# 婆羅摩笈多

## 数学

《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。

### 代數

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]

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

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

### 運算

#### 級數

12.20. 平方和是[头几个整数直接和]乘以两倍[项数]与1的和后再除以3的结果。立方和是这直接和的平方。[注 3][6]

#### 零

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]

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]

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]

### 幾何

#### 婆羅摩笈多公式

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]

#### 圆周率

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]

## 原文引注

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

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].