**婆羅摩笈多**（梵文：**ब्रह्मगुप्त**，國際梵文轉寫字母：**Brahmagupta**，598年—668年）係印度一位數學家同天文學家，响印度嘅拉賈斯坦邦賓瑪律出世，一生可能大多數時間都係响嗰度度過。當時上面講嘅地區屬於哈爾沙帝國。婆羅摩笈多係烏賈因天文台台長，响佢做嘢嗰陣，寫咗兩本關於數學同天文學嘅書，當中包括响628年寫成嘅《婆羅摩曆算書（英文：Brāhmasphuṭasiddhānta）》。

婆羅摩笈多係第一個講有關0嘅計算規則嘅數學家。婆羅摩笈多同當時好多嘅印度數學家一樣，會將啲字排成橢圓形嘅句子，而且最後會有一個環狀咁排嘅詩。由於無講到證明，唔知到佢嘅數學推導過程。

生平同佢作嘅嘢
-------

响《婆羅摩曆算書》第十四篇嘅第7句同第8句講到婆羅摩笈多係响三十歲嗰年作呢本書，亦即係西元628年，因此可以推到婆羅摩笈多係响西元598年出世
。婆羅摩笈多寫咗四本有關數學同天文學嘅書，分別係624年嘅《Cadamekela》、628年嘅《婆羅摩曆算書》、665年嘅《Khandakhadyaka》及672年嘅《Durkeamynarda》，其中最出名嘅係《婆羅摩曆算書》。波斯歷史學家比魯尼响佢作嘅《Tariq al-Hind》講到阿拉伯帝國阿拔斯王朝嘅哈里發馬蒙曾經派大使去印度，並將一本「算書」帶到巴格達翻譯做阿拉伯文，一般覺得嚟本算書就係《婆羅摩曆算書》。

數學
--

《婆羅摩曆算書》中有四章半講嘅係純數學，第12章講嘅係演算系列同少少幾何學。第18章係關於代數，婆羅摩笈多响呢度引入咗一個解二次丟番圖方程例如*nx*² + 1 = *y*²嘅方法。

婆羅摩笈多重提供咗計任何四條邊已知嘅圓內接四邊形嘅面積嘅公式。海倫公式係婆羅摩笈多比嘅公式一個特殊嘅形式（一邊係零）。婆羅摩笈多公式同海倫公式之間嘅關係好似餘弦定理擴展咗畢氏定理。

### 代數

婆羅摩笈多响《婆羅摩曆算書》第十八章畀咗線性方程嘅解：

> 
> 色之間嘅數交換咗嘅差除未知數嘅差，就係方程嘅解。
> 
> 

當中方程{\displaystyle bx+c=dx+e}嘅解係{\displaystyle x={\tfrac {e-c}{b-d}}}，而色係指常數項c同e。佢然後進一步畀咗二次方程兩個解：

> 
> 18.44：色同二次項同4相乘嘅積加一次項嘅二次方嘅數，將呢個數開方後減一次項，再將全個數除一次項嘅2倍，就係方程嘅解。  
> 
> 18.45：色同二次項嘅積加一次項一半嘅二次方嘅數，將呢個數開方之後減一次項嘅一半，再約全個數除一次項就係方程嘅解。
> 
> 
> 

其實佢哋分別講咗方程{\displaystyle ax^{2}+bx=c}恆等於

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

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

#### 級數

婆羅摩笈多畀咗頭n個平方或者立方和嘅計法：

> 
> 個平方嘅和係平方嘅項數乘以項數加1乘項數加1嘅兩倍除6。立方嘅和係本身數嘅和嘅二次方。
> 
> 
> 

呢個婆羅摩笈多嘅方法比較接近依加嘅樣。

呢度婆羅摩笈多畀咗頭n個自然數嘅平方同立方計嘅方法，分別係{\displaystyle {\tfrac {n(n+1)(2n+1)}{6}}}同{\displaystyle ({\tfrac {n(n+1)}{2}})^{2}}

#### 零

婆羅摩笈多畀咗數學入面一個好重要嘅概念：0。《婆羅摩曆算書》係到依加為止知到咗嘅第一部將0當作一個普通嘅數字嚟用嘅書。除此之外呢本書重講咗負數同0嘅運算規則。呢啲規則同今日嘅規則好接近。

婆羅摩笈多响《婆羅摩曆算書》嘅第十八章入面咁樣講：

> 
> 18.30：正數加正數係正數，負數加負數係負數。正數加負數係佢哋之間嘅差，如果佢哋一樣，結果就係零。負數加係負數，正數加零係正數，零加零係零  
> 
> 18.32：負數減零係負數，正數減零係正數，零減零係零，正數減負數係佢哋之間嘅和。
> 
> 
> 

佢咁樣講乘法：

> 
> 18.33：正乘負係負，負乘負係負，正乘正係正，正數乘零、負數乘零同零乘零都係零。
> 
> 
> 

最大嘅區別係响婆羅摩笈多試吓定義「除零」，响現代數學入面呢個運算係唔知嘅。

> 
> 18.34：正數除正數或者負數除負數係正數，正數除負數或者負數除正數係負數，零除零係零  
> 
> 18.35：正數或者負數除零會有零做嗰個數嘅除數，零除正數或者負數會有正數或者負數做嗰個數嘅除數。正數或者負數嘅平方係正數，零嘅平方係零。
> 
> 
> 

婆羅摩笈多嘅定義唔實用，例如佢覺得{\displaystyle {\tfrac {0}{0}}=0}。而佢並無保證{\displaystyle {\tfrac {a}{0}}}且{\displaystyle a\neq 0}嘅講法係啱嘅。

### 幾何

#### 婆羅摩笈多公式

圖用嚟參考
內文：婆羅摩笈多公式
婆羅摩笈多响《婆羅摩曆算書》第十二章入面咁講

> 
> 12.21：一個四邊形或者三角形嘅大約面積係邊同對邊嘅和嘅一半。四邊形嘅準確面積係每一條邊分別咁俾另外三條邊減嘅和嘅一半嘅開方。
> 
> 
> 

設一個圓內接四邊形嘅四條邊係*p*﹑*q*﹑*r*﹑*s*，大約面積係{\displaystyle ({\tfrac {p+r}{2}})({\tfrac {q+s}{2}})}，設{\displaystyle t={\tfrac {p+q+r+s}{2}}}，準確面積就係{\displaystyle {\sqrt {(t-p)(t-q)(t-r)(t-s)}}.}。

雖然婆羅摩笈多無點樣講四邊形係圓內接四邊形，但其實依個係明顯嘅。

#### 圓周率

婆羅摩笈多重提供咗一個化圓為方嘅幾何方法：

> 
> 12.40：直徑同半徑嘅二次方每個乘3分別係圓形大約嘅周界同面積。而準確值就係直徑同半徑嘅二次方乘開方10。
> 
> 
> 

呢個方法唔係好啱，跟佢嘅計算得到嘅圓周率係{\displaystyle \pi ={\sqrt {10}}\approx 3.162}。

天文學
---

婆羅摩笈多係最早用代數去解掂天文問題嘅人。一般覺得阿拉伯人係通過《婆羅摩曆算書》知到印度天文學嘅。770年阿拔斯王朝第二代哈里發曼蘇爾去請烏賈因嘅學者去巴格達用《婆羅摩曆算書》介紹印度嘅代數天文學。佢重請人將婆羅摩笈多嘅書翻譯做阿拉伯語。

婆羅摩笈多其它重要嘅天文成就响：計星曆表、天體出生同下降嘅時間、合相、日食同月食嘅方法。婆羅摩笈多話往世書入面大地係平嘅或者好似碗一樣入面無嘢嘅理論唔啱。相反咁佢嘅觀察覺得大地同天空係圓嘅，不過佢錯咗咁話大地唔郁。

睇埋
--

* 婆羅摩笈多－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
----

1. ↑  英文原文係："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. ↑  英文原文係："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. ↑  英文原文係："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. ↑  英文原文係："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. ↑  英文原文係："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. ↑  英文原文係："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. ↑  英文原文係："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. ↑  英文原文係："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. ↑  英文原文係："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. ↑  英文原文係："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
--

1. 1 2  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. 原著喺2013-09-15歸檔. 喺2013-07-15搵到.
3. ↑  David Pingree. *Census of the Exact Sciences in Sanskrit (CESS)*. American Philosophical Society. pp. p254. `{{cite book}}`: `|pages=` has extra text (help)
4. ↑  Boyer (1991). "The Arabic Hegemony". p. 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. `{{cite book}}`: Missing or empty `|title=` (help)
5. 1 2 3 4 5 6  （Plofker 2007，pp.428–434）
6. 1 2 3  （Plofker 2007，pp.421–427）
7. ↑  Boyer (1991). "China and India". p. 220. However, here again Brahmagupta spoiled matters somewhat by asserting that {\displaystyle 0\div 0=0}, and on the touchy matter of {\displaystyle a\div 0}, he did not commit himself: `{{cite book}}`: Missing or empty `|title=` (help)
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. 原著喺2013-09-15歸檔. 喺2013-07-15搵到.`{{cite web}}`: CS1 maint: date format (link)