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Riemannian geometry, also called elliptic geometry, one of the non-Euclidean
geometries that completely rejects the validity of Euclids fifth postulate and
modifies his second postulate. Simply stated, Euclids fifth postulate is: through a
point not on a given line there is only one line parallel to the given line. In
Riemannian geometry, there are no lines parallel to the given line. Euclids
second postulate is: a straight line of finite length can be extended continuouslywithout bounds. In Riemannian geometry, a straight line of finite length can be
extended continuously without bounds, but all straight lines are of the same ...
(100 of 248 words).
Unlike other branches of math, geometry has been connected with two purposes since the
ancient Greeks. Not only is it an intellectual discipline, but also, it has been considered an
accurate description of our physical space. However in order to talk about the different types
of geometries, we must not confuse the term geometry with how physical space really works.
Geometry was devised for practical purposes such as constructions, and land surveying.Ancient Greeks, such as Pythagoras (around 500 BC) used geometry, but the various
geometric rules that were being passed down and inherited were not well connected. So
around 300 BC, Euclid was studying geometry in Alexandria and wrote a thirteen-volume
book that compiled all the known and accepted rules of geometry called The Elements, and
later referred to as Euclids Elements. Because math was a science where every theorem is
based on accepted assumptions, Euclid first had to establish some axioms with which to use
as the basis of other theorems. He used five axioms as the 5 assumptions, which he needed to
prove all other geometric ideas. The use and assumption of these five axioms is what it means
for something to be categorized as Euclidean geometry, which is obviously named after
Euclid, who literally wrote the book on geometry.
The first four of his axioms are fairly straightforward and easy to accept, and no
mathematician has ever seriously doubted them. The first four of Euclids axioms are:
1.) One straight line may be drawn from any two points.
2.) Any terminated straight line may be extended indefinitely.
3.) A circle may be drawn with any given center and any given radius.
4.) All right angles are congruent.
With no concern over the first four axioms, they are regarded as the axioms of all geometries
or basic geometry for short. The fifth and last axiom listed by Euclid stands out a little bit.
It is a bit less intuitive and a lot more convoluted. It looks like a condition of the geometry
more than something fundamental about it. The fifth axiom is:
5.) If two straight lines lying in a plane are met by another line, and if the sum of he internal
angles on one side is less than two right angles, then the straight lines will meet if the
extended on the side on which the sum of the angles is less than two right angles.
The fifth axiom, also known as Euclids parallel postulate deals with parallel lines, and it is
equivalent to this slightly more clear statement: For a given line and point there is only one
line parallel to the first line passing through the point (This statement was first proved to be
equivalent to Euclids fifth axiom by John Playfair in the 18th century). This seems obviousto us because of what we have been taught, but it is far less as intuitive as the first four. Later
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mathematicians, and even Euclid himself were not comfortable with axiom five; it is quite a
complicated statement and axioms are meant to be small, simple and straightforward. Axiom
five looked more like a theorem than an axiom, and as such it should have to be proved to be
true and not assumed. The problem is that mathematicians were not comfortable using the
fifth axiom (Even Euclid did not use it in The Elements until his 29th example). However,
mathematicians found no way of showing that this problematic axiom it could be provenfrom the first four 4 axioms. However, all the theorems that can be proved from it worked
and many mathematicians were happy just to leave it. So they came up with a version of
geometry that included the fifth postulate and one that excluded it. Basic geometry was
defined as being based on the first 4 axioms alone. However, Euclidean geometry was
defined as using all five of the axioms.
The type of geometry we are all most familiar with today is called Euclidean geometry.
Euclidean geometry consists basically of the geometric rules and theorems taught to kids in
todays schools. Such as the Pythagorean theorem, rules about triangles and congruency and
most other rules concerning shapes, areas, and angles. It is amazing to consider that Euclids
axioms still form the basis of our practical understanding of geometry over two thousandyears later. The Elements had been the most widely purchased non-religious work in the
world.
Introducing non-Euclidean Geometries
The historical developments of non-Euclidean geometry were attempts to deal with the fifth
axiom. Mathematicians first tried to directly prove that the first 4 axioms could prove the
fifth. However, mathematicians were becoming frustrated and tried some indirect methods.
Girolamo Saccheri (1667-1733) tried to prove a contradiction by denying the fifth axiom. He
started with quadrilateral ABCD (later called the Saccheri Quadrilateral) with right angles at
A and B and where AD = BC. Since he is not using the fifth axiom, he concludes there are
three possible outcomes. Angles at C and D are right angles, C and D are both obtuse, or C
and D are both acute. Saccheri knew that the only possible solution was right angles. Saccheri
said this was enough to claim a contradiction and he stopped. His reasoning to stop was based
on faulty logic. He was going on the presumption that lines and parallel lines worked like
those in flat geometry. So his contradiction was only applicable in Euclidean geometry,
which was not a contradiction to what he was actually trying to prove. Of course Saccheri did
not realize this at the time and he died thinking he had proved Euclids fifth axiom from the
first four. A contemporary of Saccheri, Johann Lambert (1728-1777), picked up where
Saccheri left off and took the problem just a few steps further. Lambert considered the three
possibilities that Saccheri had concluded as consequences of the first four axioms. Instead offinding a contradiction, he found two alternatives to Euclidean geometry. The first option
represented Euclidean geometry and while the other two appeared silly, they could not be
proven wrong. Through time (and quite a lot of criticism), these two other possibilities were
now being considered as alternative geometries to Euclids geometry. Eventually these
alternate geometries were scholarly acknowledged as geometries, which could stand alone to
Euclidean geometry. The two non-Euclidean geometries were known as hyperbolic and
elliptic. Hyperbolic geometry was explained by taking the acute angles for C and D on the
Saccheri Quadrilateral while elliptic assumed them to be obtuse. Lets compare hyperbolic,
elliptic and Euclidean geometries with respect to Playfairs parallel axiom and see what role
parallel lines have in these geometries:
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1.) Euclidean: Given a line L and a point P not on L, there is exactly one line passing through
P, parallel to L.
2.) Hyperbolic: Given a line L and a point P not on L, there are at least two lines passing
through P, parallel to L.
3.) Elliptic: Given a line L and a point P not on L, there are no lines passing through P,
parallel to L.
It is important to realize that these statements are like different versions of the parallel
postulate and all these types of geometries are based on a root idea of basic geometry and that
the only difference is the use of the altering versions of the parallel postulate. Similar to the
way variations of a game are played, non-Euclidean geometries are geometries that use
varying rules. Of course, when you change the rules of a game, the consequences are
different and of course using varying axioms leads to different geometries. Acknowledging
that there are different types of geometries is the reason we can no longer view geometry as
an accurate description of our physical space. To say our space is Euclidean, is to say our
space is not curved, which seems to make a lot of sense regarding our drawings on paper,
however non-Euclidean geometry is an example of curved space.
Although mathematicians showed the possibility of non-Euclidean space, people were still
reluctant to reject Euclids fifth postulate. In fact, German philosopher, Immanuel Kant,
argued that since space is largely a creation of our minds, and since we cannot imagine non-
Euclidean space, that Euclids fifth postulate must necessarily be true. However, not much
later, people were giving examples (models) of the non-Euclidean axiomatic systems. For
instance, mathematicians long regarded a straight line as the shortest route between two
points no matter what type of geometry we are considering, but when they tried this on the
surface of a sphere, the arc connecting two points did not appear straight. However
appearance was not important because the lines were only curved extrinsically (as part of our
perception, and not as part of a different type of geometry). Ultimately, the surface of a
sphere became the prime example of elliptic geometry in 2 dimensions (although all
positively curved surfaces such as a football-shaped and other elliptical objects are examples
too).
Elliptic Geometry
Elliptic geometry also says that the shortest distance between two points is an arc on a great
circle (the greatest size circle that can be made on a spheres surface). As part of the
revised parallel postulate for elliptic geometries, we learn that there are no parallel lines in
elliptical geometry. This means that all straight lines on the spheres surface intersect(specifically, they all interesect in two places). A famous non-Euclidean geometer, Bernhard
Riemann, who dealt mostly with and is credited with the development of elliptical
geometries, theorized that the space (we are talking about outer space now) could be
boundless without necessarily implying that space extends forever in all directions. This
theory suggests that if we were to travel one direction in space for a really long time, we
would eventually come back to where we started! This theory involves the existence of four-
dimensional space similar to how the surface of a sphere (which is three dimensional)
represents an elliptic 2 dimensional geometry. Einstein addressed the idea that space could be
unbounded without being in finite in his theory of relativity. However, it should be noted that
this idea raises some issues regarding Euclids second axiom, which says a line segment can
be extended indefinitely.
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There are many practical uses for elliptical geometries. Elliptical geometry, which describes
the surface of a sphere, is used by pilots and ship captains as they navigate around the
spherical Earth, which we live. In fact, working in elliptical geometry has some non-intuitive
results. For example, the shortest flying distance from Florida to the Philippine Islands is a
path across Alaska. The Philippines are South of Florida so it is not apparent why flying
North to Alaska would be shorter. The answer is that Florida, Alaska, and the Philippines arecollinear locations in elliptical geometry. Another odd property of elliptical geometry is that
the sum of the angles of a triangle is always greater then 180. Relatively small triangles,
such as a triangle that is formed by three intersecting roads have angle sums very close to
180. In order to notice the effect of triangles with larger angle sums, we have to consider
much larger triangles such as the triangle formed by New York, Los Angeles and Miami.
Because a triangles angle sum distortion is proportional to the size of the triangle, we also
can deduce that a triangles area is related to its angle sum! Furthermore, this notion destroys
our idea of similar triangles, because similar triangles are triangles with the same angle
measurments but different areas but since area is related to angle sum in elliptic geometry,
there are no similar triangles (just congruent ones)!
Hyperbolic Geometry:
The other type of non-Euclidean geometry we have yet to examine is hyperbolic geometry.
Recalling the corresponding Playfairs axiom for hyperbolic geometry, we see that in
hyperbolic geometry, there is more than one parallel line to L, passing through point P, not on
L. Furthermore, hyperbolic geometries comes with some more restrictions about parallel
lines. In Euclidean geometry, we can show that parallel lines are always equidistant, but in
hyperbolic geometries, of course, this is not the case. Therefore, in hyperbolic geometries, we
merely can assume that parallel lines carry only the restriction that they dont interesect.
Furthermore, the parallel lines dont seem straight in the conventional sense. They can even
approach each other in an asymptotically fashion. The surfaces on which these rules on lines
and parallels hold true are on negatively curved surfaces.
When compared with the other geometrys triangle angle sums, we see that in hyperbolic
geometry, the triangles angle sum is less than 180 degrees whereas elliptic geomtry has more
than 180 degrees. Similarly, the larger the sides of the triangle, the greater the distortion of
the angle sums on both elliptic and hyperbolic geometries. Much like elliptic geometries, the
area of a triangle is proportional to its angle sum and of course this implies that there are no
similar triangles as well. In some ways, hyperbolic geometry is simpler than elliptic so
technically hyperbolic was discovered first. Gauss, Schweikart, Lobachevsky and Jnos
Bolyai all separately and all in the first half of the 19th century, are credited with thediscovery of hyperbolic geometry. Now that we see what the nature of a hyperbolic
geometry, we probably might wonder what some models of hyperbolic surfaces are. Some
traditional hyperbolic surfaces are that of the saddle (hyperbolic parabola) where the surface
curves in two different directions and more scholarly, the Poincar Disc. The Poincar Disc is
a model of hyperbolic geometry envisioned by French mathematician/philosopher, Poincar
(1854 1912). His model is a sort of 2 dimensional model, which makes it appealing to those
who are working on paper. In one of his philosophical writings, Science and Hypothesis
1901, he wrote of his model as an imaginary universe occupying the interior of a disc (or
circle) in the Euclidean plane. And we as observers get to watch the inhabitants move around.
However, they appear to shrink as they approach the infinitely distant horizon (the
boundary of the disc). Furthermore, the inhabitants do not notice the effect because their rulershrinks with them as they move. They think that they live in a normal Euclidean space, but
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we see them in a non-Euclidean space with their dimensions behaving strangely. Since the
edge of the disc represents infinity, their universe still contains infinite space, however large
line segments appear to grow smaller as they get closer to the circles edge. Straight lines, in
the Poincar Disc, intersect the discs edge at 90-degree angles. Like many of our examples
of non-Euclidean geometries, measurements on the Poincar Disc become more distorted
when we are looking at larger areas and line segments. In fact, if you were to draw a trianglewith vertices close to the edge of the disc (infinity), the triangles area would be near zero!
Applications of non-Euclidean Geometries:
Practically, non-Euclidean geometries have long been regarded as curiousities because they
seemed to have little to do with our real universe. Thanks to Einstein and subsequent
cosmologists, non-Euclidean geometries began to replace the use of Euclidean geometries in
many contexts. For example, physics is largely founded upon the constructs of Euclidean
geometry but was turned upside-down with Einstein's non-Euclidean "Theory of Relativity"
(1915). Newtonian physics, based upon Euclidean geometry, failed to consider the curvature
of space, and that this constituted for major errors in the equations of planetary motion andgravity.
Einstein's general theory of relativity proposes that gravity is a result of an intrinsic curvature
of spacetime (as opposed to a Newtonian action-at-a-distance explanation). Intrinsic
curvature, explains how straight lines could have the properties associated with curvature
without actually being curved in the ordinary sense, is now used to explain how something
which is obviously curved, like the orbit of a planet, is really straight. Also, it means that if
we accept an intrinsic curvature of spacetime (a curve of spacetime, as opposed to a curve
within spacetime), then these curved lines in three dimensional space (such as those used to
describe gravity), then we must assume that the lines are curved in a higher dimension, into
which the straight lines are curved in the conventional sense. In laymans terms, this explains
that the phrase curved space is not a curvature in the usual sense but a curve that exists of
spacetime itself and that this curve is in the direction of the fourth dimension.
So if our space has a non-conventional curvature in the direction of the fourth dimension, that
that means our universe is not flat in the Euclidean sense and finally we realize our
universe is probably best described by a non-Euclidean geometry! The future of the universe
will be determined by whatever is the geometry of the universe happens to be. According to
current theories in cosmology, if the geometry is hyperbolic, the universe will expand
indefinitely; if the geometry is Euclidean, the universe will expand indefinitely at escape
velocity; and if the geometry is elliptic, the expansion of the universe will coast to a halt, andthen the universe will start to shrink back to a singularity and possibly to explode again with
a whole new big bang.
Conclusion:
It would be fallacious to assume that because math works that we understand what is
happening in our real space. However there are those that speak as though math somehow
explains what is going on in the real world. The mathematics of Euclid were were simple and
straightforward, but it did not confer an understanding about what the nature of the universe
was. Many people who, with an almost religious fervor, proclaimed that Euclidean geometry
was the one and only geometry resisted the recognition of the existence of the non-Euclideangeometries as mathematical systems. Such attitudes reflect a failure to recognize that
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geometry is a mathematical system that is determined by its assumptions. The most we can
assume about our universe now is that Euclidean geometry provides an excellent
representation for the localized part of the universe that we inhabit. Poincar added some
insight to the debate between Euclidean and non-Euclidean geometries when he said, One
geometry cannot be more true than another; it can only be more convenient.
http://www.reocities.com/CapeCanaveral/7997/noneuclid.html
Elliptic geometry
Recall that one model for the Real projective plane is the unit sphere S2 with opposite points
identified.
On this model we will take "straight lines" (the shortest routes between points) to be great
circles (the intersection of the sphere with planes through the centre).
We may then measure distance and angle and we can then look at the elements ofPGL(3, R)
which preserve his distance. This is a groupPO(3) which is in fact the quotient group ofO(3)by the scalar matrices. (In fact, since the only scalars in O(3) are Iit is isomorphic to
SO(3)).
This geometry then satisfies all Euclid's postulates except the 5th. Since any two "straight
lines" meet there are no parallels.
This geometry is called Elliptic geometry and is a non-Euclidean geometry.
Some properties
1. All lines have the same finite length .
2. The area of the elliptic plane is 2.
3. The sum of the angles of a triangle is always > .
In fact one has the following theorem (due to the French mathematician Albert Girard(1595
to 1632) who proved the result for spherical triangles).
Girard's theorem
The sum of the angles of a triangle - is the area of the triangle.
Proof
Take the triangle to be a spherical triangle lying in one hemisphere.
The lines b and c meet in antipodal points
A andA' and they define a lune with area 2 .
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We get a picture as on the right of the sphere divided into 8 pieces
with ' the antipodal triangle to and 1 the above lune, etc.
The area = area ', 1 = '1,etc.
Then + 1 = area of the lune = 2
and + 2 = 2and + 1 = 2
Also 2 + 2 1 + 2 2 + 2 3 = 4 2 = 2 + 2 + 2 - 2 as
required.
What is elliptic
geometry?An elliptic geometry is a non-Euclidean geometry with positive curvature whichreplaces the parallel postulate with the statement through any point in the plane,
there exists no line parallel to a given line.
dianne18, Answers Expert
Euclidean geometry
Euclidean geometry is a mathematical system attributed to theGreekmathematicianEuclid ofAlexandria. Euclid's textElements is the earliest known systematic discussion of
geometry. It has been one of the most influential books in history, as much for its method as
for its mathematical content. The method consists of assuming a small set of intuitively
appealing axioms, and then proving many otherpropositions( theorems) from those axioms.
Although many of Euclid's results had been stated by earlier Greek mathematicians, Euclid
was the first to show how these propositions could be fit together into a comprehensive
deductive and logical system.
TheElements begin withplane geometry, still taught in secondary schoolas the first
axiomatic system and the first examples offormal proof. TheElements goes on to the solid
geometryof three dimensions, and Euclidean geometry was subsequently extended to anyfinite number ofdimensions. Much of theElements states results of what is now called
number theory, proved using geometrical methods.
For over two thousand years, the adjective "Euclidean" was unnecessary because no other
sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious that any
theorem proved from them was deemed true in an absolute sense. Today, however, many
otherself-consistentnon-Euclidean geometries are known, the first ones having been
discovered in the early 19th century. It also is no longer taken for granted that Euclidean
geometry describes physical space. An implication ofEinstein's theory ofgeneral relativity is
that Euclidean geometry is a good approximation to the properties of physical space only if
the gravitational fieldis not too strong.
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Axiomatic approach
Euclidean geometry is an axiomatic system, in which all theorems ("true
statements") are derived from a finite number of axioms. Near the beginning of
the first book of theElements
, Euclid gives five postulates (axioms):1. Any two points can be joined by a straight line.2. Any straight line segment can be extended indefinitely in a straight line.
3. Given any straight line segment, a circle can be drawn having thesegment as radius and one endpoint as center.
4. All right angles are congruent.
5. Parallel postulate. If two lines intersect a third in such a way that the sumof the inner angles on one side is less than two right angles, then the twolines inevitably must intersect each other on that side if extended far
enough.
These axioms invoke the following concepts: point, straight line segment and line, side of a
line, circle with radius and center, right angle, congruence, inner and right angles, sum. The
following verbs appear: join, extend, draw, intersect. The circle described in postulate 3 is
tacitly unique. Postulates 3 and 5 hold only for plane geometry; in three dimensions, postulate
3 defines a sphere.
Postulate 5 leads to the same geometry as the following statement, known asPlayfair's
axiom, which also holds only in the plane:
Postulates 1, 2, 3, and 5 assert the existence and uniqueness of certain geometric figures, and
these assertions are of a constructive nature: that is, we are not only told that certain things
exist, but are also given methods for creating them with no more than a compass and an
unmarked straightedge. In this sense, Euclidean geometry is more concrete than many
modern axiomatic systems such as set theory, which often assert the existence of objects
without saying how to construct them, or even assert the existence of objects that cannot be
constructed within the theory.
Strictly speaking, the constructs of lines on paper etc are modelsof the objects defined within
the formal system, rather than instances of those objects. For example a Euclidean straight
line has no width, but any real drawn line will.
TheElements also include the following five "common notions":
1. Things that equal the same thing also equal one another.2. If equals are added to equals, then the wholes are equal.
3. If equals are subtracted from equals, then the remainders are equal.
4. Things that coincide with one another equal one another.
5. The whole is greater than the part.
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Euclid also invoked other properties pertaining to magnitudes. 1 is the only part of the
underlying logic that Euclid explicitly articulated. 2 and 3 are "arithmetical" principles; note
that the meanings of "add" and "subtract" in this purely geometric context are taken as given.
1 through 4 operationally define equality, which can also be taken as part of the underlying
logic or as anequivalence relationrequiring, like "coincide," careful prior definition. 5 is a
principle ofmereology. "Whole", "part", and "remainder" beg for precise definitions.
In the 19th century, it was realized that Euclid's ten axioms and common notions do not
suffice to prove all of theorems stated in the Elements . For example, Euclid assumed
implicitly that any line contains at least two points, but this assumption cannot be proved
from the other axioms, and therefore needs to be an axiom itself. The very first geometric
proof in theElements, shown in the figure on the right, is that any line segment is part ofa
triangle; Euclid constructs this in the usual way, by drawing circles around both endpoints
and taking their intersection as the thirdvertex. His axioms, however, do not guarantee that
the circles actually intersect, because they are consistent with discrete, rather than continuous,
space. Starting with Moritz Pasch in 1882, many improved axiomatic systems for geometry
have been proposed, the best known being those ofHilbert, George Birkhoff, and Tarski.
To be fair to Euclid, the first formal logic capable ofsupporting his geometry was that of
Frege's 1879Begriffsschrift, little read until the 1950s. We now see that Euclidean geometry
should be embedded in first-order logic with identity, a formal system first set out in
Hilbert and Wilhelm Ackermann's 1928Principles of Theoretical Logic. Formal
mereologybegan only in 1916, with the work ofLesniewski and A. N. Whitehead.
Tarski and his students did major work on the foundations of elementary geometryas
recently as between 1959 and his death in 1983.
The parallel postulate
To the ancients, the parallel postulate seemed less obvious than the others; verifying it
physically would require us to inspect two lines to check that they never intersected, even at
some very distant point, and this inspection could potentially take an infinite amount of time.
Euclid himself seems to have considered it as being qualitatively different from the others, as
evidencedby the organization of theElements : the first 28 propositions he presents are those
that can be proved without it.
Many geometers tried in vain to prove the fifth postulate from the first four. By 1763 at least
28 different proofs had beenpublished, but all were found to be incorrect. In fact the parallel
postulate cannot be proved from the other four: this was shown in the 19th century by theconstruction of alternative ( non-Euclidean) systems of geometry where the other axioms are
still truebut the parallel postulate is replaced by a conflicting axiom. One distinguishing
aspect of these systems is that the three angles of a triangle do not add to 180: in hyperbolic
geometrythe sum of the three angles is always less than 180 and can approach zero, while in
elliptic geometry it is greater than 180. If theparallel postulate is dropped from the list of
axioms without replacement, the result is the more general geometry called absolute
geometry.
Treatment using analytic geometry
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The development ofanalytic geometry provided an alternative method for
formalizing geometry. In this approach, a point is represented by its
Cartesian (x,y) coordinates, a line is represented by its equation, and so on. In
the 20th century, this fit into David Hilbert's program of reducing all of
mathematics to arithmetic, and then proving the consistency of arithmetic using
finitistic reasoning. In Euclid's original approach, the Pythagoreantheorem follows from Euclid's axioms. In the Cartesian approach, the axioms are
the axioms of algebra, and the equation expressing the Pythagorean theorem is
then a definition of one of the terms in Euclid's axioms, which are now
considered to be theorems. The equation
|PQ|=sqrt{(p-r)^2+(q-s)^2}
defining the distance between two points P=(p,q) and Q=(r,s) is then known as the
Euclidean metric, and other metrics definenon-Euclidean geometries.
As a description of physical reality
Euclid believed that his axioms were self-evident statements about physical reality.
This led to deep philosophical difficulties in reconciling the status of knowledge from
observation as opposed to knowledge gained by the action of thought and reasoning. A major
investigation of this area was conducted by Immanuel KantinThe Critique of Pure Reason.
However, Einstein's theory ofgeneral relativity shows that the true geometry of spacetime is
non-Euclidean geometry. For example, if a triangle is constructed out of three rays of light,
then in general the interior angles do not add up to 180 degrees due to gravity. A relatively
weak gravitational field, such as the Earth's or the sun's, is represented by a metric that is
approximately, but not exactly, Euclidean. Until the 20th century, there was no technology
capable of detecting the deviations from Euclidean geometry, but Einstein predicted that such
deviations would exist. They were later verified by observations such as the observation of
the slight bending of starlight by the Sun during a solar eclipse in 1919, andnon-Euclidean
geometryis now, for example, an integral part of the software that runs the GPS system. It is
possible to object to the non-Euclidean interpretation of general relativity on the grounds that
light rays might be improper physical models of Euclid's lines, or that relativity could be
rephrased so as to avoid the geometrical interpretations. However, one of the consequences of
Einstein's theory is that there is no possible physical test that can do any better than a beam oflight as a model of geometry. Thus, the only logical possibilities are to accept non-Euclidean
geometryas physically real, or to reject the entire notion of physical tests of the axioms of
geometry, which can then be imagined as a formal system without any intrinsic real-world
meaning.
Because of the incompatibility of theStandard Model with general relativity, and because of
some recent empirical evidence against the former, both theories are now under increased
scrutiny, and many theories have been proposed to replace or extend the former and, in many
cases, the latter as well. The disagreements between the two theories come from their claims
about space-time, and it is now accepted that physical geometry must describe space-time
rather than merely space. While Euclidean geometry, the Standard Model and generalrelativity are all in principle compatible with any number of spatial dimensions and any
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specification as to which of these if any are compactified (see string theory), and while all but
Euclidean geometry (which does not distinguish space from time) insist on exactly one
temporal dimension, proposed alternatives, none of which are yet part ofscientific consensus,
differ significantly in their predictions or lack thereof as to these details of space-time. The
disagreements between the conventional physical theories concern whether space-time is
Euclidean (since quantum field theoryin the standard model is built on the assumption that itis) and on whether it is quantized. Few if any proposed alternatives deny that space-time is
quantized, with the quanta of length and time are respectively thePlanck lengthand the
Planck time. However, which geometry to use - Euclidean, Riemannian, de Stitter, anti de
Stitter and some others - is a major point of demarcationbetween them. Many physicists
expect some Euclidean string theory to eventually become the Theory Of Everything, but
their view is by no means unanimous, and in any case the future of this issue is unpredictable.
Regarding how if at all Euclidean geometry will be involved in future physics, what is
uncontroversial is that the definition of straight lines will still be in terms of the path in a
vacuum of electromagnetic radiation (including light) until gravity is explained with
mathematical consistency in terms of a phenomenon other than space-time curvature, and that
the test of geometrical postulates (Euclidean or otherwise) will lie in studying how thesepaths are affected by phenomena. For now, gravity is the only known relevant phenomenon,
and its effect is uncontroversial (see gravitational lensing).
Conic sections and gravitational theory
Apollonius and other Ancient Greek geometers made an extensive study of the conic sections
curves created by intersecting a cone and a plane. The (nondegenerate) ones are the
ellipse, theparabolaand the hyperbola, distinguished by having zero, one, or two
intersections with infinity. This turned out to facilitate the work ofGalileo,Keplerand
Newton in the 17th Century, as these curves accurately modeled the movement of bodiesunder the influence of gravity. UsingNewton's law of universal gravitation, the orbit of a
cometaround the Sun is
an ellipse, if it is moving too slowly for its position (below escapevelocity), in which case it will eventually return;
a parabola, if it is moving with exact escape velocity (unlikely), and willnever return because the curve reaches to infinity; or
a hyperbola, if it is moving fast enough (above escape velocity), andlikewise will never return.
In each case the Sun will be at one focus of the conic, and the motion will sweep out equal
areas in equal times.
Galileo experimented with objects falling small distances at the surface of the Earth, and
empirically determined that the distance travelled was proportional to the square of the time.
Given his timing and measuring apparatus, this was an excellent approximation. Over such
small distances that the acceleration of gravity can be considered constant, and ignoring the
effects ofair(as on a falling feather) and the rotation of theEarth, the trajectoryof a
projectilewill be a parabolic path.
Later calculations of these paths for bodies moving undergravity would be performed usingthe techniques of analytical geometry (using coordinates and algebra) and differential
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calculus, which provide straightforward proofs. Of course these techniques had not been
invented at the time that Galileo investigated the movement of falling bodies. Once he found
that bodies fall to the earth with constant acceleration (within the accuracy of his methods),
he proved that projectiles will move in a parabolic path using the procedures of Euclidean
geometry.
Similarly, Newton used quasiEuclidean proofs to demonstrate the derivation of Keplerian
orbital movements from his laws of motion and gravitation.
Centuries later, one of the first experimental measurements to support Einstein'sgeneral
theory of relativity, which postulated a non-Euclidean geometryfor space, was the orbit of
the planet Mercury. Kepler described the orbit as a perfect ellipse.Newtonian theory
predicted that the gravitational influence of other bodies would give a more complicated
orbit. But eventually all such Newtonian corrections fell short of experimental results; a small
perturbation remained. Einstein postulated that the bending ofspace would precisely account
for that perturbation.
Logical status
Euclidean geometry is a first-order theory. That is, it allows statements such as
those that begin as "for all triangles ...", but it is incapable of forming statements
such as "for all sets of triangles ...". Statements of the latter type are deemed to
be outside the scope of the theory.
We owe much of our present understanding of the properties of the logical and
metamathematical properties of Euclidean geometry to the work ofAlfred Tarski and his
students, beginning in the 1920s. Tarski proved his axiomatic formulation of Euclideangeometry to be complete in a certain sense: there is an algorithm which, for every
proposition, can show it to be either true or false.Gdel's incompleteness theorems showed
the futility of Hilbert's program of proving the consistency of all of mathematics using
finitistic reasoning. Tarski's findings do not violate Gdel's theorem, because Euclidean
geometry cannot describe a sufficient amount ofarithmetic for the theorem to apply.
Although complete in the formal senseused in modern logic, there are things that Euclidean
geometry cannot accomplish. For example, the problem oftrisecting an angle with a compass
and straightedge is one that naturally occurs within the theory, since the axioms refer to
constructive operations that can be carried out with those tools. However, centuries of efforts
failed to find a solution to this problem, until Pierre Wantzelpublished a proof in 1837 that
such a construction was impossible.
Absolute geometry, first identified by Bolyai, is Euclidean geometry weakened by omission
of the fifth postulate, that parallel lines do not meet. Of strength intermediate between
absolute geometry and Euclidean are geometries derived from Euclid's by alterations of the
parallel postulate that can be shown to be consistent by exhibiting models of them. For
example, geometry on the surface of a sphere is a model ofelliptical geometry. Another
weakening of Euclidean geometry is affine geometry, first identified by Euler, which retains
the fifth postulate unmodified while weakening postulates three and four in a way that
eliminates the notions of angle (whence right triangles become meaningless) and of equalityof length of line segments in general (whence circles become meaningless) while retaining
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the notions of parallelism as an equivalence relation between lines, and equality of length of
parallel line segments (so line segments continue to have a midpoint).
Classical theorems
Ceva's theorem Heron's formula
Nine-point circle
Pythagorean theorem
Tartaglia's formula
Menelaus's theorem
Angle bisector theorem
The butterfly theorem
Parallel Postulate
See also
Analytic geometry Interactive geometry software
Non-Euclidean geometry
Ordered geometry
Incidence geometry
Birkhoff's axioms
Hilbert's axioms
Tarski's axioms
Parallel postulate
Schopenhauer's criticism of the proofs of the Parallel Postulate
Notes
References
Ball, W.W. Rouse (1960).A Short Account of the History of Mathematics .4th ed. [Reprint. Original publication: London: Macmillan & Co., 1908],New York: Dover Publications. ISBN 0-486-20630-0.
Boyer, Carl B. (1991).A History of Mathematics . Second Edition, John
Wiley & Sons, Inc..
http://www.reference.com/browse/wiki/Ceva's_theoremhttp://www.reference.com/browse/wiki/Heron's_formulahttp://www.reference.com/browse/wiki/Nine-point_circlehttp://www.reference.com/browse/wiki/Pythagorean_theoremhttp://www.reference.com/browse/wiki/Niccolo_Fontana_Tartagliahttp://www.reference.com/browse/wiki/Menelaus's_theoremhttp://www.reference.com/browse/wiki/Angle_bisector_theoremhttp://www.reference.com/go/http:/wikipedia.org/wiki/The_butterfly_theoremhttp://www.reference.com/browse/wiki/Parallel_Postulatehttp://www.reference.com/browse/wiki/Analytic_geometryhttp://www.reference.com/browse/wiki/Interactive_geometry_softwarehttp://www.reference.com/browse/wiki/Non-Euclidean_geometryhttp://www.reference.com/browse/wiki/Ordered_geometryhttp://www.reference.com/browse/wiki/Incidence_geometryhttp://www.reference.com/browse/wiki/Birkhoff's_axiomshttp://www.reference.com/browse/wiki/Hilbert's_axiomshttp://www.reference.com/browse/wiki/Tarski's_axiomshttp://www.reference.com/browse/wiki/Parallel_postulatehttp://www.reference.com/browse/wiki/Schopenhauer's_criticism_of_the_proofs_of_the_Parallel_Postulatehttp://www.reference.com/browse/wiki/W._W._Rouse_Ballhttp://www.reference.com/browse/wiki/Carl_Benjamin_Boyerhttp://www.reference.com/browse/wiki/Carl_Benjamin_Boyerhttp://www.reference.com/browse/wiki/Ceva's_theoremhttp://www.reference.com/browse/wiki/Heron's_formulahttp://www.reference.com/browse/wiki/Nine-point_circlehttp://www.reference.com/browse/wiki/Pythagorean_theoremhttp://www.reference.com/browse/wiki/Niccolo_Fontana_Tartagliahttp://www.reference.com/browse/wiki/Menelaus's_theoremhttp://www.reference.com/browse/wiki/Angle_bisector_theoremhttp://www.reference.com/go/http:/wikipedia.org/wiki/The_butterfly_theoremhttp://www.reference.com/browse/wiki/Parallel_Postulatehttp://www.reference.com/browse/wiki/Analytic_geometryhttp://www.reference.com/browse/wiki/Interactive_geometry_softwarehttp://www.reference.com/browse/wiki/Non-Euclidean_geometryhttp://www.reference.com/browse/wiki/Ordered_geometryhttp://www.reference.com/browse/wiki/Incidence_geometryhttp://www.reference.com/browse/wiki/Birkhoff's_axiomshttp://www.reference.com/browse/wiki/Hilbert's_axiomshttp://www.reference.com/browse/wiki/Tarski's_axiomshttp://www.reference.com/browse/wiki/Parallel_postulatehttp://www.reference.com/browse/wiki/Schopenhauer's_criticism_of_the_proofs_of_the_Parallel_Postulatehttp://www.reference.com/browse/wiki/W._W._Rouse_Ballhttp://www.reference.com/browse/wiki/Carl_Benjamin_Boyer7/23/2019 bahan makalah geometri
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Franzn, Torkel (2005). Gdel's Theorem: An Incomplete Guide to its Useand Abuse . AK Peters.
Heath, Thomas L. (1956). The Thirteen Books of Euclid's Elements . 2nded. [Facsimile. Original publication: Cambridge University Press, 1925],New York: Dover Publications. ISBN 0-486-60088-2 (vol. 1), ISBN 0-486-60089-0 (vol. 2), ISBN 0-486-60090-4 (vol. 3). Heath's authoritativetranslation of Euclid's Elements plus his extensive historical research anddetailed commentary throughout the text.
Hofstadter, Douglas R. (1979). Gdel, Escher, Bach: An Eternal GoldenBraid . New York: Basic Books.
Nagel, E. and Newman, J.R. (1958). Gdel's Proof. New York UniversityPress.
Alfred Tarski (1951)A Decision Method for Elementary Algebra andGeometry. Univ. of California Press.
http://www.reference.com/browse/Euclidean_geometry
2.7.3 Elliptic Parallel PostulateThe postulate on parallels...was in antiquity the final solution of a problem that must have
preoccupied Greek mathematics for a long period before Euclid.
Hans Freudenthal(19051990)
Elliptic Parallel Postulate.Any two lines intersect in at least one point.
An important note is how elliptic geometry differs in an important way from eitherEuclidean geometry or hyperbolic geometry. Whereas, Euclidean geometry and hyperbolic
geometry are neutral geometries with the addition of a parallel postulate, elliptic geometry
cannot be a neutral geometry due to Theorem 2.14, which stated that parallel lines exist in a
neutral geometry. Hence, the Elliptic Parallel Postulate is inconsistentwith the axioms of a
neutral geometry. Elliptic geometry requires a different set of axioms for the axiomatic
system to be consistent and contain an elliptic parallel postulate.
Georg Friedrich Bernhard Riemann (18261866) was the first to recognize that the
geometry on the surface of a sphere, spherical geometry, is a type of non-Euclidean
geometry. This is the reason we name the spherical model for elliptic geometry after him, the
Riemann Sphere. (To help with the visualization of the concepts in this section, use a ball or a
globe with rubber bands or string.) Click hereto download Spherical Easel a java
exploration of the Riemann Sphere model. In a spherical model:
Two lines, which are great circles, intersect in two points calledpoles orantipodal
points.
A line is unbounded, by this it is meant that a line has no endpoints. A great circle has
no beginning and no end.
A line has finite length.
The concept of betweenness of points does not make sense. What does it mean for a
point to be between two other points on a line (great circle)?
http://www.reference.com/browse/wiki/http://www.reference.com/browse/wiki/http://www.reference.com/browse/wiki/T._L._Heathhttp://www.reference.com/browse/wiki/T._L._Heathhttp://www.reference.com/browse/wiki/Douglas_R._Hofstadterhttp://www.reference.com/browse/wiki/Douglas_R._Hofstadterhttp://www.reference.com/browse/wiki/http://www.reference.com/browse/wiki/Alfred_Tarskihttp://www.reference.com/browse/wiki/Alfred_Tarskihttp://www.reference.com/browse/Euclidean_geometryhttp://en.wikipedia.org/wiki/Hans_Freudenthalhttp://en.wikipedia.org/wiki/Hans_Freudenthalhttp://web.mnstate.edu/peil/geometry/c2euclidnoneuclid/6ExteriorAngleR.htm#T2_14http://web.mnstate.edu/peil/geometry/c2euclidnoneuclid/6ExteriorAngleR.htm#T2_14http://web.mnstate.edu/peil/geometry/C1AxiomSystem/AxiomaticSystems.htm#consisthttp://web.mnstate.edu/peil/geometry/C1AxiomSystem/AxiomaticSystems.htm#consisthttp://en.wikipedia.org/wiki/Georg_Friedrich_Bernhard_Riemannhttp://web.mnstate.edu/peil/geometry/c2euclidnoneuclid/1Models.htm#riemannhttp://merganser.math.gvsu.edu/easel/http://www.reference.com/browse/wiki/http://www.reference.com/browse/wiki/T._L._Heathhttp://www.reference.com/browse/wiki/Douglas_R._Hofstadterhttp://www.reference.com/browse/wiki/http://www.reference.com/browse/wiki/Alfred_Tarskihttp://www.reference.com/browse/Euclidean_geometryhttp://en.wikipedia.org/wiki/Hans_Freudenthalhttp://web.mnstate.edu/peil/geometry/c2euclidnoneuclid/6ExteriorAngleR.htm#T2_14http://web.mnstate.edu/peil/geometry/C1AxiomSystem/AxiomaticSystems.htm#consisthttp://en.wikipedia.org/wiki/Georg_Friedrich_Bernhard_Riemannhttp://web.mnstate.edu/peil/geometry/c2euclidnoneuclid/1Models.htm#riemannhttp://merganser.math.gvsu.edu/easel/7/23/2019 bahan makalah geometri
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From these properties of a sphere, we see that in order to formulate a consistent axiomatic
system, several of the axioms from a neutral geometry need to be dropped or modified,
whether using either Hilbert's or Birkhoff's axioms. The incidence axiom that "any two points
determine a unique line,"needs to be modified to read "any two points determine at least one
line."Hilbert's Axioms of Order (betweenness of points) may be replaced with axioms of
separation that give the properties of how points of a line separate each other. (For a listing ofseparation axioms seeEuclidean and Non-Euclidean Geometries Development and History
by Greenberg.) With these modifications made to the axiom system, the Elliptic Parallel
Postulate may be added to form a consistent system. Often spherical geometry is called
double ellipticgeometry, since two distinct lines intersect in two points.
One problem with the spherical geometry model is that two lines intersect in more than
one point. Felix Klein (18491925) modified the model by identifying each pair of
antipodal points as a single point, see theModified Riemann Sphere. With this model, the
axiom that any two points determine a unique line is satisfied. Often an elliptic geometry that
satisfies this axiom is called asingle elliptic geometry. Note that with this model, a line no
longer separates the plane into distinct half-planes, due to the association of antipodal points
as a single point.
Klein formulated another model for elliptic geometry through the use
of a circle. The model is similar to the Poincar Disk. Given a Euclidean
circle, a point in the model is of two types: a point in the interior of the
Euclidean circle or a point formed by the identification of two antipodal
points which are the endpoints of a diameter of the Euclidean circle. The
lines are of two types: diameters of the Euclidean circle or arcs of
Euclidean circles that intersect the given Euclidean circle at the
endpoints of diameters of the given circle. The model on the left
illustrates four lines, two of each type. The model can be viewed as taking the Modified
Riemann Sphere and flattening onto a Euclidean plane. Click here for a javasketchpadconstruction that uses the Klein model.
Exercise 2.75. In the Riemann Sphere, what properties are true about all lines perpendicular
to a given line?
Exercise 2.76.How does a Mbius strip relate to the Modified Riemann Sphere?
Exercise 2.77.Describehow it is possible to have a triangle with three right angles.
Exercise 2.78. Find an upper bound for the sum of the measures of the angles of a triangle in
the Riemann Sphere.
Exercise 2.79. Use a ball to represent the Riemann Sphere, construct a Saccheri quadrilateral
on the ball. Are the summit angles acute, right, or obtuse? Is the length of the summit more or
less than the length of the base? (Remember the sides of the quadrilateral must be segments
of great circles.)
Non-Euclidean space is the false invention of demons, who gladly furnish the dark
understanding of the non-Euclideans with false knowledge... The non-Euclideans, like the
ancient sophists, seem unaware that their understandings have become obscured by the
promptings of the evil spirits.
Matthew Ryan (1905)
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http://web.mnstate.edu/peil/geometry/c2euclidnoneuclid/7elliptic.htm
Geometri Non Euclid
Non-Euclidean geometri adalah salah satu
dari dua geometri tertentu yang, longgar berbicara, diperoleh dengan meniadakan Euclidean
paralel postulat , yaitu hiperbolik dan geometri eliptik . Ini adalah satu istilah yang, untuk
alasan sejarah, memiliki arti dalam matematika yang jauh lebih sempit dari yang terlihat
untuk memiliki dalam bahasa Inggris umum. Ada banyak sekali geometri yang tidak
geometri Euclidean , tetapi hanya dua yang disebut sebagai non-Euclidean geometri.
Perbedaan penting antara geometri Euclidean dan non-Euclidean adalah sifat paralelbaris.Euclid s kelima mendalilkan, yangparalel mendalilkan, setara dengan yang Playfair postulat
yang menyatakan bahwa, dalam bidang dua dimensi, untuk setiap garis yang diketahui dan
A titik, yang tidak pada , ada tepat satu garis melalui A yang tidak berpotongan . Dalam
geometri hiperbolik, sebaliknya, ada tak terhingga banyak baris melalui A tidak
berpotongan, sementara dalam geometri eliptik, setiap baris melalui A memotong (lihat
entri pada geometri hiperbolik , geometri berbentuk bulat panjang , dan geometri mutlak
untuk informasi lebih lanjut).
Cara lain untuk menggambarkan perbedaan antara geometri adalah mempertimbangkan dua
garis lurus tanpa batas waktu diperpanjang dalam bidang dua dimensi yang baik tegak lurus
ke saluran ketiga:
Dalam geometri Euclidean garis tetap konstan jarak dari satu sama lainbahkan jika diperpanjang hingga tak terbatas, dan dikenal sebagai paralel.
Dalam geometri hiperbolik mereka kurva pergi satu sama lain,peningkatan jarak sebagai salah satu bergerak lebih jauh dari titikpersimpangan dengan tegak lurus umum, garis-garis ini sering disebutultraparallels.
Dalam geometri berbentuk bulat panjang garis kurva ke arah satu samalain dan akhirnya berpotongan.
Sejarah
Sejarah awal
Sementara geometri Euclidean, dinamaimatematikawan YunaniEuclid, termasuk beberapa
dari matematika tertua, non-Euclidean geometri tidak secara luas diterima sebagai sah sampai
abad ke-19.
Perdebatan yang akhirnya menyebabkan penemuan non-Euclidean geometri mulai segera
setelah karya Euclid s Elemen ditulis. Dalam Elemen, Euclid dimulai dengan sejumlahasumsi (23 definisi, lima pengertian umum, dan lima postulat) dan berusaha untuk
http://web.mnstate.edu/peil/geometry/c2euclidnoneuclid/7elliptic.htmhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Parallel_postulate&usg=ALkJrhh-0UswTEH9fnQluSpHy-6IkQwKwAhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Parallel_postulate&usg=ALkJrhh-0UswTEH9fnQluSpHy-6IkQwKwAhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Hyperbolic_geometry&usg=ALkJrhh62gFF8FgB0Xdqc_qNTAqHvMiWIAhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Hyperbolic_geometry&usg=ALkJrhh62gFF8FgB0Xdqc_qNTAqHvMiWIAhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Elliptic_geometry&usg=ALkJrhgQJ4ieOSY7vlm72saNAUFibiWUCAhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Euclidean_geometry&usg=ALkJrhhtCzpuDk76eHdfNHmRtNWaIqYhOwhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Parallel_(geometry)&usg=ALkJrhg3DtfirdfjaluEQmvcZy2oq-uQpwhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Parallel_(geometry)&usg=ALkJrhg3DtfirdfjaluEQmvcZy2oq-uQpwhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Euclid&usg=ALkJrhh-5CMR7iPm-k7mGPNucmFS4Gr17Qhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Parallel_postulate&usg=ALkJrhh-0UswTEH9fnQluSpHy-6IkQwKwAhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Parallel_postulate&usg=ALkJrhh-0UswTEH9fnQluSpHy-6IkQwKwAhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Playfair%27s_Postulate&usg=ALkJrhg7JB15qAyv8PRQdW1-lJF84g2QNAhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Infinite_set&usg=ALkJrhinC2I4NAyZfHZUu6UAyOSLDam36Qhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Hyperbolic_geometry&usg=ALkJrhh62gFF8FgB0Xdqc_qNTAqHvMiWIAhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Hyperbolic_geometry&usg=ALkJrhh62gFF8FgB0Xdqc_qNTAqHvMiWIAhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Elliptic_geometry&usg=ALkJrhgQJ4ieOSY7vlm72saNAUFibiWUCAhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Absolute_geometry&usg=ALkJrhhhOdEPAl0feuyEsdRuMzpO74L1dQhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Perpendicular&usg=ALkJrhj5sYjMwEwj1S0tsK0Nueao_8ufZghttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Distance&usg=ALkJrhiayDSMP_uKbt54JhaiiS2kUX86ZAhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Euclidean_geometry&usg=ALkJrhhtCzpuDk76eHdfNHmRtNWaIqYhOwhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Euclidean_geometry&usg=ALkJrhhtCzpuDk76eHdfNHmRtNWaIqYhOwhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Greek_mathematics&usg=ALkJrhhpu9Qa6QAYrtLLUaDh_9EsltTTcAhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Greek_mathematics&usg=ALkJrhhpu9Qa6QAYrtLLUaDh_9EsltTTcAhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Euclid&usg=ALkJrhh-5CMR7iPm-k7mGPNucmFS4Gr17Qhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Euclid&usg=ALkJrhh-5CMR7iPm-k7mGPNucmFS4Gr17Qhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Euclid%27s_Elements&usg=ALkJrhhYJZK2-JS7HV3m48nvf5f_QJAwZQhttp://existsbox.files.wordpress.com/2012/02/nona1.gifhttp://web.mnstate.edu/peil/geometry/c2euclidnoneuclid/7elliptic.htmhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Parallel_postulate&usg=ALkJrhh-0UswTEH9fnQluSpHy-6IkQwKwAhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Hyperbolic_geometry&usg=ALkJrhh62gFF8FgB0Xdqc_qNTAqHvMiWIAhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Elliptic_geometry&usg=ALkJrhgQJ4ieOSY7vlm72saNAUFibiWUCAhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Euclidean_geometry&usg=ALkJrhhtCzpuDk76eHdfNHmRtNWaIqYhOwhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Parallel_(geometry)&usg=ALkJrhg3DtfirdfjaluEQmvcZy2oq-uQpwhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Euclid&usg=ALkJrhh-5CMR7iPm-k7mGPNucmFS4Gr17Qhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Parallel_postulate&usg=ALkJrhh-0UswTEH9fnQluSpHy-6IkQwKwAhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Playfair%27s_Postulate&usg=ALkJrhg7JB15qAyv8PRQdW1-lJF84g2QNAhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Infinite_set&usg=ALkJrhinC2I4NAyZfHZUu6UAyOSLDam36Qhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeometri%2Bnon%2Beuclid%26hl%3Did%26biw%3D1024%26bih%3D422%26prmd%3Dimvns&rurl=translate.google.co.id&sl=en&u=http://en.wikipedia.org/wiki/Hyperbolic_geometry&usg=ALkJrhh62gFF8FgB0Xdqc_qNTAqHvMiWIAhttp://translate.googleusercontent.com/translate_c?hl=id&prev=/search%3Fq%3Dgeome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bahan makalah geometri
17/28
7/23/2019 bahan makalah geometri
18/28
Giordano Vitale , dalam bukunya Euclide restituo (1680, 1686), menggunakan Saccheri
segiempat untuk membuktikan bahwa jika tiga poin adalah jarak yang sama di pangkalan AB
dan CD KTT, maka AB dan CD di mana-mana berjarak sama.
Dalam sebuah karya berjudul Euclides ab Omni Naevo Vindicatus (Euclid Dibebaskan dari
Semua Cacat), yang diterbitkan tahun 1733, Saccheri geometri eliptik cepat dibuang sebagaikemungkinan (beberapa orang lain dari aksioma Euclid harus dimodifikasi untuk geometri
berbentuk bulat panjang untuk bekerja) dan mulai bekerja membuktikan besar jumlah hasil
dalam geometri hiperbolik. Dia akhirnya mencapai titik di mana ia percaya bahwa hasil
menunjukkan ketidakmungkinan geometri hiperbolik. Klaimnya tampaknya telah didasarkan
pada pengandaian Euclidean, karena tidak ada kontradiksi logis hadir. Dalam upaya untuk
membuktikan geometri Euclidean ia malah tidak sengaja menemukan sebuah geometri baru
yang layak, tapi tidak menyadarinya.
Pada 1766 Johann Lambert menulis, tetapi tidak mempublikasikan, Theorie der
Parallellinien di mana ia mencoba, sebagai Saccheri lakukan, untuk membuktikan postulat
kelima. Dia bekerja dengan angka yang hari ini kita sebut segiempat Lambert, suatusegiempat dengan tiga sudut kanan (dapat dianggap setengah dari segiempat Sacch