High schoolers know that calculus has something to do with integration and differentiation. If you ask a modern encyclopedia, it will likely tell you that calculus is a branch of mathematics concerned with the study of continuous change. But what precisely did “the calculus” mean to its original creators? What was so important about the new calculus, and how did it distinguish itself from all the predecessors who also analyzed continuous change? After all, it was sufficiently novel and important to ignite the most famous priority dispute in the history of science, the dispute between Newton and Leibniz.
For millennia, mathematicians have been interested in various properties of curves. One important property is area (or relatedly volume and center of mass). Archimedes, the greatest mathematician of antiquity, proved that the area enclosed by a circle was πr². He was even able to find the volume of a sphere, his proudest achievement.
Many other properties of curves have long been studied, including length, concavity (whether they bend upwards or downwards), and tangents. Here, once again, Archimedes analyzed a spiral and was able to find both its length and its tangent at any given point. Mathematicians have also been interested in extrema of curves, that is finding the position where the curve attains its maximum or minimum. In his treatise on Conics, the ancient geometer Apollonius explicitly tackled such questions of maxima and minima as they pertained to conic sections.
While such properties could easily be determined for specific curves, for other curves it proved difficult. Many seemingly basic questions of area, length and tangents could not be solved by the ancient geometers. Even when solutions could be found, they were always ad hoc. As with the circle, Archimedes was also able to find the area enclosed by a line and a parabola. But his solution required an ingenious setup that was entirely different from his solution for the circle. Every property of every curve needed its own distinct solution.
This was the problem that 17th century mathematicians were aiming to address. They wanted a general method for tackling problems of this kind. Before calculus had been invented, René Descartes described in a 1619 letter his ambitious goal to create an “entirely new science, by which all problems that can be posed, concerning any kind of quantity, continuous or discrete, can be generally solved.”
Newton’s mentor Isaac Barrow demonstrated the fundamental theorem of calculus and came close to developing a true general calculus. In describing his unified approach to analyzing curves, he wrote: “I have previously proved a number of general properties of curves of continuous curvature, deducing them from a certain mode of construction common to all.”1
The first publication on (differential) calculus proper is credited to Gottfried Leibniz in 1684; it is from here the name “calculus” originates. To understand how Leibniz perceived his invention and its importance, we need only read the title of the paper: “A new method for maxima and minima, and for tangents, that is not hindered by fractional or irrational quantities, and a singular kind of calculus for the above mentioned.” In other words, it was one method (“singular kind of calculus”) from which multiple properties (maxima, minima, and tangents) could be obtained, applicable even to complicated curves (“not hindered by…”).
The key was clearly its general-purpose, algorithmic nature. Leibniz gave for the first time the world a tool that made the calculation of these properties easy, systematic and general. In the paper he provided not only algorithms for determining tangents, minima and maxima for general curves, but also a corresponding notation and general rules of differentiation (e.g., the addition rule, product rule, quotient rule and power rule). A few years later, Leibniz also published papers on integral calculus and on the fundamental theorem of calculus. In fact, Leibniz’s ambitions went beyond just curves—he ultimately hoped “to create a kind of universal symbolistic in which all truths of reason would be reduced to a kind of calculus.”
Independently and before Leibniz, Newton also invented a calculus he called the Method of Fluxions, though it remained unpublished until after his death. The two methods differed in important ways—certainly in terminology and symbols used—and yet generality was also key to the Method of Fluxions. Newton spoke in a letter of generalizing a tangent method developed by the French mathematician Pierre de Fermat:
“I had the hint of this method [of fluxions] from Fermat’s way of drawing tangents, and by applying it to abstract equations, directly and invertedly [i.e., differentiation and its inverse, integration], I made it general.”2
The first ever textbook on calculus was published in 1696 by Guillaume de l’Hôpital (1661–1704), building upon the work of Leibniz.3 In the preface of the book, l’Hôpital explicitly contrasts the new calculus with its predecessors. He writes that the ancient methods were inadequate precisely because they weren’t general:
“What we have from the Ancients on these matters, principally from Archimedes, is surely worthy of admiration. However, in addition to the fact that they touched upon very few curves, which they touched only lightly upon, almost all of what we have from them are particular propositions, without order, in which we cannot see any regular and coherent method.”4
Whereas the new calculus distinguished itself by its general applicability:
“The scope of this calculus is immense. It applies to mechanical curves as well as to geometric curves;5 it handles radical signs and often even conveniently so; it extends to as many indeterminates as we wish; and it easily compares infinitely small quantities of all orders equally well.”
When Bernard de Fontenelle (1657–1757) wrote about the importance of l’Hôpital’s book, he also emphasized the general-purpose nature of the new calculus:
“Such is the effect of the general Methods, once one has sought to discover them. One is at the fountain, and one needs only to follow the peaceful current of the consequences. A single Rule from the Book of Mr. de l’Hôpital gives the Tangents of all imaginable curves; another, all of the greatest and least Ordinates, or all the Inflection points and Cusps, or all of the Evolutes, or all Catoptrics at once, or all Dioptrics. Entire Treatises written by the great Authors sometimes are reduced to a few Corollaries that we meet along the road, and we have trouble distinguishing among the multitude. Everything is related by the kinds of Systems that Mr. de l’Hôpital began to put into Geometry, and which will become widespread in the future.”6
With just one sentence—“a single rule gives the tangents of all imaginable curves”—Fontenelle gets to the crux of how the new calculus distinguished itself from earlier approaches. And as he goes on to explain, the same applies to other properties of curves. Similarly, Siméon Poisson (1781–1840) would later write that differential calculus “consists in a system of rules proper for finding the differentials of all functions, rather than in the use which may be made of these infinitely small variations in the solution of one or two isolated problems.”7
Thus if we ask why the invention of calculus was so important that it could ignite a vicious priority dispute, the answer is clear. Of central importance was not its beauty, rigor, or even yet its scientific applications. Instead, calculus was important because it was the first systematic and analytical tool that could find almost all properties of almost all curves. From the very first generation of practitioners, the new calculus was defined by its generality and ease of use. The new calculus not only opened the way for analyzing a whole host of curves that stumped the ancients, it also replaced the long and subtle arguments they used with short routine calculations.
This brings us to the question of what precisely the ‘invention of calculus’ entails (or ‘discovery’ depending on your philosophical bent). Briefly stated, calculus must be understood as a general, operational, analytical tool with standardized rules, notation, and broad applicability. Most obviously, it necessarily entails systematic methods for differentiating and integrating any suitably nice curves, not just specific examples or a limited class of curves. It must also be organized into a broader system based on the recognition and demonstration of differentiation and integration being inverse operations. Lastly, it must be analytical; that is, it must be equipped with a suitable notation, and it must have transcended pure geometric reasoning by means of expressing mathematical rules and relationships in the language of symbolic algebra.8
It should therefore be clear why historians of mathematics consider calculus a late-17th century European invention rather than an earlier one. Though ancient Greek mathematicians as well as medieval European, Muslim and especially Hindu mathematicians tackled and solved related problems of continuous change, they never invented a true calculus. The fundamental distinction between all predecessors and calculus proper is that the latter is systematic, analytical and general. Before the 17th century, no one had developed methods of differentiation and integration that were applicable to general curves, let alone having stated and proved the fundamental theorem of calculus.9
J. M. Child, The Geometrical Lectures of Isaac Barrow, p. 66.
Because Barrow’s approach was entirely geometrical instead of analytical, and it lacked a corresponding notation, the consensus is that his work cannot quite be called the invention of calculus.
See The Correspondence of Isaac Newton, Vol. 5, p. 214.
This book heavily borrowed from work done by Johann Bernoulli, but that’s a story for another time. See Bradley et al. (2015). L’Hôpital’s Analyse des infiniments petits: An Annotated Translation with Source Material by Johann Bernoulli.
See “L’Hôpital’s Preface” in Bradley et al. (2015). L’Hôpital’s Analyse des infiniments petits: An Annotated Translation with Source Material by Johann Bernoulli.
By “geometric curves,” l’Hôpital means what we would today call “algebraic curves.” It is based on a distinction due to Descartes. A curve that is not geometrical was called “mechanical” by Descartes and is called “transcendental” in modern usage.
Bradley et al. (2015). L’Hôpital’s Analyse des infiniments petits: An Annotated Translation with Source Material by Johann Bernoulli, p. 299.
Quoted from Florian Cajori’s A History of Mathematics (1894).
See Florian Cajori’s “Who was the first Inventor of the Calculus?” (1919) for a short discussion on what the invention of calculus entails.
See Katz (1995) for a discussion on predecessors of calculus achieved in the Muslim and Indian world.
Fascinating article! It really highlights how the shift from ad hoc ancient methods to a general, systematic, and algorithmic tool was the true breakthrough of 17th-century mathematics.
Wonderful historical perspective on intellectual breakthroughs and mathematical thought. Foundational concepts become intuitive only after generations of careful formalization and empirical testing. Reflected on testability and the growth of knowledge in this dialogue: https://gzambrano.substack.com/p/the-limit-of-the-testable-a-dialogue-in-three-acts