An interactive, chronological history of calculus, built around one question: how do we calculate with infinitely many, infinitely small things? It follows that question for four thousand years, from Babylonian clay tablets and Zeno's paradoxes to Cauchy's limits and Robinson's infinitesimals, with 25 hands-on labs and 56 step-by-step proofs, worked examples and source readings along the way.
- A true timeline. 61 moments sit in date order on a vertical line marked in millennia, then centuries, then decades. The line stretches where history gets crowded, and a rail on the right of wide screens shows true time and lets you jump.
- Cards that open. Each compact card opens in place to show the full story, the key mathematics, a quote, its artwork and readings from the original texts. Its lab, or its first proof, opens with it.
- Symbols show what kind of moment it was: measurement, beyond Europe, key text, breakthrough, proof and rigour, the infinite, critique.
- Threads link ideas across centuries (area, limits, rates of change, infinitesimals, series). Hover over a card to see what it builds on and what it leads to, or follow one thread at a time.
- Where it happened. Each open card has a map with the places involved (a soft glow over the region, dots where the place is known, arcs for people who moved). On very wide screens a second map beside the timeline follows you as you scroll.
- Colour-coded kinds. Measurement pink, beyond Europe green, key text peach, breakthrough yellow, proof and rigour blue, the infinite purple, critique red; a card with several kinds gets a ring of colours on the timeline.
- Three themes run through the whole story, each with its own way in from the top of the page: the limit (a road of six stages, from Zeno's question to Dedekind's real numbers, shown on every card it touches), the infinite (Zeno, Aristotle's potential infinity, Galileo's paradox, Cantor), and methods used long before they were justified.
- Twenty questions narrate the story. The central question box lists them, and each card that raises one shows it.
| Date | Card | Lab |
|---|---|---|
| c. 450 BC | The Paradox of the Infinite | Achilles vs. the Tortoise |
| c. 430 to 250 BC | Squeezing the Circle | Squeezing the circle |
| c. 350 to 50 BC | Jupiter’s Path as an Area | Jupiter on a clay tablet |
| c. 500 | Equal Slices, Equal Volumes | Zu Gengzhi’s sphere |
| c. 1000 | Sums of Powers | al-Haytham’s staircase |
| 1150 | Instantaneous Speed | Bhāskara’s instantaneous motion |
| c. 1400 | Infinite Series for Pi | Madhava’s series for π |
| 1615 | Wine Barrels | Kepler’s wine barrels |
| 1615 | Wine Barrels | Unrolling the circle |
| 1635 | Indivisibles | Cavalieri’s principle |
| 1643 | Gabriel’s Horn | Gabriel’s horn |
| 1647 to 1649 | The Area Under the Hyperbola | Three roads to e |
| 1665 to 1666 | Fluxions | The derivative |
| 1670 | The Fundamental Theorem | Area is anti-differentiation |
| 1687 | The Laws of Motion | Differential equations from nature |
| c. 1683, published 1690 | Compound Interest and e | Three roads to e |
| 1696 | The First Calculus Textbook | Indeterminate limits |
| 1696 to 1697 | The Brachistochrone | The brachistochrone race |
| 1715 | Taylor Series | Taylor and Maclaurin series |
| 1768 | Euler’s Method | Euler, Heun and Runge–Kutta |
| 1798 | Population Growth | Fitting population models |
| 1807 to 1822 | The Flow of Heat | Fourier series |
| 1817 | Continuity Defined | Continuity, the epsilon-delta way |
| 1826 | Cauchy’s Mistake | Abel’s counterexample |
| 1838 | Limits to Growth | Fitting population models |
| 1854 | The Integral Defined | Accumulation: Riemann sums to Simpson |
| c. 1861 | Epsilon and Delta | The epsilon-delta limit game |
| 1925 to 1926 | Predators and Prey | Predators, prey and competitors |
56 step-by-step proofs, worked examples and source readings open inside their cards. A slider (or Back and Next) reveals one line at a time, each with a note explaining the step.
- Historical first. Arguments are shown as the people on each card made them, labelled by person and date (Archimedes balancing slices on a lever, Oresme's figure, Newton's fluxions, Leibniz's differentials, Euler's Basel argument), with notes on where the original reasoning was informal.
- Diagrams that follow the argument. Geometric proofs redraw their figure at every step.
- Modern versions where they help, stated with their hypotheses and every step justified.
- Key texts on the page. Thirteen cards show a page from the original document (the Rhind papyrus, Archimedes' palimpsest, Aristotle's Physics, Kepler, Cavalieri, Descartes, Galileo, Leibniz's first calculus paper, the Principia, The Analyst, Maclaurin, Euler and Cauchy) beside a short quotation, in the original language where it helps, with a plain note on what it means.
- Source readings quote the original texts: Berkeley's "ghosts of departed quantities", Newton on ultimate ratios, Lagrange's title page, Cauchy's false theorem and Abel's footnote.
| Card | Type | Title | Versions |
|---|---|---|---|
| Babylonian Circles (c. 1800 BC) | Worked example | The scribe’s calculation on YBC 7302 | Worked example |
| The Rhind Papyrus (c. 1650 BC) | Worked example | Problem 50: the round field | Worked example |
| The Paradox of the Infinite (c. 450 BC) | Proof | Summing a geometric series | Modern |
| Democritus’ Cone (c. 400 BC) | Proof | The volume of a cone | Modern |
| The Secret Method (c. 250 BC, found 1906) | Proof | The area of a parabolic segment | Found by balancing; Proved by exhaustion |
| The Secret Method (c. 250 BC, found 1906) | Story | The journey of the Method | The story |
| Cut Until It Cannot Be Cut (AD 263) | Worked example | Liu Hui doubles the polygon | Liu Hui’s method |
| The Best Pi for 900 Years (c. 480) | Worked example | Why |
A modern lens |
| Instantaneous Speed (1150) | Proof | The derivative of sine | Bhāskara’s instantaneous motion; Modern |
| The Highest Point (c. 1200) | Worked example | When does a cubic have a positive root? | Worked example |
| Motion as a Shape (c. 1335 to 1350) | Proof | The mean speed theorem | Oresme’s figure |
| Shrinking Is Not Enough (c. 1350) | Proof | Why the harmonic series diverges | Oresme’s grouping; Modern |
| Adequality (c. 1636) | Proof | Maxima and the derivative | Fermat’s adequality; Modern |
| Adequality (c. 1636) | Worked example | Descartes’ challenge: the folium | Fermat’s method, in modern notation |
| Analytic Geometry (1637) | Worked example | Tangents by double roots | Double roots |
| Two New Sciences (1638) | Worked example | Galileo’s odd-number rule | Galileo |
| Gabriel’s Horn (1643) | Proof | Gabriel’s horn: finite volume, infinite surface | Torricelli’s indivisibles; Modern |
| The Area Under the Hyperbola (1647 to 1649) | Worked example | Areas under the hyperbola are logarithms | Saint-Vincent and Sarasa; Modern |
| Arithmetica Infinitorum (1655 to 1656) | Proof | Wallis’s product for π | Wallis’s interpolation; Modern |
| Fluxions (1665 to 1666) | Proof | The power rule | Newton’s way; Modern; By induction |
| Tangents Undo Areas (1668 to 1671) | Proof | Tangents undo areas | Gregory’s theorem, in modern dress |
| The Fundamental Theorem (1670) | Proof | The Fundamental Theorem of Calculus | Intuitive; Modern; Modern |
| Nova Methodus (1684) | Proof | The sum rule | Modern |
| Nova Methodus (1684) | Proof | Stretching a graph | Vertical stretch; Horizontal stretch; Translations |
| Nova Methodus (1684) | Proof | The product rule | Leibniz’s differentials; Modern |
| Nova Methodus (1684) | Proof | The quotient rule | From first principles; Using the product and chain rules |
| Nova Methodus (1684) | Proof | The chain rule | Leibniz’s notation; Modern |
| Compound Interest and e (c. 1683, published 1690) | Worked example | Compound interest and the number e | Jacob Bernoulli |
| Compound Interest and e (c. 1683, published 1690) | Worked example | Why e^x is its own derivative | Euler’s infinitely small numbers; Modern |
| The First Calculus Textbook (1696) | Proof | Why L’Hôpital’s rule works | Bernoulli’s idea; Modern |
| The Priority Dispute (1699 to 1716) | Story | The dispute, year by year | The story |
| Taylor Series (1715) | Proof | Where the Taylor coefficients come from | Taylor’s derivation (1715); Matching derivatives; Uniqueness of power series |
| Calculus Without Calculus (before 1680 to 1722) | Worked example | Seki’s π and Takebe’s series | Seki; Takebe |
| Ghosts of Departed Quantities (1734) | Source reading | Where Newton’s argument breaks | Newton (1704); The modern repair |
| The Basel Problem (1734 to 1735) | Proof | The Basel problem | Euler’s argument (1735); Modern |
| Treatise of Fluxions (1742) | Worked example | Maclaurin series of |
Worked example |
| The Vibrating String (1747) | Proof | d’Alembert’s solution of the wave equation | d’Alembert’s formula; Modern |
| The Function (1748) | Worked example | Euler’s formula |
Euler’s series (1748) |
| Instituzioni Analitiche (1748) | Worked example | The curve of Agnesi and its area | Worked example |
| Principia in French (1759) | Proof | Why energy grows with |
Modern |
| Euler’s Method (1768) | Proof | Where Euler’s method comes from | Euler’s idea (1768); Modern |
| Calculus from Series (1797) | Proof | Taylor’s theorem with remainder | Modern; Worked use; Worked use |
| Population Growth (1798) | Proof | Solving |
Separating the variables; Modern |
| The Flow of Heat (1807 to 1822) | Worked example | Fourier coefficients of a square wave | Orthogonality |
| Continuity Defined (1817) | Proof | The Intermediate Value Theorem | Modern |
| Cours d’analyse (1821) | Proof | The limit of a sum is the sum of the limits | Modern; Differentiable implies continuous |
| Cauchy’s Mistake (1826) | Proof | Abel’s counterexample | The counterexample; Modern |
| Limits to Growth (1838) | Worked example | Solving the logistic equation | Separation and partial fractions |
| The Integral Defined (1854) | Worked example |
|
Right-endpoint sums; Squeezed from both sides |
| Epsilon and Delta (c. 1861) | Worked example | Proving a limit with |
|
| Cutting the Line (1872) | Proof | The gap at |
Modern |
| When Equations Have Solutions (1874) | Worked example | Solving differential equations with power series |
|
| Sizes of Infinity (1874 and 1891) | Proof | Cantor’s diagonal argument | The rationals can be listed; The reals cannot |
| Area, Part Two (1902) | Proof | A function Riemann cannot integrate | Riemann’s integral fails; Lebesgue’s answer |
| Predators and Prey (1925 to 1926) | Proof | Why predator–prey orbits close up | Worked consequence; Modern |
| Infinitesimals Return (1961) | Worked example | Differentiating with infinitesimals, rigorously | The derivative of |
Built for IB Mathematics (Analysis and Approaches, and Applications and Interpretation) and A-level students meeting limits, differentiation from first principles, integration and differential equations. Open a card on the board and step through its proof or lab together, or let students explore on their own devices. Some ideas:
-
The derivative (Newton): drag
$h$ towards 0 and discuss why the secant gradient is undefined at exactly$h = 0$ but has a limit; then step through the power rule, Newton's way and the modern way. - Euler, Heun and Runge–Kutta: press "Halve h" and compare how the errors shrink (by about 2, 4 and 16): the order of each method. Then step through the Euler error proof.
- Predators, prey and competitors: build the model term by term (logistic prey, predators that get full) and predict the behaviour before running it; try to beat the fitted squared error on the Hudson's Bay data by hand; switch to competition and find the four outcomes.
- Population models: fit an exponential to the US census up to 1860 and see what it predicts for 2020; repeat Pearl and Reed's 1920 logistic fit; then fit Carlson's yeast.
- Fourier and Abel: watch the Gibbs overshoot refuse to shrink, then use Abel's example to discuss why "every partial sum is continuous" does not make the limit continuous.
-
Epsilon-delta (Weierstrass): challenge students to find the largest
$\delta$ that works for a given$\varepsilon$ .
It is a single self-contained HTML file. Download index.html and open it in any modern browser. An internet connection is needed for the fonts and for MathJax, which renders the formulas. Linking to a card works too: add its id, for example index.html#newton.
Every card has AI-generated artwork in a consistent style, made from the prompts in image-prompts.md. The page uses web-sized copies in images/ (about 70 KB each, .webp) and small square thumbnails in images/thumbs/. Full-size originals are kept locally in images/originals/, which .gitignore leaves out of the repository.
- HTML Canvas and vanilla JavaScript for the timeline and labs
- Tailwind CSS (compiled and inlined, no build step needed to run)
- MathJax for the mathematics
- Google Fonts: Cinzel (headings), Playfair Display (the central question), IBM Plex Sans and IBM Plex Mono (text and figures), Noto Sans Cuneiform and Material Symbols
Designed by Liam Yardley, Curriculum Leader for Mathematics, as a teaching resource on the history and foundations of calculus. Developed iteratively with AI assistance (Google Gemini, with refinement by Claude).
- Code (HTML, CSS and JavaScript): MIT Licence.
- Content (the written history, questions, proofs, worked examples, lab text, diagrams and the AI-generated artwork): Creative Commons Attribution-NonCommercial-ShareAlike 4.0. Teachers and students are welcome to use, share and adapt it for teaching and learning, with credit to Liam Yardley and a link to this repository; adaptations must be shared under the same licence, and it may not be used commercially.
- Historical images and quotations keep their own licences and copyright, listed below and credited on each card.
images/sources/: page images of the thirteen key texts, made from public-domain scans and photographs on Wikimedia Commons (one page, the Archimedes Palimpsest, is from the Walters Art Museum and the Archimedes Palimpsest Project under CC BY 3.0, and the Cauchy title page is CC0). Each card credits its scan and links to the Commons file; the full list is in images/sources/CREDITS.md.
images/jupiter-tablet.jpg: clay tablet BM 34757, astronomical procedure text for Jupiter. © The Trustees of the British Museum. Shared under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 licence. Collection record. This image is not covered by the MIT Licence or by the content licence above.