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When a geometry is described by a set of axioms, the notion of a line is usually left undefined (a so-called primitive object). The properties of lines are then determined by the axioms which refer to them. One advantage to this approach is the flexibility it gives to users of the geometry.
Euclidean geometry is an example of synthetic geometry, ... For example, a Euclidean straight line has no width, but any real drawn line will have. ... or undefined ...
For example, the imaginary number is undefined within the set of real numbers. So it is meaningless to reason about the value, solely within the discourse of real numbers. However, defining the imaginary number i {\displaystyle i} to be equal to − 1 {\displaystyle {\sqrt {-1}}} , allows there to be a consistent set of mathematics referred to ...
Thus, a line segment AB defined as the points A and B and all the points between A and B in absolute geometry, needs to be reformulated. A line segment in this new geometry is determined by three collinear points A, B and C and consists of those three points and all the points not separated from B by A and C. There are further consequences.
Non-Euclidean geometry is an example of a scientific revolution in the history of science, in which mathematicians and scientists changed the way they viewed their subjects. [24] Some geometers called Lobachevsky the "Copernicus of Geometry" due to the revolutionary character of his work. [25] [26]
However, an open line segment is an open set in V if and only if V is one-dimensional. More generally than above, the concept of a line segment can be defined in an ordered geometry. A pair of line segments can be any one of the following: intersecting, parallel, skew, or none of these. The last possibility is a way that line segments differ ...
This is the same as Euclid's method of treating point and line as undefined primitive notions and axiomatizing their relationships. Great circles in many ways play the same logical role in spherical geometry as lines in Euclidean geometry, e.g., as the sides of (spherical) triangles.
In other words, the elements of geometry form a system which is not susceptible of extension, if we regard the five groups of axioms as valid. The old axiom V.2 is now Theorem 32. The last two modifications are due to P. Bernays. Other changes of note are: The term straight line used by Townsend has been replaced by line throughout.