Timeline of calculus and mathematical analysis
Encyclopedia
A timeline of calculus
and mathematical analysis
.
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
and mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
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1000 to 1500
- 1020 — Abul WáfaAbul WáfaAbū al-Wafāʾ, Muḥammad ibn Muḥammad ibn Yaḥyā ibn Ismāʿīl ibn al-ʿAbbās al-Būzjānī was a Persian mathematician and astronomer who worked in Baghdad...
— Discussed the quadrature of the parabolaParabolaIn mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...
and the volume of the paraboloidParaboloidIn mathematics, a paraboloid is a quadric surface of special kind. There are two kinds of paraboloids: elliptic and hyperbolic. The elliptic paraboloid is shaped like an oval cup and can have a maximum or minimum point....
. - 1021 — Ibn al-Haytham completes his Book of OpticsBook of OpticsThe Book of Optics ; ; Latin: De Aspectibus or Opticae Thesaurus: Alhazeni Arabis; Italian: Deli Aspecti) is a seven-volume treatise on optics and other fields of study composed by the medieval Muslim scholar Alhazen .-See also:* Science in medieval Islam...
, which formulated and solved “Alhazen's problem” geometrically, and developed and proved the earliest general formula for infinitesimalInfinitesimalInfinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
and integralIntegralIntegration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
calculus using mathematical inductionMathematical inductionMathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers...
. - 12th century — Bhāskara II conceives differential calculusDifferential calculusIn mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....
, and also develops Rolle's theoremRolle's theoremIn calculus, Rolle's theorem essentially states that a differentiable function which attains equal values at two distinct points must have a point somewhere between them where the first derivative is zero.-Standard version of the theorem:If a real-valued function ƒ is continuous on a closed...
, Pell's equationPell's equationPell's equation is any Diophantine equation of the formx^2-ny^2=1\,where n is a nonsquare integer. The word Diophantine means that integer values of x and y are sought. Trivially, x = 1 and y = 0 always solve this equation...
, a proof for the Pythagorean TheoremPythagorean theoremIn mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
, computes πPi' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
to 5 decimal places, and calculates the time taken for the earth to orbit the sun to 9 decimal places - 14th century — MadhavaMadhava of SangamagramaMādhava of Sañgamāgrama was a prominent Kerala mathematician-astronomer from the town of Irińńālakkuţa near Cochin, Kerala, India. He is considered the founder of the Kerala School of Astronomy and Mathematics...
is considered the father of mathematical analysisMathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
, who also worked on the power series for p and for sine and cosine functions, and along with other Kerala school mathematicians, founded the important concepts of CalculusCalculusCalculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem... - 14th century — ParameshvaraParameshvaraVatasseri Parameshvara Nambudiri was a major Indian mathematician and astronomer of the Kerala school of astronomy and mathematics founded by Madhava of Sangamagrama. He was also an astrologer...
, a Kerala school mathematician, presents a series form of the sine function that is equivalent to its Taylor seriesTaylor seriesIn mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
expansion, states the mean value theoremMean value theoremIn calculus, the mean value theorem states, roughly, that given an arc of a differentiable curve, there is at least one point on that arc at which the derivative of the curve is equal to the "average" derivative of the arc. Briefly, a suitable infinitesimal element of the arc is parallel to the...
of differential calculusDifferential calculusIn mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....
, and is also the first mathematician to give the radius of circle with inscribed cyclic quadrilateralCyclic quadrilateralIn Euclidean geometry, a cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. Other names for these quadrilaterals are chordal quadrilateral and inscribed... - 1400 — MadhavaMadhava of SangamagramaMādhava of Sañgamāgrama was a prominent Kerala mathematician-astronomer from the town of Irińńālakkuţa near Cochin, Kerala, India. He is considered the founder of the Kerala School of Astronomy and Mathematics...
discovers the series expansion for the inverse-tangent function, the infinite series for arctan and sin, and many methods for calculating the circumference of the circle, and uses them to compute πPi' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
correct to 11 decimal places
16th century
- 1501 — Nilakantha SomayajiNilakantha SomayajiKelallur Nilakantha Somayaji was a major mathematician and astronomer of the Kerala school of astronomy and mathematics. One of his most influential works was the comprehensive astronomical treatise Tantrasamgraha completed in 1501...
writes the “Tantra Samgraha”, which lays the foundation for a complete system of fluxions (derivativeDerivativeIn calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
s), and expands on concepts from his previous text, the “Aryabhatiya Bhasya”. - 1550 — JyeshtadevaJyeshtadevaJyeṣṭhadeva was an astronomer-mathematician of the Kerala school of astronomy and mathematics founded by Sangamagrama Madhava . He is best known as the author of Yuktibhāṣā,...
, a Kerala school mathematician, writes the “Yuktibhāṣā”, the world's first calculusCalculusCalculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
text, which gives detailed derivations of many calculus theorems and formulae.
17th century
- 1629 - Pierre de Fermat develops a rudimentary differential calculusDifferential calculusIn mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....
, - 1634 - Gilles de RobervalGilles de RobervalGilles Personne de Roberval , French mathematician, was born at Roberval, Oise, near Beauvais, France. His name was originally Gilles Personne or Gilles Personier, that of Roberval, by which he is known, being taken from the place of his birth.Like René Descartes, he was present at the siege of La...
shows that the area under a cycloidCycloidA cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line.It is an example of a roulette, a curve generated by a curve rolling on another curve....
is three times the area of its generating circle, - 1658 - Christopher WrenChristopher WrenSir Christopher Wren FRS is one of the most highly acclaimed English architects in history.He used to be accorded responsibility for rebuilding 51 churches in the City of London after the Great Fire in 1666, including his masterpiece, St. Paul's Cathedral, on Ludgate Hill, completed in 1710...
shows that the length of a cycloidCycloidA cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line.It is an example of a roulette, a curve generated by a curve rolling on another curve....
is four times the diameter of its generating circle, - 1665 - Isaac NewtonIsaac NewtonSir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
works on the fundamental theorem of calculusFundamental theorem of calculusThe first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation...
and develops his version of infinitesimal calculusInfinitesimal calculusInfinitesimal calculus is the part of mathematics concerned with finding slope of curves, areas under curves, minima and maxima, and other geometric and analytic problems. It was independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s...
, - 1671 - James GregoryJames Gregory (astronomer and mathematician)James Gregory FRS was a Scottish mathematician and astronomer. He described an early practical design for the reflecting telescope – the Gregorian telescope – and made advances in trigonometry, discovering infinite series representations for several trigonometric functions.- Biography :The...
develops a series expansion for the inverse-tangent function (originally discovered by MadhavaMadhava of SangamagramaMādhava of Sañgamāgrama was a prominent Kerala mathematician-astronomer from the town of Irińńālakkuţa near Cochin, Kerala, India. He is considered the founder of the Kerala School of Astronomy and Mathematics...
) - 1673 - Gottfried LeibnizGottfried LeibnizGottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....
also develops his version of infinitesimal calculusInfinitesimal calculusInfinitesimal calculus is the part of mathematics concerned with finding slope of curves, areas under curves, minima and maxima, and other geometric and analytic problems. It was independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s...
, - 1675 - Isaac Newton invents a Newton's methodNewton's methodIn numerical analysis, Newton's method , named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots of a real-valued function. The algorithm is first in the class of Householder's methods, succeeded by Halley's method...
for the computation of functional roots, - 1691 - Gottfried Leibniz discovers the technique of separation of variables for ordinary differential equationDifferential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s, - 1696 - Guillaume de L'Hôpital states his ruleL'Hôpital's ruleIn calculus, l'Hôpital's rule uses derivatives to help evaluate limits involving indeterminate forms. Application of the rule often converts an indeterminate form to a determinate form, allowing easy evaluation of the limit...
for the computation of certain limitsLimit (mathematics)In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...
, - 1696 - Jakob Bernoulli and Johann BernoulliJohann BernoulliJohann Bernoulli was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family...
solve brachistochrone problemBrachistochrone curveA Brachistochrone curve , or curve of fastest descent, is the curve between two points that is covered in the least time by a point-like body that starts at the first point with zero speed and is constrained to move along the curve to the second point, under the action of constant gravity and...
, the first result in the calculus of variationsCalculus of variationsCalculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
,
18th century
- 1712 - Brook TaylorBrook TaylorBrook Taylor FRS was an English mathematician who is best known for Taylor's theorem and the Taylor series.- Life and work :...
develops Taylor seriesTaylor seriesIn mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
, - 1730 - James StirlingJames Stirling (mathematician)James Stirling was a Scottish mathematician. The Stirling numbers and Stirling's approximation are named after him.-Biography:...
publishes The Differential Method, - 1734 - Leonhard EulerLeonhard EulerLeonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
introduces the integrating factorIntegrating factorIn mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calculus, in this case often multiplying through by an...
technique for solving first-order ordinary differential equationDifferential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s, - 1735 - Leonhard Euler solves the Basel problemBasel problemThe Basel problem is a famous problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1644 and solved by Leonhard Euler in 1735. Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate...
, relating an infinite series to π, - 1739 - Leonhard Euler solves the general homogeneous linear ordinary differential equationOrdinary differential equationIn mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
with constant coefficientsConstant coefficientsIn mathematics, constant coefficients is a term applied to differential operators, and also some difference operators, to signify that they contain no functions of the independent variables, other than constant functions. In other words, it singles out special operators, within the larger class of...
, - 1748 - Maria Gaetana AgnesiMaria Gaetana AgnesiMaria Gaetana Agnesi was an Italian linguist, mathematician, and philosopher. Agnesi is credited with writing the first book discussing both differential and integral calculus. She was an honorary member of the faculty at the University of Bologna...
discusses analysis in Instituzioni Analitiche ad Uso della Gioventu Italiana, - 1762 - Joseph Louis LagrangeJoseph Louis LagrangeJoseph-Louis Lagrange , born Giuseppe Lodovico Lagrangia, was a mathematician and astronomer, who was born in Turin, Piedmont, lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics...
discovers the divergence theoremDivergence theoremIn vector calculus, the divergence theorem, also known as Gauss' theorem , Ostrogradsky's theorem , or Gauss–Ostrogradsky theorem is a result that relates the flow of a vector field through a surface to the behavior of the vector field inside the surface.More precisely, the divergence theorem...
,
19th century
- 1807 - Joseph FourierJoseph FourierJean Baptiste Joseph Fourier was a French mathematician and physicist best known for initiating the investigation of Fourier series and their applications to problems of heat transfer and vibrations. The Fourier transform and Fourier's Law are also named in his honour...
announces his discoveries about the trigonometric decomposition of functionsFourier seriesIn mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
, - 1811 - Carl Friedrich Gauss discusses the meaning of integrals with complex limits and briefly examines the dependence of such integrals on the chosen path of integration,
- 1815 - Siméon-Denis Poisson carries out integrations along paths in the complex plane,
- 1817 - Bernard BolzanoBernard BolzanoBernhard Placidus Johann Nepomuk Bolzano , Bernard Bolzano in English, was a Bohemian mathematician, logician, philosopher, theologian, Catholic priest and antimilitarist of German mother tongue.-Family:Bolzano was the son of two pious Catholics...
presents the intermediate value theoremIntermediate value theoremIn mathematical analysis, the intermediate value theorem states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is at least one point in its domain that the function maps to that value....
---a continuous functionContinuous functionIn mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
which is negative at one point and positive at another point must be zero for at least one point in between, - 1822 - Augustin-Louis Cauchy presents the Cauchy integral theorem for integration around the boundary of a rectangle in the complex planeComplex planeIn mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
, - 1825 - Augustin-Louis Cauchy presents the Cauchy integral theorem for general integration paths—he assumes the function being integrated has a continuous derivative, and he introduces the theory of residueResidue (complex analysis)In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities...
s in complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
, - 1825 - André-Marie AmpèreAndré-Marie AmpèreAndré-Marie Ampère was a French physicist and mathematician who is generally regarded as one of the main discoverers of electromagnetism. The SI unit of measurement of electric current, the ampere, is named after him....
discovers Stokes' theoremStokes' theoremIn differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Lord Kelvin first discovered the result and communicated it to George Stokes in July 1850...
, - 1828 - George Green proves Green's theoremGreen's theoremIn mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C...
, - 1831 - Mikhail Vasilievich OstrogradskyMikhail Vasilievich OstrogradskyMikhail Vasilyevich Ostrogradsky was an Russian / Ukrainian mathematician, mechanician and physicist...
rediscovers and gives the first proof of the divergence theorem earlier described by Lagrange, Gauss and Green, - 1841 - Karl WeierstrassKarl WeierstrassKarl Theodor Wilhelm Weierstrass was a German mathematician who is often cited as the "father of modern analysis".- Biography :Weierstrass was born in Ostenfelde, part of Ennigerloh, Province of Westphalia....
discovers but does not publish the Laurent expansion theorem, - 1843 - Pierre-Alphonse Laurent discovers and presents the Laurent expansion theorem,
- 1850 - Victor Alexandre Puiseux distinguishes between poles and branch points and introduces the concept of essential singular pointsMathematical singularityIn mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...
, - 1850 - George Gabriel Stokes rediscovers and proves Stokes' theoremStokes' theoremIn differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Lord Kelvin first discovered the result and communicated it to George Stokes in July 1850...
, - 1873 - Georg Frobenius presents his method for finding series solutions to linear differential equations with regular singular points,
20th century
- 1908 - Josip PlemeljJosip PlemeljJosip Plemelj was a Slovene mathematician, whose main contributions were to the theory of analytic functions and the application of integral equations to potential theory.- Life :...
solves the Riemann problem about the existence of a differential equation with a given monodromic group and uses Sokhotsky - Plemelj formulae, - 1966 - Abraham RobinsonAbraham RobinsonAbraham Robinson was a mathematician who is most widely known for development of non-standard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were incorporated into mathematics....
presents Non-standard analysisNon-standard analysisNon-standard analysis is a branch of mathematics that formulates analysis using a rigorous notion of an infinitesimal number.Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson. He wrote:...
. - 1985 - Louis de Branges de BourciaLouis de Branges de BourciaLouis de Branges de Bourcia is a French-American mathematician. He is the Edward C. Elliott Distinguished Professor of Mathematics at Purdue University in West Lafayette, Indiana. He is best known for proving the long-standing Bieberbach conjecture in 1984, now called de Branges' theorem...
proves the Bieberbach conjecture,