This write up below is verbatim version of the above link . The author of this blog post has not contributed towards this article in any form. Objective of this post is to disseminate any knowledge about ancient Indian thought
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 compositions 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 constructed 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
particular 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 1 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
constructions, 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 should be borne in mind that 2 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 mathematics. Their motivation came not from any
rituals, which they abhorred, but from contemplation of the cosmos, of which
they had evolved an elaborate conception. 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 philosophers 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 to the Digambara
tradition of Jainas. 3 One of the notable features of the Jaina tradition is
the departure from old 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), Nemicandra (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
combination of the native Jaina tradition together with Indo-Persian
literature. Some of 4 the geometry discussed, involving domes, arches etc., has
close connections with the development of Islamic architecture in India. A
Jaina astronomer Mahendra S¯uri who was at the court of a monarch of the
Tughluq dynasty during the late 14th century wrote on the astrolabe. Various
links between Sanskrit and Islamic science in Jaina astronomical works have been
discussed in [26] The text of Gan. ita-s¯ara-sam. graha with English
translation was brought out by Rangacharya in 1912, and has been recently
reprinted [31]. More recently an edition with English and Kannada translations
together with the original text has been brought out by Padmavathamma [22]. An
edition of T. hakkura Pher¯u’s Gan. ita-s¯ara-kaumudi has been brought out very
recently through a collaborative effort [34]. A collection of papers giving an
overview of various aspects of Jaina mathematics may be found in [14]; see also
[5], an older exposition on the topic. On the whole however there has not been
adequate systematic study of the Jaina works from a mathematical point of view
(even from a comparative point of view within the context of studies in ancient
Indian mathematics). With regard to the older Jaina works, from BCE and the
early centuries of CE, there is considerable lack of understanding. Adequate
information needs to be pooled up on available resources, in the first place.
Development of the number system and numerals
Study of development of the number system in
India cuts across the Vedic, Jaina and Buddhist traditions. From the early
times one sees a fascination for large numbers in India, as we noted in the
earlier sections on the Vedic and Jaina traditions. Large numbers are also
found in the Buddhist tradition, and Buddha himself was renowned for his
prowess with numbers; Tallaks.han. a, a term from the Buddhist tradition,
represented 1053. The names of powers of 10 however differed over traditions
and also over period; e.g. Par¯ardha, which literally means “halfway to
heaven”, meant 1012 in early literature, it stood for 1017 in later works such
as of Bh¯askara II. The oral tradition of usage of several powers of 10 is
likely to have played a crucial role in the emergence of decimal representation
in written form at later stage, apparently in the early centuries of the common
era. This connection is however not very straightforward; there was a long
period, of several hundred years, in between when written form of numbers did
not follow the place value notation (see below); besides, even after the
decimal place value system with zero came into vogue the other systems seem to
have continued to be used for quite a while. One may wonder about the reasons
for this in the context of 5 the frequent references to the powers of 10 in the
oral tradition, and the apparent convenience and elegance of the decimal place
value system. On the other hand the Chinese seem to have used decimal place value
system for representation of numbers, without a symbol for zero in place of
which they left a blank space, from very early times and at least from the 3rd
century BCE. Introduction of zero as a place holder paved the way for the
writing the numbers as we do now, as far as the whole numbers are concerned;
the full decimal representation system as we now use, extending also to the
fractional part, with a separating decimal point, had its beginnings in the 15
th century Europe, though it is noted to have been first used by Arabs in the
10 th century. Conceptualisation of zero as a number, integrated into the
number system, happened in the early centuries of the common era in India, and
in Brahmagupta’s work Br¯ahmasphut.asiddh¯anta we find a systematic exposition,
in the seventh century, which includes also arithmetic with negative numbers.
The development of the numerals is a parallel topic. Written numerals in
various forms have been studied. The earliest of these could go back to the
Indus seals, with strokes representing numbers. Kharos. t.hi numerals which
were used between 3rd century BCE to 3rd century CE and are found in the
inscriptions from Kalderra, Taks.a´sila and Lorian, and Br¯ahmi numerals from
N¯an. eghat (first century BCE) are some of the ancient numerals; incidentally
they did not use the place value system. The earliest extant inscriptions
involving the decimal numeral system is said to be from Gujarat, dated 595 CE;
it has however been argued by R.Saloman in [32] that this is a spurious inscription.
The oldest known zero in an inscription in India is from 876 CE and is found in
a temple in Gwalior (an image of this may be viewed online, thanks to Bill
Casselman). Much research was done by Bhagwanlal Indraji in the late nineteenth
century, an account of which may be found in the book of George Ifrah [16]; (a
good deal of what Ifrah says has been contradicted by various reviewers - see
[8] for details - one may nevertheless suppose that what he reports from the
work of Bhagwanlal Indraji would be reliable). Apart from the inscriptions in
stone, copper plates that were legal documents from around the 7 th to 10th
centuries, recording grants of gifts by kings or rich persons to Br¯ahmanas,
have been examined for numerals presented in decimal system. There have been
some objections to this source, as the plates are susceptible to forgery, on
account of attempts to misappropriate the properties involved, 6 but while this
may apply to a few plates, as a whole the plates may not be discounted as a source;
see [7], pages 44-48, for a discussion on this, where the author tries to rebut
the objections. Numerals appearing in ancient manuscripts are another source in
this respect. Mathematical works in all the traditions involve large numbers
and the way the numbers are represented in the extant manuscripts from
different times would be an interesting aspect of study. It may be observed
that various Indian languages have their own symbols for the individual digits,
and the genesis of these systems would also be a related issue. The author is
not aware of any comprehensive study on the topic. A systematic archiving of
the material in this respect from various sources is very much called for,
followed by an analysis of the path of development of ideas as may be discerned
from the sources.
The mathematical astronomy
tradition
The Siddh¯anta or mathematical
astronomy tradition has been the dominant stream of mathematics in India, with
an essentially continuous tradition that flourished for close to a thousand
years, starting from about the third or fourth centuries. Aryabhat ¯ .a (476 -
550) is the first major figure from the tradition and is regarded as the
founder of scientific astronomy in India. The Siddh¯anta tradition indeed
continued until Bh¯askara II (1114 - 1185), and also beyond, though he is
viewed as the last major exponent in the continuity. The Aryabhat ¯ . ¯iya,
written in 499, is the earliest completely surviving composition from among the
Siddh¯anta works and is basic to the tradition, and also to the later works of
the Kerala school of M¯adhava which I shall discuss below. It consists of 121
verses divided into four chapters G¯itikap¯ada, Ganitap¯ada, Kalakriyap¯ada and
Golap¯ada. The first one sets out the cosmology and contains also a table of 24
sine differences at intervals of 225 minutes of arc, in a single verse. The
second chapter, as the name suggests, is devoted to mathematics, and includes
in particular procedures for finding square roots and cube roots, an
approximate expression for π, formulae for areas and volumes of various
geometric figures, formulae for sums of consecutive integers, sums of squares,
sums of cubes and computation of interest; see [40] and [41] for various
details. The other two chapters are concerned with astronomy, dealing with
distances and relative motions of planets, eclipses etc. (we shall not go into
the details here). Var¯ahamihira (505 - 587), Bh¯askara I (600 - 680),
Brahmagupta (598 - 668), Govindaswam¯i (800 - 860), Sankaran¯ar¯ayana (840 -
900), ´ Aryabhat ¯ .a II (920 - 7 1000), Vijay¯anand¯i (940 - 1010), Sr¯ipati
(1019 - 1066), Brahmadeva (1060 - 1130) were some of the major figures during
the period until Bh¯askara II; N¯ar¯ayana Pan. dit and Gan. e´sa may be named
from the later centuries (14 th and 16 th) as directly from the tradition. It
should be mentioned that many of the dates quoted here are approximate, as
there is no reliable historical information available on them and the dating is
based on various indirect inferences. The bulk of the work presented in
Brahmagupta’s Br¯ahmasphutasiddh¯anta is on astronomy. There are however two
chapters, 12th and 18th, devoted to general mathematics. Also, the 21st chapter
has verses dealing with trigonometry, which in Siddhanta astronomy literature
used to be combined with astronomy, rather than the mathematical topics
discussed in the works. Another special feature of the work is chapter 11,
which is a critique on earlier works including Aryabhat ¯ . ¯iya; like in other
scientific communities this tradition had also many internal controversies; the
strong language used would however seem disconcerting by contemporary
standards. Chapter 12 is known for a sytematic formulation of arithmetic
operations, including with negative numbers, which eluded European mathematics
until the middle of the second millennium. The chapter also contains geometry,
including in particular his famous formula for the area of a quadrilateral
generalisizing Heron’s formula for the area of a triangle; it is however stated
without the condition of cyclicity of the quadrilateral that is needed for its
validity - a point criticised by later mathematicians in the tradition. The
18th chapter is devoted to the kut. t.aka and other methods for solving
second-degree indeterminate equations. The reader is referred to [39] for the
original text, and [25], for instance, for a summary of the contents. The
Br¯ahmsphutasiddh¯anta considerably influenced mathematics in the Arab world.
Bh¯askara II is the author of the famous mathematical texts L¯il¯avat¯i and
B¯ijagan. ita. Apart from being an accomplished mathematician he was a great
teacher and populariser of mathematics. L¯il¯avat¯i, which literally means
“playful”, presents mathematics in a playful way, with several verses directly
addressing a pretty young woman, and examples presented with reference to
various animals, trees, ornaments, etc. (Legend has it that the book is named
after his daughter after her wedding failed to materialise on account of an
accident with the clock, but there is no historical evidence to that effect.).
The book presents apart from various introductory aspects of arithmetic,
geometry of triangles and quadrilaterals, examples of applications of the
Pythagoras theorem, trir¯asika and kut. t.aka 8 methods, problems on permutations
and combinations etc. The B¯ijagan. ita is at an advanced level. It is a
treatise on Algebra, the first known independent work of its kind in Indian
tradition. Operations with unknowns, kut. t.aka and cakrav¯ala methods for
solutions of indeterminate equations are some of the topics discussed, together
with examples. Bh¯askara’s works on astronomy, Siddh¯anta ´siroman. i and
Karan. a kut¯uhala, contain several important results in trigonometry, and also
some ideas of calculus; see [43], a nice recent account of trigonometry from
ancient times in various cultures. The works in the Siddh¯anta tradition have
been edited and there are various commentaries, including many from the earlier
centuries, and works by European authors such as Colebrook [3], and many Indian
authors including Sudh¯akara Dvived¯i, Kuppan. n. a Sastri and K.V. Sarma; see
[20] for an overview of the works. ´ The 2-volume book of Datta and Singh [7]
and the books of Saraswati Amma [33] and A.K. Bag [1] serve as convenient
references for many results known in this tradition. Various details have been
described in the book of Plofker [25]; see also [24]. Nevertheless, more
detailed accounts suitable for a modern context are called for, in the context
of the scope of the subject both from mathematical and historical point of
view. Cataloguing of the extant manuscripts and making them accessible is also
a necessary task in this respect. A census of the Sanskrit manuscripts on exact
sciences was produced by David Pingree during 1970-94; the role and proportion
of mathematics in the body as whole seems to be difficult to ascertain and
calls for serious study; a good beginning may be said to have been made in
[20], where it is noted in particular that only a small proportion of the
listed manuscripts from the census pertain directly to mathematics - the census
encompasses all manuscripts with some connection with jyotis.a. A similar
compendium on the Sanskrit texts in the repositories in Kerala and Tamilnadu
was produced by K.V. Sarma in 2002. A bibliography of Sanskrit works on
astronomy and mathematics was also brought out by S.N. Sen, A.K. Bag and R.S.
Sarma in 1966 [38]. The related theme that needs to be pursued is to understand
the instruments used, such as the clocks, astrolabes and other instruments
involved in astrnomy and the mathematics underlying them. Considerable work has
been done in this respect by S.R. Sarma. Interplay between observational and
mathematical aspects is another theme on which work needs been done, it being
generally believed that mathematical astronomy overshadowed the study of
astronomy in India.
Pat¯igan. ita and the Bakhsh¯al¯i
manuscript
While a majority of the works and
authors dealing with mathematics conceened themselves with astronomy, there
have indeed been works dealing exclusively with arithmetic. The term p¯at¯igan.
ita seems to have come into use for this in Sanskrit works from around the 7 th
century. Some later authors referred to it as vyaktagan. ita (calculation with
the “known”) contrasting it with avyaktagan. ita (calculation with the
“unknown”, which referred to algebra). Sr ´ ¯idhara’s Tri´satik¯a (ca. 750),
Mah¯avira’s Gan. ita-s¯ara-sam. graha (ca. 850), Gan. ita-tilaka (1039) of Sr ´
¯ipati, L¯il¯avat¯i (1150) of Bh¯askara, Gan. ita-kaumudi (1356) of N¯arayan. a
Pan.d. ita and some of the major works of this genre. While individually they
have many special features in terms of a variety of detail, there is also an
underlying thread of unity in presenting arithmetic. There is also another
rather unique manuscript which broadly falls in this general category of
pat¯igan. ita, viz. the Bakhsh¯al¯i manuscript, which has been a crucial but
enigmatic source in the study of ancient Indian mathematics, with many open
issues and controversies around it. The manuscript was found in 1881, buried in
a field in the village Bakhshali, near Peshawar, from which it derives its
name. It was acquired by the Indologist A.F.R. Hoernle who studied and
published a short account on it, and later in 1902 presented the manuscript to
the Bodleian library at Oxford, where it has been since then. The manuscript
consists of 70 folios of bh¯urjapatra (birch bark); of these 51 folios retained
a fair proportion of the original - of the rest one folio is blank while others
are either much damaged or blank. Birch bark (unlike palm leaf which is another
material that was extensively used for manuscripts) is generally known to be a
rather fragile material that tends to deteriorate relatively fast and is
vulnerable to crumbling on being handled, when it is more than two or three
hundred years old; see [45] for a discussion on ancient manuscripts.
Fortunately the manuscript was in a condition suitable enough for the early
studies, but unluckily certain steps taken for preservation of the leaves are
said have apparently made the folios stick together and it is now difficult to
separate them (orally received information - needs confirmation). Facsimile
copies of all the folios were brought out by Kaye in 1927 [19], which has since
then been the source for the subsequent studies. The date of the manuscript has
been a subject of much controversy since the early years. Hoernle dated it to
be from the 3rd or 4th century while Kaye argued it to be from the twelfth
century. Various dates have since then been proposed by many subsequent
authors, ranging from the early centuries of 10 CE to the 10th century. The
mathematical contents of the manuscript may be expected to pre-date the
manuscript itself and there have been many suggestions in that respect as well.
T. Hayashi who produced what is perhaps the most comprehensive account [15] so
far, examining various issues in detail, concludes that the manuscript may be
assigned some time between the 8th and the 12th century, while the work may
most probably be from the 7th century. One way of settling the issue of the
date of the manuscript would of course be to use carbon dating techniques.
Though some attempts were made by Bill Casselman towards getting this done,
they have not borne fruit. One hopes that eventually this will be done, and it
may clarify the historical issues around the manuscript. An approximate formula
for extraction of square-roots of non-square numbers, used systematically in
many problems in the manuscript, dealing with quadratic equations, has
attracted much attention. Some calculations in the manuscript involve
computations with fractions with large numerator and denominator (each
expressed in decimal representation). One of the verifications of a solution of
a quadratic equation involves a fraction whose numerator has 23 digits and the
denominator has 19 digits!
The Kerala school of mathematics
In the 1830s Charles Whish, an
English civil servant in the Madras establishment of the East India Company,
brought to light a collection of manuscripts from a mathematical school that
flourished in the central part of Kerala, between Kozikode and Kochi, starting
from late fourteenth century and continuing at least into the beginning of the
seventeenth century. The school is seen to have originated with M¯adhava, to
whom his successors have attributed many results presented in their texts.
Since the middle of the 20th century Indian scholars have worked on these
manuscripts and the contents of the manuscripts have been studied. Apart from
M¯adhava, N¯ilakantha Somay¯aji was another leading personality from the
School. There are no extant works of M¯adhava on mathematics, (though some work
on astronomy is known). N¯ilakant.ha authored a book called Tantrasam. graha
(in Sanskrit) in 1500 CE. There was a long teacher-student lineage during the
period of over two hundred years, and there have been expositions and
commentaries by many of them, notable among them being Yukti d¯ipika and
Kriy¯akramakar¯i by Sankara and ´ Gan. ita-yukti-bh¯as.¯a by Jyes. t.hadeva
which is in Malayalam. An edited English translation of the latter was produced
by K.V. Sarma and it has recently been published with explanatory notes by K.
Ramasubramanian, M.D. 11 Srinivas and M.S. Sriram [35]. An edited translation
of Tantrasam. graha has been brought out, more recently, by K. Ramasubramanian
and M.S. Sriram [30]. The Kerala works contain mathematics at a considerably
advanced level than the earlier works; see [23] and [29] for a mathematical
overview and also [10], [11], [12] and [17] for discussions on various aspects
of the Kerala works and issues related to them. They include a series expansion
for π and the arc-tangent series, and the series for sine and cosine functions
that were obtained in Europe by Gregory, Leibnitz and Newton, over two hundred
years later. Some numerical values for π that are accurate to 11 decimals are
also a highlight of the work. The work of the Kerala mathematicians anticipated
the calculus as it developed in Europe later, and in particular it involves manipulations
with indefinitely small quantities (in the determination of circumference of
the circle etc.) reminiscent of the infinitesimals in calculus; it has also
been argued by some authors that the work is indeed calculus already. The
overall context raises a question of possible transmission of ideas from Kerala
to Europe, through some intermediaries. No definitive evidence has emerged in
this respect, but there have been discussions on the issue based on
circumstantial evidence; see [17] for instance. References [1] A.K. Bag,
Mathematics in Ancient and Medieval India, Chaukhambha Orientalia, Varanasi and
Delhi, 1979. [2] Bhagyashree Bavare, The Origin of decimal counting: analysis
of number names in the R. gveda, Ganita Bharati (to appear). [3] H.T. Colebrook,
Algebra with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and
Bhascara, John Murray, London, 1817. [4] S.G. Dani, Geometry in Sulvas¯utras,
in ´ Studies in the history of Indian mathematics, pp. 9-37, Cult. Hist. Math.
5, Hindustan Book Agency, New Delhi, 2010. [5] B.B. Datta, The Jaina School of
Mathematics, Bull. Cal. Math. Soc., Vol 21 (1929), No. 2, 115-145. 12 [6] B.B.
Datta, The Science of the Sulba, a Study in Early Hindu Geometry, University of
Calcutta, 1932. [7] B.B. Datta and A.N. Singh, History of Hindu Mathematics, A
Source Book (2 volumes), Motilal Banarasidas, Lahore, 1935 (Part I) and 1938
(Part II), Asia Publishing House, Bombay, 1962 (reprint), Bharat¯iya Kala
Prakashan, Delhi, 2001 (reprint). [8] Joseph Dauben, Book Review of “The
Universal History of Numbers and the Universal History of Computing” by Georges
Ifrah, Notices of the Amer. Math. Soc. 49 (2002), 32-38. [9] Vers une ´edition
critique des Sulbam ´ ¯im¯am. s¯a, commentaries du Baudh¯ayana Sulbas¯utra ´ :
Contribution `a l’histoire des math´ematiques sanskrites, Doctoral dissertion,
Universit´e Libre de Bruxelles, 2001-2002. [10] P.P. Divakaran, The first
textbook of calculus: Yukti-bh¯as.¯a, Journal of Indian Philosophy 35, 2007,
417-443. [11] P.P. Divakaran, Notes on Yuktibh¯as.¯a: recursive methods in
Indian mathematics. Studies in the history of Indian mathematics, 287-351,
Cult. Hist. Math., 5, Hindustan Book Agency, New Delhi, 2010. [12] P.P.
Divakaran, Calculus under the coconut palms: the last hurrah of medieval Indian
mathematics. Current Science 99 (2010), no. 3, 293-299. [13] R.C. Gupta, New
Indian values of π from the M¯anava Sulba s¯utra, Centaurus ´ 31 (1988), no. 2,
114-125. [14] R.G. Gupta, Ancient Jain Mathematics, Jain Humanities Press, Tempe,
Arizona, USA, 2004. [15] Takao Hayashi, The Bakhsh¯al¯i Manuscript, An ancient
Indian mathematical treatise, Groningen Oriental Studies, XI, Egbert Forsten,
Groningen, 1995. [16] George Ifrah, Universal History of Numbers, Translated
from the 1994 French original by David Bellos, E. F. Harding, Sophie Wood and
Ian Monk. John Wiley & Sons, Inc., New York, 2000. 13 [17] George G.
Joseph, Crest of the Peacock, Non-European roots of mathematics, Third edition,
Princeton University Press, Princeton, NJ, 2011. [18] M.C. Joshi (Ed.), Indian
Archaelogy 1988-89 - A review, Archaelogical Survey of India, New Delhi, 1993;
available online. [19] G.R. Kaye, The Bakhshali Manuscript: A Study In Medieval
Mathematics, 3 Parts, Calcutta 1927; reprinted Aditya Prakashan, 2004. [20]
Agathe Keller, On Sanskrit commentaries dealing with mathematics (fifthtwelfth
century), in Looking at it from Asia: the Processes that Shaped the Sources of
History of Science, Boston Studies in the Philosophy of Science, 2010, Volume
265, Part 2, 211-244, available at
http://www.springerlink.com/content/u459618708745v1p/ [21] Raghunath P.
Kulkarni, Char Shulbas¯utra (in Hindi), Maharshi Sandipani Rashtriya Vedavidya
Pratishthana, Ujjain, 2000. [22] Padmavathamma, Gan. ita-s¯ara-sam. graha of Mah¯av¯ira,
Original text with Kannada and English translation, Siddhnthak¯irthi
Grantham¯ala of Sri Hombuja Jain Math, Shimoga District, Karnataka, India,
2000. [23] S. Parameswaran, The Golden Age of Indian Mathematics, Swadeshi
Science Movement, Kerala, 1998. [24] Kim Plofker, Mathematics in India, in: The
Mathematics of Egypt, Mesopotemia, China, India and Islam - A Sourcebook, Ed:
Victor J. Katz, pp. 385-514, Princeton University Press, 2007. [25] Kim
Plofker, Mathematics in India; Princeton University Press, Princeton, NJ, 2008.
[26] Kim Plofker, Links between Sanskrit and Muslim science in Jaina
astronomical works, International Journal of Jaina Studies, Vol. 6, No. 5
(2010), 1-13. [27] Satya Prakash and Ram Swarup Sharma, Baudh¯ayana
Sulbas¯utram, with ´ Sanskrit Commentary by Dvarakanatha Yajvan and English
Translation and Critical Notes by G. Thibaut, The Research Institute of Ancient
Scientific Studies, New Delhi, 1968. 14 [28] Satya Prakash and Ram Swarup
Sharma, Apastamba ¯ Sulvas¯utra, with com- ´ mentaries from Kapardswamin,
Karvinda and Sundararaja and English translation by Satya Prakash, The Research
Institute of Ancient Scientific Studies, New Delhi, 1968. [29] K.
Ramasubramanian and M.D. Srinivas, Development of Calculus in India, in:
Studies in the history of Indian mathematics, pp. 201-286, Cult. Hist. Math. 5,
Hindustan Book Agency, New Delhi, 2010. [30] K. Ramasubramanian and M.S.
Sriram, Tantrasagraha of Nlakan. t.ha Somay¯aj¯i, with a foreword by B. V.
Subbarayappa, Culture and History of Mathematics, 6. Hindustan Book Agency, New
Delhi, 2011. [31] M. Rangacharya, The Ganita-Sara-Sangraha of Mahaviracarya,
Cosmo Publications, 2011. [32] R. Salomon, Indian Epigraphy, A Guide to the
Study of Inscriptions in Sanskrit, Prakrit, and the Other Indo-Aryan Languages,
Oxford University Press, 1999. [33] Saraswati Amma, Geometry in Ancient and
Medieval India, Motilal Banarasidas, Delhi, 1979. [34] SaKHYa, Gan.
itas¯arakaumud¯i; the Moonlight of the Essence of Mathematics, by T. hakkura
Pher¯u; edited with Introduction, Translation, and Mathematical Commentary,
Manohar Publishers, New Delhi 2009. [35] K.V. Sarma, Gan. ita-yukti-bh¯as.¯a
(Rationales in mathematical astronomy) of Jyes. t.hadeva (Vol. I. Mathematics
and Vol. II. Astronomy), with explanatory notes by K. Ramasubramanian, M. D.
Srinivas and M. S. Sriram, Sources and Studies in the History of Mathematics
and Physical Sciences. Springer, New York, Hindustan Book Agency, New Delhi,
2008. [36] A. Seidenberg, The ritual origin of geometry, Archive for History of
Exact Sciences (Springer, Berlin) Vol. 1, No. 5, January 1975. (available at:
http://www.springerlink.com/content/r6304ku830258l85/) [37] S.N. Sen and A.K.
Bag, The Sulbas¯utras ´ , Indian National Science Academy, New Delhi 1983. 15
[38] S.N. Sen, A.K. Bag and S.R. Sarma, A bibliography of Sanskrit works on
astronomy and mathematics, National Institute of Sciences of India, 1966. [39]
Ram Swarup Sharma (Ed.), Br¯ahmasphut.asiddh¯antah. , Indian Institute of
Astronomical and Sanskrit Research, New Delhi, 1966. [40] Kripa Shankar Shukla
and K.V. Sarma (Ed.), Aryabhat ¯ ¯iya of Aryabhata, ¯ Critically Edited with
Translation, Indian National Science Academy, New Delhi, 1976. [41] Kripa
Shankar Shukla and K.V. Sarma (Ed.),Aryabhat ¯ ¯iya of Aryabhata, ¯ with the
commentary of Bhaskara 1 and Some´svara, Indian National Science Academy, New
Delhi, 1976. [42] G. Thibaut, The Sulvas¯utras ´ , The Journal, Asiatic Society
of Bengal, Part I, 1875, Printed by C.B. Lewis, Baptist Mission Press, Calcutta,
1875; Edited with an introduction by Debiprasad Chattopadhyaya and published by
K P Bagchi and Company, Calcutta, 1984; the original also published as a book,
Cosmo Publications, 2012. [43] Glen van Brummelen, The mathematics of the
heavens and the earth - The early history of trigonometry, Princeton University
Press, Princeton, NJ, 2009. [44] J.M. van Gelder, M¯anava Srautasutra,
belonging to the Maitr¯ayan¯i Samhita, International Academy of Indian Culture,
New Delhi, 1963. [45] Dominik Wujastyk, Indian manuscripts, in Manuscript
Cultures: Mapping the Field, De Gruyter, Berlin (to appear). S.G. Dani School
of Mathematics Tata Institute of Fundamental Research Homi Bhabha Road Mumbai
400005
No comments:
Post a Comment