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.

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 takes cases throughout the state of Hawai. Those who are injured in motor cycle and pedestrian accidents, he can be of immense help. Also clients injured in truck or car accidents. You can call today for a free consultation with David W. Barlow, Personal Injury Lawyer Honolulu

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 !

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

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 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

India should remember catches win matches !

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 Lunar Model Of Munjala

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

Ancient Indian Mathematics

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 model found in the Almagest, and furthermore, that analysis of the level of development of Indian astronomy contemporary to their planetary schemes strongly suggests, but does not rigorously prove, that the planetary bisected equant model is pre-Ptolemaic. "

The mutli step algorithms of Indian Astronomy never approximated any Greek geometrical model. Ptolemy's Almagest was the first book, according to Western Astronomy. We have now the information that Ptolemy did not invent the equant.

Bhaskara II was an astronomer-mathematician par excellence and his magnum opus, the Siddhanta Siromani (" Crown of Astronomical Treatises") , is a treatise on Astronomy and Mathematics. His book deals with arithemetic, algebra, computation of celestial longitudes of planets and spheres. His work on Kalana ( Calculus ) predates Liebniz and Newton by half a millenium.

The Siddanta Siromani is divided into four parts


1)The Lilavati - ( Arithmetic ) wherein Bhaskara gives proof of c^2 = a^ + b^2. The solutions to cubic, quadratic and quartic indeterminate equations are explained.

2)The Bijaganitha ( Algebra )- Properties of Zero, estimation of Pi, Kuttaka ( indeterminate equations ), integral solutions etc are explained.

3)The Grahaganitha ( Mathematics of the planets ).


For both Epicycles

The Manda Argument , Mean Longitude of Planet - Aphelion = Manda Anomaly

The Sheegra Argument, Ecliptic Longitude - Long of Sun = Sheegra Anomaly

and the computations from there on are explained in detail.

4)The Gola Adhyaya ( Maths of the spheres )


Bhaskara is known for in the discovery of the principles of Differential Calculus and its application to astronomical problems and computations. While Newton and Liebniz had been credited with Differential Calculus, there is strong evidence to suggest that Bhaskara was the pioneer in some of the principles of differential calculus. He was the first to conceive the differential coefficient and differential calculus.

Sachin, the best in the world

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 started playing (Test cricket) at the age of 16. And at 38 now, we have not seen a better player. Sachin is the best batsman in the world."

"I am going to the Lord's to see Sachin make his 100th century," he said.

Dravid, the master batsman, described as the Wall of India, said "In India we have many Gods and he is one of them."

When interviewed by cricket commentator Henry Blofeld, Lara said "While Sachin is special, Dravid is the wall."

"Dravid is always going to be there - he is a tremendous player."

Khagola, The Celestial Coordinate System

( 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 from 0 degrees Beta Arietis, Ashwinyadi, which is the beginning point of the Sidereal Zodiac. This is the Nirayana System, sidereal. The Tropical System, Sayana, also has its adherents in India and starts from Meshadi, 0 degree Aries.



The Galactic Center, the Vishnu Nabhi, lies in Sagittarius. NEP is the North Ecliptic Pole, NGP is the North Galactic Pole and NCP is the North Celestial Pole.

It is raining cats and dogs in Kerala

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.

The Double Epicyclic Model of India

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 are those books, prior to the Almagest, which had influenced the Indian system? The answer is "unknown sources".




Remarks Duke " Indeed, since the very earliest investigation of the Indian models by Western scholars it has been presumed that the models are somehow related to a double epicycle system, with one epicycle accounting for the zodiacal anomaly, and the other accounting for the solar anomaly (retrograde motion) This perception was no doubt reinforced by the tendency of some Indian texts to associate the manda and sighra corrections with an even older Indian tradition of some sort of forceful cords of air tugging at the planet and causing it to move along a concentric deferent . Since our goal in this paper is to investigate the nature of any connection with ancient Greek planetary models, it is only important to accept that the models appear in Indian texts that clearly pre-date any possible Islamic influences, which could, at least in principle, have introduced astronomical elements that Islamic astronomers might have derived from Greek sources. ( "The Equant in India: the Mathematical Basis of Ancient Indian Planetary Models" By Dennis Duke, Florida State University )

Computation of Geocentric Distance, Sheegra Karna

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.

Calculation of the geocentric longitude of Mercury

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.

Indian Astronomy Pre-Ptolemaic

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 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 model found in the Almagest, and furthermore, that analysis of the level of development of Indian astronomy contemporary to their planetary schemes strongly suggests, but does not rigorously prove, that the planetary bisected equant model is pre-Ptolemaic" says he.

The earliest Indian Planetary Models are two sets from the writer Aryabhata, both dating from 6th Century AD.

1) The Sunrise System , after the Epoch, which is taken from the sunrise of 18th Feb 3102 ( Arya Paksha ). It appears first in Aryabhatiya

2) The Midnight System, after the Epoch, which is taken from the midnight of 17/18 FEB 3102 ( Ardha Ratri Paksha ). It appears first in Latadeva's Soorya Siddhanta


The Local Meridien is taken as Lanka, Longitude 76 degrees, Latitude 0 degrees.

Of Manda and Sheegra Epicycles

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.

Vyasardha, the Radius of the Circle

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 of the values of trignometric sines expressed in arcminutes. Because of this, this Table is referred to as the Table of Sine Differences.

Aryabhata's Sine Tables

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 Chapa.

Bhujajya is radius multiplied by modern sine and bhujachapa is the arcsine. Kotijya is radius multiplied by modern cosine and Kotichapa is arccos. Sparshjya is tan and sparshachapa is arctan.

Aryabhata's Sine Table was the first ever constructed sine table in the History of Maths.


This is Aryabhata's Sine Table given for different Kakshyas ( One Kakshya is 3 degrees 45 mins, one eighth of 30 degrees Zodiacal Sign )

Sl. No Angle ( A ) (in degrees, arcminutes) Value in Āryabhaṭa's numerical notation
(in Devanagari) Value in Āryabhaṭa's numerical notation (in ISO 15919 transliteration) Value in Arabic numerals Āryabhaṭa's value of jya (A) Modern value of jya (A)
(3438 × sin (A))

1 03° 45′ मखि makhi 225 225′ 224.8560
2 07° 30′ भखि bhakhi 224 449′ 448.7490
3 11° 15′ फखि phakhi 222 671′ 670.7205
4 15° 00′ धखि dhakhi 219 890′ 889.8199
5 18° 45′ णखि ṇakhi 215 1105′ 1105.1089
6 22° 30′ ञखि ñakhi 210 1315′ 1315.6656
7 26° 15′ ङखि ṅakhi 205 1520′ 1520.5885
8 30° 00′ हस्झ hasjha 199 1719′ 1719.0000
9 33° 45′ स्ककि skaki 191 1910′ 1910.0505
10 37° 30′ किष्ग kiṣga 183 2093′ 2092.9218
11 41° 15′ श्घकि śghaki 174 2267′ 2266.8309
12 45° 00′ किघ्व kighva 164 2431′ 2431.0331
13 48° 45′ घ्लकि ghlaki 154 2585′ 2584.8253
14 52° 30′ किग्र kigra 143 2728′ 2727.5488
15 56° 15′ हक्य hakya 131 2859′ 2858.5925
16 60° 00′ धकि dhaki 119 2978′ 2977.3953
17 63° 45′ किच kica 106 3084′ 3083.4485
18 67° 30′ स्ग sga 93 3177′ 3176.2978
19 71° 15′ झश jhaśa 79 3256′ 3255.5458
20 75° 00′ ङ्व ṅva 65 3321′ 3320.8530
21 78° 45′ क्ल kla 51 3372′ 3371.9398
22 82° 30′ प्त pta 37 3409′ 3408.5874
23 86° 15′ फ pha 22 3431′ 3430.6390
24 90° 00′ छ cha 7 3438′ 3438.0000

Sine Table by courtesy www.wikipedia.org

The Epicyclic Theory of Indian Astronomy

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 Jya Ganitha Kriya, a trignometric reduction of the mean longitudes and distances of the planets to their heliocentric longitudes and distances.

India emerge triumphant !

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 to form of Ishant, who bowled with fire to take 22 wickets in the Test series.

Nobody thought that the World Champions would reign supreme in both the formats and that too for the first time in the Carribean. With the Test at Dominica ending in a draw, India ended her Carribean cruise, emerging victors in both formats of the game.

Strauss has warned that England will give India the baptism of fire and Indians are relieved that the veterans are back in the game. India now take on England in England ! Will Dhoni's men rise up to the challenge ?

India comes back with a bang !

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 and took three of the four Indian wickets that fell, including the prize scalp of Dhoni who was out for 74.

Kranti, the Sun's declination

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 Eccentric Theory of Indian Astronomy

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 Equation of Centre at both perigee and apogee becomes zero.

When the Mean Anomaly lies between 0 and 180 degrees, the Equation of Center is negative and this is known in Vedic Astronomy as Meshadi Rinam, Rinam meaning minus. And when it is between 180 and 360, it is additive and it is known in Indian Astronomy as Thuladi Dhanam, dhanam meaning additive.

Anomaly has been defined in Western Astronomy as the angle between position and perihelion. Manda Kendra here is the angle between Position and Aphelion, aphelion being mandoccha. The term manda explains that planets move slower at Aphelion !

Dhoni redeems India

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 Raina's wicket. Dhoni and Raina added 103 runs for the sixth wicket, before Raina was trapped lbw by the pacer Fidel Edwards.

Ishant & Co skittle out WI for 204

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 haul in Tests. Fidel Edwards pulled injudiciously and WI were 204/9, and Bishoo dragged one without opening his account and Windies were skittled out for 204.

Chara, the Ascensional Difference

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.

Ishant Sharma strikes again !

Ishant Sharma struck twice, removing Kirk Edwards and Adrian Barath and WI were reeling at three down for 40 !

It was the swerve of Praveen Kumar which removed debutant Keiron Powell. Powell dabbed at a Praveen delivery to be caught in the second slip by Laxman. Dhoni asked the WI to bat first and got the opportunity to bowl first in the series. The reason was simple. There was moisture in the pitch. Powell was out for three and paid the price for fishing outside the off stump ! A Sharma bouncer accounted for Barath, as Barath tried to pull, but only succeeded in dragging the ball onto his wicket. He made 12 off 38 balls. Then Sharma induced Edward to hook and was caught by Dhoni.

WI finished at 75/3, with the south paws, Chanderpaul and Darren Bravo on 17 and 22 respectively.

After 100 years, Dominica got her First Test match and the Dominican PM was delighted about it.

Reduction to Geocentric Coordinates

In the above diagram,

m is the Sheegra Anomaly
and the angle EJS is the Sheegra Phalam

where E = Earth, S = Sun and J = Jupiter


In order to get the geocentric longitude, the longitude of the Sun is deducted from the Ecliptic longitude and then we get the Sheegra Kendra, the angle between the planet and the Earth Sun plane.

Ecliptic longitude - Long Sun = Sheegra Kendra

Arka Sphutnoniham Kheda Manda Sphutamihodhitham

Sheegra Phalam is the angle EJS ( Earth, Jupiter, Sun ) and this Sheegra Phalam is deducted to get the true, geocentric longitude. ( Added if Sheegra Anomaly > 180 and deducted if Sheegra Anomaly < 180 ).

Like Kepler, the Indian astronomers may not have said that all planets move fastest at perihelion but this principle was known to them as they called perihelion Sheegrochha. Sheegra in Sanskrit means fast. Similarly they called Aphelion Mandochcha, manda meaning slow, and it implies that all planets move slowly at Aphelion !

Sheegra Kriya, reduction to geocentric longitude

The Vedic astronomers never said like Kepler that planets traverse in elliptical orbits. But then they talked about Eccentric and Epicyclic theories. They said for the luminaries, the Sun and the Moon, reduction to the geocentric (Sheegra Kriya ) coordinate system was not necessary. But for the Tara Grahas, like Mercury, Venus, Jupiter, Mars and Saturn, both the jya samskaras ( trignometric corrections ) are necessary.

The Vedic astronomical texts say that the planets go in an epicycle, whose center moves along the mean circular orbit from west to east. This theory came to be known as the Epicyclic Theory. There is another theory called the Eccentric Theory, which states that a planet goes in a circle whose center is not the Earth, but a different point other than the Earth.

Bhaskara says " The Center of the Celestial Sphere coincides with the center of the Earth. The circle in which a planet goes does not have its center coinciding with the Center of the Earth". Hence the Vedic astronomers like Bhaskara, Brahmagupta and Aryabhata, prescribed bhujaphalam ( otherwise called Equation of Center for mandaphala, whereas in the case of the five Tara Grahas, this bhujaphala, despite connoting Equation of Center or mandaphala, also stands for corrections required to reduce their longitudes from the Heliocentric Coordinate System to the Geocentric Coordinate System ).

Sreeshant is in !

Sreeshant is back in the Indian Test team to tour England. Sehwag will miss the first two weeks due to his injury, but will play after that.

Zaheer, Munaf, Sreeshant join the triumphant duo of Praveen and Ishant Sharma. The spin attack is in the hands of Bajji.

Yuvraj, Gambhir, Sachin, Sehwag all return to the India squad. It is a full strength squad, rising to the challenge of England, who said they will beat India in their home ground.

Sachin came back after skipping the Carribean tour because of badly needed rest, Yuvraj also came back after missing the WI tour due to chest infection while Gambhir, Gautam and Sreeshant came back after injury breaks. Kohli and Vijay were dropped, while Raina's good performance helped him to retain his place.

The reserve opener Abhinav Mukund goes to England and Saha edged out Parthiv Patel. Cheteswar Pujara missed out again due to injury.

Munaf came back after injury problems and he along with Zaheer and Sreeshant and the in form Ishant and Praveen will pose problems for the English batsmen, particularly in a country where the ball swings from morn till eve.

Sreeshant said that he will waiting for the Test at Lords, to see Sachin crack his 100th century !

Squad

Dhoni ( wk and cpt ), Yuvraj, Sachin, Sehwag, Dravid, Laxman, Gambhir, Raina, Mukund, Saha ( reserve wk ), Amit Mishra, Harbhajan, Zaheer, Munaf, Sreeshant, Praveen Kumar and Ishant Sharma.

Ishant Sharma picks up ten wickets

Ishant took 4/53, as West Indies struggled to draw the Second Test with India. Ishant took his haul to ten wickets in the match. He will be spearheading the Indian attack for years to come.

Earlier Edwards swung the ball and removed Kohli (27), Laxman (87) and Dhoni (6) and India declared at 270/6. A challenging 281 runs to get, but the WI batsmen made heavy weather of the situation and lost seven wickets for 202.

Ishant took 10/108 in the match, signalling his return to the Indian team. Baugh played a solid 46 not out and Rampaul (0) were at the crease, when stumps were drawn.

India leads the series 1-0, having won the First Test.

Kohli edged Edwars to Sammy in the slips, and Laxman was removed in the same fashion. And then Edwards shot out Dhoni, who gave a simple catch to midoff. Raina with 16 and Bajji with 6 were at the crease, when Dhoni declared the innings closed. setting 281 for the WI to win.

Ishant removed Simmons early and Praveen accounted for the scalp of Sarwan, who was caught in the slips. The early loss of Simmons and Sarwan put WI on the defensive. Even though Darren Bravo played well for his 73, none of the WI batsmen could support him. Baugh with 46 not out saw to that WI drew the match !

After lunch, Ishant removed Barath for 27. Barath edged one to Raina. Bravo and Chanderpaul consolidated the WI innings and the visitors could not make any breakthrough. WI were 108/3 at tea. Chanderpaul was adjudged lbw to Bajji and WI were in the doldrums ! The left hander tried to cut Bajji and the ball was leaving the off stump, but the umpire thought otherwise and gave him out ! With a glorious inswinger, Ishant trapped Samuels lbw and WI were five down ! Then Dhoni dropped Baugh. If he had held on, history would have been otherwise. Mithun removed Bravo. Now all eyes are riveted on Jul 6th, when the Third Test starts at Dominica.

Laxman powers India to respectability

Laxman scored 72 not out and India recovered to 229/3, after the early loss of M Vijay. Rahul Dravid was at his scintillating best and scored 55 and steadied the Indian innings, like a true Wall ! India lead by 240 runs.

Laxman with 72 not and Kohli wiht 26 not out were at the crease when stumps were drawn. Rampaul dismissed Vijay (3), when the right hander lunged forward to a well pitched delivery that kissed the otuside edge of his bat, before going into the hands of Baugh.

Mukund drove Sammy for four and Dravid steered Edwards for his first four and their partnership blossomed. Dravid survived a catch appeal, when Barath plucked the ball out of the ground. Dravid stood his ground and the umpires after a lenghty consultation gave him not out.

Mukund followed an Edwards delivery into the hands of Baugh and India had lost two wickets. Then Laxman and Dravid pulled India out of the danger zone, both scoring brilliant half centuries.

In the post tea session, India scored 81 runs, thanks to Dravid and Laxman. The immense experience of both paved the way for a huge Indian score. Dravid then steered Edwards into the safe hands of Baugh. Virat Kohli joined the in form Laxman, who has now hit two half centuries in this Test Match.

Virat Kohli had to undergo the baptism of fire, a flurry of fast, short pitched deliveries from the fiery Edwards. He pulled Edwards for six, but looked a bit fidgety in his short, unconquered innnigs. He ensured that India did not lose more wickets yesterday.