Indian Spherical Trignometry
Sine ), Kotijya (
Cosine ) and Utkram Jya (
Versine ) are the three trignometric functions introduced by the Indian
astronomers and mathematicians.
order to compute the celestial longitudes of planets, these functions were
used by the trinity of Indian Astronomy, Bhaskara, Brahnmagupta and Aryabhata.
the Radius or R
Circumference = 2 Pi r
360 = 2 Pi r
r = 360/2Pi in degrees
r = 180/Pi
minutes = 21600 minutes
= 21600 /2 Pi = 3438 minutes = 3438*60 = 206265 seconds
degree is 60 minutes and one minute is 60 seconds and hence one degree is 3600
in seconds will be ((360/2pi)* 3600 ) Vikalas or 206265 seconds. This figure
206265 is known as the Magic Figure of Astronomy.
Arybhata based his famous Sine tables on R = 3438 minutes and he gave the
Jya ( modern Sine ) values of Kakshyas of 3 degrees and 45 minuts
( 30 degrees / 8 ) .
= R Sin
Kotijya = R Cos
arc sine of the angle is Bhujachapa
The arc cosine of the angle is Kotichapa
The arc tangent of the angle is Sparshachapa
Bramhmagupta meant 5 degrees of a sign of 30 degrees. Hence a Zodiacal Sign
consists of 6 Jyas ( 30 degrees ) . The Zodiac of 360 degrees was divided into
4 quarters of 90 degrees each. Three Jyas of
30 degrees each becomes a quadrant of the Zodiac and was called Thrijya. Thrijya is
also the Radius, theVyasardha.
magnum opus, the Brahmasphuta
Siddhanta was translated
by the Arabs as As Sind Hind. Jya became jiba and
Kotijya became kojiba in
Arabic. It was translated into Latin as sinus ( meaning ” bosom ” ). So
Sinus and Co-sinus when translated into English became Sine and Cosine !
The Aryabhateeyam of
Aryabhata was translated by the Arabs as Al
Arjabhat.Trignometry is derived from the Sanskrit Thrikonamithi and
Geometry from Jyamithi
are four major methods of calculation in Astronomy
calculated along the Zodiac or Ecliptic - The Ecliptic System
Longitudes calculated along the Celestial Equator - The Equatorial System
Longitudes calculated along the Celestial Horizon - The Horizontal System
Longitudes calculated along the Celestial Meridian - The Meridian System
measured along the Kranti
Vritta, the Ecliptic or Bha
Chakra , the Zodiac is known as Kranti
Vritteeya Sphuta, the true longitude of the planet.
measured along the Vishuvat
Vritta, the Celestial
Equator is known asVishuvat
Vritteeya Sphuta, Right Ascension.
Ascendent and Astha
Lagna, the Descendent are measured along the Celestial Horizon,The Kshitija
Lagna, the MC and the Patala
Lagna, the IC are
measured along the Celestial Meridian, the Nadi
order to compute the celestial longitudes of planets, first the Graha
Madhyam, the mean longitude of the planet is computed.
have to understand that the planets traverse in elliptical orbits. If their
orbits are circular, then there is no need for jya
samskaras ( trignometric
the mean longitudes of the planets are ascertained, then we first start with
the First Jya Samskara, the first trignometric correction. Manda
Jya means Sin M in
Kepler Equation is M = E - e Sin E, where e is eccentricity and E is the
Eccentric Anomaly, an auxilary angle in Kepler’s equations.
Kepler who brought in an auxiliary angle ( E, the Eccentric Anomaly ), Indian
Astronomy uses Vikshepa
Vritta, an auxiliary
circle. The mean longitude of a planet reduced by Manda Kriya is the Vikshepa
Vritteeya Sphuta, the once corrected longitude of the planet.
Western astronomers compute the celestial longitudes using the formula Theta =
v + w ( Celestial Longitude = True Anomaly + the Argument of Perihelion ),
Indian astronomers use the Triune Trignometric Method. Longitudes are
corrected thrice using Manda
Kriya and Sheegra
perturbations of planets
planets have perturbations. Moon has 300 perturbations, of which 14 are major.
Hence for 14 perturbations, 14 jya
samskaras have to be done
Dasa Jya Samskara). The largest of them is the Evection. There are others
like the Variation, the Annual Equation and the Parallactic Equation. When 14
trignometric corrections are done, we get the Reduced Longitude of the Moon,
has five major perturbations ( guror
pancha kendrani bhavanthi ) and
Saturn has six. So five jya
samskaras and six jya
samskaras have to be done
for Jupiter and Saturn, before commencing the Triune Trignometric Method.