EUCLIDEAN GEOMETRY is a mathematical system attributed to the
Alexandrian Greek mathematician
Euclid
Euclid , which he described in his
textbook on geometry : the Elements . Euclid's method consists in
assuming a small set of intuitively appealing axioms , and deducing
many other propositions (theorems ) from these. Although many of
Euclid's results had been stated by earlier mathematicians, Euclid
was the first to show how these propositions could fit into a
comprehensive deductive and logical system . The Elements begins with
plane geometry, still taught in secondary school as the first
axiomatic system and the first examples of formal proof . It goes on
to the solid geometry of three dimensions . Much of the Elements
states results of what are now called algebra and number theory ,
explained in geometrical language.
For more than two thousand years, the adjective "Euclidean" was
unnecessary because no other sort of geometry had been conceived.
Euclid's axioms seemed so intuitively obvious (with the possible
exception of the parallel postulate ) that any theorem proved from
them was deemed true in an absolute, often metaphysical, sense. Today,
however, many other self-consistent non-Euclidean geometries are
known, the first ones having been discovered in the early 19th
century. An implication of
Albert Einstein
Albert Einstein 's theory of general
relativity is that physical space itself is not Euclidean, and
Euclidean space is a good approximation for it only where the
gravitational field is weak.
Euclidean geometry
Euclidean geometry is an example of synthetic geometry , in that it
proceeds logically from axioms to propositions without the use of
coordinates . This is in contrast to analytic geometry , which uses
coordinates.
CONTENTS
* 1 The Elements
* 1.1
Axioms
* 1.2
Parallel postulate
* 2 Methods of proof
* 3 System of measurement and arithmetic
* 4 Notation and terminology
* 4.1 Naming of points and figures
* 4.2 Complementary and supplementary angles
* 4.3 Modern versions of Euclid\'s notation
* 5 Some important or well known results
* 5.1
Pons Asinorum
* 5.2 Congruence of triangles
* 5.3
Triangle
Triangle angle sum
* 5.4
Pythagorean theorem
Pythagorean theorem
* 5.5 Thales\' theorem
* 5.6 Scaling of area and volume
* 6 Applications
* 7 As a description of the structure of space
* 8 Later work
* 8.1
Archimedes
Archimedes and Apollonius
* 8.2 17th century: Descartes
* 8.3 18th century
* 8.4 19th century and non-
Euclidean geometry
Euclidean geometry
* 8.5 20th century and general relativity
* 9 Treatment of infinity
* 9.1 Infinite objects
* 9.2 Infinite processes
* 10 Logical basis
* 10.1
Classical logic
* 10.2 Modern standards of rigor
* 10.3 Axiomatic formulations
* 10.4 Constructive approaches and pedagogy
* 11 See also
* 11.1 Classical theorems
* 12 Notes
* 13 References
* 14 External links
THE ELEMENTS
Main article: Euclid\'s Elements
The Elements is mainly a systematization of earlier knowledge of
geometry. Its improvement over earlier treatments was rapidly
recognized, with the result that there was little interest in
preserving the earlier ones, and they are now nearly all lost.
There are 13 total books in the Elements:
Books I–IV and VI discuss plane geometry. Many results about plane
figures are proved, for example "In any triangle two angles taken
together in any manner are less than two right angles." (Book 1
proposition 17 ) and the
Pythagorean theorem
Pythagorean theorem "In right angled
triangles the square on the side subtending the right angle is equal
to the squares on the sides containing the right angle." (Book I,
proposition 47)
Books V and VII–X deal with number theory, with numbers treated
geometrically via their representation as line segments with various
lengths. Notions such as prime numbers and rational and irrational
numbers are introduced. The infinitude of prime numbers is proved.
Books XI–XIII concern solid geometry . A typical result is the 1:3
ratio between the volume of a cone and a cylinder with the same height
and base.
AXIOMS
The parallel postulate (
Postulate 5): If two lines intersect a
third in such a way that the sum of the inner angles on one side is
less than two right angles, then the two lines inevitably must
intersect each other on that side if extended far enough.
Euclidean geometry
Euclidean geometry is an axiomatic system , in which all theorems
("true statements") are derived from a small number of axioms. Near
the beginning of the first book of the Elements,
Euclid
Euclid gives five
postulates (axioms) for plane geometry, stated in terms of
constructions (as translated by Thomas Heath):
"Let the following be postulated":
* "To draw a straight line from any point to any point."
* "To produce a finite straight line continuously in a straight
line."
* "To describe a circle with any centre and distance ."
* "That all right angles are equal to one another."
* The parallel postulate : "That, if a straight line falling on two
straight lines make the interior angles on the same side less than two
right angles, the two straight lines, if produced indefinitely, meet
on that side on which are the angles less than the two right angles."
Although Euclid's statement of the postulates only explicitly asserts
the existence of the constructions, they are also taken to be unique.
The Elements also include the following five "common notions":
* Things that are equal to the same thing are also equal to one
another (formally the Euclidean property of equality, but may be
considered a consequence of the transitivity property of equality).
* If equals are added to equals, then the wholes are equal (Addition
property of equality).
* If equals are subtracted from equals, then the remainders are
equal (Subtraction property of equality).
* Things that coincide with one another are equal to one another
(Reflexive Property).
* The whole is greater than the part.
PARALLEL POSTULATE
Main article:
Parallel postulate
To the ancients, the parallel postulate seemed less obvious than the
others. They were concerned with creating a system which was
absolutely rigorous and to them it seemed as if the parallel line
postulate should have been able to be proven rather than simply
accepted as a fact. It is now known that such a proof is impossible.
Euclid
Euclid himself seems to have considered it as being qualitatively
different from the others, as evidenced by the organization of the
Elements: the first 28 propositions he presents are those that can be
proved without it.
Many alternative axioms can be formulated that have the same logical
consequences as the parallel postulate. For example, Playfair\'s axiom
states: In a plane , through a point not on a given straight line, at
most one line can be drawn that never meets the given line. A
proof from
Euclid's Elements
Euclid's Elements that, given a line segment, an
equilateral triangle exists that includes the segment as one of its
sides. The proof is by construction: an equilateral triangle ΑΒΓ is
made by drawing circles Δ and Ε centered on the points Α and Β,
and taking one intersection of the circles as the third vertex of the
triangle.
METHODS OF PROOF
Euclidean
Geometry
Geometry is constructive . 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
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 lines on paper are models of 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. Though nearly all modern mathematicians
consider nonconstructive methods just as sound as constructive ones,
Euclid's constructive proofs often supplanted fallacious
nonconstructive ones—e.g., some of the Pythagoreans' proofs that
involved irrational numbers, which usually required a statement such
as "Find the greatest common measure of ..."
Euclid
Euclid often used proof by contradiction .
Euclidean geometry
Euclidean geometry also
allows the method of superposition, in which a figure is transferred
to another point in space. For example, proposition I.4,
side-angle-side congruence of triangles, is proved by moving one of
the two triangles so that one of its sides coincides with the other
triangle's equal side, and then proving that the other sides coincide
as well. Some modern treatments add a sixth postulate, the rigidity of
the triangle, which can be used as an alternative to superposition.
SYSTEM OF MEASUREMENT AND ARITHMETIC
Euclidean geometry
Euclidean geometry has two fundamental types of measurements: angle
and distance . The angle scale is absolute, and
Euclid
Euclid uses the right
angle as his basic unit, so that, e.g., a 45-degree angle would be
referred to as half of a right angle. The distance scale is relative;
one arbitrarily picks a line segment with a certain nonzero length as
the unit, and other distances are expressed in relation to it.
Addition of distances is represented by a construction in which one
line segment is copied onto the end of another line segment to extend
its length, and similarly for subtraction.
Measurements of area and volume are derived from distances. For
example, a rectangle with a width of 3 and a length of 4 has an area
that represents the product, 12. Because this geometrical
interpretation of multiplication was limited to three dimensions,
there was no direct way of interpreting the product of four or more
numbers, and
Euclid
Euclid avoided such products, although they are implied,
e.g., in the proof of book IX, proposition 20. An example of
congruence. The two figures on the left are congruent, while the third
is similar to them. The last figure is neither. Congruences alter some
properties, such as location and orientation, but leave others
unchanged, like distance and angles . The latter sort of properties
are called invariants and studying them is the essence of geometry.
Euclid
Euclid refers to a pair of lines, or a pair of planar or solid
figures, as "equal" (ἴσος) if their lengths, areas, or volumes
are equal, and similarly for angles. The stronger term "congruent "
refers to the idea that an entire figure is the same size and shape as
another figure. Alternatively, two figures are congruent if one can be
moved on top of the other so that it matches up with it exactly.
(Flipping it over is allowed.) Thus, for example, a 2x6 rectangle and
a 3x4 rectangle are equal but not congruent, and the letter R is
congruent to its mirror image. Figures that would be congruent except
for their differing sizes are referred to as similar . Corresponding
angles in a pair of similar shapes are congruent and corresponding
sides are in proportion to each other.
NOTATION AND TERMINOLOGY
NAMING OF POINTS AND FIGURES
Points are customarily named using capital letters of the alphabet.
Other figures, such as lines, triangles, or circles, are named by
listing a sufficient number of points to pick them out unambiguously
from the relevant figure, e.g., triangle ABC would typically be a
triangle with vertices at points A, B, and C.
COMPLEMENTARY AND SUPPLEMENTARY ANGLES
Angles whose sum is a right angle are called complementary .
Complementary angles are formed when a ray shares the same vertex and
is pointed in a direction that is in between the two original rays
that form the right angle. The number of rays in between the two
original rays is infinite.
Angles whose sum is a straight angle are supplementary .
Supplementary angles are formed when a ray shares the same vertex and
is pointed in a direction that is in between the two original rays
that form the straight angle (180 degree angle). The number of rays in
between the two original rays is infinite.
MODERN VERSIONS OF EUCLID\'S NOTATION
In modern terminology, angles would normally be measured in degrees
or radians .
Modern school textbooks often define separate figures called lines
(infinite), rays (semi-infinite), and line segments (of finite
length). Euclid, rather than discussing a ray as an object that
extends to infinity in one direction, would normally use locutions
such as "if the line is extended to a sufficient length," although he
occasionally referred to "infinite lines." A "line" in
Euclid
Euclid could be
either straight or curved, and he used the more specific term
"straight line" when necessary.
SOME IMPORTANT OR WELL KNOWN RESULTS
*
The PONS ASINORUM or BRIDGE OF ASSES THEOREM states that in an
isosceles triangle, α = β and γ = δ.
*
The TRIANGLE ANGLE SUM THEOREM states that the sum of the three
angles of any triangle, in this case angles α, β, and γ, will
always equal 180 degrees.
*
The PYTHAGOREAN THEOREM states that the sum of the areas of the two
squares on the legs (a and b) of a right triangle equals the area of
the square on the hypotenuse (c).
*
THALES\\' THEOREM states that if AC is a diameter, then the angle at
B is a right angle.
PONS ASINORUM
The Bridge of Asses (Pons Asinorum) states that in isosceles
triangles the angles at the base equal one another, and, if the equal
straight lines are produced further, then the angles under the base
equal one another. Its name may be attributed to its frequent role as
the first real test in the Elements of the intelligence of the reader
and as a bridge to the harder propositions that followed. It might
also be so named because of the geometrical figure's resemblance to a
steep bridge that only a sure-footed donkey could cross.
CONGRUENCE OF TRIANGLES
Congruence of triangles is determined by specifying two sides
and the angle between them (SAS), two angles and the side between them
(ASA) or two angles and a corresponding adjacent side (AAS).
Specifying two sides and an adjacent angle (SSA), however, can yield
two distinct possible triangles unless the angle specified is a right
angle.
Triangles are congruent if they have all three sides equal (SSS), two
sides and the angle between them equal (SAS), or two angles and a side
equal (ASA) (Book I, propositions 4, 8, and 26). Triangles with three
equal angles (AAA) are similar, but not necessarily congruent. Also,
triangles with two equal sides and an adjacent angle are not
necessarily equal or congruent.
TRIANGLE ANGLE SUM
The sum of the angles of a triangle is equal to a straight angle (180
degrees). This causes an equilateral triangle to have 3 interior
angles of 60 degrees. Also, it causes every triangle to have at least
2 acute angles and up to 1 obtuse or right angle .
PYTHAGOREAN THEOREM
The celebrated
Pythagorean theorem
Pythagorean theorem (book I, proposition 47) states
that in any right triangle, the area of the square whose side is the
hypotenuse (the side opposite the right angle) is equal to the sum of
the areas of the squares whose sides are the two legs (the two sides
that meet at a right angle).
THALES\' THEOREM
Thales\' theorem , named after
Thales of Miletus states that if A, B,
and C are points on a circle where the line AC is a diameter of the
circle, then the angle ABC is a right angle. Cantor supposed that
Thales proved his theorem by means of
Euclid
Euclid Book I, Prop. 32 after
the manner of
Euclid
Euclid Book III, Prop. 31. Tradition has it that Thales
sacrificed an ox to celebrate this theorem.
SCALING OF AREA AND VOLUME
In modern terminology, the area of a plane figure is proportional to
the square of any of its linear dimensions, A L 2
{displaystyle Apropto L^{2}} , and the volume of a solid to the cube,
V L 3 {displaystyle Vpropto L^{3}} .
Euclid
Euclid proved these
results in various special cases such as the area of a circle and the
volume of a parallelepipedal solid.
Euclid
Euclid determined some, but not
all, of the relevant constants of proportionality. E.g., it was his
successor
Archimedes
Archimedes who proved that a sphere has 2/3 the volume of
the circumscribing cylinder.
APPLICATIONS
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(March 2009)
Because of Euclidean geometry's fundamental status in mathematics, it
would be impossible to give more than a representative sampling of
applications here.
*
A surveyor uses a level
*
Sphere
Sphere packing applies to a stack of oranges .
*
A parabolic mirror brings parallel rays of light to a focus.
As suggested by the etymology of the word, one of the earliest
reasons for interest in geometry was surveying , and certain
practical results from Euclidean geometry, such as the right-angle
property of the 3-4-5 triangle, were used long before they were proved
formally. The fundamental types of measurements in Euclidean geometry
are distances and angles, and both of these quantities can be measured
directly by a surveyor. Historically, distances were often measured by
chains such as Gunter\'s chain , and angles using graduated circles
and, later, the theodolite .
An application of Euclidean solid geometry is the determination of
packing arrangements , such as the problem of finding the most
efficient packing of spheres in n dimensions. This problem has
applications in error detection and correction .
Geometric optics uses
Euclidean geometry
Euclidean geometry to analyze the focusing of
light by lenses and mirrors.
*
Geometry
Geometry is used in art and architecture.
*
The water tower consists of a cone, a cylinder, and a hemisphere. Its
volume can be calculated using solid geometry.
*
Geometry
Geometry can be used to design origami.
Geometry
Geometry is used extensively in architecture .
Geometry
Geometry can be used to design origami . Some classical construction
problems of geometry are impossible using compass and straightedge ,
but can be solved using origami .
AS A DESCRIPTION OF THE STRUCTURE OF SPACE
Euclid
Euclid believed that his axioms were self-evident statements about
physical reality. Euclid's proofs depend upon assumptions perhaps not
obvious in Euclid's fundamental axioms, in particular that certain
movements of figures do not change their geometrical properties such
as the lengths of sides and interior angles, the so-called Euclidean
motions, which include translations, reflections and rotations of
figures. Taken as a physical description of space, postulate 2
(extending a line) asserts that space does not have holes or
boundaries (in other words, space is homogeneous and unbounded );
postulate 4 (equality of right angles) says that space is isotropic
and figures may be moved to any location while maintaining congruence
; and postulate 5 (the parallel postulate ) that space is flat (has no
intrinsic curvature ).
As discussed in more detail below,
Einstein
Einstein 's theory of relativity
significantly modifies this view.
The ambiguous character of the axioms as originally formulated by
Euclid
Euclid makes it possible for different commentators to disagree about
some of their other implications for the structure of space, such as
whether or not it is infinite (see below) and what its topology is.
Modern, more rigorous reformulations of the system typically aim for
a cleaner separation of these issues. Interpreting Euclid's axioms in
the spirit of this more modern approach, axioms 1-4 are consistent
with either infinite or finite space (as in elliptic geometry ), and
all five axioms are consistent with a variety of topologies (e.g., a
plane, a cylinder, or a torus for two-dimensional Euclidean geometry).
LATER WORK
ARCHIMEDES AND APOLLONIUS
A sphere has 2/3 the volume and surface area of its
circumscribing cylinder. A sphere and cylinder were placed on the tomb
of
Archimedes
Archimedes at his request.
Archimedes
Archimedes (ca. 287 BCE – ca. 212 BCE), a colorful figure about
whom many historical anecdotes are recorded, is remembered along with
Euclid
Euclid as one of the greatest of ancient mathematicians. Although the
foundations of his work were put in place by Euclid, his work, unlike
Euclid's, is believed to have been entirely original. He proved
equations for the volumes and areas of various figures in two and
three dimensions, and enunciated the
Archimedean property of finite
numbers.
Apollonius of Perga
Apollonius of Perga (ca. 262 BCE–ca. 190 BCE) is mainly known for
his investigation of conic sections. René Descartes. Portrait
after
Frans Hals
Frans Hals , 1648.
17TH CENTURY: DESCARTES
René Descartes (1596–1650) developed analytic geometry , an
alternative method for formalizing geometry which focused on turning
geometry into algebra.
In this approach, a point on a plane is represented by its Cartesian
(x, y) coordinates, a line is represented by its equation, and so on.
In Euclid's original approach, the
Pythagorean theorem
Pythagorean theorem follows from
Euclid's axioms. In the Cartesian approach, the axioms are the axioms
of algebra, and the equation expressing the
Pythagorean theorem
Pythagorean theorem is
then a definition of one of the terms in Euclid's axioms, which are
now considered theorems.
The equation P Q = ( p x q x ) 2 + ( p
y q y ) 2 {displaystyle PQ={sqrt
{(p_{x}-q_{x})^{2}+(p_{y}-q_{y})^{2}}},}
defining the distance between two points P = (px, py) and Q = (qx,
qy) is then known as the Euclidean metric , and other metrics define
non-Euclidean geometries .
In terms of analytic geometry, the restriction of classical geometry
to compass and straightedge constructions means a restriction to
first- and second-order equations, e.g., y = 2x + 1 (a line), or x2 +
y2 = 7 (a circle).
Also in the 17th century,
Girard Desargues , motivated by the theory
of perspective , introduced the concept of idealized points, lines,
and planes at infinity. The result can be considered as a type of
generalized geometry, projective geometry , but it can also be used to
produce proofs in ordinary
Euclidean geometry
Euclidean geometry in which the number of
special cases is reduced. Squaring the circle: the areas of this
square and this circle are equal. In 1882, it was proven that this
figure cannot be constructed in a finite number of steps with an
idealized compass and straightedge .
18TH CENTURY
Geometers of the 18th century struggled to define the boundaries of
the Euclidean system. Many tried in vain to prove the fifth postulate
from the first four. By 1763 at least 28 different proofs had been
published, but all were found incorrect.
Leading up to this period, geometers also tried to determine what
constructions could be accomplished in Euclidean geometry. For
example, the problem of trisecting 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 Wantzel published a proof in 1837 that
such a construction was impossible. Other constructions that were
proved impossible include doubling the cube and squaring the circle .
In the case of doubling the cube, the impossibility of the
construction originates from the fact that the compass and
straightedge method involve equations whose order is an integral power
of two, while doubling a cube requires the solution of a third-order
equation.
Euler discussed a generalization of
Euclidean geometry
Euclidean geometry called affine
geometry , 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
equality of length of line segments in general (whence circles become
meaningless) while retaining 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).
19TH CENTURY AND NON-EUCLIDEAN GEOMETRY
In the early 19th century, Carnot and Möbius systematically
developed the use of signed angles and line segments as a way of
simplifying and unifying results.
The century's most significant development in geometry occurred when,
around 1830,
János Bolyai and Nikolai Ivanovich Lobachevsky
separately published work on non-
Euclidean geometry
Euclidean geometry , in which the
parallel postulate is not valid. Since non-
Euclidean geometry
Euclidean geometry is
provably relatively consistent with Euclidean geometry, the parallel
postulate cannot be proved from the other postulates.
In the 19th century, it was also realized that Euclid's ten axioms
and common notions do not suffice to prove all of the theorems stated
in the Elements. For example,
Euclid
Euclid assumed implicitly that any line
contains at least two points, but this assumption cannot be proved
from the other axioms, and therefore must be an axiom itself. The very
first geometric proof in the Elements, shown in the figure above, is
that any line segment is part of a triangle;
Euclid
Euclid constructs this in
the usual way, by drawing circles around both endpoints and taking
their intersection as the third vertex. His axioms, however, do not
guarantee that the circles actually intersect, because they do not
assert the geometrical property of continuity, which in Cartesian
terms is equivalent to the completeness property of the real numbers.
Starting with
Moritz Pasch in 1882, many improved axiomatic systems
for geometry have been proposed, the best known being those of Hilbert
, George Birkhoff , and Tarski .
20TH CENTURY AND GENERAL RELATIVITY
A disproof of
Euclidean geometry
Euclidean geometry as a description of physical
space. In a 1919 test of the general theory of relativity, stars
(marked with short horizontal lines) were photographed during a solar
eclipse . The rays of starlight were bent by the Sun's gravity on
their way to the earth. This is interpreted as evidence in favor of
Einstein's prediction that gravity would cause deviations from
Euclidean geometry.
Einstein\'s theory of general relativity shows that the true geometry
of spacetime is not 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
Einstein predicted that such
deviations would exist. They were later verified by observations such
as the slight bending of starlight by the Sun during a solar eclipse
in 1919, and such considerations are now an integral part of the
software that runs the GPS system. It is possible to object to this
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 distinguish
between a beam of light as a model of a geometrical line and any other
physical model. Thus, the only logical possibilities are to accept
non-
Euclidean geometry
Euclidean geometry as 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.
TREATMENT OF INFINITY
INFINITE OBJECTS
Euclid
Euclid sometimes distinguished explicitly between "finite lines"
(e.g.,
Postulate 2) and "infinite lines" (book I, proposition 12).
However, he typically did not make such distinctions unless they were
necessary. The postulates do not explicitly refer to infinite lines,
although for example some commentators interpret postulate 3,
existence of a circle with any radius, as implying that space is
infinite.
The notion of infinitesimal quantities had previously been discussed
extensively by the
Eleatic School , but nobody had been able to put
them on a firm logical basis, with paradoxes such as Zeno\'s paradox
occurring that had not been resolved to universal satisfaction. Euclid
used the method of exhaustion rather than infinitesimals.
Later ancient commentators such as
Proclus (410–485 CE) treated
many questions about infinity as issues demanding proof and, e.g.,
Proclus claimed to prove the infinite divisibility of a line, based on
a proof by contradiction in which he considered the cases of even and
odd numbers of points constituting it.
At the turn of the 20th century,
Otto Stolz
Otto Stolz ,
Paul du Bois-Reymond ,
Giuseppe Veronese , and others produced controversial work on
non-Archimedean models of Euclidean geometry, in which the distance
between two points may be infinite or infinitesimal, in the Newton
–Leibniz sense. Fifty years later,
Abraham Robinson provided a
rigorous logical foundation for Veronese's work.
INFINITE PROCESSES
One reason that the ancients treated the parallel postulate as less
certain than the others is that 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.
The modern formulation of proof by induction was not developed until
the 17th century, but some later commentators consider it implicit in
some of Euclid's proofs, e.g., the proof of the infinitude of primes.
Supposed paradoxes involving infinite series, such as Zeno\'s paradox
, predated Euclid.
Euclid
Euclid avoided such discussions, giving, for
example, the expression for the partial sums of the geometric series
in IX.35 without commenting on the possibility of letting the number
of terms become infinite.
LOGICAL BASIS
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See also: Hilbert\'s axioms ,
Axiomatic system , and Real closed
field
CLASSICAL LOGIC
Euclid
Euclid frequently used the method of proof by contradiction , and
therefore the traditional presentation of
Euclidean geometry
Euclidean geometry assumes
classical logic , in which every proposition is either true or false,
i.e., for any proposition P, the proposition "P or not P" is
automatically true.
MODERN STANDARDS OF RIGOR
Placing
Euclidean geometry
Euclidean geometry on a solid axiomatic basis was a
preoccupation of mathematicians for centuries. The role of primitive
notions , or undefined concepts, was clearly put forward by Alessandro
Padoa of the Peano delegation at the 1900 Paris conference:
...when we begin to formulate the theory, we can imagine that the
undefined symbols are completely devoid of meaning and that the
unproved propositions are simply conditions imposed upon the undefined
symbols.
Then, the system of ideas that we have initially chosen is simply one
interpretation of the undefined symbols; but..this interpretation can
be ignored by the reader, who is free to replace it in his mind by
another interpretation.. that satisfies the conditions...
Logical questions thus become completely independent of empirical or
psychological questions... The system of undefined symbols can then be
regarded as the abstraction obtained from the specialized theories
that result when...the system of undefined symbols is successively
replaced by each of the interpretations... — Padoa, Essai d'une
théorie algébrique des nombre entiers, avec une Introduction logique
à une théorie déductive quelconque
That is, mathematics is context-independent knowledge within a
hierarchical framework. As said by
Bertrand Russell
Bertrand Russell :
If our hypothesis is about anything, and not about some one or more
particular things, then our deductions constitute mathematics. Thus,
mathematics may be defined as the subject in which we never know what
we are talking about, nor whether what we are saying is true.
— Bertrand Russell, Mathematics and the metaphysicians
Such foundational approaches range between foundationalism and
formalism .
AXIOMATIC FORMULATIONS
Geometry
Geometry is the science of correct reasoning on incorrect figures.
— George Polyá, How to Solve It, p. 208
* Euclid's axioms: In his dissertation to Trinity College,
Cambridge,
Bertrand Russell
Bertrand Russell summarized the changing role of Euclid's
geometry in the minds of philosophers up to that time. It was a
conflict between certain knowledge, independent of experiment, and
empiricism, requiring experimental input. This issue became clear as
it was discovered that the parallel postulate was not necessarily
valid and its applicability was an empirical matter, deciding whether
the applicable geometry was Euclidean or non-Euclidean .
* Hilbert\'s axioms :
Hilbert's axioms had the goal of identifying a
simple and complete set of independent axioms from which the most
important geometric theorems could be deduced. The outstanding
objectives were to make
Euclidean geometry
Euclidean geometry rigorous (avoiding hidden
assumptions) and to make clear the ramifications of the parallel
postulate.
* Birkhoff\'s axioms : Birkhoff proposed four postulates for
Euclidean geometry
Euclidean geometry that can be confirmed experimentally with scale and
protractor. This system relies heavily on the properties of the real
numbers . The notions of angle and distance become primitive
concepts.
* Tarski\'s axioms :
Alfred Tarski (1902–1983) and his students
defined elementary
Euclidean geometry
Euclidean geometry as the geometry that can be
expressed in first-order logic and does not depend on set theory for
its logical basis, in contrast to Hilbert's axioms, which involve
point sets. Tarski proved that his axiomatic formulation of
elementary
Euclidean geometry
Euclidean geometry is consistent and complete in a certain
sense : there is an algorithm that, for every proposition, can be
shown either true or false. (This doesn't violate Gödel\'s theorem ,
because
Euclidean geometry
Euclidean geometry cannot describe a sufficient amount of
arithmetic for the theorem to apply. ) This is equivalent to the
decidability of real closed fields , of which elementary Euclidean
geometry is a model.
CONSTRUCTIVE APPROACHES AND PEDAGOGY
The process of abstract axiomatization as exemplified by Hilbert\'s
axioms reduces geometry to theorem proving or predicate logic . In
contrast, the Greeks used construction postulates, and emphasized
problem solving. For the Greeks, constructions are more primitive
than existence propositions, and can be used to prove existence
propositions, but not vice versa. To describe problem solving
adequately requires a richer system of logical concepts. The contrast
in approach may be summarized:
* Axiomatic proof: Proofs are deductive derivations of propositions
from primitive premises that are ‘true’ in some sense. The aim is
to justify the proposition.
* Analytic proof: Proofs are non-deductive derivations of hypotheses
from problems. The aim is to find hypotheses capable of giving a
solution to the problem. One can argue that Euclid's axioms were
arrived upon in this manner. In particular, it is thought that Euclid
felt the parallel postulate was forced upon him, as indicated by his
reluctance to make use of it, and his arrival upon it by the method
of contradiction.
Andrei Nicholaevich
Kolmogorov proposed a problem solving basis for
geometry. This work was a precursor of a modern formulation in terms
of constructive type theory . This development has implications for
pedagogy as well.
If proof simply follows conviction of truth rather than contributing
to its construction and is only experienced as a demonstration of
something already known to be true, it is likely to remain meaningless
and purposeless in the eyes of students. — Celia Hoyles, The
curricular shaping of students' approach to proof
SEE ALSO
*
Analytic geometry
* Birkhoff\'s axioms
*
Cartesian coordinate system
* Hilbert\'s axioms
*
Incidence geometry
*
List of interactive geometry software
*
Metric space
*
Non-Euclidean geometry
*
Ordered geometry
*
Parallel postulate
*
Type theory
CLASSICAL THEOREMS
*
Angle
Angle bisector theorem
*
Butterfly theorem
* Ceva\'s theorem
* Heron\'s formula
* Menelaus\' theorem
*
Nine-point circle
*
Pythagorean theorem
Pythagorean theorem
NOTES
* ^ Eves, vol. 1., p. 19
* ^ Eves (1963), vol. 1, p. 10
* ^ Eves, p. 19
* ^ Misner, Thorne, and Wheeler (1973), p. 47
* ^ The assumptions of
Euclid
Euclid are discussed from a modern
perspective in Harold E. Wolfe (2007). Introduction to Non-Euclidean
Geometry. Mill Press. p. 9. ISBN 1-4067-1852-1 .
* ^ tr. Heath, pp. 195–202.
* ^ Florence P. Lewis (Jan 1920), "History of the Parallel
Postulate", The American Mathematical Monthly, The American
Mathematical Monthly, Vol. 27, No. 1, 27 (1): 16–23, doi
:10.2307/2973238 ,
JSTOR
JSTOR 2973238 .
* ^ Ball, p. 56
* ^ Within Euclid's assumptions, it is quite easy to give a formula
for area of triangles and squares. However, in a more general context
like set theory, it is not as easy to prove that the area of a square
is the sum of areas of its pieces, for example. See Lebesgue measure
and
Banach–Tarski paradox .
* ^ Daniel Shanks (2002). Solved and Unsolved Problems in Number
Theory. American Mathematical Society.
* ^ Coxeter, p. 5
* ^ Euclid, book I, proposition 5, tr. Heath, p. 251
* ^ Ignoring the alleged difficulty of Book I,
Proposition 5, Sir
Thomas L. Heath mentions another interpretation. This rests on the
resemblance of the figure's lower straight lines to a steeply inclined
bridge that could be crossed by an ass but not by a horse: "But there
is another view (as I have learnt lately) which is more complimentary
to the ass. It is that, the figure of the proposition being like that
of a trestle bridge, with a ramp at each end which is more practicable
the flatter the figure is drawn, the bridge is such that, while a
horse could not surmount the ramp, an ass could; in other words, the
term is meant to refer to the sure-footedness of the ass rather than
to any want of intelligence on his part." (in "Excursis II," volume 1
of Heath's translation of The Thirteen Books of the Elements.)
* ^ Euclid, book I, proposition 32
* ^ Heath, p. 135. Extract of page 135
* ^ Heath, p. 318
* ^ Euclid, book XII, proposition 2
* ^ Euclid, book XI, proposition 33
* ^ Ball, p. 66
* ^ Ball, p. 5
* ^ Eves, vol. 1, p. 5; Mlodinow, p. 7
* ^ Tom Hull. "
Origami
Origami and Geometric Constructions".
* ^ Richard J. Trudeau (2008). "Euclid's axioms". The Non-Euclidean
Revolution. Birkhäuser. pp. 39 ff. ISBN 0-8176-4782-1 .
* ^ See, for example: Luciano da Fontoura Costa; Roberto Marcondes
Cesar (2001). Shape analysis and classification: theory and practice.
CRC Press. p. 314. ISBN 0-8493-3493-4 . and Helmut Pottmann; Johannes
Wallner (2010). Computational Line Geometry. Springer. p. 60. ISBN
3-642-04017-9 . The group of motions underlie the metric notions of
geometry. See
Felix Klein
Felix Klein (2004). Elementary Mathematics from an
Advanced Standpoint:
Geometry
Geometry (Reprint of 1939 Macmillan Company ed.).
Courier Dover. p. 167. ISBN 0-486-43481-8 .
* ^ Roger Penrose (2007). The Road to Reality: A Complete Guide to
the Laws of the Universe. Vintage Books. p. 29. ISBN 0-679-77631-1 .
* ^ A B Heath, p. 200
* ^ e.g., Tarski (1951)
* ^ Eves, p. 27
* ^ Ball, pp. 268ff
* ^ Eves (1963)
* ^ Hofstadter 1979, p. 91.
* ^
Theorem 120, Elements of Abstract Algebra, Allan Clark, Dover,
ISBN 0-486-64725-0
* ^ Eves (1963), p. 64
* ^ Ball, p. 485
* ^ *
Howard Eves , 1997 (1958). Foundations and Fundamental
Concepts of Mathematics. Dover.
* ^ Birkhoff, G. D., 1932, "A Set of Postulates for Plane Geometry
(Based on Scale and Protractors)," Annals of Mathematics 33.
* ^ A B Tarski (1951)
* ^ Misner, Thorne, and Wheeler (1973), p. 191
* ^ Rizos, Chris.
University of New South Wales . GPS Satellite
Signals. 1999.
* ^ Ball, p. 31
* ^ Heath, p. 268
* ^ Giuseppe Veronese, On Non-Archimedean Geometry, 1908. English
translation in Real Numbers, Generalizations of the Reals, and
Theories of Continua, ed.
Philip Ehrlich , Kluwer, 1994.
* ^ Robinson, Abraham (1966). Non-standard analysis.
* ^ For the assertion that this was the historical reason for the
ancients considering the parallel postulate less obvious than the
others, see Nagel and Newman 1958, p. 9.
* ^ Cajori (1918), p. 197
* ^ A B A detailed discussion can be found in James T. Smith
(2000). "Chapter 2: Foundations". Methods of geometry. Wiley. pp. 19
ff. ISBN 0-471-25183-6 .
* ^ Société française de philosophie (1900). Revue de
métaphysique et de morale,
Volume 8. Hachette. p. 592.
* ^
Bertrand Russell
Bertrand Russell (2000). "Mathematics and the metaphysicians".
In James Roy Newman. The world of mathematics. 3 (Reprint of Simon and
Schuster 1956 ed.). Courier Dover Publications. p. 1577. ISBN
0-486-41151-6 .
* ^
Bertrand Russell
Bertrand Russell (1897). "Introduction". An essay on the
foundations of geometry. Cambridge University Press.
* ^ George David Birkhoff; Ralph Beatley (1999). "Chapter 2: The
five fundamental principles". Basic
Geometry
Geometry (3rd ed.). AMS Bookstore.
pp. 38 ff. ISBN 0-8218-2101-6 .
* ^ James T. Smith. "Chapter 3: Elementary Euclidean Geometry".
Cited work. pp. 84 ff.
* ^ Edwin E. Moise (1990). Elementary geometry from an advanced
standpoint (3rd ed.). Addison–Wesley. ISBN 0-201-50867-2 .
* ^ John R. Silvester (2001). "§1.4 Hilbert and Birkhoff".
Geometry: ancient and modern. Oxford University Press. ISBN
0-19-850825-5 .
* ^
Alfred Tarski (2007). "What is elementary geometry". In Leon
Henkin; Patrick Suppes; Alfred Tarski. Studies in Logic and the
Foundations of Mathematics – The Axiomatic Method with Special
Reference to
Geometry
Geometry and Physics (Proceedings of International
Symposium at Berkeley 1957–8; Reprint ed.). Brouwer Press. p. 16.
ISBN 1-4067-5355-6 . We regard as elementary that part of Euclidean
geometry which can be formulated and established without the help of
any set-theoretical devices
* ^ Keith Simmons (2009). "Tarski's logic". In Dov M. Gabbay, John
Woods. Logic from Russell to Church. Elsevier. p. 574. ISBN
0-444-51620-4 . CS1 maint: Uses editors parameter (link )
* ^ Franzén, Torkel (2005). Gödel's Theorem: An Incomplete Guide
to its Use and Abuse. AK Peters. ISBN 1-56881-238-8 . Pp. 25–26.
* ^ A B Petri Mäenpää (1999). "From backward reduction to
configurational analysis". In Michael Otte; Marco Panza. Analysis and
synthesis in mathematics: history and philosophy. Springer. p. 210.
ISBN 0-7923-4570-3 .
* ^ Carlo Cellucci (2008). "Why proof? What is proof?". In Rossella
Lupacchini; Giovanna Corsi. Deduction, Computation, Experiment:
Exploring the Effectiveness of Proof. Springer. p. 1. ISBN
88-470-0783-6 .
* ^ Eric W. Weisstein (2003). "Euclid's postulates". CRC concise
encyclopedia of mathematics (2nd ed.). CRC Press. p. 942. ISBN
1-58488-347-2 .
* ^ Deborah J. Bennett (2004). Logic made easy: how to know when
language deceives you. W. W. Norton & Company. p. 34. ISBN
0-393-05748-8 .
* ^ AN Kolmogorov; AF Semenovich; RS Cherkasov (1982). Geometry: A
textbook for grades 6–8 of secondary school (3rd ed.). Moscow:
"Prosveshchenie" Publishers. pp. 372–376. A description of the
approach, which was based upon geometric transformations , can be
found in Teaching geometry in the USSR Chernysheva, Firsov, and
Teljakovskii
* ^ Viktor Vasilʹevich Prasolov; Vladimir Mikhaĭlovich Tikhomirov
(2001). Geometry. AMS Bookstore. p. 198. ISBN 0-8218-2038-9 .
* ^ Petri Mäenpää (1998). "Analytic program derivation in type
theory". In Giovanni Sambin; Jan M. Smith. Twenty-five years of
constructive type theory: proceedings of a congress held in Venice,
October 1995. Oxford University Press. p. 113. ISBN 0-19-850127-7 .
* ^ Celia Hoyles (Feb 1997). "The curricular shaping of students'
approach to proof". For the Learning of Mathematics. FLM Publishing
Association. 17 (1): 7–16.
JSTOR
JSTOR 40248217 .
REFERENCES
* Ball, W.W. Rouse (1960). A Short Account of the History of
Mathematics (4th ed. ed.). New York: Dover Publications. pp. 50–62.
ISBN 0-486-20630-0 .
* Coxeter, H.S.M. (1961). Introduction to Geometry. New York: Wiley.
* Eves, Howard (1963). A Survey of Geometry. Allyn and Bacon.
* Heath, Thomas L. (1956). The Thirteen Books of Euclid's Elements
(2nd ed. ed.). New York: Dover Publications.
(3 vols.): ISBN 0-486-60088-2 (vol. 1), ISBN 0-486-60089-0 (vol. 2),
ISBN 0-486-60090-4 (vol. 3). Heath's authoritative translation of
Euclid's Elements
Euclid's Elements plus his extensive historical research and detailed
commentary throughout the text.
* Misner, Thorne, and Wheeler (1973). Gravitation. W.H. Freeman. CS1
maint: Multiple names: authors list (link )
* Mlodinow (2001). Euclid's Window. The Free Press.
* Nagel, E.; Newman, J.R. (1958). Gödel's Proof. New York
University Press.
*
Alfred Tarski (1951) A Decision Method for Elementary
Algebra
Algebra and
Geometry. Univ. of California Press.
EXTERNAL LINKS
* Hazewinkel, Michiel , ed. (2001) , "Euclidean geometry",
Encyclopedia of Mathematics ,