部將0當作一個普通的數字來使用的著作。除此之外這部書還闡述了負數和0的運算規則。這些規則與今天的規則非常接近。

婆羅摩笈多在《婆羅摩曆算書》第十八章中這樣提到:

18.30:正數加正數為正數,負數加負數為負數。正數加負數為他們彼此的差,如果它們相等,結果就是零。負數加零為負數,正數加零為正數,零加零為零[注 4]
18.32:負數減零為負數,正數減零為正數,零減零為零,正數減負數為他們彼此的和。[注 5][5]

他這樣描述乘法:

18.33:正負得負,負負得負,正正得正,正數乘零﹑負數乘零和零乘零都是零。[注 6][5]

最大的區別在於婆羅摩笈多試圖定義除以零,在現代數學中這個運算是不確定的。

18.34:正數除正數或負數除負數為正數,正數除負數或負數除正數為負數,零除零為零[注 7][5]
18.35:正數或負數除零有零作為該數的除數,零除正數或負數有正數或負數作為該數的除數。正數或負數的平方為正數,零的平方為零。[注 8][5]

婆羅摩笈多的定義不實用,比如他認為\tfrac{0}{0} = 0。而他並沒有保證\tfrac{a}{0}a \neq 0的說法是對的。[7]

幾何[编辑]

婆羅摩笈多公式[编辑]

圖供參考

婆羅摩笈多在《婆羅摩曆算書》第十二章中這樣提到

12.21:一個四邊形或三角形的大約面積是邊和對邊的和的一半。四邊形的準確面積是每一個邊分別地被另外三條邊減的和的一半的開方。[注 9][6]

設一個圓內接四邊形的四條邊為pqrs,大約面積為(\tfrac{p + r}{2}) (\tfrac{q + s}{2}),設t = \tfrac{p + q + r + s}{2},準確面積則為\sqrt{(t - p)(t - q)(t - r)(t - s)}.

雖然婆羅摩笈多並沒有說四邊形為圓內接四邊形,但其實這是明顯的。[8]

圆周率[编辑]

婆羅摩笈多還提供了一個化圓為方的幾何方法:

12.40:直徑和半徑的二次方每個乘3分別地為圓形大約的周界和面積。而準確值則為直徑和半徑的二次方乘開方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\pi=\sqrt{10}\approx3.162

天文学[编辑]

婆羅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。他还请人将婆羅摩笈多的著作译成阿拉伯语。

婆羅摩笈多其它重要的天文成就在于:计算星曆表、天体出生和下降的时间、合相日食月食的方法。婆羅摩笈多批评往世书中大地是平的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,不过他错误地认为大地不运动。

相關條目[编辑]

原文引注[编辑]

  1. ^ 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].”
  2. ^ 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.”
  3. ^ 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]”
  4. ^ 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. [...]”
  5. ^ 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 [...]”
  6. ^ 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.”
  7. ^ 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.”
  8. ^ 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.”
  9. ^ 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.”
  10. ^ 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[编辑]

  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]. 
  3. ^ David Pingree. Census of the Exact Sciences in Sanskrit (CESS). American Philosophical Society. : p254. 
  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." 
  5. ^ 5.0 5.1 5.2 5.3 5.4 5.5 Plofker 2007,pp.428–434)
  6. ^ 6.0 6.1 6.2 Plofker 2007,pp.421–427)
  7. ^ 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:" 
  8. ^ 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.
  9. ^ Brahmagupta, and the influence on Arabia. School of Mathematical and Computational Sciences University of St Andrews. 2002-05 [2013-07-15].