Siddhanta Darpana by Samanta

Siddhanta Darpana by Samanta is a great treatise on Astronomy. The placing of the five major planets in the Tychonic Model of the solar system is in agreement with the Titius-Bode Law and even with Kepler's third law. This is a great contribution from Samanta. Epicycles for Solar Anomaly, known as Ugra Phala give correct distances of planets, in both Indian and Greek Astronomy.

Here we give the modern values of the radii of the orbits of the planets and those given in his magnum opus, the Siddhanta Darpana.


Planetary distances in astronomical unit


Planet Distance according to Bode’s law Actual distance

Mercury 0.4 0.387

Venus 0.7 0.732

Earth 1.0 1.0

Mars 1.6 1.524

Asteroid belt 2.8 2.68

Saturn 10.0 9.539

Uranus 19.6 19.19

Neptune 38.8 30.1

Pluto 77.2 39.5



Siddhanta Darpana



Even Odd

quadrant quadrant

Sun 1.00 1.00

Mercury 0.386 0.388

Venus 0.725 0.727

Mars 1.5126 1.5184

Jupiter 5.1428 5.2173

Saturn 9.230 9.4773

Siddhanta Darpana by Samanta

Siddhanta Darpana by Samanta is a great treatise on Astronomy. The placing of the five major planets in the Tychonic Model of the solar system is in agreement with the Titius-Bode Law and even with Kepler's third law. This is a great contribution from Samanta. Epicycles for Solar Anomaly, known as Ugra Phala give correct distances of planets, in both Indian and Greek Astronomy.

Here we give the modern values of the radii of the orbits of the planets and those given in his magnum opus, the Siddhanta Darpana.


Planetary distances in astronomical unit


Planet Distance according to Bode’s law Actual distance

Mercury 0.4 0.387

Venus 0.7 0.732

Earth 1.0 1.0

Mars 1.6 1.524

Asteroid belt 2.8 2.68

Saturn 10.0 9.539

Uranus 19.6 19.19

Neptune 38.8 30.1

Pluto 77.2 39.5



Siddhanta Darpana



Even Odd

quadrant quadrant

Sun 1.00 1.00

Mercury 0.386 0.388

Venus 0.725 0.727

Mars 1.5126 1.5184

Jupiter 5.1428 5.2173

Saturn 9.230 9.4773

Adverse Jupiter Strikes for Dhoni and India !

Adverse Jupiter for both India ( Jove in the 10th ) and Dhoni ( 8th ) wreacked havoc as India slumped to a huge defeat.

A defeat by 319 runs and England moved over to top place in ICC Rankings. What happened to India is a huge slump.

Jupiter moved over to the 10th in India's national chart on May 8. Dhoni is under 8th Jupiter ( Ashtame Haritho Bali ).

England holds absolute sway !

Bell's century ensured that England is in control of this Test.

Indian bowlers could not achieve much success as Bell plundered 159, Morgan 70, KP 63 and Prior 64 not out. England were in absolute command at 441/6.

India will have to fight it hard to save this Test. It will need a century of centuries from the Little Master, if India is to hold on to a draw.

Maybe this is the occasion we are waiting for !

The Obliquity of the Ecliptic !

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 Obliquity of the Ecliptic !

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

The Wall scores 34th Test century !

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 my innings. But I would say I am not tired but pleasantly weary. At the back of my mind, I think there is a lot of play left in the Test. We need strong performance in the remaining three days" said the dour batsman.

Yuvraj passed the Nottingham "Test". Playing in his first Test in England, he was a bit jittery in the initial stages. Taking the Wall's advice, he became confident and played a superb innings when India needed it most !

Indian swing bowlers strike !

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.

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Reduction to the Heliocentric Coordinate System

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 swings out India

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.

Ishant's quadruple strike saves not India

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 be in a commanding positon, but you can never tell in cricket !

Ishant's quadruple strike saves not India

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 be in a commanding positon, but you can never tell in cricket !

The Computation of Lunar Longitude

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

Computation of Lunar Longitude

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 saves India from ignominy

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 England owes a lot to its dominance in this Test match. Even though Luck had graced England, still they could contain the best ranked Test team in the world.

Meanwhile, Kapil has criticised Zaheer. If he was not fully fit, why was he taken ? There are other equally good swing bowlers and India owes a lot to its pitiable state to its poor selection.

Boycott wrote in the "Daily Telegraph" "The only blot on the day was the poor decision by the square leg umpire, Asad Rauf, when he should have given Pietersen out caught at leg slip when he was on 49.

"He had an uninterrupted clear view from 20 yards away and chickened out of making the correct decision. Anybody who works in television will tell you that low catches will always look as if the ball has bounced before being caught,"

"Batsmen throughout the game know that if a low catch is referred to television umpire there is every chance they will get away with it. I don't blame the batsman for standing. I blame the umpire for not having the courage of his convictions and making the decision without television."

Boycott critised India's bowling performance in the absence of Zaheer.

"It is not Pietersen's fault that India were one bowler short and their other bowling wasn't special. Harbhajan Singh made no impact whatsoever, Ishant Sharma tried his best but his inexperience and inability to put the ball in awkward areas showed up time and again," said the dour ex England batsman.

"Praveen Kumar is military medium but swung the ball and fully deserved his successful haul of five wickets which shows that in Test cricket, provided you move the ball off straight and narrow, you have a chance of getting people out.

"But with this bowling line up you wonder how the hell India are going to take England wickets without Zaheer Khan. Even more so, you wonder how they got to number one in the world," he added.

The Epicycle Model of Bhaskara

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 Epicycle Model of Bhaskara

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 True Equant

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 lunar anomaly was the Evection and the second, the Variation. The first inequality was the Equation of Center and the Evection and the Variation became the second and third inequalities. Actually Indian Astronomy recognised 14 major perturbations of the Moon and 14 corrections are therefore given to get the Cultured Longitude of the Moon, the Samskrutha Chandra Madhyamam. Then Reduction to Ecliptic is done to get the true longitude of Luna !



This diagram is by courtesy of Jean-Pierre Lacroix and Robert Baywater, www.ancientcartography.net

The True Equant

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 lunar anomaly was the Evection and the second, the Variation. The first inequality was the Equation of Center and the Evection and the Variation became the second and third inequalities. Actually Indian Astronomy recognised 14 major perturbations of the Moon and 14 corrections are therefore given to get the Cultured Longitude of the Moon, the Samskrutha Chandra Madhyamam. Then Reduction to Ecliptic is done to get the true longitude of Luna !



This diagram is by courtesy of Jean-Pierre Lacroix and Robert Baywater, www.ancientcartography.net

The Concentric Equant Model of Aryabhata

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