The Alpha Numeric System of Mathematica Indica

Known as Paral Sankhya, the Alpha Numeric System was developed by Vararuchi. It is also known as Katapayadi, as 1 represents the letters Ka, Ta, Pa and Ya, whose corresponding digit is 1.


Number Letters

1 ka ta pa ya
2 kha tta pha ra
3 hu da bha la
4 gha da ba va
5 nga na ma sha
6 cha tha sha
7 chha dha sa
8 ja dha ha
9 jha ddha
0 nja na ksha

The Value of Kaa ki ku ke and ko was 1


31 was represented by Kala

Kala is ka + la = 31 ( The digits are counted from the right )

and 351 by Kamala
Kamala = ka + ma + la = 351

Elemental Numbers in Mathematica Indica

Sanskrit poets used Elemental Numbers in their poems. As eyes are 2, two represents eyes. As the Vedas are four, four represents the Vedas. Netra Netra ( eyes , eyes ) means 22 and Veda Veda means 44.

Here we give the correspondence between numbers and the Elements


1 - Earth, Moon
2. Eyes, hands, feet, ears
3. Worlds, trinitarian unity, three yogas, three vedas.
4. Vedas
5. Sense organs.
6 Rithus or the six seasons.
7. Rishies
8 Directions, eightfold prosperity
9 Digits, numbers
0 Ether


10 - Ego and its ten heads.
11 - Rudras
12 Months, Zodiacal Signs
13 Universe
14 Vidyas, Manus
15 Lunations, thidhis
16 King
17 Athushti
18 Treatises astronomical, mythological
19 Athi Dhruthi
20 Nails
21 Uthkruthi
22 Akrithi
23 Vikruthi
24 Gayathri
25 Principles
27 Constellations
33 Deities


The digits for the letters are written, counting from right to left

For example,

Bhanandagni

Bham = Stars = 27
Nandan = 9
Agni = 3

Bhanandagni = 3927

Of Scholarship

Scholarship is great and is like an expanding flower

Pandityam nava pallavam vikasitham pushpam prayogajnatha

In order to have scholarism graced, one should strive hard to gain it. It is said that the hard work done by the scholar is known only to the scholar ! How can others know it ?

The scholar and the world, the endless strife
The discord in the harmonies of Life
The love of Learning, the sequestered nooks
And the sweet serenity of Books
The marketplace, eager love for gain
Whose aim is vanity, whose end is pain !


Vidwaneva Vijanathi
Vidwajjana parishramam
Na hi vandhya vijanathi
Gurveem prasava vedanam !

How can others know the work hard
Which the scholars have indulged in ?
Like a barren woman who cannot know
The pain of delivering !

Names of big Indian numbers ( contd )

There are greater numbers, than the ones mentioned below.

Maha Padma = 10^15
Kshoni = 10^16
Parardha = 10^17
Sankha = 10^18
Maha Sankha = 10^19
Kshithi = 10^20
Maha Kshiti = 10^21
Kshobha = 10^22
Maha Kshobha = 10^23

Neethi Lamba = 10^27
Sarva Bala = 10^45
Thallakshana = 10^83

Indian names for big numbers

Ganitha Kerala, the Kerala School of Astronomy and Maths, has given a lot of names for big numbers.

Like the Western

Myriad = 10^4
Million = 10^6

Indian names are

Shata = 10^2
Sahasra = 10^3
Ayutha = 10^4
Laksha = 10^5
Dasa Laksha = 10^6 ( Million )
Koti = 10^7
Dasa Koti = 10^8
Vrinda = 10^9
Kharva = 10^10
Nikharva = 10^11
Mahapadma = 10^12
Mahakharva = 10^13
Padma = 10^14

In the Western, we have

Million = 1000^2
Billion = 1000^3
Trillion = 1000^4

or

Nillion = 1000^(n+1)

where n is the number. Tri meaning 3 and trillion = 1000^(3+1)!

Of Mathematica Vedica

By one more than the one before

Ekadhikena Purvecha is Sanskrit for " One more than the previous one". This
sutra can be used for multiplying or dividing algorithms.

You can use this formula to compute the squares of numbers.

For example.

35×35 = ((3×3)+3),25 = 12,25 and 125×125 = ((12×12)+12),25 = 156,25

By the sūtra, multiply "by one more than the previous one."

35×35 = ((3×4),25 = 12,25 and 125×125 = ((12×13),25 = 156,25

India will overtake Japan by 2012

India may overtake Japan by 2012 and become the third largest economy by 2012. According to Pricewatherhouse Coopers, India's GDP PPP will overtake Japan by 2012. Such a forecast seems to be probable, as Jupiter will be in the House of Gains for India next year. Jove in the 11th will bring handsome gains.

China will become the world's largest economy by 2020. Within one year, Indian GDP PPP will become $4.32 trillion, overtaking Japan's economy, estimated at $1000^4 ( 4.32 tr to be exact ).

“While the exact date is open to doubt, it seems highly likely that, by 2030, China will clearly be the largest economy in the world on PPP,” John Hawksworth, head of macroeconomics at PwC, wrote in the report.

The Decoupling theory, that India and China will not be affected by the global meltdown, still holds true. China will overtake Japan as the second largest economy this year, as Japan is facing a severe crisis !.

Yesterday the Sensex grew by 353 points and closed about 17 K. Despite the secondary reactions, the Stock Market seems to be on a growth track.

The Churning of the Milk Ocean at Bangkok Airport

We were amazed to see Palazhi Madhanam, the Churning of the Ocean, depicted at
Thai Suvarnabhoomi Airport.

The Churning of the Ocean is allegorical, symbolising the great fight in the human bosom between Virtue and Vice. The forces of Virtue, the celestials, the devas fight the forces of Vice, the demons, the asuras. In this fight Immortality is the end result and Immortality is nothing but Self Actualisation !

P N Oak's theory that the base Civilization was Vedic from which everything began seems to be proved right !




Dr Kenneth Chandler averrs that the Vedic Civilizations is 10,000 years old.

He writes in his thesis " The Origins of Vedic Civilization" ....

How Ancient is Vedic Civilization?

Astronomical References in the Rig Veda and Other Evidence

Evidence from other sources known since the late nineteenth century also tends to
confirm the great antiquity of the Vedic tradition. Certain Vedic texts, for example,
refer to astronomical events that took place in ancient astronomical time. By calculating
the astronomical dates of these events, we thus gain another source of evidence that can
be used to place the Rig Veda in a calculable time-frame.


A German scholar and an Indian scholar simultaneously discovered in 1889 that the
Vedic Brahmana texts describe the Pleiades coinciding with the spring equinox. Older
texts describe the spring equinox as falling in the constellation Orion. From a
calculation of the precision of the equinoxes, it has been shown that the spring equinox lay in Orion in about 4,500 BC.



The German scholar, H. Jacobi, came to the conclusion that the Brahmanas are from a
period around or older than 4,500 BC. Jacobi concludes that “the Rig Vedic period of
culture lies anterior to the third pre-Christian millennium.”22 B. Tilak, using similar astronomical calculations, estimates the time of the Rig Veda at 6,000 BC.23
More recently, Frawley has cited references in the Rig Veda to the winter solstice
beginning in Aries. On this basis, he estimates that the antiquity of these verses of the Veda must go back at least to at least 6,500 BC.24 The dates Frawley gives for Vedic civilization are:

Period 1. 6500-3100 BC, Pre-Harappan, early Rig Vedic
Period 2. 3100-1900 BC, Mature Harappan 3100-1900, period of the Four Vedas
Period 3. 1900-1000 BC, Late Harappan, late Vedic and Brahmana period

Professor Dinesh Agrawal of Penn State University reviewed the evidence from a
variety of sources and estimated the dates as follows:

• Rig Vedic Age - 7000-4000 BC
• End of Rig Vedic Age - 3750 BC
• End of Ramayana-Mahabharat Period - 3000 BC
• Development of Saraswati-Indus Civilization - 3000-2200 BC
• Decline of Indus and Saraswati Civilization - 2200-1900 BC
• Period of chaos and migration - 2000-1500 BC
• Period of evolution of syncretic Hindu culture - 1400-250 BC

The Taittiriya Samhita (6.5.3) places the constellation Pleiades at the winter solstice, which correlates with astronomical events that took place in 8,500 BC at the earliest.

The Taittiriya Brahmana (3.1.2) refers to the Purvabhadrapada nakshatra as rising due
east—an event that occurred no later than 10,000 BC, according to Dr. B.G.Siddharth
of India’s Birla Science Institute. Since the Rig Veda is more ancient than the
Brahmanas, this would put the Rig Veda before 10,000 BC.

Ramayana Garden in Thailand

This is the photo of the Ramayana Garden in Thailand.

The first poem and the greatest was Ramayana, opined Aurobindo. When the poet Valmiki saw two birds mating and a native killing one of the two said " Ma Nishada ", meaning " Dont do that, for it is Sin, it is Evil, it is against the Law of Conscience"!

Ma nishada pratishtantva nigama shashvathi sama
Yat krouncha midhunath eva avadhim kamamohitam

Ramayana is an allegorical and epic poem. As the Aryan Invasion Theorists argue, it is the not onslaught of the " Aryans" against the " Dravidians". The Aryan Invasion Theory has been proved wrong by scholars like Dr Konrad Elst, Dr Dennis Chandler, Aurobindo and Vivekananda.



Man's dual aspect is highlighted in this poem. The Ego is the imperfect term in Man and Self or the Divine in the perfect term. The Ego is the lower imperfect term of our being; Self is the higher, perfect term.

The Ego is symbolised as Ravana, the ten headed demon king. His ten heads represent Mind, Intelligence, Processing Mind, Ego, Lust, Greed, Anger, Avarice, Jealousy, Gluttony and Sloth.The Self is symbolised by Rama, who goes in search of Peace, Seetha, his beloved consort. The great struggle for supremacy between good and evil, light and darkness, prosperity and adversity, happening in the human bosom, is the Ramayana, where the forces of the Ego clash with the forces of the Self.

Ramayana is part of Buddhist lore and the Ramayana Garden is a testimony to this fact. This strengthens P N Oak's theory that the mother civilization of all was Vedic.

Saturn in Thailand

Shani, the deity of Justice, Destiny and Retribution, has His presence in Thailand. In ancient Siam, one can see his statue.

Saturn, the melancholy planet, was rising at the birth of Dante. He is prominent in the horoscopes of Goethe and other poets and philosophers like Sankara. Without His Grace, liberation is not possible.

The most ancient Civilization was the Vedic and Dr Kenneth Chandler puts the origin at 4900 BC. According to him, the Aryan Invasion Theory is not correct.

Will Durant opined that India " is the mother of us all, through Sanskrit, the mother of Europe's languages".

Dr Kenneth Chandler writes

"The original theory proposed by the early historical linguistics who considered these issues was that Vedic Sanskrit conserved the original sound system of the “proto-Indo-European” language most closely, and that Iranian and European languages underwent a systematic sound shift, creating break-away or daughter languages spoken by the people who populated India and Europe. According to this theory, Vedic Sanskrit was put at near the trunk of the proto-Indo-European language tree, if not the trunk itself.

This theory has been challenged and hotly debated in recent years, most especially by
computer linguists.

Since the 1990s, it is now common for computer linguists to hold
that Sanskrit is not so near the root of the Indo-European language tree, but a
subsequent branch. A currently dominant theory is that the original Indo-European
language stemmed from an Indo-European proto-language that has since been lost.
The first languages to break off from the proto-Indo-European root, according to the
dominant contemporary linguistic theories, was Anatolian (the language of what is now
central Turkey), followed by Celtic (a language found in nearby Thrace in northeastern Greece, and also Ireland suggesting that there was a commerce or colonization between Ireland and early Thrace), then Greek, and then Armenian. According to these theories, the Indian and Iranian language groups are still later branches off the proto-Indo-European “root.”

The linguistic evidence appears to imply migrations of people from the Black Sea area
into India, and yet there is no anthropological evidence to support either a migration into northern India, or an invasion. Evidence from skeletal remains, as we saw, as well as pottery and other artifacts, show no cultural replacement at any time in north Indian Thomas V. Gamkrelidze and V. V. Ivanov, “Family Tree of the Indo-European Languages,” Scientific American, March, 1990, p. 110 and following.

Dr. Don Ringe and Dr. Ann Taylor, two linguists at the University of Pennsylvania, with the help of computer scientist Dr. Tandy Warnow, developed a computer algorithm to sift through the Indo- European languages and look for grammatical and phonetic similarities between them. Their work, published in 1996, has thrown up four possible family trees. “We have come up with a favorite,” says Dr. Warnow. The tree shows that the first breakaway language was Anatolian, an ancient group of languages once spoken in Turkey. Celtic was quick to follow, spawning Irish, Gaelic, Welsh and Breton. Armenian and Greek then developed from proto-Into-European. Strangely enough, one of the later branches to emerge, according to the runs of the computer programs, was Sanskrit.

It is interesting that the Celts settled in Thrace in northern Greece, just a short distance from Anatolia. Thrace was the birthplace of the Orphic mysteries which swept into Greece in the sixth century BC. Celtic is one of the earliest languages, along with Anatolian and Greek, to break off from the Indo European proto-language. The technique for self-knowledge described by Socrates were said to have come from Thrace. The Anatolians of central Turkey occupied the area near where the pre-Socratic tradition sprang up in the sixth century BC. This suggests that a technique was passed from India into the Celtic language.

Of Intercalary months, Adhi Masas

The solar month is 30.438030202068 days

A lunar month = 29.5305881 days

It need not be added that a lunation or synodic month means the interval between two consecutive full moons or new moons. Conjunction ( New Moon ) is 0 degrees and Opposition ( Full Moon ) is 180 degrees

Hence a solar year does not have a whole number of lunar months ( about 12.37 lunations ) So a thirteenth embolismic or intercalary month is inserted.

It was observed that 19 solar years or 19*12 = 228 solar months = 235 lunations and hence 7 Adhi Masas were found in every 19 years. An intercalary or 13th month had to be inserted in a 19 year cycle and 19/7 was the ratio. .

They are called Adhi Masas in Indian Astronomy and they were computed using the Theory of continued fractions. The Theory of contiued Fractions is attributed to Euler. This 19 year old cycle is called the Metonic Cycle, named after the Greek astronomer, Meton.

But then the Indian mathematicians correctly computed the Adhi Masas, centuries before Meton or Euler ! The Indian National Calender is lunisolar, whose dates both indicate the solar year and the moon phases and the next date when the New Moon or Full Moon will occur. The length of the synodic month is given as 29.5305879 days in the Surya Siddhanta, which is correct to six decimals. Surya Siddhanta stated that there are 15933396 Adhi Masas in 51840000 solar months !

Of Vedic Maths

Consisting of 16 basic aphorisms or Sutras, Vedic Mathematics is a system of Maths which prevailed in ancient India. Composed by Bharati Krishna Thirtha, these 16 sutras help one to do faster maths.

The first aphorism is this

"Whatever the extent of its deficiency, lessen it still further to that very extent; and also set up the square (of that deficiency)"

When computing the square of 9, as the nearest power of 10 is 9, let us take 10 as our base. As 9 is 1 less than 10, we can decrease it by the deficiency = 9-1 =8. This is the leftmost digit
On the right hand put deficiency^2, which is 1^2.

Hence the square of nine is 81.

For numbers above 10, instead of looking at the deficit we look at the surplus.



For example:


11^2 = (11+1)*10+1^2 = 121

12^2 = (12+2)*10+2^2 = 144

14^2 = ( 14+4)*10+4^2 = 196

25^2 = ((25+5)*2)*10+5^2 = 625

35^2= ((35+5)*3)*10+5^2 = 1225

Indian Maths, on Argive heights divinely sang !

In India, mathematics is related to Philosophy. We can find mathematical
concepts like Zero ( Shoonyavada ), One ( Advaitavada ) and Infinity
(Poornavada ) in Philosophia Indica.

The Sine Tables of Aryabhata and Madhava, which gives correct sine values or values of
24 R Sines, at intervals of 3 degrees 45 minutes and the trignometric tables of
Brahmagupta, which gives correct sine and tan values for every 5 degrees influenced
Christopher Clavius, who headed the Gregorian Calender Reforms of 1582. These
correct trignometric tables solved the problem of the three Ls, ( Longitude, Latitude and
Loxodromes ) for the Europeans, who were looking for solutions to their navigational
problem ! It is said that Matteo Ricci was sent to India for this purpose and the
Europeans triumphed with Indian knowledge !

The Western mathematicians have indeed lauded Indian Maths & Astronomy. Here are
some quotations from maths geniuses about the long forgotten Indian Maths !

In his famous dissertation titled "Remarks on the astronomy of Indians" in 1790,
the famous Scottish mathematician, John Playfair said

"The Constructions and these tables imply a great knowledge of
geometry,arithmetic and even of the theoretical part of astronomy.But what,
without doubt is to be accounted,the greatest refinement in this system, is
the hypothesis employed in calculating the equation of the centre for the
Sun,Moon and the planets that of a circular orbit having a double
eccentricity or having its centre in the middle between the earth and the
point about which the angular motion is uniform.If to this we add the great
extent of the geometrical knowledge required to combine this and the other
principles of their astronomy together and to deduce from them the just
conclusion;the possession of a calculus equivalent to trigonometry and
lastly their approximation to the quadrature of the circle, we shall be
astonished at the magnitude of that body of science which must have
enlightened the inhabitants of India in some remote age and which whatever
it may have communicated to the Western nations appears to have received
another from them...."

Albert Einstein commented "We owe a lot to the Indians, who taught us how to count,
without which no worthwhile scientific discovery could have been made."

The great Laplace, who wrote the glorious Mechanique Celeste, remarked

"The ingenious method of expressing every possible number
using a set of ten symbols (each symbol having a place value and an absolute
value) emerged in India. The idea seems so simple nowadays that its
significance and profound importance is no longer appreciated. Its
simplicity lies in the way it facilitated calculation and placed arithmetic
foremost amongst useful inventions. The importance of this invention is more
readily appreciated when one considers that it was beyond the two greatest
men of antiquity, Archimedes and Apollonius."

The Infinite Series of the Pi function of Madhava

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.

Arctangent series of Madhava, Gregory and Liebniz

The inverse tangent series of Madhava is given in verse 2.206 – 2.209 in Yukti-dipika commentary (Tantrasamgraha-vyakhya) by Sankara Variar . It is also given by Jyeshtadeva in Yuktibhasha and a translation of the verses is given below.


Now, by just the same argument, the determination of the arc of a desired sine can be (made). That is as follows: The first result is the product of the desired sine and the radius divided by the cosine of the arc. When one has made the square of the sine the multiplier and the square of the cosine the divisor, now a group of results is to be determined from the (previous) results beginning from the first. When these are divided in order by the odd numbers 1, 3, and so forth, and when one has subtracted the sum of the even(-numbered) results from the sum of the odd (ones), that should be the arc. Here the smaller of the sine and cosine is required to be considered as the desired (sine). Otherwise, there would be no termination of results even if repeatedly (computed).

Rendering in modern notations

Let s be the arc of the desired sine, bhujajya, y. Let r be the radius and x be the cosine (kotijya).

The first result is y.r/x
From the divisor and multiplier y^2/x^2
From the group of results y.r/x.y^2/x^2, y.r/x. y^2/x^2.y^2/x^2
Divide in order by number 1,3 etc
1 y.r/1x, 1y.r/3x y^2/x^2, 1y.r/5x.y^2/x^2.Y^2/x^2

a = (Sum of odd numbered results) 1 y.r/1x + 1y.r/5x.y^2/x^2.y^2/x^2+......

b= ( Sum of even numbered results) 1y.r/3x.y^2/x^2 + 1 y.r/7x.y^2/x^2.y^2/x^.y^2/x^2+.....

The arc is now given by
s = a - b

Transformation to current notation

If x is the angle subtended by the arc s at the Center of the Circle, then s = rx and kotijya = r cos x and bhujajya = r sin x. And sparshajya = tan x

Simplifying we get

x = tan x - tan^3x/'3 + tan^5x/5 - tan^7x/7 + .....

Let tan x = z, we have

arctan ( z ) = z - z^3/3 + z^5/5 - z^7/7

We thank www.wikipedia.org for publishing this on their site.

The Madhava cosine series

Madhava's cosine series is stated in verses 2.442 and 2.443 in Yukti-dipika commentary (Tantrasamgraha-vyakhya) by Sankara Variar. A translation of the verses follows.



Multiply the square of the arc by the unit (i.e. the radius) and take the result of repeating that (any number of times). Divide (each of the above numerators) by the square of the successive even numbers decreased by that number and multiplied by the square of the radius. But the first term is (now)(the one which is) divided by twice the radius. Place the successive results so obtained one below the other and subtract each from the one above. These together give the śara as collected together in the verse beginning with stena, stri, etc.



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

The following numerators are formed first:

s.s^2,
s.s^2.s^2
s.s^2.s^2.s^2

These are then divided by quantities specified in the verse.

1)s.s^2/(2^2-2)r^2,
2)s. s^2/(2^2-2)r^2. s^2/4^2-4)r^2
3)s.s^2/(2^2-2)r^2.s^2/(4^2-4)r^2. s^2/(6^2-6)r^2


As per verse,

sara or versine = r.(1-2-3)

Let x be the angle subtended by the arc s at the center of the Circle. Then s = rx and sara or versine = r(1-cosx)

Simplifying we get the current notation

1-cosx = x^2/2! -x^4/4!+ x^6/6!......

which gives the infinite power series of the cosine function.

The Madhava Trignometric Series

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

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

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

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


Multiply the arc by the square of the arc, and take the result of repeating that (any number of times). Divide (each of the above numerators) by the squares of the successive even 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:

s.s^2,
s.s^2.s^2
s.s^2.s^2.s^2

These are then divided by quantities specified in the verse.

1)s.s^2/(2^2+2)r^2,
2)s. s^2/(2^2+2)r^2. s^2/4^2+4)r^2
3)s.s^2/(2^2+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 www.wikipedia.org and we thank Wikipedia for publishing this on their site.

Calculus, India's gift to Europe

The Jesuits took the trignometric tables and planetary models from the Kerala School of Astronomy and Maths and exported it to Europe starting around 1560 in connection with the European navigational problem, says Dr Raju.

Dr C K Raju was a professor Mathematics and played a leading role in the C-DAC team which built Param: India’s first parallel supercomputer. His ten year research included archival work in Kerala and Rome and was published in a book called " The Cultural Foundations of Mathematics". He has been a Fellow of the Indian Institute of Advanced Study and is a Professor of Computer Applications.

“When the Europeans received the Indian calculus, they couldn’t understand it properly because the Indian philosophy of mathematics is different from the Western philosophy of mathematics. It took them about 300 years to fully comprehend its working. The calculus was used by Newton to develop his laws of physics,” opines Dr Raju.



The Infinitesimal Calculus: How and Why it Was Imported into Europe

By Dr C.K. Raju

It is well known that the “Taylor-series” expansion, that is at the heart of calculus, existed in India in widely distributed mathematics / astronomy / timekeeping (“jyotisa”) texts which preceded Newton and Leibniz by centuries.

Why were these texts imported into Europe? These texts, and the accompanying precise sine values computed using the series expansions, were useful for the science that was at that time most critical to Europe: navigation. The ‘jyotisa’ texts were specifically needed by Europeans for the problem of determining the three “ells”: latitude, loxodrome, and longitude.

How were these Indian texts imported into Europe? Jesuit records show that they sought out these texts as inputs to the Gregorian calendar reform. This reform was needed to solve the ‘latitude problem’ of European navigation. The Jesuits were equipped with the knowledge of local languages as well as mathematics and astronomy that were required to understand these Indian texts.

The Jesuits also needed these texts to understand the local customs and how the dates of traditional festivals were fixed by Indians using the local calendar (“panchânga”). How the mathematics given in these Indian ancient texts subsequently diffused into Europe (e.g. through clearing houses like Mersenne and the works of Cavalieri, Fermat, Pascal, Wallis, Gregory, etc.) is yet another story.

The calculus has played a key role in the development of the sciences, starting from the “Newtonian Revolution”. According to the “standard” story, the calculus was invented independently by Leibniz and Newton. This story of indigenous development, ab initio, is now beginning to totter, like the story of the “Copernican Revolution”.

The English-speaking world has known for over one and a half centuries that “Taylor series” expansions for sine, cosine and arctangent functions were found in Indian mathematics / astronomy / timekeeping (‘jyotisa’) texts, and specifically in the works of Madhava, Neelkantha, Jyeshtadeva, etc. No one else, however, has so far studied the connection of these Indian developments to European mathematics.

The connection is provided by the requirements of the European navigational problem, the foremost problem of the time in Europe. Columbus and Vasco da Gama used dead reckoning and were ignorant of celestial navigation. Navigation, however, was both strategically and economically the key to the prosperity of Europe of that time.

Accordingly, various European governments acknowledged their ignorance of navigation while announcing huge rewards to anyone who developed an appropriate technique of navigation. These rewards spread over time from the appointment of Nunes as Professor of Mathematics in 1529, to the Spanish government’s prize of 1567 through its revised prize of 1598, the Dutch prize of 1636, Mazarin’s prize to Morin of 1645, the French offer (through Colbert) of 1666, and the British prize legislated in 1711.

Many key scientists of the time (Huygens, Galileo, etc.) were involved in these efforts. The navigational problem was the specific objective of the French Royal Academy, and a key concern for starting the British Royal Society.

Prior to the clock technology of the 18th century, attempts to solve the European navigational problem in the 16th and 17th centuries focused on mathematics and astronomy. These were (correctly) believed to hold the key to celestial navigation. It was widely (and correctly) held by navigational theorists and mathematicians (e.g. by Stevin and Mersenne) that this knowledge was to be found in the ancient mathematical, astronomical and time-keeping (jyotisa) texts of the East.

Though the longitude problem has recently been highlighted, this was preceded by the latitude problem and the problem of loxodromes. The solution of the latitude problem required a reformed calendar. The European calendar was off by ten days. This led to large inaccuracies (more than 3 degrees) in calculating latitude from the measurement of solar altitude at noon using, for example, the method described in the Laghu Bhâskarîya of Bhaskara I.

However, reforming the European calendar required a change in the dates of the equinoxes and hence a change in the date of Easter. This was authorised by the Council of Trent in 1545. This period saw the rise of the Jesuits. Clavius studied in Coimbra under the mathematician, astronomer and navigational theorist Pedro Nunes. Clavius subsequently reformed the Jesuit mathematical syllabus at the Collegio Romano. He also headed the committee which authored the Gregorian Calendar Reform of 1582 and remained in correspondence with his teacher Nunes during this period.

Jesuits such as Matteo Ricci who trained in mathematics and astronomy under Clavius’ new syllabus were sent to India. In a 1581 letter, Ricci explicitly acknowledged that he was trying to understand the local methods of time-keeping (‘jyotisa’) from the Brahmins and Moors in the vicinity of Cochin.

Cochin was then the key centre for mathematics and astronomy since the Vijaynagar Empire had sheltered it from the continuous onslaughts of Islamic raiders from the north. Language was not a problem for the Jesuits since they had established a substantial presence in India. They had a college in Cochin and had even established printing presses in local languages like Malayalam and Tamil by the 1570’s.

In addition to the latitude problem (that was settled by the Gregorian Calendar Reform), there remained the question of loxodromes. These were the focus of efforts of navigational theorists like Nunes and Mercator.

The problem of calculating loxodromes is exactly the problem of the fundamental theorem of calculus. Loxodromes were calculated using sine tables. Nunes, Stevin, Clavius, etc. were greatly concerned with accurate sine values for this purpose, and each of them published lengthy sine tables. Madhava’s sine tables, using the series expansion of the sine function, were then the most accurate way to calculate sine values.

Madhava's sine series

sin x = x - x^3/3! + x^5/5! - x^7/7!+......


The Europeans encountered difficulties in using these precise sine values for determining longitude, as in the Indo-Arabic navigational techniques or in the Laghu Bhâskarîya. This is because this technique of longitude determination also required an accurate estimate of the size of the earth. Columbus had underestimated the size of the earth to facilitate funding for his project of sailing to the West. His incorrect estimate was corrected in Europe only towards the end of the 17th century CE.

Even so, the Indo-Arabic navigational technique required calculations while the Europeans lacked the ability to calculate. This is because algorismus texts had only recently triumphed over abacus texts and the European tradition of mathematics was “spiritual” and “formal” rather than practical, as Clavius had acknowledged in the 16th century and as Swift (of ‘Gulliver’s Travels’ fame) had satirized in the 17th century. This led to the development of the chronometer, an appliance that could be mechanically used without any application of the mind.

The Idea of Planetary Mass in India

Many ancient cultures have contributed to the development of Astro Physics.

Some examples are

The Saros cycles of eclipses discovered by Egyptians
The classification of stars by the Greeks
Sunspot observations of the Chinese
The phenomenon of Retrogression discovered by Babylonians

In this context the Indian contribution to Astro Physics ( which includes Astronomy, Maths and Astrology ) is the the development of the ideas of planetary forces and differential equations to calculate the geocentric planetary longitudes, several centuries before the European Renaissance.

Natural Strength is one of the Sixfold Strengths, Shad Balas and goes by the name Naisargika Bala. It is directly proportional to the size of the celestial bodies and inversely proportional to the geocentric distance. ( Horasara ).

Naisargika Bala or Natural Strength is used to compare planetary physical forces. When two planets occupy the same, identical position in the Zodiac at a given instant of time, such a phenomenon goes by the name of planetary war or Graha Yuddha,happening when two planets are in close conjunction. The Karanaratna written by Devacharya explains that the planet with the larger diameter is the victor in this planetary war. This implies Naisargika Bala.

The Surya Siddhanta says " The dynamics or quantity of motion produced by the action of a fixed force to different planetary objects is inversely related to the quantity of matter in these objects"

This definition more or less equals the statement of Newton’s second law of motion

M = Fa
or
a = F/M

So it strongly suggests that the idea of planetary mass was known to the ancient Indian astronomers and mathematicians.

Differential Equations used in Siddhantas

Motional strength is one the sixfold strengths, known as Cheshta Bala. This motional strength is computed by the formula

Motional Strength = 0.33 ( Sheegrocha or Perigee - geocentric longitude of the planet ). This motional strength is known as Cheshta Bala.


Differential Calculus is the science of rates of the change. If y is the longitude of the planet and t is time, then we have the differential equation ,dy/dt.

During direct motion, we find that dy/dt > 0 and during retrogression dy/dt < 0. During backward motion of the planet ( retrogression) y decreases with time and during direct motion y increases with time. When there are turning points known as Vikalas or stationary points, we have dy/dt = 0 ( where planets like Mars will appear to be stationary for an observer on Earth ).

The quantity in bracket is the Sheegra Anomaly, the Anomaly of Conjuction, the angular distance of the planet from the Sun. This Anomaly or Cheshta Bala is maximum at the center of the Retrograde Loop. Cheshta Kendra is defined as the Arc of Retrogression and is the same as Sheegra Kendra, Kendra being an angle in Sanskrit. During Opposition, when the planet is 180 degrees from the Sun, Cheshta Bala is maximum and during Conjunction, when the planet is 0 degrees from the Sun, it is minimum

The Motional Strength is given in units of 60s, Shashtiamsas.

Direct motion ( Anuvakra ) 30
Stationary point ( Vikala ) 15
Very slow motion ( Mandatara ) 7.5
Slow motion ( Manda ) 15
Average speed ( Sama ) 30
Fast motion ( Chara ) 30
Very fast motion ( Sheegra Chara ) 45
Max orbital speed ( Vakra ) 60
(Centre of retrograde)

The Nine Oribtal Elements

Mean and true planetary longitudes in the Zodiac is computed by Nine Orbital Elements, in Indian Astronomy.


Mean longitude of Planet, Graha Madhyama , M
Daily Motion of the Mean Longitude, Madhyama Dina Gathi, Md
Aphelion, Mandoccha, Ap
Daily Motion of Aphelion, Mandoccha Dina Gathi, Apd
Ascending Node, Patha, N
Daily Motion of Ascending Node, Patha Dina Gathi, Nd
Heliocentric Distance, Manda Karna, radius vector, mndk
Maximum Latitude, L, Parama Vikshepa
Eccentricity, Chyuthi,e


In Western Astronomy, we have six orbital elements

Mean Anomaly, m
Argument of Perihelion, w
Eccentricity, e
Ascending Node, N
Inclination, i, inclinent of orbit
Semi Major Axis, a

With the Nine Orbital Elements, true geocentric longitude of the planet is computed, using multi step algorithms.

There is geometrical equivalence between both the Epicycle and the Eccentric Models.

The radius of the Epicycle, r = e, the distance of the Equant from the Observer.

Astronomical Units of Time Measurement

We find Yuga cylces mentioned not only in astronomical works, but also in mythological works in India.

Kali Yuga began on the midnight of 17th Feb 3102 BCE and the duration of this Kali Yuga is said to be 4.32 K solar years. Dwapara is 2*Kali Yuga years. Treta is 3*K Y and Krita Yuga is 4*K Y.


Krita Treta Dwaparascha Kalischaiva Chaturyugam
Divya Dwadasabhir varshai savadhanam niroopitham


Thus an Equinoctial Cycle, Mahayuga is equal to 4+3+2+1 = 10 KYs.

E C = 10 KYs.

A Greater Equinoctial Cycle ( Manvantara ) = 71 Equinoctial Cycles

There are cusps happening in between Manvantaras, each equal to a Krita Yuga in duration. A Krita is equal to 4 KYs or 2/5 of a Maha Yuga. Since there will 15 such cusps happening amongst the Fourteen Manvantaras, they are equal to 15*2/5 = 6 Mahayugas.

Hence 14*71+6 = 1000 Mahayugas = 4.32 Billion Years

Sahasra yuga paryantham
Aharyal brahmano vidu
Ratrim yugah sahasrantham
The Ahoratra vido janah ( The Holy Geetha ).

This is one Cosmological Cycle, called Brahma Kalpa.

Chaturyuga sahasram indra harina dinam uchyathe


From one second, it can be logarithmically shown, upto 10^22 seconds. This is what the above diagram shows. This diagram is by courtesy of Wikipedia.

From 10^0 it goes upto 10^22 seconds. One day of Brahma is 4.32 billion years and 100 years of Brahma therefore is 311.04 trillion years, which is shown logarithmically above.

One Asu is 4 seconds, one Vinadi is 24 seconds and one Nadi is 24 minutes. 60 such Nadis make up one day. This is the Sexagesimal division of a day into 60 Nadis ! In Astronomy, one degree is sixty minutes and one minute is sixty seconds. Hence sexagesimal division is justified ! 365.25 such days constitute a year and Hindu calculation goes upto 311.04 trillion years !

The Geometric Model of Paramesvara

The Indian astronomers were interested in the computation of eclipses, of geocentric longitudes, the risings and settings of planets,which had relevance to the day to day activities of people.

Did not Emerson say?

"Astronomy is excellent, it should come down and give life its full value, and not rest amidst globes and spheres ".

They were not bothered about proposing Models of the Universe and gaining publicity. But then they did discuss the geometrical model, the rationale of their computations.

The above diagram explains the Geometric Model of Parameswara, another Kerala astronomer. Paramesvara and Nilakanta modified the Aryabhatan Model.

By Sheegroccha, he meant the longitude of the Sun." Sheegrocham Sarvesham Ravir bhavathi ", he says is his book Bhatadeepika . For the interior planets, the longitude of the Sheegra correction is to be deducted from the Sun's longitude, Ravi Sphuta to get the Anomaly of Conjunction.

The Manda Prathimandala is the mean angular motion of the Planet, from which the trignometric corrections are given to get the true, geocentric longitude.

Vikshepa Koti, the cosine of celestial latitude

Jyeshtadeva was a Kerala astronomer who helped in the calculation of longitudes, when there is latitudinal deflection. In his Yukti Bhasa, he calculates correctly the cos l, the cosine of latitude, which is important in the Reduction to the Ecliptic.

There is a separate section in the Yukti Bhasa, which deals with the effects of the inclination of a planet's orbit on its latitude. He describes how to find the true longitude of a planet, Sheegra Sphutam, when there is latitudinal deflection.


"Now calculate the Vikshepa Koti, cos l, by subtracting the square of the Vikshepa from the square of the Manda Karna Vyasardha and calculating the root of the difference."

In the above diagram,

N is the Ascending Node
P is the planet on the Manda Karna Vritta, inclined to the Ecliptic


Vikshepa Koti = OM = SQRT( OP^2 - PM^2 )

Taking this Vikshepa Koti and assuming it to be the Manda Karna, sheegra sphuta, the true longitude, has to be calculated as before.

Vikshepa, the Celestial Latitude

l, Vikshepa, is the Celestial Latitude, the latitude of the planet, the angular distance of the planet from the Ecliptic.

i is the inclination, inclinent of Orbit.


Sin l = Sin i Sin( Heliocentric Long - Long of Node ).

Celestial Latitude is calculated from this equation.

The longitude of the Ascending Node, pata, is minussed from the heliocentric longitude and this angle is called Vipata Kendra.

Sidereal Periods in the Geocentric Model

In the last post we said that Angle AES is Sheegroccha, which is the longitude of the Sun. ( Sheegrocham Sarvesham Ravir Bhavathi ). The Angle AEK is the Heliocentric longitude of the planet.

Sidereal Periods of superior Planets in the Geocentric = Sidereal periods in the Heliocentric.

Sidereal Periods of Mercury and Venus = Mean Sun in the Geocentric

In the Planetary Model of Aryabhata, we find the equation

Heliocentric Longitude - Longitude of Sun = The Anomaly of Conjunction ( Sheegra Kendra ).

As Astronomy is Universal, we are indebted to these savants who made astro calculation possible. Even the word " genius " is an understatement of their brilliant IQ !


Development of the Planetary Models in Astronomy

Hipparchus 150 BCE
Claudious Ptolemy 150 ACE
Aryabhata 499 ACE
Varaha 550 ACE
Brahmagupta 628 ACE
Bhaskara I 630 ACE
Al Gorismi 850 ACE
Munjala 930 ACE
Bhaskara II 1150 ACE
Madhava 1380 ACE
Ibn al Shatir 1350 ACE
Paramesvara 1430 ACE
Nilakanta 1500 ACE
Copernicus 1543 ACE
Tycho Brahe 1587 ACE
Kepler 1609 ACE
Laplace 1700 ACE
Urbain Le Verrier 1850 ACE
Simon Newcomb 1900 ACE
E W Brown 1920 ACE

Reduction to the Geocentric for superior planets in the Eccentric Model

In the above diagram,


A = Starting Point, 0 degree Aries
P = Planet
S = Sun
E = Earth

Angle AEK = Manda Sphuta, heliocentric longitude, after manda samskara
Angle AES = Sheegroccha, mean Sun, mean longitude of Sol
Angle AEP = True geocentric longitude of planet
Angle KEP = The Sheegra Correction or sheegra phalam

The Anomaly of Conjunction = Sheegra Kendra = Angle AES - Angle AEK

x = Angle AES - Angle AEK

Sin ( x ) =

r sin (x)
_______________________
((R + r cos x)^2 + rsin x^2 ))^1/2

which is the Sheegra correction formula given by the Indian astronomers to calculate the geocentric position of the planet.

Aslesha Njattuvela brings rains !

It was Monsoon Tourism, as Aslesha Njattuvela was on. It was raining heavily, cats and dogs in Kerala. I got the rains when I reached Kochi. I had some work at the Passport Office and I finished the work at Noon. Then I went on a tour of the famous Goshree Islands.

I went by boat yesterday to the beautiful Bolgatty Island. A two minutes walk saw me entering the lovely Bolgatty Palace, a resort by the Kerala Tourism Development Corporation.





I walked to the Bolgatty Bus Stand and took a bus to Vallarpadam International Container Terminal. Now everything is in place and one ship, OEL Dubai, was unloading. The progress of the ICTT is slow, but steady.





The Bolgatty Palace is beautiful and well situated in the Mulavukad Island. This island is connected to Vallarpadam by a bridge. Vallarpadam is in turn connected to Vypin by a bridge. In fact these bridges are known as Goshree Bridges, as these beateous islands are known as Goshree Islands. In Vypin, one can see the GAIL LNG terminals, which adorn Puthuvypin.




A new bridge, parallel to the existing Vallarpadam bridge, is being built to ease the traffic. I saw a barge jetty at Bolgatty and a barge carrying containers there.




Kochi is a cauldron of world cultures. A versatile land where visitors from abroad, right from Arabs and Phoenicians to the Chinese, Italians, Portugese, Dutch and British have left indelible marks. A great Port, universally known as the Queen of the Arabian Sea. The newly renovated Bolgatty Palace has 4 palace suits, 6 waterfront cottages, 16 well maintained rooms and one can enjoy four star faciliites and a range of leisure options.





Said a honeymooner, Asmita, Calcullat about Bolgatty " We went to Kerala for our honeymoon and Kochi was our first stop. We reached the resort at around 3:30pm after a long flight and were famished. Since we reached post lunch timings none of the restaurants were open, however the room service was very prompt and we had an amazing keralite food. The rooms are large, and built in a princely way. The property is equipped with all the modern facilities and the stay was really comfortable.
We enjoyed the Kerala body message in the resort . The location is also great. Overall its a great place to stay ".

The Bolgatty Palace was built in 1744 by the Dutch and is a short boat ride away from the Ernakulam Mainland. This is one of the oldest Dutch Palaces outside Holland the only Palace Hotel of its kind in Kerala. Now she has a Palace block and a resort block, called Bolgatty Island Resort. Amenities here comprise Swimming Pool, 9 hole Golf Course and is a destination of choice for select Indian corporates for their conference. It is a favourite destination for Indian elite and overseas tourists. The Kochi Airport is just 32 kms away and the rail and bus terminals just 2 km away.

Kochi International Marina is a KTDC venture located on the eastern coast of Bolgatty Island in the Bolgatty Palace Heritage Hotel. It is the first full fledged marina of international standards in Bharat. It provides berthing facilities to 37 yachts and also offers services like electricity, water and fuel for boats. It is close to the international sea route at the South West Coast of peninsular India , with minimum tidal variations and favorable conditions.






The Bolgatty Event Center overlooks the backwaters of Cochin Seaport and is an exotic venue for conducting Conferences, Exhitions, Wedding Receptions, Conventions and theme dinners. Imbued with resplendent greenery, the Arabian Sea and the ICTT at Vallarpadam gives an easy access to the Center.

Another Indian Astronomical Treatise

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

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

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

The Munjala Coefficients

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

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

Corrections to the Equation of Center for Moon

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

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

Indian Astronomy Original

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.

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 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.

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.