Where does division collide?

Christopher Simpson's 1659 The division-violist set out the art of playing divisions: taking a slow repeating bass line, called a ground, and spinning rapid melodic figures above it so that each long note is broken into many shorter ones. In this seventeenth-century English practice a division is an ornamented variation, improvised or written out, over that fixed ground. The referent is a musical technique of melodic elaboration, counted in notes over a bass, and it has nothing to do with splitting work among laborers, an algebraic operation, or a cell separating into two.[1]

Unit

A passage of divisions played over one repeating ground bass

Recovery means

Not applicable

Measured by

Described by note values, the ground it ornaments, and whether it is improvised or composed

Adam Smith's 1776 An inquiry into the nature and causes of the wealth of nations opened with the division of labor: breaking a job such as making a pin into many specialized steps performed by different workers, which raises output. Emile Durkheim's 1893 De la division du travail social later recast it as a source of social solidarity, binding people together through their interdependence. The referent is the organization of productive work into specialized roles, and it is unrelated to ornamenting a bass line, an algebraic structure, or a biological cell dividing.[2][3]

Unit

A task or role assigned to a specialized worker within a production process

Recovery means

Not applicable

Measured by

Assessed by the number of distinct tasks, the degree of specialization, and output per worker

William Rowan Hamilton's 1844 paper on quaternions built the first number system beyond the real and complex numbers in which division still works, meaning every nonzero quantity has an inverse to divide by. Joseph Wedderburn's 1905 A theorem on finite algebras later proved that any finite such system is in fact commutative, sharpening the modern idea of a division ring. The referent is an algebraic structure in which division is always defined, and it shares nothing with a musician's variations, the specialized splitting of labor, or a living cell splitting in two.[4][5]

Unit

A ring in which every nonzero element has a multiplicative inverse

Recovery means

Not applicable

Measured by

Established by proving each nonzero element is invertible and stating the structure's axioms

Walther Flemming's 1882 Zellsubstanz, Kern und Zelltheilung described in detail how a cell splits in two and coined the term mitosis for the orderly separation of its threadlike chromosomes. Paul Nurse's 2000 A long twentieth century of the cell cycle and beyond reviewed how this division is controlled and timed across the life of a cell. The referent is the biological process by which one cell becomes two daughter cells, and it is not a musical variation, the specialization of work, or an algebraic operation.[6][7]

Unit

One parent cell dividing into two or more daughter cells

Recovery means

Chromosomes decondense and nuclear envelopes reform in each daughter cell after separation

Measured by

Counted as a division rate and observed by microscopy of the stages from one cell into two