What is Mathematics? What are the History of Mathematics? Sir Isaac Newton and Gottfried Wilhelm Leibniz have such great contribution to the History of Mathematics as of 17th Century.

1. 17TH CENTURY
MATHEMATICS
Mary Jane R. De Ade
BSED II

2. 17TH CENTURY
MATHEMATICS
SIR ISAAC NEWTON
GOTTFRIED LEIBNIZ

3. 17TH CENTURY
MATHEMATICS
SIR ISAAC
NEWTON
(1643-1727)

4. SIR ISAAC NEWTON
In the heady atmosphere of 17th Century
England, with the expansion of the British
Empire in full swing, grand old universities
like Oxford and Cambridge were producing
many great scientists and mathematicians.
But the greatest of them all was
undoubtedly Sir Isaac Newton.

5. SIR ISAAC NEWTON
Physicist, mathematician, astronomer,
natural philosopher, alchemist and
theologian, Newton is considered by
many to be one of the most influential
men in human history.

6. SIR ISAAC NEWTON
His 1687 publication, the "Philosophiae
Naturalis Principia Mathematica" (usually
called simply the "Principia"), is considered
to be among the most influential books in
the history of science, and it dominated the
scientific view of the physical universe for
the next three centuries.

7. SIR ISAAC NEWTON
Although largely synonymous in the minds of
the general public today with gravity and the
story of the apple tree, Newton remains a
giant in the minds of mathematicians
everywhere (on a par with the all-time greats
like Archimedes and Gauss), and he greatly
influenced the subsequent path of
mathematical development.

8. SIR ISAAC NEWTON
Over two miraculous years, during the time
of the Great Plague of 1665-66, the young
Newton developed a new theory of light,
discovered and quantified gravitation, and
pioneered a revolutionary new approach to
mathematics: infinitesimal calculus.

9. SIR ISAAC NEWTON
His theory of calculus built on earlier work
by his fellow Englishmen John Wallis and
Isaac Barrow, as well as on work of such
Continental mathematicians as Rene
Descartes, Pierre de Fermat, Bonaventura
Cavalieri, Johann van Waveren Hudde and
Gilles Personne de Roberval.

10. SIR ISAAC NEWTON
Unlike the static geometry of the
Greeks, calculus allowed mathematicians
and engineers to make sense of the
motion and dynamic change in the
changing world around us, such as the
orbits of planets, the motion of fluids, etc.

11. SIR ISAAC NEWTON
The initial problem Newton was confronting was
that, although it was easy enough to represent
and calculate the average slope of a curve (for
example, the increasing speed of an object on a
time-distance graph), the slope of a curve was
constantly varying, and there was no method to
give the exact slope at any one individual point
on the curve i.e. effectively the slope of a tangent
line to the curve at that point.

12. SIR ISAAC NEWTON
Intuitively, the slope at a particular point can
be approximated by taking the average slope
(“rise over run”) of ever smaller segments of
the curve. As the segment of the curve being
considered approaches zero in size (i.e. an
infinitesimal change in x), then the calculation
of the slope approaches closer and closer to
the exact slope at a point.

13. Differentiation (derivative) approximates the
slope of a curve as the interval approaches zero
SIR ISAAC NEWTON

14. SIR ISAAC NEWTON
Without going into too much complicated
detail, Newton (and his contemporary
Gottfried Leibniz independently) calculated
a derivative function f „(x) which gives the
slope at any point of a function f(x ).

15. SIR ISAAC NEWTON
This process of calculating the slope or
derivative of a curve or function is called
differential calculus or differentiation (or, in
Newton‟s terminology, the “method of
fluxions” - he called the instantaneous rate
of change at a particular point on a curve
the "fluxion", and the changing values of x
and y the "fluents").

16. SIR ISAAC NEWTON
For instance, the derivative of a straight
line of the type f(x) = 4x is just 4; the
derivative of a squared function f(x) = x2 is
2x; the derivative of cubic function f(x) = x3
is 3x2, etc. Generalizing, the derivative of
any power function f(x) = xr is rxr-1 .

17. SIR ISAAC NEWTON
Other derivative functions can be stated,
according to certain rules, for exponential and
logarithmic functions, trigonometric functions
such as sin(x), cos(x), etc. so that a derivative
function can be stated for any curve without
discontinuities. For example, the derivative of
the curve f(x) = x4 - 5p3 + sin(x2) would be f
‟(x) = 4x3 - 15x2 + 2xcos(x2).

18. SIR ISAAC NEWTON
Having established the derivative function
for a particular curve, it is then an easy
matter to calculate the slope at any
particular point on that curve, just by
inserting a value for x. In the case of a
time-distance graph, for example, this
slope represents the speed of the object at
a particular point.

19. SIR ISAAC NEWTON
The “opposite” of differentiation is
integration or integral calculus (or, in
Newton‟s terminology, the “method of
fluents”), and together differentiation and
integration are the two main operations of
calculus.

20. SIR ISAAC NEWTON
Newton‟s Fundamental Theorem of
Calculus states that differentiation and
integration are inverse operations, so
that, if a function is first integrated and
then differentiated (or vice versa), the
original function is retrieved.

21. SIR ISAAC NEWTON
The integral of a curve can be thought of
as the formula for calculating the area
bounded by the curve and the x axis
between two defined boundaries.

22. SIR ISAAC NEWTON
For example, on a graph of velocity
against time, the area “under the curve”
would represent the distance travelled.
Essentially, integration is based on a
limiting procedure which approximates the
area of a curvilinear region by breaking it
into infinitesimally thin vertical slabs or
columns.

23. SIR ISAAC NEWTON
In the same way as for differentiation, an
integral function can be stated in general
terms: the integral of any power f(x) = xr is
xr+1⁄r+1, and there are other integral
functions for exponential and logarithmic
functions, trigonometric functions, etc, so
that the area under any continuous curve
can be obtained between any two limits.

24. Integration
approximates
the area under
a curve as the
size of the
samples
approaches
zero
SIR ISAAC NEWTON

25. SIR ISAAC NEWTON
Newton chose not to publish his
revolutionary mathematics straight away,
worried about being ridiculed for his
unconventional ideas, and contented
himself with circulating his thoughts among
friends.

26. SIR ISAAC NEWTON
After all, he had many other interests such
as philosophy, alchemy and his work at the
Royal Mint. However, in 1684, the German
Leibniz published his own independent
version of the theory, whereas Newton
published nothing on the subject until
1693.

27. SIR ISAAC NEWTON
Although the Royal Society, after due
deliberation, gave credit for the first discovery to
Newton (and credit for the first publication to
Leibniz), something of a scandal arose when it
was made public that the Royal Society‟s
subsequent accusation of plagiarism against
Leibniz was actually authored by none other
Newton himself, causing an ongoing controversy
which marred the careers of both men.

28. SIR ISAAC NEWTON
Despite being by far his best known
contribution to mathematics, calculus was
by no means Newton‟s only contribution.
He is credited with the generalized
binomial theorem, which describes the
algebraic expansion of powers of a
binomial (an algebraic expression with two
terms, such as a2 - b2);

29. SIR ISAAC NEWTON
he made substantial contributions to the
theory of finite differences (mathematical
expressions of the form f(x + b) - f(x + a));
he was one of the first to use fractional
exponents and coordinate geometry to
derive solutions to Diophantine equations
(algebraic equations with integer-only
variables);

30. SIR ISAAC NEWTON
he developed the so-called “Newton's
method” for finding successively better
approximations to the zeroes or roots of a
function;
he was the first to use infinite power series
with any confidence; etc.

31. SIR ISAAC NEWTON
In 1687, Newton published his “Principia” or “The
Mathematical Principles of Natural Philosophy”,
generally recognized as the greatest scientific
book ever written. In it, he presented his theories
of motion, gravity and mechanics, explained the
eccentric orbits of comets, the tides and their
variations, the precession of the Earth's axis and
the motion of the Moon.

32. SIR ISAAC NEWTON
Later in life, he wrote a number of religious
tracts dealing with the literal interpretation
of the Bible, devoted a great deal of time to
alchemy, acted as Member of Parliament
for some years, and became perhaps the
best-known Master of the Royal Mint in
1699, a position he held until his death in
1727.

33. SIR ISAAC NEWTON
In 1703, he was made President of the
Royal Society and, in 1705, became the
first scientist ever to be knighted. Mercury
poisoning from his alchemical pursuits
perhaps explained Newton's eccentricity in
later life, and possibly also his eventual
death.

34. Newton's Method for approximating the roots of a
curve by successive interactions after an initial guess
SIR ISAAC NEWTON

35. 17TH CENTURY
MATHEMATICS
GOTTFRIED
LEIBNIZ
(1646-1716)

36. GOTTFRIED LEIBNIZ
The German polymath Gottfried Wilhelm Leibniz
occupies a grand place in the history of
philosophy. He was, along with René Descartes
and Baruch Spinoza, one of the three great 17th
Century rationalists, and his work anticipated
modern logic and analytic philosophy. Like many
great thinkers before and after him, Leibniz was
a child prodigy and a contributor in many
different fields of endeavor.

37. GOTTFRIED LEIBNIZ
But, between his work on philosophy and
logic and his day job as a politician and
representative of the royal house of
Hanover, Leibniz still found time to work on
mathematics.

38. GOTTFRIED LEIBNIZ
He was perhaps the first to explicitly employ the
mathematical notion of a function to denote
geometric concepts derived from a curve, and he
developed a system of infinitesimal calculus,
independently of his contemporary Sir Isaac
Newton. He also revived the ancient method of
solving equations using matrices, invented a
practical calculating machine and pioneered the
use of the binary system.

39. GOTTFRIED LEIBNIZ
Like Newton, Leibniz was a member of the
Royal Society in London, and was almost
certainly aware of Newton‟s work on
calculus. During the 1670s (slightly later
than Newton‟s early work), Leibniz
developed a very similar theory of
calculus, apparently completely
independently.

40. GOTTFRIED LEIBNIZ
Within the short period of about two
months he had developed a complete
theory of differential calculus and integral
calculus (
see the section on Newton for a brief
description and explanation of the
development of calculus).

41. GOTTFRIED LEIBNIZ
Unlike Newton, however, he was more than
happy to publish his work, and so Europe first
heard about calculus from Leibniz in 1684, and
not from Newton (who published nothing on the
subject until 1693). When the Royal Society was
asked to adjudicate between the rival claims of
the two men over the development of the theory
of calculus, they gave credit for the first
discovery to Newton, and credit for the first
publication to Leibniz.

42. GOTTFRIED LEIBNIZ
However, the Royal Society, by then under the
rather bias presidency of Newton himself, later
also accused Leibniz of plagiarism, a slur from
which Leibniz never really recovered.
Ironically, it was Leibniz‟s mathematics that
eventually triumphed, and his notation and his
way of writing calculus, not Newton‟s clumsier
notation, is the one still used in mathematics
today.

43. Leibniz’s and
Newton’s
notation for
Calculus
GOTTFRIED LEIBNIZ

44. GOTTFRIED LEIBNIZ
In addition to calculus, Leibniz re-
discovered a method of arranging linear
equations into an array, now called a
matrix, which could then be manipulated to
find a solution.

45. GOTTFRIED LEIBNIZ
A similar method had been pioneered by
Chinese mathematicians‟ almost two
millennia earlier, but had long fallen into
disuse. Leibniz paved the way for later
work on matrices and linear algebra by
Carl Friedrich Gauss.

46. GOTTFRIED LEIBNIZ
He also introduced notions of self-similarity
and the principle of continuity which
foreshadowed an area of mathematics
which would come to be called topology.

47. GOTTFRIED LEIBNIZ
During the 1670s, Leibniz worked on the
invention of a practical calculating
machine, which used the binary system
and was capable of multiplying, dividing
and even extracting roots, a great
improvement on Pascal‟s rudimentary
adding machine and a true forerunner of
the computer.

48. GOTTFRIED LEIBNIZ
He is usually credited with the early
development of the binary number system
(base 2 counting, using only the digits 0
and 1), although he himself was aware of
similar ideas dating back to the I Ching of
Ancient China.

49. GOTTFRIED LEIBNIZ
Because of the ability of binary to be
represented by the two phases "on" and
"off", it would later become the foundation
of virtually all modern computer
systems, and Leibniz's documentation was
essential in the development process.

50. Binary Number
System
GOTTFRIED LEIBNIZ

51. GOTTFRIED LEIBNIZ
Leibniz is also often considered the most
important logician between Aristotle in Ancient
Greece and George Boole and Augustus De
Morgan in the 19th Century. Even though he
actually published nothing on formal logic in his
lifetime, he enunciated in his working drafts the
principal properties of what we now call
conjunction, disjunction, negation, identity, set
inclusion and the empty set.

52. 17TH CENTURY
MATHEMATICS
“I can calculate the motion of heavenly bodies but not
the madness of people”
-Isaac Newton
“Music is the pleasure the human mind experiences from
counting without being aware that it is counting.”
-Gottfried Wilhelm Leibniz
THANK YOU!! 