Friday, 10 February 2012

Archaeoastronomy in India by Subhash Kak Oklahoma State University


 This article is a verbatim reproduction of article by Subhash Kak. As his contact details not known I am not able to contact him. this post will be deleted if there is an written objection by him. 



Archaeoastronomy in India


Subhash Kak
Oklahoma State University, Stillwater



Our  understanding  of  archaeoastronomical  sites  in  India  is  based  not  only  on  a  rich  archaeological record and texts that go back thousands of years, but also on a living tradition that is connected to the past. Conversely,  India  has  much  cultural  diversity  and  a  tangled  history  of  interactions  with neighboring regions that make the story complex. The texts reveal to us the cosmological ideas that lay behind  astronomical  sites  in  the  historical  period  and  it  is  generally  accepted  that  the  same  idea  also apply to the Harappan era of the third millennium BCE (Kenoyer, 1998: 52‐53).
In the historical period, astronomical observatories were part of temple complexes where the king was
consecrated. Such consecration served to confirm the king as foremost devotee of the chosen deity, who
was taken to be the embodiment of time and the universe (Kak, 2002a: 58). For example, Udayagiri is an
astronomical site connected with the Classical age of the Gupta dynasty (320‐500 CE), which is located a
few kilometers from Vidisha in central India (Willis, 2001; Dass and Willis, 2002). The imperial Guptas
enlarged  the  site,  an  ancient  hilly  observatory  going  back  at  least  to  the  2
nd  century  BCE  at  which
observations  were  facilitated  by  the  geographical  features  of  the  hill,  into  a  sacred  landscape  to  draw
royal authority.
Indian  astronomy  is  characterized  by  the  concept  of  ages  of  successive  larger  durations,  which  is  an
example  of  the  pervasive  idea  of  recursion,  or  repetition  of  patterns  across  space, scale  and  time.    An
example of this is the division of the ecliptic into 27 star segments (nakshatras), with which the moon is
conjoined  in  its  monthly  circuit,  each  of  which  is  further  sub‐divided  into  27  sub‐segments  (
upa‐
nakshatras),  and  the  successive  divisions  of  the  day  into  smaller  measures  of  30  units.  The  idea  of
recursion underlies the concept of the sacred landscape and it is embodied in Indian art, providing an
archaeoastronomical  window on sacred and monumental  architecture.  It appears that this  was an old
idea  because  intricate  spiral  patterns,  indicating  recursion,  are  also  found  in  the  paintings  of  the
Mesolithic period. Tyagi (1992) has claimed that they are unique to Indian rock art.

According to the Vāstu Shāstra, the structure of the building mirrors the emergence of cosmic order out of  primordial  chaos  through  the  act  of  measurement.    The  universe  is  symbolically  mapped  into  a square that emphasizes the four cardinal directions. It is represented by the square vāstu‐mandala, which in its various forms is the basic plan for the house and the city.  There exist further elaborations of this plan, some of which are rectangular.
It  is  significant  that  yantric  buildings  in  the  form  of  mandalas  have  been  discovered  in  North
Afghanistan that belong to a period that corresponds to the late stage of the Harappan tradition (Kak,
2000a;  Kak,  2005b)    providing  architectural  evidence  in  support  of  the  idea  of  recursion  at  this  time.
Although  these  building  are  a  part  of  the  Bactria‐Margiana  Archaeological  Complex  (BMAC),  their
affinity  with  ideas  that  are  also  present  in  the  Harappan  system  shows  that  these  ideas  were  widely
spread.







                                                                                                                                                                                            






Contents

1.     Chronology and Overview
2.     Pre‐historical and Harappan Period
3.     Neolithic and Megalithic Sites
4.     The Plan of the Temple
5.     Observatory in Udayagiri
6.     Pilgrimage Complexes
7.     Sacred Cities
8.     Conclusions


1.  Chronology and Overview

India’s archaeological record in the northwest has unbroken continuity going back to about 7500 BCE at
Mehrgarh (Kenoyer, 1998; Lal, 2002), and it has an rock art tradition, next only to that of Australia and
Africa  in  abundance,  that  is  much  older  (Pandey,  1993;  Bednarik,  2000).  Some  rock  art  has  been
assigned to the Upper Paleolithic period. There is surprising uniformity, both in style and content, in
the rock art paintings of the Mesolithic period (10,000 - 2500 BCE) (Tyagi, 1992; Wakankar, 1992).

The archaeological phases of the Indus (or Sindhu‐Sarasvati) tradition have been divided into four eras:
early  food‐producing  era  (c.  6500‐  5000  BCE),  regionalization  era  (5000  -  2600  BCE),  integration  era  (2600  -
1900 BCE),  and  localization  era  (1900  -  1300  BCE)  (Shaffer,  1992).  The  early  food‐producing  era  lacked
elaborate ceramic technology. The regionalization era was characterized by styles in ceramics, lapidary
arts, glazed faience and seal making that varied across regions. In the integration era, there is significant
homogeneity in material culture over a large geographical area and the use of the so‐called Indus script,
which  is  not  yet  deciphered.  In  the  localization  era,  patterns  of  the  integration  era  are  blended  with
regional  ceramic  styles,  indicating  decentralization  and  restructuring  of  the  interaction  networks.  The
localization  era  of  the  Sindhu‐Sarasvati  tradition  is  the  regionalization  era  of  the  Ganga‐Yamuna
tradition which transforms into the integration era of the Magadha and the Mauryan dynasties. There is
also continuity in the system of weights and lengths between the Harappan period and the later historic
period (Mainkar, 1984).
The  cultural  mosaic  in  the  third  millennium  BCE  is  characterized  by  the  integration  phase  of  the Harappan civilization of northwest India, copper and copper/bronze age cultures or central and north India, and Neolithic cultures of south and east India (Lal, 1997). Five large cities of the integration phase are  Mohenjo‐Daro,  Harappa,  Ganweriwala,  Rakhigarhi,  and  Dholavira.  Other  important  sites  of  this period are Kalibangan, Rehman Dheri, Nausharo, Kot Diji, and Lothal.
A majority of the towns and settlements of the Harappan period were in the Sarasvati valley region. Hydrological changes, extended period of drought, and the drying up of the Sarasvati River due to its major tributaries being captured by the Sindh and Ganga Rivers after an earthquake in 1900 BCE led to the abandonment of large areas of the Sarasvati valley (Kak, 1992). The Harappan phase went through various  stages  of  decline  during  the  second  millennium  BCE.  A  second  urbanization  began  in  the Ganga and Yamuna valleys around 900 BCE. The earliest surviving records of this culture are in Brahmi script.  This  second  urbanization  is  generally  seen  at  the  end  of  the  Painted  Gray  Ware  (PGW)  phase (1200‐ 800 BCE) and with the use of the Northern Black Polished Ware (NBP) pottery. Late Harappan was  partially  contemporary  with  the  PGW  phase.  In  other  words,  a  continuous  series  of  cultural developments link the two early urbanizations of India.
The setting for the hymns of the Rigveda, which is India’s most ancient literary text, is the area of Sapta
Saindhava,  the  region  of  north  India  bounded  by  the  Sindh  and  the  Ganga  rivers  although  regions


                                                                                                                                                                                            






around this heartland are also mentioned. The Rigveda describes the Sarasvati River to be the greatest of the rivers and going from the mountains to the sea. The archaeological record, suggesting that this river had turned dry by1900 BCE, indicates that the Rigveda is prior to this epoch. The Rigveda and other early Vedic  literature  have  astronomical  references  related  to  the  shifting  astronomical  frame  that  indicate epochs of the fourth and third millennium BCE which is consistent with the hydrological evidence. The nakshatra lists are found in the Vedas, either directly or listed under their presiding deities, and it one may conclude that their names have not changed. Vedic astronomy used a luni‐solar year in which an intercalary month was employed as adjustment with solar year.
The shifting of seasons through the year and the shifting of the northern axis allow us to date several
statements in the Vedic books (Sastry, 1985). Thus the 
Shatapatha Brāhmana (2.1.2.3) has a statement that
points to an earlier epoch where it is stated that the Krittikā (Pleiades) never swerve from the east. This
corresponds to 2950 BCE. The 
Maitrāyanīya Brāhmana Upanishad (6.14) refers to the winter solstice being
at the mid‐point of the Shravishthā segment and the summer solstice at the beginning of Maghā. This
indicates 1660 BCE.  The 
Vedānga  Jyotisha  mentions  that  winter  solstice  was  at  the  beginning  of
Shravishthā and the summer solstice at the mid‐point of Ashleshā. This corresponds to about 1300 BCE.
The nakshatras in the Vedānga Jyotisha are defined to be 27 equal parts of the ecliptic.  The nakshatra
list of the late Vedic period begin with Krittikā (Pleiades) whereas that of the astronomy texts after 200
CE begin with Ashvini (α and β Arietis), indicating a transition through 2 nakshatras, or a time span of
about 2,000 years.
The foundation of Vedic cosmology is the notions of bandhu (homologies or binding between the outer
and  the  inner).  In  the  Ayurveda,  medical  system  associated  with  the  Vedas,  the  360  days  of  the  year
were taken to be mapped to the 360 bones of the developing fetus, which later fuse into the 206 bones of
the  person.  It  was  estimated  correctly  that  the  sun  and  the  moon  were  approximately  108  times  their
respective diameters from the earth (perhaps from the discovery that the angular size of a pole removed
108 times its height is the same as that of the sun and the moon), and this number was used in sacred
architecture.  The distance to the sanctum sanctorum of the  temple from the gate and the perimeter  of
the  temple  were  taken  to  be  54  and  180  units,  which  are  one‐half  each  of  108  and  360  (Kak,  2005a).
Homologies at many levels are at the basis of the idea of 
recursion, or repetition in scale and time. The
astronomical  basis  of  the  Vedic  ritual  was  the  reconciliation  of  the  lunar  and  solar  years  (Kak,  2000a;
Kak, 2000b).
Texts of the Vedic and succeeding periods provide us crucial understanding of the astronomy and the archaeoastronomy of the historical period throughout India. The medieval period was characterized by pilgrimage  centers  that  created  sacred  space  mirroring  conceptions  of  the  cosmos.  Sacred  temple architecture served religious and political ends.

The  instruments  that  were  used  in  Indian  astronomy  include  the  water  clock  (ghati  yantra),  gnomon
(shanku), cross‐staff (yasti yantra), armillary sphere (gola‐yantra), board for sun’s altitude (phalaka yantra),
sundial (
kapāla yantra), and astrolabe (Gangooly, 1880). In early 18th century, Maharaja Sawai Jai Singh II
of  Jaipur  (r.  1699‐1743)  built  five  masonry  observatories  called  Jantar  Mantar  in  Delhi,  Jaipur,  Ujjain,
Mathura, and Varanasi. The Jantar Mantar  consists of  the Ram Yantra (a cylindrical structure with an
open top and a pillar in its center to measure the altitude of the sun), the Rashivalaya Yantra (a group of
twelve  instruments  to  determine  celestial  latitude  and  longitude),  the  Jai  Prakash (a  concave
hemisphere), the Laghu Samrat Yantra (small sundial), the Samrat Yantra (a huge equinoctial dial), the
Chakra  Yantra  (upright  metal  circles  to  find  the  right  ascension  and  declination  of  a  planet),  the
Digamsha  Yantra  (a  pillar  surrounded  by  two  circular  walls),  the  Kapali  Yantra (two  sunken
hemispheres  to  determine  the  position  of  the  sun  relative  to  the  planets  and  the  zodiac),  and  the Narivalaya Yantra (a cylindrical dial).



                                                                                                                                                                                            







2.  Pre‐historical and Harappan Period

The city of Mohenjo‐Daro (2500 BCE), like most other Harappan cities (with the exception of Dholavira
as  far  as  we  know  at  this  time)  was  divided  into  two  parts:  the  acropolis  and  the  lower  city.  The
Mohenjo‐Daro  acropolis,  a  cultural  and  administrative  centre,  had  as  its  foundation  a  12  meter  high
platform of 400 m 
¯ 200 m. The lower city had streets oriented according to the cardinal directions and
provided  with  a  network  of  covered  drains.  Its  houses  had  bathrooms.  The  city’s  wells  were  so  well
constructed with tapering bricks that they have not collapsed in 5000 years. The Great Bath (12 m 
¯ 7
m) was built using finely fitted bricks laid on with gypsum plaster and made watertight with bitumen.
A high corbelled outlet allowed it to be emptied easily. Massive walls protected the city against flood
water.

The absence of monumental buildings such as palaces and temples makes the Harappan city strikingly different from its counterparts of Mesopotamia and Egypt, suggesting that the polity of the Harappan state was de‐centralized and based on a balance between the political, the mercantile, and the religious elites. The presence of civic amenities such as wells and drains attests to considerable social equality. The  power  of  the  mercantile  guilds  is  clear  in  the  standardization  of  weights  of  carefully  cut  and polished chart cubes that form a combined binary and decimal system.

Mohenjo‐Daro and other sites show slight divergence of 1° to 2° clockwise of the axes from the cardinal
directions (Wanzke, 1984). It is thought that this might have been due to the orientation of Aldebaran
(
Rohinī in Sanskrit) and the Pleiades (Krtikkā in Sanskrit) that rose in the east during 3000 BCE to 2000
BCE at the spring equinox; the word “rohinī” literally means rising. Furthermore, the slight difference
in  the  orientations  amongst  the  buildings  in  Mohenjo‐Daro  indicates  different  construction  periods
using the same traditional sighting points that had shifted in this interval (Kenoyer, 1998).

Mohenjo‐Daro’s  astronomy  used  both  the  motions  of  the  moon  and  the  sun  (Maula,  1984).  This  is attested by the use of great calendar stones, in the shape of ring, which served to mark the beginning and end of the solar year.


Dholavira
Dholavira  is  located  on  an  island  just  north  of  the  large  island  of  Kutch  in  Gujarat.  Its  strategic
importance  lay  in  its  control  of  shipping  between  Gujarat  and  the  delta  of  the  Sindh  and  Sarasvati
rivers.
The layout of Dholavira is unique in that it comprises of three “towns,” which is in accord with Vedic
ideas (Bisht, 1997; Bisht, 1999a; Bisht, 1999b). The feature of recursion in the three towns, or repeating
ratios at different scales, is significant. Specifically, the design is characterized by the nesting proportion
of 9:4 across the lower and the middle towns and the castle. The proportions of 5/4, 7/6, and 5/4 for the
lower  town,  the  middle  town,  and  the  castle  may  reflect  the  measures  related  to  the  royal  city,  the
commander’s quarter, and the king’s quarter, respectively, which was also true of Classical India (Bhat,
1995).
A  Dholavira  length,  D,  has  been  determined  by  finding  the  largest  measure  which  leads  to  integer
dimensions for the various parts of the city. This measure turns out be the same as the Arthaśāstra (300
BCE) measure of 
dhanus (arrow) that equals 108 angulas (fingers). This scale is confirmed by a terracotta
scale  from  Kalibangan  and  the  ivory  scale  found  in  Lothal.  The  Kalibangan  scale  (Joshi, 2007;
Balasubramaniam and Joshi, 2008) corresponds to  units  of 17.5 cm, which is substantially the same  as the Lothal scale and the small discrepancy may be a consequence of shrinkage upon firing.



                                                                                                                                                                                            





The  analysis  of  the  unit  of  length  at  Dholavira  is  in  accord  with  the  unit  from  the  historical  period
(Danino, 2005; Danino, 2008). The unit that best fits the Dholavira dimensions is 190.4 cm, which when
divided by 108 gives the Dholavira 
angula of 1.763 cm. The subunit of angula is confirmed when one
considers that the bricks in Harappa follow ratios of 1:2:4 with the dominating size being 7 ¯ 14 ¯ 28
cm (Kenoyer, 1998). These dimensions can be elegantly expressed as 4 
¯ 8 ¯ 16 angulas, with the unit of
angula taken as 1.763 cm. It is significant that the ivory scale at Lothal has 27 graduations in 46 mm, or
each graduation is 1.76 mm.















Figure 1. Map of Dholavira (Bisht, 1997)
With the new Dhloavira unit of D, the dimensions of Mohenjo‐Daro’s acropolis turn out to be 210 ¯ 105 D; Kalibangan’s acropolis turn out to be 126 ¯ 63 D. The dimensions of the lower town of Dholavira are 405 ¯ 324 D; the width of the middle town is 180 D; and the inner dimensions of the castle are 60 ¯ 48 D. The sum of the width and length of the lower town comes to 729 which is astronomically significant since it is 27 ¯ 27, and the width 324 equals the nakshatra year 27 ¯ 12.
Continuity has been found between the grid and modular measures in the town planning of Harappa and  historical  India,  including  that  of  Kathmandu  Valley  (Pant  and  Funo,  2005).  The  measure  of  19.2 meters  is  the unit in quarter‐blocks  of Kathmandu; this is nearly the same as the  unit characteristic of the dimensions of Dholavira. It shows that the traditional architects and town planners have continued the use of the same units over this long time span.






















Figure 2. Astronomical seal from Rehman Dheri


                                                                                                                                                                                            







Rehman Dheri
A 3rd millennium seal from Rehman Dheri, showing a pair of scorpions on one side and two antelopes
on the other, that suggests knowledge of Vedic themes. It has been suggested that this seal represents
the  opposition  of  the  Orion  (Mrigashiras,  or  antelope  head)  and  the  Scorpio  (Rohini  of  the  southern
hemisphere which is 14 nakshatras from the Rohini of the northern hemisphere) nakshatras. The arrow
near the head of one of the antelopes could represent the decapitation of Orion. It is generally accepted
that  the  myth  of  Prajapati  being  killed  by  Rudra  represents  the  shifting  of  the  beginning  of  the  year
away from Orion and it places the astronomical event in the fourth millennium BCE (Kak, 2000a).

3.  Neolithic and Megalithic Sites
Interesting sites of archaeoastronomical interest include the Neolithic site of Burzahom from Kashmir in
North India, and megalithic sites from Brahmagiri and Hanamsagar from Karnataka in South India.

Burzahom, Kashmir
This  Neolithic  site  is  located  about  10  km  northeast  of  Srinagar  in  the  Kashmir  Valley  on  a  terrace  of
Late  Pleistocene‐Holocene  deposits.  Dated  to  around  3000  ‐  1500  BCE,  its  deep  pit  dwellings  are
associated with ground stone axes, bone tools, and gray burnished pottery. A stone slab of 48 cm ¯ 27
cm, obtained from a phase dated to 2125 BCE shows two bright objects in the sky with a hunting scene
in the foreground. These have been assumed to be a depiction of a double star system (Kameshwar Rao,
2005).















Figure 3. Burzahom sky scene

Brahmagiri, Karnataka
The  megalithic  stone  circles  of  Brahmagiri  (latitude  14o  73,  longitude  76 77),  Chitradurga  district  of
Karnataka  in  South  India,  that  have  been  dated  to 900  BCE,  show  astronomical  orientations.
Kameswara  Rao  (1993)  has  argued  that  site  lines  from  the  centre  of  a  circle  to  an  outer  tangent  of another circle point to the directions of the sunrise and full moon rise at the time of the solar and lunar solstices and equinox.










                                                                                                                                                                                            
























Figure 4. Megalithic stone circles of Brahmagiri

Hanamsagar, Karnataka.
Hanamsagar is a megalithic site with stone alignments pointing to cardinal directions. It is located on a
flat area between hills about 6 km north of the Krishna river at latitude 16
o 19 18 and longitude 7627
10.    The  stones,  which  are  smooth  granite,  are  arranged  in  a  square  of    side  that  is  about  600  meters
with 50 rows and 50 column (for a total of 2,500 stones), with a separation between stones of about 12
m.  The  stones  are  between  1  to  2.5  m  in  height  with  a  maximum  diameter  of  2  to  3  m.  The  lines  are
oriented in cardinal directions. There is a squarish central structure known as 
chakri katti.
It has been argued that the directions of summer and winter solstice can be fixed in relation to the outer and  the  inner  squares.  Kameswara  Rao  (2005)  suggests  that  it  could  have  been  used  for  several  other kind of astronomical observations such as use of shadows to tell the time of the day, the prediction of months, seasons and passage of the year.























Figure 5. Alignments at Hanamsagar




                                                                                                                                                                                            





4.  The Plan of the Temple

The  sacred  ground  for  Vedic  ritual  is  the  precursor  to  the  temple.  The  Vedic  observances  were connected with the circuits of the sun and the moon (Kak, 1993; Kak, 1995; Kak, 1996). The altar ritual was associated with the east‐west axis and we can trace its origins to priests who maintained different day  counts  with  respect  to  the  solstices  and  the  equinoxes.    Specific  days  were  marked  with  ritual observances that were done at different times of the day.














Figure 6. The three altars of the Vedic house: circular
(earth, body), half‐moon (atmosphere, prāna), square (sky, consciousness)

In  the  ritual  at  home,  the  householder  employed  three  altars  that  are  circular  (earth),  half‐moon
(atmosphere),  and  square  (sky),  which  are  like  the  head,  the  heart,  and  the  body  of  the  Cosmic  Man
(Purusha).  In  the  Agnichayana,  the  great  ritual  of  the  Vedic  times  that  forms  a  major  portion  of  the
narrative of the Yajurveda, the atmosphere and the sky altars are built afresh in a great ceremony to the
east. This ritual is based upon the Vedic division of the universe into three parts of earth, atmosphere,
and sky that are assigned numbers 21, 78, and 261, respectively. The numerical mapping is maintained
by  placement  of  21  pebbles  around  the  earth  altar,  sets  of  13  pebbles  around  each  of  6  intermediate
(13
¯6=78) altars, and 261 pebbles around the great new sky altar called the Uttara‐vedi, which is built
in the shape of a falcon; these numbers add up to 360, which is symbolic representation of the year. The
proportions related to these three numbers, and others related to the motions of the planets, and angles
related to the sightings of specific stars are reflected in the plans of the temples of the historical period
(Kak, 2002b; Kak, 2006a; Kak, 2009; Kaulācara, 1966).













Figure 7. The falcon altar of the Agnichayana altar
The Agnichayana altar is the prototype of the temple and of the tradition of architecture (Vāstu).  The altar is first built of 1,000 bricks in five layers (that symbolically represent the five divisions of the year, the  five  physical  elements,  as  well  as  five  senses)  to  specific  designs.  The  altar  is  constructed  in  a sequence  of  95  years,  whose  details  are  matched  to  the  reconciliation  of  the  lunar  and  solar  years  by means of intercalary months.


                                                                                                                                                                                            







In the ritual ground related to the Agnichayana ceremony, the Uttara‐vedi is 54 units from the entrance in  the  west  and  the  perimeter  of  the  ritual  ground  is  180  units  (Kak,  2005a).  These  proportions characterize many later temples.

The Temple Complex at Khajuraho
The  town  of  Khajuraho  extends  between  79°  54’  30”  to  79°  56’  30”  East  and  24°  50’  20”  to  24°  51’  40”
North,  in  Chhatarpur  district,  in  Madhya  Pradesh.  The  temples  of  Khajuraho  were  built  in  9th  ‐12th
century CE by the Chandela kings. Originally there were 84 temples, of which 23 have survived. Of the
surviving temples, 6 are associated with Shiva, 8 with Vishnu, and 5 with the goddess (Singh, 2009b).


Apabh       Asvini
Krttika               1            27         Revati
Rohini           2                                                26       U. Prosth.
3                                                                                 25
Mrga                             Vaisakha                      Caitra                                Prosthap.
4                                                                                                               24

Ardra           Jyaistha                                 I                      XII                         Phalguna
5                                                                                                                                 Satabhisaj23

II                                                               XI
Punarvasu                                                                                                                                            Sravishtha
6                                                                                                                                                          22
Asadha                                                                                                                       Magha
Pusya                               III                                                                                          X                             Srona
7                                                                                                                                                              21



Asresa
8       Sravana

Magha
      9


P. Phal
      10




IV




V


Bhadrapada
                                         VI




IX                             U. Asadh
20
Pausa
P. Asadh
    19

VIII
                              Mula
18
VII                      Margasirsa

U. Phal
        11
Hasta
     12




Asvayuja

Citra
    13





Svati
    14



Rohini
       17
Kartika
                           Anuradha
16
Visakha
       15

Figure 8. Mapping of the nakshatras to the solar months


At the eastern edge of the temple complex are the Dantla hills, with a peak of 390 m at which is located a  shrine  to  Shiva,  which  is  a  reference  point  for  the  temple  entrances.  All  the  temples  excepting  the Chaturbhuja face the east. The southeastern edge has the Lavanya hill that is separated from the Dantla hills by the eastward flowing river Khudar. At the foothills of the Lavanya hill at a height of 244m is the shrine of goddess Durga as Mahishasurmardini.

The shrines to Shiva and Durga on the Dantla and Lavanya hills span the polarities of spirit (Purusha)
and  matter  (Prakriti),  which  are  bridged  by  the  river  between  the  hills.  The  temples  of  Khajuraho  are
popular  pilgrimage  centers  during  two  spring  festivals:  Shivaratri  that  falls  on  the  new  moon  of
Phalguna (February/March), and Holi, which falls on the full moon of Chaitra (March/April).
The Lakshmana temple, one of the oldest of the complex, is considered the axis mundi of the site. It was built by  the king Yashovarman  (925‐950) as symbol of the Chandela victory over the Pratiharas and a record of supremacy of their power. This temple is oriented to the sunrise on Holi.




                                                                                                                                                                                            





The groups of temples form three overlapping mandalas, with centers at the Lakshmana (Vishnu), the Javeri (Shiva), and the Duladeva (Shiva) temples. Their deviation from true cardinality is believed to be due to the direction of sunrise on the day of consecration (Singh, 2009).

The temple, as a representation of the cosmos and its order, balances the asuras (demons) and the devas (gods),  as  well  as  inheres  in  itself  other  polarities  of  existence.  In  the  Lakshmana  Temple,  Vishnu  is depicted in a composite form with the usual calm face bracketed by the faces of lion and boar. The conception of the sanctum is as a mandala (Desai, 2004).

The planetary deities, the grahas, encircle the temple in the following arrangement:

Surya (Sun)
Soma (Moon)                            Mangala (Mars)
Brhaspati (Jupiter)                             Budha (Mercury)
Shani (Saturn)                          Shukra (Venus)
Ganesha                                       Durga
Ganesha  and  Durga  are  the  deities  of  the  ascending  and  the  descending  nodes  of  the  moon,
respectively. The temple is envisioned like Mount Meru, the axis of the universe, and the planets move
around it.

5.  The Udayagiri Observatory

Udayagiri (“hill of [sun]‐rise’’] is one of the principal ancient astronomical observatories of India. It is
located  at  23
o31’  N  latitude  on  the  Tropic  of  Cancer  in  Madhya  Pradesh,  about  50  kilometers  from
Bhopal, near Vidisha, Besnagar and Sanchi. An ancient site that goes back to at least the second century
BCE,  it  was  substantially  enlarged  during  the  reign  of  the  Gupta  Emperor  Chandragupta  II
Vikramaditya  (r.  375‐414).  This  site  is  associated  with  20  cave  temples  that  have  been  cut  into  rock;
nineteen of these temples are from the period of Chandragupta’s reign (Dass and Willis, 2002).














Figure 9. Udayagiri layout (Balasubramaniam , 2008)

It  appears  that  the  ancient  name  of  Udayagiri  was  Vishnupadagiri,  or  the  “hill  of  the  footprint  of
Vishnu,” and the name Udayagiri is after the Paramara ruler Udayaditya (c. 1070‐93). The hill is shaped
like a foot. A saddle connects the northern and southern hills, and a passageway is located at the place
where the northern hill meets the saddle. The Gupta period additions and embellishments at Udayagiri
were concentrated around this passage. Most of the cave temples are located around the passageway.

On the summer solstice day, there was an alignment of the sun’s movement with the passageway. The
day  mentioned  in  the  dated  Chandragupta  II  Vikramaditya  period  inscription  in  cave  6  has  been
calculated to be very close to the summer solstice of the year 402 CE. On this day, the shadow of the


                                                                                                                                                                                          





Iron Pillar of Delhi, which was originally located at the entrance of the passageway, fell in the direction of the reclining Vishnu panel (Balasubramaniam, 2008).
On  the  northern  hilltop,  there  exists  a  flat  platform  commanding  a  majestic  view  of  the  sky.  Several astronomical marks have been identified at this platform, indicating that this was the site of the ancient astronomical observatory.

6.  Medieval Pilgrimage Complexes

Medieval pilgrimage centers fulfilled many functions including that of trade and business. They were important  to  the  jyotishi  (astrologer)  who  would  make  and  read  the  pilgrims’  horoscope.    The  better astrologers  were  also  interested  in  astronomy  and  this  knowledge  was  essential  for  the  alignment  of temples and palaces.

Every  region  of  India  has  important  pilgrimage  centers,  some  of  which  are  regional  and  others  pan‐
Indic. The  most  famous  of  the  pan‐Indic  centers  are  associated  with  Shiva  (Varanasi),  Krishna
(Mathura,  Dwarka),  Rama  (Ayodhya),  Vishnu  (Tirupati),  and  the  12‐yearly  rotation  of  the  Kumbha
Mela  at  Prayag,  Haridwar,  Ujjain,  and  Nashik.  For  pilgrimage  centers  such  as  Chitrakut,  Gaya,
Madurai,  Varanasi,  Vindhyachal,  and  Khajuraho,  the  question  of  alignments  of  temples  to  cardinal
directions or to direction of the sun on major festivals has been studied by scholars (Singh, 2009b). Here
we will consider the sun temples of Varanasi (Malville, 1985; Singh, 2009a and 2009b).





























Figure 10. Khajuraho: Landscape Geometry and Topography (Singh, 2009b).


The Sun Temples of Varanasi
Varanasi is an ancient city dating from the beginning of the first millennium BCE, whose Vedic name is
Kashi  (Sanskrit  for  “radiance”),  a  name  that  continues  to  be  used  together  with  Banaras.  Of  its  many
temples,  the  most  important  is  Kashi  Vishvanath  Temple,  or  “Golden  Temple,”  dedicated  to  Lord


                                                                                                                                                                                          





Shiva,  the  presiding  deity  of  the  city.  Because  of  repeated  destruction  by  the  sultans  and  later  by Aurangzeb,  the  current  Vishvanath  is  a  relatively  modern  building.  It  was  built  in  1777  by  Maharani Ahilyabai of Indore, and its shikhara (spire) and ceilings were plated with of gold in 1839, which was a gift from Maharaja Ranjit Singh (Singh, 2009a and 2009b).






















































Figure 11. Sun Shrines: Cosmic Order and Cyclic orientation of Time (Singh, 2009a).



                                                                                                                                                                                          







Shiva  represents  both  the  axis  of  the  universe  as  well  as  that  of  one’s  inner  being.  One  of  the  great
festivals celebrated in Varanasi is Shivaratri which is celebrated on the 13th day of the dark fortnight of
the Phalguna month (February‐March). On that day you can see the sun rise in the east with the  new
moon just above it, which is represented iconographically by Shiva (as the sun) wearing the moon on
his head.
There  are  several  pilgrimage  circuits  in  Varanasi  for  circumambulating  the  city.  The  Panchakroshi
circuit has 108 shrines on it, and the four inner circuits have a total of 324 shrines. It is also known for
the  circuit  of  the  Aditya  shrines.  The  Adityas  are  the  7  or  8  celestial  gods,  although  their  number  is
counted  to  12  in  later  books.  In  Puranic  India,  they  are  taken  to  be  the  deities  of  the  twelve  solar
months.    The  Aditya  temples  were  also  razed  during  the  centuries  of  Muslim  rule,  but  have  been  re‐
established at the same sites and are now part of the active ritualscapes (Singh, 2009a).
Several  Aditya  shrines  have  been  located  with  the  aid  of  descriptions  in  the  Kashi  Khanda  and
pilgrimage guides (Singh and Malville, 1995; Singh, 2009a and 2009b). Six of these lie along one sides of
an  isosceles  triangle  with  a  base  of 2.5km.  The  triangle  surrounds  the  former  temple  of
Madhyameshavara,  which  was  the  original  center  of  Kashi.  Pilgrims  walking  along  the  triangle  are symbolically circumambulating the cosmos.
7.  Sacred Cities

There  are  numerous  sacred  cities  in  the  Indian  sub‐continent  that  were  either  built  to  an  archetypal master plan or grew organically by virtue of being connected to a specific celestial deity.  Some of the important sacred cities are:

1.     Varanasi
2.     Vijayanagara
3.     Ayodhya
4.     Mathura
5.     Bhaktapur
6.     Tirupati
7.     Kanchipuram
8.     Dwarka
9.     Ujjain

Robert  Levy  viewed  the  Indian  sacred  city  as  a  structured      “mesocosm”,  situated  between  the
microcosm  of  the  individual  and  the  macrocosm  of  the  culturally  conceived  larger  universe  (Levy, 1991). Such  a  city  is  constructed  of  spatial  connected  mandalas,  each  of  which  is  sustained  by  its  own culture  and  performance.  The  movements  of  the  festival  year  and  rites  of  passage  constitute  a  “civic dance”, which defines the experience of its citizens.

The  life‐cycle  passages  and  festivals  dedicated  to  the  gods  affirm  the  householders’  moral  compass, identities and relationships. But there also exist other deities, represented generally by goddesses, who point  to  the  forces  of  nature  outside  of  moral  order.  These  are  brought  into  the  larger  order  through tantric  invocations  and  amoral  propitiatory  offerings.  Performances  invoking  the  goddess  are  the responsibility of the king and the merchants.

Sacrality and Royal Power at Vijayanagara
The city of Vijayanagara (also known as Hampi) was founded in the 14th century and sacked in 1565.
The best known kings associated with Vijayanagara are Harihara I and II and Bukka Raya I (ca. 1336‐
1404), and  Krishnadevaraya  and  his  half‐brother  Achyutadevaraya  (1509‐42).    From  the  mid‐14th
century  to  1565,  the  city  was  the  capital  of  the  Vijayanagara  Empire.  According  to  the  Persian

                                                                                                                                                                                          





ambassador  Abdur  Razaaq  (1442  CE):  “The  City  of  Vijayanagara  is  such  that  the  pupil  of  the  eye  has never  seen  such  a  place  like  it,  and  the  ear  of  intelligence  has  never  been  informed  that  there  existed anything to equal it in the world.”


















































Figure 12. Vijayanagara City

Hampi  had  for  centuries  been  an  important  pilgrimage  city  due  to  its  mythic  association  with  river
Goddess  Pampā  and  her  consort  Virupaksha,  or  Pampāpati.  An  inscription  dated  1163  CE  records  a
mahādāna,  a  religious  offering  in  the  presence  of  Lord  Virupaksha  of  Hampi  by  the  Kalachuri  King
Bijjala. The region was part of the kingdom of Kampiladeva until 1326 when the armies of Mohammed
Bin  Tughlaq  defeated  the  king  and  imprisoned  the  two  sons  of  Sangama,  Hukka  and  Bukka.  Some

                                                                                                                                                                                          






years later the Sultan sent the two as governors of the province. In 1336 they broke free from Tughlaq allegiance and established the Sangama dynasty with its capital at Vijayanagara.

The  destruction  of  Vijayanagara  in  1565  was  captured  vividly  in  the  account  of  Robert  Sewell  (1900):
“They slaughtered the people without mercy; broke down the temples and palaces; and wreaked such
savage vengeance on the abode of the kings that, with the exception of a few great stone built temples
and walls, nothing now remains but a heap of ruins to mark the spot where once the stately buildings
stood…  They  lit  huge  fires  in  the  magnificently  decorated  buildings  forming  the  temple  of
Vitthalaswami  near  the  river,  and  smashed  its  exquisite  stone  sculptures.  With  fire  and  sword,
crowbars and axes they carried on day after day their work of destruction. Never perhaps in the history
of the world has such havoc been wrought so suddenly on so splendid a city; teeming with a wealthy
and industrious population in the full plenitude of prosperity one day and on the next seized, pillaged
and reduced to ruins amid scenes of savage massacre and horrors beggaring description.”

Hampi has a strong association with the Ramayana and the names of many sites in the area bear names
mentioned in the epic. These include Rishimukha, Malyavanta hill and Matanga hill along with a cave
where  Sugriva  is  said  to  have  kept  the  jewels  of  Sita.  The  site  of  Anegundi  is  associated  with  the
kingdom  of  Angad,  son  of  Vali.  The  Anjaneya  Parvata,  a  hill  to  the  west  of  Anegundi,  is  the  fabled
birthplace of Hanuman.

Hampi is also linked with the river goddess Pampā and the legend of her marriage to Lord Virupaksha or Shiva. Each year, in the month of Chaitra (March‐April), this marriage is re‐enacted, with the priests of  Virupaksha  temple  devoutly  performing  every  ritual  from  Phalapūjā  (betrothal)  to  Kalyānotsava (marriage) in the temple.

The  Sacred  Center  of  the  city  lies  south  of  the  Tungabhadra  River,  and  it  is  dominated  by  four  large complexes  of  the  Virupaksha,  Krishna,  Tiruvengalanatha  (Achyutaraya)  and  Vitthala  temples.  The major  temples  are  either  close  to  cardinality,  departing  by  an  average  of  10’,  or  are  oriented  to  major features of the sacred landscape.

Further  south  of  the  Sacred  Center  is  the  Royal  Center,  which  is  divided  into  the  public  and  private realms. The division is achieved by a north‐south axis, which passes almost precisely between the kingʹs 100‐column audience hall in the east and the queenʹs large palace in the west. The Ramachandra temple pierces the axis by connecting the private and the public domains. In the homology of the king and the deity, the king is able to inhere in him the royalty and divinity of Rama.

The Virabhadra temple is on the summit of Matanga hill, which is the center of the vāstu‐mandala and
the symbolic source of protection that extended outward from it along radial lines.  As viewed from a
point midway between the audience hall and the queen’s palace, the 
shikhara of the Virabhadra lies only
4 minutes  of  arc  (4’)  from  true  north.  The  ceremonial  gateway  in  the  corridor  west  of  Ramachandra temple  joined  with  the  summit  of  Matanga  hill  departs  from  true  north  by  0.6  minutes  of  arc  (0.6’) (Malville, 2000).

The  orientations  of  the  major  axes  of  the  small  temples,  shrines,  and  palaces  of  the  urban  core  are  in
marked  contrast  to  those.  The  smaller  structures  are  rotated  away  from  cardinality  for  the  four
directions by 17o, suggesting that they were influenced by the position of the rising sun on the morning when it crosses the zenith.
The bazaar streets of the Virupaksha, Vitthala and Krishna temples are set between 13 and 15 degrees south of east. Malville (2000) speculates that there may be some link between these orientations and the rising point of the star Sirius.



                                                                                                                                                                                          








8.  Conclusions
Interest in archaeoastronomy and art, as connected to temples and ancient monuments, has increased in
India as the country’s prosperity has increased. This increase is also owing to the major archaeological
discoveries that have been made in the past few decades and the importance of temple tourism.
The principal authority over significant sites is the Indian Archaeological Survey of India (ASI) and its
sister institutions that function at the state level as Departments of Archaeology and Museums. In 1976,
the Indian Government initiated projects to excavate three great medieval cities: Fatehpur Sikri in Uttar
Pradesh,  Champaner  in  Gujarat,  and  Vijayanagara  in  Karnataka,  which  are  UNESCO  World  Heritage
sites.  The  wealth  of  discoveries  made  in  these  cities  is  strengthening  the  movement  to  expose  and
preserve  other  sites  in  the  country.  The  efforts  at  excavation,  conservation,  and  research  can  only  be
expected to increase. In particular, greater attention will be given to the archaeoastronomical aspects of
the monuments.
Acknowledgements. I am thankful to R. Balasubramaniam, Michel Danino, McKim Malville, and Rana Singh for their advice. The essay is dedicated to the memory of R. Balasubramaniam who passed away in December 2009.

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© Subhash Kak, December 2009

















                                                                                                                                                                                          


Tuesday, 31 January 2012

BHASKARACHARYA

Thanks to: http://veda.wikidot.com.

This post is just an introduction to one of the greatest astronomer of India Bhaskarachārya! I am in search for his works in detail, will post them here later. I am taking a small step to bring out the works by Indians on field of Astronomy especially solar astronomy. 

Life Cycles of the Universe

The Indians view that the Universe has no beginning or end, but follows a cosmic creation and dissolution. Indians are the one who propounds the idea of life-cycles of the universe. It suggests that the universe undergoes an infinite number of deaths and rebirths. Indians views the universe as without a beginning (anadi = beginning-less) or an end (ananta = end-less). Rather the universe is projected in cycles. Hindu scriptures refer to time scales that vary from ordinary earth day and night to the day and night of the Brahma that are a few billion earth years long.

According to Carl Sagan,


"Millenniums before Europeans were willing to divest themselves of the Biblical idea that the world was a few thousand years old, the Mayans were thinking of millions and the Indians billions".

Continues Carl Sagan,

    "… is the only religion in which the time scales correspond… to those of modern scientific cosmology."

Its cycles run from our ordinary day and night to a day and night of the Brahma, 8.64 billion years long, longer than the age of the Earth or the Sun and about half the time since the Big Bang". One day of Brahma is worth a thousand of the ages (yuga) known to humankind; as is each night." Thus each kalpa is worth one day in the life of Brahma, the God of creation. In other words, the four ages of the mahayuga must be repeated a thousand times to make a "day to Brahma", a unit of time that is the equivalent of 4.32 billion human years, doubling which one gets 8.64 billion years for a Brahma day and night. This was later theorized (possibly independently) by Aryabhata in the 6th century. The cyclic nature of this analysis suggests a universe that is expanding to be followed by contraction… a cosmos without end. This, according to modern physicists is not impossibility.

Bhaskara II or Bhaskarachārya was an Indian mathematician and astronomer who extended Brahmagupta's work on number systems. He was born near Bijjada Bida (in present day Bijapur district, Karnataka state, South India) into the Deshastha Brahmin family. Bhaskara was head of an astronomical observatory at Ujjain, the leading mathematical centre of ancient India. His predecessors in this post had included both the noted Indian mathematician Brahmagupta (598–c. 665) and Varahamihira. He lived in the Sahyadri region. It has been recorded that his great-great-great-grandfather held a hereditary post as a court scholar, as did his son and other descendants. His father Mahesvara was as an astrologer, who taught him mathematics, which he later passed on to his son Loksamudra. Loksamudra's son helped to set up a school in 1207 for the study of Bhāskara's writings

Bhaskara (1114 – 1185) (also known as Bhaskara II and Bhaskarachārya)

Bhaskaracharya's work in Algebra, Arithmetic and Geometry catapulted him to fame and immortality. His renowned mathematical works called Lilavati" and Bijaganita are considered to be unparalleled and a memorial to his profound intelligence. Its translation in several languages of the world bear testimony to its eminence. In his treatise Siddhant Shiromani he writes on planetary positions, eclipses, cosmography, mathematical techniques and astronomical equipment. In the Surya Siddhant he makes a note on the force of gravity:

    "Objects fall on earth due to a force of attraction by the earth. Therefore, the earth, planets, constellations, moon, and sun are held in orbit due to this attraction."

Bhaskaracharya was the first to discover gravity, 500 years before Sir Isaac Newton. He was the champion among mathematicians of ancient and medieval India. His works fired the imagination of Persian and European scholars, who through research on his works earned fame and popularity. Some say Mayans and Chinese too know of gravity some 2000 years before him.

Birth and Education of Bhaskaracharya

Ganesh Daivadnya has bestowed a very apt title on Bhaskaracharya. He has called him ‘Ganakchakrachudamani’, which means, ‘a gem among all the calculators of astronomical phenomena.’ Bhaskaracharya himself has written about his birth, his place of residence, his teacher and his education, in Siddhantashiromani as follows, ‘A place called ‘Vijjadveed’, which is surrounded by Sahyadri ranges, where there are scholars of three Vedas, where all branches of knowledge are studied, and where all kinds of noble people reside, a brahmin called Maheshwar was staying, who was born in Shandilya Gotra (in Hindu religion, Gotra is similar to lineage from a particular person, in this case sage Shandilya), well versed in Shroud (originated from ‘Shut’ or ‘Vedas’) and ‘Smart’ (originated from ‘Smut’) Dharma, respected by all and who was authority in all the branches of knowledge. I acquired knowledge at his feet’.

From this verse it is clear that Bhaskaracharya was a resident of Vijjadveed and his father Maheshwar taught him mathematics and astronomy. Unfortunately today we have no idea where Vijjadveed was located. It is necessary to ardently search this place which was surrounded by the hills of Sahyadri and which was the centre of learning at the time of Bhaskaracharya. He writes about his year of birth as follows,
‘I was born in Shake 1036 (1114 AD) and I wrote Siddhanta Shiromani when I was 36 years old.’

Bhaskaracharya has also written about his education. Looking at the knowledge, which he acquired in a span of 36 years, it seems impossible for any modern student to achieve that feat in his entire life. See what Bhaskaracharya writes about his education,

    ‘I have studied eight books of grammar, six texts of medicine, six books on logic, five books of mathematics, four Vedas, five books on Bharat Shastras, and two Mimansas’.

Bhaskaracharya calls himself a poet and most probably he was Vedanti, since he has mentioned ‘Parambrahman’ in that verse.
Siddhanta Shriomani

Bhaskaracharya wrote Siddhanta Shiromani in 1150 AD when he was 36 years old. This is a mammoth work containing about 1450 verses. It is divided into four parts, Lilawati, Beejaganit, Ganitadhyaya and Goladhyaya. In fact each part can be considered as separate book. The numbers of verses in each part are as follows, Lilawati has 278, Beejaganit has 213, Ganitadhyaya has 451 and Goladhyaya has 501 verses.
One of the most important characteristic of Siddhanta Shiromani is it consists of simple methods of calculations from Arithmetic to Astronomy. Essential knowledge of ancient Indian Astronomy can be acquired by reading only this book. Siddhanta Shiromani has surpassed all the ancient books on astronomy in India. After Bhaskaracharya nobody could write excellent books on mathematics and astronomy in lucid language in India. In India, Siddhanta works used to give no proofs of any theorem. Bhaskaracharya has also followed the same tradition.

Lilawati is an excellent example of how a difficult subject like mathematics can be written in poetic language. Lilawati has been translated in many languages throughout the world. When British Empire became paramount in India, they established three universities in 1857, at Bombay, Calcutta and Madras. Till then, for about 700 years, mathematics was taught in India from Bhaskaracharya’s Lilawati and Beejaganit. No other textbook has enjoyed such long lifespan.
Bhaskara's contributions to mathematics

Lilawati and Beejaganit together consist of about 500 verses. A few important highlights of Bhaskar's mathematics are as follows:
Terms for numbers

In English, cardinal numbers are only in multiples of 1000. They have terms such as thousand, million, billion, trillion, quadrillion etc. Most of these have been named recently. However, Bhaskaracharya has given the terms for numbers in multiples of ten and he says that these terms were coined by ancients for the sake of positional values. Bhaskar's terms for numbers are as follows:

eka(1), dasha(10), shata(100), sahastra(1000), ayuta(10,000), laksha(100,000), prayuta (1,000,000=million), koti(107), arbuda(108), abja(109=billion), kharva (1010), nikharva (1011), mahapadma (1012=trillion), shanku(1013), jaladhi(1014), antya(1015=quadrillion), Madhya (1016) and parardha(1017).
Kuttak

Kuttak is nothing but the modern indeterminate equation of first order. The method of solution of such equations was called as ‘pulveriser’ in the western world. Kuttak means to crush to fine particles or to pulverize. There are many kinds of Kuttaks. Let us consider one example.

In the equation, ax + b = CY, a and b are known positive integers. We want to also find out the values of x and y in integers. A particular example is, 100x +90 = 63y

Bhaskaracharya gives the solution of this example as, x = 18, 81, 144, 207… And y=30, 130, 230, 330…
Indian Astronomers used such kinds of equations to solve astronomical problems. It is not easy to find solutions of these equations but Bhaskara has given a generalized solution to get multiple answers.
Chakrawaal

Chakrawaal is the “indeterminate equation of second order” in western mathematics. This type of equation is also called Pell’s equation. Though the equation is recognized by his name Pell had never solved the equation. Much before Pell, the equation was solved by an ancient and eminent Indian mathematician, Brahmagupta (628 AD). The solution is given in his Brahmasphutasiddhanta. Bhaskara modified the method and gave a general solution of this equation. For example, consider the equation 61x2 + 1 = y2. Bhaskara gives the values of x = 22615398 and y = 1766319049

There is an interesting history behind this very equation. The Famous French mathematician Pierre de Fermat (1601-1664) asked his friend Bessy to solve this very equation. Bessy used to solve the problems in his head like present day Shakuntaladevi. Bessy failed to solve the problem. After about 100 years another famous French mathematician solved this problem. But his method is lengthy and could find a particular solution only, while Bhaskara gave the solution for five cases. In his book ‘History of mathematics’, see what Carl Boyer says about this equation,

‘In connection with the Pell’s equation ax2 + 1 = y2, Bhaskara gave particular solutions for five cases, a = 8, 11, 32, 61, and 67, for 61x2 + 1 = y2, for example he gave the solutions, x = 226153980 and y = 1766319049, this is an impressive feat in calculations and its verifications alone will tax the efforts of the reader’

Henceforth the so-called Pell’s equation should be recognized as ‘Brahmagupta-Bhaskaracharya equation’.
Simple mathematical methods

Bhaskara has given simple methods to find the squares, square roots, cube, and cube roots of big numbers. He has proved the Pythagoras theorem in only two lines. The famous Pascal Triangle was Bhaskara’s ‘Khandameru’. Bhaskara has given problems on that number triangle. Pascal was born 500 years after Bhaskara. Several problems on permutations and combinations are given in Lilawati. Bhaskar. He has called the method ‘ankapaash’. Bhaskara has given an approximate value of PI as 22/7 and more accurate value as 3.1416. He knew the concept of infinity and called it as ‘khahar rashi’, which means ‘anant’. It seems that Bhaskara had not notions about calculus, One of his equations in modern notation can be written as, d (sin (w)) = cos (w) dw.
A Summary of Bhaskara's contributions
    A proof of the Pythagorean Theorem by calculating the same area in two different ways and then canceling out terms to get a² + b² = c².

    In Lilavati, solutions of quadratic, cubic and quartic indeterminate equations.

    Solutions of indeterminate quadratic equations (of the type ax² + b = y²).

    Integer solutions of linear and quadratic indeterminate equations (Kuttaka). The rules he gives are (in effect) the same as those given by the Renaissance European mathematicians of the 17th century

    A cyclic Chakravala method for solving indeterminate equations of the form ax² + bx + c = y. The solution to this equation was traditionally attributed to William Brouncker in 1657, though his method was more difficult than the chakravala method.

    His method for finding the solutions of the problem x² − ny² = 1 (so-called "Pell's equation") is of considerable interest and importance.

    Solutions of Diophantine equations of the second order, such as 61x² + 1 = y². This very equation was posed as a problem in 1657 by the French mathematician Pierre de Fermat, but its solution was unknown in Europe until the time of Euler in the 18th century.

    Solved quadratic equations with more than one unknown, and found negative and irrational solutions.

    Preliminary concept of mathematical analysis.

    Preliminary concept of infinitesimal calculus, along with notable contributions towards integral calculus.

    Conceived differential calculus, after discovering the derivative and differential coefficient.

    Stated Rolle's theorem, a special case of one of the most important theorems in analysis, the mean value theorem. Traces of the general mean value theorem are also found in his works.

    Calculated the derivatives of trigonometric functions and formulae. (See Calculus section below.)

    In Siddhanta Shiromani, Bhaskara developed spherical trigonometry along with a number of other trigonometric results.
Arithmetic

Bhaskara's arithmetic text Lilavati covers the topics of definitions, arithmetical terms, interest computation, arithmetical and geometrical progressions, plane geometry, solid geometry, the shadow of the gnomon, methods to solve indeterminate equations, and combinations.

Lilavati is divided into 13 chapters and covers many branches of mathematics, arithmetic, algebra, geometry, and a little trigonometry and mensuration. More specifically the contents include:

    Definitions.
    Properties of zero (including division, and rules of operations with zero).
    Further extensive numerical work, including use of negative numbers and surds.
    Estimation of π.
    Arithmetical terms, methods of multiplication, and squaring.
    Inverse rule of three, and rules of 3, 5, 7, 9, and 11.
    Problems involving interest and interest computation.
    Arithmetical and geometrical progressions.
    Plane (geometry).
    Solid geometry.
    Permutations and combinations.
    Indeterminate equations (Kuttaka), integer solutions (first and second order). His contributions to this topic are particularly important, since the rules he gives are (in effect) the same as those given by the renaissance European mathematicians of the 17th century, yet his work was of the 12th century. Bhaskara's method of solving was an improvement of the methods found in the work of Aryabhata and subsequent mathematicians.

His work is outstanding for its systemisation, improved methods and the new topics that he has introduced. Furthermore the Lilavati contained excellent recreative problems and it is thought that Bhaskara's intention may have been that a student of 'Lilavati' should concern himself with the mechanical application of the method.
Algebra

His Bijaganita ("Algebra") was a work in twelve chapters. It was the first text to recognize that a positive number has two square roots (a positive and negative square root). His work Bijaganita is effectively a treatise on algebra and contains the following topics:

    Positive and negative numbers.
    Zero.
    The 'unknown' (includes determining unknown quantities).
    Determining unknown quantities.
    Surds (includes evaluating surds).
    Kuttaka (for solving indeterminate equations and Diophantine equations).
    Simple equations (indeterminate of second, third and fourth degree).
    Simple equations with more than one unknown.
    Indeterminate quadratic equations (of the type ax² + b = y²).
    Solutions of indeterminate equations of the second, third and fourth degree.
    Quadratic equations.
    Quadratic equations with more than one unknown.
    Operations with products of several unknowns.

Bhaskara derived a cyclic, chakravala method for solving indeterminate quadratic equations of the form ax² + bx + c = y. Bhaskara's method for finding the solutions of the problem Nx² + 1 = y² (the so-called "Pell's equation") is of considerable importance.

He gave the general solutions of:

    Pell's equation using the chakravala method.
    The indeterminate quadratic equation using the chakravala method.

He also solved:

    Cubic equations.
    Quartic equations.
    Indeterminate cubic equations.
    Indeterminate quartic equations.
    Indeterminate higher-order polynomial equations.

Trigonometry

The Siddhanta Shiromani (written in 1150) demonstrates Bhaskara's knowledge of trigonometry, including the sine table and relationships between different trigonometric functions. He also discovered spherical trigonometry, along with other interesting trigonometrical results. In particular Bhaskara seemed more interested in trigonometry for its own sake than his predecessors who saw it only as a tool for calculation. Among the many interesting results given by Bhaskara, discoveries first found in his works include the now well known results for \sin\left(a + b\right) and \sin\left(a - b\right) :
Calculus

His work, the Siddhanta Shiromani, is an astronomical treatise and contains many theories not found in earlier works. Preliminary concepts of infinitesimal calculus and mathematical analysis, along with a number of results in trigonometry, differential calculus and integral calculus that are found in the work are of particular interest.

Evidence suggests Bhaskara was acquainted with some ideas of differential calculus. It seems, however, that he did not understand the utility of his researches, and thus historians of mathematics generally neglect this achievement. Bhaskara also goes deeper into the 'differential calculus' and suggests the differential coefficient vanishes at an extremum value of the function, indicating knowledge of the concept of 'infinitesimals'.

    There is evidence of an early form of Rolle's theorem in his work:
        If f\left(a\right) = f\left(b\right) = 0 then f'\left(x\right) = 0 for some \ x with \ a < x < b

    He gave the result that if x \approx y then \sin(y) - \sin(x) \approx (y - x)\cos(y), thereby finding the derivative of sine, although he never developed the general concept of differentiation.
        Bhaskara uses this result to work out the position angle of the ecliptic, a quantity required for accurately predicting the time of an eclipse.

    In computing the instantaneous motion of a planet, the time interval between successive positions of the planets was no greater than a truti, or a 1⁄33750 of a second, and his measure of velocity was expressed in this infinitesimal unit of time.

    He was aware that when a variable attains the maximum value, its differential vanishes.

    He also showed that when a planet is at its farthest from the earth, or at its closest, the equation of the centre (measure of how far a planet is from the position in which it is predicted to be, by assuming it is to move uniformly) vanishes. He therefore concluded that for some intermediate position the differential of the equation of the centre is equal to zero. In this result, there are traces of the general mean value theorem, one of the most important theorems in analysis, which today is usually derived from Rolle's theorem. The mean value theorem was later found by Parameshvara in the 15th century in the Lilavati Bhasya, a commentary on Bhaskara's Lilavati.

Madhava (1340-1425) and the Kerala School mathematicians (including Parameshvara) from the 14th century to the 16th century expanded on Bhaskara's work and further advanced the development of calculus in India.
Astronomy

Using an astronomical model developed by Brahmagupta in the 7th century, Bhaskara accurately defined many astronomical quantities, including, for example, the length of the sidereal year, the time that is required for the Earth to orbit the Sun, as 365.2588 days which is same as in Suryasiddhanta. The modern accepted measurement is 365.2563 days; it means that Bhaskaracharya was off by only 0.0002%.

His mathematical astronomy text Siddhanta Shiromani is written in two parts: the first part on mathematical astronomy and the second part on the sphere.

The twelve chapters of the first part cover topics such as:

    Mean longitudes of the planets.
    True longitudes of the planets.
    The three problems of diurnal rotation.
    Syzygies.
    Lunar eclipses.
    Solar eclipses.
    Latitudes of the planets.
    Sunrise equation
    The Moon's crescent.
    Conjunctions of the planets with each other.
    Conjunctions of the planets with the fixed stars.
    The patas of the Sun and Moon.

The second part contains thirteen chapters on the sphere. It covers topics such as:

    Praise of study of the sphere.
    Nature of the sphere.
    Cosmography and geography.
    Planetary mean motion.
    Eccentric epicyclic model of the planets.
    The armillary sphere.
    Spherical trigonometry.
    Ellipse calculations.[citation needed]
    First visibilities of the planets.
    Calculating the lunar crescent.
    Astronomical instruments.
    The seasons.
    Problems of astronomical calculations.

Ganitadhyaya and Goladhyaya of Siddhanta Shiromani are devoted to astronomy. All put together there are about 1000 verses. Almost all aspects of astronomy are considered in these two books. Some of the highlights are worth mentioning.
Earth’s circumference and diameter

Bhaskara has given a very simple method to determine the circumference of the Earth. According to this method, first find out the distance between two places, which are on the same longitude. Then find the correct latitudes of those two places and difference between the latitudes. Knowing the distance between two latitudes, the distance that corresponds to 360 degrees can be easily found, which the circumference of is the Earth. For example, Satara and Kolhapur are two cities on almost the same longitude. The difference between their latitudes is one degree and the distance between them is 110 kilometers. Then the circumference of the Earth is 110 X 360 = 39600 kilometers. Once the circumference is fixed it is easy to calculate the diameter. Bhaskara gave the value of the Earth’s circumference as 4967 ‘yojane’ (1 yojan = 8 km), which means 39736 kilometers. His value of the diameter of the Earth is 1581 yojane i.e. 12648 km. The modern values of the circumference and the diameter of the Earth are 40212 and 12800 kilometers respectively. The values given by Bhaskara are astonishingly close.
Aksha kshetre

For astronomical calculations, Bhaskara selected a set of eight right angle triangles, similar to each other. The triangles are called ‘aksha kshetre’. One of the angles of all the triangles is the local latitude. If the complete information of one triangle is known, then the information of all the triangles is automatically known. Out of these eight triangles, complete information of one triangle can be obtained by an actual experiment. Then using all eight triangles virtually hundreds of ratios can be obtained. This method can be used to solve many problems in astronomy.
Geocentric parallax

Ancient Indian Astronomers knew that there was a difference between the actual observed timing of a solar eclipse and timing of the eclipse calculated from mathematical formulae. This is because calculation of an eclipse is done with reference to the center of the Earth, while the eclipse is observed from the surface of the Earth. The angle made by the Sun or the Moon with respect to the Earth’s radius is known as parallax. Bhaskara knew the concept of parallax, which he has termed as ‘lamban’. He realized that parallax was maximum when the Sun or the Moon was on the horizon, while it was zero when they were at zenith. The maximum parallax is now called Geocentric Horizontal Parallax. By applying the correction for parallax exact timing of a solar eclipse from the surface of the Earth can be determined.
Yantradhyay

In this chapter of Goladhyay, Bhaskar has discussed eight instruments, which were useful for observations. The names of these instruments are, Gol yantra (armillary sphere), Nadi valay (equatorial sun dial), Ghatika yantra, Shanku (gnomon), Yashti yantra, Chakra, Chaap, Turiya, and Phalak yantra. Out of these eight instruments Bhaskara was fond of Phalak yantra, which he made with skill and efforts. He argued that ‘ this yantra will be extremely useful to astronomers to calculate accurate time and understand many astronomical phenomena’. Bhaskara’s Phalak yantra was probably a precursor of the ‘astrolabe’ used during medieval times.
Dhee yantra

This instrument deserves to be mentioned specially. The word ‘dhee’ means ‘ Buddhi’ i.e. intelligence. The idea was that the intelligence of human being itself was an instrument. If an intelligent person gets a fine, straight and slender stick at his/her disposal he/she can find out many things just by using that stick. Here Bhaskara was talking about extracting astronomical information by using an ordinary stick. One can use the stick and its shadow to find the time, to fix geographical north, south, east, and west. One can find the latitude of a place by measuring the minimum length of the shadow on the equinoctial days or pointing the stick towards the North Pole. One can also use the stick to find the height and distance of a tree even if the tree is beyond a lake.

A GLANCE AT THE ASTRONOMICAL ACHIEVEMENTS OF BHASKARACHARYA

    The Earth is not flat, has no support and has a power of attraction.
    The north and south poles of the Earth experience six months of day and six months of night.
    One day of Moon is equivalent to 15 earth-days and one night is also equivalent to 15 earth-days.
    Earth’s atmosphere extends to 96 kilometres and has seven parts.     There is a vacuum beyond the Earth’s atmosphere.
    He had knowledge of precession of equinoxes. He took the value of its shift from the first point of Aries as 11 degrees. However, at that time it was about 12 degrees.

    Ancient Indian Astronomers used to define a reference point called ‘Lanka’. It was defined as the point of intersection of the longitude passing through Ujjaini and the equator of the Earth. Bhaskara has considered three cardinal places with reference to Lanka, the Yavakoti at 90 degrees east of Lanka, the Romak at 90 degrees west of Lanka and Siddhapoor at 180 degrees from Lanka. He then accurately suggested that, when there is a noon at Lanka, there should be sunset at Yavkoti and sunrise at Romak and midnight at Siddhapoor.

 Bhaskaracharya had accurately calculated apparent orbital periods of the Sun and orbital periods of Mercury, Venus, and Mars. There is slight difference between the orbital periods he calculated for Jupiter and Saturn and the corresponding modern values.

Engineering

The earliest reference to a perpetual motion machine date back to 1150, when Bhāskara II described a wheel that he claimed would run forever.

Bhāskara II used a measuring device known as Yasti-yantra. This device could vary from a simple stick to V-shaped staffs designed specifically for determining angles with the help of a calibrated scale.
References

 Pingree, David Edwin. Census of the Exact Sciences in Sanskrit. Volume 146. American Philosophical Society, 1970. ISBN 9780871691460
    BHASKARACHARYA, Written by Prof. Mohan Apte

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The Madras Quartet — Radha and Her Circle of Physics

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