望“婆羅摩笈多”个源码

{{Infobox person
|name        = 婆羅摩笈多
|image       = Hindu astronomer, 19th-century illustration.jpg
|image_size  = 
|caption     = 
|birth_date  = 598年
|birth_place = [[哈尔沙帝国]][[拉贾斯坦邦]][[宾马尔]]
|death_date  = 668年
|death_place = [[瞿折羅-普羅蒂訶羅]]
|occupation  = [[印度]][[数学家]]搭[[天文学家]]
}}

'''婆羅摩笈多'''（[[梵文]]：'''ब्रह्मगुप्त'''}}，[[IAST]]: {{IAST|'''Brahmagupta'''}}，{{bd|598年||668年||catIdx=B}}），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]]<ref name="Seturo_2003"/>，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<ref>{{Cite web 
 |url=http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Brahmagupta.html 
 |title= Brahmagupta biography
 |accessdate=2013-07-15 
 |publisher=School of Mathematics and Statistics University of St Andrews, Scotland
}}</ref>。

==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<!--Śaka 550 during the reign of King Vyāghramukha-->，因此可以推得婆羅摩笈多是在598年出生<ref>{{cite book
 |title=Census of the Exact Sciences in Sanskrit (CESS) 
 |author=David Pingree
 |publisher=American Philosophical Society
 |pages=p254
}}</ref>
<ref name="Seturo_2003">{{cite book
 |title=''Brāhmasphuṭasiddhānta'' (CH. 21) of Brahmagupta with Commentary of Pṛthūdhaka, critically edited with English translation and notes 
 |author=Seturo Ikeyama
 |publisher=INSA
 |year=2003
}}</ref>。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]][[阿布·阿拔斯·阿卜杜拉·馬蒙·本·哈倫|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''nx''²&nbsp;+&nbsp;1&nbsp;=&nbsp;''y''²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：

<blockquote>
[[色 (佛教)|色]]之間个數交換後个差除以未知數个差，就是[[方程]]个解。<ref name="Plofker Chapter 18 Brahmasphutasiddhanta">{{Harv|Plofker|2007|pp=428–434}}</ref></blockquote>

當中方程<math> b x + c = d x + e</math>个解是<math>x = \tfrac{e-c}{b-d}</math>，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：

<blockquote>
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。<ref group="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].”</ref><br />
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。<ref group="注">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.”</ref><ref name="Plofker Chapter 18 Brahmasphutasiddhanta" />
</blockquote>

其實渠場分別說明方程<math>a x^2 + b x = c</math>恆等於
:<math>x = \frac{\sqrt{4ac+b^2}-b}{2a} </math>
搭
:<math>x = \frac{\sqrt{ac+\tfrac{b^2}{4}}-\tfrac{b}{2}}{a} </math>。

=== 運算 ===
==== 級數 ====
婆羅摩笈多提供了頭<math>n</math>個平方和及立方和个算法：
<blockquote>
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。<ref group="注">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]”</ref><ref name="Plofker Brahmagupta quote Chapter 12">{{Harv|Plofker|2007|pp=421–427}}</ref>
</blockquote>

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<math>n</math>個自然數个平方搭立方个算法，分別爲<math>\tfrac{n(n+1)(2n+1)}{6}</math>搭<math>[\tfrac{n(n+1)}{2}]^2</math>

==== 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：
<blockquote>
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<ref group="注">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.
[...]”</ref><br>
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。<ref group="注">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
[...]”</ref><ref name="Plofker Chapter 18 Brahmasphutasiddhanta"/>
</blockquote>

渠昰介描述乘法：
<blockquote>
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。<ref group="注">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.”</ref><ref name="Plofker Chapter 18 Brahmasphutasiddhanta"/>
</blockquote>

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。
<blockquote>
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<ref group="注">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.”</ref><ref name="Plofker Chapter 18 Brahmasphutasiddhanta"/><br>
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。<ref group="注">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.”</ref><ref name="Plofker Chapter 18 Brahmasphutasiddhanta"/>
</blockquote>
婆羅摩笈多个定義弗實用，比方佢認爲<math>\tfrac{0}{0} = 0</math>。而佢並嘸沒保證<math>\tfrac{a}{0}</math>且<math>a \neq 0</math>个說法是對个。

=== 幾何 ===
==== 婆羅摩笈多公式 ====
[[Image:Brahmaguptas formula.svg|200px|thumb|right|圖供參考]]
{{Main|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
<blockquote>
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。<ref group="注">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.”</ref><ref name="Plofker Brahmagupta quote Chapter 12"/>
</blockquote>

設一個圓內接四邊形个四條邊爲''p''﹑''q''﹑''r''﹑''s''，大約面積爲<math>(\tfrac{p + r}{2}) (\tfrac{q + s}{2})</math>，設<math>t = \tfrac{p + q + r + s}{2}</math>，準確面積嚜爲<math>\sqrt{(t - p)(t - q)(t - r)(t - s)}.</math>。

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。<ref>{{Harv|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.</ref>

==== 圆周率 ====
婆羅摩笈多還提供了一個[[化圓爲方]]个幾何方法：
<blockquote>
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。<ref group="注">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.”</ref><ref name="Plofker Brahmagupta quote Chapter 12"/>
</blockquote>

昰個方法弗是老精确，按照渠个计算得出个[[圆周率]]为<math>\pi=\sqrt{10}\approx3.162</math>。

== 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<ref>{{Cite web  |url= http://www-groups.dcs.st-and.ac.uk/~history/Projects/Pearce/Chapters/Ch8_3.html |title= Brahmagupta, and the influence on Arabia |accessdate=2013-07-15  |publisher= School of Mathematical and Computational Sciences University of St Andrews |date=2002-05}}</ref>。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。

==相關條目==
*[[婆罗摩笈多－斐波那契恒等式]]
*[[婆羅摩笈多公式]]
*[[婆羅摩笈多定理]]
*[[婆什迦羅第二]]

== 原文引注 ==
{{Reflist|2|group="注"}}

==參考資料==
{{reflist}}

[[Category:印度数学家]]
[[Category:印度天文学家]]