# Fluents and Fluxions

When calculus was first being developed, the terms "fluent" and "fluxion" appeared quite often in the Newtonian works.

I am wanting to know the etymology behind these words. I assume that "fluents" are related to the concept of "fluid" or "flowing", but (as far as I am aware) the word has gone out of usage in mathematics.

"Fluxions" are not used in modern-day texts, but I assume that "flux" (a term with which every physics student will be familiar) derives from "fluxion".

If context helps, one would usually talk of a "fluent" as a moving point, generating a curve, and a "fluxion" as its velocity (direction + speed). I am interested in knowing why these terms were used, and what they referred to.

EDIT: On a related note, Newton writes "Fluxions of quantities are in the first ratio of their nascent parts or, what is exactly the same, in the last ratio of those parts as they vanish by defluxion." What would "defluxion" refer to?

I have found this:

Fluxion, in mathematics, the original term for derivative , introduced by Isaac Newton in 1665. Newton referred to a varying (flowing) quantity as a fluent and to its instantaneous rate of change as a fluxion. Newton stated that the fundamental problems of the infinitesimal calculus were: (1) given a fluent (that would now be called a function), to find its fluxion (now called a derivative); and, (2) given a fluxion (a function), to find a corresponding fluent (an indefinite integral). Thus, if y = x3, the fluxion of the quantity y equals 3x2 times the fluxion of x; in modern notation, dy/dt = 3x2(dx/dt). Newton’s terminology and notations of fluxions were eventually discarded in favour of the derivatives and differentials that were developed by G.W.

Etymology

Fluent (adj.) 1580s, "flowing freely" (of water, also of speech), from Latin fluentem(nominative fluens) "lax, relaxed," figuratively "flowing, fluent," present participle of fluere "to flow, stream, run, melt," from PIE *bhleugw-, extended form of *bhleu- "to swell, well up, overflow" (cognates: Latin flumen "river;" Greek phluein "to boil over, bubble up," phlein "to abound"), an extension of root *bhel- (2) "to blow, inflate, swell;" see bole. Used interchangeably with fluid in Elizabethan times. Related: Fluently.

Flux (n.) late 14c., from Old French flus "flowing, rolling, bleeding," or directly from Latin fluxus "flowing, loose, slack," past participle of fluere "to flow" (see fluent). Originally "excessive flow" (of blood or excrement); an early name for "dysentery;" sense of "continuous succession of changes" is first recorded 1620s. The verb is early 15c., from the noun.

• Interesting, thank you for adding the etymology. "Flux" in mathematics typically refers to flow across a region, or through a region. Do you see any reasons that this might make sense etymologically? Commented May 16, 2014 at 22:25
• Well, it appears that mathematicians adopted this term for its original meaning.
– user66974
Commented May 16, 2014 at 22:32

According to the Encyclopaedia Britannica, fluxion was the original name for derivatives:

fluxion, in mathematics, the original term for derivative, introduced by Isaac Newton in 1665. Newton referred to a varying (flowing) quantity as a fluent and to its instantaneous rate of change as a fluxion.

A fluent was what we would today call an integral:

In the doctrine of fluxions, the variable or flowing quantity in fluxions which is continually increasing or decreasing; an integral. See fluxion. In fluxions, enlarging (or diminishing) continuously, that is, by infinitesimal increments (+ or —).

As for their etymology, fluxion derives from the Latin fluxus (flowing; fluid; loose) via the French fluxion; and fluent :

from the Latin fluēns, fluent-, present participle of fluere, to flow; see bhleu- in Indo-European roots.

• So they basically derive from the same words? Commented May 16, 2014 at 21:58
• @user76407 my understanding (based on my non-existent knowledge of Latin and 5 minutes worth of googling) is that they both derive from the same root but not necessarily the same form of the Latin word. Commented May 16, 2014 at 22:01