INTRODUCTION
Indian
at
Mathematics
least Indus
in
Indian
500
Mathematics
B.C.)
broad
sense
Valley Civilization which
which
are
are
the
Sulba
compilations
is
old
texts
Sutras
mathematical
as
in the
The earliest
(or Sulva)
of
as
is also true
(V~stuvidya).
case of Indian Architecture
on
its
(800-
principles
that have developed in India during ancient times.
Several
or
mentioned
platforms,
geometrical
implied
sacrificial
principles
in
used
for
measuring
geometry
was
length
called
.
Indian Mathematics
and
Rajju
(gaoita)
other
and
measuring rod and thread or rope
( 11)
.
in
explicitly
altars,
of
with
things
(raj ju) .
breadth,
s8stra
(17)
either
construction
the
halls
are
in
the
a
Since rope was
course
of
early
The dev elopmen t
time,
period
of
of geomet ry
_ay just as well have been stimulated by the practical needs
of construction and
surveying
between
mathematics
geometry
or
and
this
and
explains
the
architecture
relation
in
ancient
century A. D.
to l7 th
India.
The period extend ing from the 5
century
was
development
among
he
but
had
the
of
general
~ryabhata
in India.
top
not
also
one
ranking
only
anticipated
prosperity
I
many
discoveries
for
scientific
(b. A.D.476) was the first
mathematicians
introduced
th
new
of
of
ancient
theories
modern
and
times.
India,
as
formulae
He
has
2
foundation
the
laid
also
Var~hamihira
mathematics.
Sridhara
(A.D.7S0),
A. D.1114)
are
notable
some
(A.D.SOS),
Mahavlra,
of
contributions
the
to
of
Bhaskara
and
th is
of
(A.D.628),
Brahmagupta
Sripati
scholars
the
branches
different
of
per iod
twin disciplines
of
(b.
who
made
Mathematics
and Astronomy.
It
and
was
the
an
evolutionary
historical
traceable
mainly
period
evolution
from
of
manuscripts
of
are
of
about
every
theory
and
treatises,
300
an
houses,
"
authoritative
in
two
some
of
the
(Vastuviaya).
Mayamata
Manas3.ra
India,
regional
~carya
P.K.
in
India
of
there
different
century
building
temples
A.D.
and
Is~nagurudevapaddhati,
chapters.
Mayamata,
on
and
Indian
M3.nasdra
architecture
All these works are comprehensive and masterly
in
over
on
compilations
compilations
and
in
is
from the surviving
to
Brhatsamhita
treatment
separate
and
architecture
Samaranga~asutradhara,
'Klmik8gama,
are
on
Varahamihira's
languages.
gives
texts
published
According
epoch.
architecture
architectural
from critical essays and commentaries,
buildings
Indian
highly
some
texts
technical
were
treated
variations
because
of
Sanskrit.
in
the
as
Even
reference
details
were
geographical
though,
texts
adopted
and
all
in
climatic
differences of different regions.
In
Kerala
the
A.D. to 12 th century)
Mah~dayapuram(2)
marked an all
period
pervasive
(9 th
century
transformation
3
in the pol it i cal,
known structural
of the 9
th
early
a
and
Stella
Kerala
attained
style
rose
t hough
during
century
them
calls
the
first quarter
the
that
architecture
perfection
Kramrisch(38)
in
principles
of
periods
fields and the earliest
Even
to v:"!!!"stu
unique
modern
in
(A. D. 823) .
according
India,
Kerala
temples
centu ry
constructed
in
social and cuI tu ral
the
temples were
was
was
evolved
late
A.D.
palaces
and
mathematical,
of
investigations
large
this
shapes
As
were
concentrate
our
an
effusion
architectural
result
a
with
in temples,
been
and
period.
treatises
of
We
disciplines.
had
astronomical
during
number
There
churches.
and
century) .
to
their high sloping roofs and angular silhouet tes,
mosques,
in
medieval
Kerala
purely
prevalent
of
formulated
investigations
this,
a
in
these
on
Indian
mathematics related to architecture to this period only,
with
special reference to Kerala.
V@ovaroha
(GOlavid)
and
other
(1340-1425),
DrggaJ;li ta
Tantrasaugraha of Nilakar;t tha
or
are
GaoitanyayasaQgraha
some
astronomers
(Rationale
Malayalam
mathematics
traditional
of
of
in
the
and
Parame~wara
of
S~may~j
i
during
deals
astronomy
architecture
this
of
and
some of the mathematical results(66)
and
period.
which
even
Kerala
etc,
mathematicians
several
with
Yukt ibhaliSa
(appr.1500-1610)
of
language)
Madhava
(1360-1455),
(1443-1545),
contributions
Malayalam
Sa~gamagrama
of
Jy~~tadeva
of
Kerala
language
works
for
ci tes
the
is
composed
branches
examples
in
under
from
verification
of
4
Further,
formulated
N!r~yanan
(16 th
several
during
this
treatises
centu ry) ,
Neelakant;.han
the
architectural
compilations of vastu
The
first
other
explains
Kerala.
Tantrasamuccaya
versions
century)
texts
we
with
245
verses
which
was
published
Cenn~s
of trikumara
etc,
are
Kerala.
details
of
domestic
consider
mainly
arranged
by
in
the
and
These
seven
in 1125 M.E(56)contains
There
classification.
This
of
may
be
the
of
texts
are
many
contain
But
the
one
Pari~kara~a
verses
173
and
them
chapters.
only
are
verses.
two
Ma1ayalabh~9a
Kochi
of
archi tecture
the
most
of
some
architecture
Manu~yalayacandrika.
and
about
chapterwise
of
temple
Manu~yalayacandrika
of
Committee
deals
the
Here
th
of
principles composed in Sanskr i t
two. texts
two
(16
were
Manu ~yalayacandr i ka
(Anm. ) ,
Thirumangalath
we1lknown
Si lpara tna
(A. D .1428),
Vastuv idya
architecture
Tantrasamuccaya
period.
Namboothir ippad
on
without
considered
any
as
the
"\
earliest of all the commentaries on
One
of
architecture
In v3stu
a
the
numeral
Indian
the
method
of
representation
texts,
the
system
of
representation
of
Sanskrit
(BhutasaQkhya
Sanskrit names for
catur,
of
is
combination
numerals
contributions
Manu~yalayacandrika.
pafica,
~at,
system,
system)
the first
sapta,
numbers
names
numbers
with
place
of
of
numbers
and
the
by
means
of
is
word
The
value.
dvi,
and nava respectively.
expressed
to
numbers.
nine numbers are eka,
a~ta
are
for
mathematics
tri,
In word
words
as
in the place value notation which was developed and perfected
5
in
India
in
the
early
centuries
of Christ ian
era.
In
system the numerals are expressed by names of things,
or
concepts
therefore
which
the
system.
are
system
very
is
also
Thus the words,
numbers
zero,
system
is
The words
nine
'bh~ga'
and
in
the
defining
bhGta
The
The
meaning
padOnam
and
saQkhya
ka tapayad i
terms
part
are
used
or
used
proportionate
the
numbers
The fractions
1/4, 1/2 and 3/4 respectively.
employed
the
people
representing
Kerala.
lamsa l
arddha~,
padam,
as
the
beings
baQa and nanda represen t
for
of
and
to
respectively.
employed
architecture
are
known
!kaf:la,
and
also
traditional
fraction
five
familiar
this
in
for
portion.
for
denoting
are frequent ly
measures
of
the
elements of the building.
The ratio and trairasika or Rule of Three
play
e"
an
important
role
in
traditional
(proportion)
architecture.
The
different parts of a building are proportionate to each other
and
hence,
known,
if
the
measure
the measures
at by proportion.
ratio',
belongs
of
of
other
elements
to
the
I
to
is
approximately
where F
This
n
is
denotes
the
'golden
equal
limiting value of
elements
easily
the roof
ratio of
to
the
(amippu)
the
ratio
arrived
traditional
1.618:1
of
length
or
simply
the fraction
run
by
is def ined
considering
an
in
is
'the golden
in
to
be
F n+l/ F n I
the nth term of a F ibonacci series (4) •
inclination of
rise
the
ratios
rectangle I,
a
1.618.
can
arddh~dhikam'
In
width
of
The celebrated ratio known as
architecture.
its
anyone
The
terms of
elemental
the
right
6
angled
when
triangle
the
of
base
sides
proportionately,
of unit
of
the
a
length.
right
ratio
of
It
is
triangle
height
to
implied
are
that
increased
base
remains
the
same.
Geometr ical
determining
the
vastumat;lQala
a
are
imp 1 ici tly
directions
si te.
foundation
respect
Yamasutra
cardinal
on
intellectual
with
principles
and
made
forming
the
the
in
square
the
is derived graphically
to two perpendicular axes named
at
of
vc§:stupuru~amat;l9ala,
The
of a building,
intersecting
use
centre
of
Brahmasutra and
the
vastumat;l9ala.
The grid system and the vlthi systems are geometrical methods
of
determining
(foundation).
rectangle,
the
The
exact
position
geometrical
circle,
hexagon
of
constructions
and
an
octagon
grhav~dika
the
of
a
whose
triangle,
perimeters
are the same as that of a given square have contributed many
major results to mathematics.
in
these
geometrical
It is significant to note that
constructions
perimeter is kept as a constant
is due
to
the
fact
that
the
(or
instead of
perimeter
is
conversions)
their area.
considered
prime dimension
of a vastu which defines the yOni,
air
a
of
(pr!Qa)
Kerala
Further,
of
building.
architecture
an
independent
length of diagonal,
that
method
~f
Several
is
of
is
approximately,
the Pythagoras theorem.
by implication for
from
This
one
other
derived
of a
the
of
the
parts
for
as
the
the vital
difference
of
India.
finding
square wi thout
approximations
This
are
the
using
obtained
an irrational number which is the ratio
7
of
the
circumference
contribution
irrational
is
these,
the
number
construct ion
of
of
circle
The
image
idea
of
its
of
of
value
I imi t
is
principles
man
of
through
architects
vastusastra
these
local
of Kerala
were
made
'thaccans'
the
Apart
from
for
types of rafters
(perumtaccans).
to
the
The
common
therefore vastusastra
"thaccus~stram"
was often known by the name
in
formulated
available
and
another
impl ied
are
determining accurate dimensions of different
by the traditional
of
(sival inga) .
methods
Another
diameter.
the
I inga
practical
various
to
implication
12.
the
a
in Kerala.
The chapterwise content of the thesis is given below
Chapter
Kerala.
The
I,
deals
with
selection
of
of the cardinal directions,
the
site,
domestic
geometrical
formation
grid system and the vithi systems for
the
concept
measurement,
and
of
vertical
elements
of square mar;tdala,
fixing
of yon i,
of
a
different
building
of
determination
the
the grhavedika,
Vastupuru~amal)9ala,
marma,
the concept
architecture
un its
types of
are
of
"salas'
explained
with
appropriate figures.
In
chapter
of
Kerala
is
of
temples,
of
temples
11, ·an
given.
of
the
The characteristics
construction
depending
outline
on
of
their
the
temple
of
different
garbhagrha,
plan-shape,
architecture
tal a
types
classification
(storey)
and
8
perimeter,
pancaprakaras,
vertical
components
of
a
temple,
natyamaQQapam (kuthampalam) etc, are explained briefly.
Chapter
I I I,
~
irrational number
value
~
of
Another
various
implied
by the
(approximate)
vrttapr~sada
is the
gives
and
(~sanna'
value
assumed
~
of
in
the
in
in
explaines
IV
traditional
of
the
uddhidhikam
ratios
its
I
(b.
of
which
3.1416
A.D.476).
is
3.125
gajaprstba
its
shape.
to
between
rectangle,
rat i
method
width
construction
relation
golden
'golden
The
length
ratio',
properties,
of
Kerala.
and
and 'golden
the Fibonacci series and
The
construction
is
ex istan ce
of
natyamaodapa
the
texts.
Tan trasamuccaya'
of
to
are 3.2 and 22/7.
architecture
construction
••etangle .and
I
~
the
Bh~~a"
construction
Some other values adopted for
Chapter
in
by ~ryabhata
given
impl ied
approximately,
architectural
"Thaccus~stram
in
value
approximations
of
ratio,
of
the
0'
golden
terms
application
of
in
the
(t raira-si ka)
and
idol construction etc, are given in this chapter.
Chapter V explains
its
applications.
trairlsika)
right
are
triangles
inclination of
Trairasika
given
and
the
Ru le of
and
in detail.
their
the collarpin
explained with figures.
determining
the
heights
Theorems
applications
(vala)
the
trair!~ika
inverse
in
Application
of
three
ridge
in
roof
of
from
on
(vyasta
similarity
of
determining
the
construction
are
traira§ika
the
level
in
of
the
9
(u t taram)
wallplate
for
of
different
the
and
aviccil
side
of
an
the
corresponding
(pitch),
octagon,
EttampramaQam (sth_Postu la te)
various proportions of
leng ths
determination
the
and
lengths
of
of
of
ra f ters
the
length
rafters
using
NaTampramal)am (4th_poS tu 1 ate)
ridgeheight to semiwidth etc.
I
are also
included in this chapter.
VI
Chapter
various
and
plan
shapes
deductions
implied
shapes
in
of
a
The
detail.
circle,
and
The
approximate length of
Further,
triangle,
which
method
of
included
the
rectangle,
having
the
'Sricakram',
this
close
methods
hexagon,
same
an
of
different
types
of
and
vi t'!na
(vertical
a
and
method
rafters
and
and
are
determining
the
a
given
of
and
the
square
are
construction
1 ine)
a
of
of
demarcation
horizontal
of
are
each
square
determination
the
using
J2
to
circle
given
the
circle),
construction
of
The
of
of
to
or
construction
approximation
in
method
1 ine
of
ass igned
square without
octagon
chapter.
geometrical
a
octagon
perimeter
inscribing
in
very
of
method
of
used
(elongated
methods
practical
a
val ues
of
archi tecture
are
which
vrttctyatam
the
construction
tradi tional
values
the diagonal
Pythagoras theorem and
explained.
the
arch i tectural
(apsidal)
in
in
mathematical
square,
gajaprstl;ta
given
adopted
of
them.
geometrical
explains
on
of
lengths
of
lamba
them
are
also explained in this chapter.
A
translation
of
'Manu~yalayacandrika'
in
English
language is given at the end of this work as an appendix.