Departure question
Catch change and accumulationPort 3 of 18Where two intuitions collide
An infinitely small change cannot be held directly, yet limits reveal differentiation and accumulation as inverse processes.
This voyage is an editorial path for understanding, not a claim of direct historical influence or sole invention.
Differentiation studies instantaneous change near a point, while integration studies accumulation across an interval. Under suitable conditions such as continuity, the Fundamental Theorem of Calculus connects the two processes. Building on earlier methods from several mathematical traditions, Newton and Leibniz independently developed general procedures and notation in the seventeenth century.
The mathematics of change (derivatives) and accumulation (integrals).
Q.d/dx(x²)
Q.∫ x dx
Ports in time
Follow the scenes to see problems, notation, standards of proof, and applications changing across different times and places.
In Syracuse, Archimedes found the area between a parabola and a chord through a geometric series of successively smaller triangles and a bounding proof. It is a powerful precedent, not the modern definition of an integral.
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Continue this scene on the mapDuring the Cambridge closure of 1665–1666, Newton developed ideas about series, tangents, areas, fluxions, optics, and gravitation at Woolsthorpe and nearby. Their exact dates, locations, completion, and publication do not collapse into one miracle day.
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Continue this scene on the mapLeibniz tested d and ∫ in private Paris manuscripts in 1675. His differential method appeared in Leipzig in 1684, followed by printed integral notation in 1686; the Bernoullis and teaching networks helped the notation spread.
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Continue this scene on the mapThe Cours d’analyse made limits, continuity, sequences, and convergence more explicit. It was a decisive stage of rigor, not the final modern foundation of the real numbers or every epsilon–delta quantifier.
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Continue this scene on the mapHis 1859–1864 lectures rebuilt analysis around arithmetic conditions; in 1872, a continuous nowhere-differentiable function exposed the limits of assuming that every curve is locally smooth.
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Continue this scene on the mapGradient descent in machine learning, tomographic reconstruction in medical imaging, marginal analysis in economics, rocket trajectories, and differential equations for heat, fluids, and electromagnetic fields all use calculus to model continuous change and accumulation.
Curated sources and problems. Bring one discovery back from OEIS, Project Euler, MathOverflow, or arXiv.
No concept belongs to one person
These are not inventor credits. They are different ports: opening a problem, sharpening a language, or carrying it into another world.
Number lenses
These numbers are editorial lenses for the voyage, not required prerequisites.
The completeness of the reals gives approaching limits a place to land.
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Extending calculus to complex numbers reveals deep symmetries in waves, rotations, and analysis.
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Follow the languages built to calculate a world that will not stand still, from planets and fluids to waves, optimization, and chaos.
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Concept genealogy
Concepts arriving from before
Current port
Calculus
Only direct editorial links are shown; this is not a complete learning order or historical influence line.
