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).
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.