page_title: Curve
section_title: History
hierarchical_section_title: Curve / History
caption_attribution_description: The entrance slab to Newgrange tomb, Ireland. This Irish megalithic stone, about 4500-5500 years old, is 10 feet by 4 feet in size and 5 tons in weight. It is one of the most famous rocks in Ireland and stands--about an hour from Dublin--at the entrance to Newgrange, a “passage tomb” 300 ft. in diameter and 40 ft. high that is older than the Egyptian pyramids and Stonehenge. The carved spiral design, though more ancient than the coming of the Celts to Ireland, was often copied by them and became one of the legendary Celtic designs.
context_page_description: In mathematics, a curve is an object similar to a line, but that does not have to be straight.
  Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that appeared more than 2000 years ago in Euclid's Elements: "The [curved] line is […] the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which […] will leave from its imaginary moving some vestige in length, exempt of any width."
  This definition of a curve has been formalized in modern mathematics as: A curve is the image of a continuous function from an interval to a topological space. In some contexts, the function that defines the curve is called a parametrization, and the curve is a parametric curve. In this article, these curves are sometimes called topological curves to distinguish them from more constrained curves such as differentiable curves. This definition encompasses most curves that are studied in mathematics; notable exceptions are level curves, and algebraic curves.
context_section_description: Interest in curves began long before they were the subject of mathematical study.  This can be seen in numerous examples of their decorative use in art and on everyday objects dating back to prehistoric
  times. Curves, or at least their graphical representations, are simple to create, for example by a stick in the sand on a beach.
  Historically, the term line was used in place of the more modern term curve. Hence the terms straight line and right line were used to distinguish what are today called lines from curved lines. For example, in Book I of Euclid's Elements, a line is defined as a "breadthless length" (Def. 2), while a straight line is defined as "a line that lies evenly with the points on itself" (Def. 4). Euclid's idea of a line is perhaps clarified by the statement "The extremities of a line are points," (Def. 3). Later commentators further classified lines according to various schemes. For example:
  Composite lines (lines forming an angle)
  Incomposite lines
  Determinate (lines that do not extend indefinitely, such as the circle)
  Indeterminate (lines that extend indefinitely, such as the straight line and the parabola)
  The Greek geometers had studied many other kinds of curves. One reason was their interest in solving geometrical problems that could not be solved using standard compass and straightedge construction.
  These curves include:
  The conic sections, studied in depth by Apollonius of Perga
  The cissoid of Diocles, studied by Diocles and used as a method to double the cube.
  The conchoid of Nicomedes, studied by Nicomedes as a method to both double the cube and to trisect an angle.
  The Archimedean spiral, studied by Archimedes as a method to trisect an angle and square the circle.
  The spiric sections, sections of tori studied by Perseus as sections of cones had been studied by Apollonius.
  A fundamental advance in the theory of curves was the introduction of analytic geometry by René Descartes in the seventeenth century. This enabled a curve to be described using an equation rather than an elaborate geometrical construction. This not only allowed new curves to be defined and studied, but it enabled a formal distinction to be made between algebraic curves that can be defined using polynomial equations, and transcendental curves that cannot. Previously, curves had been described as "geometrical" or "mechanical" according to how they were, or supposedly could be, generated.
  Conic sections were applied in astronomy by Kepler.
  Newton also worked on an early example in the calculus of variations. Solutions to variational problems, such as the brachistochrone and tautochrone questions, introduced properties of curves in new ways (in this case, the cycloid). The catenary gets its name as the solution to the problem of a hanging chain, the sort of question that became routinely accessible by means of differential calculus.
  In the eighteenth century came the beginnings of the theory of plane algebraic curves, in general. Newton had studied the cubic curves, in the general description of the real points into 'ovals'. The statement of Bézout's theorem showed a number of aspects which were not directly accessible to the geometry of the time, to do with singular points and complex solutions.
  Since the nineteenth century, curve theory is viewed as the special case of dimension one of the theory of manifolds and algebraic varieties. Nevertheless, many questions remain specific to curves, such as space-filling curves, Jordan curve theorem and Hilbert's sixteenth problem.
caption: Megalithic art from Newgrange showing an early interest in curves
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