EUCLIDEAN GEOMETRY is a mathematical system attributed to the
Alexandrian Greek mathematician
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
CONTENTS * 1 The _Elements_ * 1.1
* 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
* 6 Applications * 7 As a description of the structure of space * 8 Later work * 8.1
* 9 Treatment of infinity * 9.1 Infinite objects * 9.2 Infinite processes * 10 Logical basis * 10.1
* 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
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 (
"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:
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.
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_ 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
SYSTEM OF MEASUREMENT AND ARITHMETIC
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
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 .
Angles whose sum is a straight angle are supplementary .
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
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
THALES\' THEOREM Thales\' theorem , named after
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}} .
APPLICATIONS _ THIS SECTION NEEDS EXPANSION. You can help by adding to it . (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 *
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 .
*
The water tower consists of a cone, a cylinder, and a hemisphere. Its volume can be calculated using solid geometry. *
AS A DESCRIPTION OF THE STRUCTURE OF SPACE
As discussed in more detail below,
The ambiguous character of the axioms as originally formulated by
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
17TH CENTURY: DESCARTES
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
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_ = 2_x_ + 1 (a line), or _x_2 + _y_2 = 7 (a circle). Also in the 17th century,
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
Euler discussed a generalization of
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,
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,
20TH CENTURY AND GENERAL RELATIVITY A disproof of
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
TREATMENT OF INFINITY INFINITE OBJECTS
The notion of infinitesimal quantities had previously been discussed
extensively by the
Later ancient commentators such as
At the turn of the 20th century,
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.
LOGICAL BASIS _ 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 2010)_ See also: Hilbert\'s axioms ,
CLASSICAL LOGIC
MODERN STANDARDS OF RIGOR Placing
...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
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
* Euclid's axioms: In his dissertation to Trinity College,
Cambridge,
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
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 *
CLASSICAL THEOREMS *
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
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
* 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. ( |