Of course Robert, "invented" is the wrong word; yours is much better: "contributed". What we can do, is to name those who contributed in a decisive manner - but drawing a line is always difficult. In addition, such list and line will certainly upset some people, especially Gauss, Princeps mathematicorum, who's got "decisive" contributions in many many areas of mathematics et al., but only indirect contributions to Calculus...

Leibniz (Nova Methodus pro Maximis et Minimis) and Newton (Method of Fluxions) are usually both given credit for independently inventing and developing calculus in the 17th, early 18th, century. Yet many of the ideas of calculus had been developed earlier in Greece, China, India, Iraq, Persia, and Japan; and many ideas came after Leibniz and Newton. We owe much of the modern calculus to Cauchy, Weierstrass and Riemann, but the list is much longer, with names that we remember (or not) today.

George Stoica,

Thank you for your succinct and helpful reply.

The little we know about Eudoxus suggests he had a rough idea about limit; the method of exhaustion used in Greek mathematics is an antecedent to integral calculus, at least. About 50 years ago I read (I can't remember where) that there were about 100,000 notable mathematicians in history. I guess that could be an upper bound for the number of mathematicians who contributed to development of calculus, at least as of around 1970.

You are correct, Robert: a recorded history of those many names will popularize their contributions, many of which might be unknown to quite a lot of us.

In the Islamic era, Ibn al-Haytham was able to use an integrative method to derive the general formula for the sum of an arithmetic sequence of the fourth degree. Then the Chinese invented equations dealing with integration, and in India the derivation began to appear at the hand of a mathematician Indian who described infinitesimal changes as others reached sequences similar to Taylor series.

With the advent of the Renaissance, the West began to learn and translate ancient books from Arabic and to develop the sciences of mathematics, physics, and some other sciences, and the science of calculus in particular developed at the hands of Isaac Newton. The well-known world historian (Urant Wol) said that Thabit Ibn Qurrah, the greatest of Muslim engineering scholars, had contributed a great deal to the advancement of geometry, and that he who paved the way for finding the science of calculus and was able to solve algebraic equations by engineering methods.

With an absurd oversimplification, the "invention" of the calculus is sometimes ascribed to two men, Newton and Leibniz. In reality, the calculus is the product of a long evolution that was neither initiated nor terminated by Newton and Leibniz, but in which both played a decisive part.

From the beginning of chapter VIII, What is Mathematics, Courant and Robbins, Second edition, page 398 Oxford U Press, 1996.

Of course Robert, "invented" is the wrong word; yours is much better: "contributed". What we can do, is to name those who contributed in a decisive manner - but drawing a line is always difficult. In addition, such list and line will certainly upset some people, especially Gauss, Princeps mathematicorum, who's got "decisive" contributions in many many areas of mathematics et al., but only indirect contributions to Calculus...

So much was said before, but indeed, Calculus neither started not terminated with Leibniz/Newton. As to the claim by GS that Gauss contributions to Calculus were indirect, let me only remind you about Gauss-Ostrogradsky theorem and work on curvature.

Might it be the case that most mathematicians contributed to developing calculus. The different number systems, integers, rational, reals, and complex numbers play a role. Analytical geometry enables algebraic equations of geometric curves. I am inclining to treating the total number of notable mathematicians, 100,000 or so, as a working upper bound for a back of the envelope, order of magnitude, kind of estimate of the energy content of calculus.

I know Roman, but I did not want to go into the timeline of the divergence theorem: it was first discovered by Lagrange in 1762; then later independently rediscovered by Gauss in 1813; by Ostrogradsky, who also gave the first proof of the general theorem, in 1826; by Green in 1828; Simeon-Denis Poisson in 1824; and Frédéric Sarrus in 1828 (the story is even longer than what I wrote here - a good example to Robert's question on people who have contributed to the development of Calculus).

I suspect there is no authoritative ball park estimate of the number of mathematicians who contributed to the development of the calculus. And what about teachers of calculus, popularizers and popular accounts?

There are some histories specifically about calculus, as distinguished from general histories of mathematics, including:

Baron, Margaret E. (1969) The Origins of the Infinitesimal Calculus. New York: Dover

Boyer, Carl B. (1949) The History of the Calculus and Its Conceptual Development. New York: Dover

Edwards, C.H., Jr. (1979) The Historical Development of the Calculus. New York: Springer-Verlag

PS. To George Stoica.

I was interested in your inclusion of Frédéric Sarrus in your reply above about the divergence theorem.

My only previous encounter with Sarrus is in connection with the origins of metabolic scaling, that began in an obscure 1838 paper by Sarrus and Rameaux, which took me almost a year to complete in 2017 translating from French to English.

Article Sarrus and Rameaux, 1838, translated into English

The problem of metabolic scaling is for me almost endlessly fascinating. The most interesting thing about the 1838 paper is the underlying problem solving involving scaling and dimension, which also connects to Galileo's 1638 Two New Sciences, and with some effort reveals (what I think are) fundamental principles of physics.

I think the same methods used by Sarrus and Rameaux in the 1838 account of their work, generalized both as to the number of dimensions and as to viewpoint (a dimensional viewpoint is more powerful than a scaling viewpoint), and somewhat adapted can account for the astronomical observations that are described as dark energy.

I have posted several articles on ResearchGate relating to the 1838 paper collected in a Project on Metabolic Scaling, including:

Preprint Metabolic scaling, cosmological inflation and dimensional capacity

Preprint Size, scaling, and invariant ratios

Preprint The Kozlowski Konarzewski - West Brown Enquist Debate

The article on size, scaling, ratios attempts to get the fundamentals, but I am pretty sure the quality of the ideas is superior to the quality of the writing (though it is possible that some reviewers might, out of a habit of dismissiveness, insist that the reverse is true).

Robert - very well explained, I am looking forward to reading the 1838 paper by Sarrus and Rameaux, together with your 3 papers listed above.

As for Sarrus' contribution to the divergence theorem, I can offer the following reference:

Frédéric Sarrus, Hydro-dynamique. Mémoire sur les oscillations des corps flottans. Annales de mathématiques pures et appliquées 19 (1828), 185-210,

available in e-format:

http://www.numdam.org/article/AMPA_1828-1829__19__185_0.pdf

Interesting question. However, the foundations that allowed calculus to evolve started long before Newton and Leibniz. The foundation is not calculus but the concept of the limit. Archimedes (287-212 BC) was probably the first to recognize what became the concept of the limit in is estimation pi and the area of the circle by taking inscribed and circumscribed polygons bounding the circle and using the simple fact that one approaches pi from below while the other from above. Of each sequence defines a Cauchy sequence - concept unknown at that time and the completion of the reals (also unknown at that time) show the limit of each sequence is the same and equal a the number pi. In reality the concept of infinitesimal - goes back to Archimedes although the formal concept of "infinity" was not accepted until long afterwards.

Roll backwards to the Greeks when faced with the proposition of an infinite number of prime number, was a problem as they believed the universe was finite. Infinity was not something the Greeks wanted to accept and Aristotle (385-348 BC). But Archimedes had just shown that infinity and infinitesimals had a role in mathematics - in fact a central role. It was not to the 1600's that mathematicians attack the problem of infinity to try to understand what it mean as they developed the concept of numbers that are used today.

https://www.math.tamu.edu/~don.allen/history/infinity.pdf

As we better understood our real number system, the concept of point set or general topology was defined to abstract and better understand the structure. In general topology concepts like nets (generalizations of sequences required to define integrals for example), convergence, close to, in a neighborhood and limits are all defined through the concept of open sets which are used to define a topology on a set which now allows for the definition and study of the concept of limits and continuity and handle infinity. While the formulations of general topology came along after the "birth of the calculus" and known as Analysis Situs a term coined by Henri Poincaré through the work of Poincaré , Euler, Cantor, Lefschetz, Courant, Hilbert, etc., a firm foundation was laid not only to the real number system, but to limits, continuity all the foundations of what we now know as 'calculus."

Topology is so important to the foundations of calculus and the concept of a limit that in his classic text, "General Topology," John Kelley writes in the first paragraph of the preface he writes, "...I have, with difficulty, been prevented by my friends from labeling it: What Every Young Analyst Should Know." No truer words have been spoken or written as the foundations of topology has allowed the concept of calculus to be expanded far beyond its original intent.

Regarding "And what about teachers of calculus, popularizers and popular accounts?", I think that Bernoulli-L'Hopital Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes, and Euler's Institutiones calculi differentialis and

Institutionum calculi integralis deserve mention. You can read these books today and they are still useful! In particular, Euler set up many of the notations we use nowadays (and he was the first in writing the master equation for applications of Calculus: F= dp/dt =ma, when m is constant).

If you are interested in precursors, we have ancient Greeks: Eudoxus of Cnidus and his method of exhaustion, Archimedes (the most famous); in the Italian Renaissance, Niccolò Cusano and his use of the infinite on the circle; Bonaventura Cavalieri and his method of indivisibles; Torricelli, Gregory and Barrow for the integral calculus; and many others.

The official history begins with Newton and Leibniz; then we have a huge list of contributors, both in results and in foundations... Cauchy, Weierstrass... (limit, the method epsilon-delta...) and so on (see other answers)

It is possible to consider Zeno of Elea as a negative precursor, a sort of: Achilles and the tortoise, the arrow... He had some ideas about infinitesimals - how to use them in a paradoxical way.

The list is very lengthy. Can't be enlisted

I think we drifted away from the title of this tread, but it is fine. No one knows how many mathematicians contributed to the development of Calculus.

We need to wonder what calculus really is. Formally, we can use the different name for it: Mathematical Analysis (MA). This name is used, e.g., in former satellite countries of the former USSR. It is also used in the USA, as a higher level of Calculus. Higher level means precise formulation of theorems, rigorous proofs based on modern foundations of math. MA subject exists also next to Calculus at an undergrad level. All the so-called STEM majors are supposed to take some calculus classes. Such Calculus does not pretend to be called MA. It is enough that students understand some technicalities (like differentiation, integration, diff eqns) to live their professional lives. For mathematicians such a calculus is far not sufficient. They need a deeper, complete understanding and it is what they find in MA. There is also term Advanced Calculus that means nothing alse but MA.