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2. Publication of precise trigonometry tables,
improvement of surveying methods using
trigonometry, and mathematical analysis of
trigonometric relationships. (approx. 1530 – 1600)
• Logarithms introduced by Napier in 1614 as a
calculation aid. This advances science in a
manner similar to the introduction of the
computer.

3. Explosion of mathematical and scientific ideas
across Europe, a period sometimes called the Age
of Reason.
The Logarithm of a number is the exponent when
the number is expressed as a power of 10 (or any
other base). It is effectively the inverse of
exponentiation.

5. Development of symbolic algebra,
principally by the French mathematicians
Francois Viete and Rene Descartes
The cartesian coordinate system and
analytic geometry developed by Rene
Descartes and Pierre Fermat (1630 – 1640)

6. François Viète(1540-1603)
In his influential treatise In Artem
Analyticam Isagoge (Introduction to the
Analytic Art, published in 1591) Viète
demonstrated the value of symbols. He
suggested using letters as symbols for
quantities, both known and unknown.

7. The development of analytical geometry
and Cartesian coordinates of Descartes in
the mid-17th century soon allowed the
orbits of the planets to be plotted on a
graph, as well as laying the foundations for
the later development of calculus (and
much later multi-dimensional geometry)

8. Two other French mathematicians were
close contemporaries of Descartes: Pierre
de Fermat and Blaise Pascal. Fermat
formulated several theorem which greatly
extended the knowledge in number theory.
Pascal is most famous for Pascal’s Triangle
of binomial coefficients.

9. French mathematicians and engineer
Girard Desagues is considered one of the
founders of the field of projective
geometry, later developed further by Jean
Victor Poncelet and Gaspard Monge.
He developed the pivotal concept of “point
of infinity” where parallels actually meet.

10. Johannes Kepler(1571-1630)
Kepler’s first attempt to describe
planetary orbits used a model of nested
regular polyhedral (Platonic solids).

11. By “standing on the shoulders of giants”,
the Englishman Sir Isaac Newton was able
to pin down the laws of physics, laid the
groundwork for classical mechanics, almost
single-handledly.
Newton along with Archimedes and Gauss,
as one of the greatest mathematicians of
all time.

12. Calculus co-invented by Isaac Newton and
Gottfried Leibniz. Major ideas of the calculus
expanded and refined by others, especially
the Bernoulli family and Leonhard Euler.
(approx. 1660 – 1750).
A powerful tool to solve scientific and
engineering problems, it opened the door to
a scientific and mathematical revolution.

13. Both Newton and Leibniz contributed greatly
in areas of mathematics, including Newton’s
contributions to a generalized binomial
theorem, the theory of finite differences and
the use of infinite series, and Leibniz’s
development of a mechanical forerunner to
the computer and the use of a matrices to
solve linear equations.

14. Newton is considered by many to be one of the
most influential men in human history. The
“Philosophiae Naturalis Principia Mathematica”
(simply “Principia”) is considered to be among
the most influential books in the history of
science, it dominated the scientific view of the
physical universe for the next three centuries.
“Mathematics Principle of Natural Philosophy”

16. The period was dominated, though, by one
family, the Bernoulli’s of Basel in Switzerland,
which boasted two or three generations of
exceptional mathematicians, particularly the
brothers, Jacob and Johann.
There were largely responsible for developing
Leibniz’s infinitesimal calculus known as
“calculus of variations” as well as Pascal and
Fermat’s probability and number theory.

17. Unusually in the history of math, a single family,
the Bernoulli’s produced half a dozen
outstanding mathematicians over a couple of
generations at the end of the 17th and start of
the 18th Century.
James Bernoulli (1654-1705) carrying out his
father’s wish for him to enter the ministry, took
a degree in theology at the University of Basel in
1676. (Jacques, Jacob)

18. John Bernoulli (1667-1748), like his brother
James, ran counter to his father’s plan regarding
his work in life. He was being privately tutored
by his brother in the mathematical sciences.
When Leibniz’s papers began to appear in the
Acta Eruditorum, John mastered the new
methods and followed in Jame’s footsteps as one
of the leading exponents of the calculus. (Jean,
Johann)

19. After Johann graduated, the two developed a
rather jealous and competitive relationship.
Johann in particular was jealous of the elder
Jacob’s position as professor at Basel University.
Johann merely shifted his jealousy toward his own
talented son, Daniel. Johann received a taste of his
own medicine, though, when his student Guillaume
de l’Hopital published a book in his own name
consisting almost all of Johann’s lectures.

20. Despite their competitive and combative
relationship, though, the brothers had constantly
challenged and inspired each other. They
established an early correspondence with Gottfried
Leibniz, and were among the first mathematicians
to not only study and understand infinitesimal
calculus but to apply it to various problems.

21. Contributions:
Designing a sloping ramp, a less steeped curved ramp
- brachistochrone curve (upside-down cycloid)
Calculus of variations
Book: The Art of Conjecture (1713)
Bernoulli Numbers Sequence
Technique for solving separable differential equations
Polar coordinates & Integrals (angles and distances)
Approximate value of the irrational number e.

22. Pierre-Simon Laplace, “the French Newton”, was an
important mathematician and astronomer, whose
monumental work “Celestial Mechanics”. His early
work was mainly on differential equations and finite
differences, he was already thinking about the
mathematical and philosophical concepts of
probability and statistics in the 1770’s and he
developed his own version of the so-called Bayesian
Interpretation of probability (Thomas Bayes)

23. Another Frenchman, Gaspard Monge was the
inventor of descriptive geometry, a clever method
of representing three dimensional objects by
projections on the two-dimensional plane using a
specific set of procedures, a technique which would
later become important in the fields of engineering,
architecture and design. His orthographic projection
became the graphical method used in almost all
modern mechanical drawing.

24. After many centuries of increasingly accurate
approximations, Johann Lambert, a swiss mathematician
and prominent astronomer, finally provided a rigorous
proof in 1761 that pi (𝜋) is irrational, i.e. it can not be
expressed as a simple fraction using integers only or as a
terminating or repeating decimal
He also first introduce hyperbolic functions into
trigonometry and made some prescient conjectures
regarding non-Euclidean space and the properties of
hyperbolic triangles.

25. One of the giants of the 18th Century mathematics. Born
in Basel Switzerland, and he studied for a while under
Johann Bernoulli at Basel University but he spent most of
his time in Russia and Germany.
He produced one mathematical paper every week – as he
compensated for it with his mental calculation skills and
photographic memory (for example he could repeat the
Aeneid of Virgil from beginning to end without hesitation,
and for every page in the edition he could indicate which
line was the first and which the last.

26. Today, Euler is considered one of the greatest
mathematicians of all time. His interests covered almost
all aspects of mathematics, from geometry to calculus to
trigonometry to algebra to number theory, as well as
optics, astronomy, cartography, mechanics, weights and
measures and even the theory of music.
The list of theorems and methods pioneered by Euler is
immense, and largely outside the scope of an entry-level
study but to mention, these are just some of them:

27. The definition of Euler Characteristics (Chi) for the
surface of polyhedral
A new method for solving quadratic equations
The Prime Number theorem
Proofs (and disproofs) of some of Fermat’s theorems and
conjectures.
A method of calculating integrals with complex limits
The calculus of variations
ETC.

29. Joseph Fourier's study, at the beginning of the
19th Century, of infinite sums in which the terms
are trigonometric functions were another
important advance in mathematical analysis.
Periodic functions that can be expressed as the
sum of an infinite series of sines and cosines are
known today as Fourier Series, and they are still
powerful tools in pure and applied mathematics.

30. In 1806, Jean-Robert Argand published his
paper on how complex numbers (of the
form a + bi, where i is √-1) could be
represented on geometric diagrams and
manipulated using trigonometry and
vectors.

31.  The Frenchman Évariste Galois proved in the late 1820s that
there is no general algebraic method for solving polynomial
equations of any degree greater than four, going further than
the Norwegian Niels Henrik Abel who had, just a few years
earlier, shown the impossibility of solving quintic equations,
and breaching an impasse which had existed for centuries.
 Galois' work also laid the groundwork for further
developments such as the beginnings of the field of abstract
algebra, including areas like algebraic geometry, group
theory, rings, fields, modules, vector spaces and non-
commutative algebra.

32. Later in life, Gauss also claimed to have
investigated a kind of non-Euclidean geometry
using curved space but, unwilling to court
controversy, he decided not to pursue or publish
any of these avant-garde ideas. This left the field
open for János Bolyai and Nikolai Lobachevsky
(respectively, a Hungarian and a Russian) who
both independently explored the potential of
hyperbolic geometry and curved spaces.

33. British mathematics also saw something of a resurgence
in the early and mid-19th century. Although the roots of
the computer go back to the geared calculators of
Pascal and Leibniz in the 17th Century, it was Charles
Babbage in 19th Century England who designed a
machine that could automatically perform
computations based on a program of instructions stored
on cards or tape. His large "difference engine" of 1823
was able to calculate logarithms and trigonometric
functions, and was the true forerunner of the modern
electronic computer.

34. Another 19th Century Englishman, George
Peacock, is usually credited with the
invention of symbolic algebra, and the
extension of the scope of algebra beyond the
ordinary systems of numbers. This
recognition of the possible existence of non-
arithmetical algebras was an important
stepping stone toward future developments
in abstract algebra.

35. In the mid-19th Century, the British mathematician
George Boole devised an algebra (now called
Boolean algebra or Boolean logic), in which the only
operators were AND, OR and NOT, and which could
be applied to the solution of logical problems and
mathematical functions. He also described a kind of
binary system which used just two objects, "on" and
"off" (or "true" and "false", 0 and 1, etc), in which,
famously, 1 + 1 = 1. Boolean algebra was the starting
point of modern mathematical logic and ultimately
led to the development of computer science.

36. The concept of number and algebra was
further extended by the Irish
mathematician William Hamilton, whose
1843 theory of quaternions (a 4-
dimensional number system, where a
quantity representing a 3-dimensional
rotation can be described by just an angle
and a vector).

37. the Frenchman Augustin-Louis Cauchy,
completely reformulated calculus in an even
more rigorous fashion, leading to the
development of mathematical analysis, a branch
of pure mathematics largely concerned with the
notion of limits (whether it be the limit of a
sequence or the limit of a function) and with the
theories of differentiation, integration, infinite
series and analytic functions.

38. August Ferdinand Möbius is best known for his
1858 discovery of the Möbius strip, a non-
orientable two-dimensional surface which has
only one side when embedded in three-
dimensional Euclidean space (actually a
German, Johann Benedict Listing, devised the
same object just a couple of months before
Möbius, but it has come to hold Möbius'
name).

39. Many other concepts are also named after
him, including the Möbius configuration,
Möbius transformations, the Möbius
transform of number theory, the Möbius
function and the Möbius inversion formula.
He also introduced homogeneous
coordinates and discussed geometric and
projective transformations.

40. The Norwegian mathematician Marius
Sophus Lie also applied algebra to the
study of geometry. He largely created the
theory of continuous symmetry, and
applied it to the geometric theory of
differential equations by means of
continuous groups of transformations
known as Lie groups.

41. In an unusual occurrence in 1866, an unknown
16-year old Italian, Niccolò Paganini,
discovered the second smallest pair of
amicable numbers (1,184 and 1210), which
had been completely overlooked by some of
the greatest mathematicians in history
(including Euler, who had identified over 60
such numbers in the 18th Century, some of
them huge).

42. Georg Cantor established the first foundations of
set theory, which enabled the rigorous treatment
of the notion of infinity, and which has since
become the common language of nearly all
mathematics. In the face of fierce resistance from
most of his contemporaries and his own battle
against mental illness, Cantor explored new
mathematical worlds where there were many
different infinities, some of which were larger than
others.

43. A great friend of David Hilbert and teacher of
the young Albert Einstein, developed a branch
of number theory called the "geometry of
numbers" late in the 19th Century as a
geometrical method in multi-dimensional space
for solving number theory problems, involving
complex concepts such as convex sets, lattice
points and vector space.

44. Later, in 1907, it was Minkowski who
realized that the Einstein’s 1905 special
theory of relativity could be best
understood in a four-dimensional
space, often referred to as Minkowski
space-time.

45. “Begriffsschrift” (roughly translated as
“Concept-Script”) broke new ground in the
field of logic, including a rigorous treatment
of the ideas of functions and variables. In his
attempt to show that mathematics grows out
of logic, he devised techniques that took him
far beyond the logical traditions of Aristotle
(and even of George Boole).

46. He was the first to explicitly
introduce the notion of variables in
logical statements, as well as the
notions of quantifiers, universals
and existentials.

47. came to prominence in the latter part of the
19th Century with at least a partial solution to
the “three body problem”, a deceptively simple
problem which had stubbornly resisted
resolution since the time of Newton, over two
hundred years earlier. Although his solution
actually proved to be erroneous, its
implications led to the early intimations of what
would later become known as chaos theory.

48. He is sometimes referred to as the
“Last Univeralist” as he was perhaps
the last mathematician able to shine in
almost all of the various aspects of
what had become by now a huge,
encyclopedic and incredibly complex
subject.

50. Hardy and Ramanujan
Bertrand Russell and Alfred North Whitehead
David Hilbert
Kurt Godel
Alan Turing
Andre Weil
Paul Cohen
Julia Robinson & Matiyasevich

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Contributions : Equations, Functions, Applications
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2 pages ONLY (Compiled - Soft Copies)
Deadline: October 16, 2019

52. The 20th Century continued the trend of
the 19th towards increasing generalization
and abstraction in mathematics, in which
the notion of axioms as “self-evident
truths” was largely discarded in favour of
an emphasis on such logical concepts as
consistency and completeness.

53. It also saw mathematics become a major profession,
involving thousands of new Ph.D.s each year and jobs
in both teaching and industry, and the development
of hundreds of specialized areas and fields of study,
such as group theory, knot theory, sheaf theory,
topology, graph theory, functional analysis,
singularity theory, catastrophe theory, chaos theory,
model theory, category theory, game theory,
complexity theory and many more.

54.  The eccentric British mathematician G.H. Hardy and his
young Indian protégé Srinivasa Ramanujan, were just two of
the great mathematicians of the early 20th Century who
applied themselves in earnest to solving problems of the
previous century, such as the Riemann hypothesis. Although
they came close, they too were defeated by that most
intractable of problems, but Hardy is credited with
reforming British mathematics, which had sunk to
something of a low ebb at that time, and Ramanujan proved
himself to be one of the most brilliant (if somewhat
undisciplined and unstable) minds of the century.

55. The early 20th Century also saw the beginnings of
the rise of the field of mathematical logic, building
on the earlier advances of Gottlob Frege, which
came to fruition in the hands of Giuseppe Peano,
L.E.J. Brouwer, David Hilbert and, particularly,
Bertrand Russell and A.N. Whitehead, whose
monumental joint work the “Principia
Mathematica” was so influential in mathematical
and philosophical logicism.

56. The century began with a historic
convention at the Sorbonne in Paris in the
summer of 1900 which is largely
remembered for a lecture by the young
German mathematician David Hilbert in
which he set out what he saw as the 23
greatest unsolved mathematical problems
of the day.

57. These “Hilbert problems” effectively set the
agenda for 20th Century mathematics, and laid
down the gauntlet for generations of
mathematicians to come. Of these original 23
problems, 10 have now been solved, 7 are
partially solved, and 2 (the Riemann hypothesis
and the Kronecker-Weber theorem on abelian
extensions) are still open, with the remaining 4
being too loosely formulated to be stated as
solved or not.

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