Appendix I
Where each story goes in your two textbooks
This page is Appendix I turned into an interface: every place your Derivatives or Integrals book touches this history, with the hook to use and the link that lands on the exact section.
This appendix is keyed to the two live course books, read on 14 August 2026:
- Derivatives (Aqua):
https://derivatives.megan-warren.com/ - Integrals (Berry):
https://integrals.megan-warren.com/
Every section number and title below was taken from those two tables of contents, not from a generic syllabus. Updated 14 August 2026 with material from three sources obtained through Boston Public Library: Katz on Islam and India (S061), Whitman on the cycloid (S064), and a Zinsser essay on biography (S065). Then a second batch on the same day added the two primary sources behind the natural logarithm (Sarasa 1649 and Saint-Vincent's Opus geometricum) and two on du Châtelet. A fourth batch then brought Mazzotti and Cupillari on Agnesi and Del Centina on Newton's Lemma 12. Between them those added fourteen rows and rewrote Integrals 2.4 and 3.6 and the whole Agnesi section from the ground up. Both books are single-page documents with anchor navigation, and every anchor now lives in registers/placements.csv, one per row. Six patterns are confirmed from live URLs: #s-13 for a numbered section, #u1 for a unit opener, #mp-1 for Mixed Practice, #recap-1 for a Recap, and #a10 for Appendix A. Appendix B does not follow them: it uses semantic slugs, read off the live pages on 17 August 2026 (B.1 is #notation, B.2 is #glossary, B.5 is #properties). Every link in this appendix was checked against the live books on that date; the check reruns before every publish.
I.1 How to read these tables
The Box column names a container from the course books' own Legend, so each row is a drop-in rather than a research note:
| Your box | What it is for here |
|---|---|
| Picture This | A story that sets up the idea before the mathematics arrives. |
| Why this matters in a world that moves | A live, present-day payoff. |
| ⚠ Watch out | A historical mistake that is the same mistake students make today. |
| ⛏ Dig In | Rigor: why the definition had to be tightened. |
| Dig Deeper (collapsed) | Optional depth for the student who wants it. |
| Definitions | Where the word or symbol came from. |
| Unit opener | Framing for the whole unit. |
Practice and Recap sections (x.P and x.R) are not mapped, except Derivatives 1.R where a story earns its place. Those sections do not need history and padding them would dilute the rest.
One thing to know before you start. Integrals 1.0 already opens with Archimedes trapping a circle between two ladders of polygons, and already tells students that Newton and Leibniz found the area problem and the tangent problem to be one subject. That opener is the spine of this entire compendium, already in your book. Chapters 1, 2, and 5 here are the long version of the two paragraphs already in the course books.
I.2 Derivatives (Aqua), section by section
| Section | Box | The hook | Where it is in this book | Sources |
|---|---|---|---|---|
| A.1 to A.3 Algebra Review (signed numbers) | Picture This | Agnesi introduced positive and negative numbers through debits and credits in her 1748 textbook, and wrote the whole thing in Italian rather than Latin, producing the first systematic Italian vocabulary for calculus. She was building a language, not just a book. | Ch. 9 | Source 77, Mazzotti |
| 1.0 Unit 1 Opener | Unit opener | Calculus is Latin for pebble, a limestone counter on a Roman reckoning board. A teacher of arithmetic in Rome was a calculo if he was a slave and a calculator if he was freeborn. The subject is named after a slave's pebble. | Ch. 1 | Source 453, Miller and Aldrich; Source 455, Lewis and Short |
| 1.1 Rates of Change | Picture This | Bhaskara II introduced instantaneous velocity (tatkalika gati) around 1150 because the Moon moves too fast for a daily average to predict when a religious date ends. He calls the daily average "only a rough or approximate rate". The derivative enters Indian mathematics for the same reason it enters Newton's. | Ch. 3 | Source 50, Ramasubramanian |
| 1.1 Rates of Change | Dig Deeper | The mean speed theorem, that uniformly changing speed covers the same distance as the steady average, was stated at Merton College, Oxford in 1335 and proved by Oresme at Paris. It is buried in a book of logic-class brainteasers. | Ch. 4 | Source 101, Jung; Source 104, Lovell-Read; Source 107, Oresme |
| 1.2 The Notion of a Limit | Picture This | Zeno's Arrow paradox asks whether an instant can contain motion. You cannot answer it without defining velocity at a point by a limit. He posed it around 450 BCE; the mathematics arrives in the 1820s. | Ch. 2 | Source 7, Huggett |
| 1.2 The Notion of a Limit | ⛏ Dig In | Bolzano, 1817: everyone agreed a curve that starts below the axis and ends above it must cross. He said that is a picture, not a proof. His title is the argument: Purely analytic proof, where "purely" means no pictures, no motion, no geometry. | Ch. 10 | Source 363, Bolzano; Source 387, Ruch |
| 1.2 The Notion of a Limit | Definitions | Epsilon is the e of erreur, delta the d of différence. The symbol of mathematical precision started life as the symbol for being wrong. Cauchy, 1821 and 1823. | Ch. 10 | Source 360, Grabiner; Source 451, Cajori |
| 1.3 What Is a Derivative | ⚠ Watch out | Fermat's method, c. 1636: divide by a small quantity e, which requires e ≠ 0, then delete it, which requires e = 0. Berkeley called the result "the ghosts of departed quantities" in 1734, and he was mathematically right. This is the exact step students find slippery, and they are in good company. | Ch. 5, Ch. 9 Appendix E.6 |
Source 130, Katz; Source 310, Berkeley |
| 1.3 What Is a Derivative | Dig Deeper | Berkeley's objection stood for 232 years. In 1966 Abraham Robinson gave infinitesimals a rigorous footing, and Keisler's textbook put the payoff in one line: "Leibniz was on the right track, but 300 years too soon!" | Ch. 10 | Source 381, O'Connor; Source 432, OConnor and Robertson; Source 440, Keisler |
| 1.4 The Derivative as a Function | Definitions | The word "derivative" is Lagrange's, 1797, from fonction dérivée, and so is the f'(x) prime notation, introduced "pour plus de simplicité et d'uniformité". His book's subtitle promises no infinitesimals, no limits, no fluxions, just algebra. | Ch. 9 | Source 317, Lagrange |
| 1.4 The Derivative as a Function | Dig Deeper | Newton called it a fluxion: a quantity flows in time, and its fluxion is the rate of flow. And the famous dot over the letter is from late 1691, not 1666. The dots in the printed text are William Jones's 1710 tidying-up. | Ch. 6 | Source 185, Newton; Source 192, Whiteside |
| 1.5 The Power Rule | Dig Deeper | Ibn al-Haytham needed the exact sum of fourth powers to find a paraboloid's volume, because he had no limit concept. We only need the leading term because we do. The formula is the price of not having limits. He stopped at the fourth power for the plainest reason: nothing he wanted to measure needed a fifth. | Ch. 3 Appendix E.7, E.13 |
Source 54, Dennis; Source 61, Katz |
| 1.5 The Power Rule | Dig Deeper | Newton's first great result was extending the binomial theorem to fractional and negative exponents, by interpolating from Wallis. Twelve terms give the square root of 1.2 to ten decimal places. | Ch. 6 Appendix E.8 |
Source 190, Whiteside |
| 1.6 Derivatives and Their Graph Shapes | Picture This | Oresme invented the graph as a tool of proof around 1350. Time along the bottom, speed up the side, and then the sentence that makes calculus possible: the area of the figure is the total effect. A student can redo his whole proof with a ruler in ninety seconds. | Ch. 4 | Source 107, Oresme |
| 1.6 Derivatives and Their Graph Shapes | Dig Deeper | Graph intuition has a limit. Weierstrass presented a function that is continuous everywhere and differentiable nowhere in 1872. Hermite: "I turn away with dread and horror from this lamentable plague of continuous functions that have no derivatives." Sixty years later Banach and Mazurkiewicz proved almost every continuous function is one of them. | Ch. 10 | Source 365, Hermite; Source 386, Johansson; Source 388, Kesavan |
| 1.R Recap | Picture This | Liu Hui, 263 CE, could not compute the volume he needed, said so in print, and closed with a verse: "I dare to let the doubtful points stand, waiting for one who can expound them." Two hundred and fifty years later, someone answered. | Ch. 3 | Source 22, O'Connor; Source 23, Wagner |
| 2.0 Unit 2 Opener | Unit opener | The rules in this unit exist because Leibniz's 1684 paper gave no proofs at all. Jacob and Johann Bernoulli had to reverse-engineer it, then built most of the early calculus from what they worked out. | Ch. 7, Ch. 9 | Source 231, Leibniz |
| 2.1 The Product Rule | ⚠ Watch out | The best teaching artifact in the whole history of calculus. On 11 November 1675 Leibniz guessed d(xy) = dx·dy, checked it on one example, got the right answer by accident, and refuted himself the same afternoon. He left the error and the correction on the same page. | Ch. 7 Appendix E.9 |
Source 230, Child |
| 2.1 The Product Rule | ⛏ Dig In | Why the guess fails, in numbers: with x = 3, y = 5, dx = dy = 1/1000, the actual change is 0.008001 and dx·dy is 0.000001. The wrong answer and the right answer differ by exactly the second-order term you are entitled to discard. | Appendix E.9 | Source 230, Child |
| 2.2 Derivatives of Trig Functions | Definitions | "Sine" is a mistranslation nobody ever fixed. Sanskrit ardha-jya, half-bowstring, became Arabic jiba, which is written identically to jaib, "bay". A twelfth-century translator in Toledo picked the real word and rendered it sinus. Gunter abbreviated it to "sin" in 1624. | Ch. 3 | Source 60 |
| 2.2 Derivatives of Trig Functions | Dig Deeper | Bhaskara II used the Rcosine as the rate of change of the Rsine around 1150, and Madhava had the sine and cosine series by about 1400, roughly two and a half centuries before Newton. | Ch. 3 | Source 50, Ramasubramanian; Source 55, O'Connor; Source 62, Roy |
| 2.3 The Chain Rule | Why this matters in a world that moves | This rule trains every chatbot on Earth. Backpropagation is the chain rule run backwards. It has to run backwards for a reason a student can check: a network has one output and billions of inputs, so going forwards costs one pass per input and going backwards costs one pass, full stop. | Ch. 12 Appendix E.18 |
Source 436, Baydin, Pearlmutter, Radul and Siskind; Source 437, Schmidhuber |
| 2.3 The Chain Rule | Dig Deeper | Reverse mode was published in 1970 by a Finnish master's student who was not thinking about learning machines at all. He was worried about rounding errors. The paper is titled "Taylor expansion of the accumulated rounding error". It was then reinvented at least four times. | Ch. 12 | Source 436, Baydin, Pearlmutter, Radul and Siskind; Source 438, Griewank |
| 2.3 The Chain Rule | Dig Deeper | The chain rule looks obvious in Leibniz's notation because dy/dx behaves like a ratio, and painful in Newton's. Britain taught dots for a century and fell behind. Cambridge undergraduates finally campaigned for "D-ism against Dot-age" in 1812. | Ch. 7, Ch. 8 | Source 232, Leibniz (trans. Walker); Source 239, Babbage |
| 2.4 Derivatives of Exponential Functions | Definitions | e was born in a paper about cannon. Euler used it in a 1727 manuscript and printed it in 1736; no source records why he chose the letter. | Ch. 9 | Source 325, Miller; Source 451, Cajori |
| 2.5 The Quotient Rule | ⚠ Watch out | Which rule did Leibniz guess wrong on 11 November 1675, and what was the guess? You met the manuscript at Derivatives 2.1. Show itThe product rule. He guessed d(xy) = dx·dy, checked it on one example, and the example agreed with him. |
Ch. 7 | Source 230, Child |
| 2.6 The First Derivative Test | Dig Deeper | Sharaf al-Din al-Tusi found the maximum of a cubic around 1209, roughly 450 years before Fermat, and used the discriminant to ask when a solution exists rather than what it is. Whether he "had the derivative" is disputed: Rashed says yes, Hogendijk says the evidence does not support it. | Ch. 3 | Source 53, O'Connor |
| 2.6 The First Derivative Test | Dig Deeper | Rolle's theorem is the load-bearing lemma under the whole differential calculus, and Rolle called the calculus "a collection of ingenious fallacies" in print for five years. His 1691 statement was unproved and only about polynomials. | Appendix D | Source 459, OConnor and Robertson |
| 2.7 The Second Derivative Test | ⛏ Dig In | Agnesi tests concavity in 1748 with no calculus at all. At Instituzioni p. 361 she takes $a^{3} - zy^{2} = 0$, and compares the height of a chord against the height of the curve at the midpoint, reducing the whole question to the inequality 9a²/3 > a². That is midpoint convexity, argued by hand, and it agrees with the sign of the second derivative. A genuinely good "before we had the test, here is what you had to do" opener. | Ch. 9 | Source 75, Cupillari |
| 2.7 The Second Derivative Test | Picture This | The founding paper of calculus, October 1684, shipped with a typo that swapped concavity and convexity. Seven pages, no proofs, printer's errors, and it started a subject. | Ch. 7 | Source 231, Leibniz; Source 232, Leibniz (trans. Walker) |
| 3.0 Unit 3 Opener | Unit opener | Every application in this unit is older than the notation. Kepler was optimizing wine barrels in 1615 and Huygens was building clocks that kept time on a cycloid in the 1650s, decades before anyone wrote dy/dx. | Ch. 5 | Source 150, O'Connor; Source 159, Kepler |
| 3.1 Implicit Differentiation | Picture This | Descartes' La Géométrie (1637) is the enabling technology for this entire unit. Once a curve is an equation, "find the tangent" becomes an algebra problem you can hand to anybody. He wrote it as an appendix, and made it deliberately hard because he wanted readers to work. | Ch. 5 | Source 148, O'Connor |
| 3.2 Related Rates | Definitions | Newton's own vocabulary is built for this section: a fluent is a quantity that flows with time, and its fluxion is the rate at which it flows. Related rates is fluxions, in the original sense. | Ch. 6 | Source 185, Newton |
| 3.2 Related Rates | Dig Deeper | The idealized setups in this section have a medieval licence. The Oxford Calculators defended their artificial thought experiments in one line: the scenario does not have to be physically real, only conceivable, and "for purposes of the sophisma, that is enough." | Ch. 4 | Source 102, Jung |
| 3.3 Optimization | Picture This | The best origin story in the subject. Kepler remarried, bought wine, watched the merchant shove a rod into every barrel and name one price "without reasoning or calculation", got annoyed, and wrote a book computing about ninety volumes of revolution to check whether he was being cheated. | Ch. 5 | Source 159, Kepler; Source 154, O'Connor |
| 3.3 Optimization | ⛏ Dig In | Kepler noticed the thing that makes optimization useful: near a maximum, a quantity is insensitive to error. That is why the merchant's crude rod worked at all, and it is why the second derivative is small there. | Ch. 5 | Source 134, Cardil; Source 159, Kepler |
| 3.3 Optimization | Dig Deeper | Johann Bernoulli posed the brachistochrone in 1696 partly to embarrass his brother. Jacob solved it better. Johann spent years insisting otherwise. The family feud produced the calculus of variations as a by-product. | Ch. 9 | Source 330, O'Connor |
I.3 Integrals (Berry), section by section
| Section | Box | The hook | Where it is in this book | Sources |
|---|---|---|---|---|
| 1.0 Unit 1 Opener | Extends what you already have | Your opener already has Archimedes and the two ladders of polygons. The payoff you can add: in 1906 we found out how he really worked. A recovered letter shows he found his results by slicing figures and balancing them on an imaginary lever, then built the rigorous proof afterwards. He tells Eratosthenes that publishing only the polished proof wastes everyone's time. | Ch. 2 | Source 2, Heath; Source 3, Netz |
| 1.0 Unit 1 Opener | Picture This | Your tagline, "from a stack of rectangles to one exact total", is literally what the symbol means. ∫ is a long letter S for summa, "sum". | Ch. 7 | Source 230, Child; Source 238, Miller |
| 1.1 Approximating Area | Dig Deeper | Every student meets equal-width strips. Thabit ibn Qurra used unequal partitions in ninth-century Baghdad, a thousand years before Riemann formalized the general partition. Unequal widths can be the right tool. | Ch. 3 | Source 57, O'Connor |
| 1.1 Approximating Area | ⛏ Dig In | Archimedes squeezed the parabola with a geometric series and refused to take a limit in print. He proved the finite identity, then argued by contradiction. That caution is exactly what the epsilon-delta definition later made unnecessary. | Ch. 2 Appendix E.1 |
Source 1, Heath; Source 2, Heath |
| 1.1 Approximating Area | Picture This | Liu Hui's instruction for the circle, 263 CE: "divide the circle until it cannot be divided any more." That is either a limit or a contradiction depending on how charitable you are, and he knew it. | Ch. 3 | Source 20, Guo |
| 1.2 Antiderivatives | Definitions | The word "integral" was negotiated between three people who could not stand each other. Jacob Bernoulli printed it, Johann claimed it, and Leibniz had to be talked out of his own preferred name, calculus summatorius. | Ch. 9 | Source 324, Miller |
| 1.3 The Definite Integral | Definitions | 29 October 1675. Three days after writing a formula so clogged with the word omnia that it was unusable, Leibniz wrote: "It will be useful to write ∫ for omn." He invented the most recognizable symbol in mathematics to save ink. | Ch. 7 | Source 230, Child; Source 451, Cajori |
| 1.3 The Definite Integral | Dig Deeper | Cauchy told his students the same thing in 1823: it is a letter S, stretched, for somme. He also gave the first real definition of the definite integral as a limit of sums. Fourier added the limits above and below the sign in 1822. | Ch. 10 | Source 362, Cauchy; Source 451, Cajori |
| 1.3 The Definite Integral | ⛏ Dig In | Riemann defined his integral in 1854 only because he could not state his question without it. His own sentence ends: "If it does not have this property, then ∫f(x)dx has no meaning." And what students learn as the Riemann integral is usually Darboux's 1875 reformulation. | Ch. 10 Appendix D |
Source 391, Riemann |
| 1.4 The Fundamental Theorem | Picture This | Huygens tested a 26-year-old diplomat in Paris in 1672 by asking him to sum the reciprocals of the triangular numbers. Leibniz noticed every term is a difference, that the sum collapses, and that the answer is exactly 2. Sums and differences, the two operations he would later call ∫ and d, are the whole architecture already in that afternoon. | Ch. 7 | Source 237, Arthur |
| 1.4 The Fundamental Theorem | ⚠ Watch out | Two corrections at once. Gregory proved the inverse relationship in 1668 and Barrow stated it in 1670, both before Newton published, and four distinguished scholars have read the same evidence and reached four different verdicts about what Barrow gave Newton. And the name "Fundamental Theorem of Calculus" dates only to 1942. | Ch. 5 Appendix D |
Source 136, O'Connor; Source 138, Nauenberg; Source 158, Barrow |
| 1.4 The Fundamental Theorem | Dig Deeper | Leibniz stated the inverse relationship in print in 1686 as an analogy students already understand: "for as powers and roots in common calculation, so with us sums and differences or ∫ and d, are reciprocals." | Ch. 7 | Source 232, Leibniz (trans. Walker); Source 233, Leibniz |
| 1.5 The Net Change Theorem | Picture This | Time along the bottom, speed up the side. What did Oresme say the area of that figure means? You met his graph in Derivatives 1.6. Show itThe total effect. That sentence, written around 1350, is the theorem this section is named after. |
Ch. 4 | Source 107, Oresme |
| 1.5 The Net Change Theorem | Why this matters in a world that moves | Kepler's second law is not "planets speed up near the Sun". It is "equal areas in equal times", which says area measures elapsed time. Using it means adding up infinitely many thin slivers of an ellipse: an integral, in 1609, sixty-six years before Leibniz wrote the sign. | Ch. 5 | Source 152, Davis |
| 2.0 Unit 2 Opener | Unit opener | Your tagline is "the toolkit that reaches every function". The toolkit exists because the Bernoulli brothers reverse-engineered a proof-free seven-page paper and then spent thirty years arguing about who owned the result. | Ch. 9 | Source 231, Leibniz; Source 324, Miller |
| 2.1 Integral Properties and the Average Value | Picture This | Uniformly changing speed covers the same distance as one steady speed. Which steady speed, and who proved that? You met the Merton mean speed theorem in Derivatives 1.1. Show itThe average of the start and end speeds. Merton College, Oxford, 1335, and Oresme proved it at Paris with a picture. |
Ch. 4 | Source 101, Jung; Source 104, Lovell-Read; Source 107, Oresme |
| 2.2 Substitution | Why this matters in a world that moves | This section is the single best argument for Leibniz's notation. The dx carries the "of what": change variables and the dx tells you exactly what to do. In Newton's dot notation the same move is painful. That difference is why the Continent pulled ahead of Britain for a century. | Ch. 7, Ch. 8 | Source 232, Leibniz (trans. Walker); Source 239, Babbage; Source 241, Leibniz (ed. Gerhardt) |
| 2.3 Trigonometric Integrals and Substitution | Definitions | "Sine" records a mistranslation. Which step went wrong, and what did the word mean first? You met the chain in Derivatives 2.2. Show itArabic into Latin, and it meant bowstring: Sanskrit ardha-jya to Arabic jiba, read as jaib, "bay", and rendered sinus by a twelfth-century translator in Toledo. |
Ch. 3 | Source 60 |
| 2.3 Trigonometric Integrals and Substitution | Dig Deeper | The Kerala derivation of the sine series runs on ibn al-Haytham's summation identity, and names nobody. Katz locates the exact step. The mathematics travelled from Baghdad to the Malabar coast; the credit did not. A cleaner case study in citation than any modern example, and the evidence is a single line of a proof. | Ch. 3 | Source 61, Katz |
| 2.3 Trigonometric Integrals and Substitution | Dig Deeper | Pascal's Traité des sinus du quart de cercle contains the characteristic triangle, the little right triangle with legs dx and dy. Leibniz read it and later wrote that subito magna lux oborta est, "a great light suddenly appeared". Not a falling apple: a lemma about the area of a sphere. | Ch. 5, Ch. 7 | Source 143, [author unverified]; Source 153, O'Connor; Source 232, Leibniz (trans. Walker) |
| 2.4 Exponential and Logarithmic Integrals | Picture This | The natural logarithm was found during an argument. Mersenne set a challenge problem partly to attack Saint-Vincent's claimed quadrature of the circle. Saint-Vincent's pupil Sarasa answered it in 1649, and partway through he stops and says: "But to what end, you will say, all this roundabout? I am leading to logarithms." Then: "in place of the numbers 6, 7, 8, 9, 10, 11, which were the logarithms of the magnitudes, we shall be able to take the hyperbolic quantities." In place of the logarithms, take the areas. | Ch. 5 Appendix E.17 |
Source 66, Sarasa |
| 2.4 Exponential and Logarithmic Integrals | ⛏ Dig In | Why it works, in one line students can check: multiply x by a fixed factor and you add a fixed amount of area. Geometric progression in x, arithmetic progression in area. Doubling from 1 to 2 to 4 to 8 adds 0.693147 every time. Products become sums, which is the whole reason anyone wanted logarithms. | Appendix E.17 | Source 66, Sarasa |
| 2.4 Exponential and Logarithmic Integrals | Definitions | "Natural logarithm" was coined for a curve, not a number. It is natural because it falls out of the area under the plainest hyperbola there is, xy = 1, with no base chosen by anybody. And Saint-Vincent's own index calls the relevant theorems admirandae, "to be wondered at". | Ch. 5 | Source 67, Gregorius a Sancto Vincentio; Source 141, O'Connor; Source 155, Coolidge |
| 2.4 Exponential and Logarithmic Integrals | Dig Deeper | Mercator's 1668 series for log(1+x) is the first infinite series for a logarithm ever printed, and it is what panicked Newton into writing up De analysi to establish that he had got there first. | Ch. 5, Ch. 6 | Source 149, O'Connor; Source 184, Newton |
| 3.0 Unit 3 Opener | Unit opener | Your tagline is "the shapes and totals of the moving world". Every application in this unit predates the notation, and most of them predate Newton. | Ch. 5 | Source 150, O'Connor; Source 152, Davis; Source 159, Kepler |
| 3.1 Arc Length | Picture This | Christopher Wren, 1658. During Pascal's cycloid contest he sent in, without proof, the length of one arch: exactly four times the diameter of the generating circle. A curved length exactly equal to a straight one, with no pi in it. Shown it, Roberval reportedly proved it on sight and said he had known it for years. He had not published that either. | Ch. 5 Appendix E.16 |
Source 64, Whitman |
| 3.1 Arc Length | Definitions | The old word for this is rectification, literally "straightening". For most of the seventeenth century mathematicians doubted that a curved length could be expressed by a straight one at all, which is why Wren's result landed the way it did. | Appendix B | Source 64, Whitman; Source 451, Cajori; Source 453, Miller and Aldrich |
| 3.2 Areas between Curves | ⛏ Dig In | Archimedes proved a parabolic segment is exactly four thirds of its inscribed triangle, by summing 1 + 1/4 + 1/16 + ... and never taking a limit. He proves the finite identity, then rules out every other answer by contradiction. | Ch. 2 Appendix E.1 |
Source 1, Heath |
| 3.2 Areas between Curves | ⚠ Watch out | Galileo tried to find the area under a cycloid in 1599 by weighing it. He cut an arch and its generating circle from the same material, balanced them, and got about three times. Then he threw the result away, because he believed the ratio was irrational. It is exactly 3. A correct experiment, discarded because the answer looked too clean. | Ch. 5 Appendix E.15 |
Source 64, Whitman |
| 3.3 Surface Area of a Revolution | ⚠ Watch out | The most valuable correction in this compendium for your book. Torricelli's horn is real: an infinitely long solid with finite volume, which he had by 1641. But he never proved the infinite surface area. The painter's paradox every textbook prints is a later overlay on a seventeenth-century theorem. The shock in 1644 was simpler: a solid that goes on forever and fits in a cup. | Ch. 5 Appendix G |
Source 131, O'Connor; Source 157, Wijeratne |
| 3.4 Volume Methods | Picture This | Cavalieri's principle is Zu Geng's principle, about 1,100 years earlier: "If blocks are piled up to form volumes, and corresponding areas are equal, then the volumes cannot be unequal." It was written to answer a problem Liu Hui had published as an unsolved failure 250 years before. | Ch. 3 Appendix D |
Source 22, O'Connor; Source 23, Wagner |
| 3.4 Volume Methods | Dig Deeper | Kepler computed about ninety solids of revolution in 1615. What made him do it? You met the reason in Derivatives 3.3. Show itHe suspected his wine merchant: one rod shoved into every barrel, one price, "without reasoning or calculation". |
Ch. 5 | Source 151, [attributed to Jourdain, P. E. B. - UNVERIFIED]; Source 159, Kepler |
| 3.4 Volume Methods | ⛏ Dig In | Ibn al-Haytham needed the exact sum of fourth powers, not just the leading term. What was he short of? You met that in Derivatives 1.5. Show itA limit concept. The exact formula is the price of not having one. |
Ch. 3 Appendix E.14 |
Source 61, Katz |
| 3.5 Mass and Density | Why this matters in a world that moves | How did Archimedes find his results, before he wrote the proofs? You met the recovered letter in Integrals 1.0. Show itHe sliced figures into infinitely thin sections and balanced them on an imaginary lever, using mechanics to get the answer, then built the geometric proof afterwards. That letter is The Method. |
Ch. 2 | Source 2, Heath; Source 3, Netz |
| 3.6 Force and Work | Picture This | The eighteenth century fought over what "force" even meant. Newtonians measured it by mv, Leibnizians by mv². Émilie du Châtelet argued the mv² side and was pointing at what we now call energy. She was 42, pregnant, knew the danger, and "rushed on with her translation and commentary, hoping to finish the work by the time her child was born". She was right about the danger. | Ch. 9 | Source 332, Project Vox; Source 69, Allen |
| 3.6 Force and Work | Dig Deeper | Two of her three major works went out with no name on them. The 1738 essay on fire was submitted anonymously because a submission known to be by a woman would not have been taken seriously; it lost, and the Académie published it anyway. The Institutions de physique was also anonymous. The Académie admitted no women at all, except as spectators twice a year. | Ch. 9 | Source 69, Allen |
| 3.6 Force and Work | Why this matters in a world that moves | Her Principia is still the only complete French translation that exists. It came out ten years after she died, and not through neglect: Clairaut, the mentor who had checked her calculations while she was alive, pressed the publisher to bring it out and had the illustrations finished. | Ch. 9 | Source 68, Zinsser; Source 69, Allen |
| 3.7 Probability | ⛏ Dig In | The Radon-Nikodym derivative is why "probability density" means anything. Every time this section writes p(x) dx, the reason that expression is legitimate is a theorem finished in 1930. | Ch. 12 | Source 435, OConnor and Robertson |
| 3.7 Probability | Dig Deeper | Oresme, around 1350, argued against astrology on probabilistic grounds: if celestial ratios are almost certainly irrational, no exact conjunction ever repeats, so astrological prediction fails in principle. One of the earliest uses of "most probable" as a mathematical argument in European science. | Ch. 4 | Source 108, Caroti |
I.4 Both books, Appendix B: Study Tools
Three of your Study Tools sections map onto whole appendices here. This is the part of the alignment that pays back fastest, because it is reference material you already maintain and it needs no new pedagogy.
| Your section | What to pull from here | Size |
|---|---|---|
| B.1 Notation & Symbols | Appendix B of this book, first-use dates for every symbol on the page: ∫ (29 Oct 1675, a long S for summa), d and dx (11 Nov 1675, and d began life as a divisor), f'(x) (Lagrange 1797), the dot (1691, not 1666), ∞ (Wallis 1655, first used as a denominator), π (William Jones 1706, "it simply came, unheralded"), e, Σ and Δ (Euler 1755, twins in one sentence), lim (L'Huilier 1786), the arrow (Leathem 1905, not Hardy), ε and δ, and the integral limits (Fourier 1822). | 52 symbols |
| B.2 Glossary | The Glossary and Appendix B of this book. Etymologies for the words your glossary already defines: calculus (a pebble), tangent (to touch), secant (to cut), asymptote (not falling together), quadrature (squaring), rectification (straightening), limit (limes, a boundary fence), integral, derivative, function, infinitesimal. | 75 words |
| B.5 Rules & Theorems | Appendix D of this book, whose name is on it and who did the work. L'Hopital's rule is Johann Bernoulli's, bought under contract for 300 pounds a year. Maclaurin's series is Stirling's, 1717. Cavalieri's principle is Zu Geng's. Newton's method, as taught, is Simpson's, 1740. Cauchy-Schwarz is Bunyakovsky's, 1859. Every row is sourced, and the whole table is a lesson on its own. | 20 corrections |
I.5 The prerequisite appendices
| Book and section | The hook | Sources |
|---|---|---|
| Derivatives A.7 to A.9, Integrals A.1 to A.3 (trig, unit circle, identities) | The ardha-jya to jiba to jaib to sinus chain. A chord looks like a drawn bowstring and the arc is the bow, which is why the whole subject is named after archery equipment. | Source 60 |
| Derivatives A.6, Integrals A.4 to A.5 (exponentials and logarithms) | The natural logarithm was named for the area under xy = 1, not for a base. Napier built the word from logos plus arithmos. | Source 141, O'Connor; Source 155, Coolidge |
| Derivatives A.1 to A.3 (exponent and fractional-exponent rules) | Fractional exponents entered calculus through Newton's generalized binomial theorem in 1665, which he got by interpolating from Wallis's tables. | Source 190, Whiteside |
| Integrals A.7 (area, surface area, volume) | Cavalieri's principle has an earlier owner. Who, and by how long? You met him in Integrals 3.4. Show itZu Geng, by about 1,100 years. |
Source 22, O'Connor; Source 23, Wagner; Source 159, Kepler |
| Integrals A.8 (basic physics) | The Merton mean speed theorem and Oresme's velocity graph are the origin of every distance-from-velocity problem in the section. | Source 101, Jung; Source 107, Oresme |
| Integrals A.9 (basic probability) | Oresme's probabilistic argument against astrology, and the 1930 theorem that makes "probability density" mean something. | Source 108, Caroti; Source 435, OConnor and Robertson |
I.6 History in this compendium with no home in your current scope
Your two books cover derivatives through optimization and integrals through probability. They do not cover sequences, series, convergence tests, Taylor or Maclaurin series, formal real analysis, or measure theory. A large and good part of this compendium therefore has nowhere to land right now. It is listed here so nothing gets quietly lost, and so you know what is already written if you ever build a third volume.
| Topic with no section | What is already written here | Where |
|---|---|---|
| Sequences and series | Oresme proving the harmonic series diverges around 1350, with the block-grouping proof and a checked table of partial sums. | Ch. 4, Appendix E.5 |
| Power series and pi | Madhava's series c. 1400, his error-analyzed correction terms, and his pi recorded as a poem about gods and elephants that decodes to eleven correct decimal places. | Ch. 3, Appendix E.3, E.4 |
| Taylor and Maclaurin series | Stirling had "Maclaurin's series" in 1717, twenty-five years early, and Maclaurin credited Taylor anyway. Gregory and Johann Bernoulli had Taylor's earlier. | Ch. 9, Appendix D |
| Convergence tests | Cauchy's false theorem on page 131 of the Cours d'analyse, and Abel's 1826 footnote citing that page number while calling the book indispensable. You can watch the error being caught in print, on facing pages. | Ch. 10 |
| Formal real analysis | Bolzano silenced by an emperor, Dedekind's cuts worked out on a specific Wednesday in 1858 and then sat on for fourteen years, Weierstrass teaching gymnasium for years before one paper made him famous. | Ch. 10 |
| Measure and the Lebesgue integral | Why the Riemann integral was not enough, and the whole chain from Fourier's heat problem to Cantor's set theory. | Ch. 10 |
| Partial derivatives | The ∂ symbol's three-stage history: Condorcet 1770, Legendre 1786 who then dropped it, Jacobi 1841 who revived it after a fifty-five-year gap. | Appendix B |
Two of these are worth a "Dig Deeper" in a book you already have. Madhava's series fits naturally in Integrals 2.3 next to the trigonometric material, and the Cauchy and Abel exchange fits in Derivatives 1.2 as an example of what the formal definition of a limit is protecting against.
I.7 Fitting the voice
Three notes so the history sits inside your Legend rather than beside it.
- Your books already use history sparingly and well. Integrals 1.0 names three mathematicians in two paragraphs and then gets on with the mathematics. Keep that ratio. The material in this compendium is a supply depot, not a script. One box per section is plenty, and several sections above list three options so you can pick.
- "Dig Deeper" is collapsed by default, which makes it the right home for most of this. A student who wants the Leibniz manuscript story opens it; a student who wants to finish the problem set does not have to scroll past it.
- The "⚠ Watch out" rows are the ones to prioritize. Derivatives 2.1 (Leibniz's product-rule error), Derivatives 1.3 (Berkeley on dividing then deleting), and Integrals 3.3 (Torricelli's horn) are all cases where a historical mistake is the same mistake a student makes, or where the standard textbook account is simply wrong. Those three earn their space immediately.