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
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 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[edit]
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
The parallel postulate (
"To draw a straight line from any point to any point." "To produce [extend] a finite straight line continuously in a straight line." "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 (Reflexive Property). The whole is greater than the part. Parallel postulate[edit]
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.[7]
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
Methods of proof[edit]
Euclidean
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.
The
The
The
Pons Asinorum[edit] 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.[12] 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.[13] Congruence of triangles[edit] 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.
A ∝ L 2 displaystyle Apropto L^ 2 , and the volume of a solid to the cube, V ∝ L 3 displaystyle Vpropto L^ 3 .
This section needs expansion. You can help by adding to it. (March 2009) Because of Euclidean geometry's fundamental status in mathematics, it is impractical 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,[20] 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.[21] 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.
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.
René Descartes. Portrait after Frans Hals, 1648. 17th century: Descartes[edit]
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
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[edit]
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.[31]
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
A disproof of
Einstein's theory of general relativity shows that the true geometry
of spacetime is not Euclidean geometry.[38] 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.[39] 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-
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, Axiomatic system, and Real closed field
Classical logic[edit]
...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:[48] 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[edit]
Euclid's axioms: In his dissertation to Trinity College, Cambridge,
Constructive approaches and pedagogy[edit] 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.[57] 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.[57] The contrast in approach may be summarized:[58] 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,[59] and his arrival upon it by the method of contradiction.[60] 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[edit] 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[edit]
Notes[edit] ^ 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[edit] 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.
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
External links[edit] Hazewinkel, Michiel, ed. (2001) [1994], "Euclidean geometry",
Encyclopedia of Mathematics, Springer Science+Business Media B.V. /
Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Hazewinkel, Michiel, ed. (2001) [1994], "Plane trigonometry",
Encyclopedia of Mathematics, Springer Science+Business Media B.V. /
Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Kiran Kedlaya,
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