Origins of Calculus

Study Notes

Ancient civilizations (Egypt, Babylonia) developed rudimentary concepts related to integral calculus, including area and volume estimations.
The Moscow Papyrus demonstrates formulas for calculating volumes and areas, some of which are approximate and not deduced rigorously.
Babylonian astronomers potentially discovered the trapezoidal rule.

Eudoxus and Archimedes, through the method of exhaustion, calculated areas and volumes, foreshadowing limits in calculus.
Archimedes also innovated heuristics comparable to modern calculus concepts.
Democritus was the first to consider dividing objects into infinite cross-sections.
Archimedes refined this method, eventually devising techniques akin to modern calculus.
He also tackled the tangent problem, anticipating some aspects of differential calculus.

In China, Liu Hui used the method of exhaustion (4th century AD) to determine the area of a circle.
Zu Chongzhi (5th century AD) established Cavalieri's principle for calculating the volume of a sphere.
In the Middle East, Hasan Ibn al-Haytham derived a formula for the sum of fourth powers that was used to determine the volume of a paraboloid.
Bhaskara II (India) demonstrated knowledge of differential calculus and propounded Rolle's theorem.
The Kerala School of Astronomy and Mathematics made further contributions to calculus components like infinite series expansions and Taylor series, albeit without full synthesis into a unified framework.

The 17th century witnessed the independent development of calculus by Isaac Newton and Gottfried Wilhelm Leibniz.
Sir Isaac Newton, an English physicist and mathematician, developed calculus in the 1660s, primarily for solving problems in physics, as seen in his seminal work Principia Mathematica (1687).
He viewed calculus as the "scientific description of the generation of motion and magnitudes," focusing on the concept of fluxions - instantaneous rates of change.
Gottfried Wilhelm Leibniz, a German mathematician and philosopher, also developed calculus independently around the same time.
He published his findings in the 1680s, introducing a more systematic notation and rules for working with infinitesimals, focusing on differentials, infinitesimally small increments.
Leibniz's approach emphasized the tangent problem and conceptualized calculus as a metaphysical explanation of change.

Both their approaches lacked complete rigor, sparking debates about calculus's foundations.
During the 18th and 19th centuries, mathematicians like Augustin-Louis Cauchy, Karl Weierstrass, and Bernhard Riemann reformulated calculus using limits, eliminating the need for infinitesimals.
This rigorous and consistent framework solidified the mathematical foundation for modern calculus.

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