TOPIC : MATHEMATICIANS AND THEIR

CONTRIBUTIONS

SUBMITTED BY : BIBASWAN HAZRA

AISSCE
MATHS PRACTICAL EXAMINATION
2019-20

BOARD ROLL NUMBER :

DELHI PUBLIC SCHOOL, RUBY PARK

KOLKATA
INDEX M
 CERTIFICATE
 ACKNOWLEDGEMENT
 FIVE INDIAN MATHEMATICIANS
 FIVE FOREIGN MATHEMATICIAN
 CERTIFICATE
This is to certify that Bibaswan Hazra a bonafide
student of Delhi public school, Ruby Park Kolkata
class 12 success fully completed his Mathematics
project “ List of Mathematicians and their
contribution in Mathematics “ during the year
2019-20 as per CBSE guidelines for the AISSCE
Practical Examination 2020.
 ACKNOWLEDGEMENT
I express my sincere gratitude towards my chemistry
teacher who had given us the opportunity to explore
new avenues for the Mathematics project; CBSE for
providing us with a platform to illustrate our
creativity; my friends for unconditional support and
cooperation. The book and websites need special
mention here as they laid the foundation to our
projects. And of course ‘ my parents, without whose
guidance and perseverance the project would not
have been possible.
INTRODUCTION
Our World is unthinkable without mathematics. From the
ancient times, from the earliest civilisations of Egypt, Indus
valley and Babylonia special emphasis had been laid on the
subject of mathematics. Great thinkers, philosophers and
mathematicians were considered priced gems of the ancient
civilisations. They made it possible to build mammoth
structures of those times which has now become symbols of
those civilisations through their intricate knowledge of
geometry. Also smooth functioning was possible due to
highly developed statistical knowledge of those times. From
ancient mathematicians like Pythagoras and Aryabhatta to
most recent ones like Newton and Ramanujan modern
mathematics have evolved a lot.
Mathematics have evolved such that modern world is
immobile without it. It lays the basis for modern physics to
proper functioning of the government through various
statistical data handling. It is also an integral part of
economics.
Thus detailed knowledge about various mathematicians
along with their contributions are a much required
knowledge in today’s world.
 LIST OF FIVE INDIAN MATHEMATICIANS
AND THEIR CONTRIBUTION
ARYABHATTA :
Aryabhata was one of the first Indian mathematicians and
astronomers belonging to the classical age. He was born in
476 BC in Tarenaga, a town in Bihar, India. It is however
definite that he travelled to Kusumapara (modern day
Patna) for studies and even resided there for some time. It
is mentioned in a few places that Aryabhata was the head
of the educational institute in Kusumapara. The University
of Nalanda had an observatory in its premises so it is
hypothesized that Aryabhata was the principal of the
university as well. On the other hand some other
commentaries mention that he belonged to Kerala.
Mathematical Work
Aryabhata wrote many mathematical and astronomical
treatises. His chief work was the ‘Ayrabhatiya’ which was a
compilation of mathematics and astronomy. The name of
this treatise was not given to it by Aryabhata but by later
commentators. A disciple by him called the ‘Bhaskara’
names it ‘Ashmakatanra’ meaning ‘treatise from the
Ashmaka’. This treatise is also referred to as ‘Ayra-shatas-
ashta’ which translates to ‘Aryabhata’s 108’. This is a very
literal name because the treatise did in fact consist of 108
verses. It covers several branches of mathematics such as
algebra, arithmetic, plane and spherical trigonometry.
Also included in it are theories on continued fractions,
sum of power series, sine tables and quadratic equations.
Aryabhata worked on the place value system using letters
to signify numbers and stating qualities. He also came up
with an approximation of pi (π) and area of a triangle. He
introduced the concept of sine in his work called ‘Ardha-
jya’ which is translated as ‘half-chord’.

SRINIVASA RAMANUJAN :
Srinivasa Ramanujan, an Indian mathematician was born
in 22nd December, 1887 in Madras, India. Like Sophie
Germain, he received no formal education in mathematics
but made important contributions to advancement of
mathematics.

Contribution to Mathematics
His chief contribution in mathematics lies mainly in
analysis, game theory and infinite series. He made in depth
analysis in order to solve various mathematical problems
by bringing to light new and novel ideas that gave impetus
to progress of game theory. Such was his mathematical
genius that he discovered his own theorems. It was
because of his keen insight and natural intelligence that he
came up with infinite series for π
This series made up the basis of certain algorithms that are
used today. One such remarkable instance is when he
solved the bivariate problem of his roommate at spur of
moment with a novel answer that solved the whole class
of problems through continued fraction. Besides that he
also led to draw some formerly unknown identities such as
by linking coefficients of and providing identities for
hyperbolic secant.
He also described in detail the mock theta function, a
concept of mock modular form in mathematics. Initially,
this concept remained an enigma but now it has been
identified as holomorphic parts of maass forms. His
numerous assertions in mathematics or concepts opened
up new vistas of mathematical research for instance his
conjecture of size of tau function that has distinct modular
form in theory of modular forms. His papers became an
inspiration with later mathematicians such as G. N.
Watson, B. M. Wilson and Bruce Berndt to explore what
Ramanujan discovered and to refine his work. His
contribution towards development of mathematics
particularly game theory remains unrivaled as it was based
upon pure natural talent and enthusiasm. In recognition
of his achievements, his birth date 22 December is
celebrated in India as Mathematics Day. It would not be
wrong to assume that he was first Indian mathematician
who gained acknowledgment only because of his innate
genius and talent.
P.C.MAHALANOBIS :
Prasanta Chandra Mahalanobis OBE, FNA, FASc, FRS
(29 June 1893 – 28 June 1972) was an Indian Bengali
scientist and applied statistician. He is best remembered
for the Mahalanobis distance, a statistical measure, and for
being one of the members of the first Planning
Commission of free India. He made pioneering studies in
anthropometry in India. He founded the Indian Statistical
Institute, and contributed to the design of large-scale
sample surveys. For his contributions, Mahalanobis has
been considered the father of modern statistics.
Contributions to statistics
Mahalanobis Distance
Mahalanobis Distance is one of the most widely used
metric to find how much a point diverges from a
distribution, based on measurements in multiple
dimensions. It is widely used in the field of cluster analysis
and classification. It was first proposed by Mahalanobis in
1930 in context of his study on racial likeness. From a
chance meeting with Nelson Annandale, then the director
of the Zoological Survey of India, at the 1920 Nagpur
session of the Indian Science Congress led to Annandale
asking him to analyse anthropometric measurements of
Anglo-Indians in Calcutta. Mahalanobis had been
influenced by the anthropometric studies published in the
journal Biometrika and he chose to ask the questions on
what factors influence the formation of European and
Indian marriages. He wanted to examine if the Indian side
came from any specific castes. He used the data collected
by Annandale and the caste-specific measurements made
by Herbert Risley to come up with the conclusion that the
sample represented a mix of Europeans mainly with
people from Bengal and Punjab but not with those from
the Northwest Frontier Provinces or from Chhota Nagpur.
He also concluded that the intermixture more frequently
involved the higher castes than the lower ones. This
analysis was described by his first scientific paper in 1922.
During the course of these studies he found a way of
comparing and grouping populations using a multivariate
distance measure. This measure, denoted "D2" and now
eponymously named Mahalanobis distance, is
independent of measurement scale. Mahalanobis also took
an interest in physical anthropology and in the accurate
measurement of skull measurements for which he
developed an instrument that he called the "profiloscope".
Sample surveys
His most important contributions are related to large-
scale sample surveys. He introduced the concept of pilot
surveys and advocated the usefulness of sampling
methods. Early surveys began between 1937 and 1944 and
included topics such as consumer expenditure, tea-
drinking habits, public opinion, crop acreage and plant
disease. Harold Hotelling wrote: "No technique of random
sample has, so far as I can find, been developed in the
United States or elsewhere, which can compare in
accuracy with that described by Professor Mahalanobis"
and Sir R. A. Fisher commented that "The ISI has taken the
lead in the original development of the technique of
sample surveys, the most potent fact-finding process
available to the administration".
He introduced a method for estimating crop yields which
involved statisticians sampling in the fields by cutting
crops in a circle of diameter 4 feet. Others such as P. V.
Sukhatme and V. G. Panse who began to work on crop
surveys with the Indian Council of Agricultural Research
and the Indian Agricultural Statistics Research Institute
suggested that a survey system should make use of the
existing administrative framework. The differences in
opinion led to acrimony and there was little interaction
between Mahalanobis and agricultural research in later
years.
BRAHMAGUPT(598‐668 AD)
Brahmagupta was born in 598 A.D.in Bhinmal city in the
state of Rajasthan. He was a mathematician and
astronomer, who wrote many important works on
mathematics and astronomy. His best known work is the
“Brahmasphuta‐siddhanta”, written in 628 AD in
He was the first to use zero as a number. He gave rules to
compute with zero. He gave four methods of
multiplication. He gave following formulae, used in G.P.
series a+ar+ar2+ar3 +……….+arn‐1 =a(rn ‐1)/( r‐1)
He gave the following formulae(Brahmagupta’s formula):
Area of a cyclic quadrilateral with side
a,b,c,d =√(s‐a)(s‐b)(s‐c)(s‐d), where 2s= a+b+c+d.
Length of its diagonals=√ 𝑏𝑐+𝑎𝑑
𝑎𝑏+𝑐𝑑
(ac+bd)

SHAKUNTALA DEVI
She was born in 1939.She is an Indian calculating
prodigy. By age 6,She demonstrated her calculation
and memorization abilities at university of Mysore.
At the age of 8,she had successes at Annamalai
University by doing the same.
CONTRIBUTION :
On June 18, 1980,She demonstrated the
multiplication of two 13‐digit numbers
7,686,369,774,870X2,465,099.745,779 picked at
random by the Computer Department of Imperial
College, London. She answered the question in 28
seconds. However, the time is more likely the time
for dictating the answer (a 26‐digit number) than the
time for mental calculation(the time of 28 seconds
was quoted on her website).Her answer was
18,947,668,177,995,426,773,730.This event is
mentioned on page 26 of the 1995 Guinness Book of
Records. In Dallas, she competed with a computer to
see who give the cube of 188138517 faster, she won.
At University of USA she was asked to give the 23rd
root of
91674867692003915809866092758538016248310668
014430862240712651642793465
70408670965932792057674808067900227830163549
248523803357453169351119035
965775473400756881868830562082101612913284556
4895780158806771. She answered in 50 seconds. The
answer is 546372891.It took a UNIVAC 1108
computer, full one minute (10 seconds more) to
confirm that she was right after it was fed with 13000
instructions. Now she is known to be Human
Computer.
FIVE FOREIGN MATHEMATECIANS

BLAISE PASCAL :
Blaise Pascal was a French mathematician, physicist,
inventor, writer and Catholic philosopher. He did
pioneering works in calculating machines and came
up with the mechanical calculator. He was an
influential mathematician and made significant
researches in areas like projective geometry and
probability theory. He also made important
contribution in arithmetic triangle and cycloid.
Along with Fermat, Pascal came up with the calculus
of probabilities, which later led to the foundation of
mathematical theory of probabilities. He also
worked in natural and applied sciences and made
important contributions in concepts like fluids,
pressure and vacuum. Honoring his works and
contributions, his name Pascal has been made the SI
unit of pressure and also a programming language.
His other notable contributions include Pascal's law,
Pascal's triangle and Pascal's wager. His religious
works, "Lettres provinciales and the Pensees" had a
religious influence all over France and created a new
level of style in French prose.
Contribution In Mathematics & Science: Pascal
always remained an influential mathematician
throughout his life. His convenient tabular
presentation of binomial coefficients described in
his Traité du triangle arithmétique, released in 1653,
later became famous as Pascal’s triangle. In 1654,
following a friend,the Chevalier de Méré’s interest in
gambling problem, Pascal discussed this subject
with Fermat, which later led to the foundation of
mathematical theory of probabilities. One of the
gambling problems was of two players who wanted
to finish a game early, and given the then condition
of the game, wanted to share the stakes fairly, based
on the fact that each player had equal chances of
winning the match from that point. In this context,
Pascal used a probabilistic argument also known as
Pascal's wager. The work done by Pascal and Fermat
later helped Leibniz formulate infinitesimal
calculus. Pascal also made important contribution to
the philosophy of mathematics with his works like
De l'Esprit géométrique and De l'Art de persuader.
Pascal contribution to the physical sciences includes
his works in fields of hydrodynamics and
hydrostatics which were mostly based on hydraulic
principles. He also had the credit of inventing
syringe and hydraulic press. Following the views of
Galileo and Torricelli, he opposed the Aristotelian
notion which says that a creation is a thing of
substance, whether visible or invisible. He advocated
the presence of vacuum in substances. He said that
it is the vacuum which keeps the mercury floating in
a barometer and even fills the space above the
mercury in the tube. In his work in 1647,
“Experiences nouvelles touchant le vide” he gave
more experiments regarding his statement on
vacuum. These experiments performed by Pascal
were praised throughout the Europe and established
his principle and also the value of barometer.

BERNOULLI:
Jacob Bernoulli can be rightly called the initiator of
the Bernoulli family's mathematical dynasty. A class
of his own, Jacob was bright and intelligent right
from the very beginning. His well-researched
concepts brought about a revolution in Swiss
mathematics. Jacob Bernoulli is credited for being
the first person to develop the technique for solving
separable differential equations. He is also
responsible for the Bernoulli numbers, by which he
derived the exponential series. In addition to that,
the subject of probability, which is now called the
Bernoulli law of large numbers, also falls in the kitty
of this prominent and renowned mathematician.
With this article, know more about the life and
history of Jacob Bernoulli.

Contribution to Mathematics Jacob Bernoulli’s

foremost significant contributions to mathematics
were noted in the pamphlets which accounted
information on the parallels of logic and algebra and
his work on probability. These were published in the
year 1685. Two years later, Bernoulli’s work on
geometry was published, which gave a construction
to divide any triangle into four equal parts with two
perpendicular lines. In 1689, Jacob Bernoulli
published an important work on infinite series and
his law of large numbers in probability theory.
According to his interpretation, if an experiment is
repeated a number of times, then the relative
frequency with which an event occurs equals the
probability of the event. The law of large numbers is
a mathematical interpretation of this result. From
1682 to 1704, Jacob Bernoulli published five treatises
on infinite series. He also researched on the
exponential series which came out of examining
compound interest. In the year 1690, Jacob Bernoulli
proved that the problem of determining the
isochrone is equivalent to solving a first-order
nonlinear differential equation, in the paper
published in Acta Eruditorum. According to it, the
isochrone, or simply the curve of a constant descent,
is a curve along which a particle descends under
gravity from any point to the bottom in exactly the
same time, regardless of what the starting point is.
After finding the differential equation, Bernoulli
then solved it by what we now know as separation of
variables. His paper of 1690 is exceptionally essential
in the history of calculus, since it was then that the
term integral appeared for the first time with its
integration meaning. Other than these, Jacob also
ascertained a general method to determine evolutes
of a curve as the envelope of its circles of curvature.
In 1692, he investigated on the caustic curves,
particularly, those associated with the curves of the
parabola, the logarithmic spiral and epicycloids. Two
years later, he first conceived the lemniscate of
Bernoulli and later the next year, he carried out a
research wherein he applied calculus in the building
of suspension bridges. In 1696, Bernoulli solved the
equation, now known as "the Bernoulli equation".
However, Jacob Bernoulli’s most notable work was
the Ars Conjectandi (The Art of Conjecture), which
was published posthumously in 1713, by his nephew
Nicholas. Herein he went on to brief about the
known results in probability theory and in
enumeration, often providing alternative proofs of
known results. The work also included the
application of probability theory to games of chance
and his introduction of the theorem known as the
law of large numbers. The terms “Bernoulli trial” and
“Bernoulli numbers” are the products from this
work.

JOHN NAPPIER:
David Hume's personification of the title "a great
man" more than aptly describes the prominence and
distinction of John Napier. A distinguished Scottish
mathematician and theological writer, John Napier
is famously credited as the man who originated the
concept of logarithm. With his innovative
discoveries and research, Napier created a storm in
the field of mathematical calculations. While his
concept of logarithms gained most limelight,
Napier's other contributions in the field of spherical
trigonometry, the invention of the divining rods and
pressing forward the use of decimal fraction are
second to none. It was due to his ground-breaking
inventions that Napier earned the respect of some of
the most illustrious astronomers and scientists of the
age. Know more about the life and contributions of
this ace mathematician through the following lines.

Contributions in Mathematics: Napier’s interest in

astronomy led way to his interest in mathematics.
His long hours of leisure were spent exploring and
devising new methods of computation that could
help astronomers during their research. Logarithm,
as we now know it, was a seed of thought of Napier
who, from his research, brought out a newer and
simpler way for performing large number
calculations. He found out that through the use of
exponentials, the operations which involved
multiplication and division of very large numbers
reduced to being just an addition of the exponents.
He gradually went on to elaborate his computational
system wherein roots, products and quotients could
be easily found out from tables showing powers of a
fixed number at the base. Napier’s finding was first
made public in the year 1614, through his book,
Mirifici logarithmorum canonis descriptio
(Description of the Marvellous Canon of Logarithm).
The book, though, only had brief details of the steps
that led to the discovery, importantly contained his
first set of logarithmic table. Not only did the table
found an instant acceptance from the astronomers
and scientists around the world, it paved way to the
cooperative movement including the development
of Base 10. In his second book, Mirifici
Logarithmorum Canonis Constructio (Construction
of the Marvelous Canon of Logarithms), which was
published posthumously, Napier further advanced
the concept of decimal fraction, which was first
introduced by Simon Stevin, a Flemish
mathematician. His research, which suggested that
a simple point could separate a whole number and
fractional parts of the number, was an instant hit in
Great Britain. The advances in the field of
computation through logarithms not only made
calculations by hand quicker, but it also opened
doors to further scientific advancements done in the
field of astronomy, dynamics, physics and even
astrology. Although his invention of logarithm
outshined his other mathematical findings, Napier,
nevertheless, holds the credit for his inventions in
the arena of spherical trigonometry. Two formulae
known as "Napier's analogies" used in solving
spherical triangles and an invention called "Napier's
bones" used for mechanically multiplying, dividing
and taking square roots and cube roots also fall in
the kitty of this ace mathematician. The latter
concepts were published in his book, Rabdologiæ
seu Numerationis per Virgulas libri duo.

PYTHAGORAS:
Pythagoras of Samos was a well-known
mathematician, scientist and a religious teacher. He
was born in Samos and is often hailed as the first
great mathematician. Pythagoras is remembered
today for his famous theorem in geometry, the
'Pythagoras Theorem'. His mentors were Thales,
Pherekydes and Anaximander, who inspired him to
pursue mathematics and astronomy. Pythagoras also
made important discoveries in music, astronomy
and medicine. He accepted priesthood and
performed the rites that were required in order to
enter one of the temples in Egypt, known as
Diospolis. He set up a brotherhood with some of his
followers, who practiced his way of life and pursued
his religious ideologies. He became one of the most
distinguished teachers of religion in ancient Greece.
Read on to know more about the childhood and
career of this ancient Greek philosopher and
mathematician.
Mathematical Concepts

Pythagoras studied properties of numbers and

classified them as even numbers, odd numbers,
triangular numbers and perfect numbers etc. The
‘Pythagoras theorem’ is one of the earliest theorems
in geometry, which states that in right-angle
triangles, the square of the hypotenuse is equal to
the sum of square the other two sides. This theorem
was already proposed during the reign of the
Babylonian King Hammurabi, but Pythagoras
applied it to mathematics and science and refined
the concept. Pythagoras also asserted that dynamics
of the structure of the universe lies on the
interaction of the contraries or the opposites, such
as, light and darkness, limited and unlimited, square
and oblong, straight and crooked, right and left,
singularity and plurality, male and female,
motionless and movement and good and bad.
RENE DESCARTES:
Rene Descartes was an eminent French
Mathematician, philosopher and writer, who has
been popularly referred to as 'Father of Modern
Philosophy'. Descartes was the foremost amongst all
to highlight the importance of reason for the growth
of natural sciences. He regarded philosophy as a
belief system that contained immense knowledge.
To this day, his work on philosophy " Meditations on
First Philosophy" is taught as a standard text in many
universities. His philosophical statement "Cogito
ergo sum" meaning "I think, therefore I am",
mentioned in his book 'Discourse on the Method'
took him to fame. In his natural philosophy he
refuted the 'analysis of corporeal substance into
matter and form' and rejected any appeal to divine
or natural ends in explaining natural phenomena.
His contribution in mathematics was immense that
he has been called the 'father of analytical geometry'.
Descartes was also proponent of continental
rationalism along with Leibniz, Gottfried and
Spinoza in the seventeenth century.

Career
Descartes came back to France in 1622. It was during
his stay in Paris that he wrote his first essay—
Regulae ad Directionem Ingenii (Rules for the
Direction of the Mind). In 1628 René Déscartes
moved to Dutch Republic and got himself enrolled
in the University of Franeker and the Leiden
University to study mathematics. He lived in Dutch
Republic for over 20 years, during which he
published many works on philosophy and
mathematics. Descartes withheld the publication of
his work “Treatise on the World” following
censorship of Galileo works by Catholic Church in
1633. However, he produced part of his writings in
his essays namely La Géométrie, La Dioptrique and
Les Météores. He presented his work such as
Meditations on First Philosophy (1641) and
Principles of Philosophy (1644) on metaphysics.
After Cartesian philosophy faced criticism at the
University of Utrecht in 1643, Descartes established
contact with Princess Elisabeth of Bohemia through
correspondence, writing topics on psychology and
morality, which he compiled in Passions of the Soul
(1649) with dedication to the Princess. He argued
that moral philosophy must include the study of the
body as well. He dealt with this in his books “The
Description of the Human Body” and “Passions of
the Soul”, where he argues that human body is more
like a machine and therefore, it has material
properties. The King of France rewarded Déscartes a
pension in 1647. However, his books were banned by
the Pope in 1663.
Legacy
Descartes left rich legacy in mathematics by ideas on
Cartesian geometry and creation of XYZ as
representation for unknown equation. His works
became foundation for development of calculus
theory by Leibinz and Newton. Besides, he also made
contribution in the field of optics.