Ancient Indian Mathematics :
The aim of this article is to present an overview of ancient Indian mathematics
together with a discussion on the sources and directions for future studies. For this
purpose it would be convenient to divide the material broadly into the following
groups:
1. Vedic mathematics
2. Mathematics from the Jaina tradition
3. Development of the number system and numerals
4. The mathematical astronomy tradition
5. Pat.
¯igan.
ita and the Bakhsh¯al¯i manuscript
6. The Kerala school of M¯adhava
We shall discuss these topics individually with the points as above in view.
Vedic mathematics
A major part of the body of mathematical knowledge from the Vedic period
that has come down to us is from the Sulvas¯utras ´ . The Sulvas¯utras ´ are composi-
tions aimed at providing instruction on the principles involved and procedures of
construction of the vedis (altars) and agnis (fireplaces) for the performance of the
yajnas, which were a key feature of the Vedic culture. The fireplaces were con-
structed in a variety of shapes such as falcons, tortoise, chariot wheels, circular
trough with a handle, pyre, etc (depending on the context and purpose of the par-
ticular yajna) with sizes of the order of 20 to 25 feet in length and width, and there
is a component of the Sulvas¯utras ´ describing the setting up of such platforms with
tiles of moderate sizes, of simple shapes like squares, triangles, and occasionally
special ones like pentagons. Many of the vedis involved, especially for the yajnas
for special occasions had dimensions of the order of 50 to 100 feet, and making the
overall plan involved being able to draw perpendiculars in that setting. This was
accomplished both through the method that is now taught in schools, involving
perpendicularity of the line joining the centres of two intersecting circles with the
line joining the two points of intersection, as also via the use of the converse of
"Pythagoras theorem"; they were familiar with the "Pythagoras theorem", and
explicit statement of the theorem is found in all the four major Sulvas¯utras ´ . The
Sulvas¯utras ´ also contain descriptions of various geometric principles and construc-
tions, including procedures for converting a square into a circle with equal area,
and vice versa, and a good approximation to the square root of 2 (see [4] for some
details).
The Sulvas¯utras ´ , like other Vedic knowledge, were transmitted only orally over
a long period. There have also been commentaries on some of the Sulvas¯utras ´ in
Sanskrit, but their period remains uncertain. When the first written versions of
the Sulvas¯utras ´ came up is unclear. The text versions with modern commentaries
were brought out by European scholars (Thibaut, B¨urk, van Gelder and others)
starting from the second half of the nineteenth century (see [42], [27]. [28], [37],
[21], [6], [44]). With regard to genesis of his study of the Sulvas¯utras ´ Thibaut
mentions that the first to direct attention to the importance of the Sulvas¯utras ´ was
Mr. A.C. Burnell, who in his Catalogue of a Collection of Sanscrit Manuscripts,
p 29, remarks that "we must look to the Sulva ´ portions of the Kalpas¯utras for the
earliest beginnings among the Br¯ahman. as.".
While the current translations are reasonably complete, some parts have eluded
the translators, especially in the case of M¯anava Sulvas¯utra ´ which turns out to be
more terse than the others. New results have been brought to light by R.G. Gupta
[13], Takao Hayashi and the present author (see [4], § 3), and perhaps also by
others, not recognised by the original translators. Lack of adequate mathematical
background on the part of the translators could be one of the factors in this
respect. There is a case for a relook on a substantial scale to put the mathematical
knowledge in the Sulvas¯utras ´ on a comprehensive footing. There is also scope for
work in the nature of interrelating in a cohesive manner the results described in
the various Sulvas¯utras ´ . The ritual context of the Sulvas¯utras ´ lends itself also
to the issue of interrelating the ritual and mathematical aspects, and correlating
with other similar situations from other cultures; for a perspective on this the
reader may refer Seidenberg [36].
Another natural question that suggests itself in the context of the Sulvas¯utras ´
is whether there are any of the fireplaces from the old times to be found. From
the description of the brick construction it would seem that they would have been
too fragile to withstand the elements for long; it
the purpose involved did not warrant a long-lasting construction. Nevertheless,
excavations at an archaeological site at Singhol in Panjab have revealed one large
brick platform in the traditional shape of a bird with outstretched wings, dated
to be from the second century BCE ([18], p.79-80 and [25], footnote on page
18); it however differs markedly from the numerical specifications described in
the Sulvas¯utras ´ . This leaves open the possibility of finding other sites, though
presumably not a very promising one.
Apart from the Sulvas¯utras ´ , mathematical studies have also been carried out
in respect of the Vedas, mainly concerning understanding of the numbers. For a
composition with a broad scope, including spiritual and secular, the R. gveda shows
considerable preoccupation with numbers, with numbers upto 10,000 occurring,
and the decimal representation of numbers is seen to be rooted there; see [2]. (It
should be borne in mind however that the numbers were not written down, and
the reference here is mainly to number names.) The Yajurveda introduces names
for powers of 10 upto 1012 and various simple properties of numbers are seen to
be involved in various contexts; [25] for instance. There is scope for further work
in understanding the development as a whole; this would involve familiarity with
mathematics on the one hand and knowledge of Vedic sanskrit on the other hand.
Mathematics from the Jaina tradition
There has been a long tradition among the Jainas of engaging with mathe-
matics. Their motivation came not from any rituals, which they abhorred, but
from contemplation of the cosmos, of which they had evolved an elaborate con-
ception. In the Jaina cosmography the universe is supposed to be a flat plane
with concentric annular regions surrounding an innermost circular region with a
diameter of 100000 yojanas known as the Jambudv¯ipa (island of Jambu), and the
annular regions alternately consist of water and land, and their widths increasing
twofold with each successive ring; it may be mentioned that this cosmography is
also found in the Pur¯anas. The geometry of the circle played an important role in
the overall discourse, even when the scholars engaged in it were primarily philoso-
phers rather than practitioners of mathematics. Many properties of the circle
have been described in S¯uryapraj˜n¯apti which is supposed to be from the fourth
or fifth century BCE (earliest extant manuscript is from around 1500, on paper)
and in the work of Um¯asv¯ati, who is supposed to have lived around 150 BCE
according to the Svet¯ambara ´ tradition and in the second century CE according
belief of 3 as the ratio of the circumference to the diameter; S¯uryapraj˜n¯apti recalls
the then traditional value 3 for it, and discards it in favour of √
10. The Jainas
were also aware from the early times that the ratio of the area of the circle to the
square of its radius is the same as the ratio of the circumference to the diameter.
They had also interesting approximate formulae for the lengths of circular arcs
and the areas subtended by them together with the corresponding chord. How
they arrived at these formula is not understood. Permutations and combinations,
sequences, categorisation of infinities are some of the other mathematical topics
on which elaborate discussion is found in Jaina literature.
Pronounced mathematical activity in the Jaina tradition is seen again from
the 8th century, and it may have continued until the middle of the 14 th century.
Gan.
ita-s¯ara-sangraha of Mah¯av¯ira, written in 850, is one of the well-known works
in this respect. V¯irasena (8th century), Sr¯idhara (between 850 and 950), Nemi-
candra (around 980 CE), T. hakkura Pher¯u (14 th century) are some of the other
names that may be mentioned with regard to development of mathematics in the
Jaina canon.
An approximation for π was given by V¯irasena by: "sixteen times the diameter,
together with 16, divided by 113 and thrice the diameter becomes a fine value
(of the circumference)". There is something strange about the formula that it
prescribes "together with 16" - surely it should have been known to the author that
the circumference is proportional to the diameter and that adding 16, irrespective
of the size of the diameter, would not be consistent with this. If one ignores
that part (on what ground?) we get the value of π as 3 + 16
113 =
355
113 , which
is indeed a good approximation, as the author stresses with the phrase "a fine
value (s¯ukshmadapi s¯ukshamam)", accurate to seven significant places. The same
formula was given by Chong-Zhi in China in the 5th century, (and I was told by
a Jaina scholar that the latter also involved the same error mentioned above).
This suggests the issue of exploring the mathematical contact with China and the
channels through which it may have occurred if it did. Specifically how such a
value may have been found (wherever it was found independently) would also be
worth exploring from a mathematical point of view.
In the work of T. hakkura Pher¯u from the early 14 th century one sees a combi-
nation of the native Jaina tradition together with Indo-Persian literature. Some of
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