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 !

Big names for Indian numbers

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

Big names for Indian numbers

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

Commodities Astrology !

Commodities Astrology ? Gold has risen to $1600, after touching $1921.

We are in the midst of a commodity bull market cycle, which started in
2000. We have been advocating buying gold, from $900 onwards. Silver
has gone upto $32 and lead and others are following suit. Increased
demand from China and India and the bearish nature of the stock market,
global cues all lead to this commodity bull market cycle.

Gold price is inversely proportional to the Dollex, the USDX. Jupiter rules gold. More info at Zodiac Astrology and Stock Market Astrology

It was observed that a Bull Market Cycle lasts for 18 years. In 1982, we had the stock market boom and the bust of 2000. Not only the stock market, but the dot com market also went bust. This started the commodities bull market cycle and experts say it will last till 2016/18. So the stock market is on a decline, along with Realty and dot com. On the contrary, commodities have gone up.

There are many reasons cited for the commodities boom. A falling US Dollar, the China Effect ( more than 700 million Chinese will be added to the consumer class by 2020), the India Effect ( more than 300 million Indians will be added to the consumer culture), tight supplies, demand more than supply and the interest rate cuts by the Fed.

Astrologically Jupiter in Aries, about to enter Taurus in May 2012, is responsible for the commodities boom. He rules Gold and other precious metals. ( Gurum Kanchana Sannibham ).

These are the Correspondence between Planets and Metals

Sun - Copper and gold.
Moon -Silver.
Mars - Copper.
Mercury - Brass.
Jupiter - Gold.
Venus - Silver & Aluminium.
Saturn - Iron, Minerals & Crude.



It has been observed that when the Stock Market is in Recession, commodities go up and vice verse. Hence these years will seem a boom in commodities.

Some of the reasons cited for the commodities boom

The Fed will lower interest rates
Supplies of oil, grains and metals are very low and the possibility of producing more in the next few years is low
The Dollex, the USDX, may decline furthur
As the standard of living rises in Chindia ( China and India ), more consumption will be the order of the day, leading to heavy demand.

Gold Astrology

Gold is one of the best precious metals and we have been advocating buying gold at declines. We had started saying BUY Gold, when the price was $900 onwards. Now Gold is at $1628 for an ounce and still going strong.

The Oil Ministers decided in 2009 to trade oil in gold, yuan, yen and euro, as the Dollar was showing signs of weakness. This is one of the reasons why Gold is going up.

We have reports that billionaires like Soros and Bufett etc are taking up great positions in gold and buying mining companies.

It was observed that the Gold Price is inversely proportional to the Dollex, the USDX. Yesterday USDX went up by one point and gold declined from 1674 to 1645. Now it is at 1628 per ounce.

Jupiter represents Gold in Astrology. Now Jupiter in Aries is responsible for the secondary reaction happening in Gold. From a high of 1900 dollars, it is now at 1628.
You can draw your own conclusions !

Sensex drift lower, as Jove takes a negative stance

We had warned in our columns that the markets are headed southwards.

The Sensex is now at 16084. What a drastic fall. On global cues, it fell 300 odd points yesterday. The Return of the Bear ?

Weak sentiments prevailed on Dalal Steet. It is not surprising to the men that see, as the planet of Finance, Jupiter, transits the adverse 10th for India.

L & T fell heavily to 1312 and Siemens to 829.

It is better to hold on for the time being. Provisionally down at 1.73% on Greek default fears , the Sensex may take time to recover. The market is falling for the second, straight day. All the 13 sectoral indices on the BSE is in the red and index heavyweight, Reliance, is edging lower.

After good auto sales in September, auto shares were mixed. Metal shares fell across the board, as global commodity prices fell on global economic worries. JSW Steel fell, as reports of a CBI raid was rumoured.

Deficit in August was 14 billion, as 28 billion exports were achieved against a 38 billion import.

Blood bath on Dalal Street

We have said in our sites and columns that Jove in the 10th is detrimental to the Indian economy. It was proved right as the stock market plunged 704 points.

It was virtually blood bath on Dalal Street. The Dow fell and so did all indices.

Even though India may become the third largest economy by 2012 ( by GDP PPP ), now for seven months ( during Jupiter's transit of Aries ), she is in for a turbulent phase.
It is better to hold on. only at the bottom should one invest.

Speculation is taboo as anything can happen. Gold is also falling, may be temporarily. So is crude.

The problem can be compounded when Saturn leaves Virgo on Nov 15th. Only in May, when Jupiter moves over to Taurus, in India's 11th can we hope to see a bullish curve !

A 4.2 percent drop was the least desirable. The Federal Reserve warned of significant risks to global economic growth. All other Asian markets fell with key indices in various countries losing between 2-5 percent. Selling pressure was seen all over the bourses, as all the broader markets and sectoral indices closed deep in the red. Realty, metals, oil, gas and banking indices of the BSE closed 4-6 percent down. Nothing to panic, say the technical analysts !

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.

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

Maths & Philosophy

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 Pi series 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.

The Infinite Pi series 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.

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.

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