Showing posts with label MADHVA. Show all posts
Showing posts with label MADHVA. Show all posts

Sunday, April 5, 2015

Madhavacharya and Trignometry

The Madhava Trignometric series 

The Madhava Trignometric series is one of a series in a collection of infinite series expressions discovered by Madhava of Sangramagrama ( 1350-1425 ACE ), the founder of the Kerala School of Astronomy and Mathematics. These are the infinite series expansions of the Sine, Cosine and the ArcTangent functions and Pi. The power series expansions of sine and cosine functions are called the Madhava sine series and the Madhava cosine series.

The power series expansion of the arctangent function is called the Madhava- Gregory series.

The power series are collectively called as Madhava Taylor series. The formula for Pi is called the Madhava Newton series.

One of his disciples, Sankara Variar had translated his verse in his Yuktideepika commentary on Tantrasamgraha-vyakhya, in verses 2.440 and 2.441

Multiply the arc by the square of the arc, and take the result of repeating that (any number of times). Divide (each of the above numerators) by the squares of the successive even numbers increased by that number and multiplied by the square of the radius. Place the arc and the successive results so obtained one below the other, and subtract each from the one above. These together give the jiva, as collected together in the verse beginning with "vidvan" etc.

Rendering in modern notations

Let r denote the radius of the circle and s the arc-length.

The following numerators are formed first:


These are then divided by quantities specified in the verse.

2)s. s^2/(2^2+2)r^2. s^2/4^2+4)r^2
3)s.s^2/(2^2+2)r^2.s^2/(4^2+4)r^2. s^2/(6^2+6)r^2

Place the arc and the successive results so obtained one below the other, and subtract each from the one above to get jiva:

Jiva = s-(1-2-3)

When we transform it to the current notation

If x is the angle subtended by the arc s at the center of the Circle, then s = rx and jiva = r sin x.

Sin x = x - x^3/3! + x^5/5! - x^7/7!...., which is the infinite power series of the sine function.

By courtesy and we thank Wikipedia for publishing this on their site.

By means of the same argument, the circumference can be computed in another way too. That is as (follows): The first result should by the square root of the square of the diameter multiplied by twelve. From then on, the result should be divided by three (in) each successive (case). When these are divided in order by the odd numbers, beginning with 1, and when one has subtracted the (even) results from the sum of the odd, (that) should be the circumference. ( Yukti deepika commentary )

This quoted text specifies another formula for the computation of the circumference c of a circle having diameter d. This is as follows.

c = SQRT(12 d^2 - SQRT(12 d^2/3.3 + sqrt(12 d^2)/3^2.5 - sqrt(12d^2)/3^3.7 +.......

As c = Pi d , this equation can be rewritten as

Pi = Sqrt(12( 1 - 1/3.3 + 1/3^2.5 -1/3^3.7 +......

This is obtained by substituting z = Pi/ 6 in the power series expansion for arctan (z).

Pi/4 = 1 - 1/3 +1/5 -1/7+.....

This is Madhava's formula for Pi, and this was discovered in the West by Gregory and Liebniz.

Source :

Thursday, December 11, 2014


Now,people are blinded by falsehood of Newton as father of calculus.Infact he stole it from Indian,who practiced calculus in a school of Astronomy in Malabar,Kerla 2000 years before  Jesus came. It is same port where Vasco De Gama landed in 1498 for christian conversion of Hindus. Stupid King at that time let Vasco de Gama come but did not realize that it will be like East India Company which came to dupe Indians and Hindus like what Germans did ,stole Swastika and turned this pious sign to wrong end. Now with internet, people are understanding and started researching.

Astronomy school of Kerla and also Ujjain in Madhya Pradesh produced a student,namedAryabhatta who calculated value of Pi of 3.1416 and the solar year of 365.358 days . He is the one who produced heliocentric universe 4200 years before Copernicus, with elliptically orbiting planets and a spherical earth spinning on its axis explaining the motion of the heavens. Aryabhatta is father of Trigonometry and Algebra,when Europe was in the dark ages.. Westerners, both Germans and British stole his theory and propogated as their, but truth get revealed slowly but sure. Germans are destroyed ,so west heading that way because of stolen sanctity of pios ancient Indian scriptures and misusing it in wrong way.
Aryabhatta was the first to determine the circumference of the earth, with an error of 64 miles 4000 yrs ago. .Aryabhatta gave square, cube, triangle, trapezium, circle and sphere in geometry.
He was called Arjehir by the Arabs.
Remember that Galileo was killed by church when he told world what he learned from  Indian Scriptures that ,that it is earth  that circles the sun .(Aryabhatta explained this 4 millenuims years before)
Calculus was developed and many books were written ,some of them are here-
Parameshwara's book of Calculus , including Drigganita was available even to the Arabs. The East India company was based in Calicut. Several Europeans like Fillippo Sasetti who came to Kerala to study Sanskrit in the end of the 15th century. It was this Italian who revealed the vicious and secret Portuguese Inquisition ordered by St Francis Xavier , in Goa, to the Western world.  Finally it was the British who stopped this dutch Francis Xavior to stop but they took over from Dutch and ended of controlling India for 300 years. Time is ripe to pay it back now.
In 1580 Matteo Ricci borrowed Calculus Malayalam texts from Calicut kings in 1580, never to return it.
They took Calculus to Europe , from where the likes of Gottfried Wilhelm Von Liebniz , Isaac Newton and Robert Hooke raced with each other to translate , re-invent and market it in their own names, in a acrimonious manner.
Newton copied  laws of gravity from "Surya Sidhanta" the great Sanskrit astronomical work written in the Vedic age . Reproduced in another written text by Bhaskara , 1200 years before Newton it clearly explains gravity without an apple. However Vedic gravity was a push ( after observing the solar eclipse ) and NOT a pull. 

Before thete, it was a Greeck-Pythagoras" who copied Bhaskara's theorem from the great Sanskrit mathematics text Baudhayana Sulba Sutra, published thousands of years earlier.
Thanks to John Wallis , while he was the keeper of Oxford Univeristy archives who first started pondering over translated Mathematics stolen from India.
John Wallis patented Vedic Math infinity and infinitesimal ( Vishnu reclining in horizontal 8 position ) in his own name. Rest he could NOT understand and passed it to Newton.
You all know late Shakuntala Devi ,who has beaten all supercomputers in mathmatical calculation.
Also India produced great mathmatician-Ramanujan,but during British regime, he was not terated properly and was not given Noble Prize.

Newtons laws of motion were lifted from the Sanskrit texts of 4000 BC and Aryabhatta’s written work in 2700 BC in Sanskrit.
Newtons gravity laws of explaining gravity as pull or attraction  is all wrong.In fact  It is explained in GURUTVAKARSHAN that gravity is a property of matter and akasha warp. If Magnet is reason for gravity, how come SUN HAS IMMENSE GRAVITY WITHOUT MAGNET ,IRON PER SE AND MOON WITH MASS .
Newton's third law-about every action has an equal and opposite reaction is infact law of KARMA of Gita, explained that you get what you do.
Calculus was written in Malayalam atleast 200 years before Newton was born, by the Kerala school of Calculus. Calculus in Sanskrit was written 4800 years ago.
“When Vedic ideas are proved correct, it is just dreaming , come right. When Western work ( lifted from ancient Vedanta ) is proved right, it is scientific knowledge ” -- Nikola Tesla .

Pythogorus stole his theorem from Vedic texts 2500 years ago . Pythagoras was a student of Mathematics in India .
The 6000 year old Vedic texts taken to Palestine by King David's mistress the attractive dusky long haired Bathsheba ( Solomon's mother ) from Calicut in 3000 years ago. A lot of these ancient palm leaf texts were recovered from the Solomon's temple at Jerusalem by the Templar Knights .
P. Johnstone: “Gravitation was known to the Hindus (Indians) before the birth of Newton. The system of blood circulation was discovered by them centuries before Harvey was heard of.”

Aryabhatta’s (2700 BC), formula giving the tat-kalika-gati (instantaneous motion) is given by the following -

u'- v' = v' - v ± e (sin w' - sin w) (i)
where u, v, w are the true longitude, mean longitude, mean anomaly respectively at any particular time and u', v', w' the values of the respective quantities at a subsequent instant; and e is the eccentricity or the sine of the greatest equation of the orbit

"True motion in minutes is equal to the cosine (of the mean anomaly) multiplied by the difference (of the mean anomalies) and divided by the cheda, added or subtracted contrarily (to the mean motion)."
δu = δv ± e cos θ δ θ.

Proof of the Differential Formula

Let a point P (See Fig. 1) move on a circle. Let its position at two successive intervals be denoted by P and Q. Now, if P and Q are taken very near each other, the direction of motion in the interval PQ is the same as that of the tangent at P.

Let PT be measured along the tangent at P equal to the arc PQ. Then PT would be the motion of the point P if its velocity at P had not changed direction.
"The difference between the longitudes of a planet found at any time on a certain day and at the same time on the following day is called its (sphuta) gati (true rate of motion) for that interval of time."

The tatkalika-gati (instantaneous motion) of a planet is the motion which it would have, had its velocity during any given interval of time remained uniform."

In the figure given above, let the arc PQ = A. Then
R (sin
BOQ - sin BOP) = QN - PM = Qn
which is the Bhogya Khanda
Now from the similar triangles PTr and PMO
R : PT : : R cos w : Tr ................ (iii)
Tr = PT cos w.
But Tr = R(sin w' - sin w) and PT = R(w' - w)
(sin w' - sin w) = (w' - w) cos w.

Thus the Tat-kalika Bhogya Khanda (the instantaneous sine difference) in modern notation is
δ (sin θ) = cos θ δ θ.

This formula has been used by Bhaskara to calculate the ayana-valana ("angle of position").
the epithet Tat-kalika (instantaneous) gati (motion) to denote these differentials.
Theory of proportion (similar triangles) –
(1) The sine-difference sin (θ + δ θ) - sin θ varies as the cosine and decreases as θ increases.
(2) The cosine-difference cos (θ + δ θ) - cos θ varies as the sine negatively and numerically increases as θ increases.
(1) The difference of the sine-difference varys as the sine negatively and increases (numerically) with the angle.
(2) The difference of the cosine-difference varys as the cosine negatively and decreases (numerically) with the angle.

For Δ2 (sin θ),

the differential of an inverse sine function. This result--

if Tn denotes the nth jya (or Rsine), Cn the circumference of the corresponding circle, An the area of the nth annulus and S the area of the surface of the sphere, then we shall have

the summation being taken so as to include all the Rsines in a quadrant of the circle. Since there are ordinarily 24 Rsines in a Hindu trigonometrical table, we have

Hence approximately
S = 21600 x 2 x 3437
Area of the surface = circumference x diameter.

If ln denotes the length of the nth transverse arc, we have


the summation being taken so as to include all the Rsines. 

Hence the area of a lune is numerically equal to the diameter of the sphere. As the number of limes is equal to the number of parts of the circumference of the sphere, we get
Area of the surface = circumference x diameter.
It pays to remember that 4600 years back, half this planet was doing grunt grunt for language and wearing animal skins!

Rothschild , who owned British East India company gave a lot of stolen Vedic Maths and Astronomy papers in Sanskrit to their bloodline represented by German Jew Sir Frederick William Herschel ( 1738-1822 ) from Hanover. But this man was NOT smart enough. So Rothschild made a observatory for him, so that he could atleast patent the vedic Astronomical data in his name. The British made a big hue and cry when William "discovered " Uranus on March 13th 1781-- hi hi the stupid Indians never knew all this. . They were jubilant as Indian Vedic astrology does NOT use Uranus ( Shweta ) Neptune ( Shyama ) Pluto ( Teevra ), just because they are too far away to affect your DNA and they stay in one single rashi for too long. Vedic rishi astrologists did NOT need a telescope, they read off from Aakashik records.

Ptolemy came to India in 155 AD, and he stole from the astronomical data from Surya Siddhanta (12.85-90) , the most significant being the diameters of Mercury, Venus, Mars, Jupiter and Saturn . You must understand that these diameters were calculated accurately more than 6 millenniums ago when even the atmospheric refraction of earth was different.

Much before in 500 BC Pythagoras came to India and stole his theorem.

William passed over the stolen ( translated to English by Kashmiri Pandits ) papers to his son, Sir John Frederick Hershel, ( 1792-1871 ) an English citizen. He made full use of the Chemical and Botanical Vedic papers too. He also dabbled with Kerala Calculus. This man is buried next to Isaac Newton and Charles Darwin, in Westminister Abbey-- next to the English Kings--probably to ironically reveal, that these men were NOT scientists , but thieving politicians.

John send his son Sir William James Herschel ( 1833 -1917 ) to steal more, which he did --and how! He patented the ancient Indian finger printing method . The Indian Panchatantra stories, which were written 5000 years ago, have episodes of written contracts , signed by a indelible Indian ink thumb print. James was Rothschilds represenative in India, overseeing the change over after the First war of Independence ( Sepoy's mutiny ) in 1857--merrily taking finger prints all all and sundry--this man really got a kick out of all this.
Gravity thief Isaac Newton stole everything from GURUTWAKARSHANA the pioneering work in Sankrit of astronomer Mihira Muni ( sage Varahamihira ) in 2660 BC. Mihira Muni was the disciple of Mathematician Aryabhatta. Mihira Muni's observatory was at Sultan Bathery , Western Ghat mountains , Kerala --where today you can see a 3800 year old Jain temple --converted by Muslim Invader Tipu Sultan as his fort. He stayed at Kapletta at a lake by the name of Pookode lake.
Without understanding the concept of Akasha , written in Vedic texts -- he hastily propounded the " Aether Wave theory"-- and fell flat on his face when asked to explain refraction of light and diffusion. ( the same way his gravitation theory is bullshit too- little knowledge is a dangerous thing !.
 When super genius Indian Mathematician Srinivasan Ramanujan arrived at London, he was greeted by Professor Hardy. Hardy made a innocent remark that the number of the taxi , he came in is 1729-- a boring number.

Ramanujan looked at the number plate himself and replied casually in a knee jerk manner "No, actually it is a very interesting number. .It is the smallest natural number representable in two different ways as the sum of two cubes" This is known as equation HARDY-RAMANUJAN NUMBER,(Hardy has nothing to do except he was British,white).


BBC documentary on Madhavan, Aryabhatta ( they say 600 AD while it is 2700).

The sages who gave this planet the Vedas , Upanishads did not care to leave their names; the truths they set down were eternal, and the identity of those who arranged the words irrelevant.
The mathematical value pi has always been a curious number throughout history.ratio of a circle's circumference to its diameter,
Here is Aryabhatta's version of the value of Pi in 2700 BC:
Lets split the words and understand the meaning:
चतुरधिकम्शतम्- Four more than hundred (=104)
अष्टगुणम्- multiplied by 8 (104 x 8 = 832)
द्वाषष्टि= 62
तथासहस्राणाम्= of 1000 as such (=62000; totalling 62832)
अयुतद्वय= 10,000 x 2 (=20,000)
विष्कम्भस्य= of the diameter
आसन्न:- approximately
वृत्तपरिणाह:- to the circumference.
In effect, 62832/20000 = 3.1416 !

It interesting to note the large numbers he has used to arrive at Pi and the remark that pi is only an approximate value.

And we have also coded it in Mantras by the KATAPAYADI SYSTEM
Vararuchi was a Mathematician from Kerala who taught at Bhoj Shala University inside the Saraswati Temple in 2860 BC . Today this ancient Hindu university is a Muslim mosque.

Gopibhagya madhuvrata srngisodadhisandhiga|
Khalajivitakhatava galahalarasandhara||
Kaṭapayādi system dictates that:
* As the first digit is 3, the first consonant must be one of ga, ḍa, ba, la
* As the second digit is 1, the second consonant must be one of ka, ṭa, pa, ya
* For the third digit to be 4, the third consonant must be one of gha, ḍha, bha, va…
So to fit3.141592653589793…, the list of consonants in the verse must satisfy the regex
ga - 3 pii - 1 bhaa - 4 gya - 1 ma - 5 dhu - 9 ra - 2 ta -6 shru - 5 ga - 3 sho - 5 da - 8 dhi - 9 sa -7 dha - 9 ga - 3 kha - 2 la - 3 jii - 8 vi - 4 ta - 6 kha - 2 ta - 6 va - 4 ga- 3 la - 3 ra - 2 sa - 7 dha - 9 ra – 2

pi = 3.1415926535897932384626433832792

From Vadakail- Thanks for great knowledge.


Tuesday, January 28, 2014


Madhava of Sangamagramma was born near Cochin on the coast in the Kerala state in southwestern India. It is only due to research into Keralese mathematics over the last twenty-five years that the remarkable cont...ributions of Madhava have come to light. In Rajagopal and Rangachari put his achievement into context when they write:-

[Madhava] took the decisive step onwards from the finite procedures of ancient mathematics to treat their limit-passage to infinity, which is the kernel of modern classical analysis.

All the mathematical writings of Madhava have been lost, although some of his texts on astronomy have survived. However his brilliant work in mathematics has been largely discovered by the reports of other Keralese mathematicians such as Nilakantha who lived about 100 years later.

Madhava discovered the series equivalent to the Maclaurin expansions of sin x, cos x, and arctan x around 1400, which is over two hundred years before they were rediscovered in Europe. Details appear in a number of works written by his followers such as Mahajyanayana prakara which means Method of computing the great sines. In fact this work had been claimed by some historians such as Sarma (see for example [2]) to be by Madhava himself but this seems highly unlikely and it is now accepted by most historians to be a 16th century work by a follower of Madhava. This is discussed in detail in [4].

Jyesthadeva wrote Yukti-Bhasa in Malayalam, the regional language of Kerala, around 1550. In [9] Gupta gives a translation of the text and this is also given in [2] and a number of other sources. Jyesthadeva describes Madhava's series as follows:-

The first term is the product of the given sine and radius of the desired arc divided by the cosine of the arc. The succeeding terms are obtained by a process of iteration when the first term is repeatedly multiplied by the square of the sine and divided by the square of the cosine. All the terms are then divided by the odd numbers 1, 3, 5, .... The arc is obtained by adding and subtracting respectively the terms of odd rank and those of even rank. It is laid down that the sine of the arc or that of its complement whichever is the smaller should be taken here as the given sine. Otherwise the terms obtained by this above iteration will not tend to the vanishing magnitude.

This is a remarkable passage describing Madhava's series, but remember that even this passage by Jyesthadeva was written more than 100 years before James Gregory rediscovered this series expansion. Perhaps we should write down in modern symbols exactly what the series is that Madhava has found. The first thing to note is that the Indian meaning for sine of θ would be written in our notation as r sin θ and the Indian cosine of would be r cos θ in our notation, where r is the radius. Thus the series is

r θ = r(r sin θ)/1(r cos θ) - r(r sin θ)3/3r(r cos θ)3 + r(r sin θ)5/5r(r cos θ)5- r(r sin θ)7/7r(r cos θ)7 + ...

putting tan = sin/cos and cancelling r gives

θ = tan θ - (tan3θ)/3 + (tan5θ)/5 - ...

which is equivalent to Gregory's series

tan-1θ = θ - θ3/3 + θ5/5 - ...

Now Madhava put q = π/4 into his series to obtain

π/4 = 1 - 1/3 + 1/5 - ...

and he also put θ = π/6 into his series to obtain

π = √12(1 - 1/(3×3) + 1/(5×32) - 1/(7×33) + ...

We know that Madhava obtained an approximation for π correct to 11 decimal places when he gave

π = 3.14159265359

which can be obtained from the last of Madhava's series above by taking 21 terms. In [5] Gupta gives a translation of the Sanskrit text giving Madhava's approximation of π correct to 11 places.

Perhaps even more impressive is the fact that Madhava gave a remainder term for his series which improved the approximation. He improved the approximation of the series for π/4 by adding a correction term Rn to obtain

π/4 = 1 - 1/3 + 1/5 - ... 1/(2n-1) ± Rn

Madhava gave three forms of Rn which improved the approximation, namely

Rn = 1/(4n) or
Rn = n/(4n2 + 1) or
Rn = (n2 + 1)/(4n3 + 5n).

There has been a lot of work done in trying to reconstruct how Madhava might have found his correction terms. The most convincing is that they come as the first three convergents of a continued fraction which can itself be derived from the standard Indian approximation to π namely 62832/20000.

Madhava also gave a table of almost accurate values of half-sine chords for twenty-four arcs drawn at equal intervals in a quarter of a given circle. It is thought that the way that he found these highly accurate tables was to use the equivalent of the series expansions

sin θ = θ - θ3/3! + θ5/5! - ...

cos θ = 1 - θ2/2! + θ4/4! - ...

Jyesthadeva in Yukti-Bhasa gave an explanation of how Madhava found his series expansions around 1400 which are equivalent to these modern versions rediscovered by Newton around 1676. Historians have claimed that the method used by Madhava amounts to term by term integration.

Rajagopal's claim that Madhava took the decisive step towards modern classical analysis seems very fair given his remarkable achievements. In the same vein Joseph writes in :-

We may consider Madhava to have been the founder of mathematical analysis. Some of his discoveries in this field show him to have possessed extraordinary intuition, making him almost the equal of the more recent intuitive genius Srinivasa Ramanujan, who spent his childhood and youth at Kumbakonam, not far from Madhava's birthplace.