婆羅摩笈多

婆羅摩笈多
婆羅摩笈多
出生	598年; 哈爾沙帝國拉賈斯坦邦賓馬爾
逝世	670年; 瞿折羅-普羅蒂訶羅
職業	印度數學家和天文學家

婆羅摩笈多（梵語：ब्रह्मगुप्त，IAST: Brahmagupta，598年—668年），印度數學家和天文學家，生於印度拉賈斯坦邦賓馬爾^[1]，一生可能大多數時間都在生地度過。當時，該地隸屬於哈爾沙帝國。婆羅摩笈多為烏賈因天文台台長，於任職期間著書二部，乃關於數學和天文學，當中包括於628年寫成的《婆羅摩歷算書（英語：Brāhmasphuṭasiddhānta）》。

婆羅摩笈多係首位提出0計算規則的數學家。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。由於未提出證明，不知其中的數學推導過程^[2]。

生平和著作

在《婆羅摩歷算書》第十四篇的第7句及第8句，提及婆羅摩笈多於三十歲著作此書，也是628年，便可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《婆羅摩歷算書（英語：Brāhmasphuṭasiddhānta）》、665年的《Khandakhadyaka》及672年的《Durkeamynarda》，其中最著名的是《婆羅摩歷算書》。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的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《婆羅摩歷算書》^[4]。

數學

《婆羅摩歷算書》中，有四章半講述純數學，第12章講述演算系列和少許幾何學。第18章是關於代數，婆羅摩笈多在這裏引入了一個解二次丟番圖方程如nx² + 1 = y²的方法。

婆羅摩笈多還提供了計算任何四邊已知的圓內接四邊形的面積的公式。海倫公式是婆羅摩笈多給出的公式的一個特殊形式（一邊為零）。婆羅摩笈多公式與海倫公式之間的關係，類似餘弦定理擴展了勾股定理。

代數

婆羅摩笈多在《婆羅摩曆算書》第十八章給了線性方程的解：

色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 $bx+c=dx+e$ 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 $ax^{2}+bx=c$ 的解恆等於

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

和

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 $n$ 個平方和及立方和的算法：

12.20. 平方和是[頭幾個整數直接和]乘以兩倍[項數]與1的和後再除以3的結果。立方和是這直接和的平方。^{[注 3]}^[6]

婆羅摩笈多的方法較近似於現代形式。

這裏，婆羅摩笈多所給的頭 $n$ 個自然數的平方和立方的算法，分別為 ${\tfrac {n(n+1)(2n+1)}{6}}$ 和 $[{\tfrac {n(n+1)}{2}}]^{2}$

零

婆羅摩笈多普及了數學中的重要概念：0。《婆羅摩曆算書》是至今為止，已知首部將0視為普通數字使用之著作。除此之外，這部書還闡述了負數和0的運算規則。這些規則與現代規則非常接近。

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，負負得正，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無解。

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：正數或負數除以零有零作為該數的除數，零除以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 ${\tfrac {a}{0}}$ 且 $a\neq 0$ 的說法是對的。^[7]

幾何

婆羅摩笈多公式

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p﹑q﹑r﹑s，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]

圓周率

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]

這個方法不十分精確，按照它的計算得出的圓周率為 $\pi ={\sqrt {10}}\approx 3.162$ 。

天文學

婆羅摩笈多是最早使用代數解決天文問題的人。一般認為，阿拉伯人是通過《婆羅摩歷算書》了解到印度天文學的^[9]。770年，阿拔斯王朝第二代哈里發曼蘇爾邀請烏賈因的學者赴巴格達，使用《婆羅摩歷算書》介紹印度代數天文學。他還請人將婆羅摩笈多的著作譯成阿拉伯語。

婆羅摩笈多其它重要的天文成就在於：計算星曆表、天體出生和下降的時間、合相、日食和月食的方法。婆羅摩笈多批評往世書中大地平坦或者像碗一樣中空的理論。相反地，他的觀察認為大地和天空是圓的，不過他誤認為大地不運動。

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. 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]. （原始內容存檔於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=為空 (幫助)
^ ^5.0 ^5.1 ^5.2 ^5.3 ^5.4 ^5.5 (Plofker 2007，第428–434頁)
^ ^6.0 ^6.1 ^6.2 (Plofker 2007，第421–427頁)
^ 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=為空 (幫助)
^ (Plofker 2007，第424頁) 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]. （原始內容存檔於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]. （原始內容存檔於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=為空 (幫助)

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

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

[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=為空 (幫助)

[17] (Plofker 2007，第424頁) 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]. （原始內容存檔於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]

權威控制資料庫
國際	FAST ISNI VIAF 2 3 4 5
各地	西班牙法國 BnF data 加泰羅尼亞德國以色列美國澳大利亞荷蘭
學術	CiNii MathSciNet zbMATH
人物	Trove
其他	IdRef

生平和著作

數學

代數

運算

級數

零

幾何

婆羅摩笈多公式

圓周率

天文學

相關條目

原文引注

參考資料