1 History of mathematics

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

4