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History of Mathematics - Facets of India : Ancient and Modern
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History of Ganit (Mathematics)
Introduction
Ganit (Mathematics) has been considered a very important subject since ancient times. We find very
elaborate proof of this in Vedah (which were compiled around 6000 BC). The concept
of division, addition et-cetera
was used even that time. Concepts of zero and infinite were there. We also find roots of algebra in Vedah. When
Indian
Beez Ganit reached Arab, they called it Algebra. Algebra was name of the Arabic book that described Indian
concepts. This knowledge reached to Europe from there. And thus ancient Indian Beez Ganit is currently referred to
as Algebra.
The book Vedang jyotish (written 1000 BC) has mentioned the importance of Ganit as follows-
Meaning:
Just as branches of a peacock and jewel-stone of a snake are placed at the highest place of body (forehead),
similarly position of Ganit is highest in all the branches of Vedah and Shastras
Famous Jain Mathematician Mahaviracharya has said the following-
Meaning:
What is the use of much speaking. Whatever object exists in this moving and nonmoving world, can not be understood
without the base of Ganit(Mathematics).
This fact was well known to intellectuals of India that is why they gave special importance to the development of
Mathematics, right from the beginning. When this knowledge was negligible in Arab and Europe, India had acquired
great achievements.
People from Arab and other countries used to travel to India for commerce. While doing commerce, side by side, they
also learnt easy to use calculation methods of India. Through them this knowledge reached to Europe. From time to
time many inquisitive foreigners visited India and they delivered this matchless knowledge to their countries.
This will not be exaggeration to say that till 12th century India was the World Guru in the area of Mathematics.
The auspicious beginning on Indian Mathematics is in Aadi Granth (ancient/eternal book) Rigved. The
history of Indian Mathematics can be divided into 5 parts, as following.
1) Ancient Time (Before 500 BC)
a)Vedic Time (1000 BC-At least 6000 BC)
a)Later Vedic Time (1000 BC-500BC)
2) Pre Middle Time (500 BC- 400 AD)
3) Middle Time or Golden Age (400 AD - 1200 AD)
4) Later Middle Time (1200 AD - 1800 AD)
5) Current Time (After 1800 AD)
1) Ancient Time (Before 500 BC)
Ancient time is very important in the history of Indian Mathematics. In this time different branches of Mathematics,
such as Numerical Mathematics; Algebra; Geometrical Mathematics, were properly and strongly established.
There are two main divisions in Ancient Time. Numerical Mathematics developed in Vedic Time and Geometrical
Mathematics developed in Later Vedic Time.
1a) Vedic Time (1000 BC-At least 6000 BC)
Numerals and decimals are cleanly mentioned in Vedah (Compiled at lease 6000 BC). There is a Richa in Veda,
which says the following-
In the above Richa , Dwadash (12), Treeni (2), Trishat (300) numerals have been used.
This indicates the use of writing numerals based on 10.
In this age the discovery of ZERO and "10th place value method"(writing number based on 10) is great contribution to world by India
in the arena of Mathematics.
If "zero" and "10 based numbers" were not discovered, it would not have been possible today to write big numbers.
The great scholar of America Dr. G. B. Halsteed has also praised this. Shlegal has also accepted that this is the
second greatest achievement of human race after the discovery of Alphabets.
This is not known for certain that who invented "zero" and when. But it has been in use right from the "vedic" time.
The importance of "zero" and "10th place value method" is manifested by their wide spread use in today's world.
This discovery is the one that has helped science to reach its current status.
In the second section of earlier portion of Narad Vishnu Puran (written by Ved Vyas) describes
"mathematics" in the context of Triskandh Jyotish. In that numbers have been described which are ten times
of each other, in a sequence (10 to the power n). Not only that in this book, different methods of "mathematics"
like Addition, Subtraction, Multiplication, Addition, Fraction, Square, Square root, Cube root et-cetera have been
elaborately discussed. Problems based on these have also been solved.
This proves at that time various mathematical methods were not in concept stage, rather those were getting used in a
methodical and expanded manner.
"10th place value method" dispersed from India to Arab. From there it got transferred to Western countries. This is
the reason that digits from 1-9 are called "hindsa" by the people of Arab. In western countries 0,1,2,3,4,5,6,7,8,9
are called Hindu-Arabic Numerals.
1b) Later Vedic Time (1000 BC - 500 BC)
1b.1) Shulv and Vedang Jyotish Time
Vedi was very important while performing rituals. On the top of "Vedi" different type of
geomit(geometry: as you notice this word is derived from a Sanskrit word)) were made.
To measure those geometry properly, "geometrical mathematics" was developed. That knowledge was available in form
of Shulv Sutras (Shulv Formulae). Shulv means rope. This rope was used in measuring geometry
while making vedis.
In that time we had three great formulators-Baudhayan, Aapstamb and Pratyayan.
Apart from them Manav, Matrayan, Varah and Bandhul are also famous mathematician
of that time.
The following excerpt from "Baudhayan Sulv Sutra (1000 BC)" is today known as Paithogorus Theorem (amazing, isn't it
?)
In the above formula , the following has been said. In a Deerghchatursh (Rectangle) the Chetra
(Square) of Rajju (hypotenuse) is equal to sum of squares of Parshvamani (base) and
Triyangmani (perpendicular).
In the same book Baudhayan has discussed the method of making a square equal to difference of two squares. He has
also described method of making a square shape equal to addition of two squares.
He has also mentioned the formula to find the value (upto five decimal places) of a root (square root, cube root
...) a number, according to
that the square root of 2 can be found as below-
While Geometric Mathematics was developed for making Vedi in Yagya , in parallel there was a
need
to find appropriate timing for Yagya. This need led to development of Geotish Shastra
(Astrology)
In Geotish Shastra (Astrology) they calculated time, position and motion of stars. By reading the book
Vedanga Jyotish (At least 1000 BC) we find that astrologers knew about addition, multiplication,
subtraction et-cetera.
For example please read below-
Meaning:
Multiply the date by 11, then add to it the "Bhansh" of "Parv" and then divide it by "Nakshatra" number. In this way
the "Nakshtra" of date should be told.
1b.2) Surya Pragyapti Time
We find elaborated description of Mathematics in the Jain literature. In fact the clarity and elaboration
by which Mathematics is described in Jain literature, indicates the tendency of Jain philosophy to convey the
knowledge to
the language and level of common people (This is in deviation to the style of Veda which told the facts indirectly).
Surya Pragyapti and Chandra Pragyapti (At least 500 BC) are two famous scriptures of Jain branch
of Ancient India. These describe the use of Mathematics.
Deergha Vritt (ellipse) is clearly described in the book titled Surya
Pragyapti. "Deergha Vritt" means the
outer circle (Vritta) on a rectangle(Deergha), that was also known as Parimandal.
This is clear that Indians had discovered this at least 150 years before Minmax (150 BC). As this history was not
known to the West so they consider Minmax as the first time founder of ellipse.
This is worth mentioning that in the book Bhagvati Sutra (Before 300 BC) the word Parimandal has
been used for Deergha Vritt (ellipse). It has been described to have two types 1)
Pratarparimandal and 2)Ghanpratarparimandal.
Jain Aacharyas contributed a lot in the development of Mathematics. These gurus have described different
branches of mathematics in a very through and interesting manner. They are examples too.
They have described fractions, algebraic equations, series, set theory, logarithm, and exponents ....
Under the set theory they have described with examples- finite, infinite, single sets.
For logarithm they have used terms like Ardh Aached , Trik Aached, Chatur Aached. These terms mean log
base 2, log base 3 and log base 4 respectively. Well before Joan Napier (1550-1617 AD),
logarithm had been invented and used in India which is a universal truth.
Buddha literature has also given due importance to Mathematics. They have divided Mathematics under two categories-
1) Garna (Simple Mathematics) and 2)Sankhyan (Higher Mathematics). They have described numbers
under three categories-1)Sankheya(countable),2)Asankheya(uncountable) and
3)Anant(infinite). Which clearly indicates that Indian Intellectuals knew "infinite number" very well.
2) Pre Middle Time (500 BC- 400 AD)
This is unfortunate that except for the few pages of the books Vaychali Ganit, Surya Siddhanta and Ganita Anoyog
of this time, rest of the writings of this time are lost. From the remainder pages of this time and the
literature of Aryabhatt, Brahamgupt et-cetera of Middle Time, we can conclude that in this time too Mathematics
underwent sufficient development.
Sathanang Sutra, Bhagvati Sutra and Anoyogdwar Sutra are famous books of this time. Apart from these the
book titled Tatvarthaadigyam Sutra Bhashya of Jain philosopher Omaswati (135 BC) and the book titled
Tiloyapannati of Aacharya (Guru) Yativrisham (176 BC) are famous writings of this time.
The book titled Vaychali Ganit discusses in detail the following -the basic calculations of mathematics,
the numbers based on 10, fraction, square, cube, rule of false position, interest methods, questions on purchase and
sale...
The book has given the answers of the problems and also described testing methods. Vachali Ganit is a
proof of the fact that even at that time (300 BC) India was using various methods of the current Numerical
Mathematics. This is noticeable that this book is the only written Hindu Ganit book of this time that was found
as a few survived pages in village Vaychat Gram (Peshawar) in 1000 AD.
Sathanang Sutra has mentioned five types of infinite and Anoyogdwar Sutra has mentioned four types
of Pramaan (Measure). This Granth(book) has also described permutations and
combinations which are termed as Bhang and Vikalp .
This is worth mentioning that in the book Bhagvati Sutra describes the following. From
n types taking 1-1,2-2 types together the combinations such made are termed as Akak, Dwik Sanyog and the
value of such combinations is mentioned as n(n-1)/2 which is used even today.
Roots of the Modern Trignometry lie in the book titled Surya Siddhanta .
It mentions Zya(Sine), Otkram Zya(Versesine),
and Kotizya(Cosine). Please remember that the same word (Zia) changed to "Jaib" in
Arab.
The translation of Jaib in Latin was done as "Sinus". And this "Sinus" became "Sine" later on.
This is worth mentioning that Trikonmiti word is pure Indian and with the time it
changed to Trignometry. Indians used Trignometry in deciding the position , motion et-cetera of the spatial
planets.
In this time the expansion of Beezganit (When this knowledge reached Arab from
India it became Algebra)was revolutionary. The roots of Modern Algebra lie in the book
Vaychali Ganit. In this book while describing Isht Karma |
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