The development of calculus and infinite series spans from the Kerala School's trigonometric series in the 14th century to the rigorous foundations laid by Cauchy and Weierstrass in the 19th century. Key figures include Madhava, Newton, Leibniz, Euler, and Cauchy, whose work transformed mathematics and enabled modern science.
Madhava, founder of the Kerala School of astronomy and mathematics, develops infinite series expansions for trigonometric functions, including the sine and cosine series. These are among the earliest known infinite series in mathematics, predating European developments by centuries. #mathematics #calculus #series
Madhava discovers the infinite series π/4 = 1 - 1/3 + 1/5 - 1/7 + ..., later known as the Leibniz series. He also provides a correction term for faster convergence, demonstrating advanced understanding of series. #mathematics #pi #series
Parameshvara, a student of Madhava, writes commentaries that preserve and expand upon Madhava's series expansions. His works help transmit the Kerala School's mathematical discoveries to later generations. #mathematics #history
Nilakantha Somayaji compiles the Tantrasangraha, which includes series expansions for trigonometric functions and the arctangent series. This work represents a high point of the Kerala School's mathematical achievements. #mathematics #astronomy
Jyeshthadeva writes the Yuktibhāṣā, a comprehensive treatise on the mathematics of the Kerala School. It provides proofs and derivations for infinite series, including the sine and cosine series, and is considered one of the first calculus texts. #mathematics #calculus
Simon Stevin publishes tables of trigonometric functions, contributing to the practical use of trigonometry. His work influences later mathematicians in the development of calculus. #mathematics #trigonometry
John Napier publishes Mirifici Logarithmorum Canonis Descriptio, introducing logarithms as a computational aid. Logarithms simplify multiplication and division, and later become integral to calculus. #mathematics #logarithms
Fermat develops a method for finding maxima and minima using infinitesimals, a precursor to differential calculus. He also works on tangents and quadrature, laying groundwork for Newton and Leibniz. #mathematics #calculus
Cavalieri introduces the method of indivisibles, a precursor to integral calculus. His work allows calculation of areas and volumes by summing infinitely many indivisible elements. #mathematics #calculus
Descartes' La Géométrie introduces analytic geometry, linking algebra and geometry. This provides a foundation for calculus by allowing curves to be represented by equations. #mathematics #geometry
Wallis introduces the concept of infinite series and the symbol ∞ for infinity. He also develops the Wallis product for π, influencing Newton's work on series. #mathematics #series
During the plague years, Newton develops his method of fluxions, a form of differential calculus. He applies it to find tangents, areas, and solve problems in physics, but does not publish immediately. #mathematics #calculus
Newton writes a manuscript on infinite series and fluxions, circulating it among colleagues. This work contains the binomial theorem for fractional exponents and methods for solving differential equations. #mathematics #series
James Gregory independently discovers the arctangent series, later known as the Gregory series. He also works on the fundamental theorem of calculus, linking differentiation and integration. #mathematics #series
Barrow's lectures contain methods for finding tangents and areas, and he recognizes the inverse relationship between them. His work influences Newton's development of calculus. #mathematics #calculus
Newton completes a treatise on fluxions and series, but it remains unpublished until 1736. The work systematizes his calculus methods and includes applications to geometry and physics. #mathematics #calculus
Gottfried Wilhelm Leibniz independently develops differential calculus, introducing the notation dx and dy. He also discovers the product rule and chain rule, and begins corresponding with other mathematicians. #mathematics #calculus
Leibniz introduces the integral sign ∫ as an elongated S for summa, and develops notation for integration. His notation becomes standard due to its convenience. #mathematics #notation
Leibniz publishes the first paper on differential calculus, detailing his method for finding maxima, minima, and tangents. This marks the first public presentation of calculus. #mathematics #calculus
Leibniz publishes a paper on integral calculus, introducing the integral sign and discussing the fundamental theorem. He also begins the notation for higher differentials. #mathematics #calculus
Newton's Principia uses geometric methods with infinitesimals to derive laws of motion and universal gravitation. Although not using his fluxional notation, it demonstrates the power of calculus in physics. #physics #calculus
Rolle publishes a critique of calculus, but his name becomes associated with Rolle's theorem, a fundamental result in calculus. The theorem states that a differentiable function has a stationary point between two equal values. #mathematics #calculus
L'Hôpital publishes the first textbook on differential calculus, which includes the rule for evaluating limits of indeterminate forms (L'Hôpital's rule). The rule was actually discovered by Johann Bernoulli. #mathematics #calculus
Newton's Opticks includes an appendix on quadrature (integration) and fluxions, providing more details on his calculus. This helps disseminate his methods. #physics #calculus
Taylor introduces the Taylor series, which represents functions as infinite sums of their derivatives. This becomes a fundamental tool in analysis and approximation theory. #mathematics #series
Berkeley criticizes the foundations of calculus, questioning the logic of infinitesimals. His critique spurs efforts to put calculus on a rigorous basis, leading to the development of limits. #philosophy #calculus
Euler proves that the sum of the reciprocals of squares equals π²/6, establishing his reputation. He uses infinite series and introduces the Euler–Mascheroni constant. #mathematics #series
Euler's work on the calculus of variations lays the foundation for this field. He derives the Euler–Lagrange equation, essential for optimization problems in physics and engineering. #mathematics #calculus
Euler's textbook systematizes analysis, covering functions, infinite series, and the exponential function. It introduces many notations still used today, such as f(x) for functions. #mathematics #analysis
Euler's comprehensive textbook on differential calculus formalizes the subject. He treats differentials as zeros and develops the calculus of finite differences. #mathematics #calculus
Euler's three-volume work on integral calculus covers methods of integration, differential equations, and applications. It becomes a standard reference for generations. #mathematics #calculus
Lagrange reformulates mechanics using calculus of variations, introducing Lagrangian mechanics. His work emphasizes analytical methods over geometric ones, advancing mathematical physics. #physics #calculus
Lagrange attempts to base calculus on algebraic series, avoiding infinitesimals. He introduces the notion of derived functions and the Lagrange remainder for Taylor series. #mathematics #analysis
Gauss's work on number theory includes contributions to analysis, such as the hypergeometric series. His rigorous approach influences the development of real analysis. #mathematics #number theory
Gauss's memoir on the hypergeometric series provides convergence criteria and establishes the series as a unifying concept in analysis. It influences later work by Riemann and others. #mathematics #series
Cauchy's textbook rigorously defines limits, continuity, and derivatives, putting calculus on a solid foundation. He introduces the epsilon-delta definition of limit, ending reliance on infinitesimals. #mathematics #analysis
Cauchy further develops integral calculus, defining the definite integral as a limit of sums. He also proves the fundamental theorem of calculus using his rigorous approach. #mathematics #calculus
Abel's work on algebraic equations also includes contributions to analysis, such as the Abel–Ruffini theorem. His work on elliptic functions influences later developments in complex analysis. #mathematics #algebra
Riemann's work on Fourier series introduces the Riemann integral and conditions for integrability. His ideas lay the groundwork for modern real analysis and the theory of integration. #mathematics #analysis
Riemann's lecture introduces Riemannian geometry, which uses calculus on manifolds. This work profoundly influences general relativity and modern differential geometry. #mathematics #geometry
Weierstrass develops the epsilon-delta definition of limit and continuity, eliminating infinitesimals entirely. He also introduces the concept of uniform convergence, crucial for analysis. #mathematics #analysis
Weierstrass shocks the mathematical community by constructing a function that is continuous everywhere but differentiable nowhere. This example highlights the need for rigorous analysis. #mathematics #analysis
Dedekind introduces Dedekind cuts to define real numbers rigorously, providing a foundation for analysis. His work clarifies the concept of continuity and the real number line. #mathematics #analysis
Lindemann proves that π is transcendental, using calculus and the properties of the exponential function. This result settles the ancient problem of squaring the circle. #mathematics #number theory
Poincaré's work on celestial mechanics uses calculus and dynamical systems theory. He discovers chaos and develops qualitative methods for differential equations. #mathematics #physics
Weierstrass's collected lectures on complex analysis and real analysis solidify the rigorous foundations of calculus. His work influences the next generation of mathematicians. #mathematics #analysis
Hilbert's problems include foundational issues in analysis, such as the continuum hypothesis and the axiomatization of real numbers. These problems guide mathematical research in the 20th century. #mathematics #foundations
