Reference
Glossary
Terms a 15-to-19-year-old reader, or a teacher from another subject, may not know. Mathematical terms are defined as this book uses them, not in full generality.
| Term | Definition |
|---|---|
| adequality (adaequalitas) | Fermat's operation of setting two nearly-equal expressions against each other, then suppressing the small terms. From Diophantus' Greek παρισότης by way of Bachet's Latin. Fermat's own French: "a feigned comparison, or an adequality" |
| anni mirabiles | "Wonderful years", the plague years 1665 to 1667 in which Newton later said he did his best work |
| automatic differentiation (AD) | Computing exact derivatives of a program by applying the chain rule mechanically to its operations. Forward mode costs one pass per input; reverse mode costs one pass per output. Backpropagation is reverse mode |
| backpropagation | The algorithm that trains neural networks. It is reverse-mode automatic differentiation, which is the chain rule applied backwards through a computation |
| bhuta-sankhya | The Indian word-numeral system, in which digits are written as words for things that come in known quantities (eyes = 2, Vedas = 4). Used because verses survive copying better than digits do |
| brachistochrone | The curve of fastest descent between two points under gravity. Posed by Johann Bernoulli in 1696; the answer is a cycloid |
| calculus of variations | The branch that optimizes over whole functions rather than over numbers. Grew out of the brachistochrone problem |
| catenary | The curve a hanging chain makes under its own weight. Not a parabola |
| characteristic triangle | The tiny right triangle with legs dx and dy whose hypotenuse lies along a curve. Pascal's diagram; Leibniz said reading it produced "a great light" |
| codex / palimpsest | A codex is a bound book, as opposed to a scroll. A palimpsest is a manuscript whose original text was scraped off so the parchment could be reused |
| colophon | A note at the end of a manuscript recording who copied it, where, and when |
| cycloid | The curve traced by a point on the rim of a rolling wheel. The subject of so many seventeenth-century quarrels that it acquired a nickname about the Trojan war |
| diplomatic transcription | An edition reproducing a manuscript exactly as written, including errors, deletions, and spelling |
| éloge | A formal memorial address delivered at a French learned academy on the death of a member |
| enri (円理) | "Circle principle", the Japanese series methods for arcs and areas developed in the wasan tradition |
| epsilon-delta | The modern definition of a limit, stated with inequalities and no appeal to motion or pictures. Epsilon from erreur, delta from différence |
| evanescent quantity | Newton's term for a quantity in the act of vanishing. Berkeley's target |
| exhaustion, method of | The Greek technique of bracketing an unknown area between inscribed and circumscribed figures and ruling out every other answer by contradiction. A 17th-century name; it brackets rather than exhausts |
| floruit | "He or she flourished". Used when someone's birth and death dates are unknown but their period of activity is not |
| fluxion / fluent | Newton's terms. A fluent is a quantity that flows with time; its fluxion is its rate of flow. Modern equivalent: function and derivative |
| folio (f. / ff.), recto, verso | A folio is a manuscript leaf. Recto is the front, verso is the back. "f. 15r" means the front of leaf 15 |
| geoid | The Earth's true gravitational shape: not a sphere, not an ellipsoid, but a lumpy surface, because gravity varies over mountains and trenches. GPS depends on it |
| holograph | A document written entirely in the hand of its author |
| imprimatur | Official permission to print, from a church or state censor. Agnesi's 1748 book carries one |
| indivisibles | The pre-calculus idea that an area is made of lines and a solid of planes. Cavalieri's method; theologically contentious in Counter-Reformation Italy |
| infinitesimal | A quantity smaller than any positive number but not zero. Used freely for two centuries, banished in the nineteenth, made rigorous by Robinson in 1966 |
| measure theory | The rigorous theory of size (length, area, volume) that lets you integrate functions too badly behaved for Riemann. Lebesgue, 1902 |
| nonstandard analysis | Abraham Robinson's 1966 framework giving infinitesimals a rigorous foundation via model theory. The standard part of a number is the nearest real number to it |
| Old Style / New Style (OS / NS) | Julian and Gregorian calendar dates. England used Julian until 1752 and started its civil year on 25 March |
| priority dispute | An argument about who discovered something first. The Newton-Leibniz one is the most famous in science |
| quadrature | Literally "squaring": finding the area of a region, historically by constructing a square of equal area. The old word for integration |
| rectification | Finding the length of a curve. Literally "straightening" |
| sakoku (鎖国) | "Closed country", Japan's policy of near-total isolation from about 1639 to 1853, under which wasan mathematics developed independently |
| sophisma (pl. sophismata) | A deliberately puzzling sentence used as a training exercise in medieval logic. The mean speed theorem appears inside one |
| tatkalika gati | Sanskrit: "instantaneous motion". Bhaskara II's term for what we call instantaneous velocity |
| tithi | A lunar day in the Hindu calendar. Predicting when one ends is why Bhaskara II needed instantaneous velocity |
| uniform convergence | A stronger kind of convergence for a sequence of functions, in which the rate does not depend on the point. Its absence is why Cauchy's 1821 sum theorem is false |
| versiera | Italian for the rope that turns a sail, from Latin versoria. Mistranslated as avversiera, she-devil, giving "the Witch of Agnesi" |
| vis viva | "Living force", the eighteenth-century quantity mv². The Newtonian rival was mv. Both were right: they are kinetic energy and momentum |
| wasan (和算) | Traditional Japanese mathematics, developed in isolation during the Edo period |
| Yuktibhasa | Jyeshthadeva's c. 1530 Malayalam prose text giving derivations, not just results, for the Kerala school's series |