In Newton’s calculus, a fluxion is a quantity’s instantaneous rate of change over time. The modern equivalent is a derivative (Swetz, 2013). A fluent is the equivalent of a function.
Fluxion Notation
Newton’s notation for a fluxion is usually reported as a dot over a letter. For example:
This is equivalent in meaning to Liebniz’s “d” notation, which is more common: (dx/dy).
Historical Note
Newton’s manuscript on his theory of calculus, De Methodis Serierum et Fluxionum (On the Method of Series and Fluxions), was written in 1671. It was unseen until he made a modified version public, 40 years later (Mumford).
Newton described a fluent (now called a function) as the area under a curve from a to t and its fluxion (the rate of change of the area as the left side of the graph is moved) as the height of the curve at a point t (Mumford).
Newton wrote the following concerning his “Method of Fluxions”:
“…I shall propose, concerning a Space describ’d by local Motion, any how accelerated or retarded.
I. The length of the space describ’d being continually (that is, at all times) given; to find the velocity of the motion at any time propos’d.
II. The velocity of the motion being continually given; to find the length of the Space describ’d at any time propos’d (Whiteside, p. 50).
References
Kitcher, P. Fluxions, Limits, and Infinite Littlenesse: A Study of Newton’s Presentation of the Calculus. Isis. Vol. 64, No. 1 (Mar., 1973), pp. 33-49
Mumford, D. Chapter Five: Newton, fluxions and forces. Retrieved June 4, 2020 from: https://www.dam.brown.edu/people/mumford/beyond/coursenotes/2006PartIIa.pdf
Frank J. Swetz (Pennsylvania State University), “Mathematical Treasure: Newton’s Method of Fluxions,” Convergence (September 2013)
Whiteside, D. The Mathematical Works of Isaac Newton: Volume 1 Hardcover – January 1, 1964.