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0.
Euclid
(circa 300 BC) produced the definitive
treatment of Greek geometry and number theory
in the 13 volume Elements.
1.
Ptolemy
(circa 130 AD) assumed that there was at
least one line parallel to a line through a
given point which is equivalent to Euclid's
postulate - circular reasoning
2.
Proclus
(410 - 485) assumed parallel lines are always
equidistance which is an added assumption
about parallel lines.
3.
Wallis
(1616 - 1703) proved the Parallel Postulate
assuming a postulate about Similar Triangles
which is equivalent to Euclid's postulate -
circular reasoning.
4. Saccheri
(1667 -1733) worked with
quadrilaterals, now called Saccheri
quadrilaterals, where the base angles are
rights angles and the sides adjacent to the
base are congruent.
The question is: what can be
proven about the summit angles, <D and
<C? Without assuming the Parallel
Postulate, it can be proven that the two
summit angles are congruent. Then, there
are three distinct possibilities:
1) The summit angles are acute
angles.
2) the summit angles are right
angles.
3) the summit angles are obtuse angles.
What Saccheri finally wrote was: "The
hypothesis of the acute angle is
absolutely false, because [it is]
repugnant to the nature of the straight
line!" (Greenberg, p.155)
5.
Clairaut
(1713 -1765) proved the Parallel Postulate
assuming a postulate about the Existence of
Rectangles which is equivalent to Euclid's
postulate - circular reasoning.
6.
Legendre
(1752 -1833) worked with the Parallel
Postulate assuming a postulate about angle
sum of a triangle being equal to
180o which is equivalent to
Euclid's postulate - circular reasoning.
7.
Lambert
(1728 -1777) worked with quadrilaterals, now
called Lambert quadrilaterals, which have
three right angles. The question is what can
be said about the fourth angle?
8. Since so many mathematicians had tried
to prove Euclid's Parallel Postulate,
Klügel
did his doctoral thesis in 1763
finding the flaws in 28 different proofs of
this postulate. The thesis led
d'Alembert
to call Euclid's Parallel Postulate "the
scandal of geometry." (Greenberg, p.161)
9. The Hungarian
Farkas Bolyai
wrote to his son János:
You must not attempt this
approach to parallels. I know this way to
its very end. I have traversed this
bottomless night, which extinguished all
light and joy in my life. I entreat you,
leave the science of parallels alone... I
thought I would sacrifice myself for the
sake of truth. I was ready to become a
martyr who would remove the flaw from
geometry and return it purified to
mankind.... I turned back when I saw that
no man can reach the bottom of the night.
I turned back unconsoled, pitying myself
and all mankind.
..... I have traveled past all reefs of
this infernal Dead Sea and have always
come back with broken mast and torn sail.
The ruin of my disposition and my fall
date back to this time. I thoughtlessly
risked my life and happiness.
(Greenberg, pp. 161-162)
10. The son
János Bolyai
(1802 -1860)wrote back:
It is now my definite plan to
publish a work on parallels as soon as I
can complete and arrange the material....
When you, my dear Father, see them, you
will understand; at present I can only say
nothing except this: that out of
nothing I have created a strange new
universe. All that I have sent you
previously is like a house of cards in
comparison to a tower.
(Greenberg, p. 163)
11. When János's father send his
work to
Gauss
(1777 - 1855), Gauss wrote back that he, in
essence, had done this work but would never
publish it since:
Most people have not the insight
to understand our conclusions and I have
encountered only a few who received with
any particular interest what I
communicated to them.
(Greenberg, p. 178)
12.
Lobachesky (1792-1656) was the
mathematician first to publish an account of
non-Euclidean geometry in 1829. However, the
original was published in Russian. It was not
until 1840 that the work was published in
German and received some recognition. Since
his work openly challenged Kant's view of
space as "a priori" knowledge, he was fired
1846 from his university post.
13. In 1868,
Beltrami
settled the question about
Euclid's Parallel Postulate by proving that
no proof was possible.
14.
Riemann
(1826 - 1866) developed elliptic geometry
starting in 1854.
15. Klein,
Beltrami,
and
Poincaré worked in the last
half of the 19th century in developing models
for hyperbolic geometry.
16. In 1882,
Pasch
developed one of the first modern set of
axioms for Euclidean geometry.
17. In 1902,
Hilbert,
a great champion of the axiomatic method,
published a set of axioms which filled the
gaps for Euclidean geometry.
18. In 1932,
Birkhoff
developed a new set of axioms for
geometry, based totally on the connections
between geometry and real numbers and include
distance and angle as undefined
terms.
19.
Gödel,
in 1940, proved that no mathematical system
can be complete.
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and a point P not on