Transcendence

The Earth's Axial tilt is called the Obliquity of the Ecliptic and the is angle between the perpendicular to Orbit and the North Celestial Pole. The Equatorial coordinate system is based on the 360 degree Celestial Equator Circle. The Ecliptic coordinate system is based on the 360 degree Ecliptic circle. The mathematical conversion from Equatorial to Ecliptic is effectuated by the equation for the Ascendant Lagna = arctan ( Sin E / Cos E Cos w - Sin w Tan A ) where Lagna is the Lagna on the Ecliptic andE is the Lagna on the Celestial Equator, the Sayana Kala Lagna. The Sayana Kala Lagna , E, is reduced to the Ecliptic by this equation. The Lagna is the intersecting point between the Eastern Celestial Horizon, the Kshitija with the Ecliptic. w is the Sun's maximum declination and A is the latitude of the place. The above diagram is by courtesy of Wikipedia

The Earth's Axial tilt is called the Obliquity of the Ecliptic and the is angle between the perpendicular to Orbit and the North Celestial Pole. The Equatorial coordinate system is based on the 360 degree Celestial Equator Circle. The Ecliptic coordinate system is based on the 360 degree Ecliptic circle. The mathematical conversion from Equatorial to Ecliptic is effectuated by the equation for the Ascendant Lagna = atan ( Sin E / Cos E Cos w - Sin w Tan A ) L is the Lagna on the Ecliptic. E is the Lagna on the Celestial Equator, the Sayana Kala Lagna. The Sayana Kala Lagna , E, is reduced to the Ecliptic by this equation. The Lagna is the intersecting point between the Eastern Celestial Horizon, the Kshitija with the Ecliptic. w is the Sun's maximum declination and A is the latitude of the place. The above diagram is by courtesy of Wikipedia

Dravid scored a century again. India were in a strong, commanding position at 267/4 but then were bowled out for 288, thanks to a hat trick from Stuart Broad. Broad took his revenge by dismissing Yuvraj for 62 ( Yuvraj had hit him for six sixes in one over before ) and then dismissed Dhoni, Bajji and Kumar and took a rich six wicket haul The Wall was in his inimitable best, but rued the advantage which India had lost ! "It was sort of frustrating at the way we played. Look we were 267 for four but were reduced to 273 for eight," said the Wall, who became the joint second Indian century maker in Test cricket with Gavaskar. ( after Sachin, of course ). "It was not easy for the guys though. The pitch was difficult to bat and the guys tried their best. We need to pull our socks and come up with some strong performance in the remaining three days," said the solid batsman. "It was tough conditions out there. It was a hot day. I had slight cramps towards the end of m

Had it not been for a brilliant 64 by Stuart Broad, India would have skittled out England for less than 150. Sreeshant, Praveen and Ishant took three wickets each, as India reduced England to 124/8 at tea. Then the tail wagged and England finished at 221. India were 14 for the loss of Mukund. Meanwhile, Gavaskar has criticised the Indian selection for not taking three openers for the England tour. Because of this factor, Dravid had to come in as opener. There is a lot of difference being an opener and a No 3 batsman ! Indians held on to their catches, unlike at Lord's. All three swing bowlers swung the ball well, pitching the ball in the right areas. England were in dire straits at tea, having lost 8 wickets to the swing maestros. Sreeshant proved to be an effective replacement for Zaheer and he got the prize scalp of KP. Honolulu Injury Lawyer David W Barlow is a Honolulu Injury Lawyer , reputed for his skill set. He has an office in Downtown Honolulu and also in Leeward Oahu.He

In the Concentric Equant Model, reduction to the Heliocentric was easy. In the above diagram, OM was the Radius of the Epicycle. And q is the Equation of the Center. The word Equation in Astronomy is the angle between mean planet and true planet. It is got by the formula q = arcsine ( e* sin (M) ) In the above diagram, BOM is the mean Centrum or the Mean Anomaly and SOM is the true Centrum or the True Anomaly. t - m = q, the Equation of Center. This diagram is by courtesy of Jean-Pierre Lacroix and Robert Baywater, www.ancientcartography.net

Anderson, with superb swing bowling, skittled out the Indians at Lord. He grabbed his 11th 5 wicket Test haul, when he took 5/69 yesterday. He bowled Praveen Kumar with a beauty and then removed Raina ( 78) and Ishant ( 1) with consecutive deliveries. The Indians were expectant of a century of centuries from the Little Master, but had to contend with defeat. KP was declared Man of the Match. The Nottingham Test starts on Friday and England has already taken a lead of 1-0 in the four match Test series. Despite the loss, Dhoni said he was happy the way the Indians performed. Lots of injuries was the problem and he said the Indians rose to the occasion ! Tremlett achieved the breakthrough after tea, as he removed the dangerous Dhoni. The Indians lost their seventh wicket as Bajji tried to pull Anderson and was caught by Tremlett.

The quadruple strike of Ishant sent England reeling at 107/6, but then Prior and Broad redeemed England with some scintillating batting. Prior was not out 103 and Broad on 74, when England declared at at 269/6. India were 80/1 in their second innings, still needing 378 runs to win the game. Mukund was removed by Broad for the second time in the series, but then the doggedness of Laxman and Dravid saw to that India did not lose any furthur wickets. Dravid opened the innings instead of Gambhir who was hospitalised because of an injury on his left arm. Prior and Broad put on 162 for the seventh wicket. Ishant shocked England by sending back Pietersen ( 1), Ian Bell (0) and Trott ( 22). Striking like a bolt of lightning, Sharma proved to be a different bowler than the Sharma of the first innings! Pietersen was done in by a snorter from Sharma, which he gloved helplessly to Dhoni. Bell was done in by the swing of Sharma and Trott was beaten and bowled by Sharma. England, therefore, seems to

w was an important angle in the Munjala Model and the solution to the problem of a difference of 2.5 degrees in the lunar longitude had to be solved. So Munjala brought in an angle, w, angle between the Mean Sun and the Moon's Apogee. The angle n is the elongation of the Sun from the Mean Moon and so the Manda Anomaly, Alpha = w + n This diagram is by courtesy of Jean-Pierre Lacroix and Robert Baywater, www.ancientcartography.net

w was an important angle in the Munjala Model and the solution to the problem of a difference of 2.5 degrees in the lunar longitude had to be solved. So Munjala brought in an angle, w, angle between the Mean Sun and the Moon's Apogee. The angle n is the elongation of the Sun from the Mean Moon and so the Manda Anomaly, Alpha = w + n This diagram is by courtesy of Jean-Pierre Lacroix and Robert Baywater, www.ancientcartography.net

Dravid's ton at Lords, saved India from the ignominy of following on. India went past the 274 mark, thanks to his superb century. With this milestone, Dravid has gone past Ponting, as No 2 Test scorer in the history of the game. India were all out for 284 and just avoided the follow on. England are 5/0, after declaring at 474/8 in the first innings. He was behind 49 runs behind Ponting when the match started. Here are the stats Sachin Tendulkar: 14726 Rahul Dravid: 12373 Ricky Ponting: 12363 Brian Lara: 11953 Jacques Kallis: 11947 Alan Border: 11174 Steve Waugh: 10927 Sunil Gavaskar: 10122 Fifteen years ago, the Wall fell in the nervous nineties and now he has proved to the world that he is not over the hill. The briliant double ton scored by KP was superb. Boycott says he was out at 49, but umpire Asad Rauf's lacked of conviction saved him. He was out caught in the leg slip, but Rauf referred it to the third umpire. This however does not diminish the brilliance of KP ! To KP

The Model propounded by Aryabhata is an algorithm. The Khmers drew the diagrams of the Sun by using the epicyle equivalent of the model developed by Bhaskara in the seventh century. Eccentricity is variable in this Epicycle Model. You can see a relevant animation of Dennis Duke at http://people.scs.fsu.edu/~dduke/pingree2.html

The Model propounded by Aryabhata is an algorithm. The Khmers drew the diagrams of the Sun by using the epicyle equivalent of the model developed by Bhaskara in the seventh century. Eccentricity is variable in this Epicycle Model. You can see a relevant animation of Dennis Duke at http://people.scs.fsu.edu/~dduke/pingree2.html

The Indian astronomers could calculate the Manda Kendra ( The Equation of Center of Western Astronomy ) and the Manda Phala , but a problem presented itself when calculating the lunar longitude. The Concentric Model and the Epicylic Model could not calculate Moon's longitude at quadrature, even though they could calculate the lunar longitude at the times of New Moon and Full Moon. There was a difference of 2.5 degrees between the longitude computed by the Concentric Equant and Epicyclic Models. So the ancients had to give a correction to the Equation of Center, which reached a maximum of 2.5 degrees. So the Indian astronomers came out with a solution. They created a new Equant (E'), the true Equant, which moves on an epicycle, whose center is the Mean Equant, E. The epicycle has a radius e, equal to EoE., on the Line of Apsis, OA. q1 = Equation of Center, first lunar inequality q2 = Correction, second lunar inequality. True Longitude = Mean long + Eq of Center + q2 The first l

The Indian astronomers could calculate the Manda Kendra ( The Equation of Center of Western Astronomy ) and the Manda Phala , but a problem presented itself when calculating the lunar longitude. The Concentric Model and the Epicylic Model could not calculate Moon's longitude at quadrature, even though they could calculate the lunar longitude at the times of New Moon and Full Moon. There was a difference of 2.5 degrees between the longitude computed by the Concentric Equant and Epicyclic Models. So the ancients had to give a correction to the Equation of Center, which reached a maximum of 2.5 degrees. So the Indian astronomers came out with a solution. They created a new Equant (E'), the true Equant, which moves on an epicycle, whose center is the Mean Equant, E. The epicycle has a radius e, equal to EoE., on the Line of Apsis, OA. q1 = Equation of Center, first lunar inequality q2 = Correction, second lunar inequality. True Longitude = Mean long + Eq of Center + q2 The first l

Aryabhata developed a Concentric Equant Model, in the sixth century. The Sun moves on a circle of radius R, called a deferent, whose center is the Observer on Earth. The distance between the Earth and the Sun, the Ravi Manda Karna, is constant. The motion of the Sun is uniform from a mathematical point, called the " Equant", which is located at a distance R x e from the observer in the direction of the Apogee ( e = eccentricity ). All Indian computations are based on this Concentric Equal Model. The normal equation for computing the Manda Anomaly is R e Sin M and resembles the Kepler Equation, M = E - e Sin E. This diagram is by courtesy of Jean-Pierre Lacroix and Robert Baywater, www.ancientcartography.net

Aryabhata developed a Concentric Equant Model, in the sixth century. The Sun moves on a circle of radius R, called a deferent, whose center is the Observer on Earth. The distance between the Earth and the Sun, the Ravi Manda Karna, is constant. The motion of the Sun is uniform from a mathematical point, called the " Equant", which is located at a distance R x e from the observer in the direction of the Apogee ( e = eccentricity ). All Indian computations are based on this Concentric Equal Model. The normal equation for computing the Manda Anomaly is R e Sin M and resembles the Kepler Equation, M = E - e Sin E. This diagram is by courtesy of Jean-Pierre Lacroix and Robert Baywater, www.ancientcartography.net

India missed three chances. Trott gave two and Pietersen one. If Dravid had held on to that catch offered by Trott off Bajji or if Dhoni had held onto that catch off Trott, the story would have been otherwise. One remembers the catch missed by Kiran More off Gooch at 33 and Gooch went on to score 333. Earlier, Zaheer Khan removed the dangerous Strauss and Cook. And England were reeling at 62/2. Then Trott and Pietersen did the retrieving work and England are on their way to recovery at 127/2. In a day curtailed by rain, only 49.2 overs were possible. Ishant missed a Strauss run out at 2. Dhoni had won the toss and opted to bowl first. Pietersen batted nervously, trying to negotiate the swinging ball and edging and missing most of the time. Let us see what England does today.

The Concentric Equant theory was developed by the Indian astronomer, Munjala ( circa 930 CE ). The Geocentric theory of the ancient astronomers had the ability to produce true Zodiacal Longitudes for the Moon. But the perturbations of the Moon were so complex, that the early Indian and Greek astronomers had to give birth to complicated theories. The simplest model is a concentric Equant Model to compute the lunar true longitude. In the above diagram M = Moon O = Observer Eo = Equant , located at a distance r from the observer , drawn on the Line of Apsis and the Apogee. A = Apogee, Luna's nearest point to Earth Angle Alpha = Angle between Position and Apogee Angle q1 = Equation of Center . Angle subtended at Luna between Observer and Equant Equation in Astronomy = The angle between true and mean positions. These diagrams are by courtesy of Jean-Pierre Lacroix and Robert Baywater, www.ancientcartography.net

The Concentric Equant theory was developed by the Indian astronomer, Munjala ( circa 930 CE ). The Geocentric theory of the ancient astronomers had the ability to produce true Zodiacal Longitudes for the Moon. But the perturbations of the Moon were so complex, that the early Indian and Greek astronomers had to give birth to complicated theories. The simplest model is a concentric Equant Model to compute the lunar true longitude. In the above diagram M = Moon O = Observer Eo = Equant , located at a distance r from the observer , drawn on the Line of Apsis and the Apogee. A = Apogee, Luna's nearest point to Earth Angle Alpha = Angle between Position and Apogee Angle q1 = Equation of Center . Angle subtended at Luna between Observer and Equant Equation in Astronomy = The angle between true and mean positions. These diagrams are by courtesy of Jean-Pierre Lacroix and Robert Baywater, www.ancientcartography.net

The Physics Professor of Florida State University, Dennis Duke remarks "The planetary models of ancient Indian mathematical astronomy are described in several texts. These texts invariably give algorithms for computing mean and true longitudes of the planets, but are completely devoid of any material that would inform us of the origin of the models. One way to approach the problem is to compare the predictions of the Indian models with the predictions from other models that do have, at least in part, a known historical background. Since the Indian models compute true longitudes by adding corrections to mean longitudes, the obvious choices for these latter models are those from the Greco-Roman world. In order to investigate if there is any connection between Greek and Indian models, we should therefore focus on the oldest Indian texts that contain fully described, and therefore securely computable, models. We shall see that the mathematical basis of the Indian models is the equant

The Physics Professor of Florida State University, Dennis Duke remarks "The planetary models of ancient Indian mathematical astronomy are described in several texts. These texts invariably give algorithms for computing mean and true longitudes of the planets, but are completely devoid of any material that would inform us of the origin of the models. One way to approach the problem is to compare the predictions of the Indian models with the predictions from other models that do have, at least in part, a known historical background. Since the Indian models compute true longitudes by adding corrections to mean longitudes, the obvious choices for these latter models are those from the Greco-Roman world. In order to investigate if there is any connection between Greek and Indian models, we should therefore focus on the oldest Indian texts that contain fully described, and therefore securely computable, models. We shall see that the mathematical basis of the Indian models is the equant

Lara define Sachin as the best in the world. Former Eng captain Alex Stewart called him "the modern day Bradman". The world is waiting with tremendous angst and the question on everybody's lips is " Can Sachin do the impossible, score his 100th international hundred at Lords ?". His record at Lords had been dismal. In the last seven innings at Lords, his scores have been 10, 27, 31, 16, 12, 37 and 16. And Lords seems to be the only Test ground where he has not scored a fifty. But he is in tremendous form and his average against England is 61.42, the highest against any standard Test side. This may be his last tour to England. Strauss has already warned India that this will be a baptism of fire. If Eng beat India in two Tests, they can surpass South Africa and go to the top of the table, in ICC Rankings. Lara, participating in a panel discussion at Dhoni's "East Meets West" Gala Dinner at Hilton Park Lane in London, said " "Sachin starte

( Above diagram by courtesy of www.wikipedia.org ) A 360 degree Circle is a coordinate System. And Khagola is the Celestial Coordinate System. Like the Geographical Coordinate System, the Celestial Coordinate System is another coordinate system, which computes the coordinates of the Khagola, the Celestial Sphere. The Ascending Sign is called the Ascendant ( Raseenam udayo lagnam ) and is the intersecting point of the Ecliptic ( at the East Point) with the Celestial Horizon. The Descending Sign is the Descendant ( Astha Lagna ) and lies 180 degrees West on the Khshithija , the Celestial Horizon. Like the Geographical Meridien ( the Prime Meridien ) and the Geographic Equator, the Celestial Coordinate System has a Celestial Meridien ( Nadi Vritta ) and a Celestial Equator. The Vernal Equinox and the Autumnal Equinox are two intersecting points of the Ecliptic with the Vishu vat Vritta , the Celestial Equator, known as Meshadi and Thuladi. The Hindu Zero Point of the Ecliptic starts

( Above diagram by courtesy of www.wikipedia.org ) A 360 degree Circle is a coordinate System. And Khagola is the Celestial Coordinate System. Like the Geographical Coordinate System, the Celestial Coordinate System is another coordinate system, which computes the coordinates of the Khagola, the Celestial Sphere. The Ascending Sign is called the Ascendant ( Raseenam udayo lagnam ) and is the intersecting point of the Ecliptic ( at the East Point) with the Celestial Horizon. The Descending Sign is the Descendant ( Astha Lagna ) and lies 180 degrees West on the Khshithija , the Celestial Horizon. Like the Geographical Meridien ( the Prime Meridien ) and the Geographic Equator, the Celestial Coordinate System has a Celestial Meridien ( Nadi Vritta ) and a Celestial Equator. The Vernal Equinox and the Autumnal Equinox are two intersecting points of the Ecliptic with the Vishu vat Vritta , the Celestial Equator, known as Meshadi and Thuladi. The Hindu Zero Point of the Ecliptic starts

On Saturday, the heavens brimmed with pessimistic prophecies and then came the downpour. ( Today is 19th Jul 2011 ) The Sun has disappeared and it is now raining cats and dogs here. As a concomitant result, I got cold ! This SW Monsoon, defined as a failure this season, may perk up, compensating for the lack of rains during the earlier Mrigasira and Aridra Solar Periods ( Njattuvelas ). Now Punarvasu Njattuvela is on, as the Sun transits Beta Geminorum. Now the paddy fields are full of water and it rained heavily at night day before yesterday. The ocean became hostile on Chavakkad Beach and surrounding areas, wreaking destruction.

On Saturday, the heavens brimmed with pessimistic prophecies and then came the downpour. ( Today is 19th Jul 2011 ) The Sun has disappeared and it is now raining cats and dogs here. As a concomitant result, I got cold ! This SW Monsoon, defined as a failure this season, may perk up, compensating for the lack of rains during the earlier Mrigasira and Aridra Solar Periods ( Njattuvelas ). Now Punarvasu Njattuvela is on, as the Sun transits Beta Geminorum. Now the paddy fields are full of water and it rained heavily at night day before yesterday. The ocean became hostile on Chavakkad Beach and surrounding areas, wreaking destruction.

This diagram is by courtesy of Jean-Pierre Lacroix and Robert Baywater, www.ancientcartography.net We have the Double Epicyclic Model - that of Manda Epicycle and Sheegra Epicycles - in Indian Astronomy, which explain the Zodiacal and Solar anomalies. One Epicycle explains the Zodiacal Anomaly and the other the Solar Anomaly. ( Zodiacal Anomaly - That all planets move slower at Aphelion and faster at Perihelion. Solar Anomaly - The astronomical phenomenon of Retrogression. Backward Motion. When a planet changes its course from perihelion to aphelion, it retrogrades in order to gain the Sun's celestial gravity ) Dennis Duke, of Florida State University, says " We have only to conclude that Ptolemy did not invent the equant. " If Ptolemy did not invent the equant, as Westerners widely believe, then who did ? "The bisected Indian equant model is pre-Ptolemaic' says he. Other Greek books, prior to Ptoemy, may have influenced Indian Astronomy,says he. Then what a

This diagram is by courtesy of Jean-Pierre Lacroix and Robert Baywater, www.ancientcartography.net We have the Double Epicyclic Model - that of Manda Epicycle and Sheegra Epicycles - in Indian Astronomy, which explain the Zodiacal and Solar anomalies. One Epicycle explains the Zodiacal Anomaly and the other the Solar Anomaly. ( Zodiacal Anomaly - That all planets move slower at Aphelion and faster at Perihelion. Solar Anomaly - The astronomical phenomenon of Retrogression. Backward Motion. When a planet changes its course from perihelion to aphelion, it retrogrades in order to gain the Sun's celestial gravity ) Dennis Duke, of Florida State University, says " We have only to conclude that Ptolemy did not invent the equant. " If Ptolemy did not invent the equant, as Westerners widely believe, then who did ? "The bisected Indian equant model is pre-Ptolemaic' says he. Other Greek books, prior to Ptoemy, may have influenced Indian Astronomy,says he. Then what a

In the diagram above, the geocentric distance, EQ called X here , the distance of the planet from the Earth is calculated by the equation X^2 = EQ^2(EP+PL)^2 + QL^2 or = EN^2 + QN^2 In a trignometric correction, called Sheegra Sphashteekarana, this equation is given by Bhaskara. where E = Earth P = Planet in its Orbit Q = Planet on the Epicycle QL = Sin PL = Cos We have said that Sheegra Kriya reduces the heliocentric postions to the geocentric. According to this oscillating Epicyclic Model of Bhaskara, EP = R ( Called Thrijya ), PQ is the Sheegra Phala , QL is the Bhujaphala and PL is Kotiphala. The Hindu algorithms for the computation of mean and true celestial longitudes seems to be totally different from the Western, from the methods adopted by Kepler, Laplace and Co. Hence the Hindu Planetary Model is original and not influenced by Greco Roman sources, as some Western scholars believe.

In the diagram above, the geocentric distance, EQ called X here , the distance of the planet from the Earth is calculated by the equation X^2 = EQ^2(EP+PL)^2 + QL^2 or = EN^2 + QN^2 In a trignometric correction, called Sheegra Sphashteenarana, this equation is given by Bhaskara. where E = Earth P = Planet in its Orbit Q = Planet on the Epicycle QL = Sin PL = Cos We have said that Sheegra Kriya reduces the heliocentric postions to the geocentric. According to this oscillating Epicyclic Model of Bhaskara, EP = R ( Called Thrijya ), PQ is the Sheegra Phala , QL is the Bhujaphala and PL is Kotiphala. The Hindu algorithms for the computation of mean and true celestial longitudes seems to be totally different from the Western, from the methods adopted by Kepler, Laplace and Co. Hence the Hindu Planetary Model is original and not influenced by Greco Roman sources, as some Western scholars believe.

Different equations have been given for superior planets ( Mars, Jupiter and Saturn ) and inferior planets ( Mercury and Venus ) in Astronomia Indica. In the case of Mercury, an inferior planet in the diagram above, the center of the Sheegra Epicycle is located on the straight line running through the Sun and the observer, on the geographical parallel of the observer. The above diagram is by courtesy of Jean-Pierre Lacroix and Robert Baywater, www.ancientcartography.net The Sheegra Phalam, x, in the equation 1/2 Tan ( A -x ), where A is the Elongation or Sheegra Kendra, obtained is deducted from the Sun's longitude, to get the geocentric longitudes of Mercury and Venus.

Different equations have been given for superior planets ( Mars, Jupiter and Saturn ) and inferior planets ( Mercury and Venus ) in Astronomia Indica. In the case of Mercury, an inferior planet in the diagram above, the center of the Sheegra Epicycle is located on the straight line running through the Sun and the observer, on the geographical parallel of the observer. The above diagram is by courtesy of Jean-Pierre Lacroix and Robert Baywater, www.ancientcartography.net The Sheegra Phalam, x, in the equation 1/2 Tan ( A -x ), where A is the Elongation or Sheegra Kendra, obtained is deducted from the Sun's longitude, to get the geocentric longitudes of Mercury and Venus.

This diagram is by courtesy of Jean-Pierre Lacroix and Robert Baywater, www.ancientcartography.net In the above diagram, Saturn, a superior planet, is on the circumference of the Sheegra Epicycle, where it is met by a radius drawn parallel to the direction of the Sun from the observer. To the Western scholars, Indian Astronomy is mysterious. Let us see what astro scholars have said about IA. Dennis Duke, of Florida State University suggests that Indian Astronomy predates Greek Astronomy "The planetary models of ancient Indian mathematical astronomy are described in several texts.1 These texts invariably give algorithms for computing mean and true longitudes of the planets, but are completely devoid of any material that would inform us of the origin of the models. One way to approach the problem is to compare the predictions of the Indian models with the predictions from other models that do have, at least in part, a known historical background. Since the Indian models compute true

This diagram is by courtesy of Jean-Pierre Lacroix and Robert Baywater, www.ancientcartography.net In the above diagram, Saturn, a superior planet, is on the circumference of the Sheegra Epicycle, where it is met by a radius drawn parallel to the direction of the Sun from the observer. To the Western scholars, Indian Astronomy is mysterious. Let us see what astro scholars have said about IA. Dennis Duke, of Florida State University suggests that Indian Astronomy predates Greek Astronomy "The planetary models of ancient Indian mathematical astronomy are described in several texts.1 These texts invariably give algorithms for computing mean and true longitudes of the planets, but are completely devoid of any material that would inform us of the origin of the models. One way to approach the problem is to compare the predictions of the Indian models with the predictions from other models that do have, at least in part, a known historical background. Since the Indian models compute true

This diagram is by courtesy of Jean-Pierre Lacroix and Robert Baywater, www.ancientcartography.net In the above diagram, both the theories of Manda Kriya and Sheegra Kriya are given. In the case of a superior planet, a deferent is drawn from an earth based observer. The Center of the Manda Epicyle rotates around the terrestrial observer, travelling around the deferent. The peripheral end of one radius of this Manda Epicycle determines the center of another epicyle called the Sheegra Epicycle.

This diagram is by courtesy of Jean-Pierre Lacroix and Robert Baywater, www.ancientcartography.net In the above diagram, both the theories of Manda Kriya and Sheegra Kriya are given. In the case of a superior planet, a deferent is drawn from an earth based observer. The Center of the Manda Epicyle rotates around the terrestrial observer, travelling around the deferent. The peripheral end of one radius of this Manda Epicycle determines the center of another epicyle called the Sheegra Epicycle.

Aryabhata, one of the earliest mathematicians and astronomers, ( circa 476-550 CE ) postulated that Vysasardha , the Radius of the Circle is 3438 minutes and Arc is 5400 minutes. Circumference = 2 Pi R R = 360/2 Pi R = 57.3 degrees R = 57.3 * 60 = 3438 arcminutes R = 3438 * 60 = 206265 arcseconds Half Chord of 90 degrees = 90*60 = 5400 arcminutes. In his astronomical treatise, the Aryabhatiya , he postulated that the Circumference of the Circle is 360*60 = 21600 minutes. All these formulae are useful for the computation of half chords of certain sets of arcs in a circle and became the base of Hindu Trignometry. In his Sine Tablest, he called 3 degrees 45 minutes divisions by many Sanskrit names, given below. मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व | घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कला-अर्ध-ज्यास् || Aryabhata's Sine Table is not a set of values of the trignometric sine functions, but rather is a table of the first differences

Aryabhata, one of the earliest mathematicians and astronomers, ( circa 476-550 CE ) postulated that Vysasardha , the Radius of the Circle is 3438 minutes and Arc is 5400 minutes. Circumference = 2 Pi R R = 360/2 Pi R = 57.3 degrees R = 57.3 * 60 = 3438 arcminutes R = 3438 * 60 = 206265 arcseconds Half Chord of 90 degrees = 90*60 = 5400 arcminutes. In his astronomical treatise, the Aryabhatiya , he postulated that the Circumference of the Circle is 360*60 = 21600 minutes. All these formulae are useful for the computation of half chords of certain sets of arcs in a circle and became the base of Hindu Trignometry. In his Sine Tablest, he called 3 degrees 45 minutes divisions by many Sanskrit names, given below. मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व | घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कला-अर्ध-ज्यास् || Aryabhata's Sine Table is not a set of values of the trignometric sine functions, but rather is a table of the first differences

In Hindu Trignometry ( which is derived from Trikonamithi , trikona = triangle and trignon = triangle ), Jya resembles the modern Sine and Koti Jya, the cosine. But in actuality, Jya is R Sin, that is Radius multiplied by modern sine. By Jya , Brahmagupta meant 5 degrees of a circle. In Hindu Sine Tables and Tan Tables, the values are given for 5 degrees, 10 degrees, 15 degrees etc so that the Astro Maths students need not bother about using the Indian trignometric and inverse functions. Aryabhata's sine tables are found to be accurate, when compared to modern sine tables. In other words, one Zodiacal Constellation, which is 30 degrees is made up of 6 jyas and a total of 72 Jyas constitute the Zodiac. Koti Jya is R Cos, that is Radius multiplied by modern cosine. Utkram Jya is the reverse sine, defined as 1- cos x. Since the Reverse sine resembled an arrow, Brahmagupta called it Sara . And since the Arcsine resembled a bow, he called it Chap a. Bhujajya is radius multiplied b

All planets traverse in ellipses and epicycles and this came to be known as the Epicycles Theory. In the above diagram, the circle A is the mean orbit of the planet. P is the mean Position of the planet and the small circle P is the epicycle. The small epicycles traversed by a planet are calculated and the mandaphala , the equation of center is computed and added if the Manda Kendra is in between 180 and 360 and subtracted if the M K is < 180. Manda Kendra is the angle between Position and Aphelion. From the perspective of the Epicyclic theory & the Hindu astronomers, the radius of the epicycle was given instead of PQ and the circumferences of the epicycles. Both circumferences and radii are given in degrees, minutes and seconds, so that the equation of the center may be computed in deg min and secs. The epicycle in the case of the Equation of Center is given as Manda Nicha Uccha Vritta. Manda - Manda Phala or Equation of Center Uccha - Apogee Nicha - Perigee Manda Kriya is a

All planets traverse in ellipses and epicycles and this came to be known as the Epicycles Theory. In the above diagram, the circle A is the mean orbit of the planet. P is the mean Position of the planet and the small circle P is the epicycle. The small epicycles traversed by a planet are calculated and the mandaphala , the equation of center is computed and added if the Manda Kendra is in between 180 and 360 and subtracted if the M K is < 180. Manda Kendra is the angle between Position and Aphelion. From the perspective of the Epicyclic theory & the Hindu astronomers, the radius of the epicycle was given instead of PQ and the circumferences of the epicycles. Both circumferences and radii are given in degrees, minutes and seconds, so that the equation of the center may be computed in deg min and secs. The epicycle in the case of the Equation of Center is given as Manda Nicha Uccha Vritta. Manda - Manda Phala or Equation of Center Uccha - Apogee Nicha - Perigee Manda Kriya is a

Chanderpaul's ton denied India victory, but India emerged triumphant in the series, beating the WI 1-0 in Tests and 3-2 in ODIs. Chanderpaul was unconquered on 122, when WI crashed to 322 all out. India scored 94/3, with Vijay scoring 45. Dravid was not out 34 and India drew the Dominica Test. Bajji took 4/75. India fielded a second XI, with Sachin, Sehwag, Gauti and Zaheer conspicous by their absence, but still kept the numero uno position in Tests. The man with the Midas touch, the brilliant Dhoni helped India win the Test series easily and India under Raina performed well in the ODIs. Despite the mediocrity of Indian batting, certain knocks stood above the rest. Dravid's century in the First Test and Laxman's glorious half centuries in the second, were the redeeming grace for India. Dhoni was also resting on his laurels with only a brilliant 74 run knock to remember. Bajji's 400 wickets and Dravid's 15th year in Test cricket was also accompanied by the return

Despite a century by Kirk Edwards, West Indies were in trouble, as India reduced them to 224/6. One more day to go and anything can happen. Chanderpaul and Kirk Edwards pulled the WI to respectability with a brilliant 161 run partnership. Chanderpaul is still there with 73 not out. WI were 40/3 at one stage and this partnership pulled them to a seemingly high score. Edwards hit nine boundaries and a six during his glorious innings of 110. He was, however, out caught behind by Dhoni and then Samuels followed suit for a duck and the home team lost their fifth wicket. WI suffered a jolt, when Darren Bravo tried to hit Bajji, but holed out to Praveen Kumar for 14. The Indian new ball bowlers, Ishant and Praveen struck early by removing the WI openers. Barath was scalped by Kumar and Powell was snapped by Sharma. Earlier Fidel Edwards grabbed his 10th five wicket haul in Tests, as India folded up for 347. India could only add 39 more runs to take a 143 run lead. Fidel Edwards took 5/103 a

The declination of the Sun is computed by the formula Sin J = Sin L Sin w where J is the Sun's declination of that particular date and time, L is the tropical longitude of the Sun and w, the Sun's maximum declination, which is 23 degrees and 27 minutes. The Sun's maximum declination will be reached during Karkyadi ( The First Point of Cancer ) and Makaradi ( The First Point of Capricorn ). At Karkyadi, it will be +23 d 23 m and at Makaradi, it will be -23 d 27 minutes. And at Meshadi ( The First Point of Aries ) and Thuladi ( The First Point of Libra ), the solar declination will be zero. Days and nights will be of equal duration and hence they are called Equinoxes. Yada Mesha Thulayo varthathe thada ahoratranam samanani bhavanthi

The declination of the Sun is computed by the formula Sin J = Sin L Sin w where J is the Sun's declination of that particular date and time, L is the tropical longitude of the Sun and w, the Sun's maximum declination, which is 23 degrees and 27 minutes. The Sun's maximum declination will be reached during Karkyadi ( The First Point of Cancer ) and Makaradi ( The First Point of Capricorn ). At Karkyadi, it will be +23 d 23 m and at Makaradi, it will be -23 d 27 minutes. And at Meshadi ( The First Point of Aries ) and Thuladi ( The First Point of Libra ), the solar declination will be zero. Days and nights will be of equal duration and hence they are called Equinoxes. Yada Mesha Thulayo varthathe thada ahoratranam samanani bhavanthi

We have said in our columns that the Indian astronomers said that the planets revolve around a point different from that of the Earth and Indian Astronomy is heliocentric. The above diagram depicts the Eccentric Theory and Reduction of the longitude of the planet to the heliocentric coordinate system ( known as Manda Kriya ). In the above diagram The angle NAP = Manda Kendra or Mean Anomaly The Circle A is the mean orbit of the planet The Circle B is the true orbit The mean planet moves on the mean orbit, known as the deferent. When the planet is at N in the mean orbit, he is at M in the eccentric and this is called Mandoccha ( Aphelion ). When the planet is at R in the mean, he is at S in the eccentric. M and S correspond to Apogee and Perigee in the case of Manda Phala correction ( otherwise known as the Equation of Center ). When the planet is at M in the eccentric, the position of the true planet coincides with N, the mean planet and so the Manda Phala correction is Zero.The E

A brilliant 65 not out by Dhoni saw India score 308/6. Half centuries by Mukund ( 62), Laxman ( 56) and Raina ( 50 ) saw India battle it out for supremacy and is poised to take a huge lead. Dhoni on 65 not out and Bajji on 12 were at the crease. Earlier, the West Indies fought back magnificently to reduce India to 182/5 at tea. A brilliant piece of glove by Baugh off Chanderpaul saw Laxman depart, after a well compiled 56. Laxman left a delivery outside his off stump and Baugh whipped off the bails in a flash ! Promising youngster Kohli was caught down the legside off Sammy and India lost a valuable wicket at 168. Then Bishoo dismissed Mukund, who made a good 62. Mukund was involved in a 84 run partnership with Laxman for the third wicket. India started at 8/0, but lost the wickets of Vijay and Dravid. Dravid was bowled ingloriously through the gate and Vijay dabbed at a delivery outside the offstump. Both scored five runs. India added 126 runs after tea for the loss of only Suresh Rai

Ishant delivered the goods again by taking 5/77 and triggered another WI collapse. WI was going great guns at 190/6, when Ishant shot out the rest. Earlier, Bajji became the 11th bowler in the world and the fourth spinner to take 400 wickets. He is the fourth spinner after Murali, Warne and Kumble. India were 8/0, and Mukund ( 6 ) and Vijay (1) were at the crease at the close of day 1. A downpour after tea curtailed play. West Indies started the day at 75/3 disastrously, as Chanderpaul edged a Munaf delivery into the safe hands of Dhoni. Then Praveen Kumar clean bowled Samuels, who left after making nine runs. Darren Bravo along with Baugh put on 29 for the sixth wicket, before Ishant removed Bravo for 50. Bajji got the WI captain Sammy caught at forward short leg by Mithun for 20 and then grabbed his 400th wicket, as he went through the defense of Baugh, who fell for a gutsy innings of 60 off 79 balls. Ishant then removed the last two WI batsmen and took 5/77, his third five wicket h

In Indian Astronomy, Chara is the Ascensional Difference and is the difference between Right Ascension and Oblique Ascension. And is calculated by the equation Sin ( Chara ) = tan ( decl ) tan ( latitude ). Here Theta is Chara. Sin C is called Chara Jya Right Ascension means the longitudes measured along the Celestial Equator, the Vishu vat Vritta. Oblique Ascension means the longitudes measured along the Ecliptic, the Kranti Vritta. Chara is used in the computation of Sunrise and Sunset. The formula for Sunrise = 6 H + Equation of Time - Chara - Refraction Correction The formula for Sunset = 18 H + Equation of Time + Chara + Refraction Correction.

In Indian Astronomy, Chara is the Ascensional Difference and is the difference between Right Ascension and Oblique Ascension. And is calculated by the equation Sin ( Chara ) = tan ( decl ) tan ( latitude ). Here Theta is Chara. Sin C is called Chara Jya Right Ascension means the longitudes measured along the Celestial Equator, the Vishu vat Vritta. Oblique Ascension means the longitudes measured along the Ecliptic, the Kranti Vritta. Chara is used in the computation of Sunrise and Sunset. The formula for Sunrise = 6 H + Equation of Time - Chara - Refraction Correction The formula for Sunset = 18 H + Equation of Time + Chara + Refraction Correction.