Euclid's axioms are five postulates about the behavior of geometric objects; they constitute the foundation upon which Euclid built the entire edifice of geometry that is known today as Euclidean geometry. Before Euclid wrote his famous book The Elements around 300 BC, many geometric ideas were well understood, but in a disorganized way that obscured their logical structure. It was often unclear which facts depended on which others for their proofs, and this vagueness opened the door to circular reasoning and other logical errors. To systematize the study of geometry, Euclid formulated five axioms, statements so simple he considered them self-evident, and then attempted to prove all other geometric facts using only these five axioms and the principles of logical reasoning. Euclid's analysis was so definitive and far-reaching that it laid the foundation for the study of geometry for the next 2000 years. Still more, it took the groundbreaking step of subjecting mathematics to the rigors of logic, making The Elements one of the milestones in the history of human thought.
Euclid realized that it is impossible to prove anything without starting with a few basic assumptions, or axioms; the ideal in mathematics is to start with as few and simple axioms as possible, and prove as many statements from the axioms as possible. Euclid limited himself to the following five axioms:
First, any two points can be connected by one and only one straight line.
Second, any line segment is contained in a full (infinitely long) line.
Third, given a point and a line segment starting at the point, there is a circle that has the given point as its center and the given line segment as a radius.
Fourth, all right angles are equal to each other (Euclid defines a right angle to be the angle formed when two lines intersect each other perpendicularly, that is, forming equal angles on both sides of the intersection).
Fifth (known as Euclid's parallel postulate), given a line and a point P that is not on the line, there is one and only one line through P that never meets the original line.
The fifth is the most controversial of his assumptions, and it has been framed in many different ways; first by the mathematician Proclus in the 5th century.
Using these five axioms, Euclid was able to prove, for example, that two triangles that have all equal side-lengths are congruent; that two triangles that have all equal angles are similar; that a tangent line to a circle is perpendicular to the diameter that it intersects; and that in a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse, the famous Pythagorean theorem.
Of Euclid's five axioms, the fifth is conspicuously more complicated than the others. Through the centuries many believed, therefore, that the fifth axiom was not fundamental enough to be one of the basic axioms, and should instead be provable using the other four. Euclid himself avoided any use of the fifth axiom in the proofs of the first 28 propositions of The Elements. This evasion has led some historians to the conclusion that even Euclid felt the fifth axiom to be less natural than the others. Many mathematicians, first the Greeks, then the Arabs in the Middle Ages, then the Europeans in the Renaissance, tried to show that the fifth axiom was a consequence of the others, but with no success. In the middle of the 19th century, the mathematicians Nicolay Lobachevsky, János Bolyai and Eugenio Beltrami independently made a further attempt, using a technique that mathematicians call reductio ad absurdum (see proof by reductio ad absurdum): to prove that something is true, assume the opposite and follow a logical line of arguments until you reach something ridiculous. So they assumed that, given a line and a point P not on the line, it is possible to have more than one line through P that never meets the original line. With this assumption, they built up a framework of logical consequences, waiting for an absurd conclusion to emerge. There was just one problem: the absurd conclusion failed to appear. Using the strict reasoning of pure logic, they constructed an alternative geometry in which Euclid's first four axioms are true, but the fifth is not. Thus, the quest to prove that the fifth axiom was a consequence of the first four led instead to a discovery that astonished the mathematical world: an entirely new, counterintuitive geometry known as hyperbolic geometry. The geometry of Euclid's five axioms is now called Euclidean geometry.
The existence of hyperbolic geometry in no way undermines Euclid's geometry. Euclidean geometry remains as consistent as it ever was; it is simply not the only consistent geometry, as was originally believed. This consistency is not quite perfect, however. Mathematicians have long been aware that Euclid's systemization contains many small flaws--uses of vaguely defined terms and hidden propositions that had not yet been proven. In 1899 the famous mathematician David Hilbert set out to correct these gaps with a more complete system of axioms; his system has the defect, however, that it has many axioms and is very complex, making the proofs of even the simplest propositions extremely cumbersome. Euclid's system remains unrivaled in its simplicity and power. The elegance and logical strength of his arguments have led many to regard The Elements as the pinnacle of pure reasoning.
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