1. Mathematics Development in
Europe
Overview

2. Overview
• Transmission period
• Fibonacci and the 13th century
• Cubic and quadratic equations
• Euclidean postulate
• Non Euclidean pioneers
• Discovery of non Euclidean
• Contribution: Gauss, Newton, Fermat and
Euler

3. Transmission period
• The collapse of Rome and the general chaos
that followed has no great advancements in
the mathematical community in it.
• The Dark ages and then the Middle Ages
were upon the land and civilization let alone
the science of mathematics was having
trouble surviving the times.

4. Transmission period

5. Fibonacci and the 13th century
• Born-died 1170-1240
• He helped introduce the Hindy – Arabic
numerals.

6. Fibonacci Series
• Also known as arithmetical sequence,
invented in 1225 by Leonardo Fibonacci.
• Each number in the series which begins 1, 1,
2, 3, 5, 8, 13, 21, 34 is, after the first two
figures, merely the sum of the previous two
numbers.
• These numbers have occurred in nature and
the Arts.

7. Fibonacci Series
• rabbits, bees,
sunflowers,pinecones,...
• reasons for seed-
arrangement
(mathematical)
• connections to the
Golden number

8. Cubic and quadratic equations
• Girolamo Cardano (1501 – 1576) who wrote
cubic equation in Ars magna in 1545.

9. The Cubic Equations
• A particular equation
• The general case

10. A particular equation

11. A particular equation

12. A particular equation

13. A particular equation

14. A particular equation

15. A particular equation

16. The general case

17. The general case

18. The solution of a quadratic equation

19. The solution of a quadratic equation

20. The solution of a quadratic equation

21. The solution of a quadratic equation

22. The solution of a quadratic equation

23. Pierre de Fermat (1601- 1665)
• Fermat was a French mathematician
who is best known for his work on
number and theory
• One of his last theorem’s was proven
by Andrew Wiles in 1994.

24. Pierre de Fermat (1601- 1665)
• Whilst in Bordeaux, Fermat
produced work on maxima and
minima, which was important. His
methods of doing this were similar to
ours, however as he has not a
professional mathematician his work
was very awkward.

25. Pierre de Fermat (1601- 1665)
• Fermat’s last theorem was that if you
had the equation, xn + yn = zn
• This equation has no nonzero integer
solutions x, y, and z when the
integer exponent n can be no more
than two.
• When n is more than two, the
equation does not work.

26. Fermat’s Method
• One of the ideas of calculus is to find the
tangent line to a given curve.
 the tangent line to a circle.
 in classical geometry, the tangent line to a
circle C at a point P is the line passes through
P and is perpendicular to the radius P.

27. Fermat’s Method
• The tangent line to the curve is interpreted as
the line passes through P and touches the
curve at P.
• But, to Fermat, the tangent line has the
special feature; it only intersect the curve at
one point.

28. Fermat’s Method

29. Example

30. Solution

31. Solution

32. The derivative
• Through Fermat, differential calculus
emerged. This idea led to Descartes, Newton,
Leibniz, and others develop mathematical
knowledge for calculating tangents, finding
maxima and minima functions, and
performing operations in analysis and
mechanics.

33. the tangent line

34. the tangent line
• However, the slope of the line at (c, f(c)) can
be determined by the slopes of secant line (a
line connecting two different points nearby on
the curve – here (c, f(c)) and (c + h, f(c + h))

35. the tangent line

36. The theory of limit
• It is deep and subtle. Discovered more than
2000 years ago, and never got it right. Even
Isaac Newton used limits (with trepidation),
but never really understood them.

37. Example

38. Solution

39. Fermat’s Lemma
• It is based on a geometric observation about
differentiable functions.
• P is vertically higher than points nearby, and is
called a local maximum, whereas Q P is
vertically lower than points nearby, and is
called a local minimum.

40. Fermat’s Lemma
• Here, the point P on the graph goes neither
uphill nor downhill or in other words, the
tangent line is horizontal (its slope is zero).
• Thus, the derivative of a function at a point of
differentiability where the function assumes a
local maximum is 0.

41. Fermat’s Lemma
• Here, the point Q on the graph goes neither
uphill nor downhill or in other words, the
tangent line is horizontal (its slope is zero).
• Thus, the derivative of a function at a point of
differentiability where the function assumes a
local minimum is 0.

42. Fermat’s Lemma

43. Example

44. Solution

45. Solution

46. Solution

47. Solution

48. Solution

49. Euler (1707 –1783)
• important discoveries in calculus… graph theory.
• introduced much of modern mathematical
terminology and notation, particularly for
mathematical analysis,
• renowned for his work in mechanics, optics, and
astronomy.
• Euler is considered to be the preeminent
mathematician of the 18th century and one of
the greatest of all time

50. Euler (1707 –1783)
• son of Protestant minister. Was minister but
studied mathematics. Renowned for
Algebra,Calculus