Where does tangent collide?

In geometry a tangent is a straight line that meets a curve at a single point and, near that point, grazes alongside it rather than crossing it. Euclid's Elements (about 300 B.C.E.) gave the first definition, for a line that touches a circle without cutting it (Book III, Definition 2), and the idea was later extended to any smooth curve through the tangent methods of Fermat and Leibniz, a history traced by Coolidge (1951). One such line exists at each point of a well-behaved curve, and its slope is the curve's rate of change at that point. This referent is a geometric object, a line in the plane, not a numerical ratio in a triangle, a striking blade in a keyboard, or a pattern of off-point speech.[1][2]

Unit

One line touching a given curve at a single point

Recovery means

Not applicable

Measured by

Constructed geometrically or found as the limit of secant lines, with its slope given by the derivative

In trigonometry the tangent is a ratio attached to an angle: in a right triangle it is the length of the side opposite the angle divided by the length of the side next to it. Thomas Fincke's Geometriae rotundi libri XIIII (1583) introduced the Latin name tangens for this quantity, which earlier writers had called the turned shadow, and Van Brummelen (2009) traces its longer history in astronomy and surveying. The value has no upper bound and repeats every half turn, and it equals the length of the segment cut on the line touching a unit circle, which is why it carries the same name as the geometric tangent. This referent is a number that varies with an angle, not a line drawn against a curve, a part of a musical instrument, or a way of speaking.[3][4]

Unit

A pure ratio for a given angle, opposite over adjacent

Recovery means

Not applicable

Measured by

Computed from an angle's coordinates or side lengths, or read from tables and calculators

In the clinical assessment of thought and language, tangentiality is a pattern of speech in which a person answers a question in an oblique or irrelevant way, drifting off the point so that the reply never comes back to it. Andreasen's Thought, Language, and Communication Disorders (1979) gave the term a standardized definition within a rated set of speech features used to study conditions such as schizophrenia. It is judged from what a person actually says in reply, and it describes how the reply relates to the question rather than any single word. This referent is a described feature of human communication, not a line touching a curve, a trigonometric ratio, or a blade in a clavichord.[5]

Unit

One rated instance of an off-point reply in speech

Recovery means

Not applicable

Measured by

Rated by trained clinicians from observed or recorded speech using defined criteria