“A differential is the difference between differences — Latin differentia meant distinction or difference, and the calculus differential measures how a quantity changes across an infinitely small interval.”
Latin differentia combined dis (apart) with ferre (to carry, to bear). Differentia was what carried two things apart — the distinction, the characteristic that made one thing different from another. Aristotle used the Greek equivalent (diaphora) in his logical system for the defining characteristic that distinguished species within a genus. The medieval scholastics adopted the Latin differentia as a technical logical term.
Gottfried Wilhelm Leibniz, one of the two independent inventors of calculus in the 1670s-1680s, developed the differential as a central concept: an infinitely small increment of a variable, denoted dx or dy. The ratio dy/dx — the differential of y with respect to x — measured the rate of change, the slope at a point. Leibniz's notation, rather than Newton's, became the standard because it made the algebraic manipulation of differentials intuitive.
The differential equation — an equation involving rates of change — became the fundamental language of physics after Newton. Newtonian mechanics, electromagnetism (Maxwell's equations), quantum mechanics, general relativity: all are formulated as differential equations. The universe, it turns out, is described more naturally in terms of how things change than in terms of what they are at any instant.
Today differential appears in calculus (the differential of a function), mechanics (differential gears, which allow wheels to turn at different speeds), and everyday language (the pay differential, the differential between prices). All uses carry the Latin distinction: the difference between things, and the measurement of that difference.
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The differential equation is the language in which the universe speaks to physicists. From the orbit of satellites to the spread of epidemics to the flow of heat, the natural world is described most precisely by how things change with respect to other things. The differential is not a description of what is; it is a description of what is becoming.
Leibniz's notation — dx, dy, d/dt — was a gift to physics precisely because it treated the differential as a manipulable quantity. The infinitely small difference between moments is the hinge on which all of mathematical physics turns.
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