Where does line collide?

In Euclid's Elements (c. 300 BCE) a line is defined as breadthless length, an idealized figure that has extension in one direction only, and a straight line is one that lies evenly with the points on itself. Hilbert's 1899 Grundlagen der Geometrie recast the line as an undefined primitive fixed only by axioms about how points and lines relate, rather than by a picture. The referent is a purely geometric object with length and no thickness, living in an abstract space. That is unlike a dark band in a spectrum, an income threshold in economics, or a drawn mark on a surface.[1][2]

Unit

An idealized one-dimensional figure with length but no breadth

Recovery means

Not applicable

Measured by

Defined axiomatically rather than measured; specified by two points or by an equation

A line here is a narrow feature in a spectrum, a dark or bright band at one wavelength where a substance absorbs or emits light. Fraunhofer (1817) mapped hundreds of such dark lines in sunlight and labeled the strongest of them, and Kirchhoff and Bunsen (1860) showed that each line corresponds to a chemical element, founding the method of spectral analysis. The referent is a measurable feature of radiation, located by its wavelength and carrying information about the matter that produced it. It is not the length-without-breadth figure of geometry, a threshold of income, or an element of drawing.[3][4]

Unit

A feature at a specific wavelength in a spectrum

Recovery means

Not applicable

Measured by

Recorded by a spectrograph and located by wavelength, with its width and depth read from the spectrum

A line here is the poverty line, a threshold of income or consumption below which a person or household is counted as poor. Rowntree's 1901 Poverty: A Study of Town Life set an early operational line from the cost of a minimum diet and other necessities, and Ravallion, Chen, and Sangraula (2009) refined the international line used for counting global poverty. The referent is a monetary cutoff, stated in a given currency and period and used to classify and count people. It shares nothing with a geometric figure, a band in a spectrum, or a mark in a drawing beyond the word.[5][6]

Unit

A currency threshold per person per period

Recovery means

Not applicable

Measured by

Set from the cost of a minimum basket of goods, or as a fraction of median income, in a stated currency and time

A line here is a fundamental element of visual art, a mark that has length and direction and that the eye follows across a surface. Kandinsky's 1926 Punkt und Linie zu Fläche (translated as Point and Line to Plane) gave the classic theoretical account of the line at the Bauhaus, treating it as the trace of a moving point and analyzing its tensions and directions. The referent is a perceptual and compositional mark, described by its length, direction, weight, and character rather than read off a scale. That differs from a geometric primitive, a spectral feature, or an income threshold.[7]

Unit

A single drawn or implied mark, taken by its length and direction

Recovery means

Not applicable

Measured by

Described qualitatively by length, direction, weight, and character rather than by a fixed scale