婆什迦罗第二Bhaskara II

婆什迦罗第二也被称为婆什迦罗第二或Bhaskaracharya，后一个名字意为“婆什迦罗第二老师”。由于他在印度以Bhaskaracharya闻名，我们在本文中将始终用这个名字称呼他。Bhaskaracharya的父亲是一位名叫Mahesvara的婆罗门。Mahesvara本人作为占星家而闻名。这在印度社会中经常发生，一个家族连续几代都是杰出的数学家，并且常常担任其他家庭成员的老师。

Bhaskaracharya成为乌贾因天文台台长，乌贾因是当时印度领先的数学中心。伐罗诃密希罗和婆罗摩笈多等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。

在许多方面，婆什迦罗代表了12世纪数学知识的顶峰。他对数系和解方程的理解，是欧洲在数百年后才达到的。

已知婆什迦罗有六部著作，但据称属于他的第七部著作，被许多历史学家认为是晚期的伪作。这六部著作是：Lilavati（《美丽》）是关于数学的；Bijaganita（《种子计数或求根》）是关于代数的；Siddhantasiromani分为两部分，第一部分是关于数学天文学的，第二部分是关于球面的；Mitaksara的Vasanabhasya，这是婆什迦罗自己对Siddhantasiromani的注释；Karanakutuhala（《天文奇观的计算》）或Brahmatulya，是Siddhantasiromani的简化版本；以及Vivarana，是对拉拉的Shishyadhividdhidatantra的注释。其中前三部著作最为有趣，当然是从数学的角度来看，我们将集中讨论这些著作的内容。

鉴于他是在婆罗摩笈多的知识和理解基础上发展，Bhaskaracharya理解零和负数就不足为奇了。然而他的理解甚至比婆罗摩笈多更进一步。在我们更详细地考察他的工作之前，举一些例子，我们注意到他知道$x^{2} = 9$有两个解。他还给出了公式

$\sqrt{a \pm \sqrt{b}}=\sqrt \frac{a+\sqrt{a^{2}-b}}{2} \pm \sqrt \frac{a-\sqrt{a^{2}-b}}{2}$

Bhaskaracharya研究了约翰·佩尔的方程$px^{2} + 1 = y^{2}$，其中$p$ = 8、11、32、61和67。当$p = 61$时，他找到了解$x = 226153980, y = 1776319049$。当$p = 67$时，他找到了解$x = 5967, y = 48842$。他研究了许多Diophantine problems。

让我们首先考察Lilavati。首先值得重复Fyzi讲述的故事，他于1587年将此著作翻译成波斯文。我们给出Joseph在[5]中所述的故事：-

Lilavati是Bhaskaracharya女儿的名字。通过占星，他发现她婚礼的吉时将是某一天的特定时辰。他在一个装满水的容器底部放了一个带小孔的杯子，安排得使杯子在吉时开始时下沉。当一切准备就绪，杯子放入容器时，Lilavati突然出于好奇弯身靠近容器，她衣服上的一颗珍珠掉入杯中，堵住了杯孔。吉时过去，杯子没有下沉。Bhaskaracharya认为，安慰他如今永远无法结婚的沮丧女儿的方式，就是为她写一本数学手册！

这是一个迷人的故事，但很难看出有任何证据表明它是真实的。甚至不能确定Lilavati是Bhaskaracharya的女儿。还有一种理论认为Lilavati是Bhaskaracharya的妻子。该书十三章涵盖的主题有：定义；算术术语；利息；算术和几何级数；平面几何；立体几何；日晷的阴影；kuttaka；组合。

在处理数字时，婆什迦罗像他之前的婆罗摩笈多一样，有效地处理了涉及负数的算术。他在涉及零的加法、减法和乘法上是正确的，但意识到婆罗摩笈多的除以零的想法存在问题。Madhukar Mallayya在[14]中认为，婆什迦罗在Lilavati中给出的规则$(a.0)/0 = a$中所使用的零，等价于现代非零“无穷小”的概念。尽管这一说法并非没有根据，但也许它看到了超出婆什迦罗本意的思想。

Bhaskaracharya在他的Lilavati中给出了两种乘法方法。我们遵循Ifrah在[4]中解释的Bhaskaracharya的这两种方法。要将325乘以243，Bhaskaracharya这样写数字：

 243 243 243
 3 2 5
-------------------

现在处理三个和中最右边的一个，他计算了5乘以3，然后5乘以2，跳过了5乘以4，最后做这个，并写在其他数字下面向左移一位。注意这避免了在头脑中进行“进位”。

 243 243 243
 3 2 5
-------------------
 1015
 20


-------------------

现在把按此位置摆放的 1015 与 20 相加，并把答案写在第二行下方、和数旁边靠左的位置。

 243 243 243
 3 2 5
-------------------
 1015
 20
-------------------
 1215

像右边那个一样算出中间的和，同样避免“进位”，再把它们相加，把答案写在 1215 下方，但向左错开一位。

 243 243 243
 3 2 5
-------------------
 4 6 1015
 8 20
-------------------
 1215
 486

最后用同样的方法算出最左边的和，并再次把所得的加法结果向左错开一位写在 486 下方。

 243 243 243
 3 2 5
-------------------
 6 9 4 6 1015
 12 8 20
-------------------
 1215
 486
 729
-------------------

最后把第二行下方的三个数相加，得到答案 78975。

 243 243 243
 3 2 5
-------------------
 6 9 4 6 1015
 12 8 20
-------------------
 1215
 486
 729
-------------------
 78975

尽管在前几个阶段避免了“进位”，当然在最后这次加法中仍然会遇到“进位”。

⟦N1⟧的第二种方法如下：

 325
 243
--------

从最左边的数字开始，将下面的数乘以最上面的数，然后向右进行。每一行都比上一行向右移动一位。第一步

 325
 243
--------
 729

第二步

 325
 243
--------
 729
 486

第三步，然后相加

 325
 243
--------
 729
 486
 1215
--------
 78975

Bhaskaracharya 和许多印度数学家一样，把数的平方视为乘法的特殊情形，认为它值得采用特殊方法。他在Lilavati中给出了四种这样的平方方法。

下面是一个关于反比例解释的例子，取自Lilavati第3章。Bhaskaracharya 写道：-

在反演法中，运算被反过来进行。即，果实应乘以增量并除以需求量。当果实随需求量的增加或减少而增加或减少时，使用正比例法则。否则使用反演法。

反三率法：若果实随需求增加而减少，或随需求减少而增加，则精通算术者认为三率法应反用。当果实减少而需求增加，或果实增加而需求减少时，就采用反三率法。

与三率法一样，⟦N1⟧也讨论了一些例子来说明复合比例法则，如五率法（Pancarasika）、七率法（Saptarasika）、九率法（Navarasika）等。⟦N1⟧使用这些法则的例子在15中有所讨论。

第5章关于算术级数和几何级数的一个例子如下：-

例子：一位国王出征夺取敌人的大象，第一天行进了两由旬。请说，聪明的计算者，他每天行进的增加率是多少，因为他用一周时间到达了距离八十由旬的敌城？

⟦N1⟧表明，他每天必须比前一天多行进$\large\frac{22}{7}\normalsize$由旬，才能在7天内到达敌城。

第12章关于求解不定方程的kuttaka方法的一个例子如下：-

例子：数学家，快说，那个乘数是什么，使得二百二十一乘以它，再加上六十五，所得之和除以一百九十五后恰好除尽。

Bhaskaracharya 正在求 $195x = 221y + 65$ 的整数解。他得到解 $(x, y) = (6, 5)$ 或 (23, 20) 或 (40, 35) 等等。

在关于组合的最后一章中，Bhaskaracharya 考虑了以下问题。设一个 $n$ 位数用通常的十进制形式表示为

$d_{1}d_{2}... d_{n}$ (*)

其中每个数字满足 $1 ≤ d_{j} ≤ 9, j = 1, 2, ... , n$。那么 Bhaskaracharya 的问题是求形如 (*) 且满足

$d_{1} + d_{2} + ... + d_{n} = S$。

在 Lilavati 的结论中，Bhaskaracharya 写道：-

对于将⟦N1⟧紧贴喉部的人来说，这个世界上的欢乐与幸福确实在不断增长，其肢体因分数的约简、乘法与乘方而装饰整齐，其解答纯粹而完美，其言语如所例示的那样富有品味。

Bijaganita是一部十二章的著作。主题包括：正数与负数；零；未知量；根式；kuttaka；不定二次方程；一次方程；二次方程；含多个未知量的方程；含多个未知量的二次方程；多个未知量乘积的运算；以及作者及其著作。

在解释了如何进行负数算术之后，⟦N2⟧给出问题来测试读者在负数和正数计算方面的能力：-

例：若你懂得正数与负数的加法，请迅速说出三和四这两个数，无论负或正，合在一起的结果；即，正与负，或两者皆负，或两者皆正，作为单独的例子。

负数通过在它们上方放置一个点来表示：-

表示已知量和未知量的符号，应首先写出以一般地指示它们；而那些变为负数的，则应在其上方标上一个点。

例如：三减二，正减正，负减负，或者相反，快告诉我结果……

在Bijaganita中，婆什迦罗试图改进婆罗摩笈多除以零的尝试（以及他自己在Lilavati中的描述），他写道：-

一个量除以零成为一个分数，其分母为零。这个分数被称为无限量。在这个由以零为除数的量组成的量中，没有变化，尽管可以插入或提取许多；正如当世界被创造或毁灭时，无限不变的上帝没有变化，尽管无数种类的存在被吸收或发出。

所以婆什迦罗试图通过写$n$/0 = ∞来解决这个问题。乍一看，我们可能会倾向于相信婆什迦罗是正确的，但当然他并不正确。如果这是真的，那么0乘以∞必须等于每个数$n$，所以所有数都相等。印度数学家无法让自己承认不能除以零。

导致多于一个解的方程由婆什迦罗给出：-

例如：在森林里，一群猿猴中总数的八分之一的平方数量的猿猴正在玩吵闹的游戏。剩下的十二只猿猴，性格更为严肃，在附近的山上，被来自森林的尖叫声激怒。这群猿猴的总数是多少？

这个问题导出一个二次方程，Bhaskaracharya说两个解，即16和48，同样可取。

用于求解不定方程的kuttaka方法被应用于含三个未知量的方程。问题是求形如$ax + by + cz = d$的方程的整数解。他给出的一个例子是：-

例：属于四个人的马分别为5、3、6和8匹。属于同样这些人的骆驼分别为2、7、4和1头。属于他们的骡子分别为8、2、1和3头，牛分别为7、1、2和1头。这四个人拥有相等的财产。快告诉我每匹马、每头骆驼、每头骡子和每头牛的价格。

当然，这类问题没有唯一解，Bhaskaracharya对此完全清楚。他求出一个解，即最小的解，马85，骆驼76，骡子31，牛4。

Bhaskaracharya对Bijaganita的结论引人入胜，因为它让我们得以洞察这位伟大数学家的思想：-

点滴的教导能把知识传给一个领悟力强的头脑；而一旦到达那里，它便凭自身的冲力扩展开来，如同倾在水上的油，如同托付给卑劣者的秘密，如同施予配得之人的施舍，无论多么少，注入智慧头脑中的知识也照样凭内在的力量传播开来。

对于理解清晰的人来说显而易见，三项法则构成算术，而睿智构成代数。因此我说过……三项法则是算术；无瑕的理解是代数。对于智者还有什么未知的呢？因此它只为愚钝者而设。

Siddhantasiromani是一部数学天文学著作，其布局与这一时期及更早时期的许多印度天文学著作相似。第一部分的十二章涵盖的主题包括：行星的平均经度；行星的真经度；周日旋转的三个问题；朔望；月食；日食；行星的纬度；升落；月牙；行星彼此之间的合；行星与恒星的合；以及太阳和月亮的影。

第二部分包含十三章关于球面的内容。它涵盖的主题包括：对球面研究的赞扬；球面的性质；宇宙志与地理学；行星平均运动；行星的偏心本轮模型；浑天仪；球面三角学；椭圆计算；行星的首次可见性；计算月牙；天文仪器；季节；以及天文计算问题。

这部著作中有关于三角学的有趣结果。特别是，Bhaskaracharya似乎比他的前辈更出于三角学本身而对它感兴趣，而他的前辈只把它视为计算的工具。在Bhaskaracharya给出的许多有趣结果中包括：

$\sin(a + b) = \sin a \cos b + \cos a \sin b$

和

$\sin(a - b) = \sin a \cos b - \cos a \sin b$。

Bhaskaracharya因其卓越贡献而理所当然地赢得了杰出声誉。1207年，建立了一个教育机构来研究Bhaskaracharya的著作。印度一座寺庙中的中世纪铭文写道：

光辉的Bhaskaracharya胜利了，他的功绩受到智者和学者的崇敬。他是一位兼具名声与宗教功德的诗人，犹如孔雀的冠羽。

约瑟夫的著作5的书名正是出自这段引文。

Bhaskara is also known as Bhaskara II or as Bhaskaracharya, this latter name meaning "Bhaskara the Teacher". Since he is known in India as Bhaskaracharya we will refer to him throughout this article by that name. Bhaskaracharya's father was a Brahman named Mahesvara. Mahesvara himself was famed as an astrologer. This happened frequently in Indian society with generations of a family being excellent mathematicians and often acting as teachers to other family members.

Bhaskaracharya became head of the astronomical observatory at Ujjain, the leading mathematical centre in India at that time. Outstanding mathematicians such as Varahamihira and Brahmagupta had worked there and built up a strong school of mathematical astronomy.

In many ways Bhaskaracharya represents the peak of mathematical knowledge in the 12th century. He reached an understanding of the number systems and solving equations which was not to be achieved in Europe for several centuries.

Six works by Bhaskaracharya are known but a seventh work, which is claimed to be by him, is thought by many historians to be a late forgery. The six works are: Lilavati (The Beautiful) which is on mathematics; Bijaganita (Seed Counting or Root Extraction) which is on algebra; the Siddhantasiromani which is in two parts, the first on mathematical astronomy with the second part on the sphere; the Vasanabhasya of Mitaksara which is Bhaskaracharya's own commentary on the Siddhantasiromani ; the Karanakutuhala (Calculation of Astronomical Wonders) or Brahmatulya which is a simplified version of the Siddhantasiromani ; and the Vivarana which is a commentary on the Shishyadhividdhidatantra of Lalla. It is the first three of these works which are the most interesting, certainly from the point of view of mathematics, and we will concentrate on the contents of these.

Given that he was building on the knowledge and understanding of Brahmagupta it is not surprising that Bhaskaracharya understood about zero and negative numbers. However his understanding went further even than that of Brahmagupta. To give some examples before we examine his work in a little more detail we note that he knew that $x^{2} = 9$ had two solutions. He also gave the formula

$\sqrt{a \pm \sqrt{b}}=\sqrt \frac{a+\sqrt{a^{2}-b}}{2} \pm \sqrt \frac{a-\sqrt{a^{2}-b}}{2}$

Bhaskaracharya studied Pell's equation $px^{2} + 1 = y^{2}$ for $p$ = 8, 11, 32, 61 and 67. When $p = 61$ he found the solutions $x = 226153980, y = 1776319049$. When $p = 67$ he found the solutions $x = 5967, y = 48842$. He studied many Diophantine problems.

Let us first examine the Lilavati. First it is worth repeating the story told by Fyzi who translated this work into Persian in 1587. We give the story as given by Joseph in [5]:-

Lilavati was the name of Bhaskaracharya's daughter. From casting her horoscope, he discovered that the auspicious time for her wedding would be a particular hour on a certain day. He placed a cup with a small hole at the bottom of the vessel filled with water, arranged so that the cup would sink at the beginning of the propitious hour. When everything was ready and the cup was placed in the vessel, Lilavati suddenly out of curiosity bent over the vessel and a pearl from her dress fell into the cup and blocked the hole in it. The lucky hour passed without the cup sinking. Bhaskaracharya believed that the way to console his dejected daughter, who now would never get married, was to write her a manual of mathematics!

This is a charming story but it is hard to see that there is any evidence for it being true. It is not even certain that Lilavati was Bhaskaracharya's daughter. There is also a theory that Lilavati was Bhaskaracharya's wife. The topics covered in the thirteen chapters of the book are: definitions; arithmetical terms; interest; arithmetical and geometrical progressions; plane geometry; solid geometry; the shadow of the gnomon; the kuttaka; combinations.

In dealing with numbers Bhaskaracharya, like Brahmagupta before him, handled efficiently arithmetic involving negative numbers. He is sound in addition, subtraction and multiplication involving zero but realised that there were problems with Brahmagupta's ideas of dividing by zero. Madhukar Mallayya in [14] argues that the zero used by Bhaskaracharya in his rule $(a.0)/0 = a$, given in Lilavati, is equivalent to the modern concept of a non-zero "infinitesimal". Although this claim is not without foundation, perhaps it is seeing ideas beyond what Bhaskaracharya intended.

Bhaskaracharya gave two methods of multiplication in his Lilavati. We follow Ifrah who explains these two methods due to Bhaskaracharya in [4]. To multiply 325 by 243 Bhaskaracharya writes the numbers thus:

 243 243 243
 3 2 5
-------------------

Now working with the rightmost of the three sums he computed 5 times 3 then 5 times 2 missing out the 5 times 4 which he did last and wrote beneath the others one place to the left. Note that this avoids making the "carry" in ones head.

 243 243 243
 3 2 5
-------------------
 1015
 20


-------------------

Now add the 1015 and 20 so positioned and write the answer under the second line below the sum next to the left.

 243 243 243
 3 2 5
-------------------
 1015
 20
-------------------
 1215

Work out the middle sum as the right-hand one, again avoiding the "carry", and add them writing the answer below the 1215 but displaced one place to the left.

 243 243 243
 3 2 5
-------------------
 4 6 1015
 8 20
-------------------
 1215
 486

Finally work out the left most sum in the same way and again place the resulting addition one place to the left under the 486.

 243 243 243
 3 2 5
-------------------
 6 9 4 6 1015
 12 8 20
-------------------
 1215
 486
 729
-------------------

Finally add the three numbers below the second line to obtain the answer 78975.

 243 243 243
 3 2 5
-------------------
 6 9 4 6 1015
 12 8 20
-------------------
 1215
 486
 729
-------------------
 78975

Despite avoiding the "carry" in the first stages, of course one is still faced with the "carry" in this final addition.

The second of Bhaskaracharya's methods proceeds as follows:

 325
 243
--------

Multiply the bottom number by the top number starting with the left-most digit and proceeding towards the right. Displace each row one place to start one place further right than the previous line. First step

 325
 243
--------
 729

Second step

 325
 243
--------
 729
 486

Third step, then add

 325
 243
--------
 729
 486
 1215
--------
 78975

Bhaskaracharya, like many of the Indian mathematicians, considered squaring of numbers as special cases of multiplication which deserved special methods. He gave four such methods of squaring in Lilavati.

Here is an example of explanation of inverse proportion taken from Chapter 3 of the Lilavati. Bhaskaracharya writes:-

In the inverse method, the operation is reversed. That is the fruit to be multiplied by the augment and divided by the demand. When fruit increases or decreases, as the demand is augmented or diminished, the direct rule is used. Else the inverse.

Rule of three inverse: If the fruit diminish as the requisition increases, or augment as that decreases, they, who are skilled in accounts, consider the rule of three to be inverted. When there is a diminution of fruit, if there be increase of requisition, and increase of fruit if there be diminution of requisition, then the inverse rule of three is employed.

As well as the rule of three, Bhaskaracharya discusses examples to illustrate rules of compound proportions, such as the rule of five (Pancarasika), the rule of seven (Saptarasika), the rule of nine (Navarasika), etc. Bhaskaracharya's examples of using these rules are discussed in [15].

An example from Chapter 5 on arithmetical and geometrical progressions is the following:-

Example: On an expedition to seize his enemy's elephants, a king marched two yojanas the first day. Say, intelligent calculator, with what increasing rate of daily march did he proceed, since he reached his foe's city, a distance of eighty yojanas, in a week?

Bhaskaracharya shows that each day he must travel $\large\frac{22}{7}\normalsize$ yojanas further than the previous day to reach his foe's city in 7 days.

An example from Chapter 12 on the kuttaka method of solving indeterminate equations is the following:-

Example: Say quickly, mathematician, what is that multiplier, by which two hundred and twenty-one being multiplied, and sixty-five added to the product, the sum divided by a hundred and ninety-five becomes exhausted.

Bhaskaracharya is finding integer solution to $195x = 221y + 65$. He obtains the solutions $(x, y) = (6, 5)$ or (23, 20) or (40, 35) and so on.

In the final chapter on combinations Bhaskaracharya considers the following problem. Let an $n$-digit number be represented in the usual decimal form as

$d_{1}d_{2}... d_{n}$     (*)

where each digit satisfies $1 ≤ d_{j} ≤ 9, j = 1, 2, ... , n$. Then Bhaskaracharya's problem is to find the total number of numbers of the form (*) that satisfy

$d_{1} + d_{2} + ... + d_{n} = S$.

In his conclusion to Lilavati Bhaskaracharya writes:-

Joy and happiness is indeed ever increasing in this world for those who have Lilavati clasped to their throats, decorated as the members are with neat reduction of fractions, multiplication and involution, pure and perfect as are the solutions, and tasteful as is the speech which is exemplified.

The Bijaganita is a work in twelve chapters. The topics are: positive and negative numbers; zero; the unknown; surds; the kuttaka; indeterminate quadratic equations; simple equations; quadratic equations; equations with more than one unknown; quadratic equations with more than one unknown; operations with products of several unknowns; and the author and his work.

Having explained how to do arithmetic with negative numbers, Bhaskaracharya gives problems to test the abilities of the reader on calculating with negative and affirmative quantities:-

Example: Tell quickly the result of the numbers three and four, negative or affirmative, taken together; that is, affirmative and negative, or both negative or both affirmative, as separate instances; if thou know the addition of affirmative and negative quantities.

Negative numbers are denoted by placing a dot above them:-

The characters, denoting the quantities known and unknown, should be first written to indicate them generally; and those, which become negative should be then marked with a dot over them.

Example: Subtracting two from three, affirmative from affirmative, and negative from negative, or the contrary, tell me quickly the result ...

In Bijaganita Bhaskaracharya attempted to improve on Brahmagupta's attempt to divide by zero (and his own description in Lilavati ) when he wrote:-

A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth.

So Bhaskaracharya tried to solve the problem by writing $n$/0 = ∞. At first sight we might be tempted to believe that Bhaskaracharya has it correct, but of course he does not. If this were true then 0 times ∞ must be equal to every number $n$, so all numbers are equal. The Indian mathematicians could not bring themselves to the point of admitting that one could not divide by zero.

Equations leading to more than one solution are given by Bhaskaracharya:-

Example: Inside a forest, a number of apes equal to the square of one-eighth of the total apes in the pack are playing noisy games. The remaining twelve apes, who are of a more serious disposition, are on a nearby hill and irritated by the shrieks coming from the forest. What is the total number of apes in the pack?

The problem leads to a quadratic equation and Bhaskaracharya says that the two solutions, namely 16 and 48, are equally admissible.

The kuttaka method to solve indeterminate equations is applied to equations with three unknowns. The problem is to find integer solutions to an equation of the form $ax + by + cz = d$. An example he gives is:-

Example: The horses belonging to four men are 5, 3, 6 and 8. The camels belonging to the same men are 2, 7, 4 and 1. The mules belonging to them are 8, 2, 1 and 3 and the oxen are 7, 1, 2 and 1. all four men have equal fortunes. Tell me quickly the price of each horse, camel, mule and ox.

Of course such problems do not have a unique solution as Bhaskaracharya is fully aware. He finds one solution, which is the minimum, namely horses 85, camels 76, mules 31 and oxen 4.

Bhaskaracharya's conclusion to the Bijaganita is fascinating for the insight it gives us into the mind of this great mathematician:-

A morsel of tuition conveys knowledge to a comprehensive mind; and having reached it, expands of its own impulse, as oil poured upon water, as a secret entrusted to the vile, as alms bestowed upon the worthy, however little, so does knowledge infused into a wise mind spread by intrinsic force.

It is apparent to men of clear understanding, that the rule of three terms constitutes arithmetic and sagacity constitutes algebra. Accordingly I have said ... The rule of three terms is arithmetic; spotless understanding is algebra. What is there unknown to the intelligent? Therefore for the dull alone it is set forth.

The Siddhantasiromani is a mathematical astronomy text similar in layout to many other Indian astronomy texts of this and earlier periods. The twelve chapters of the first part cover topics such as: mean longitudes of the planets; true longitudes of the planets; the three problems of diurnal rotation; syzygies; lunar eclipses; solar eclipses; latitudes of the planets; risings and settings; the moon's crescent; conjunctions of the planets with each other; conjunctions of the planets with the fixed stars; and the patas of the sun and moon.

The second part contains thirteen chapters on the sphere. It covers topics such as: praise of study of the sphere; nature of the sphere; cosmography and geography; planetary mean motion; eccentric epicyclic model of the planets; the armillary sphere; spherical trigonometry; ellipse calculations; first visibilities of the planets; calculating the lunar crescent; astronomical instruments; the seasons; and problems of astronomical calculations.

There are interesting results on trigonometry in this work. In particular Bhaskaracharya seems more interested in trigonometry for its own sake than his predecessors who saw it only as a tool for calculation. Among the many interesting results given by Bhaskaracharya are:

$\sin(a + b) = \sin a \cos b + \cos a \sin b$

and

$\sin(a - b) = \sin a \cos b - \cos a \sin b$.

Bhaskaracharya rightly achieved an outstanding reputation for his remarkable contribution. In 1207 an educational institution was set up to study Bhaskaracharya's works. A medieval inscription in an Indian temple reads:-

Triumphant is the illustrious Bhaskaracharya whose feats are revered by both the wise and the learned. A poet endowed with fame and religious merit, he is like the crest on a peacock.

It is from this quotation that the title of Joseph's book [5] comes.