ℵ
0
.

Write a short outline of the important achievements and events in
geometry, from ancient times to today.

4
proved the area of a circle, surface area and volume of a sphere formulas
involving π. He was doing an early version of an integral, but as equation
s and
functions did not exist at that time, there were no calculus before the 17
th
century.
●
In around 250 BC, Apollonius wrote a book about conics to illustrate the
different cuts of a plane on a cone (i.e. circle, ellipse, parabola and hyperbola).
However, for the hyperbola, he was actually looking at a double cone.
●
For the next 1000 years, everyone did mathematics using lengths and geometry
was considered, at that time, the basis of mathematics.
●
In 1100, Umar al-Khayyami found solutions to the cubic equation using
geometry, still in lengths.
●
Between the 1200s and the 1500s were the Dark Ages of Mathematics. Nothing
new was invented…
●
In 1640, Descartes invented graphs for curves, as curves can be expressed as
equations, something that unified geometry and algebra. Calculus was invented
shortly thereafter. Cavalieri and Wallace
used Archimedes’ work to find the area
“under” a general curve, something that later become an integral.
●
In 1830, Lagrange
(and others) discovered that Euclid’s methods were not up to
the current standards. They looked back at the Parallel postulate. Later,
Lobachevsky, Bolyai and Gauss decided to prove the parallel postulate by
reductio ad absurdum and discovered non-Euclidean geometries, where
triangles no longer add up to 180° and the Pythagorean Theorem no longer true.
Consequently, there may be many or no parallels in such geometries, so the
parallel postulate becomes different in each of those geometries.
Find the foci, eccentricity and directrix of the curve given by
?
2
16
−
?
2
9
= 1.
?
2
16
−
?
2
9
= 1 is a hyperbola with
a
= 4 and
b
= 3.
Foci are located at (-
c
, 0) and (0,
c
) where
a
2
+
b
2
=
c
2
.
In this problem,
c
2
= 4
2
+ 3
2
= 25 →
c
= 5. Therefore, the foci are at (-5, 0) and (5, 0).
For each
c
, the directrix is at
x
=
?
2
?
.
Here, for
c
= -5, the directrix is at
4
2
−5
=
−16
5
. For
c
= 5, the directrix is at
4
2
5
=
16
5
.
The eccentricity is given by
√?
2
+?
2
?
=
√4
2
+3
2
4
=
√25
4
=
5
4
.