1) In the 19th century, mathematics underwent significant changes with a new emphasis on rigor, structure, and abstract concepts. 2) This included the development of non-Euclidean geometry which showed that Euclid's parallel postulate is independent of the other postulates of geometry. 3) Algebra evolved from a focus on symbols and arithmetic to studying mathematical structures in more abstract ways, such as in Boolean and quaternion algebras.

1. The Search for Certainty
The nineteenth Century

2. Learning Objectives
1.To be able to identify the mathematical
movements of this period with regards to
algebra , geometry , and analysis
2.To be able to discuss need for rigor in analysis
and name the main proponents of rigor
3.To outline the development of non Euclidean
geometry and discuss the significance of
Euclid’s Fifth Postulate in geometric thinking

3. Commentary
The newfound power of the calculus and the
pressure of the industrial movements taking
place in Europe resulted a great emphasis on
applications of mathematics in such varied
fields as thermodynamics , structural design ,
fluid dynamics ,celestial mechanics , and
studies of electricity and magnetism . The
arena of mathematical research moved from
isolation of academic and royal courts to
lecture halls of universities.

4. While there was a great momentum in
mathematics with one discovery leading to
another , some mathematicians began to
examine the basic structure of their discipline.
Rigor replaced intuition. In calculus the
concept of limit, continuity, and infinite series
were vague and often poorly conceived;

5. ideas in algebra were rapidly changing and in
geometry, the issue of parallelism and Euclids
fifth will still troublesome. Three major
movements focused on these concerns: the
extension of geometry to include non-
Euclidean geometries, the redirecting of
algebra as a study of mathematical structure,
and a refinement of analysis building on the
properties of the real number system.

6. Carl Fredrick Gauss
1777-1855

7. Carl Fredrick Gauss was a German
mathematician now recognized as
“Prince of Mathematicians” and the
greatest mathematician of the
nineteenth century. He was a child
genius noted for his ability in
mathematics and languages . In 1796
he discovered that if p=22n+1 is a

8. prime number for n a natural
number, then the p –gon is
constructible by Euclidean
methods- a problem that had
stumped mathematicians since

9. the time of the Greeks. In his
doctoral dissertation at the age
of twenty, Gauss stated and
proved the Fundamental
Theorem of Algebra-every
polynomial equation P(x)=0
has at least one root.

10. Although Gauss contributed original
results to many scientific fields-
astronomy, geodesy, differential
geometry, probability theory, complex
variable and infinite series- perhaps his
major work was on number theory,
Disquisition arithmeticae
[ Arithmetical Investigations](1801)

11. Neils Hendrik Abel

12. George Boole

13. William Rowan Hamilton

14. Herman Grassman

15. Arthur Cayley

16. Augustin Louis Cauchy

17. Georg Cantor

18. George Bernhard Reimann

19. In Disquisition written in Latin,
Gauss defined the concepts of
number congruence : If a number a
divides b and c , then b and c are
said to be congruent, otherwise
incongruent; and a itself is called
modulus.

20. The questions of independence of Euclid’s fifth
postulate did it stand by itself or was it
derived from other postulates?
Non – Euclidean Geometry

21. The fifth postulate
The fifth postulate have bothered
mathematicians for over a thousand years
Omar Khayyam and other Islamic
mathematicians had written on this subject.
Apparently the first systematic investigation of
the problem in Europe was undertaken by
Girolamo Giovanni Saccheri an Italian priest
and logician. Saccheri employed a
quadrilateral in which he varied the internal
angles to contradict this postulate.

22. Euclid’s fifth postulate

23. His fifth postulate is just a
simply two lines cut by a
transversal one of the
geometric problem commonly
encountered in high schools
today
The Euclid’s fifth Postulate

24. The investigations of the fifth postulate
that were to affirm both its
independence and the existence of a
resulting consistent non- Euclidean
geometry that would come from Gauss
and two relatively unknown young
mathematician, the Russian Nokolai
Lobachevsski(1793-1856), and the
Hungarian Janos Bolyai

25. .
Their findings published between
1829 and 1840 demonstrated the
existence of alternative geometries
from that proposed by Euclid. Now
geometry was not intuitively bound
to physical space.

26. Generation of new Geometry
-the concept of new geometry
provide the ideas to discover the
non- Euclidean Geometry

27. Euclid’s Postulate
I. To draw a straight line from any point to any point.
II. To produce a finite straight line continuously in a
straight line.
III. To describe a circle with any center and radius.
IV. That all right angles are equal one another.
V. That if a straight line is falling on two straight lines
makes the interior angles on the same side less
than two right angles, the two straight lines, if
produced indefinitely meet on the side on which
the angles less than the two right angles.

28. Plane : Euclidean
Poincarѐ : Hyperbolic
Sphere: Elliptic

29. The mathematical outlook on
algebra was also changed by a
variety of ways. The two short-
lived geniuses – the Norwegian
Neils Hendrik Abel(1802-1829) who
found that a general fifth degree
equation could not be solved using
basic algebra, and the Frence

30. University student, Evariste Galois
(1811-1832) whose work on the
solution of equations give rise to
what would be known later as
group theory- together raised the
level of algebraic investigations

31. But it was the work of British
mathematicians that altered the very
concept of algebra from arithmetic
symbols to a study of mathematical
structure. George Boole (1815-1864) in his
Treatise on Algebra (1830) presented a
formal , systematic approach to algebra
earning for himself as the title “Euclid of
Algebra”.

32. His colleagues Augustus De
Morgan(1806-1801) and
George Boole(1815-1864) saw
algebra as a form of logic.

33. William Rowan Hamilton(1805-1865)
While working on the quaternions
devised a non- commutative form
of multiplication. Certainly, the
algebra of his quaternions did not
follow the structure of arithmetic.

34. Non Commutative Algebra
By the early nineteenth century, the study of
algebra had evolve from the mere
manipulation of symbols to a mere abstract
investigations of the laws of mathematical
operations and how they combine objects.
William Rowan Hamilton( 1805-1865) was a
professor of mathematics of Trinity College ,
Dublin , in 1833 , he devised an algebra for
working with number couples.

36. This algebra could be readily applied to
complex numbers where a number of
the form a+ bi could be represented by
the ordered pair (a ,b ) . Hamilton
sought to extend his algebra further to
the consideration of triples, however,
ran into difficulty. In 1843 , he finally
extended his theory , but with number
of triples.

37. Non commutative algebra
i
j
k

38. The Rule
If the rotation is counter clockwise then :
j*k=i k*j=-i
k*i=j i*k=-j
i*j=k j*i=-k
i*j=k
i*i=-1
J*j=-1
K*k=-1

39. How is it done
When the direction is counter clockwise the
result is positive.
When the direction is clockwise the result is
negative.
When vector is paired with the same vector
the result is negative 1.

40. George Boole
The logic of George Boole is the same with the
works of De Morgan and other logicians. It
uses the numbers 1 and 0 to imply yes or no.
Now Boolean Algebra is used in electronics
and in computer controls.

41. Issues for Further thought
Given a function f(x) continuous and defined
over an interval [a , b] write the mathematical
statements using limits to define the definite
integral f(x) over the given interval.

42. •Computations
Johann Lambert devised the concept of
hyperbolic functions and defined:
Sinhu=
𝑒 𝑢 −𝑒 𝑢
2
cosh =
𝑒 𝑢 +𝑒 𝑢
2
Tanh=
𝑠𝑖𝑛ℎ𝑢
𝑐𝑜𝑠ℎ𝑢
Use the definition to show
Tanhu=
𝑒 𝑢 −𝑒 𝑢
𝑒 𝑢 +𝑒 𝑢