Project Completion Report:- File No. 47 - 488/12 (WRO) Pune
Contribution of Indian Mathematicians with special
reference to Madhava & their today’s relevance.
SUBMITTED TO
The Additional Secretary
University Grants Commission,
Western Regional office
Poona University Campus,
Ganesh Khind, Pune – 411 007
By
Sanjay Madhukarrao Deshpande
Associate Professor, Bhawabhuti Mahavidyalaya, Amgaon
And
Dr.Anant Wasudeo Vyawahare
Retired Professor, M.Mohata science college, Nagpur
Name of the College
Bhawabhuti Mahavidyalaya, Amgaon
Distt. Gondia 441 902
(Affiliated to RTM Nagpur University, Nagpur)
October 2015
Summary
Indian culture is based on Logic and Mathematics. Many
mathematical results, Baudhāyana theorem, constructions of altars,
geometrical constructions, use of number system are used by Indians very
long back (Indus Vally Mathematics has period 2300B.C. – 1700B.C.).
Indian Mathematicians made significant contributions in the world of
Mathematics from the Vedic time. Unfortunately work of Indian
mathematicians remains covered for many reasons.
Some mathematical concepts are fundamental. These concepts get
absorbed in the general thinking of all peoples of the world. So it is
difficult to pinpoint a specific period of time of birth of these concepts (or
development of mathematical concepts). One such basic concept is
introducing the place value system and the number zero, the credit for
which goes to Ancient Indian mathematicians specially Brahmagupta.
All the Indian mathematicians show interest in the Astronomy. In
most of all texts of Indian mathematicians along with Astrology a separate
chapters on measuring units and mathematics are included. There is a
gradual development of mathematics in India. In fact there are no water
tight compartment such as arithmetic, algebra, geometry and trigonometry.
This gives the fact that Indian mathematics is practically useful in daily
life. In case of many results necessary proofs are provided.
When we have a beautiful path of development of mathematics in
India. The Indus Valley civilisation (called as Harappan civilisation) dates
back to 2500 – 3000 B.C. This was city civilisation; archaeological
evidences show that the principles of city planning had been adopted. This
civilisation gives clear evidences of application of geometry in daily life.
Construction of simple geometric figures such as square, rectangle, triangle
and circle are well were known to the people. Weights used are in the form
of cubes. Plumb bobs were made with remarkable accuracy. The highlight
of this civilisation was its knowledge of baked bricks used in the
constructions. The most ideal ratio of the length, breadth and thickness
(4:2:1) of a brick, which is good for efficient bonding strength and the
principles of city planning was known to the people in the Indus Valley
civilisation. But no text of this period is available.
The Śulbasūtras is text available from the period 900 B.C. but the
name of the author is not in the text. There are nine Śulbasūtras existing.
Four of them are significant in mathematics named by Vedic scholars
Baudhāyana, Āpastamba, Mānava and Kātyāyana. The Śulbasūtras contain
rich principles of mathematics, basically of ‘geometry’. The outstanding
feature of Śulbasūtras is consistency and completeness of geometrical
results and application of these results in actual construction shows that
Śulbasūtras have deeper significance.
Āryabhata - I was the first Indian astronomer and a mathematician.
Āryabhata - I was born in 476 A.D. Āryabhata wrote two books, (1)
Āryabhatīya and (2) Āryabhata-siddhanta. He wrote Āryabhatiya when he
was 23. His name appeared in three different stanzas (verses) of
Āryabhatīya. This is the first text in which the name of author appeared.
The first part of Āryabhatīya is DaśaGītika pāda consists of 13
stanzas (of which 10 stanzas are in gītika metre). This chapter basically
states unit of time (Kalpa, Manu and yuga), Circular units of arcs (degree
and minute) and linear units (yojana, hasta and aṅgula).
The second part of Āryabhatīya is gaṇita pāda, consisting of 33
stanzas dealing with mathematics. Important mathematical topics discussed
in this section are approximate value of π, Geometrical figures and their
properties and mensuruation, Arithmetical method of finding the square
root and the cube root, Arithmetic progression, geometric progression,
Simple and compound interest, The method of solving the first order
indeterminate equations of the type ax + c = by.
The interesting part is alphabetical system of numerical notation, the
first ten notational places and the place value shows that Āryabhata knew
the decimal system. Alphabetical system of numerical notation shows that
a number however big can be written in concise way. Without knowing the
decimal number system such conversion is not possible.
Brahmagupta was born in 598 A.D. in Bhinmala, situated on the
north border of Gujarath in South Marwar. The well known works of
Brahmagupta are ‘Brahama Sphuta Siddanta (the Opening of the
Universe)’ and ‘Khanda Khadyaka’.
Brahama Sphuta Siddanta contains 1008 verses (slokas) in 25
chapters. Chapter 12 is devoted to Arithmetic and geometry consisting of
56 verses. Chapter 18 deals with Algebra in 102 verses. These are the only
two chapters on the Mathematics. The major contributions are many results
on calculating the sides of right angled triangle in different ways,
A
beautiful method to obtained infinitely many solutions of the equation Nx2
+ 1 = y2, defined a quantity having properties of modern zero and defined
some properties of modern zero.
Mahāvīra was the first Indian mathematician who wrote a separate
text on mathematics Ganitha Sāra Saṅgraha in 850 A.D. Ganitha Sāra
Saṅgraha contains nine chapter of about 1100 śhiokas and contain
elementary topics in arithmetic, algebra, geometry, mensuruation,
measurements of gold silver ornaments etc.
Elementary algebraic
equations of one variable were used for distribution of property, purchasesale transaction etc. The work of Mahāvīra is important because it is a
collection summarizing elementary mathematics of his time and provides a
rich source of information on ancient Indian Mathematics. Ganitha Sāra
Saṅgraha was widely used in South India.
The greatness of Bhāskarācārya is to make mathematics attractive
and irresistible. Bhāskarācārya is referred as Bhāskara II. He has written
three texts on mathematics Līlāvatī, Bījaganitam, Grahaganitam and
Golādhyāya. Līlāvatī is most popular work ancient Indian or Hindu
Mathematics. A good number of commentaries on Līlāvatī show the
popularity of Līlāvatī. Bījaganitam- algebra containing indeterminate
analysis, it is standard Hindu work on algebra. In Bijganitam solutions of
indeterminate quadratic equation of the type ax 2+1=y2. was solved using
the Chakravāla method. There is a chapter on cyclic method called
Cakravāla. Golādhyāya is the Jyotapatti contains trigonometry. Bhāskara
stated the important result of expansion of Sin(A+B). Bhāskara II
discussed combination and permutation by stating the results and example.
Interesting thing is number of permutation when sum of digit if fixed.
Bhāskara II has derived a beautiful method of finding the second diagonal
of a quadrilateral, where one diagonal and sides of quadrilateral are known.
Bhāskara also coated the result of finding the side of regular polygon
inscribed in a circle
Mathematicians before Mādhava were working on Arithmetic,
Algebra, Trigonometry and Combinatorics. Work of Mādhava in calculus
and infinite series motivates Mathematics from finite to infinite. We
discussed, Mādhava’s “calculus without limits” and infinite series.
Mādhava knows the Āryabhata’s method of finding the value of
by inscribing a regular hexagon in the circle. This involves square root.
Mādhava found the value of
using by the method of calculating
circumference without finding square roots is one of the contributions of
Mādhava based on concept of infinite series. This series converges slowly.
Mādhava found the series of arctan (tan-1), sine, versed sine and cosine.
Mādhava stated the foundation of calculus.
Parameśwaran quoted all the results of cyclic quadrilateral, diagonal
in term of sides, area in term of diagonals and area in terms of sides.
Parameśwaran found the formula for the radius of a circle circumscribing
the cyclic quadrilateral (circum radius) in terms of sides.
Jyeṣṭhadeva quoted all the major results from Mādhava with proof in
his major work, the Yukti-bhāṣa. He stated the wonderful result of finding
the approximate value of length of arc in terms of chord. Similarly
Nīlakaṇṭha quoted the results in Tantrasaṅgraha.
To sum up, the contributions of Indian mathematicians are notable,
deep and philosophical. Many results in Indian mathematics are coated in
the form of stanzas and śloka. The
work
on
the
trigonometry
by
Bodhayan developed using circle and not by using right angled triangle.
Hence trigonometric functions are called circular functions. Many results
on geometry, algebra, arithmetic and trigonometry and stated by most of
the Indian mathematicians.
Work on calculus was started by Bhāskara II, later developed by
Mādhava. Mādhava developed infinite series using basic concept of
integral calculus and by using theory of functions.
The report contains complete information regarding development of
Mathematics in India and put light on Indian contributions in the field of
Mathematics. The present work is also a detailed survey of Mathematical
work by ancient Indian Mathematicians.