![]() The Elements begin with plane geometry, still often taught in secondary school as the first axiomatic system and the first examples of formal proof. The Elements goes on to the solid geometry of three dimensions, and Euclidean geometry was subsequently extended to any finite number of dimensions. Much of the Elements states results of what is now called number theory, proved using geometrical methods.įor over two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious that any theorem proved from them was deemed true in an absolute sense. Many other consistent formal geometries are now known, the first ones being discovered in the early 19th century. It also is no longer taken for granted that Euclidean geometry describes physical space. Near the beginning of the first book of the Elements, Euclid gives five postulates (axioms): An implication of Einstein's theory of general relativity is that Euclidean geometry is only a good approximation to the properties of physical space if the gravitational field is not too strong.įollowing a precedent set in the Elements, Euclidean geometry has been exposited as an axiomatic system, in which all theorems ("true statements") are derived from a finite number of axioms. ![]() Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as centre.Any straight line segment can be extended indefinitely in a straight line.Īny two points can be joined by a straight line. 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. ![]() These axioms invoke the following concepts: point, straight line segment and line, side of a line, circle with radius and centre, right angle, congruence, inner and right angles, sum. The following verbs appear: join, extend, draw, intersect. ![]() The circle described in postulate 3 is tacitly unique. Postulates 3 and 5 hold only for plane geometry in three dimensions, postulate 3 defines a sphere.Ī 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. Postulate 5 leads to the same geometry as the following statement, known as Playfair's axiom, which also holds only in the plane: Through a point not on a given straight line, one and only one line can be drawn that never meets the given line. 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. Things that coincide with one another equal one another.Įuclid also invoked other properties pertaining to magnitudes.If equals are subtracted from equals, then the remainders are equal.If equals are added to equals, then the wholes are equal.Things that equal the same thing also equal one another.The Elements also include the following five "common notions": 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 probably cannot be constructed within the theory. 1 is the only part of the underlying logic that Euclid explicitly articulated. 2 and 3 are "arithmetical" principles note that the meanings of "add" and "subtract" in this purely geometric context are taken as given. "Whole," "part," and "remainder" beg for precise definitions.ġ through 4 operationally define equality, which can also be taken as part of the underlying logic or as an equivalence relation requiring, like "coincide," careful prior definition.
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