Timeline of Calculus
How calculus took shape, from Archimedes' use of the method of exhaustion in the 3rd century BCE through the Kerala school, Newton and Leibniz to the rigorous foundations of Cauchy, Weierstrass and Dedekind by 1872.
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- 25 events 3rd century BCE – 1872
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3rd century BCE: Archimedes finds a parabolic area by exhaustion
In "Quadrature of the Parabola", Archimedes of Syracuse proved that a parabolic segment has 4/3 the area of a certain inscribed triangle, summing a geometric series within the method of exhaustion that Eudoxus of Cnidus had made rigorous. His "Method of Mechanical Theorems", which showed how he first found such results by balancing thin slices of figures on a lever, was lost until it was identified in the Archimedes Palimpsest in 1906.
Wikipedia263: Liu Hui proves the circle-area rule by exhaustion
Liu Hui's commentary on "The Nine Chapters on the Mathematical Art" proved the book's rule that a circle's area is half its circumference times its radius by doubling the sides of inscribed polygons until, in his words, the polygon "will coincide and become one with the circle". He could not find a sphere's volume; Zu Chongzhi and his son Zu Gengzhi did so over two centuries later by extending his comparison of cross-sections, the principle later named in Europe after Bonaventura Cavalieri.
Wikipediaearly 11th century: Ibn al-Haytham sums fourth powers to find a volume
Ibn al-Haytham (Alhazen), a Basra-born scholar who worked in Cairo, extended Archimedes' method of exhaustion to find the volume of the solid formed by rotating a parabola about a line perpendicular to its axis. To do so he derived a formula for the sum of the fourth powers of whole numbers, by a method that generalizes to any power, and used it to perform what would now be called an integration.
Wikipediac. 1335: Oxford Calculators state the mean speed theorem
At Merton College, Oxford, William Heytesbury and the other Oxford Calculators stated that a body accelerating uniformly from rest covers the same distance as one moving for the same time at half its final speed. Nicole Oresme in Paris later proved the rule geometrically, treating distance as the area under a graph of speed against time, an early form of integration; Galileo Galilei applied the theorem to falling bodies in the 17th century.
Wikipediac. 1400: Madhava finds series for sine, cosine and arctangent
Madhava of Sangamagrama, founder of the Kerala school of astronomy and mathematics in southern India, is credited by his successors with infinite power series for the sine, cosine and arctangent, and with using the arctangent series to compute π; Jyeṣṭhadeva's Malayalam treatise "Yuktibhāṣā" (c. 1530) gave proofs. Europeans found the series in 1669–1673; no documentary evidence shows that the Kerala work was known outside India before the 19th century, though transmission through Jesuit missionaries has been proposed.
Wikipedia1635: Cavalieri publishes his method of indivisibles
Bonaventura Cavalieri's "Geometria indivisibilibus", written in 1627, compared areas and volumes through "all the lines" or "all the planes" of figures, yielding the principle that solids whose cross-sections at every height have equal areas have equal volumes. Johannes Kepler had found volumes of wine barrels from infinitely small parts in 1615; Paul Guldin attacked Cavalieri's method for its lack of rigor, but Evangelista Torricelli refined and promoted it.
Wikipediac. 1636: Fermat devises adequality for maxima and tangents
In "Methodus ad disquirendam maximam et minimam", circulated in manuscript in Paris, Pierre de Fermat found maxima and minima by setting a quantity's value at x approximately equal, or "adequal", to its value at x + e, canceling a factor of e and discarding the terms still containing it. He adapted the method to tangents; René Descartes attacked it, but Isaac Newton later credited "Fermat's way of drawing tangents" with giving him the hint for his fluxions.
Wikipedia1656: Wallis publishes "Arithmetica Infinitorum"
In "Arithmetica Infinitorum", John Wallis, Savilian Professor of Geometry at Oxford, systematized Bonaventura Cavalieri's method of indivisibles as an arithmetic of infinite sums and extended Cavalieri's formula for the areas under curves given by powers of x. By interpolation, a word he introduced, he obtained an infinite product for π, now called the Wallis product; Isaac Newton later built on his methods to find the general binomial theorem.
Wikipedia1665–1666: Newton develops the method of fluxions
Beginning at Cambridge and continuing at his family home in Woolsthorpe while the university was closed by plague, Isaac Newton worked out the calculus he later called the method of fluxions, treating quantities as flowing in time to find tangents, curvature and areas, with differentiation and integration as inverse operations. He circulated "De analysi" in 1669 but published only piecemeal from the "Principia" (1687) on; "Method of Fluxions", written in 1671, appeared in 1736.
Wikipedia1668–1670: Gregory and Barrow prove the fundamental theorem geometrically
James Gregory's "Geometriae Pars Universalis" gave the first published statement and proof of a geometric form of the fundamental theorem of calculus, which makes finding areas the inverse of finding tangents, for a restricted class of curves. Two years later Isaac Barrow, Cambridge's first Lucasian Professor of Mathematics, proved a more general version in "Lectiones Geometricae", still as geometric theorems rather than a general calculus.
Wikipedia1675: Leibniz introduces the ∫ and d notation
In private manuscripts written in Paris in October and November, Gottfried Wilhelm Leibniz began to write ∫, an elongated s for the Latin summa, for sums of infinitely many infinitely thin rectangles, and d, for differentia, for differences, treating the two operations as inverses. He reached his calculus independently of Isaac Newton, who had mastered his own methods about nine years earlier; Leibniz's symbols became the standard notation.
Wikipedia1684: Leibniz publishes the differential calculus
In the Leipzig journal "Acta Eruditorum", Gottfried Wilhelm Leibniz published "Nova Methodus pro Maximis et Minimis", the first published account of the differential calculus, giving rules for the differentials of powers, products and quotients and applying them to tangents, maxima and minima. Johann Bernoulli, who worked through it with his brother Jacob, later called the terse paper "an enigma rather than an explanation"; a paper of 1686 put the integral sign into print.
Wikipedia1696: L'Hôpital publishes the first calculus textbook
Guillaume de l'Hôpital's "Analyse des infiniment petits", the first printed textbook of the differential calculus, spread Gottfried Wilhelm Leibniz's methods across 18th-century Europe. Much of it came from Johann Bernoulli, whom l'Hôpital paid a yearly retainer from 1694 for his discoveries, among them the rule for limits of the form 0/0 now named after l'Hôpital; Bernoulli's 1691–1692 lecture notes, found in 1921, confirmed his role.
Wikipedia1712: Royal Society report backs Newton's priority claim
Answering a complaint from Gottfried Wilhelm Leibniz, the Royal Society, whose president was Isaac Newton, appointed a committee that never asked Leibniz for his account and found in Newton's favor. Newton himself wrote the report, printed with supporting letters as the "Commercium Epistolicum", and later reviewed it anonymously. The quarrel kept British mathematicians on Newton's notation until the 1820s.
Wikipedia1715: Taylor publishes the general power series expansion
In "Methodus Incrementorum Directa et Inversa", Brook Taylor published a general method for expanding a function as an infinite series whose terms come from its derivatives at a single point. James Gregory in 1671 and Isaac Newton in 1691–1692 had reached such series in unpublished work, and Joseph-Louis Lagrange later called Taylor's theorem "the main foundation of differential calculus".
Wikipedia1734: Berkeley attacks the foundations of calculus
In "The Analyst", the philosopher George Berkeley argued that Isaac Newton's fluxions and Gottfried Wilhelm Leibniz's infinitesimals rested on a fallacy: increments were treated as nonzero to divide by them, then as zero. He called them "the ghosts of departed quantities"; historian Judith Grabiner judged his criticisms of the practice "essentially correct", and Colin Maclaurin's "Treatise of Fluxions" (1742) began as a reply.
Wikipedia1748: Euler builds analysis around functions
Leonhard Euler's two-volume "Introductio in analysin infinitorum" began with variables and functions and treated the exponential, logarithmic and trigonometric functions and their infinite series without derivatives or integrals, defining e as the number whose natural logarithm is 1. Historian Carl Boyer called it "the foremost textbook of modern times"; Euler's textbooks of differential and integral calculus followed in 1755 and from 1768.
Wikipedia1797: Lagrange tries to base calculus on power series
Joseph-Louis Lagrange's "Théorie des fonctions analytiques" offered the differential calculus free of infinitely small quantities, limits and fluxions by defining derivatives through the coefficients of a function's power series. His term "fonction dérivée" gave the word "derivative", but the approach failed, as Augustin-Louis Cauchy noted, because some smooth functions differ from their Taylor series.
Wikipedia1817: Bolzano proves the intermediate value theorem
In "Rein analytischer Beweis" (Purely Analytic Proof), the Prague priest and mathematician Bernard Bolzano gave the first purely analytic proof that a continuous function taking values of opposite sign at two points has a root between them. He defined continuity with inequalities, an early form of the epsilon–delta definition, but his work remained little known until it came to the attention of Karl Weierstrass about 50 years later.
Wikipedia1821: Cauchy rebuilds analysis on limits
Written for students at the École Polytechnique in Paris, Augustin-Louis Cauchy's "Cours d'analyse" defined a limit, treated an infinitely small quantity as a variable with limit zero, defined continuity in those terms and studied the convergence of series rigorously. His 1823 "Résumé" defined the derivative as a limit of difference quotients and the integral of a continuous function as the limit of sums over ever finer subdivisions.
Wikipedia1822: Fourier represents functions by trigonometric series
In "Théorie analytique de la chaleur" (The Analytical Theory of Heat), developed from a paper he presented to the Paris Institute in 1807, Joseph Fourier solved the heat equation by expanding functions into infinite series of sines and cosines and held that any function could be so represented. Testing that claim led Peter Gustav Lejeune Dirichlet to prove convergence for a broad class of functions in 1829 and Bernhard Riemann to define the integral without assuming continuity.
Wikipedia1854: Riemann defines the integral without assuming continuity
In his habilitation thesis on trigonometric series at the University of Göttingen, Bernhard Riemann defined the integral of a function as the limit of sums over ever finer subdivisions, without requiring continuity, and gave a criterion for which functions are integrable. The thesis appeared only in 1868, after his death; the Riemann integral is still the one taught in introductory calculus, and Henri Lebesgue extended it in 1902.
Wikipedia1861: Weierstrass introduces epsilon–delta definitions
In a differential calculus course at a Berlin technical institute, recorded by his student Hermann Amandus Schwarz, Karl Weierstrass defined continuity with inequalities: f is continuous at x if for every ε > 0 there is a δ > 0 with |f(x + h) − f(x)| < ε whenever |h| < δ. He also stressed uniform convergence, the condition missing from Augustin-Louis Cauchy's claim that a convergent series of continuous functions has a continuous sum.
Wikipedia1872: Dedekind defines the real numbers by cuts
In "Stetigkeit und irrationale Zahlen" (Continuity and Irrational Numbers), Richard Dedekind characterized every real number by a cut splitting the rational numbers into a lower and an upper class, so that the limit theorems of calculus no longer rested on geometric intuition. He had found the idea in 1858 while first teaching calculus in Zürich; Eduard Heine and Georg Cantor had just published constructions from sequences of rational numbers.
Wikipedia1872: Weierstrass presents a continuous, nowhere-differentiable function
On July 18, Karl Weierstrass presented to the Prussian Academy of Sciences in Berlin a function, defined by an infinite sum of cosine waves, that is continuous everywhere but differentiable nowhere, refuting the common belief that a continuous function can lack a derivative only at isolated points; Paul du Bois-Reymond first printed it in 1875. The example overturned proofs that relied on geometric intuition; Charles Hermite called such functions a "lamentable scourge".
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