**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 12^{th} 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

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 ^{22}/_{7} 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 195*x* = 221*y* + 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 are5, 3, 6and8. The camels belonging to the same men are2, 7, 4and1. The mules belonging to them are8, 2, 1and3and the oxen are7, 1, 2and1. 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) = sinacosb+ cosasinb

and

sin(

a-b) = sinacosb- cosasinb.

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.

**Article by:** *J J O'Connor* and *E F Robertson*

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