Euclidean geometry is a mathematical system attributed to the
Alexandrian Greek mathematician 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
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'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 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
1 The Elements
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
Triangle angle sum
5.4 Pythagorean theorem
5.5 Thales' theorem
5.6 Scaling of area and volume
7 As a description of the structure of space
8 Later work
Archimedes and Apollonius
8.2 17th century: Descartes
8.3 18th century
8.4 19th century and non-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
14 External links
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 "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,
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
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 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 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 [extend] a finite straight line continuously in a straight
"To describe a circle with any centre and distance [radius]."
"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
The whole is greater than the part.
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
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
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 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
Methods of proof
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
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 often used proof by contradiction.
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
System of measurement and arithmetic
Euclidean geometry has two fundamental types of measurements: angle
and distance. The angle scale is absolute, and
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
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
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
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 could be
either straight or curved, and he used the more specific term
"straight line" when necessary.
Some important or well known results
Pons Asinorum or Bridge of Asses theorem states that in an
isosceles triangle, α = β and γ = δ.
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
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.
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 (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, named after
Thales of Miletus
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 Book I, Prop. 32 after
the manner of
Euclid Book III, Prop. 31. 
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,
displaystyle Apropto L^ 2
, and the volume of a solid to the cube,
displaystyle Vpropto L^ 3
Euclid proved these results in various special cases such as the
area of a circle and the volume of a parallelepipedal solid.
Euclid determined some, but not all, of the relevant constants of
proportionality. E.g., it was his successor
Archimedes who proved that
a sphere has 2/3 the volume of the circumscribing cylinder.
This section needs expansion. You can help by adding to it. (March
Because of Euclidean geometry's fundamental status in mathematics, it
is impractical to give more than a representative sampling of
A surveyor uses a level
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 to analyze the focusing of
light by lenses and mirrors.
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 can be used to design origami.
Geometry is used extensively in architecture.
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 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
As discussed in more detail below, Einstein's theory of relativity
significantly modifies this view.
The ambiguous character of the axioms as originally formulated by
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
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
at his request.
Archimedes (ca. 287 BCE – ca. 212 BCE), a colorful figure about whom
many historical anecdotes are recorded, is remembered along with
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
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, 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 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 theorems.
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
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 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
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
Euler discussed a generalization of
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
The century's most significant development in geometry occurred when,
János Bolyai and Nikolai Ivanovich Lobachevsky
separately published work on non-Euclidean geometry, in which the
parallel postulate is not valid. Since non-
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 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 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.
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 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
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
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 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
Treatment of infinity
Euclid sometimes distinguished explicitly between "finite lines"
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
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, 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.
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
Supposed paradoxes involving infinite series, such as Zeno's paradox,
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
This article needs attention from an expert in mathematics. Please add
a reason or a talk parameter to this template to explain the issue
with the article. WikiProject Mathematics may be able to help recruit
an expert. (December 2010)
This section needs expansion. You can help by adding to it. (June
See also: Hilbert's axioms, Axiomatic system, and Real closed field
Euclid frequently used the method of proof by contradiction, and
therefore the traditional presentation of
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
Modern standards of rigor
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
...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
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
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:
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
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 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 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 rigorous (avoiding hidden
assumptions) and to make clear the ramifications of the parallel
Birkhoff's axioms: Birkhoff proposed four postulates for 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
Alfred Tarski (1902–1983) and his students defined
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 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 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
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.
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
Cartesian coordinate system
List of interactive geometry software
Angle bisector theorem
^ 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 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,
^ 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
^ 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 and Geometric Constructions".
^ Richard J. Trudeau (2008). "Euclid's axioms". The Non-Euclidean
Revolution. Birkhäuser. pp. 39 ff.
^ 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
(2004). Elementary Mathematics from an Advanced Standpoint: Geometry
(Reprint of 1939 Macmillan Company ed.). Courier Dover. p. 167.
^ Roger Penrose (2007). The Road to Reality: A Complete Guide to the
Laws of the Universe. Vintage Books. p. 29.
^ 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,
^ Eves (1963), p. 64
^ Ball, p. 485
^ * Howard Eves, 1997 (1958). Foundations and Fundamental Concepts of
^ 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.
^ 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.
^ Société française de philosophie (1900). Revue de métaphysique
et de morale,
Volume 8. Hachette. p. 592.
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.
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 (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.
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 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.
^ Eric W. Weisstein (2003). "Euclid's postulates". CRC concise
encyclopedia of mathematics (2nd ed.). CRC Press. p. 942.
^ Deborah J. Bennett (2004). Logic made easy: how to know when
language deceives you. W. W. Norton & Company. p. 34.
^ AN Kolmogorov; AF Semenovich; RS Cherkasov (1982). Geometry: A
textbook for grades 6–8 of secondary school [Geometriya. Uchebnoe
posobie dlya 6–8 klassov srednie shkoly] (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
^ Viktor Vasilʹevich Prasolov; Vladimir Mikhaĭlovich Tikhomirov
(2001). Geometry. AMS Bookstore. p. 198.
^ 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.
^ 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 40248217.
Ball, W.W. Rouse (1960). A Short Account of the History of Mathematics
(4th ed. [Reprint. Original publication: London: Macmillan & Co.,
1908] ed.). New York: Dover Publications. pp. 50–62.
Coxeter, H.S.M. (1961). Introduction to Geometry. New York:
Eves, Howard (1963). A Survey of Geometry. Allyn and Bacon.
Heath, Thomas L. (1956). The Thirteen Books of
Euclid's Elements (2nd
ed. [Facsimile. Original publication: Cambridge University Press,
1925] ed.). New York: Dover Publications. In 3 vols.: vol. 1
ISBN 0-486-60088-2, vol. 2 ISBN 0-486-60089-0, vol. 3
ISBN 0-486-60090-4. Heath's authoritative translation of 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
Alfred Tarski (1951) A Decision Method for Elementary
Geometry. Univ. of California Press.
Hazewinkel, Michiel, ed. (2001) , "Euclidean geometry",
Encyclopedia of Mathematics, Springer Science+Business Media B.V. /
Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Hazewinkel, Michiel, ed. (2001) , "Plane trigonometry",
Encyclopedia of Mathematics, Springer Science+Business Media B.V. /
Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Geometry Unbound (a treatment using analytic geometry;
PDF format, GFDL licensed)
Major topics in Geometry
List of mathematical shapes
List of geometry topics
List of differenti