Tangent

In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that 'just touches' the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve y = f(x) at a point x = c on the curve if the line passes through the point (c, f(c)) on the curve and has slope f'(c), where f' is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space. In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that 'just touches' the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve y = f(x) at a point x = c on the curve if the line passes through the point (c, f(c)) on the curve and has slope f'(c), where f' is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space. As it passes through the point where the tangent line and the curve meet, called the point of tangency, the tangent line is 'going in the same direction' as the curve, and is thus the best straight-line approximation to the curve at that point. Similarly, the tangent plane to a surface at a given point is the plane that 'just touches' the surface at that point. The concept of a tangent is one of the most fundamental notions in differential geometry and has been extensively generalized; see Tangent space. The word 'tangent' comes from the Latin tangere, 'to touch'. Euclid makes several references to the tangent (ἐφαπτομένη ephaptoménē) to a circle in book III of the Elements (c. 300 BC). In Apollonius work Conics (c. 225 BC) he defines a tangent as being a line such that no other straight line couldfall between it and the curve. Archimedes (c.  287 – c.  212 BC) found the tangent to an Archimedean spiral by considering the path of a point moving along the curve. In the 1630s Fermat developed the technique of adequality to calculate tangents and other problems in analysis and used this to calculate tangents to the parabola. The technique of adeqality is similar to taking the difference between f ( x + h ) {displaystyle f(x+h)} and f ( x ) {displaystyle f(x)} and dividing by a power of h {displaystyle h} . Independently Descartes used his method of normals based on the observation that the radius of a circle is always normal to the circle itself. These methods led to the development of differential calculus in the 17th century. Many people contributed. Roberval discovered a general method of drawing tangents, by considering a curve as described by a moving point whose motion is the resultant of several simpler motions.René-François de Sluse and Johannes Hudde found algebraic algorithms for finding tangents. Further developments included those of John Wallis and Isaac Barrow, leading to the theory of Isaac Newton and Gottfried Leibniz. An 1828 definition of a tangent was 'a right line which touches a curve, but which when produced, does not cut it'. This old definition prevents inflection points from having any tangent. It has been dismissed and the modern definitions are equivalent to those of Leibniz who defined the tangent line as the line through a pair of infinitely close points on the curve.