<P> All definitions are ultimately circular in nature since they depend on concepts which must themselves have definitions, a dependence which cannot be continued indefinitely without returning to the starting point . To avoid this vicious circle certain concepts must be taken as primitive concepts; terms which are given no definition . In geometry, it is frequently the case that the concept of line is taken as a primitive . In those situations where a line is a defined concept, as in coordinate geometry, some other fundamental ideas are taken as primitives . When the line concept is a primitive, the behaviour and properties of lines are dictated by the axioms which they must satisfy . </P> <P> In a non-axiomatic or simplified axiomatic treatment of geometry, the concept of a primitive notion may be too abstract to be dealt with . In this circumstance it is possible that a description or mental image of a primitive notion is provided to give a foundation to build the notion on which would formally be based on the (unstated) axioms . Descriptions of this type may be referred to, by some authors, as definitions in this informal style of presentation . These are not true definitions and could not be used in formal proofs of statements . The "definition" of line in Euclid's Elements falls into this category . Even in the case where a specific geometry is being considered (for example, Euclidean geometry), there is no generally accepted agreement among authors as to what an informal description of a line should be when the subject is not being treated formally . </P> <P> When geometry was first formalised by Euclid in the Elements, he defined a general line (straight or curved) to be "breadthless length" with a straight line being a line "which lies evenly with the points on itself". These definitions serve little purpose since they use terms which are not, themselves, defined . In fact, Euclid did not use these definitions in this work and probably included them just to make it clear to the reader what was being discussed . In modern geometry, a line is simply taken as an undefined object with properties given by axioms, but is sometimes defined as a set of points obeying a linear relationship when some other fundamental concept is left undefined . </P> <P> In an axiomatic formulation of Euclidean geometry, such as that of Hilbert (Euclid's original axioms contained various flaws which have been corrected by modern mathematicians), a line is stated to have certain properties which relate it to other lines and points . For example, for any two distinct points, there is a unique line containing them, and any two distinct lines intersect in at most one point . In two dimensions, i.e., the Euclidean plane, two lines which do not intersect are called parallel . In higher dimensions, two lines that do not intersect are parallel if they are contained in a plane, or skew if they are not . </P>

A line which is not straight is called
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