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Spatial atlas

VOYAGE TEN · CONNECTING THE INSTANT AND THE WHOLE

Calculus from Many Springs — Change and Accumulation Become One Language

Walk 2,100 years as problems of area, volume, tangents, extrema, and planetary speed gather partial solutions across cultures, meet symbolic algebra and geometry, and become a language in which change and accumulation can undo one another.

QUESTION FOR THE ROUTE

Why did differentiation and integration become two directions through the same phenomenon, and why did their union become a reusable public method in the late seventeenth century?

WHAT THE LINE DOES NOT CLAIM

This line is neither the straight transmission of one manuscript from Syracuse to London nor a priority map claiming calculus was already complete. It distinguishes documented transmission, independent precedents, recurring problems, and later applications; direct transmission from Kerala to Europe is not shown as established fact.

The camera rests on each city while you read, then eases through the runway between scenes. Select any marker or scene link to travel in either direction.

The same route, four questions

A lens never hides a scene or proves a cause. It changes which places you compare first, and the URL preserves your choice.

Whole routeTranslation networksCommercial demandObservation and institutionsPatronage and power

The whole-route view keeps problem, cognitive change, place, movement, and evidence boundary at equal weight. Choose a lens when you want to test a different explanation against the same scenes.

24 scroll-controlled map scenes

Live map · 지도를 불러오는 중…

03 / 24 · c. 850 CE

Baghdad

  1. 01 · c. 250 BCE

    Syracuse · Main activity

    Filling without End while Keeping Hold of the Answer — Archimedes' Parabola

    Archimedes inscribed a triangle in a parabolic segment, filled the gaps with smaller triangles, and proved geometrically that their continuing sum was four-thirds of the first. Infinite summation and upper and lower bounds meet here, but coordinate functions, a modern definition of limit, and Riemann integration should not be projected backward onto the argument.

    PAUSE AND ASK

    How can an exact area be proved while adding indefinitely many pieces?

    How the idea changed

    Instead of stating one formula for curved area, trap the answer through a geometric series of inscribed triangles and an outer bound.

    What this place made possible

    Syracuse anchors Archimedes' career and a setting where Hellenistic geometry met mechanics, though the exact writing room of the theorem is unknown.

    How it moved

    Greek exhaustion methods → Archimedes' repeated triangles and ratio proof → Greek, Arabic, and Latin manuscript traditions → early modern rereading of area problems

    Do not overclaim

    The precedent for infinite summation is real, but it is not integration equipped with functions, coordinates, modern limits, and Riemann sums.

    Evidence sources
    • MacTutor — Archimedes

      Supports: The geometric-series argument for quadrature of the parabola and Archimedes' work at Syracuse

    Stable link to this scene
    SyracuseLuoyang

  2. 02 · 263 CE

    Luoyang · Reception

    Cut the Circle More Finely and the Error Comes into View — Liu Hui's Commentary

    In his 263 commentary on the Nine Chapters, Liu Hui repeatedly doubled the sides of regular polygons inside a circle and narrowed its area. He argued about the remaining pieces instead of reporting only a decimal. The exact writing room is unknown, so Luoyang is an editorial anchor for the Cao Wei textual world, not evidence of a modern completed integral calculus there.

    PAUSE AND ASK

    When a polygon keeps gaining sides, how do we explain the remaining error as well as the number?

    How the idea changed

    Read an approximation as narrowing upper and lower bounds, placing subdivision and error control inside one argument.

    What this place made possible

    No exact writing site survives. Luoyang is a reception pin for the Cao Wei capital's textual and administrative calculating environment.

    How it moved

    Accumulated problems of the Nine Chapters → Liu Hui's 263 commentary and geometric dissection → later Chinese commentary and teaching → repeated algorithms for pi and area

    Do not overclaim

    Luoyang is an editorial textual anchor, not a verified writing room, and circle dissection is not labeled a completed modern integral calculus.

    Evidence sources
    • MacTutor — Liu Hui

      Supports: The 263 commentary on the Nine Chapters and its circle dissection arguments for area and pi

    Stable link to this scene
    LuoyangBaghdad

  3. 03 · c. 850 CE

    Baghdad · Composition

    Translating a Figure and Proving It Again — The Banu Musa on Measurement

    The Banu Musa's Book on the Measurement of Plane and Spherical Figures moved Archimedean problems into an Arabic mathematical network while adding proofs and generalizations for circles, spheres, and conic solids. Translation was a workshop for reconstruction, not a warehouse. No evidence shows that this single book traveled in a straight line to seventeenth-century European calculus.

    PAUSE AND ASK

    When a geometric problem enters a new language, is it merely preserved or rebuilt through new proofs?

    How the idea changed

    Reprove and generalize classical areas and volumes for Arabic readers, turning translation into creative mathematical work.

    What this place made possible

    Court patronage, book collection, translators, and mathematicians in Abbasid Baghdad gathered the time and materials needed to compare geometric knowledge across languages.

    How it moved

    Greek measurement traditions → Baghdad translation and research community → the Banu Musa's Arabic treatise and new proofs → later Arabic and Latin geometric transmission

    Do not overclaim

    The Banu Musa are not reduced to preserving Greece, and one book is not turned into a documented straight route to seventeenth-century calculus.

    Evidence sources
    Stable link to this scene
    BaghdadCairo

  4. 04 · c. 1020 CE

    Cairo · Composition

    Measuring a Curved Solid through Sums of Powers — Ibn al-Haytham

    Ibn al-Haytham used sums of powers and geometric argument in problems including the volume of a paraboloid of revolution, while rebuilding the relation among light, vision, experiment, and mathematics in optics. Cairo anchors his mature career. These are major precedents in the history of integration, not a general integration algorithm or a completed fundamental theorem.

    PAUSE AND ASK

    If a curved solid cannot be cut directly, how can sums of powers approach its volume?

    How the idea changed

    Connect continuous volume to formulas for finite sums and exhaustion, making summation central to curved measurement.

    What this place made possible

    Cairo anchors a mature career joining books, optics, astronomy, and geometry, while the exact writing place and year of individual results remain approximate.

    How it moved

    Archimedean solid measurement → Arabic geometry and arithmetic → Ibn al-Haytham's power sums and paraboloid result → Latin reception of optics and geometry and later rereading

    Do not overclaim

    Cairo marks his mature workplace; the result does not include a general integration algorithm or the Fundamental Theorem of Calculus.

    Evidence sources
    Stable link to this scene
    CairoUjjain

  5. 05 · 1150 CE

    Ujjain · Main activity

    Trying to Calculate a Planet at an Instant — Bhaskara II

    In astronomical calculation Bhaskara II discussed instantaneous planetary motion and used relations involving small sine differences and vanishing change at extrema. Ujjain's astronomical tradition supplied recurring pressure from calendars and planetary positions. This should not be enlarged into the modern definition of a derivative or a differential calculus for arbitrary functions.

    PAUSE AND ASK

    When average speed is insufficient for a planet, how can its motion at this very instant be described?

    How the idea changed

    Capture motion over a tiny time interval and the vanishing of change at an extremum inside astronomical calculation.

    What this place made possible

    Ujjain's astronomical and calendrical tradition required planetary positions and sine tables to be recalculated across generations, with Bhaskara II at the center of that institutional lineage.

    How it moved

    Indian astronomy and trigonometric tables → Ujjain tradition after Brahmagupta → Bhaskara II on instantaneous motion and differences → later Sanskrit commentary and astronomical teaching

    Do not overclaim

    The language of instantaneous motion and particular difference relations matters, but it does not amount to the modern derivative definition and a complete general rule set.

    Evidence sources
    • MacTutor — Bhaskara II

      Supports: Ujjain's astronomical tradition and Bhaskara's results involving instantaneous motion, sine differences, and extrema

    Stable link to this scene
    UjjainSangamagrama

  6. 06 · c. 1400 CE

    Sangamagrama · Main activity

    Adding Corrections to an Infinite Series — Madhava

    Series for sine, cosine, and arctangent and refined computations of pi are attributed to Madhava in later Kerala texts. Correction terms improved approximations rather than merely adding more terms. Madhava's own mathematical writings do not survive, and no documented route establishes a direct transmission of these results to early modern Europe.

    PAUSE AND ASK

    If an infinite series can never be fully summed, what makes a short partial sum trustworthy?

    How the idea changed

    Express trigonometric quantities and pi through infinite series and use correction terms to shrink the remainder, turning infinity into a practical procedure.

    What this place made possible

    Sangamagrama is the traditional pin for Madhava's region. Kerala's astronomical and calendrical community and teacher–student textual network preserved results in later works.

    How it moved

    Sanskrit astronomy and trigonometry → Madhava's calculations and lost writings → later attributions and demonstrations by Nilakantha, Jyesthadeva, and others → teaching and copying within the Kerala school

    Do not overclaim

    Madhava's own mathematical writings are lost and known through later attributions. Direct transmission from Kerala to Newton or Leibniz is not established.

    Evidence sources
    Stable link to this scene
    SangamagramaPrague

  7. 07 · 1609 CE

    Prague · Main activity

    A Planet Sweeps Equal Areas in Equal Times — Kepler's Orbit

    After relentlessly calculating with Tycho Brahe's observations of Mars, Kepler recognized that planetary speed varies and that the line from Sun to planet sweeps equal areas in equal times. Prague anchors the work with observations and calculations. The area law is not itself calculus, but it forced changing speed and accumulated area into the same problem.

    PAUSE AND ASK

    How can time be translated into area along an orbit when a planet's speed keeps changing?

    How the idea changed

    Replace uniform circular motion with an observationally adequate ellipse and an equal-time–equal-area relation that makes varying speed geometric.

    What this place made possible

    The imperial mathematician's office in Prague gave Kepler Tycho's precise Mars observations and calculation time, while also producing conflict over data access and patronage.

    How it moved

    Tycho's observations, instruments, and assistants → Kepler's Prague calculations → publication of New Astronomy in 1609 → continuous-change problems in planetary motion and mechanics

    Do not overclaim

    The area law is a decisive problem joining rate and accumulation, not itself a general differentiation and integration algorithm or the Fundamental Theorem.

    Evidence sources
    Stable link to this scene
    PragueLinz

  8. 08 · 1615 CE

    Linz · Publication

    Why a Wine Barrel Resists One Simple Gauge — Kepler's Solid Geometry

    A dispute over gauging Austrian wine barrels led Kepler to publish New Solid Geometry of Wine Barrels. Comparing solids through thin cross-sections helped shape the problem landscape later addressed by Cavalieri and integration. Commerce was a catalyst, but the book did not supply the modern integral formula or its rigorous foundations.

    PAUSE AND ASK

    When does one gauging mark fail for a wine barrel that bulges through the middle?

    How the idea changed

    Compare irregular commercial solids through thin sections and combinations of known solids, turning volume into a systematic problem family.

    What this place made possible

    Wine trade in Linz and Kepler's regional mathematical office brought abstract solid measurement into recurring problems of price, tax, and inspection.

    How it moved

    Austrian wine-barrel gauging → Kepler's sections and solids of revolution → Linz publication in 1615 → Cavalieri and early modern solid measurement

    Do not overclaim

    Commercial demand sharpened the question, but one barrel method did not automatically generate modern integration as a single cause.

    Evidence sources
    Stable link to this scene
    LinzBologna

  9. 09 · 1635 CE

    Bologna · Publication

    Comparing Area by Lines and Volume by Planes — Cavalieri

    Cavalieri treated figures as collections of indefinitely many parallel lines or planes and compared corresponding sections to obtain areas and volumes. Bologna's university and print network made the controversial method teachable and criticizable. Its indivisibles were not rigorously founded in a modern sense, yet the method gathered repeated problems under a general principle.

    PAUSE AND ASK

    If a figure is viewed as indefinitely many lines or planes, can its area or volume be compared?

    How the idea changed

    Lift correspondence between sections into a ratio between whole figures, turning isolated area and volume tricks into a general comparison principle.

    What this place made possible

    A Bologna chair and Italian print and correspondence networks made indivisibles public and repeatedly testable by Galileo's circle and its critics.

    How it moved

    Solid problems around Kepler and Galileo → Cavalieri's principle of indivisibles → Bologna publication in 1635 → Torricelli, Wallis, and European quadrature

    Do not overclaim

    Indivisibles were powerful, but their ontology and conditions of summation were not rigorously settled by modern standards.

    Evidence sources
    Stable link to this scene
    BolognaToulouse

  10. 10 · c. 1636 CE

    Toulouse · Composition

    Comparing Two Almost-Equal Values to Find Extrema and Tangents — Fermat

    Fermat's method of adequality perturbed a quantity, canceled common terms, and solved problems of maxima, minima, and tangents. Manuscripts and solutions written while he served as a magistrate in Toulouse circulated through Mersenne's correspondence. The approximate 1636 date anchors circulation, not a general limit definition or a completed differential calculus.

    PAUSE AND ASK

    Why do extrema and tangents appear after a quantity is slightly changed and common terms are canceled?

    How the idea changed

    Replace a separate geometric construction for each curve with one algebraic comparison of small changes for tangency and optimization.

    What this place made possible

    Fermat calculated between legal duties in Toulouse rather than inside a research institute, and Mersenne's Paris correspondence moved manuscripts into public dispute.

    How it moved

    Viete's symbolic algebra and ancient tangent problems → Fermat's adequality manuscripts → correspondence disputes with Mersenne and Descartes → wider algorithms for extrema and tangents

    Do not overclaim

    The year 1636 approximately anchors circulation; adequality is not equated with the exact modern limit definition or a completed differential calculus.

    Evidence sources
    Stable link to this scene
    ToulouseLeiden

  11. 11 · 1637 CE

    Leiden · Publication

    Turning Curves into Equations and Equations into Curves — Descartes' Meeting

    La Geometrie, an appendix to the Discourse published at Leiden, used symbolic algebra and coordinate relations to classify curves and make tangent problems calculable. The familiar Cartesian plane was not completed by one person on one page. Still, the encounter of algebra and geometry greatly widened the possibility of treating changing curves through general rules.

    PAUSE AND ASK

    What becomes calculable when a curve drawn by moving points is captured as an equation?

    How the idea changed

    Translate geometric curves into symbolic algebraic relations, making tangents, intersections, and degree common calculable objects across curves.

    What this place made possible

    The relatively open Dutch print environment and a Leiden press connected a French text to Latin scholarly networks and readers across European correspondence.

    How it moved

    Viete's algebraic symbolism and Apollonius's conics → Descartes' Geometry of 1637 → van Schooten's Latin editions and commentary → curve calculation for Newton and Leibniz's generation

    Do not overclaim

    The encounter of coordinate reasoning and algebra was decisive, but today's entire Cartesian coordinate system was not completed by Descartes alone in one moment.

    Evidence sources
    Stable link to this scene
    LeidenCambridge

  12. 12 · 1664 CE

    Cambridge · Teaching / position

    At the Threshold where Tangents and Areas Undo One Another — Barrow's Lectures

    In Cambridge geometry lectures, Isaac Barrow showed that the tangent to a curve representing accumulated area is related to the original curve. Newton attended and helped prepare the lectures for publication. The result comes remarkably close to the fundamental theorem, but it is not the identical modern statement with today's functions, continuity hypotheses, and notation.

    PAUSE AND ASK

    Why does differentiating the new curve formed by accumulated area recover the original height?

    How the idea changed

    Reveal geometrically that tangent and area problems are not separate but two directions through one process.

    What this place made possible

    Cambridge's first Lucasian chair, lectures, students, and manuscript publication placed Barrow's geometry close to the young Newton's calculations.

    How it moved

    Area and tangent work by Cavalieri and Torricelli → Barrow's Cambridge lectures → Newton's attendance and editorial help → generalization of the inverse relation between rate and accumulation

    Do not overclaim

    It comes very close to the modern Fundamental Theorem, but is not retrospectively described as the identical theorem with today's functions, continuity assumptions, and notation.

    Evidence sources
    Stable link to this scene
    CambridgeWoolsthorpe-by-Colsterworth

  13. 13 · c. 1666 CE

    Woolsthorpe-by-Colsterworth · Commemorative scene

    Binding Flowing Quantities and Instantaneous Rates in Private Papers — Newton

    During the plague closure of Cambridge, Newton rapidly developed ideas about infinite series, tangents, areas, and fluxions at Woolsthorpe and nearby. He conceived quantities as flowing in time and fluxions as their rates. Some exact locations and dates in the 1665–1666 papers remain uncertain or were entered later, and no public textbook appeared from this moment fully formed.

    PAUSE AND ASK

    If quantities flow through time, how can instantaneous rate and total change occupy one system?

    How the idea changed

    Treat changing quantities as fluents and their instantaneous rates as fluxions, joining series, tangents, and areas in the temporal language of motion.

    What this place made possible

    When plague closed the university, the family farm at Woolsthorpe and nearby stays became refuges for concentrated work, but not every 1665–1666 paper can be fixed to one room.

    How it moved

    Barrow's lectures, Wallis's series, and Cartesian curves → Newton's private papers of 1665–1666 → limited sharing through Collins and Oldenburg → later fluxional publication and priority dispute

    Do not overclaim

    Some dates and locations may have been entered later, and discovery, systematization, publication, and diffusion of notation are different milestones.

    Evidence sources
    Stable link to this scene
    Woolsthorpe-by-ColsterworthParis

  14. 14 · 1675 CE

    Paris · Composition

    Trying Small Differences and an Elongated S on Paper — Leibniz

    While in Paris, Leibniz used d for small differences and an elongated S, the integral sign, for summation in a 1675 manuscript. Notation compressed thought and made relations such as product and chain rules manipulable. The date marks a private notebook, not public release or the instantaneous completion of a stable notation system.

    PAUSE AND ASK

    Is good notation merely shorthand for an answer, or a tool for discovering rules not yet seen?

    How the idea changed

    Arrange the small difference d and elongated summation S as manipulable relations, making differentiation and integration portable across problems.

    What this place made possible

    Paris gave Leibniz Huygens's guidance, circles around the Royal Academy, and access to letters and books, while diplomacy and patronage competition shaped his time.

    How it moved

    Huygens's guidance and Paris mathematical exchange → Leibniz's private 1675 manuscripts and notation experiments → print in 1684 and 1686 → the Bernoullis and Continental teaching networks

    Do not overclaim

    The year 1675 marks private manuscripts; the meaning and use of d and the integral sign did not instantly stabilize in their modern form that day.

    Evidence sources
    Stable link to this scene
    ParisLeipzig

  15. 15 · 1684 CE

    Leipzig · Publication

    A Six-Page Paper Opens a Calculating Community — Leibniz Goes Public

    In Leipzig's Acta Eruditorum, Leibniz published a short account of differential calculus with rules for tangents and extrema. Difficult and compressed as it was, the language of dx and dy was extendable by students and the Bernoulli brothers. The integral sign appeared in print in a follow-up paper in 1686; the 1684 article did not complete all modern calculus.

    PAUSE AND ASK

    For a discovery to become a community tool, what must be made public before a perfect textbook exists?

    How the idea changed

    Publish compact rules, notation, and examples in a journal, turning a private method into an algorithm others can learn, extend, and challenge.

    What this place made possible

    Leipzig's Acta Eruditorum joined post, print, and review in an early journal infrastructure that moved new calculations rapidly among European scholars.

    How it moved

    Leibniz's Paris and Hanover manuscripts → publication of differential calculus in 1684 and the integral sign in 1686 → Jakob and Johann Bernoulli's solutions and teaching → l'Hopital's 1696 textbook

    Do not overclaim

    The 1684 paper is pivotal for public differential calculus, but the integral sign appeared in print in 1686 and the paper did not contain all modern definitions and conditions.

    Evidence sources
    Stable link to this scene
    LeipzigLondon

  16. 16 · 1687 CE

    London · Publication

    Putting Celestial and Terrestrial Motion under One Law — The Principia

    Newton's Principia, published in London, applied laws of motion and universal gravitation to planets, the Moon, tides, and falling bodies. Fluxional and limiting reasoning stood behind parts of the work, but the book was presented mainly in classical geometric form. The year 1687 displays the public power of the mathematics, not the publication of Newton's differential-notation textbook.

    PAUSE AND ASK

    What mathematics is needed to explain terrestrial falling and the Moon's orbit through the same laws of change?

    How the idea changed

    Combine instantaneous force, changing velocity, and accumulated orbit in one mechanics for heaven and Earth, greatly widening mathematics' explanatory reach.

    What this place made possible

    The Royal Society network, Halley's editorial and financial support, and London print turned Cambridge manuscripts into a reviewable public book.

    How it moved

    Kepler's laws, Galilean motion, and Newton's fluxional background → Halley's question and encouragement → the London Principia of 1687 → analytic mechanics of Euler and Lagrange

    Do not overclaim

    The book was written mainly geometrically and 1687 was not the first full public exposition of fluxions; successful application and diffusion of notation are kept distinct.

    Evidence sources
    Stable link to this scene
    LondonGroningen

  17. 17 · 1696 CE

    Groningen · Teaching / position

    Throwing the Fastest-Descent Curve to Europe — Bernoulli's Challenge

    Johann Bernoulli, then professor at Groningen, challenged Europe's mathematicians to find the curve of fastest descent under gravity. Newton, Leibniz, Jakob Bernoulli, and l'Hopital responded, testing the new calculi through competing methods. The challenge was printed in Leipzig, so this pin marks the proposer's workplace and correspondence network rather than the publication city.

    PAUSE AND ASK

    Which curve gives the fastest descent rather than the shortest path?

    How the idea changed

    Move beyond finding the tangent to one curve toward choosing an optimal path among whole families of curves, widening calculus toward variation.

    What this place made possible

    The Groningen chair gave Johann Bernoulli students, salary, and authority to pose the problem, while a Leipzig journal handled printing; the places divided the work.

    How it moved

    Galileo's falling-body question → Johann Bernoulli's 1696 challenge → answers by Newton, Leibniz, and Jakob through Acta and correspondence → calculus of variations and optical analogy

    Do not overclaim

    The pin marks the proposer's workplace while publication occurred in Leipzig; competing solutions are not reduced to one school's exclusive victory.

    Evidence sources
    Stable link to this scene
    GroningenSaint Petersburg

  18. 18 · 1736 CE

    Saint Petersburg · Publication

    Rewriting Geometric Mechanics as a System of Differential Equations — Euler

    Euler's Mechanica organized particle motion through analytic expressions and differential equations, turning Newtonian mechanics into a general method for solving problems. The Saint Petersburg Academy combined salary, colleagues, publishing, and a supply of problems. Euler did not create analysis alone; he powerfully systematized Bernoulli networks and several calculating traditions.

    PAUSE AND ASK

    Can forces and motions be solved through one procedure without drawing a new geometric diagram each time?

    How the idea changed

    Translate motion into coordinates, functions, and differential equations, turning calculus from techniques for individual curves into reusable analysis.

    What this place made possible

    The Saint Petersburg Academy combined state patronage, international recruitment, publishing, and prize problems, sustaining Euler's immense program of calculation.

    How it moved

    Johann Bernoulli's teaching, Newtonian mechanics, and Leibnizian notation → Euler's 1736 Mechanica → differential equations, continua, and engineering calculation → European textbooks and academies

    Do not overclaim

    Euler's systematization was decisive, but it is not a solitary invention erasing the Bernoullis, Newton, Leibniz, colleagues, and assistants.

    Evidence sources
    Stable link to this scene
    Saint PetersburgMilan

  19. 19 · 1748 CE

    Milan · Publication

    Teaching One Road from Algebra to Integration — Agnesi's Textbook

    Maria Gaetana Agnesi's Italian Institutions of Analysis connected algebra, analytic geometry, differential calculus, and integral calculus in two systematic learning volumes. By comparing authors and filling explanatory gaps, it showed that making knowledge learnable can matter as much as an invention claim. It should not be advertised as the first calculus book of every kind in Europe.

    PAUSE AND ASK

    For an invented method to become the next generation's knowledge, who fills the explanatory gaps and designs the learning order?

    How the idea changed

    Connect algebra, analytic geometry, differentiation, and integration into a vernacular learning path that compares difficult papers and lets readers follow the rules themselves.

    What this place made possible

    A wealthy Milan household, learned salon, and print environment gave Agnesi access to languages and books, while women's formal university careers and public scholarship remained sharply constrained.

    How it moved

    Methods from Newton, Leibniz, the Bernoullis, and Euler → Agnesi's comparison, organization, and Italian explanation → Milan publication in 1748 → French and English translation and educational reception

    Do not overclaim

    Agnesi is neither reduced to the Witch curve anecdote nor exaggerated as author of the first calculus book of every kind in Europe.

    Evidence sources
    Stable link to this scene
    MilanParis

  20. 20 · 1788 CE

    Paris · Publication

    Building a Language for Motion without a Separate Diagram for Every Problem — Lagrange

    Lagrange's Analytical Mechanics organized levers, orbits, and oscillations through general coordinates and analytic and variational principles rather than individual geometric diagrams. Much of the work was developed in Berlin and the book was published in Paris. Its famous programmatic claim does not mean physical intuition and experiment became unnecessary.

    PAUSE AND ASK

    When different machines and orbits share one coordinate principle, what becomes visible and what gets hidden?

    How the idea changed

    Organize force and motion through generalized coordinates, energy, and variational relations, foregrounding structure and conservation over separate geometric diagrams.

    What this place made possible

    The book was published in Paris while core work matured during Lagrange's Berlin Academy years, preserving the split among research, editing, and publication sites.

    How it moved

    Eulerian analytic mechanics and variation → Lagrange's Berlin manuscripts → Paris publication of Mecanique analytique in 1788 → Hamiltonian mechanics and state spaces of modern physics

    Do not overclaim

    Paris is pinned on publication evidence; analytic unification did not make geometry, experiment, or physical interpretation unnecessary.

    Evidence sources
    Stable link to this scene
    ParisParis

  21. 21 · 1821 CE

    Paris · Publication

    After Successful Infinitesimal Calculation, Asking Again about Limits — Cauchy

    Drawing on his Ecole Polytechnique lectures, Cauchy published Cours d'analyse to control limits, continuity, sequences, and series convergence more explicitly. The question shifted from successful examples to the conditions under which a procedure is allowed. Today's complete construction of the reals and fully quantified epsilon–delta framework did not arrive finished in this one book.

    PAUSE AND ASK

    How is working in many examples different from working in every permitted case?

    How the idea changed

    Bring limits, continuity, and convergence conditions hidden behind successful infinitesimal manipulation into explicit textbook definitions and theorems.

    What this place made possible

    Post-Revolutionary Paris and the Ecole Polytechnique standardized engineering education and lecture notes; Cauchy's textbook grew inside that institutional audience and examination structure.

    How it moved

    Successes and paradoxes of eighteenth-century series and differential equations → Cauchy's lectures and 1821 Cours d'analyse → criticism of convergence conditions → Weierstrass and arithmetized analysis

    Do not overclaim

    Cauchy is a decisive stage of rigor, but one book did not complete today's real-number completeness and every quantified epsilon–delta formulation.

    Evidence sources
    Stable link to this scene
    ParisParis

  22. 22 · 1822 CE

    Paris · Publication

    Hearing the Spread of Heat as a Sum of Waves — Fourier

    Fourier expressed solutions of the heat equation through trigonometric series and opened a dispute over the senses in which even discontinuous shapes can be represented by sums of functions. Calculus expanded beyond speed and area into partial differential equations, boundary conditions, and frequency analysis. Not every function equals its Fourier series pointwise without conditions.

    PAUSE AND ASK

    How can a local law for changing heat at one point predict an entire temperature profile later?

    How the idea changed

    Connect a local partial differential equation to trigonometric series forming a global profile, extending calculus into a translator among time, space, and frequency.

    What this place made possible

    Review, dispute, and publication through the Paris Academy made heat theory a shared test of physical experiment, the concept of function, and series convergence.

    How it moved

    Heat experiments and string problems of Euler and d'Alembert → Fourier's 1807 submission and dispute → 1822 Analytical Theory of Heat → Dirichlet convergence conditions, signal processing, and PDEs

    Do not overclaim

    Fourier's insight is not turned into the unconditional claim that every function always equals its series; senses and conditions of representation are distinguished.

    Evidence sources
    • MacTutor — Joseph Fourier

      Supports: The 1807 heat memoir, publication in 1822, and disputes over trigonometric series and the heat equation

    Stable link to this scene
    ParisBerlin

  23. 23 · 1861 CE

    Berlin · Teaching / position

    Turning 'Gets Close' into a Quantified Contract — Weierstrass' Lectures

    In Berlin lectures, Weierstrass rebuilt limits, continuity, differentiation, and integration through arithmetic conditions, teaching integral calculus in 1860–1861 and foundations of real numbers soon after. His 1872 example of a continuous nowhere-differentiable function exposed the limits of curve intuition. The epsilon–delta approach was not one slogan invented by one person in one lecture.

    PAUSE AND ASK

    Without trusting a smooth-looking picture, how can 'sufficiently close' become a quantified promise?

    How the idea changed

    Separate continuity from differentiability and specify a distance responding to every error tolerance, rebuilding analysis on arithmetic conditions.

    What this place made possible

    Repeated Berlin lectures, student notes, and seminar networks spread a durable standard of rigor through European mathematics beyond any single publication.

    How it moved

    Limit work by Cauchy and Bolzano → Weierstrass's Berlin lectures of 1859–1864 → student notes and real-number foundations of Cantor and Dedekind → modern real-analysis teaching

    Do not overclaim

    The year 1861 anchors integral-calculus lectures; epsilon–delta methods and real-number foundations were not completed in one lecture by one person.

    Evidence sources
    Stable link to this scene
    BerlinLondon

  24. 24 · 1865 CE

    London · Publication

    Linking the Change of Invisible Fields to the Speed of Light — Maxwell

    Maxwell's Royal Society paper joined spatial and temporal changes of electric and magnetic fields and found an electromagnetic wave speed matching that of light. Differential equations became a predictive language for fields. The 1865 paper used many component equations and mechanical models; the compact four vector equations taught today were shaped later by Heaviside and others.

    PAUSE AND ASK

    How can local change in invisible fields predict the speed of light propagating far away?

    How the idea changed

    Extend calculus from tracking particle paths to fields valued throughout space and coupled equations for their change in time and space.

    What this place made possible

    The Royal Society in London connected telegraphy, electrical measurement, experimental physics, and Maxwell's mathematics in a public arena of review and print.

    How it moved

    Faraday's lines of force and electrical measurement → Maxwell's 1865 system of field equations → electromagnetic waves and light → Heaviside's vector reformulation, radio, and modern field theory

    Do not overclaim

    The 1865 paper used many component equations and mechanical models rather than today's four vector equations; later reformulation is not projected backward.

    Evidence sources
    Stable link to this scene

PAUSE THE FILM · TOUCH THE IDEA

Two directions through one motion

A traveler follows s(t)=t². Shrink a time interval to read local speed; split time into pieces to rebuild total displacement. Move from intuition to the exact hypotheses at your own pace.

Shrink the interval: local change

From t=2 to t=2+h, the average velocity is 4+h. Make h smaller and the secant turns toward the tangent.

Secant approaching a tangentFor position s(t)=t squared, the secant from t=2 to t=3 has slope 5, approaching tangent slope 4.0123s(t)=t²tangent slope 4secant 5

(s(2+h)−s(2)) / h = 4+h

5 4

Refine the pieces: global accumulation

Add left rectangles under v(t)=2t from zero to two. More partitions fill the missing triangular slivers.

Rectangles accumulating under velocity4 left rectangles approximate the area under v(t)=2t from zero to two as 3, approaching 4.012v(t)=2t4 left rectangles

left sum = 4−4/n

3 4

What the two directions mean

Differentiation reads velocity v=s′ from position s. Integration rebuilds the displacement s(2)−s(0) from that velocity. The Fundamental Theorem explains this inverse relation under suitable hypotheses.

Do not confuse the matching 4s

Here the derivative at t=2 is 4 m/s and the displacement from 0 to 2 is 4 m. The matching numeral is a convenient feature of s=t², not a rule that every derivative equals every integral; even their units differ.

TOUCH THE MATHEMATICS

Calculus from Many Springs — How Change and Accumulation Became One Language

Begin with area in Syracuse, a circle in Luoyang, measurement in Baghdad and Cairo, instantaneous motion at Ujjain, and series at Sangamagrama. Then cross the shoulders built by Kepler, Cavalieri, Fermat, Descartes, and Barrow before Newton and Leibniz bind change and accumulation in different languages. Bernoulli, Euler, Agnesi, Lagrange, Fourier, Cauchy, Weierstrass, and Maxwell carry that language into textbooks, mechanics, heat, rigor, and fields across twenty-four scenes.

Continue as a twenty-four-scene cinematic journey

OPEN THE FULL MAP

Reconnect change and accumulation on the concept page

Revisit derivatives, integrals, and the Fundamental Theorem through formulas and examples.

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