David Pingree
The logic of non-Western science:
mathematical discoveries in medieval India
One of the most signi½cant things one
learns from the study of the exact sci-
ences as practiced in a number of an-
cient and medieval societies is that,
while science has always traveled from
one culture to another, each culture be-
fore the modern period approached the
sciences it received in its own unique
way and transformed them into forms
compatible with its own modes of
thought. Science is a product of culture;
it is not a single, uni½ed entity. There-
fore, a historian of premodern scienti½c
texts–whether they be written in Akka-
dian, Arabic, Chinese, Egyptian, Greek,
Hebrew, Latin, Persian, Sanskrit, or any
other linguistic bearer of a distinct cul-
ture–must avoid the temptation to con-
David Pingree, a Fellow of the American Acade-
my since 1971, is University Professor in the de-
partment of the history of mathematics at Brown
University. He teaches about the transmission of
science between cultures, and his publications in-
clude many editions of astronomical, astrological,
and magical works in Akkadian, Arabic, Greek,
Latin, and Sanskrit. Most recently he has written
“Arabic Astronomy in Sanskrit” (with T. Kusuba,
2002), “Astral Sciences in Mesopotamia” (with
H. Hunger, 1999), and “Babylonian Planetary
Omens” (with E. Reiner, 1998).
© 2003 by the American Academy of Arts
& Sciences
ceive of these sciences as more or less
clumsy attempts to express modern sci-
enti½c ideas. They must be understood
and appreciated as what their practition-
ers believed them to be. The historian is
interested in the truthfulness of his own
understanding of the various sciences,
not in the truth or falsehood of the sci-
ence itself.
In order to illustrate the individuality
of the sciences as practiced in the older
non-Western societies, and their differ-
ences from early modern Western sci-
ence (for contemporary science is, in
general, interested in explaining quite
different phenomena than those that
attracted the attention of earlier scien-
tists), I propose to describe briefly some
of the characteristics of the medieval
Indian s´a¯stra of jyoti.sa. This discipline
concerned matters included in such
Western areas of inquiry as astronomy,
mathematics, divination, and astrology.
In fact, the jyoti.s¯•s, the Indian experts in
jyoti.sa, produced more literature in these
areas–and made more mathematical
discoveries–than scholars in any other
culture prior to the advent of printing. In
order to explain how they managed to
make such discoveries–and why their
discoveries remain largely unknown–I
will also need to describe briefly the gen-
eral social and economic position of the
jyoti.s¯•s.
Dædalus Fall 2003
45
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David
Pingree
on
science
‘´
S¯astra’ (‘teaching’) is the word in San-
skrit closest in meaning to the Greek
‘ ,επιστ ´ηµη’ and the Latin ‘scientia.’ The
teachings are often attributed to gods
or considered to have been composed by
divine .r.sis; but since there were many of
both kinds of superhuman beings, there
were many competing varieties of each
s´a¯stra. Sometimes, however, a school
within a s´a¯stra was founded by a human;
scientists were free to modify their
s´a¯stras as they saw ½t. No one was con-
strained to follow a system taught by a
god.
Jyoti.h is a Sanskrit word meaning
‘light,’ and then ‘star’; so that jyoti.hs´a¯stra
means ‘teaching about the stars.’ This
s´a¯stra was conventionally divided into
three subteachings: ga.nita(mathematical
astronomy and mathematics itself ), sa .m-
hit¯a (divination, including by means of
celestial omens), and hor¯a (astrology). A
number of jyoti.s¯•s (students of the stars)
followed all three branches, a larger
number just two (usually sa .mhit¯a and
hor¯a), and the largest number just one
(hor¯a).
The principal writings in jyoti.hs´a¯stra,
as in all Indian s´a¯stras, were normally in
verse, though the numerous commen-
taries on them were almost always in
prose. The verse form with its metrical
demands, while it aided memorization,
led to greater obscurity of expression
than prose composition would have en-
tailed. The demands of the poetic meter
meant that there could be no stable tech-
nical vocabulary; many words with dif-
ferent metrical patterns had to be de-
vised to express the same mathematical
procedure or geometrical concept, and
mathematical formulae had frequently
to be left partially incomplete. More-
over, numbers had to be expressible in
metrical forms (the two major systems
.
used for numbers, the bh¯utasa
nkhya¯ and
the ka.tapaya¯di, will be explained and ex-
empli½ed below), and the consequent
ambiguity of these expressions encour-
aged the natural inclination of Sanskrit
pa.n.dits to test playfully their readers’
acumen. It takes some practice to
achieve sureness in discerning the
technical meanings of such texts.
But in this opaque style the jyoti.s¯•s pro-
duced an abundant literature. It is esti-
mated that about three million manu-
scripts on these subjects in Sanskrit
and in other Indian languages still exist.
Regrettably, only a relatively small num-
ber of these has been subjected to mod-
ern analysis, and virtually the whole en-
semble is rapidly decaying. And because
there is only a small number of scholars
trained to read and understand these
texts, most of them will have disap-
peared before anyone will be able to
describe correctly their contents.
In order to make my argument clearer,
I will restrict my remarks to the ½rst
branch of jyoti.hs´a¯stra–ga.nita. Geometry,
and its branch trigonometry, was the
mathematics Indian astronomers used
most frequently. In fact, the Indian as-
tronomers in the third or fourth century,
using a pre-Ptolemaic Greek table of
chords,1 produced tables of sines and
versines, from which it was trivial to de-
rive cosines. This new system of trigo-
nometry, produced in India, was trans-
mitted to the Arabs in the late eighth
century and by them, in an expanded
form, to the Latin West and the Byzan-
tine East in the twelfth century. But, de-
spite this sort of practical innovation,
the Indians practiced geometry without
the type of proofs taught by Euclid, in
1 For a description of the table of chords, cy-
clic quadrilaterals, two-point iteration, ½xed-
point iteration, and several other mathematical
terms mentioned in this essay, please see Vic-
tor J. Katz, A History of Mathematics: An Intro-
duction (New York: HarperCollins, 1993).
46
Dædalus Fall 2003
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which all solutions to geometrical prob-
lems are derived from a small body of
arbitrary axioms. The Indians provided
demonstrations that showed that their
solutions were consistent with certain
assumptions (such as the equivalence
of the angles in a pair of similar triangles
or the Pythagorean theorem) and whose
validity they based on the measurement
of several examples. In their less rigor-
ous approach they were quite willing to
be satis½ed with approximations, such
as the substitution of a sine wave for al-
most any curve connecting two points.
Some of their approximations, like those
devised by ¯Aryabha.ta in about 500 for
the volumes of a sphere and a pyramid,
were simply wrong. But many were sur-
prisingly useful.
Not having a set of axioms from which
to derive abstract geometrical relation-
ships, the Indians in general restricted
their geometry to the solution of practi-
cal problems. However, Brahmagupta
in 628 presented formulae for solving a
dozen problems involving cyclic quadri-
laterals that were not solved in the West
before the Renaissance. He provides no
rationales and does not even bother to
inform his readers that these solutions
only work if the quadrilaterals are cir-
cumscribed by a circle (his commenta-
tor, P.rth¯udakasv¯amin, writing in about
864, follows him on both counts). In this
case, and clearly in many others, there
was no written or oral tradition that pre-
served the author’s reasoning for later
generations of students. Such disdain
for revealing the methodology by which
mathematics could advance made it dif-
½cult for all but the most talented stu-
dents to create new mathematics. It is
amazing to see, given this situation,
how many Indian mathematicians did
advance their ½eld.
I will at this point mention as exam-
ples only the solution of indeterminate
equations of the ½rst degree, described
already by ¯Aryabha.ta; the partial solu-
tion of indeterminate equations of the
second degree, due to Brahmagupta;
and the cyclic solution of the latter type
of indeterminate equations, achieved by
Jayadeva and described by Udayadiv¯a-
kara in 1073 (the cyclic solution was
rediscovered in the West by Bell and Fer-
mat in the seventeenth century). Inter-
polation into tables using second-order
differences was introduced by Brah-
magupta in his Kha.n.dakh¯adyaka of 665.
The use of two-point iteration occurs
½rst in the Pañcasiddh¯antik¯a composed by
Var¯ahamihira in the middle of the sixth
century, and ½xed-point iteration in
the commentary on the Mah¯abh¯askar¯•ya
written by Govindasv¯amin in the middle
of the ninth century. The study of com-
binatorics, including the so-called Pas-
cal’s triangle, began in India near the
beginning of the current era in theChan-
da.hs¯utras, a work on prosody composed
.
by Pi
ngala, and culminated in chapter
13 of the Ga.nitakaumud¯• completed by
N¯ar¯aya .na Pa .n .dita in 1350. The four-
teenth and ½nal chapter of N¯ar¯aya .na’s
work is an exhaustive mathematical
treatment of magic squares, whose study
in India can be traced back to the B.rhat-
sa .mhit¯a of Var¯ahamihira.
In short, it is clear that Indian mathe-
maticians were not at all hindered in
solving signi½cant problems of many
sorts by what might appear to a non-
Indian to be formidable obstacles in the
conception and expression of mathe-
matical ideas.
Nor were they hindered by the restric-
tions of ‘caste,’ by the lack of societal
support, or by the general absence of
monetary rewards. It is true that the
overwhelming majority of the Indian
mathematicians whose works we know
were Br¯ahma .nas, but there are excep-
tions (e.g., among Jainas, non-Brah-
Mathematical
discoveries
in medieval
India
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Dædalus Fall 2003
47
David
Pingree
on
science
m¯anical scribes, and craftsmen). Indian
society was far from open, but it was not
absolutely rigid; and talented mathe-
maticians, whatever their origins, were
not ignored by their colleagues. Howev-
er, astrologers (who frequently were not
Br¯ahma .nas) and the makers of calendars
were the only jyoti.s¯•s normally valued by
the societies in which they lived. The at-
traction of the former group is easily
understood, and their enormous popu-
larity continues today. The calendar-
makers were important because their job
was to indicate the times at which rituals
could or must be performed. The Indian
calendar is itself intricate; for instance,
the day begins at local sunrise and is
numbered after the tithi that is then cur-
rent, with the tithis being bounded by
the moments, beginning from the last
previous true conjunction of the Sun
and the Moon, at which the elongation
between the two luminaries had in-
creased by twelve degrees. Essentially,
each village needed its own calendar to
determine the times for performing pub-
lic and private religious rites of all kinds
in its locality.
By contrast, those who worked in the
various forms of ga.nita usually enjoyed
no public patronage–even though they
provided the mathematics used by archi-
tects, musicians, poets, surveyors, and
merchants, as well as the astronomical
theories and tables employed by astro-
logers and calendar-makers. Sometimes
a lucky mathematical astronomer was
supported by a Mah¯ar¯aja whom he
served as a royal astrologer and in whose
name his work would have been pub-
lished. For example, the popular R¯ajam.r-
.
g¯a
nka is attributed, along with dozens of
other works in many s´a¯stras, to Bhojade-
va, the Mah¯ar¯aja of Dh¯ar¯a in the ½rst
half of the eleventh century. Other jyoti-
.s¯•s substituted the names of divinities or
ancient holy men for their own as au-
thors of their treatises. Authorship often
brought no rewards; one’s ideas were
often more widely accepted if they were
presented as those of a divine being, a
category that in many men’s minds in-
cluded kings.
One way in which a jyoti.s¯• could make
a living was by teaching mathematics,
astronomy, or astrology to others. Most
frequently this instruction took place
in the family home, and, because of the
caste system, the male members of a
jyoti.s¯•’s family were all expected to fol-
low the same profession. A senior jyoti.s¯•,
therefore, would train his sons and often
his nephews in their ancestral craft. For
this the family maintained a library of
appropriate texts that included the com-
positions of family members, which
were copied as desired by the younger
members. In this way a text might be
preserved within a family over many
generations without ever being seen
by persons outside the family. In some
cases, however, an expert became well
enough known that aspirants came from
far and wide to his house to study. In
such cases these students would carry
off copies of the manuscripts in the
teacher’s collection to other family
libraries in other locales.
The teaching of jyoti.hs´a¯stra also oc-
curred in some Hindu, Jaina, and Bud-
dhist monasteries, as well as in local
schools. In these situations certain stan-
dard texts were normally taught, and the
status of these texts can be established
by the number of copies that still exist,
by their geographical distribution, and
by the number of commentaries that
were written on them.
Thus, in ga.nita the principal texts used
in teaching mathematics in schools were
clearly the L¯•lavat¯• on arithmetic and the
B¯•jaga.nita on algebra, both written by
Bh¯askara in around 1150, and, among
.
ngraha composed
Jainas, the Ga.nitas¯arasa
in about 850 by their coreligionist, Ma-
h¯av¯•ra. In astronomy there came to be
½ve pak.sas (schools): the Br¯ahmapak.sa,
whose principal text was the Siddh¯anta-
48
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s´iroma.ni of the Bh¯askara mentioned
above; the ¯Aryapak.sa, based on the ¯Arya-
bha.t¯•ya written by ¯Aryabha.ta in about
500; the ¯Ardhar¯atrikapak.sa, whose princi-
pal text was the Kha.n.dakh¯adyaka com-
pleted by Brahmagupta in 665; the
Saurapak.sa, based on the S¯uryasiddh¯anta
composed by an unknown author in
about 800; and the Ga.nes´apak.sa, whose
principal text was the Grahal¯aghava au-
thored by Ga .ne ´sa in 1520. Each region
of India favored one of these pak.sas,
though the principal texts of all of them
enjoyed national circulation. The com-
mentaries on these often contain the
most innovative advances in mathemat-
ics and mathematical astronomy found
in Sanskrit literature. By far the most
popular authority, however, was Bh¯as-
kara; a special college for the study of his
numerous works was established in 1222
by the grandson of his younger brother.
No other Indian jyoti.s¯• was ever so hon-
ored.
Occasionally, indeed, an informal
school inspired by one man’s work
would spring up. The most noteworthy,
composed of followers of M¯adhava of
.
ngamagr¯ama in Kerala in the extreme
Sa
south of India, lasted for over four hun-
dred years without any formal structure
–simply a long succession of enthusiasts
who enjoyed and sometimes expanded
on the marvelous discoveries of M¯adha-
va.
M¯adhava (c. 1360–1420), an Empr¯an-
tiri Br¯ahma .na, apparently lived all his
life on his family’s estate, Ilaññipa.l.li, in
.
Sa
ngamagr¯ama (Irinj¯alakhu .da) near
Cochin. His most momentous achieve-
ment was the creation of methods to
compute accurate values for trigonomet-
ric functions by generating in½nite se-
ries. In order to demonstrate the charac-
ter of his solutions and expressions of
them, I will translate a few of his verses
and quote some Sanskrit.
He began by considering an octant of
a circle inscribed in a square, and, after
some calculation, gave the rule (I trans-
late quite literally two verses):
Mathematical
discoveries
in medieval
India
Multiply the diameter (of the circle) by 4
and divide by 1. Then apply to this sepa-
rately with negative and positive signs
alternately the product of the diameter
and 4 divided by the odd numbers 3, 5,
and so on . . . . The result is the accurate
circumference; it is extremely accurate
if the division is carried out many times.
This describes the in½nite series:
C =
4D
1
−
4D
3
+
4D
5
−
4D
7
+
4D
9
. . . .
That in turn is equivalent to the in½nite
series for πthat we attribute to Leibniz:
π
4
= 1 − +
1
3
1
5
1
− +
7
1
9
. . . .
.
nkhy¯a system, in which numbers
M¯adhava expressed the results of this
formula in a verse employing the
bh¯utasa
are represented by words denoting
objects that conventionally occur in the
world in ½xed quantities:
vibudhanetragaj¯ahihut¯a´sanatrigu .naved-
abhav¯ara .nab¯ahava .h |
navanikharvamite v.rtivistare
paridhim¯anam ida .m jagadur budh¯a .h ||
A literal translation is:
Gods [33], eyes [2], elephants [8], snakes
[8], ½res [3], three [3], qualities [3], Vedas
[4], nak.satras [27], elephants [8], and arms
[2]–the wise say that this is the measure
of the circumference when the diameter
of a circle is nine hundred billion.
.
nkhy¯a numbers are taken in
The bh¯utasa
reverse order, so that the formula is:
π=
2827433388233
900000000000
(= 3.14159265359, which is correct to the
eleventh decimal place).
Dædalus Fall 2003
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David
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on
science
Another extraordinary verse written by
M¯adhava employs the ka.tapay¯adi system
in which the numbers 1, 2, 3, 4, 5, 6, 7, 8,
9, and 0 are represented by the consonants
that are immediately followed by a vowel;
this allows the mathematician to create a
verse with both a transparent meaning
due to the words and an unrelated nu-
merical meaning due to the consonants
in those words. M¯adhava’s verse is:
vidv¯a .ms tunnabala .h kav¯• ´sanicaya .h sar-
v¯artha ´s¯•lasthiro
.
nganarendraru
nirviddh¯a
.
n
The verbal meaning is: “The ruler whose
army has been struck down gathers to-
gether the best of advisors and remains
½rm in his conduct in all matters; then
he shatters the (rival) king whose army
has not been destroyed.”
The numerical meaning is ½ve sexa-
gesimal numbers:
0;0,44
0;33,6
16;5,41
273;57,47
2220;39,40.
These ½ve numbers equal, with R =
3437;44,48 (where R is the radius) :
540011
R1011!
54009
R89!
54007
R67!
54005
R45!
54003
R23!
These numbers are to be employed in
the formula:
sinθ= θ−(
θ
5400
)3[
54003
R23!
−(
θ
5400
)2[
54005
R45!
50
Dædalus Fall 2003
54007
R67!
θ
5400
)2[
54009
R89!
−
θ
5400
− (
(
θ
5400
)2[
)2[
540011
R1011!
−(
]]]]
and this formula is a simple transforma-
tion of the ½rst six terms in the in½nite
power series for sinθ found indepen-
dently by Newton in 1660:
sinθ=
θ3
θ−
R23!
+
θ5
R45!
−
θ7
R67!
+
θ9
R89!
−
θ11
R1011!
Not surprisingly, M¯adhava also discov-
ered the in½nite power series for the
cosine and the tangent that we usually
attribute to Gregory.
The European mathematicians of the
seventeenth century derived their
trigonometrical series from the applica-
tion of the calculus; M¯adhava in about
1400 relied on a clever combination of
geometry, algebra, and a feeling for
mathematical possibilities. I cannot here
go through his whole argument, which
has fortunately been preserved by several
of his successors; but I should mention
some of his techniques.
He invented an algebraic expansion
formula that keeps pushing an unknown
quantity to successive terms that are
alternately positive and negative; the
series must be expanded to in½nity to get
rid of this unknown quantity. Also,
because of the multiplications, as the
terms increase, the powers of the indi-
vidual factors also increase. One of these
factors in the octant is one of a series of
integers beginning with 1 and ending
with 3438–the number of parts in the
radius of the circle that is also the tan-
gent of 45°, the angle of the octant; this
means that there are 3438 in½nite series
that must be summed to yield the ½nal
in½nite series of the trigonometrical
function.
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3438
2
34382
2
It had long been known in India that
the sum of a series of integers beginning
with 1 and ending with n is:
+ 1), that is,
n(
n
2
here equals 3438, M¯adhava decided that
− + n. Since n
(n −1)
2
n2
2
n– , which equals
, is negligible
n
2
. Therefore, an
with respect to
approximation to the sum of the
. Similarly, the
series of n integers is
sums of the squares of a series of n inte-
gers beginning with 1 was known to be
2[2(n + 1)2 − (n + 1)]
n−
3
. If n is large, this
n2
2
n(n + 1)2
3
since
is approximately equal to
− (n + 1)
6
3438 × 34392
3
is negligible. But, with n = 3438,
34383
.
3
Therefore, as an approximation, the sum
of the series of the squares of 3438 inte-
is little different from
n3
3
n
2
. Finally, it
n2(n + 1)2
4
gers beginning with 1 is
was known that the sum of the cubes of
a series of n numbers beginning with 1
is: ( )2(n + 1)2 or
. If n is 3438,
there is little difference between
34382 × 34392
and
. Therefore, the
4
n4
expression
is a close approximation
4
to the sum of the cubes of a series of n
numbers beginning with 1. From these
three examples M¯adhava guessed at the
general rule that the sum of n numbers
in an arithmetical series beginning with
1 all raised to the same power, p, is ap-
34384
4
proximately equal to
n p +1
p+ 1
.
It had also been realized in India since
21,600
2π
the ½fth century–from examining the
sine table in which the radius of the cir-
(which was approximat-
cle, R, is
ed by 3438) and in which there are 24
sines in a quadrant of 90º, so that the
length of each arc whose sine is tabulat-
ed is 225′–that the sine of any tabulated
θ3
R23!
and that the versine of
θ2
2R
angle θis equal to θminus the sum of
the sums of the second differences of the
sines of the preceding tabulated angles.
M¯adhava discovered, by some very
clever geometry, that the sum of the
sums of the second differences approxi-
mately equals
θis approximately equal to
. Since
sin2θ= R2 − cos2θand versθ= R − cosθ,
M¯adhava could correct the approxima-
tion to the versine by the approximation
to the sum of the sums of the second dif-
ferences of tabulated sines; then he
could correct the approximation to the
sum of the sums of the second differ-
ences by the corrected approximation
to the versine; and he could continue
building up the two parallel series by
applying alternating corrections to
them. He ½nally arrives at two in½nite
power series, equivalent, if R = 1, to:
θ5
5!
sinθ= θ −
θ7
7!
θ3
3!
θ9
9!
. . . ,
−
+
+
and
θ8
8!
θ2
2!
θ4
4!
θ6
6!
+
+
−
. . . .
cosθ = 1 −
Subsequent members of the ‘school’ of
M¯adhava did remarkable work as well,
in both geometry (including trigonome-
try) and astronomy. This is not the occa-
sion to recite their accomplishments,
but I should remark here that, among
these members, Indian astronomers
attempted especially to use observations
to correct astronomical models and their
parameters.
Mathematical
discoveries
in medieval
India
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This began with M¯adhava’s principal
.
ngamagr¯ama. He observed eighteen
pupil, a Namp¯utiri Br¯ahma .na named
Parame ´svara, whose family’s illam was
Va.ta ´sre .ni in A ´svatthagr¯ama, a village
about thirty-½ve miles northeast of
Sa
lunar and solar eclipses between 1393
and 1432 in an attempt to correct tradi-
tional Indian eclipse theory. One pupil
of Parame ´svara’s son, D¯amodara, was
Dædalus Fall 2003
51
David
Pingree
on
science
N¯•laka .n.tha–another Namp¯utiri Br¯ah-
ma .na who was born in 1444 in the Kelal-
l¯ur illam located at Ku .n .dapura, which is
about ½fty miles northwest of A ´svattha-
gr¯ama.
N¯•laka .n.tha made a number of obser-
vations of planetary and lunar positions
and of eclipses between 1467 and 1517.
N¯•laka .n.tha presented several different
sets of planetary parameters and sig-
ni½cantly different planetary models,
which, however, remained geocentric.
He never indicates how he arrived at
these new parameters and models, but
he appears to have based them at least in
large part on his own observations. For
he proclaims in his Jyotirm¯•m¯a .ms¯a–con-
trary to the frequent assertion made by
Indian astronomers that the fundamen-
tal siddh¯antas expressing the eternal rules
of jyoti.hs´a¯stra are those alleged to have
been composed by deities such as S¯urya
–that astronomers must continually
make observations so that the computed
phenomena may agree as closely as pos-
sible with contemporary observations.
N¯•laka .n.tha says that this may be a con-
tinuous necessity because models and
parameters are not ½xed, because longer
periods of observation lead to more ac-
curate models and parameters, and be-
cause improved techniques of observing
and interpreting results may lead to su-
perior solutions. This af½rmation is al-
most unique in the history of Indian jyo-
ti.sa; jyoti.s¯•s generally seem to have mere-
ly corrected the parameters of one pak.sa
to make them closely corresponded to
those of another.
The discoveries of the successive gen-
erations of M¯adhava’s ‘school’ contin-
ued to be studied in Kerala within a
small geographical area centered on Sa
gamagr¯ama. The manuscripts of the
school’s Sanskrit and Malay¯alam treatis-
es, all copied in the Malay¯alam script,
never traveled to another region of In-
.
n-
dia; the furthest they got was Ka.tattan¯at
in northern Kerala, about one hundred
.
miles north of Sa
ngamagr¯ama, where
.
the R¯ajakum¯ara ´Sa
nkara Varman repeat-
ed M¯adhava’s trigonometrical series in a
work entitled Sadratnam¯al¯a in 1823. This
was soon picked up by a British civil ser-
vant, Charles M. Whish, who published
an article entitled “On the Hind ´u Quad-
rature of the Circle and the In½nite Se-
ries of the Proportion of the Circumfer-
ence to the Diameter in the Four Sástras,
the Tantra Sangraham, Yocti Bháshá,
Carana Paddhati and Sadratnamála” in
Transactions of the Royal Asiatic Society in
1830.2 While Whish was convinced that
the Indians (he did not know of M¯adha-
va) had discovered calculus–a conclu-
sion that is not true even though they
successfully found the in½nite series for
trigonometrical functions whose deriva-
tion was closely linked with the discov-
ery of calculus in Europe in the seven-
teenth century–other Europeans
scoffed at the notion that the Indians
could have achieved such a startling suc-
cess. The proper assessment of M¯adha-
va’s work began only with K. Mukunda
Marar and C. T. Rajagopal’s “On the
Hindu Quadrature of the Circle,” pub-
lished in the Journal of the Bombay Branch
of the Royal Asiatic Society in 1944.
So while the discoveries of Newton,
Leibniz, and Gregory revolutionized
European mathematics immediately
upon their publication, those of M¯adha-
va, Parame ´svara, and N¯•laka .n.tha, made
between the late fourteenth and early
sixteenth centuries, became known to a
handful of scholars outside of Kerala in
.
ngraha was written by
2 Note that the Tantrasa
the N¯•laka .n.tha whom we have already men-
tioned, the Yuktibh¯a.sa by his colleague and fel-
low pupil of D¯amodara, Jye.s.thadeva, and the
Kara.napaddhati by a resident of the Putumana
illam in ´Sivapura in 1723.
52
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India, Europe, America, and Japan only
in the latter half of the twentieth centu-
ry. This was not due to the inability of
Indian jyoti.s¯•s to understand the mathe-
matics, but to the social, economic, and
intellectual milieux in which they
worked. The isolation of brilliant minds
was not uncommon in premodern India.
The exploration of the millions of sur-
viving Sanskrit and vernacular manu-
scripts copied in a dozen different
scripts would probably reveal a number
of other M¯adhavas whose work deserves
the attention of historians and philoso-
phers of science. Unfortunately, few
scholars have been trained to undertake
the task, and the majority of the manu-
scripts will have crumbled in just anoth-
er century or two, before those few can
rescue them from oblivion.
Mathematical
discoveries
in medieval
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