婆罗摩笈多Brahmagupta

婆罗摩笈多的父亲是Jisnugupta，他撰写了重要的数学和天文学著作。特别是他在628年写了Brahmasphutasiddhanta Ⓣ（《婆罗摩修正体系》）。这部著作分为25章，婆罗摩笈多在文中告诉我们他是在Bhillamala（即今天的Bhinmal城）写成的。这里是Gurjara王朝统治地区的首府。

婆罗摩笈多成为Ujjain天文台台长，当时这里是古印度最重要的数学中心。像伐罗诃密希罗这样的杰出数学家曾在那里工作，并建立了一个强大的数学天文学学派。

除了Brahmasphutasiddhanta Ⓣ（《婆罗摩修正体系》）之外，婆罗摩笈多还写了第二部数学和天文学著作，即Khandakhadyaka Ⓣ（“可食之食”或“一口食物”），写于665年，当时他67岁。下面我们来看婆罗摩笈多的两部论著中包含的一些卓越思想。首先让我们概述其内容。

Brahmasphutasiddhanta Ⓣ（《婆罗摩修正体系》）包含二十五章，但其中前十章似乎构成了许多历史学家认为是婆罗摩笈多著作的第一个版本，并且存在一些只包含这些章节的手稿。这十章按主题编排，这些主题是当时印度数学天文学文本的典型主题。涵盖的主题有：行星的平均经度；行星的真经度；周日旋转的三个问题；月食；日食；升落；月牙；月影；行星之间的合；以及行星与恒星的合。

其余十五章似乎构成了第二部著作，是对原论著的重要增补。这些章节是：对先前天文学论著的考察；论数学；第一章的增补；第二章的增补；第三章的增补；第四、五章的增补；第七章的增补；论代数；论圭表；论度量；论球；论仪器；内容摘要；韵文表。

婆罗摩笈多对数系的理解远远超出了同时代的其他人。在BrahmasphutasiddhantaⓉ（《正确建立的梵天教义》）中，他将零定义为一个数减去自身的结果。他给出了如下一些性质：-

当零加到一个数上或从一个数中减去时，该数保持不变；而一个数乘以零则变为零。

他还用财富（正数）和债务（负数）给出了算术法则：-

债务减零是债务。
财富减零是财富。
零减零是零。
零减债务是财富。
零减财富是债务。
零乘以债务或财富的积是零。
零乘以零的积是零。
两个财富的积或商是一个财富。
两个债务的积或商是一个财富。
一个债务和一个财富的积或商是一个债务。
一个财富和一个债务的积或商是一个债务。

婆罗摩笈多随后试图将算术扩展到包含除以零：-

正数或负数除以零是一个以零为分母的分数。
零除以负数或正数要么是零，要么表示为一个以零为分子、有限量为分母的分数。
零除以零是零。

婆罗摩笈多说$n$除以零等于$n/0$，这几乎等于没说。他随后声称零除以零等于零，这肯定是错的。然而，这是一次将算术扩展到负数和零的出色尝试。

我们还可以描述他的乘法方法，这些方法充分利用了位值制，几乎与今天使用的方式相同。我们给出他在BrahmasphutasiddhantaⓉ（《婆罗摩笈多正确建立的教义》）中提出的三种方法的例子，在这样做时我们遵循4中Ifrah的叙述。我们描述的第一种方法被婆罗摩笈多称为“gomutrika”。Ifrah将“gomutrika”翻译为“像牛尿的轨迹”。考虑235乘以264的乘积。我们首先将和式排列如下：

2 235
6 235
4 235
----------

现在将顶行的235乘以左列顶部的2。从2 × 5 = 10开始，将0放在顶行5的下方，按通常方式进位得到

2 235
6 235
4 235
----------
 470

现在将第二行的235乘以左列中的6，将数字写在470下面一行但向右移动一位

2 235
6 235
4 235
----------
 470
 1410

现在将第三行的235乘以左列中的4，将数字写在1410下面一行但向右移动一位

2 235
6 235
4 235
----------
 470
 1410
 940

现在将线下的三个数字相加

2 235
6 235
4 235
----------
 470
 1410
 940
----------
 62040

这些变体的第一种是把第二个数写在右边，但数字的顺序颠倒如下

 235 4
 235 6
 235 2
----------
 940
 1410
 470
----------
 62040

第三种变体只是把每个数写一次，除此之外遵循第二种方法

 235
----------
 940 4
 1410 6
 470 2
----------
 62040

婆罗摩笈多提出的另一个算术结果是他计算平方根的算法。这个算法在[15]中讨论，其中证明它等价于艾萨克·牛顿-约瑟夫·拉弗森迭代公式。

婆罗摩笈多发展了一些代数记号，并给出了解二次方程的方法。他给出了解形如$ax + c = by$的不定方程的方法。Majumdar在[17]中写道：-

婆罗摩笈多也许使用了连分数方法来求形如$ax + c = by$的不定方程的整数解。

在[17]中，Majumdar给出了婆罗摩笈多的Brahmasphuta siddhantaⓉ（正确建立的梵天教义）中的原始梵文诗节及其英译和现代解释。

婆罗摩笈多还解决了类型为$ax^{2} + c = y^{2}$和$ax^{2} - c = y^{2}$的quadratic indeterminate equations。例如，他解$8x^{2} + 1 = y^{2}$得到解$(x, y) = (1, 3), (6, 17), (35, 99), (204, 577), (1189, 3363), ...$。对于方程$11x^{2} + 1 = y^{2}$，婆罗摩笈多得到解$(x, y) = (3, 10), (\large\frac{161}{5}\normalsize , \large\frac{534}{5}\normalsize ), ...$。他还解$61x^{2} + 1 = y^{2}$，这特别优雅，其最小解为$x = 226153980, y = 1766319049$。

婆罗摩笈多在BrahmasphutasiddhantaⓉ（正确建立的梵天学说）中提出并解决的一类问题的一个例子如下：-

五百德拉克马以未知利率贷出，这笔钱四个月的利息又以相同利率贷给另一个人，十个月后本息合计为78德拉克马。求利率。

书中也给出了级数求和的法则。婆罗摩笈多给出前$n$个自然数的平方和为$\large\frac{1}{6}\normalsize n(n+1)(2n+1)$，前$n$个自然数的立方和为$(\large\frac{1}{2}\normalsize n(n+1))^{2}$。没有给出证明，所以我们不知道婆罗摩笈多是如何发现这些公式的。

在BrahmasphutasiddhantaⓉ（《婆罗摩修正体系》）中，婆罗摩笈多给出了圆内接四边形的面积和对角线长度的以边表示的著名公式。这里唯一有争议的一点是婆罗摩笈多没有说明这些公式仅对圆内接四边形成立，因此一些历史学家声称这是一个错误，而另一些人则声称他明确意指这些规则仅适用于圆内接四边形。

BrahmasphutasiddhantaⓉ（《婆罗摩修正体系》）中的许多材料涉及日食和月食、行星合和行星位置。婆罗摩笈多相信地球是静止的，他在第一部著作中给出年的长度为365天6小时5分19秒，在第二本书KhandakhadyakaⓉ（“可食的一口”或“一口食物”）中将该值改为365天6小时12分36秒。当然，第二个值并不是对第一个值的改进，因为年的真实长度小于365天6小时。人们不禁要问，婆罗摩笈多的第二个年长度值是否取自阿耶波多，因为两者相差在6秒以内，却大约差了24分钟。

Khandakhadyaka Ⓣ（“可食用的一口”或“少量食物”）同样分为八章，涵盖的主题包括：行星的经度；周日旋转的三个问题；月食；日食；升落；月牙；以及行星的合。它包含一个附录，在某些版本中只有一章，在其他版本中有三章。

在婆罗摩笈多的第二部著作中，特别引起数学界兴趣的是他用来计算正弦值的插值公式。这在[13]中有详细研究，其中表明它是更一般的艾萨克·牛顿-斯特林插值公式在二阶以内的一个特例。

Brahmagupta, whose father was Jisnugupta, wrote important works on mathematics and astronomy. In particular he wrote Brahmasphutasiddhanta Ⓣ, in 628. The work was written in 25 chapters and Brahmagupta tells us in the text that he wrote it at Bhillamala which today is the city of Bhinmal. This was the capital of the lands ruled by the Gurjara dynasty.

Brahmagupta became the head of the astronomical observatory at Ujjain which was the foremost mathematical centre of ancient India at this time. Outstanding mathematicians such as Varahamihira had worked there and built up a strong school of mathematical astronomy.

In addition to the Brahmasphutasiddhanta Ⓣ Brahmagupta wrote a second work on mathematics and astronomy which is the Khandakhadyaka Ⓣ written in 665 when he was 67 years old. We look below at some of the remarkable ideas which Brahmagupta's two treatises contain. First let us give an overview of their contents.

The Brahmasphutasiddhanta Ⓣ contains twenty-five chapters but the first ten of these chapters seem to form what many historians believe was a first version of Brahmagupta's work and some manuscripts exist which contain only these chapters. These ten chapters are arranged in topics which are typical of Indian mathematical astronomy texts of the period. The topics covered are: mean longitudes of the planets; true longitudes of the planets; the three problems of diurnal rotation; lunar eclipses; solar eclipses; risings and settings; the moon's crescent; the moon's shadow; conjunctions of the planets with each other; and conjunctions of the planets with the fixed stars.

The remaining fifteen chapters seem to form a second work which is major addendum to the original treatise. The chapters are: examination of previous treatises on astronomy; on mathematics; additions to chapter 1; additions to chapter 2; additions to chapter 3; additions to chapter 4 and 5; additions to chapter 7; on algebra; on the gnomon; on meters; on the sphere; on instruments; summary of contents; versified tables.

Brahmagupta's understanding of the number systems went far beyond that of others of the period. In the Brahmasphutasiddhanta Ⓣ he defined zero as the result of subtracting a number from itself. He gave some properties as follows:-

When zero is added to a number or subtracted from a number, the number remains unchanged; and a number multiplied by zero becomes zero.

He also gives arithmetical rules in terms of fortunes (positive numbers) and debts (negative numbers):-

A debt minus zero is a debt.
A fortune minus zero is a fortune.
Zero minus zero is a zero.
A debt subtracted from zero is a fortune.
A fortune subtracted from zero is a debt.
The product of zero multiplied by a debt or fortune is zero.
The product of zero multipliedby zero is zero.
The product or quotient of two fortunes is one fortune.
The product or quotient of two debts is one fortune.
The product or quotient of a debt and a fortune is a debt.
The product or quotient of a fortune and a debt is a debt.

Brahmagupta then tried to extend arithmetic to include division by zero:-

Positive or negative numbers when divided by zero is a fraction the zero as denominator.
Zero divided by negative or positive numbers is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator.
Zero divided by zero is zero.

Really Brahmagupta is saying very little when he suggests that $n$ divided by zero is $n/0$. He is certainly wrong when he then claims that zero divided by zero is zero. However it is a brilliant attempt to extend arithmetic to negative numbers and zero.

We can also describe his methods of multiplication which use the place-value system to its full advantage in almost the same way as it is used today. We give three examples of the methods he presents in the Brahmasphutasiddhanta Ⓣ and in doing so we follow Ifrah in [4]. The first method we describe is called "gomutrika" by Brahmagupta. Ifrah translates "gomutrika" to "like the trajectory of a cow's urine". Consider the product of 235 multiplied by 264. We begin by setting out the sum as follows:

2 235
6 235
4 235
----------

Now multiply the 235 of the top row by the 2 in the top position of the left hand column. Begin by 2 × 5 = 10, putting 0 below the 5 of the top row, carrying 1 in the usual way to get

2 235
6 235
4 235
----------
 470

Now multiply the 235 of the second row by the 6 in the left hand column writing the number in the line below the 470 but moved one place to the right

2 235
6 235
4 235
----------
 470
 1410

Now multiply the 235 of the third row by the 4 in the left hand column writing the number in the line below the 1410 but moved one place to the right

2 235
6 235
4 235
----------
 470
 1410
 940

Now add the three numbers below the line

2 235
6 235
4 235
----------
 470
 1410
 940
----------
 62040

The variants are first writing the second number on the right but with the order of the digits reversed as follows

 235 4
 235 6
 235 2
----------
 940
 1410
 470
----------
 62040

The third variant just writes each number once but otherwise follows the second method

 235
----------
 940 4
 1410 6
 470 2
----------
 62040

Another arithmetical result presented by Brahmagupta is his algorithm for computing square roots. This algorithm is discussed in [15] where it is shown to be equivalent to the Newton-Raphson iterative formula.

Brahmagupta developed some algebraic notation and presents methods to solve quardatic equations. He presents methods to solve indeterminate equations of the form $ax + c = by$. Majumdar in [17] writes:-

Brahmagupta perhaps used the method of continued fractions to find the integral solution of an indeterminate equation of the type $ax + c = by$.

In [17] Majumdar gives the original Sanskrit verses from Brahmagupta's Brahmasphuta siddhanta Ⓣ and their English translation with modern interpretation.

Brahmagupta also solves quadratic indeterminate equations of the type $ax^{2} + c = y^{2}$ and $ax^{2} - c = y^{2}$. For example he solves $8x^{2} + 1 = y^{2}$ obtaining the solutions $(x, y) = (1, 3), (6, 17), (35, 99), (204, 577), (1189, 3363), ...$ For the equation $11x^{2} + 1 = y^{2}$ Brahmagupta obtained the solutions $(x, y) = (3, 10), (\large\frac{161}{5}\normalsize , \large\frac{534}{5}\normalsize ), ...$ He also solves $61x^{2} + 1 = y^{2}$ which is particularly elegant having $x = 226153980, y = 1766319049$ as its smallest solution.

A example of the type of problems Brahmagupta poses and solves in the Brahmasphutasiddhanta Ⓣ is the following:-

Five hundred drammas were loaned at an unknown rate of interest, The interest on the money for four months was loaned to another at the same rate of interest and amounted in ten mounths to 78 drammas. Give the rate of interest.

Rules for summing series are also given. Brahmagupta gives the sum of the squares of the first $n$ natural numbers as $\large\frac{1}{6}\normalsize n(n+1)(2n+1)$ and the sum of the cubes of the first $n$ natural numbers as $(\large\frac{1}{2}\normalsize n(n+1))^{2}$. No proofs are given so we do not know how Brahmagupta discovered these formulae.

In the Brahmasphutasiddhanta Ⓣ Brahmagupta gave remarkable formulae for the area of a cyclic quadrilateral and for the lengths of the diagonals in terms of the sides. The only debatable point here is that Brahmagupta does not state that the formulae are only true for cyclic quadrilaterals so some historians claim it to be an error while others claim that he clearly meant the rules to apply only to cyclic quadrilaterals.

Much material in the Brahmasphutasiddhanta Ⓣ deals with solar and lunar eclipses, planetary conjunctions and positions of the planets. Brahmagupta believed in a static Earth and he gave the length of the year as 365 days 6 hours 5 minutes 19 seconds in the first work, changing the value to 365 days 6 hours 12 minutes 36 seconds in the second book the Khandakhadyaka Ⓣ. This second values is not, of course, an improvement on the first since the true length of the years if less than 365 days 6 hours. One has to wonder whether Brahmagupta's second value for the length of the year is taken from Aryabhata I since the two agree to within 6 seconds, yet are about 24 minutes out.

The Khandakhadyaka Ⓣ is in eight chapters again covering topics such as: the longitudes of the planets; the three problems of diurnal rotation; lunar eclipses; solar eclipses; risings and settings; the moon's crescent; and conjunctions of the planets. It contains an appendix which is some versions has only one chapter, in other versions has three.

Of particular interest to mathematics in this second work by Brahmagupta is the interpolation formula he uses to compute values of sines. This is studied in detail in [13] where it is shown to be a particular case up to second order of the more general Newton-Stirling interpolation formula.