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

Of Indian Maths

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."

Madhava's Pi series

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.

Of Natural Strength, Naisargika Bala

Natural Strength, Naisargika Bala, is the inherent property of a celestial object, which possesses the following properties


1) This force is constant for a celestial object, not varying in time.

2) This force is proportional to the size of the diameter of the planets.

3) This force is inversely proportional to the distance, r, from the Sun.

4) It increases in the order from the farthest planet to the nearest planet to the Sun. From Saturn,Jupiter, Mars, Venus, Mercury, Moon and Sun.

5) This Force is a major factor when planets are involved in Planetary War ( Graha Yuddha ), when their longitudes are more or less identical in the Ecliptic.

Let F1 and F2 be the Naisargika Bala of planets 1 and 2 situated in the same distance, r, from earth. Then we have

F1 = F(D1)/r
F2 = F(D2)/r

The ratio of the planetary Naisargika Bala is

F1/F2 = F(D1)/F(D2)

The forces are given by ( according to Newtonian modern theories)

F1= ( M1M/r^2)
F2= ( M2M/r^2)

The ratio of the gravitational forces are

F1/F2 = M1/M2


M1 = v1 d1
M2 = v2 d2 ( v volume d density )


If d1 = d2

then

F1/F2 = V1/V2 = F(D1)/F(D2)


Therefore the ratio of the Naisargika Balas of two planets at the same identical position in the Zodiac, as defined by the Indian astronomers, is almost identical to the ratios of the modern gravitational forces of these planets if their mass densities are identical.