Mahavira 

Born: about 800 in possibly Mysore,
India Died: about 870 in India Mahavira
(or Mahaviracharya meaning Mahavira the Teacher) was of the Jaina
religion and was familiar with Jaina mathematics. He worked in Mysore
in southern Indian where he was a member of a school of mathematics.
If he was not born in Mysore then it is very likely that he was born
close to this town in the same region of India. We have essentially
no other biographical details although we can gain just a little of
his personality from the acknowledgement he gives in the introduction
to his only known work, see below. However Jain in [10] mentions six
other works which he credits to Mahavira and he emphasises the need
for further research into identifying the complete list of his works.
The only known book by Mahavira is Ganita Sara Samgraha,
dated 850 AD, which was designed as an updating of Brahmagupta's
book. Filliozat writes [6]: This book deals with
the teaching of Brahmagupta but contains both simplifications and
additional information. ... Although like all Indian versified texts,
it is extremely condensed, this work, from a pedagogical point of
view, has a significant advantage over earlier texts.
It consisted of nine chapters and included all mathematical knowledge
of midninth century India. It provides us with the bulk of knowledge
which we have of Jaina mathematics and it can be seen as in some
sense providing an account of the work of those who developed this
mathematics. There were many Indian mathematicians before the time of
Mahavira but, perhaps surprisingly, their work on mathematics is
always contained in texts which discuss other topics such as
astronomy. The Ganita Sara Samgraha by Mahavira is the earliest
Indian text which we possess which is devoted entirely to
mathematics. In the introduction to the work
Mahavira paid tribute to the mathematicians whose work formed the
basis of his book. These mathematicians included Aryabhata I,
Bhaskara I, and Brahmagupta. Mahavira writes:
With the help of the accomplished holy sages, who are worthy to be
worshipped by the lords of the world ... I glean from the great ocean
of the knowledge of numbers a little of its essence, in the manner in
which gems are picked from the sea, gold from the stony rock and the
pearl from the oyster shell; and I give out according to the power of
my intelligence, the Sara Samgraha, a small work on arithmetic, which
is however not small in importance. The nine
chapters of the Ganita Sara Samgraha are: 1.
Terminology 2. Arithmetical operations 3.
Operations involving fractions 4. Miscellaneous operations
5. Operations involving the rule of three 6. Mixed
operations 7. Operations relating to the calculations of
areas 8. Operations relating to excavations 9.
Operations relating to shadows Throughout the work
a placevalue system with nine numerals is used or sometimes Sanskrit
numeral symbols are used. Of interest in Chapter 1 regarding the
development of a placevalue number system is Mahavira's description
of the number 12345654321 which he obtains after a calculation. He
describes the number as: ... beginning with one
which then grows until it reaches six, then decreases in reverse
order. Notice that this wording makes sense to us
using a placevalue system but would not make sense in other systems.
It is a clear indication that Mahavira is at home with the
placevalue number system. Among topics Mahavira
discussed in his treatise was operations with fractions including
methods to decompose integers and fractions into unit fractions. For
example 2/17 = 1/12 + 1/51 + 1/68.
He examined methods of squaring numbers which, although a special
case of multiplying two numbers, can be computed using special
methods. He also discussed integer solutions of first degree
indeterminate equation by a method called kuttaka. The kuttaka (or
the "pulveriser") method is based on the use of the Euclidean
algorithm but the method of solution also resembles the continued
fraction process of Euler given in 1764. The work kuttaka, which
occurs in many of the treatises of Indian mathematicians of the
classical period, has taken on the more general meaning of "algebra".
An example of a problem given in the Ganita Sara Samgraha
which leads to indeterminate linear equations is the following:
Three merchants find a purse lying in the road. One
merchant says "If I keep the purse, I shall have twice as much money
as the two of you together". "Give me the purse and I shall have
three times as much" said the second merchant. The third merchant
said "I shall be much better off than either of you if I keep the
purse, I shall have five times as much as the two of you together".
How much money is in the purse? How much money does each merchant
have? If the first merchant has x, the second y,
the third z and p is the amount in the purse then
p + x = 2(y + z), p + y = 3(x + z), p + z = 5(x + y).
There is no unique solution but the smallest solution in positive
integers is p = 15, x = 1, y = 3, z = 5. Any solution in positive
integers is a multiple of this solution as Mahavira claims.
Mahavira gave special rules for the use of permutations and
combinations which was a topic of special interest in Jaina
mathematics. He also described a process for calculating the volume
of a sphere and one for calculating the cube root of a number. He
looked at some geometrical results including rightangled triangles
with rational sides, see for example [4]. Mahavira
also attempts to solve certain mathematical problems which had not
been studied by other Indian mathematicians. For example, he gave an
approximate formula for the area and the perimeter of an ellipse. In
[8] Hayashi writes: The formulas for a conchlike
figure have so far been found only in the works of Mahavira and
Narayana. It is reasonable to ask what a
"conchlike figure" is. It is two unequal semicircles (with diameters
AB and BC) stuck together along their diameters. Although it might be
reasonable to suppose that the perimeter might be obtained by
considering the semicircles, Hayashi claims that the formulae
obtained: ... were most probably obtained not
from the two semicircles AB and BC.
Article by: J J O'Connor and E F Robertson Source:
www.history.mcs.standrews.ac.uk/Mathematicians



