Leibniz publishes differential calculus

Seventeenth-century mathematicians investigated tangents, areas, and maximum or minimum values. Older geometrical techniques often required a fresh construction for each curve. Leibniz encountered mathematical research in Paris and developed methods for continuous change. His short paper in Acta Eruditorum in 1684 presented a new approach to extrema and tangents. It treated small differences in variables through rules that could be reused across different problems.

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Leibniz used d for differentials and later an integral sign for summation. Notation could travel with variables through products, quotients, and composed relations, helping a calculator track the source of a change. Differential relations described how one quantity changes with another, while summing contributions addressed areas. The notation made methods easier to communicate. Mathematicians including the Bernoulli brothers developed differential equations and other techniques from this growing body of work.

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Newton independently developed mathematics of change, and the two traditions later became involved in a priority dispute. Research nevertheless continued through letters, journals, and textbooks. Leibniz’s notation became widely used. Velocity, acceleration, and trajectories in mechanics could be treated with calculus, and applications expanded into probability, economics, and engineering. Continuous change acquired a shared language while new applications prompted additional methods and further examination of its theoretical foundations.

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