Interesting info that I came across during the weakend

[Guest guest]

Hari Om Sadanandaji:

 

One of the interesting verified connection between

Vedic Stotras (Divine prayers) and mathematical sutras

(formulas) have been reported in the book, Vedic

Mathematics (see the reference in the discussion

below). Vedic scholars were able to conduct high

levels of computations without the help of calculators

and computers by developing unmeasurable amount of

human memory! Modern computers have to depend on RAM

size and Harddrive capacity to determine their

capacity. The human 'mind' has limitless capacity and

has the potential to do everything that was considered

impossible!

 

In conclusion, I would declare that unless it is

proved otherwise, there exists an apparant link

between Stotras and graphical images! The extent of

mathematical knowledge in ancient India has been

recoginzied (though late) by the world of

mathematicians. There is no wonder, over 35% of high

tech experts in all western countries put together

come from India! Nearly 40% of the empolyees in

Microsoft Corporation are from Indian origin! Now let

me turn my attention to Vedic Mathematics with a list

of great ancient Indian mathematicians.

 

Vedic mathematics is a unique system of calculations

based on simple rules and principles, with which any

mathematical problem - be it arithmetic, algebra,

geometry or trigonometry - can be solved orally. It is

termed Vedic because it was derived from the ancient

Vedic texts.

 

Stotras and Sutras:

 

Stotras (prayer songs in praise of the Lord) served as

mathematical formulas (Sutras). The primary objective

of this approach is to help the students of

mathematics to memorize sutras under the guise of

devotional Stotras. Example Stotras and the associated

Sutras are illustrated in the book, Vedic Mathematics

authored by the former Shankaracharya of Puri.

Published by Motilal Banarsidass Publishers Private

Limited

41 U. A., Bungalow Road, Jawahar Nagar, Delhi 110007

ISBN: 81-208-0163-6 (cloth) (Rs. 120)

ISBN: 81-208-0164-4 (paper) (Rs. 80)

(printed originally in 1965, latest edition 1992)

 

Hidden under centuries of debris, Vedic mathematics is

now again gaining attention. Bharati Krishna

Tirthaji, the former Shankaracharya of Puri, who

delved into ancient Vedic texts and established the

techniques of this system through a pioneering work,

Vedic Mathematics, took it to the United Kingdom in

1960s. There it was hailed as a new alternative system

of mathematics and is now taught in some schools, MBA

courses and to economics students.

 

The basis of this system are sutras - 16 one-line

aphorisms originally written in Sanskrit - which can

be easily memorized. Once you have learnt them by

heart, you can solve any long problem using the sutras

orally, like Shakuntala Devi often does.

 

For instance, if you want to calculate the square of

35, you will have to use the Ekadhikena Purvena sutra.

Its literal meaning: by one more than the one before.

The rule says since the first digit is 3 and the

second one is 5, you will first have to multiply 3 (3

+1), that is 3X4 , which is equal to 12 and then

multiply 5 with 5, which is 25. The answer is 1225.

Now, you can try multiplication of all numbers ending

with five using this method. Similarly, other sutras

lay down such handy rules to arrive at answers.

 

 

Brief Biographies of Great Indian Mathematicians

 

The dates given for Panini are pure guesses. Experts

give dates in the 4th, 5th and 6th century BC. Panini

was a Sanskrit grammarian who gave a comprehensive and

scientific theory of phonetics, phonology, and

morphology. Sanskrit was the classical literary

language of the Indian Hindus. In a treatise called

Astadhyayi Panini distinguishes between the language

of sacred texts and the usual language of

communication. Panini gives formal production rules

and definitions to describe Sanskrit grammar. The

construction of sentences, compound nouns etc. is

explained as ordered rules operating on underlying

structures in a manner similar to modern theory.

Panini should be thought of as the forerunner of the

modern formal language theory used to specify computer

languages. The Backus Normal Form (a standard notation

to describe the syntax of a high level programming

language) was discovered independently by John Backus

in 1959, but Panini's notation is equivalent in its

power to that of Backus and has many similar

properties.

 

Brahmagupta was head of the astronomical observatory

at Ujjain which was the foremost mathematical centre

of ancient India. He wrote important works on

mathematics and astronomy. He wrote Brahma- sphuta-

siddhanta (The Opening of the Universe), in 21

chapters, at Bhillamala in 628. His second work on

mathematics and astronomy is Khandakhadyaka written in

665. Brahmagupta's understanding of the number systems

was far beyond others of the period. He developed some

algebraic notation. He gave remarkable formulas for

the area of a cyclic quadrilateral and for the lengths

of the diagonals in terms of the sides. Brahmagupta

also studied arithmetic progressions, quadratic

equations, theorems on right-angled triangles,

surfaces and volumes. The remaining chapters deal with

solar and lunar eclipses, planetary conjunctions and

positions of the planets. Brahmagupta believed in a

static Earth and he gave the length of the year as 365

days 6 hours 5 minutes 19 seconds in the first work,

changing the value to 365 days 6 hours 12 minutes 36

seconds in the second book. This second values os not,

of course, an improvement on the first since the true

length of the years if less than 365 days 6 hours.

 

Aryabhata wrote Aryabhatiya, finished in 499, which is

a summary of Hindu mathematics up to that time,

written in verse. It coveres astronomy, spherical

trigonometry, arithmetic, algebra and plane

trigonometry. Aryabhata gives formulas for the areas

of a triangle and a circle which are correct, but the

formulas for the volumes of a sphere and a pyramid are

wrong. Aryabhatiya also contains continued fractions,

quadratic equations, sums of power series and a table

of sines. Aryabhata gave an accurate approximation for

(equivalent to 3.1416) and was one of the first known

to use algebra. He also introduced the versine (

versin = 1 - cos) into trigonometry. Aryabhata also

wrote the astronomy text Siddhanta which taught that

the apparent rotation of the heavens was due to the

axial rotation of the Earth. The work is written in

121 stanzas. It gives a quite remarkable view of the

nature of the solar system. Aryabhata gives the radius

of the planetary orbits in terms of the radius of the

Earth/Sun orbit as essentially their periods of

rotation around the Sun. He believes that the Moon and

planets shine by reflected sunlight, incredibly he

believes that the orbits of the planets are ellipses.

He correctly explains the causes of eclipses of the

Sun and the Moon. His value for the length of the year

at 365 days 6 hours 12 minutes 30 seconds is an

overestimate since the true value is

less than 365 days 6 hours.

 

Sripati wrote on astronomy and mathematics. His

mathematical work is undertaken with applications to

astronomy in mind, for example a study of spheres. His

works include Dhikotidakarana (1039), a work on solar

and lunar eclipses, Dhruvamanasa (1056), a work on

calculating planetary longitudes, eclipses and

planetary transits, Siddhantasekhara a major work on

astronomy in 19 chapters. The titles of Chapters 13,

14, and 15 are Arithmetic, Algebra and On the Sphere.

Sripati obtained more fame in astrology than in other

areas.

 

Bhaskara represents the peak of mathematical knowledge

in the 12th Century and reached an understanding of

the number systems and solving equations which was not

to be reached in Europe for several centuries.

Bhaskara was head of the astronomical observatory at

Ujjain, the leading mathematical centre in India at

that time. He understood about 0 and negative numbers.

He knew that x2 = 9 had two solutions. Bhaskara also

studied Pell's equation x2 = 1+py2 for p=8, 11, 32, 61

and 67. When p = 61 he found the solutions x

=1776319049, y = 22615390. He studied many Diophantine

problems. Bhaskara's mathematical works include

Lilavati (The Beautiful) and Bijaganita (Seed

Counting) while he also wrote on astronomy, for

example Karanakutuhala (Calculation of Astronomical

Wonders).

 

 

==================

K. Sadananda wrote:

> I attended a conference of some like-minded people >

this Saturday and

> Sunday. One Dr. Kidambi from Toronto gave a short

> talk covering the glories of Hinduism. .....

[Guest guest]

Ram - Thanks.

 

Very interesting account of the development of Mathematics in India.

 

Few years ago my wife had interviewed Ganapati stapati for a local TV

here - the topic was about Vasu-shaastra ( Engineering) and temple

architechture. For those who do not know Ganapati stapati is well known

temple architect who runs a school for sculptures in South India and who

is in fact an architect of many Indian temples in this country. He said

they use extensive geometry in the design and construction which was passed

on from teacher to the taught. He comes from a family of stapaties and the

linage goes more than thousand years to the past, as far as he knows. One

interesting information he provided in terms of the selection of the stones

for the idols. When a stone is hit, it produces sound vibrations. Based

on the frequency, they classify the stones as male, female and neutral.

Interestingly they use only female stones for sculpturing the Goddesses and

male stones for Male gods. Neutral stones are not used for the idols.

They use them for the steps and for constructions of the base structures.

He mentioned that they can conceptualize the whole statue in complete

geometrical proportion even if one gives just a size of finger or a toe.

 

Most of the knowledge was indigenous and was developed in India and passed

on from teacher to the taught. The Saraswati Delta project ( I am not

sure how far that has progressed) seems to affirm that

the Aryans were not nomadic settlers from outside India but are original to

India itself. Many of the developments in Indian Science and technology

has been suppressed or distorted, partly due to our own fault.

 

I am, however, keenly interested in the direct implication of the subtler

impressions that these mantras chanted with appropriate intonations create

and in codifying and quantifying these using time-tested scientific methods

and techniques that are currently available. Subjective experience is the

most supreme and can never be objectified and cannot be subjected to any

tangible objective tests. However, to believe these one needs a faith in

the teacher. But objective evaluation of these using the accepted and

time-tested scientific tools will provide an objective basis for everyone

to appreciate the beauty of the mantras and care that one should exercise

in terms of proper chanting of these to get the full benefit from these.

 

Hari Om!

Sadananda

 

 

 

 

>Hari Om Sadanandaji:

>

>One of the interesting verified connection between

>Vedic Stotras (Divine prayers) and mathematical sutras

>(formulas) have been reported in the book, Vedic

>Mathematics (see the reference in the discussion

>below).

 

K. Sadananda

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