Did Newton Invent the Calculus?

Many times, at social gatherings, dinner parties and such, people ask me: “Did Newton really invent the calculus?” I always answer: “Yes! He absolutely did.”

How can I be so sure? What benchmark has to be met to be considered “inventor?”

My understanding is that while there was an existing corpus of knowledge concerning geometry and algebra stretching at least as far back as 312 B.C. when Archimedes demonstrated his “Method of Exhaustion” for determining the area of geometrical figures, especially circles, no progress had or could have been made in applying algebraic methods to geometric problems, this had to wait for Descartes in the early 1600s. Archimedes showed the area of a circle could be determined by the easier calculation of the area of an inscribed regular polygon with an ever increasing number of sides until the tiny edges pretty much coincided with the perimeter of the circle. This concept could have led directly to the calculus if the ancients had a sufficiently well developed algebra whose equations could represent geometrical shapes and curves that could be exploited to accommodate an ever increasing number of smaller and smaller polygonal edges generated by the Method of Exhaustion, but this was something not available nor even obvious to the ancients and European mathematicians of the Middle Ages right up to 1665. Newton’s first claim to “Inventor” is he realized this relationship which he called “analysis” and then identified a process for calculation of the area under a curve called “integration.” The inverse of integration would give the instantaneous motion of a particle along the curve, the “velocity” (“particles” and “motion” come from the fact Newton was studying what today would be called theoretical physics for the dynamics of moving particles--generally planetary motion--in his case). Newton’s second claim was that he was the very first person to understand and state clearly the “Fundamental Theorem of Calculus: “...the fundamental theorem is understood as a statement that the two central operations of calculus, differentiation [velocity] and integration, are inverse operations...Newton discovered the fundamental theorem in 1665.” (Niccolo Guicciardini, “Isaac Newton on Mathematical Certainty and Method,” 2009, p. 182-183).

The justification for Newton’s claim of primacy in inventing the calculus is that he simply announced the fundamental theorem. 

What makes this sweet for me is 20 years ago on one pleasant afternoon in the main library of the University of the Pacific I happened upon the complete 8 volume set of Derek Whiteside’s edited “The Mathematical Papers of Isaac Newton.” 

In volume 1 I found a table composed by Newton in 1665 (at the age of 23!) and copied it down. I could see instantly that he had written down the same differential and corresponding integral equations used and taught to this very day (vol. 1, page 305 “Calculus becomes an algorithm”). Here’s the table, see if you don’t agree:

So there you have it, incontrovertible proof that Newton was onto the calculus: a table giving the differentials and the corresponding integrals of equations for the first time in the history of mathematics; before anyone else.


Note that in 1665 a young Leibniz was still involved in law and diplomacy and didn’t even meet a real physicist, Huygens, until 1672. In later decades he became Newton’s adversary, claiming he invented the calculus first, but I think the above table and the Fundamental Theorem statement completely vindicates Newton’s claim.

Now what he did with this powerful tool is another story. Newton’s one major publication was the “Principia Mathematica.” Written in Latin it remains impenetrable even today with a modern translation. The mathematics are nothing one could learn without spending an inordinate and intolerable amount of time re-learning Newton’s method of presenting his results in geometric fashion. Why so cryptic? Well, 320 years ago people we would now call scientists were still developing ways of interacting with each other and sometimes old ways were held over for reasons we now find obscure, byzantine and prolix. To illustrate the modern view of Newton’s motivations I present a condensed paragraph from a book review in the journal “Historia Mathematica:”

“Did Newton use the calculus to find out “most of the Propositions” in his “Principia” as he once claimed? Or was his natural philosophy calculated as it was presented, in the established Archimedean tradition of geometrical physics? With a few exceptions, Newton did not work out the propositions of the “Principia” using his method of infinite series and fluxions [that is to say, using his calculus]. Newton did work out a solution to the notorious inverse problem analytically [another synonym for the calculus], and, if one combines Newton’s binomial expansion with the Taylor series expansion and takes into account his analytic expression of Taylor series from the 1690s, one arrives at the statement that the force is proportional to the second fluxion of displacement [or Newton’s famous Second Law: F = ma, where “a” is the second differential of the position with respect to time, the “second fluxion of displacement”]. Newton saw his synthetic geometric methods as continuous with ancient geometry. Classically, algebra, although useful for invention, could not be used in demonstration, which must exhibit the construction which solves the problem. Also, Newton saw geometry as grounded in mechanics; as such it fitted seamlessly with his natural philosophy. Indeed, he often based his demonstrations on physical insights which analysis [calculus] would only obscure. Analytic techniques did not constitute a unifying method for Newton’s natural philosophy, so he did not allocate them a prominent role. Yet, neither are Newton’s demonstrations in keeping with the strictures of classical geometry. The majority depend on his innovative method of first and ultimate ratios the geometrical limit procedures that constitute Newton’s “synthetic method of fluxions.”

(The preceeding paragraph was condensed from “Reading the Principia” by Niccolo Guicciardini, a book review by Richard Arthur, “Historia Mathematica” 28 (2001), 54-57.)

So while Newton had invented the calculus and could use it effectively he was more likely to resort to a geometrical method of analysis to present his results because it was in line with the accepted methods practiced by the ancient geometers like Euclid and Archimedes. Basically, Newton was caught with one foot in the past while the other foot was stepping gingerly into the future. It took about another 100 years before the calculus as we know it today was finally ready and geometric analysis ceased to be the paradigm for physics research.

And here is the complete page from Whiteside's "Mathematical Papers".