Transcendence

Consisting of 16 basic aphorisms or Sutras, Vedic Mathematics is a system of Maths which prevailed in ancient India. Composed by Bharati Krishna Thirtha, these 16 sutras help one to do faster maths. The first aphorism is this "Whatever the extent of its deficiency, lessen it still further to that very extent; and also set up the square (of that deficiency)" When computing the square of 9, as the nearest power of 10 is 9, let us take 10 as our base. As 9 is 1 less than 10, we can decrease it by the deficiency = 9-1 =8. This is the leftmost digit On the right hand put deficiency^2, which is 1^2. Hence the square of nine is 81. For numbers above 10, instead of looking at the deficit we look at the surplus. For example: 11^2 = (11+1)*10+1^2 = 121 12^2 = (12+2)*10+2^2 = 144 14^2 = ( 14+4)*10+4^2 = 196 25^2 = ((25+5)*2)*10+5^2 = 625 35^2= ((35+5)*3)*10+5^2 = 1225

Consisting of 16 basic aphorisms or Sutras, Vedic Mathematics is a system of Maths which prevailed in ancient India. Composed by Bharati Krishna Thirtha, these 16 sutras help one to do faster maths. The first aphorism is this "Whatever the extent of its deficiency, lessen it still further to that very extent; and also set up the square (of that deficiency)" When computing the square of 9, as the nearest power of 10 is 9, let us take 10 as our base. As 9 is 1 less than 10, we can decrease it by the deficiency = 9-1 =8. This is the leftmost digit On the right hand put deficiency^2, which is 1^2. Hence the square of nine is 81. For numbers above 10, instead of looking at the deficit we look at the surplus. For example: 11^2 = (11+1)*10+1^2 = 121 12^2 = (12+2)*10+2^2 = 144 14^2 = ( 14+4)*10+4^2 = 196 25^2 = ((25+5)*2)*10+5^2 = 625 35^2= ((35+5)*3)*10+5^2 = 1225

In India, mathematics is related to Philosophy. We can find mathematical concepts like Zero ( Shoonyavada ), One ( Advaitavada ) and Infinity (Poornavada ) in Philosophia Indica. The Sine Tables of Aryabhata and Madhava, which gives correct sine values or values of 24 R Sines, at intervals of 3 degrees 45 minutes and the trignometric tables of Brahmagupta, which gives correct sine and tan values for every 5 degrees influenced Christopher Clavius, who headed the Gregorian Calender Reforms of 1582. These correct trignometric tables solved the problem of the three Ls, ( Longitude, Latitude and Loxodromes ) for the Europeans, who were looking for solutions to their navigational problem ! It is said that Matteo Ricci was sent to India for this purpose and the Europeans triumphed with Indian knowledge ! The Western mathematicians have indeed lauded Indian Maths & Astronomy. Here are some quotations from maths geniuses about the long forgotten Indian Maths ! In his famou

In India, mathematics is related to Philosophy. We can find mathematical concepts like Zero ( Shoonyavada ), One ( Advaitavada ) and Infinity (Poornavada ) in Philosophia Indica. The Sine Tables of Aryabhata and Madhava, which gives correct sine values or values of 24 R Sines, at intervals of 3 degrees 45 minutes and the trignometric tables of Brahmagupta, which gives correct sine and tan values for every 5 degrees influenced Christopher Clavius, who headed the Gregorian Calender Reforms of 1582. These correct trignometric tables solved the problem of the three Ls, ( Longitude, Latitude and Loxodromes ) for the Europeans, who were looking for solutions to their navigational problem ! It is said that Matteo Ricci was sent to India for this purpose and the Europeans triumphed with Indian knowledge ! The Western mathematicians have indeed lauded Indian Maths & Astronomy. Here are some quotations from maths geniuses about the long forgotten Indian Maths ! In his famou

By means of the same argument, the circumference can be computed in another way too. That is as (follows): The first result should by the square root of the square of the diameter multiplied by twelve. From then on, the result should be divided by three (in) each successive (case). When these are divided in order by the odd numbers, beginning with 1, and when one has subtracted the (even) results from the sum of the odd, (that) should be the circumference. ( Yukti deepika commentary ) This quoted text specifies another formula for the computation of the circumference c of a circle having diameter d. This is as follows. c = SQRT(12 d^2 - SQRT(12 d^2/3.3 + sqrt(12 d^2)/3^2.5 - sqrt(12d^2)/3^3.7 +....... As c = Pi d , this equation can be rewritten as Pi = Sqrt(12( 1 - 1/3.3 + 1/3^2.5 -1/3^3.7 +...... This is obtained by substituting z = Pi/ 6 in the power series expansion for arctan (z). Pi/4 = 1 - 1/3 +1/5 -1/7+..... This is Madhava's formula for Pi, and this was disco

By means of the same argument, the circumference can be computed in another way too. That is as (follows): The first result should by the square root of the square of the diameter multiplied by twelve. From then on, the result should be divided by three (in) each successive (case). When these are divided in order by the odd numbers, beginning with 1, and when one has subtracted the (even) results from the sum of the odd, (that) should be the circumference. ( Yukti deepika commentary ) This quoted text specifies another formula for the computation of the circumference c of a circle having diameter d. This is as follows. c = SQRT(12 d^2 - SQRT(12 d^2/3.3 + sqrt(12 d^2)/3^2.5 - sqrt(12d^2)/3^3.7 +....... As c = Pi d , this equation can be rewritten as Pi = Sqrt(12( 1 - 1/3.3 + 1/3^2.5 -1/3^3.7 +...... This is obtained by substituting z = Pi/ 6 in the power series expansion for arctan (z). Pi/4 = 1 - 1/3 +1/5 -1/7+..... This is Madhava's formula for Pi, and this was disco

The inverse tangent series of Madhava is given in verse 2.206 – 2.209 in Yukti-dipika commentary (Tantrasamgraha-vyakhya) by Sankara Variar . It is also given by Jyeshtadeva in Yuktibhasha and a translation of the verses is given below. Now, by just the same argument, the determination of the arc of a desired sine can be (made). That is as follows: The first result is the product of the desired sine and the radius divided by the cosine of the arc. When one has made the square of the sine the multiplier and the square of the cosine the divisor, now a group of results is to be determined from the (previous) results beginning from the first. When these are divided in order by the odd numbers 1, 3, and so forth, and when one has subtracted the sum of the even(-numbered) results from the sum of the odd (ones), that should be the arc. Here the smaller of the sine and cosine is required to be considered as the desired (sine). Otherwise, there would be no termination of results even if repeat

The inverse tangent series of Madhava is given in verse 2.206 – 2.209 in Yukti-dipika commentary (Tantrasamgraha-vyakhya) by Sankara Variar . It is also given by Jyeshtadeva in Yuktibhasha and a translation of the verses is given below. Now, by just the same argument, the determination of the arc of a desired sine can be (made). That is as follows: The first result is the product of the desired sine and the radius divided by the cosine of the arc. When one has made the square of the sine the multiplier and the square of the cosine the divisor, now a group of results is to be determined from the (previous) results beginning from the first. When these are divided in order by the odd numbers 1, 3, and so forth, and when one has subtracted the sum of the even(-numbered) results from the sum of the odd (ones), that should be the arc. Here the smaller of the sine and cosine is required to be considered as the desired (sine). Otherwise, there would be no termination of results even if repeat

Madhava's cosine series is stated in verses 2.442 and 2.443 in Yukti-dipika commentary (Tantrasamgraha-vyakhya) by Sankara Variar. A translation of the verses follows. Multiply the square of the arc by the unit (i.e. the radius) and take the result of repeating that (any number of times). Divide (each of the above numerators) by the square of the successive even numbers decreased by that number and multiplied by the square of the radius. But the first term is (now)(the one which is) divided by twice the radius. Place the successive results so obtained one below the other and subtract each from the one above. These together give the śara as collected together in the verse beginning with stena, stri, etc. Let r denote the radius of the circle and s the arc-length. The following numerators are formed first: s.s^2, s.s^2.s^2 s.s^2.s^2.s^2 These are then divided by quantities specified in the verse. 1)s.s^2/(2^2-2)r^2, 2)s. s^2/(2^2-2)r^2. s^2/4^2-4)r^2 3)s.s^2/(2

The Madhava Trignometric series is one one of a series in a collection of infinite series expressions discovered by Madhava of Sangramagrama ( 1350-1425 ACE ), the founder of the Kerala School of Astronomy and Mathematics. These are the infinite series expansions of the Sine, Cosine and the ArcTangent functions and Pi. The power series expansions of sine and cosine functions are called the Madhava sine series and the Madhava cosine series. The power series expansion of the arctangent function is called the Madhava- Gregory series. The power series are collectively called as Madhava Taylor series. The formula for Pi is called the Madhava Newton series. One of his disciples, Sankara Variar had translated his verse in his Yuktideepika commentary on Tantrasamgraha-vyakhya, in verses 2.440 and 2.441 Multiply the arc by the square of the arc, and take the result of repeating that (any number of times). Divide (each of the above numerators) by the squares of the successive even n

The Madhava Trignometric series is one one of a series in a collection of infinite series expressions discovered by Madhava of Sangramagrama ( 1350-1425 ACE ), the founder of the Kerala School of Astronomy and Mathematics. These are the infinite series expansions of the Sine, Cosine and the ArcTangent functions and Pi. The power series expansions of sine and cosine functions are called the Madhava sine series and the Madhava cosine series. The power series expansion of the arctangent function is called the Madhava- Gregory series. The power series are collectively called as Madhava Taylor series. The formula for Pi is called the Madhava Newton series. One of his disciples, Sankara Variar had translated his verse in his Yuktideepika commentary on Tantrasamgraha-vyakhya, in verses 2.440 and 2.441 Multiply the arc by the square of the arc, and take the result of repeating that (any number of times). Divide (each of the above numerators) by the squares of the successive even n

We had warned against the adverse transits of Jupiter, adverse for both Dhoni and India. While Jove in the 10th indicates " loss of position and wandering about" for India, for Dhoni Jove transits the hostile 8th, indicating fall ! India whitewashed and humiliated, we can now understand. Sachin losing his 100th century also. Just shows how powerful the Nine Revolving Heavens are and how the physique and the psyche of the natives are affected. Now India will have to resurrect herself from the ruins! Good luck, India. Well done, England !

Natural Strength, Naisargika Bala, is the inherent property of a celestial object, which possesses the following properties 1) This force is constant for a celestial object, not varying in time. 2) This force is proportional to the size of the diameter of the planets. 3) This force is inversely proportional to the distance, r, from the Sun. 4) It increases in the order from the farthest planet to the nearest planet to the Sun. From Saturn,Jupiter, Mars, Venus, Mercury, Moon and Sun. 5) This Force is a major factor when planets are involved in Planetary War ( Graha Yuddha ), when their longitudes are more or less identical in the Ecliptic. Let F1 and F2 be the Naisargika Bala of planets 1 and 2 situated in the same distance, r, from earth. Then we have F1 = F(D1)/r F2 = F(D2)/r The ratio of the planetary Naisargika Bala is F1/F2 = F(D1)/F(D2) The forces are given by ( according to Newtonian modern theories) F1= ( M1M/r^2) F2= ( M2M/r^2) The ratio of the gravitatio

It is to the credit of Gooch that England owes its batting renaissance. Gooch gave tips to the English batsmen and you can see the result. Gooch scored 333 against India in one Test and he knows the methodology to counter Indian swing and spin. Mentally and physically he prepared the English batsmen to score massive scores. England did score heavily in Australia, where they trounced the Kangaroos in Tests. Now it is their turn to defeat India. India is now a laughing stock, despite Dravid's 35th ton, an unconquered 146 not out. India most probably will lose this Test and it is time to do some rethinking !

The Jesuits took the trignometric tables and planetary models from the Kerala School of Astronomy and Maths and exported it to Europe starting around 1560 in connection with the European navigational problem, says Dr Raju. Dr C K Raju was a professor Mathematics and played a leading role in the C-DAC team which built Param: India’s first parallel supercomputer. His ten year research included archival work in Kerala and Rome and was published in a book called " The Cultural Foundations of Mathematics". He has been a Fellow of the Indian Institute of Advanced Study and is a Professor of Computer Applications. “When the Europeans received the Indian calculus, they couldn’t understand it properly because the Indian philosophy of mathematics is different from the Western philosophy of mathematics. It took them about 300 years to fully comprehend its working. The calculus was used by Newton to develop his laws of physics,” opines Dr Raju. The Infinitesimal Calculus: How and

The Jesuits took the trignometric tables and planetary models from the Kerala School of Astronomy and Maths and exported it to Europe starting around 1560 in connection with the European navigational problem, says Dr Raju. Dr C K Raju was a professor Mathematics and played a leading role in the C-DAC team which built Param: India’s first parallel supercomputer. His ten year research included archival work in Kerala and Rome and was published in a book called " The Cultural Foundations of Mathematics". He has been a Fellow of the Indian Institute of Advanced Study and is a Professor of Computer Applications. “When the Europeans received the Indian calculus, they couldn’t understand it properly because the Indian philosophy of mathematics is different from the Western philosophy of mathematics. It took them about 300 years to fully comprehend its working. The calculus was used by Newton to develop his laws of physics,” opines Dr Raju. The Infinitesimal Calculus: How and

Many ancient cultures have contributed to the development of Astro Physics. Some examples are The Saros cycles of eclipses discovered by Egyptians The classification of stars by the Greeks Sunspot observations of the Chinese The phenomenon of Retrogression discovered by Babylonians In this context the Indian contribution to Astro Physics ( which includes Astronomy, Maths and Astrology ) is the the development of the ideas of planetary forces and differential equations to calculate the geocentric planetary longitudes, several centuries before the European Renaissance. Natural Strength is one of the Sixfold Strengths, Shad Balas and goes by the name Naisargika Bala . It is directly proportional to the size of the celestial bodies and inversely proportional to the geocentric distance. ( Horasara ). Naisargika Bala or Natural Strength is used to compare planetary physical forces. When two planets occupy the same, identical position in the Zodiac at a given instant of time

Motional strength is one the sixfold strengths, known as Cheshta Bala. This motional strength is computed by the formula Motional Strength = 0.33 ( Sheegrocha or Perigee - geocentric longitude of the planet ). This motional strength is known as Cheshta Bala. Differential Calculus is the science of rates of the change. If y is the longitude of the planet and t is time, then we have the differential equation ,dy/dt. During direct motion, we find that dy/dt > 0 and during retrogression dy/dt < 0. During backward motion of the planet ( retrogression) y decreases with time and during direct motion y increases with time. When there are turning points known as Vikalas or stationary points, we have dy/dt = 0 ( where planets like Mars will appear to be stationary for an observer on Earth ). The quantity in bracket is the Sheegra Anomaly, the Anomaly of Conjuction, the angular distance of the planet from the Sun. This Anomaly or Cheshta Bala is maximum at the center of the

Mean and true planetary longitudes in the Zodiac is computed by Nine Orbital Elements, in Indian Astronomy. Mean longitude of Planet, Graha Madhyama , M Daily Motion of the Mean Longitude, Madhyama Dina Gathi , Md Aphelion, Mandoccha , Ap Daily Motion of Aphelion, Mandoccha Dina Gathi , Apd Ascending Node, Patha , N Daily Motion of Ascending Node, Patha Dina Gath i, Nd Heliocentric Distance, Manda Karna , radius vector, mndk Maximum Latitude, L, Parama Vikshepa Eccentricity, Chyuthi ,e In Western Astronomy, we have six orbital elements Mean Anomaly, m Argument of Perihelion, w Eccentricity, e Ascending Node, N Inclination, i, inclinent of orbit Semi Major Axis, a With the Nine Orbital Elements, true geocentric longitude of the planet is computed, using multi step algorithms. There is geometrical equivalence between both the Epicycle and the Eccentric Models. The radius of the Epicycle, r = e, the distance of the Equant from the Observer.

We find Yuga cylces mentioned not only in astronomical works, but also in mythological works in India. Kali Yuga began on the midnight of 17th Feb 3102 BCE and the duration of this Kali Yuga is said to be 4.32 K solar years. Dwapara is 2*Kali Yuga years. Treta is 3*K Y and Krita Yuga is 4*K Y. Krita Treta Dwaparascha Kalischaiva Chaturyugam Divya Dwadasabhir varshai savadhanam niroopitham Thus an Equinoctial Cycle, Mahayuga is equal to 4+3+2+1 = 10 KYs. E C = 10 KYs. A Greater Equinoctial Cycle ( Manvantara ) = 71 Equinoctial Cycles There are cusps happening in between Manvantaras, each equal to a Krita Yuga in duration. A Krita is equal to 4 KYs or 2/5 of a Maha Yuga. Since there will 15 such cusps happening amongst the Fourteen Manvantaras, they are equal to 15*2/5 = 6 Mahayugas. Hence 14*71+6 = 1000 Mahayugas = 4.32 Billion Years Sahasra yuga paryantham Aharyal brahmano vidu Ratrim yugah sahasrantham The Ahoratra vido janah ( The Holy Geetha ). This

We find Yuga cylces mentioned not only in astronomical works, but also in mythological works in India. Kali Yuga began on the midnight of 17th Feb 3102 BCE and the duration of this Kali Yuga is said to be 4.32 K solar years. Dwapara is 2*Kali Yuga years. Treta is 3*K Y and Krita Yuga is 4*K Y. Krita Treta Dwaparascha Kalischaiva Chaturyugam Divya Dwadasabhir varshai savadhanam niroopitham Thus an Equinoctial Cycle, Mahayuga is equal to 4+3+2+1 = 10 KYs. E C = 10 KYs. A Greater Equinoctial Cycle ( Manvantara ) = 71 Equinoctial Cycles There are cusps happening in between Manvantaras, each equal to a Krita Yuga in duration. A Krita is equal to 4 KYs or 2/5 of a Maha Yuga. Since there will 15 such cusps happening amongst the Fourteen Manvantaras, they are equal to 15*2/5 = 6 Mahayugas. Hence 14*71+6 = 1000 Mahayugas = 4.32 Billion Years Sahasra yuga paryantham Aharyal brahmano vidu Ratrim yugah sahasrantham The Ahoratra vido janah ( The Holy Geetha ). This

We had warned in our columns that India is facing the adverse 10th transit of Jupiter. Dhoni is under the 8th Jovian transit. But that did not deter him to score 74 not out, but all his heroics were in vain, as India slid to an inglorious defeat. The 10th adverse Transit of Jove refers to "loss of position and wandering about". This is true of the One day World Champions, who will get brickbats and not the laurel, when they return home. Sending a second XI to the West Indies and a first XI to England, well, things wont work out that way. England was preparing hard and smashed India 3-0!

The Indian astronomers were interested in the computation of eclipses, of geocentric longitudes, the risings and settings of planets,which had relevance to the day to day activities of people. Did not Emerson say? "Astronomy is excellent, it should come down and give life its full value, and not rest amidst globes and spheres ". They were not bothered about proposing Models of the Universe and gaining publicity. But then they did discuss the geometrical model, the rationale of their computations. The above diagram explains the Geometric Model of Parameswara, another Kerala astronomer. Paramesvara and Nilakanta modified the Aryabhatan Model. By Sheegroccha , he meant the longitude of the Sun." Sheegrocham Sarvesham Ravir bhavathi ", he says is his book Bhatadeepika . For the interior planets, the longitude of the Sheegra correction is to be deducted from the Sun's longitude, Ravi Sphuta to get the Anomaly of Conjunction. The Manda Prathimandala i

The Indian astronomers were interested in the computation of eclipses, of geocentric longitudes, the risings and settings of planets,which had relevance to the day to day activities of people. Did not Emerson say? "Astronomy is excellent, it should come down and give life its full value, and not rest amidst globes and spheres ". They were not bothered about proposing Models of the Universe and gaining publicity. But then they did discuss the geometrical model, the rationale of their computations. The above diagram explains the Geometric Model of Parameswara, another Kerala astronomer. Paramesvara and Nilakanta modified the Aryabhatan Model. By Sheegroccha , he meant the longitude of the Sun." Sheegrocham Sarvesham Ravir bhavathi ", he says is his book Bhatadeepika . For the interior planets, the longitude of the Sheegra correction is to be deducted from the Sun's longitude, Ravi Sphuta to get the Anomaly of Conjunction. The Manda Prathimandala is th

India is dithering, with England scoring 456/3. Cook is undefeated with 182. Stauss was out for 87 and KP for 63. Eoin Morgan scored 44. Indian bowlers were made to look ordinary before the might of English batting. India is at a big psychological disadvantage, having lost both Tests. India's rating is bound to suffer because of such lack lustre performances. As we have outlined in our columns before, adverse Jupiterian transits have affected both India and Dhoni. We hope the Indian team will come out of the impasse with flying colors.

Jyeshtadeva was a Kerala astronomer who helped in the calculation of longitudes, when there is latitudinal deflection. In his Yukti Bhasa, he calculates correctly the cos l, the cosine of latitude, which is important in the Reduction to the Ecliptic. There is a separate section in the Yukti Bhasa, which deals with the effects of the inclination of a planet's orbit on its latitude. He describes how to find the true longitude of a planet, Sheegra Sphutam, when there is latitudinal deflection. "Now calculate the Vikshepa Koti, cos l, by subtracting the square of the Vikshepa from the square of the Manda Karna Vyasardha and calculating the root of the difference." In the above diagram, N is the Ascending Node P is the planet on the Manda Karna Vritta, inclined to the Ecliptic Vikshepa Koti = OM = SQRT( OP^2 - PM^2 ) Taking this Vikshepa Koti and assuming it to be the Manda Karna, sheegra sphuta, the true longitude, has to be calculated as before.

Jyeshtadeva was a Kerala astronomer who helped in the calculation of longitudes, when there is latitudinal deflection. In his Yukti Bhasa, he calculates correctly the cos l, the cosine of latitude, which is important in the Reduction to the Ecliptic. There is a separate section in the Yukti Bhasa, which deals with the effects of the inclination of a planet's orbit on its latitude. He describes how to find the true longitude of a planet, Sheegra Sphutam, when there is latitudinal deflection. "Now calculate the Vikshepa Koti, cos l, by subtracting the square of the Vikshepa from the square of the Manda Karna Vyasardha and calculating the root of the difference." In the above diagram, N is the Ascending Node P is the planet on the Manda Karna Vritta, inclined to the Ecliptic Vikshepa Koti = OM = ( OP^2 - PM^2 ) ^1/2 Taking this Vikshepa Koti and assuming it to be the Manda Karna, sheegra sphuta, the true longitude, has to be calculated as before.

Dhoni scored a gutsy 77, but that could not save India, who dithered to 224 all out. It was a pathetic display of batsmanship, as Sehwag fell for nought and Sachin for one. P Kumar gave a good stand to Dhoni, after the visitors were 7/111. Strauss scored a brilliant 50 and is still unconquered, with England 84 for no wicket. Only Gambhir (38) and Laxman (30) could stay at the wicket. The rest was a procession. Broad and Bresnan took 4 wickets each and swung India out cheap. P Kumar's 26 saved India, as he gave a good stand to the Indian skipper. Will England do a 4-0 whitewash ?

l, Vikshepa, is the Celestial Latitude, the latitude of the planet, the angular distance of the planet from the Ecliptic. i is the inclination, inclinent of Orbit. Sin l = Sin i Sin ( Heliocentric Long - Long of Node ). Celestial Latitude is calculated from this equation. The longitude of the Ascending Node, pata , is minussed from the heliocentric longitude and this angle is called Vipata Kendra.

l, Vikshepa, is the Celestial Latitude, the latitude of the planet, the angular distance of the planet from the Ecliptic. i is the inclination, inclinent of Orbit. Sin l = Sin i Sin( Heliocentric Long - Long of Node ). Celestial Latitude is calculated from this equation. The longitude of the Ascending Node, pata , is minussed from the heliocentric longitude and this angle is called Vipata Kendra.

In the last post we said that Angle AES is Sheegroccha, which is the longitude of the Sun. ( Sheegrocham Sarvesham Ravir Bhavathi ). The Angle AEK is the Heliocentric longitude of the planet. Sidereal Periods of superior Planets in the Geocentric = Sidereal periods in the Heliocentric. Sidereal Periods of Mercury and Venus = Mean Sun in the Geocentric In the Planetary Model of Aryabhata, we find the equation Heliocentric Longitude - Longitude of Sun = The Anomaly of Conjunction ( Sheegra Kendra ). As Astronomy is Universal, we are indebted to these savants who made astro calculation possible. Even the word " genius " is an understatement of their brilliant IQ ! Development of the Planetary Models in Astronomy Hipparchus 150 BCE Claudious Ptolemy 150 ACE Aryabhata 499 ACE Varaha 550 ACE Brahmagupta 628 ACE Bhaskara I 630 ACE Al Gorismi 850 ACE Munjala 930 ACE Bhaskara II 1150 ACE Madhava 1380

In the last post we said that Angle AES is Sheegroccha, which is the longitude of the Sun. ( Sheegrocham Sarvesham Ravir Bhavathi ). The Angle AEK is the Heliocentric longitude of the planet. Sidereal Periods of superior Planets in the Geocentric = Sidereal periods in the Heliocentric. Sidereal Periods of Mercury and Venus = Mean Sun in the Geocentric In the Planetary Model of Aryabhata, we find the equation Heliocentric Longitude - Longitude of Sun = The Anomaly of Conjunction ( Sheegra Kendra ). As Astronomy is Universal, we are indebted to these savants who made astro calculation possible. Even the word " genius " is an understatement of their brilliant IQ ! Development of the Planetary Models in Astronomy Hipparchus 150 BCE Claudious Ptolemy 150 ACE Aryabhata 499 ACE Varaha 550 ACE Brahmagupta 628 ACE Bhaskara I 630 ACE Al Gorismi 850 ACE Munjala 930 ACE Bhaskara II 1150 ACE Madhava 1380

In the above diagram, A = Starting Point, 0 degree Aries P = Planet S = Sun E = Earth Angle AEK = Manda Sphuta , heliocentric longitude, after manda samskara Angle AES = Sheegroccha , mean Sun, mean longitude of Sol Angle AEP = True geocentric longitude of planet Angle KEP = The Sheegra Correction or sheegra phalam The Anomaly of Conjunction = Sheegra Kendra = Angle AES - Angle AEK x = Angle AES - Angle AEK Sin ( x ) = r sin (x) _______________________ ((R + r cos x)^2 + rsin x^2 ))^1/2 which is the Sheegra correction formula given by the Indian astronomers to calculate the geocentric position of the planet.

In the above diagram, A = Starting Point, 0 degree Aries P = Planet S = Sun E = Earth Angle AEK = Manda Sphuta , heliocentric longitude, after manda samskara Angle AES = Sheegroccha , mean Sun, mean longitude of Sol Angle AEP = True geocentric longitude of planet Angle KEP = The Sheegra Correction or sheegra phalam The Anomaly of Conjunction = Sheegra Kendra = Angle AES - Angle AEK x = Angle AES - Angle AEK Sin ( x ) = r sin (x) _______________________ ((R + r cos x)^2 + rsin x^2 ))^1/2 which is the Sheegra correction formula given by the Indian astronomers to calculate the geocentric position of the planet.

It was Monsoon Tourism, as Aslesha Njattuvela was on. It was raining heavily, cats and dogs in Kerala. I got the rains when I reached Kochi. I had some work at the Passport Office and I finished the work at Noon. Then I went on a tour of the famous Goshree Islands. I went by boat yesterday to the beautiful Bolgatty Island. A two minutes walk saw me entering the lovely Bolgatty Palace, a resort by the Kerala Tourism Development Corporation. I walked to the Bolgatty Bus Stand and took a bus to Vallarpadam International Container Terminal. Now everything is in place and one ship, OEL Dubai, was unloading. The progress of the ICTT is slow, but steady. The Bolgatty Palace is beautiful and well situated in the Mulavukad Island. This island is connected to Vallarpadam by a bridge. Vallarpadam is in turn connected to Vypin by a bridge. In fact these bridges are known as Goshree Bridges, as these beateous islands are known as Goshree Islands. In Vypin, one can see the GAIL LNG terminals, which

It was Monsoon Tourism, as Aslesha Njattuvela was on. It was raining heavily, cats and dogs in Kerala. I got the rains when I reached Kochi. I had some work at the Passport Office and I finished the work at Noon. Then I went on a tour of the famous Goshree Islands. I went by boat yesterday to the beautiful Bolgatty Island. A two minutes walk saw me entering the lovely Bolgatty Palace, a resort by the Kerala Tourism Development Corporation. I walked to the Bolgatty Bus Stand and took a bus to Vallarpadam International Container Terminal. Now everything is in place and one ship, OEL Dubai, was unloading. The progress of the ICTT is slow, but steady. The Bolgatty Palace is beautiful and well situated in the Mulavukad Island. This island is connected to Vallarpadam by a bridge. Vallarpadam is in turn connected to Vypin by a bridge. In fact these bridges are known as Goshree Bridges, as these beateous islands are known as Goshree Islands. In Vypin, one can see the GAIL LNG terminals, which

It was Monsoon Tourism, as Aslesha Njattuvela was on. It was raining heavily, cats and dogs in Kerala. I got the rains when I reached Kochi. I had some work at the Passport Office and I finished the work at Noon. Then I went on a tour of the famous Goshree Islands. I went by boat yesterday to the beautiful Bolgatty Island. A two minutes walk saw me entering the lovely Bolgatty Palace, a resort by the Kerala Tourism Development Corporation. I walked to the Bolgatty Bus Stand and took a bus to Vallarpadam International Container Terminal. Now everything is in place and one ship, OEL Dubai, was unloading. The progress of the ICTT is slow, but steady. The Bolgatty Palace is beautiful and well situated in the Mulavukad Island. This island is connected to Vallarpadam by a bridge. Vallarpadam is in turn connected to Vypin by a bridge. In fact these bridges are known as Goshree Bridges, as these beateous islands are known as Goshree Islands. In Vypin, one can see the GAIL LNG terminals, which