Euclidean geometry: the only and the first in the past
Slide 2
If two lines m and l meet a third line n, so as to make the sum
of angles 1 and 2 less than 180, then the lines m and l meet on
that side of the line n on which the angles 1 and 2 lie. If the sum
is 180 then m and l are parallel THE 5 th EUCLIDS AXIOM
Slide 3
Playfairs axiom didnt satisfy mathematicians 18th century
Mathematicians : a new tack Saccheri: demonstration by
contradiction Playfairs axiom:given a line g and a point P not on
that line, there is one and only one line g on the plane of P and g
which passes through P and does not meet g
Slide 4
GAUSS A genius child Many scientific interests Challenge to
Euclids axiom: given a point P outside a line l there are more than
one parallel line through P a new kind of geometry! Fear of
publishing studies After his death his work discovered
Slide 5
FROM GAUSS TO LOBACHEVSKY & BOLYAI Gauss: the first to
discover the non Euclidean geometry but unknown Fame to Lobachevsky
& Bolyai : first to publish works about non Euclidean
geometry
Slide 6
GAUSSS NON-EUCLIDEAN GEOMETRY Gausss non Euclidean geometry:
based on contradiction of 5 th Euclidean axiom New Axiom: Given a
line l and a point P. There are infinite non secant and two
parallel lines to l through P Creating new theorems Sum of internal
angles in triangle
