bahan makalah geometri

7/23/2019 bahan makalah geometri

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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_Boyer


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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/


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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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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

Documents

bahan makalah geometri