Leibniz's Calculus Notation

Timeline

1. 1675–1676Paris · ParisLeibniz studied the geometry of infinitesimals under Huygens's guidance and corresponded with Tschirnhaus. His 1675 manuscripts used the integral sign for summation, dx for increments of the independent variable, and gave the product rule for differentiation. This notation turned finding tangents and areas into computable algorithms rather than relying solely on geometric figures.
2. 1684–1686Leipzig · LeipzigIn 1684 Leibniz published 'A New Method for Maxima and Minima' in Acta Eruditorum, making public the differential notation and algorithms for extrema and tangents; in 1686 he published a follow-up on integration. This was the first printed publication of calculus in a transmissible symbolic form, allowing European scholars to reproduce the algorithms without relying on private correspondence.
3. 1699–1713London and Hanover · EuropeNewton obtained results in calculus earlier but did not publish them in equivalent notation for a long time. The Royal Society published the Newton-favoring Commercium Epistolicum in 1713. The priority dispute damaged scholarly exchange across the English Channel, but paradoxically further consolidated Leibniz's notation among the Bernoulli school on the Continent.
4. 1713European mathematical community · EuropeIn the 18th century, Euler, the Bernoulli family, Lagrange, and others made dx, d²x, and the integral sign the everyday language of analysis. Britain did not widely adopt Leibniz's notation until the 19th century. Today calculus, differential equations, and differential forms still rest on this notation, though the rigorous interpretation of infinitesimals has been rewritten in terms of limits and axioms of analysis.