|Born: 1114 in Vijayapura, India
1185 in Ujjain, India 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.
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
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 x2 = 9 had two
solutions. He also gave the formula
Pell's equation px2 + 1 = y2 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
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
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
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;
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
 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 gave two methods of
multiplication in his Lilavati. We follow Ifrah who explains these
two methods due to Bhaskaracharya in . 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
Now add the 1015 and 20 so positioned
and write the answer under the second line below the sum next to the
243 243 243
3 2 5
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
243 243 243
3 2 5
4 6 1015
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
4 6 1015
12 8 20
the three numbers below the second line to obtain the answer 78975.
243 243 243
3 2 5
4 6 1015
12 8 20
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:
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
Third step, then add
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.
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 .
An example from Chapter 5 on arithmetical
and geometrical progressions is the following:-
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:-
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
(*) d1d2... dn
where each digit
satisfies 1 dj 9, j = 1, 2, ... , n. Then Bhaskaracharya's problem is
to find the total number of numbers of the form (*) that satisfy
d1 + d2 + ... + dn = S.
In his conclusion to Lilavati
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
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:-
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 ...
Bijaganita Bhaskaracharya attempted to improve on Brahmagupta's
attempt to divide by zero (and his own description in Lilavati ) when
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 = 8. 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 8 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
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:-
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
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.
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
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  comes.
Article by: J J
O'Connor and E F Robertson