Euclidean geometry is a mathematical system due to the Hellenistic mathematician Euclid of Egypt. Euclid's text Elements was the first complete and systematic discussion of geometry, and has been one of the most influential books in history, not just because of its mathematical content but because of its method of assuming a small set of intuitively appealing axioms, and then proving theorems from them by well-defined logical rules. Although many of Euclid's results had been given by earlier Greek mathematicians, Euclid was the first to show how they could be fitted together in a comprehensive logical system.
The Elements begin by describing plane geometry, which is the kind of geometry usually taught in secondary school, and is most young people's first introduction to the notions of axiomatic systems and formal proof. The later books of the Elements extend the system to three dimensions as solid geometry, and it can even be extended to more than three.
For over two thousand years, the adjective "Euclidean" was unnecessary, because nobody could conceive of any other system of geometry. Euclid's axioms seemed so intuitively obvious that any theorem proved from them was believed to have been established as true in an absolute sense. However, other self-consistent theories of geometry were later constructed, and we now know from Einstein's theory of general relativity that Euclid's axioms are only approximately correct as a description of physical space, in the limit of weak gravitational fields.
Axiomatic approach
The traditional presentation of Euclidean geometry is as an axiomatic system, setting out to prove all the "true statements" as theorems in geometry from a set containing a finite number of axioms.
Near the beginning of the first book of the Elements, Euclid gives five postulates (axioms):
- Any two points can be joined by a straight line.
- Any straight line segment can be extended indefinitely in a straight line.
- Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
- All right angles are congruent.
- If two lines are drawn which 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.
The third and fifth postulates fail in three dimensions (the third postulate because the circle is implied to be unique), so this is only a description of plane geometry.
A proof from Euclid's elements that, given a line segment, a 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.
The fifth postulate is called the parallel postulate, which leads to the same geometry as the following statement (note that it is formulated for two-dimensional geometry):
- Through a point not on a given straight line, one and only one line can be drawn that never meets the given line.
The first, second, third, and fifth postulates are statements of the existence and uniqueness of certain geometric figures, and these statements are of a constructive nature: that is, we are told not only that certain things exist, but we are given methods for creating them that could be carried out with a compass and an unmarked straightedge. In this sense, Euclidean geometry is more concrete than many modern axiomatic systems such as set theory, which often assert the existence of objects without saying how to construct them, or even assert the existence of objects that provably cannot be constructed within the theory.
Relatively recently, it was realized that Euclid's five axioms were incomplete. For example, Euclid assumed implicitly that any line contains at least two points, but this assumption cannot be proved from the other axioms, and therefore needs to be an axiom itself. The very first geometric proof in the Elements, shown in the figure on the right, 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. Many revised systems of axioms were constructed, the most standard ones being Hilbert's axioms, Birkhoff's axioms, and Tarski's axioms.
Euclid also had five "common notions," though he later used other properties of magnitudes.
- Things which equal the same thing also equal one another.
- If equals are added to equals, then the wholes are equal.
- If equals are subtracted from equals, then the remainders are equal.
- Things which coincide with one another equal one another.
- The whole is greater than the part.
Some of these, such as 1, are now recognized as axioms of logical reasoning; others, such as 2, as axioms of arithmetic; and still others, such as 5, as being too vague to serve as rigorous mathematical principles.
The parallel postulate
To the ancients, the parallel postulate seemed less obvious than the others; verifying it physically would require us to inspect two lines to check that they never intersected, even at some very distant point, and this inspection could potentially take an infinite amount of time.[1] 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 geometers tried in vain to prove the fifth postulate from the first four. By 1763 at least 28 different proofs of the fifth postulate had been published, but all were found to be incorrect. [2] In the 19th century it was shown that this could not be done, by constructing alternative systems of non-Euclidean geometry, in which the parallel postulate is false, while the other axioms hold. (If one simply drops the parallel postulate from the list of axioms then the result is the more general geometry called absolute geometry.) One consequence of omitting the parallel postulate is that the three angles of a triangle do not necessarily add to 180°. In hyperbolic geometry the sum of the three angles are always less than 180° and can approach zero, while in elliptic geometry the sum is greater than 180°.
Treatment using analytic geometry
The development of analytic geometry provided an alternative method for formalizing geometry. In this approach, a point is represented by its Cartesian (x,y) coordinates, a line is represented by its equation, and so on. In the 20th century, this fit into David Hilbert's program of reducing all of mathematics to arithmetic, and then proving the consistency of arithmetic using finitistic reasoning. In Euclid's original approach, the 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 to be theorems. The equation
defining the distance between two points P = (p,q) and Q = (r,s) is then known as the Euclidean metric, and other metrics define non-Euclidean geometries.
A disproof of Euclidean geometry as a description of physical space. In a 1919 test of the general theory of relativity, stars (marked with 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.
As a description of physical reality
Euclid believed that his axioms were self-evident statements about physical reality. However, Einstein's theory of general relativity shows that the true geometry of spacetime is non-Euclidean. 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 detecting the deviations from Euclidean geometry, but Einstein predicted that such deviations would exist. They were later verified by observations such as the observation of the slight bending of starlight by a solar eclipse in 1919, and non-Euclidean geometry is now, for example, an integral part of the software that runs the GPS system. It is possible to object to the non-Euclidean interpretation of general relativity on the grounds that light rays might be improper physical models of Euclid's lines, or that relativity could be rephrased so as to avoid the geometrical interpretations. However, one of the consequences of Einstein's theory is that there is no possible physical test that can do any better than a beam of light as a model of geometry. 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 meaning.
Logical status
Euclidean geometry is a first-order theory. That is, it allows statements that begin like "for all triangles ...," but it is incapable of forming statements such as "for all sets of triangles ..." Statements of the latter type are deemed to be outside the scope of the theory, just like a statement such as "chocolate is tasty."
Godel's theorem showed the futility of Hilbert's program of proving the consistency of all of mathematics using finitistic reasoning. However, Tarski used his axiomatic formulation of Euclidean geometry to show that it is not just consistent but also complete in a certain sense: that is, every proposition of Euclidean geometry can be shown to be either true or false. This does not violate Godel's theorem, since Euclidean geometry cannot describe a sufficient amount of arithmetic for the theorem to apply.[3] Thus, although Hilbert thought Euclidean geometry could be put on a firmer foundation by rewriting it in terms of arithmetic, in fact Euclidean geometry is complete and consistent in a way that Godel's theorem tells us arithmetic can never be.
Although complete in the formal sense used in modern logic, Euclidean geometry is incomplete in the sense that there are things that we would naturally like to be able to accomplish using the the theory, but that we cannot. 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, and Évariste Galois proved in the 19th century that such a construction was impossible.
Because Euclidean geometry is consistent, it follows that the removal of the parallel postulate to form absolute geometry also results in a consistent theory. If we then add an alternative to the parallel postulate to create a non-Euclidean geometry, we still have a consistent theory, because there are Euclidean models of non-Euclidean geometry. For example, geometry on the surface of a sphere is a model of an elliptical geometry, carried out within a self-contained subset of a three-dimensional Euclidean space.
Classical theorems
See also
External links
The Elements online
- a bilinguial edition (typset in PDF format, with the original Greek and an English translation on facing pages; free in PDF form, available in print)
- in English (HTML, with the figures in the form of Java applets that the user can manipulate)
- Heath's translation (HTML, without the figures, public domain)
- in ancient Greek (typeset in PDF format, public domain)
- Oliver Byrne's 1847 edition - an unusual version using color rather than labels such as ABC (scanned page images, public domain)
- Reading Euclid - a course in how to read Euclid in the original Greek, with English translations and commentaries (HTML with figures)
Notes
- ^ For the assertion that this was the historical reason for the ancients considering the parallel postulate less obvious than the others, see p. 9 of E. Nagel and J.R. Newman, Godel's Proof, New York University Press, 1958.
- ^ Douglas R. Hofstadter, Godel, Escher, Bach: An Eternal Golden Braid, New York, Basic Books, 1979, p. 91.
- ^ Torkel Franzén, Gödel's Theorem: An Incomplete Guide to its Use and Abuse, AK Peters (2005), ISBN 1568812388
The content of this page is retrieved from http://en.wikipedia.org/wiki/Euclidean_geometry under GFDL
