Taylor's theorem states that any function satisfying certain conditions may be represented
by a Taylor series,

Taylor's theorem (without the remainder term) was devised by Taylor in 1712 and published in 1715, although Gregory had actually obtained this result nearly 40 years earlier. In fact, Gregory wrote to John Collins, secretary of the Royal Society, on February 15, 1671, to tell him of the result. The actual notes in which Gregory seems to have discovered the theorem exist on the back of a letter Gregory had received on 30 January, 1671, from an Edinburgh bookseller, which is preserved in the library of the University of St. Andrews (P. Clive, pers. comm., Sep. 8, 2005).

However, it was not until almost a century after Taylor's publication that Lagrange and Cauchy derived approximations of the remainder term after a finite number of
terms (Moritz 1937). These forms are now called the Lagrange
remainder and Cauchy remainder.

Most modern proofs are based on Cox (1851), which is more elementary than that of Cauchy and Lagrange (Moritz 1937), and which Pringsheim (1900) referred to as "leaving hardly anything to wish for in terms of simplicity and strength" (Moritz 1937).

Cox, H. Cambridge and Dublin Math. J.6, 80, 1851.Dehn, M. and Hellinger, D. "Certain Mathematical Achievements
of James Gregory." Amer. Math. Monthly50, 149-163, 1943.Jeffreys,
H. and Jeffreys, B. S. "Taylor's Theorem." §1.133 in Methods
of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University
Press, pp. 50-51, 1988.Malet, A. Studies on James Gregorie (1638-1675).
Ph.D. thesis. Princeton, NJ: Princeton University, 1989.Malet, A. "James
Gregorie on Tangents and the 'Taylor' Rule for Series Expansions." Archive
for History of Exact Science46, 97-137, 1993-1994.Moritz,
R. E. "A Note on Taylor's Theorem." Amer. Math. Monthly44,
31-33, 1937.Pringsheim, A. "Zur Geschichte des Taylorschen Lehrsatzes."
Bibliotheca Math.1, 433-479, 1900.Todhunter, I. A
Treatise on the Differential Calculus with Numerous Examples, 10th ed. London:
Macmillan, p. 75, 1890.Turnbull, H. W. (Ed.). James
Gregory: Tercentenary Memorial Volume. London: Bell, 1939.