# 婆羅摩笈多 668年

## 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nx² + 1 = y²嘅方法。

### 代數

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。[注 1]
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。[注 2]

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

#### 零

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[注 4]
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。[注 5]

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。[注 6]

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[注 7]
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。[注 8]

### 幾何

#### 婆羅摩笈多公式

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。[注 9]

#### 圓周率

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。[注 10]

## 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. 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.CS1 maint: extra text (link)
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. Missing or empty |title= (help)
5. Plofker 2007，pp.428–434）
6. Plofker 2007，pp.421–427）
7. Boyer (1991). "China and India". p. 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: 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. Check date values in: |date= (help)