Appendix B
Symbols, words, and eponyms
Symbols
| Item | First attested | Introduced by | Origin | Literal meaning | What it means now | Sources |
|---|---|---|---|---|---|---|
| the dot as a hint of smallness (pre-Newton) | 1592 | Leonard Digges, Tectonicon p. 15 | English | a prick beside a fraction "betokeneth some small quantitie lesse then" it; two pricks, a little more | Not calculus, but the direct English ancestor of Newtons dot: a mark meaning "and a bit". Mercator used a dot over a letter for an infinitesimal in 1668 | Source 451, Cajori |
| sin (abbreviation) | 1624 | Edmund Gunter | English | - | Standard abbreviation for the sine function | Source 60 |
| x, y, z for variables and a, b, c for constants | 1637 | Rene Descartes, La geometrie | notation | first letters of the alphabet = known; last letters = unknown | The convention behind every "let x be ..." in mathematics. Descartes introduced it, in Cajori s words, "without comment" | Source 450, Cajori |
| infinity symbol | 1655 | John Wallis | (origin unattested) | - | infinity; Wallis's first use was as a denominator, 1/infinity = an infinitely small part | Source 139, Miller; Source 145, O'Connor |
| the infinity sign | 1655 | John Wallis, De sectionibus conicis, Prop. 1 | unknown; two conjectures | Wallis wrote "esto enim [oo] nota numeri infiniti" - let this be the sign of an infinite number. His FIRST use is as a denominator: each of infinitely many strips has height 1/oo of the whole | Infinity. Two conjectures for the shape: the late Roman numeral for 1000 (Wattenbach 1872, followed by Cajori), or the lowercase omega, last letter of the Greek alphabet. A Roman hand-abacus in the Bibliotheque Nationale has a 1000-column marked with a shape close to the lemniscate | Source 451, Cajori; Source 452, Miller |
| x-dot (fluxion) | 1665 (20 May, per Cajori/Rigaud) or 1691 (per Whiteside); 1693 in print | Isaac Newton | notation | a dot over the letter, for the velocity of a flowing quantity | The time derivative. First PRINTED in the Latin edition of Wallis Algebra (De algebra tractatus, Oxford 1693, pp. 390-96) | Source 451, Cajori; Source 192, Whiteside |
| o | 1666-1671 | Isaac Newton | Latin/algebraic convention | the letter o, used for 'an indefinitely small Quantity' | Newton's infinitesimal time-increment. The moment of a fluent is its fluxion multiplied by o. The step 'divide by o, then set o = 0' is exactly what Berkeley attacked in 1734 | Source 185, Newton; Source 192, Whiteside; Source 187, Newton |
| differential d (dx, dy) | 1675 | Gottfried Wilhelm Leibniz | Latin, from differentia | difference | the differential | Source 139, Miller |
| ∫ (integral sign) | 1675 | Gottfried Wilhelm Leibniz | Latin | An elongated letter S, for summa, "sum". It replaced the abbreviation omn. for omnia ("all the ...", as in omnia lineae, "all the lines") | The integral, i.e. the limit of a sum | Source 139, Miller; Source 230, Child; Source 233, Leibniz; Source 238, Miller |
| d (differential operator) | 1675 | Gottfried Wilhelm Leibniz | Latin | From differentia, "difference". First used as a DIVISOR (Leibniz wrote ya/d) because he thought of d as lowering dimension, the inverse of ∫ which raises it | The differential; dy/dx the derivative | Source 230, Child; Source 238, Miller |
| dx | 1675 | Gottfried Wilhelm Leibniz | Latin | "the difference of the x's"; from 11 November 1675 written as a multiplier rather than x/d | An infinitesimal or differential increment of x; in ∫f(x)dx it carries the variable of integration | Source 230, Child; Source 238, Miller; Source 231, Leibniz |
| omn. (omnia) | before 29 Oct 1675 | Gottfried Wilhelm Leibniz | Latin omnia | "all the ..." - Leibniz wrote omn. l for "all the l s" | The thing the integral sign replaced. Leibniz abandoned it on 29 Oct 1675 because nested sums produced formulas like omn. omn. omn. that were unreadable | Source 451, Cajori; Source 452, Miller |
| the integral sign as a long s | 1675 (29 Oct MS); 1686 in print | Gottfried Wilhelm Leibniz | Latin summa | a long letter s, the ordinary long s of 17th-century printing, standing for summa (sum) | The integral. NOTE the first PRINTED integral sign (Acta eruditorum 1686, pp. 297, 299) had its descender cut off and looked like a small f-shaped s; the modern tall form comes later | Source 451, Cajori; Source 452, Miller |
| d as a divisor (x/d) | 1675 (29 Oct) | Gottfried Wilhelm Leibniz | Latin differentia | Leibniz first wrote the differential UNDER a line, as ya/d, because the integral raises dimension and d lowers it: "ut [integral] augebit, ita d minuet dimensiones" | Superseded on 11 Nov 1675 by the multiplier form dx. The symbol we write a thousand times was a denominator for its first sixteen days | Source 451, Cajori; Source 452, Miller |
| dx, dy (as multipliers) | 1675 (11 Nov) | Gottfried Wilhelm Leibniz | Latin differentia | "the difference between two proximate x s" - Leibniz: "Idem est dx et x/d" | The differential; the dx in an integral that carries the variable of integration | Source 451, Cajori; Source 452, Miller |
| dy/dx (built-up quotient) | 1675 (11 Nov) in MS; NOT in the 1684 print | Gottfried Wilhelm Leibniz | Latin | the ratio of the two differentials | The derivative. Cajori S 571: the 1684 Acta paper has "dw ad dx" and "dx:dy" but not the built-up fraction | Source 451, Cajori |
| x-dot (pricked or dotted letter) | 1691 | Isaac Newton | notation | a dot placed over a letter | The fluxion (derivative) of the fluent. CAUTION: Newton did NOT use this in 1666 (he used p, q, r) or in the 1671 treatise (l, m, n, r). Whiteside dates the dot notation to late 1691. The dots printed in the 1736 English Method of Fluxions were introduced by William Jones's 1710 transcript | Source 192, Whiteside; Source 183, Newton; Source 185, Newton |
| d^(1/2) (fractional differential) | 1695/1697 | Gottfried Wilhelm Leibniz | Latin | A differential of order one-half, proposed by analogy with fractional exponents on powers | Fractional (non-integer-order) derivative; the founding notion of fractional calculus | Source 241, Leibniz (ed. Gerhardt); Source 242, Leibniz |
| ddx, d2x (higher differentials) | 1695 (numerals); ddy before that | Gottfried Wilhelm Leibniz | Latin | repeat the d | Higher derivatives. LHopital 1696 and Fontaine as late as 1764 still wrote dddd y in full | Source 451, Cajori |
| pi | 1706 | William Jones, Synopsis palmariorum mathesios, p. 263; standardised by Euler, Introductio S.126, 1748 | Greek periphereia | the first letter of the Greek for "periphery, circumference" | Ratio of circumference to diameter; Euler defines it as the semicircumference of a circle of radius 1. Jones introduced it in a beginners textbook with no fanfare; Cajori: "It simply came, unheralded." Euler used p in 1734, adopted pi in the Mechanica (1736), and made it universal with the Introductio (1748) | Source 318, Euler; Source 325, Miller; Source 451, Cajori; Source 462, OConnor and Robertson |
| e | 1727 or 1728 (MS, printed 1862); 1736 in print | Leonhard Euler, Meditatio in experimenta explosione tormentorum nuper instituta; printed in Mechanica | notation | no attested reason. Not "Euler"; not "exponential" on any evidence he left | Base of natural logarithms, 2.71828...; Euler's manuscript gives '2,7182817', wrong in the last digit. Leibniz had used b for it in letters to Huygens in 1690-91; dAlembert and others used c into the 1780s | Source 325, Miller; Source 451, Cajori |
| Delta (finite difference) | 1755 | Leonhard Euler, Institutiones calculi differentialis pp. 3-7 | Greek capital delta, for Latin differentia | the Greek capital of the same letter as Leibniz s Latin d | A finite difference: Delta y = y(x + w) - y(x). Johann Bernoulli had used Delta for a differential coefficient in 1706 | Source 451, Cajori |
| Sigma (summation) | 1755 | Leonhard Euler, Institutiones calculi differentialis p. 27 | Greek capital sigma, for Latin summa | the Greek S for "sum", exactly parallel to Leibniz s long Latin s for the integral | Summation. Euler s own sentence: "Quemadmodum ad differentiam denotandam vsi sumus signo Delta, ita summam indicabimus signo Sigma" - just as we used Delta for the difference, we shall indicate the sum by Sigma. Delta and Sigma are twins, born in one book | Source 451, Cajori |
| [x . y] (Landen s derivative) | 1758 | John Landen, Discourse concerning Residual Analysis, p. 12 | notation | a bracketed pair with a mark between | A dead rival to dy/dx. Its selling point was typographical: no fraction bar, no letters above normal type height. "Landen had no followers in the use of his symbols" | Source 451, Cajori |
| curly d (partial derivative) | 1770 Condorcet; 1786 Legendre (modern form); 1841 Jacobi (revival) | Condorcet, Memoire sur les equations aux difference partielles; Legendre for the modern form, Memoire sur ... le calcul des variations | French/Latin; the letter itself is a cursive d | a cursive/rounded d; it is the same letter as the Cyrillic cursive de | Partial differentiation. Legendre used it once, dropped it the following year, and it lay unused for 55 years until Jacobi picked it up | Source 325, Miller; Source 451, Cajori; Source 452, Miller |
| i | 1777 (presented 5 May); published 1794 | Euler, De formulis differentialibus angularibus: 'formulam sqrt(-1) littera i in posterum designabo, ita ut sit ii = -1' | Latin imaginarius | - | The imaginary unit; too late for the Introductio, which writes sqrt(-1) throughout | Source 325, Miller; Source 318, Euler |
| lim. | 1786 | Simon LHuilier, Exposition elementaire des principes des calculs superieurs, pp. 24, 31 | Latin limes | abbreviation of Latin limes, a boundary | The limit operator. LHuilier wrote "lim. DeltaP/Deltax" and said it means the same as dP/dx. The full stop was gradually dropped; Weierstrass writes "lim" without it (1841) | Source 451, Cajori; Source 452, Miller |
| f'(x), f''(x) (prime notation) | 1797 | Lagrange, Theorie des fonctions analytiques art. 16 | - | - | Successive derivatives; introduced by Lagrange expressly 'pour plus de simplicite et d'uniformite' | Source 317, Lagrange; Source 325, Miller |
| f(x) | 1797 in this form | Lagrange (art. 2) writes fx for a simple variable and f(x^2), f(a+bx), f(x,y) for compound arguments; Euler had used f(x) earlier | - | - | Function notation | Source 317, Lagrange; Source 318, Euler |
| f(x), f prime, f double prime | 1797 (prime); 1770 and 1759 for earlier accents | Joseph-Louis Lagrange | notation | a stroke over the letter, from the accent | Successive derivatives. Lagranges reason, in his own words: "pour plus de simplicite et duniformite" | Source 451, Cajori; Source 452, Miller; Source 317, Lagrange |
| the inverted rounded E (Gruson s limit) | 1798-1800 | Johann Philipp Gruson, Le calcul d exposition | notation | an inverted and rounded capital E | A dead rival notation for the limit of Deltaz/Deltax. Along with Pasquich s epsilon-y, an example of how many limit notations were tried and lost | Source 451, Cajori |
| D (as an operator) | 1800 (Arbogast); used non-operationally by Johann Bernoulli earlier | Louis Francois Antoine Arbogast, Du calcul des derivations | French/Latin derivare | D for "derivation" | The differentiation operator. Arbogast also introduced D^-1 for the inverse operation, which survives | Source 451, Cajori; Source 452, Miller; Source 462, OConnor and Robertson |
| D_x (subscripted operator) | 1800 | Arbogast (attribution from Julio Gonzalez Cabillon) | notation | D with the variable written below | Partial or total differentiation with respect to x. Cajori shows Arbogast s D but not the subscripted form, so the exact page is unconfirmed | Source 452, Miller; Source 451, Cajori |
| limits on the integral sign | 1819-20 (Memoires); 1822 (book) | Joseph Fourier, Theorie analytique de la chaleur, p. 252 (S 231) | French | "Nous designons en general par le signe [integral a to b] l integrale qui commence lorsque la variable equivaut a a, et qui est complete lorsque la variable equivaut a b" | The definite integral. Adopted at once by Plana, Fresnel and Cauchy; Poisson called it "tres-commode". Euler had written the limits in words (ab ... ad) inside brackets | Source 451, Cajori; Source 452, Miller |
| epsilon | 1821 | Augustin-Louis Cauchy, Cours danalyse (Oeuvres II.3, pp. 49-50, 54) | Greek letter for French erreur | the initial of erreur, error | The arbitrary positive tolerance. The symbol of rigor is named after being wrong | Source 451, Cajori; Source 452, Miller; Source 360, Grabiner |
| epsilon (as the symbol of rigor) | 1823 | Cauchy (in analysis); attested for 'erreur' in his 1853 probability work | Greek letter, chosen for French 'erreur' / English 'error' | the initial letter of the word 'error' | the arbitrary positive tolerance in a limit argument - the symbol of precision, born as the symbol of being wrong | Source 360, Grabiner; Source 362, Cauchy |
| delta (in epsilon-delta) | 1823 | Augustin-Louis Cauchy, Resume des lecons sur le calcul infinitesimal, Lecon 7 | Greek letter for French difference | the initial of difference | The input tolerance. Cauchy sometimes used delta where he elsewhere used epsilon; the fixed division of labor (epsilon for the output, delta for the input) settles in Weierstrass s Berlin lectures | Source 451, Cajori; Source 452, Miller; Source 360, Grabiner |
| the evaluation bar [F(x)] with limits | 1823 | Pierre Frederic Sarrus, in Gergonne s Annales XIV, p. 197 | French | a vertical rule (or bracket) with the two limits attached | The "plug in the top, subtract the bottom" bar. Used afterwards by Moigno and Cauchy | Source 451, Cajori; Source 452, Miller |
| the Laplacian written as capital delta | 1833 | Robert Murphy, Elementary Principles of the Theories of Electricity | notation | the same capital delta as finite difference, doing an entirely different job | The Laplace operator, also written nabla squared. Maxwell s own name for it was "concentration" | Source 452, Miller |
| lim with the variable underneath | 1841 and 1854 | Karl Weierstrass | notation | writing x = a below the word lim | Specifying WHAT the variable approaches. Before this, "lim" alone left it to context. W. R. Hamilton used lim{ } in 1853; Dirksen in Berlin used Gr. for German Grenze in 1832 | Source 451, Cajori; Source 452, Miller |
| absolute-value bars | 1841 | Karl Weierstrass, in a paper not printed at the time (Werke I, p. 67) | notation | two vertical rules | Absolute value / modulus. Weierstrass also used single bars for determinants and had to disambiguate in words. The competing name "module" is Argand s and Cauchy s | Source 451, Cajori |
| matrix (double vertical lines) | 1843 and 1845 | Arthur Cayley | notation | two vertical rules on each side | Matrix delimiters. W. A. Whitworth even used TRIPLE lines in 1866 to signal that three independent determinant equations could be formed | Source 451, Cajori |
| nabla (the symbol) | 1846-47 (Hamilton, rotated form); 1867 (Tait, modern form) | William Rowan Hamilton; established by P. G. Tait | notation | an inverted capital delta | The vector differential operator. Hamilton first wrote it in the modern orientation in an unpublished Southampton paper, then ROTATED it because "this more common sign has been so often used with other meanings"; Tait turned it back | Source 452, Miller; Source 451, Cajori |
| the dotted-equals sign for a limit | 1880 in print; coined by James E. Oliver | James E. Oliver (Cornell); printed in W. E. Byerly, Elements of the Differential Calculus (1880) p. 7 | notation | an equals sign with a dot above | "Approaches as a limit". The standard American limit arrow before the British arrow crossed the Atlantic. Oliver s own gloss: "Using [it] in the sense Approaches in value to, as a limit" | Source 451, Cajori |
| epsilon for set membership | 1889 | Giuseppe Peano, Arithmetices principia nova methodo exposita, Turin | Greek lunate epsilon, standing for the Latin 'est' | 'is' - Peano called it an abbreviation for 'est' | x is an element of the set A. Peano s Formulaire (1895) lists 158 signs, of which only 26 are purpose-made ideographs | Source 393, Miller; Source 451, Cajori |
| the arrow in limits | 1905 | John Gaston Leathem, Volume and Surface Integrals Used in Physics | notation | an arrow, for "tends to" | x -> a. Popularised by Bromwich (Infinite Series, 1908) and Hardy (A Course of Pure Mathematics, 1908). Hardy CREDITS Leathem and Bromwich in his Preface; Leathem complained in 1925 that the arrow had been "erroneously attributed to another writer owing to its use, with inadvertent omission of acknowledgment, in an important book published three years later" | Source 451, Cajori; Source 452, Miller |
| the integral round a closed path | 1917 | Arnold Sommerfeld, Annalen der Physik | notation | a small circle drawn through the integral sign | Contour integral. Earlier than the 1923 use Cajori knew; E. B. Wilson (1901) put a small circle BELOW the sign instead | Source 452, Miller |
Words and their origins
| Item | First attested | Introduced by | Origin | Literal meaning | What it means now | Sources |
|---|---|---|---|---|---|---|
| asymptote | before 200 BCE (Apollonius, broader sense); 1656 in English | Apollonius of Perga | Greek a- + sym- + ptotos | "not falling together" - i.e. not meeting | A line a curve approaches without meeting. In Apollonius it meant ANY lines that do not meet, in any direction. First English citation 1656, Hobbes: "Asymptotes ... come still nearer and nearer, but never touch" | Source 453, Miller and Aldrich |
| parisotes (παρισότης) | 250 | Diophantus of Alexandria | Greek | para (beside) + isotes (equality) = near-equality | approximate equality; the ancestor of 'adequality' | Source 130, Katz |
| sankalita | 499 | Aryabhata I | Sanskrit | summation | Sum of a series; vara-sankalita is repeated (higher-order) summation | Source 50, Ramasubramanian |
| asanna | 499 | Aryabhata I | Sanskrit | near, approximate | Aryabhata's own label for his pi value 62832/20000; Nilakantha built an incommensurability argument on this single word | Source 50, Ramasubramanian; Source 59, O'Connor |
| kuttaka | 499 | Aryabhata I | Sanskrit | pulveriser | The algorithm for integer solutions of linear Diophantine equations, essentially the Euclidean algorithm | Source 59, O'Connor |
| sine | 1100s | Gherardo of Cremona (c. 1114-87), translating Arabic into Latin | Sanskrit ardha-jya > jya/jiva > Arabic jiba/jaib > Latin sinus | bowstring (Sanskrit); bay or fold (Latin sinus) | The ratio opposite/hypotenuse; the modern word records a mistranslation, since the Arabic consonant string j-y-b was read as jaib ('bay') rather than the Sanskrit loanword jiba | Source 60 |
| khahara | 1150 | Bhaskara II | Sanskrit | having kha (zero) as divisor | A quantity divided by zero, which Bhaskara II says is unaltered by addition or subtraction | Source 50, Ramasubramanian |
| latitudo (latitude of a form) | 1300 | Oxford Calculators / Kilvington | Latin | breadth | the intensity or magnitude of a quality, plotted as the ordinate | Source 100, Kirschner; Source 101, Jung; Source 102, Jung; Source 107, Oresme |
| longitudo | 1355 | Nicole Oresme | Latin | length | the extension of the subject, plotted as the abscissa (for motion, time) | Source 100, Kirschner; Source 107, Oresme |
| average | c. 1500 in English | - | Old French avarie, ultimately from Arabic awariya, damaged goods | originally a maritime-trade term: the loss on damaged cargo, and then the equitable SHARING of that loss among the merchants | The arithmetic mean. The mathematics came from insurance: "average" means dividing a loss fairly, and only later dividing anything by n | Source 453, Miller and Aldrich |
| quadrature | classical Latin (a land measure); 1569 in English; used by Newton 1669 | no coiner recorded; Newton uses it in 1669 | Latin quadratura, a squaring | literally "a squaring": in classical Latin, the division of land into squares; in mathematics, the making of a square of equal area | Finding an area under a curve; computing a definite integral. To "square" a figure is to construct a square of the same area, which is why finding areas is called squaring. Newton's De analysi Rule I is a quadrature rule | Source 184, Newton; Source 187, Newton; Source 453, Miller and Aldrich |
| concave | 1571 in English | Thomas Digges, A Geometricall Practise named Pantometria | Latin concavus (com- + cavus, hollow) | "hollowed out" | Curving inwards. Its opposite, convex, is Latin convexus, "vaulted, arched" | Source 453, Miller and Aldrich |
| tangent (linea tangens) | 1583 | Thomas Fincke, Geometriae rotundi | Latin tangere | "the touching line" | A line meeting a curve and not crossing it there. In English from 1597 (Blundevil): "two other right lines belonging to a Circle, called lines Tangent, and lines Secant" | Source 453, Miller and Aldrich |
| secant (secans) | 1583 | Thomas Fincke, Geometriae rotundi | Latin secare | "the cutting line" | A line cutting a curve in two points; in trigonometry, 1/cos. Viete disliked the word for fear of confusion with the geometric sense and proposed Transsinuosa instead | Source 453, Miller and Aldrich |
| logarithm (logarithmus) | 1614 | John Napier, Mirifici logarithmorum canonis descriptio | Greek logos + arithmos | commonly read as "ratio-number"; but the OED warns that Napier never explained it and logos may simply mean "reckoning" | The exponent to which a base must be raised. Before coining the word Napier called them numeri artificiales, "artificial numbers", as against the numeri naturales they act on | Source 453, Miller and Aldrich |
| veluti ('as it were') | 1615 | Johannes Kepler | Latin | as it were, just as if | Kepler's qualifier for treating a solid as composed of infinitely many infinitely thin parts | Source 151, [attributed to Jourdain, P. E. B. - UNVERIFIED] |
| stereometria doliorum (doliometry) | 1615 | Johannes Kepler | Latin, from Greek stereos + metron; dolium = cask | solid-measurement of casks | the measurement of barrel volumes; the title subject of Kepler's 1615 book | Source 159, Kepler |
| cosine | 1620 (co.sinus); 1658 (cosinus) | Edmund Gunter proposed co.sinus; John Newton contracted it to cosinus | Latin sinus complementi | "the sine of the complement" - the sine of what is left when you take the angle from a right angle | cos. Unlike sine, cosine has no Sanskrit or Arabic ancestry: it is a European Latin abbreviation. Earlier European names for it include sinus rectus complementi (Regiomontanus c. 1463) and sinus rectus secundus (Apianus 1541) | Source 453, Miller and Aldrich |
| parameter | 1631 (Mydorge, for the latus rectum); 1673 onwards in the modern sense (Leibniz) | Claude Mydorge; then Leibniz | Greek para + metron | "measured alongside" - a quantity measured beside the others | A constant that indexes a family of curves or functions. Van Schooten (1659) is probably where Leibniz got it | Source 453, Miller and Aldrich |
| indivisibilia / indivisibles | 1635 | Bonaventura Cavalieri | Latin | things that cannot be divided | the lines or planes into which a figure is resolved for comparison of magnitudes | Source 135, O'Connor; Source 156, Andersen; Source 132, Sherry |
| adaequalitas / adequality | 1636 | Pierre de Fermat (via Bachet) | Latin, calqued from Greek parisotes | ad- (towards) + aequalitas (equality) = near-equality | the step of setting f(a+e) 'almost equal' to f(a) before dividing by e and suppressing e | Source 130, Katz |
| inflection point | c. 1638 (written); 1679 (printed) | Pierre de Fermat, appendix to Methodus ad disquirendam maximam et minimam | Latin inflectere | "a bending in" | Where a curve changes from convex to concave. Fermat: "investiganda sunt ex arte puncta inflexionum, in quibus curvatura ex convexa fit concava vel contra" - the points of inflection must be found by art, where the curvature turns from convex to concave or the reverse | Source 453, Miller and Aldrich |
| rectification | 1658 (the cycloid) | Christopher Wren for the cycloid result | Latin rectus, straight | straightening | finding the arc length of a curve | Source 64, Whitman |
| fluxion / fluent | 1665-1671 | Isaac Newton | Latin fluere, to flow | a flowing quantity (fluent) and the velocity with which it flows (fluxion) | Function of time, and its derivative. NOTE the words are older than Newton: Richard Swineshead used fluxus and fluens in the Liber calculationum c. 1350 (Boyer). The word fluxion appears once in the Principia, apparently by oversight | Source 453, Miller and Aldrich; Source 185, Newton |
| calculus (in English) | 1666 (general sense) | anonymous, Philosophical Transactions vol. 1, p. 369 | English from Latin | "a system or method of calculating" | The English noun predates Newton s and Leibniz s calculus, which is why English long said "the calculus" (i.e. THE method of reckoning) rather than "calculus". When the definite article was dropped I could not pin to a document | Source 453, Miller and Aldrich; Source 455, Lewis and Short |
| infinitesimal | 1667/68 (17/27 Feb, letter; printed 1685) | John Wallis, letter to Vincent Leotaud | Latin infinitesimus | "the infinite-th" - one part out of infinitely many, formed like decimus, centesimus, millesimus | Vanishingly small. Wallis s sentence: "Dic Angulum Contactus esse, Infinitesimam partem duorum Rectorum" - say the angle of contact is an infinitesimal part of two right angles | Source 453, Miller and Aldrich |
| limit (limes) | 1667 (terminus, Gregory of St Vincent); 1687 (Newton, ultimate ratio); 1821 (Cauchy, modern) | Cauchy for the modern definition | Latin limes | a boundary path between fields; later the fortified frontier of the Roman Empire | The value approached. The Latin word is a path, not a wall: a limes is the track along a boundary. Newton s Principia scholium argues for it with a moving body: the ultimate velocity is "that with which the body arrives at its last place" | Source 453, Miller and Aldrich; Source 451, Cajori; Source 361, Cauchy |
| convergent and divergent | 1667 | James Gregory, Vera circuli et hyperbolae quadratura | Latin convergere / divergere | to bend or lean together / to bend apart | Of a series, having or not having a limiting sum. Gregory wrote series convergens | Source 456, Cajori; Source 453, Miller and Aldrich |
| logarithmus naturalis (natural logarithm) | 1668 | Nicolaus Mercator | Latin | natural logarithm | logarithm to base e; so named because it arises with no chosen base from the area under xy=1 | Source 155, Coolidge |
| fluent | 1671 | Isaac Newton | Latin (fluens, present participle of fluere, to flow) | a flowing thing | A quantity that varies continuously with the independent variable; what we now call a function of time (or of the parameter) | Source 185, Newton; Source 192, Whiteside |
| fluxion | 1671; 1676 (concealed), 1693 (in print); 1704 (Hayes, in the first English calculus book) | Isaac Newton; Charles Hayes brought the word into English in 1704 | Latin (fluxio, a flowing); the English word comes through Latin fluere, to flow | a flowing, the rate of flow of a quantity: the rate of change of a "fluent" (a flowing quantity) | The instantaneous rate of change of a fluent; the derivative. Newton's own gloss: 'the Velocities by which every Fluent is increased by its generating Motion'. Hayes treats "Moments, Differences or Fluxions" as three names for one thing | Source 185, Newton; Source 187, Newton; Source 192, Whiteside; Source 280, [Newton]; Source 281, Newton; Source 286, Newton; Source 439, Hayes |
| moment | 1671-1704 | Isaac Newton | Latin momentum, contracted from movimentum: a movement, an instant | "a movement", hence a moment of time and a moving weight; in Newton, an infinitely small increment accruing in an infinitely small time | The infinitesimal increment of a fluent: Newton writes the moment of x as x-dot times o, roughly the modern differential dx. Newton, De quadratura curvarum 1704: "Momenta id est incrementa momentanea synchrona" | Source 185, Newton; Source 453, Miller and Aldrich |
| function (functio) | 1673 | Gottfried Wilhelm Leibniz (non-analytical); Johann Bernoulli 1698 (analytical); Euler 1748 (general) | Latin fungi, functus | the performing of a duty, an office, a task | A rule assigning one output to each input. Leibniz s 1673 use is geometric (a line "performing an office" in a figure), not analytic; Johann Bernoulli 1698 and Euler 1748 make it analytic | Source 317, Lagrange; Source 318, Euler; Source 324, Miller; Source 453, Miller and Aldrich |
| variable | 1673 onwards | Gottfried Wilhelm Leibniz | Latin variabilis, from variare | changeable | A quantity that varies. In English as an adjective 1710 (Harris), as a NOUN only in 1816, in the translation of Lacroix - meaning that for a century English speakers said "variable quantity" and never "a variable" | Source 453, Miller and Aldrich |
| rectification / to rectify | 1673 in English | William Brouncker, Phil. Trans. VIII, 6150 | Latin rectus + facere | "to make straight" | Finding the arc length of a curve, i.e. straightening it out. Brouncker: "It was easier to infer, That, if we can Rectifie the one, we may square the other" | Source 453, Miller and Aldrich |
| algorithm (algorithmus, of the calculus) | 1684 | Gottfried Wilhelm Leibniz | Latin (from Arabic al-Khwarizmi) | Leibniz calls his rules "an algorithm, so to speak, of this calculus": a set of formal rules operable without thinking about the geometry | Algorithm | Source 232, Leibniz (trans. Walker); Source 231, Leibniz |
| triangulum characteristicum (characteristic triangle) | 1686 | Gottfried Wilhelm Leibniz | Latin | "the distinguishing/defining triangle": the infinitely small triangle with sides dx, dy and the element of arc | The differential triangle underlying the derivative and arc length | Source 232, Leibniz (trans. Walker); Source 233, Leibniz |
| analysis indivisibilium atque infinitorum | 1686 | Gottfried Wilhelm Leibniz | Latin | "analysis of indivisibles and infinites" | Infinitesimal analysis | Source 233, Leibniz; Source 232, Leibniz (trans. Walker) |
| nascent | 1687 | Isaac Newton | Latin (nascens, being born) | being born, coming into being | A quantity just beginning to increase from zero; the 'first ratio of nascent quantities is that with which they begin to be' | Source 186, Newton; Source 187, Newton |
| evanescent | 1687 | Isaac Newton | Latin (evanescens, vanishing) | vanishing | A quantity in the act of shrinking to zero. Newton insists these are 'evanescent divisible quantities', not indivisibles | Source 186, Newton; Source 187, Newton |
| evanescent / nascent | 1687 | Isaac Newton, Principia Book I Section I | Latin evanescere / nasci | "vanishing" / "being born" | Quantities in the act of shrinking to zero, or of growing from it. Newton s scholium: "by the ultimate ratio of evanescent quantities is to be understood the ratio of the quantities not before they vanish, nor afterwards, but with which they vanish" | Source 453, Miller and Aldrich; Source 186, Newton |
| integral (integralis) | 1690 | Jacob Bernoulli, Acta Eruditorum May 1690, p. 218 | Latin integer / integralis | whole, entire, untouched | The total accumulated from the parts; and, as a noun, the result of the inverse of differentiation. Leibniz wanted calculus SUMMATORIUS; Johann Bernoulli wanted calculus INTEGRALIS and the capital letter I as the sign. They compromised in 1696: Bernoulli s WORD, Leibniz s SYMBOL | Source 324, Miller; Source 451, Cajori; Source 453, Miller and Aldrich |
| catenary (catenaria) / funicularia | 1690-1691 | Jacob Bernoulli posed it as the funicularia; catenaria from Huygens | Latin catena / funiculus | chain / little rope | The curve of a chain hanging under its own weight; the hyperbolic cosine, not the parabola Galileo supposed | Source 334, British Society for the History of Mathematics |
| differential (as a noun) | 1690 (10 Sept MS) | Gottfried Wilhelm Leibniz, "Methodus pro differentialibus, ponendo z = dy:dx" | Latin differentia | a difference | An infinitesimal increment. In English from 1704, in Harris s Lexicon Technicum | Source 453, Miller and Aldrich |
| brachistochrone | 1696 | Johann Bernoulli's challenge | Greek brachistos + chronos | shortest time | The curve of quickest descent under gravity; it is the cycloid | Source 330, O'Connor |
| normal | c. 1696 in English | Edmund Scarburgh, The English Euclide ("Normal Line") | Latin norma | a carpenter s square; hence "at right angles, according to the square" | Perpendicular to a curve or surface. The word for "normal" in the everyday sense - conforming to a rule - is the same carpenter s square | Source 453, Miller and Aldrich |
| analysis infinitesimalis | 1697 | Gottfried Wilhelm Leibniz | Latin | "infinitesimal analysis": Leibniz's proposed NEUTRAL name covering both his differential method and Newton's fluxions | Infinitesimal calculus | Source 241, Leibniz (ed. Gerhardt) |
| series | before 1700 as proportio; "series" standard by the 18th century | - | Latin series, from serere, to join, to link | "a row, a chain, a linked succession" - the same root as "assert" and "insert" | A sum of a sequence of terms. Smith: "The early writers often used proportio to designate a series, and this usage is found as late as the 18th century" | Source 453, Miller and Aldrich |
| exponential | 1704 in English | John Harris, Lexicon Technicum | Latin exponere | "setting out, putting forth" - the exponent is the number "set above" | Of the function a^x. Harris: "Exponential curves are such as partake both of the nature of Algebraick and Transcendent ones" | Source 453, Miller and Aldrich |
| enri (円理) | 1710 | Takebe Katahiro (long credited to Seki Takakazu) | Japanese | circle principle | The Japanese method most closely corresponding to definite integration | Source 430, OConnor and Robertson |
| Commercium Epistolicum | 1713 | Royal Society (drafted by Isaac Newton) | Latin | "exchange of letters"; short for Commercium Epistolicum D. Johannis Collinsii et aliorum de analysi promota | The Royal Society's published dossier and report on the priority dispute | Source 280, [Newton]; Source 282, Newton |
| Charta volans | 1713 | anonymous (for Leibniz; quoting Johann Bernoulli anonymously) | Latin | "flying sheet": a single loose printed sheet circulated without imprint | The anonymous Continental rebuttal to the Commercium Epistolicum | Source 284, Kollerstrom |
| versoria / la versiera | 1718 | Guido Grandi | Latin versoria, from vertere, to turn | the sheet or rope that turns a sail | The curve y = a sqrt((a-x)/x); in modern form y(x^2 + a^2) = a^3 | Source 327, O'Connor; Source 319, Agnesi |
| continuous | 1748 (Euler, in the OLD sense); 1817 Bolzano and 1821 Cauchy (modern) | Euler for the old sense; Bolzano and Cauchy for the modern | Latin continuus, from continere | hanging together, unbroken | WARNING FOR READERS OF OLD TEXTS: Euler s 1748 "continuous curve" means one given by a SINGLE formula. A curve made of two formulas was "discontinuous" even if you could draw it without lifting the pen. The modern sense is the opposite kind of criterion | Source 453, Miller and Aldrich; Source 361, Cauchy; Source 363, Bolzano |
| derivative (fonction derivee) | 1772 (term); 1797 (systematic) | Joseph-Louis Lagrange, Nouveaux Memoires de l'Academie de Berlin; systematised in Theorie des fonctions analytiques 1797 | French deriver, from Latin derivare (de + rivus, a stream) | to draw off water from a stream | The rate of change of a function: literally the function DERIVED from another. The literal image is a channel led off a river, the derived function drawn off from the primitive one | Source 317, Lagrange; Source 324, Miller; Source 451, Cajori; Source 453, Miller and Aldrich |
| D-ism / Dot-age | 1812 (recalled 1864) | Charles Babbage | English (pun) | "D-ism" puns on deism (Leibniz's d); "Dot-age" puns on dotage/senility (Newton's dot). Babbage's footnote: "Leibnitz indicated fluxions by a d, Newton by a dot." | Slogan of the Analytical Society campaign to adopt Leibnizian notation in Britain | Source 239, Babbage |
| limit | 1821 | Cauchy (modern definition); Newton 1687 'ultimate ratio'; Gregory of St Vincent used 'terminus' | Latin 'limes', a boundary path or frontier; 'terminus', a boundary stone | a boundary marker between fields, later the fortified frontier of the Roman Empire | the value a variable approaches so as to end up differing from it by as little as one wishes | Source 392, Miller; Source 361, Cauchy |
| infiniment petit / infinitesimal | 1821 | Cauchy (as a defined term in the Cours d'analyse) | French, from Latin 'infinitesimus', the infinite-th | the infinite-th part, i.e. one part of infinitely many | for Cauchy, a variable whose successive numerical values fall below any given number, i.e. a variable with limit zero - not a fixed number | Source 361, Cauchy; Source 369, Bell |
| fonction continue / continuous function | 1821 | Cauchy in print; Bolzano's equivalent definition 1817 | Latin 'continuus', from 'continere', to hold together | hanging together, unbroken | f is continuous if an infinitely small increment of the variable always produces an infinitely small increment of the function (Cauchy 1821) - now epsilon-delta | Source 361, Cauchy; Source 363, Bolzano; Source 369, Bell; Source 387, Ruch |
| serie convergente / convergent series | 1821 | Cauchy (definition by limit of partial sums) | Latin 'convergere', to incline together | to bend or lean together toward a point | a series whose partial sums approach a limit, which is called the sum of the series | Source 361, Cauchy; Source 364, Abel |
| solution de continuite (a discontinuity) | 1821 | Cauchy | French, from Latin 'solutio continuitatis', a breaking of the continuity | a breaking-apart of the holding-together; the phrase was medical, for a wound or fracture | a point where a function fails to be continuous | Source 361, Cauchy |
| integrale definie / definite integral (as a limit of sums) | 1823 | Cauchy; generalised by Riemann 1854 | Latin 'integer', whole, untouched | making whole; the integral sign is a stretched letter S for 'somme', sum | the limit of sums of the form sum f(x_i)(x_{i+1}-x_i) as the subdivisions shrink | Source 362, Cauchy; Source 391, Riemann |
| epsilon-delta | 1823 | Cauchy uses the inequalities in proofs; the notation stabilises through Weierstrass's Berlin lectures | Greek letters | - | the quantified inequality definition of limit and continuity; 'who wrote it first' is genuinely tangled because Weierstrass's lectures circulated as students' notes | Source 360, Grabiner; Source 373, O'Connor; Source 369, Bell; Source 368, Borovik |
| the first epsilon-delta proof | 1823 | Augustin-Louis Cauchy, Calcul infinitesimal, Lecon 7 (Oeuvres ser. 2 vol. 4, pp. 44-45) | - | Cauchy s proof of what is essentially the mean value theorem for derivatives | The moment the verbal definition of a limit is translated into inequalities. Judith Grabiner: in the 1820s Cauchy did not say what his delta depended on, which is why the pointwise/uniform distinction is missing | Source 452, Miller; Source 360, Grabiner |
| scientist | 1834 | Attributed by Whewell to "some ingenious gentleman" at the British Association | English (from Latin scientia, by analogy with artist) | one who does science | A person engaged in systematic study of the natural world | Source 411, Whewell |
| nature-poker / nature-peeper | 1834 | Whewell | English (rendering German Naturforscher) | one who pokes at nature | Rejected alternatives to "scientist", offered half-seriously in the same paragraph | Source 411, Whewell |
| vector | c. 1840 (Hamilton); earlier in "radius vector" in astronomy | William Rowan Hamilton | Latin vector, from vehere, to carry | "a carrier, one who conveys" | A directed magnitude. The word arrives in mathematics out of astronomy s radius vector - the "carrying radius" that sweeps a planet round - and Hamilton kept the noun after dropping the radius | Source 453, Miller and Aldrich |
| range | 1848 in mathematical use (statistics) | H. Lloyd, Proc. Royal Irish Acad. 4 | Old French rangier, to set in a row | a row, a rank; hence the spread from lowest to highest | The set of outputs of a function. NOTE the function sense and the statistical sense (max minus min) are different words in practice and students conflate them | Source 453, Miller and Aldrich |
| matrix | 1850 | James Joseph Sylvester, Philosophical Magazine, pp. 363-370 | Latin matrix, from mater | "womb; the thing from which something else is born" | A rectangular array. Sylvester meant it literally: the oblong array "is, as it were, a Matrix out of which we may form various systems of determinants". For him the matrix was the mother of the determinants, not an object in itself; Cayley made it an object in 1855 and 1858 | Source 453, Miller and Aldrich |
| Menge / set | 1851 | Bolzano, Paradoxien des Unendlichen (posthumous) | German 'Menge', a quantity, a crowd | a crowd, a multitude, an amount | a collection of objects considered as a single object | Source 371, O'Connor |
| uniform convergence | 1853 | Concept absent from Cauchy 1821; the missing hypothesis identified through Abel 1826, Bjorling, Seidel, Stokes and Weierstrass; Cauchy adds a condition in 1853 | Latin 'uniformis', of one shape | having one and the same form throughout | convergence at a rate independent of the point; whether Cauchy's 1853 condition IS uniform convergence is disputed | Source 367, Bascelli; Source 364, Abel; Source 360, Grabiner |
| nabla (the name) | c. 1870 | William Robertson Smith, suggesting it to P. G. Tait | Greek nabla, from Hebrew/Phoenician nevel | an ancient Levantine harp of ten or twelve strings | The name of the symbol, chosen by a Semitic-languages scholar purely because the inverted delta looks like a harp. Maxwell thought it a joke and used it in print exactly once: in the dedication of a comic poem, "To the Chief Musician upon Nabla: A Tyndallic Ode" | Source 454, Knott; Source 453, Miller and Aldrich |
| atled | 1870 | James Clerk Maxwell, letter to Tait, 7 Nov 1870 | English | "delta" spelled backwards | Maxwell s rival name for the nabla operator. It lost | Source 454, Knott |
| divergence | 1870 (as "convergence", Maxwell); 1892 in the modern sense | James Clerk Maxwell coined the concept-name; the sign was flipped later | Latin divergere | "bending apart" | div of a vector field. Maxwell s own word, in the letter to Tait of 7 Nov 1870, was CONVERGENCE, and modern divergence is minus Maxwell s convergence. Clifford used div u or dv u | Source 454, Knott; Source 453, Miller and Aldrich; Source 451, Cajori |
| curl | 1870 (7 Nov) | James Clerk Maxwell, letter to P. G. Tait | English | a curl, as of hair or a scroll | curl of a vector field. Maxwell s fourth attempt: he tried Twist ("suggests a screw"), then turn or version, then Twirl ("sufficiently racy ... perhaps too dynamical for pure mathematicians"), then "for Cayley s sake I might say Curl (after the fashion of Scroll)". Rival names in other languages survive: Rot (Lorentz, Abraham), Quirl (Wiechert), Vort (Voigt) | Source 454, Knott; Source 453, Miller and Aldrich |
| concentration | 1870 (7 Nov) | James Clerk Maxwell, letter to P. G. Tait | English | Maxwell s name for the Laplacian, "because it indicates the mode in which the value of the function at a point exceeds the average value of the function in a little spherical surface drawn round it" | A dead name for nabla squared - but the best one-sentence explanation of what the Laplacian MEANS that anyone has written | Source 454, Knott |
| Schnitt / Dedekind cut | 1872 | Dedekind (idea found 24 November 1858) | German 'Schnitt', a cut | a cut, a slice | a partition of the rationals into two classes with every member of the first less than every member of the second; each cut is produced by exactly one real number | Source 366, Dedekind; Source 369, Bell |
| domain | 1886 | Arthur Cayley, "On Linear Differential Equations", Quart. J. Pure Appl. Math. | Latin dominium, from dominus | "lordship, the land a lord rules" | The set of allowed inputs. Cayley: "points x within the domain of the point a" - originally a region of the complex plane, not a set of inputs | Source 453, Miller and Aldrich |
| gradient | 1897 (scalar sense); c. 1900-1909 (vector sense) | Horace Lamb, An Elementary Course of Infinitesimal Calculus | Latin gradi, to step | "a stepping, a walking" - hence a slope, as in a railway gradient | Lamb coined it for the slope of a plane curve: "It is convenient to have a name for the property of a curve which is measured by the derived function." The vector sense is later, via H. Weber s 1900 edition of Riemann s lectures. Maxwell had called it SLOPE (1870), then space-variation; Gibbs called it the derivative | Source 453, Miller and Aldrich; Source 454, Knott |
| Radon-Nikodym derivative | 1930 | Johann Radon and Otton Nikodym | English (Austrian and Polish authors) | the density of one measure with respect to another | dnu/dmu: the general form of the Fundamental Theorem of Calculus, and the reason "probability density" is well defined | Source 435, OConnor and Robertson |
| hyperreal | 1948 | Edwin Hewitt | English | beyond-real | An element of an ordered field properly extending the reals, containing infinitesimals and infinite numbers | Source 440, Keisler |
| distribution (generalised function) | 1948 | Laurent Schwartz | French (distribution) | a distribution of values over test functions | A continuous linear functional on test functions; every distribution has derivatives of all orders, so the Dirac delta becomes legitimate | Source 433, OConnor and Robertson |
| nonstandard analysis | 1961 | Abraham Robinson | English | analysis over a nonstandard model | Calculus and analysis done rigorously with infinitesimals, via the hyperreals and the Transfer Principle | Source 440, Keisler; Source 432, OConnor and Robertson |
| standard part | 1961 | Abraham Robinson | English | the standard (real) part of a finite hyperreal | The operation that rounds a finite hyperreal to the unique real infinitely close to it - the single step that makes Leibniz's dy/dx rigorous | Source 440, Keisler |
| Transfer Principle | 1961 | Abraham Robinson (anticipated by Leibniz's law of continuity and by Skolem 1934) | English | transfer of statements between models | Every first-order statement true of the reals is true of the hyperreals and conversely | Source 440, Keisler; Source 432, OConnor and Robertson |
| Wengert list | 1964 | Robert Wengert | English | - | The linear trace of elementary operations in a computation, on which automatic differentiation operates | Source 436, Baydin, Pearlmutter, Radul and Siskind |
| reverse mode / backpropagation | 1970 | Seppo Linnainmaa (reverse mode); the name backpropagation from the neural network community | English | propagating derivatives backwards | Evaluation of the gradient by traversing the computational graph backwards; the chain rule applied in reverse. Cost is a small constant multiple of one forward evaluation regardless of the number of inputs | Source 436, Baydin, Pearlmutter, Radul and Siskind; Source 437, Schmidhuber |
| Nicodemite | 16th century (Calvin); applied to Newton 1999 | Stephen Snobelen (application) | Greek/Latin via John 3:1-2 | one who follows Nicodemus, who came to Jesus by night | A person who conceals heretical belief behind outward conformity. Snobelen's term for Newton's fifty-year strategy of public Anglican conformity and private anti-Trinitarianism | Source 194, Snobelen |
| palimpsest | - | - | Greek (palimpsestos) | scraped again | A manuscript whose original text was erased so the parchment could be reused | Source 4 |
| upapatti | - | Indian mathematical tradition | Sanskrit | that which establishes, demonstration | Proof or rationale; the Indian tradition's own word for what the Yuktibhasa supplies | Source 50, Ramasubramanian |
| del | 20th century | American usage | English | a clipping of delta | The usual American name for the nabla symbol. Cajori (1929) reports "It is often read del" | Source 451, Cajori |
| calculus (the Latin word) | classical Latin | - | Latin, diminutive of calx (limestone) | a little stone, a pebble | The word that names the subject. Lewis and Short give six classical senses, all from "pebble": a pebble; a stone in the bladder or kidneys; a draughtsman in the board game duodecim scripta; a counter on a reckoning board and hence a reckoning; a voting stone (white to acquit, black to condemn); a small weight | Source 455, Lewis and Short; Source 453, Miller and Aldrich |
| calculus (the kidney stone) | 1st century CE | Celsus, De medicina 7.26; Pliny | Latin | the same pebble, lodged in the bladder | A renal calculus. It is NOT a metaphor from mathematics and mathematics is not a metaphor from it: both senses descend independently from "pebble", and both were already current in classical Latin | Source 455, Lewis and Short |
| calculator / calculo | classical and late Latin | - | Latin | one who reckons with pebbles | A teacher of arithmetic. Roman usage split by class: a slave teaching calculation was a calculo, a freeborn one a calculator or numerarius | Source 453, Miller and Aldrich |
| analysis | Greek; named as a method by Theon of Alexandria (4th c. CE) | Theon of Alexandria (per Kline); revived by Viete 1591 | Greek analysis, from analyein | "a loosening up, an undoing" - untying a knot | Now the branch of mathematics built on limits. Anciently, a method: assume the result and reduce it backwards to something known. Viete pushed "analytic art" as a replacement for "algebra"; he failed to kill algebra but succeeded in planting analysis | Source 453, Miller and Aldrich |
| maximum and minimum | classical Latin words, mathematical use from Fermat onwards | Latin writers on the calculus | Latin | the neuter superlatives of magnus (great) and parvus/minus (small): "the greatest thing", "the least thing" | The largest and smallest values. They are neuter singular nouns, which is why the plurals are maxima and minima, not maximums | Source 453, Miller and Aldrich |
| sine (the mistranslation chain) | Sanskrit 5th c.; Arabic 8th-9th c.; Latin 12th c. | Aryabhata (jya, 499) -> Arabic jiba -> misread as jaib -> Latin sinus | Sanskrit ardha-jya (half-bowstring) > jya > Arabic jiba > written unvowelled j-b > read as jaib (bosom, fold, bay) > Latin sinus (a fold in a toga, a bay) | "bowstring" (Sanskrit jya) becoming "bay, fold, bosom" (Latin sinus) | The ratio opposite/hypotenuse. Latin chorda would have been the correct translation and WAS used by Plato of Tivoli c. 1120; sinus won anyway. WHO made the misreading is genuinely unsettled: Robert of Chester (1145) and Gerard of Cremona (c. 1150) are both named in the literature | Source 453, Miller and Aldrich |
| mean | Greek theory of means, Pythagorean school | the Pythagoreans, per Heath | Latin medianus, middle | the middle one | Arithmetic, geometric or harmonic mean. Heath: the Pythagoreans had three means, "the arithmetic, the geometric and the subcontrary"; the third was later renamed the harmonic | Source 453, Miller and Aldrich |
| wasan (和算) | - | - | Japanese | Japanese calculation | The indigenous mathematics of Edo-period Japan, developed under the closed-country policy | Source 429, OConnor and Robertson; Source 430, OConnor and Robertson |
| trochoid / roulette | 17th century | Roberval used "trochoid"; French writers preferred "roulette" | Greek trochos, a wheel; French rouler, to roll | wheel-curve / little rolling thing | the cycloid and its relatives | Source 64, Whitman |
| philomath | 17th century (usage reported) | Guicciardini, describing Collins and the outer readership | Greek | lover of learning | A non-specialist enthusiast for mathematics; in Newton's case the outer ring of readers who learned his results second hand from the acolytes, sometimes in mutilated form | Source 204, Guicciardini |
Named theorems, curves, and people
| Item | First attested | Introduced by | Origin | Literal meaning | What it means now | Sources |
|---|---|---|---|---|---|---|
| Cavalieris principle | c. 480 CE (Zu Gengzhi); 1635 (Cavalieri) | Zu Gengzhi | - | - | See T-A015: the Chinese statement is about 1150 years earlier and is quoted verbatim in the Chinese sources | Source 23, Wagner; Source 135, O'Connor |
| Pascals triangle | 10th c. Halayudha (India); c. 1000 al-Karaji, c. 1100 Khayyam (Islamic world); 1050-1303 Jia Xian, Yang Hui, Zhu Shijie (China); Tartaglia and Stifel (Europe); 1654/1665 Pascal | many, independently | - | - | Cajori, writing in 1919, already says "known in the West as Pascal s arithmetical triangle" and notes it was in Chu Shih-Chieh (1303), known to Arab mathematicians in the eleventh century, and "before Pascal ... constructed by N. Tartaglia and M. Stifel". Italian still says triangolo di Tartaglia, Iranian usage says Khayyam s triangle, Chinese says Yang Hui s triangle | Source 456, Cajori |
| Alhazen | 1270 | Latin translators of Kitab al-Manazir | Latinisation of Arabic al-Hasan | - | The European name for Ibn al-Haytham; nobody in Basra ever called him that | Source 58, O'Connor |
| Calculator (as a name for Richard Swineshead) | 1350 | contemporaries of Swineshead | Latin | one who reckons | the standard medieval nickname for Swineshead; later the name of a machine | Source 106, Gale |
| Gregory s series | c. 1400 (Madhava); 1671 (Gregory); 1674 (Leibniz) | Madhava of Sangamagrama | - | - | The arctan series, and the pi/4 case. Miller: "the result had been obtained around three hundred years earlier by the Indian mathematician Madhava. Western mathematicians only learnt of Madhava s contribution in the twentieth century" | Source 453, Miller and Aldrich; Source 55, O'Connor; Source 50, Ramasubramanian |
| Simpsons rule | 1639 Cavalieri; 1615 Kepler (the barrel rule); 1668 James Gregory; 1743 Simpson | Simpson himself said he had it from Newton | - | - | MacTutor: "the numerical method known today as Simpson s rule, although it did appear in his work, was something he learned from Newton as Simpson himself acknowledged". Whittaker and Robinson: "first given (in a geometrical form) by Cavalieri [1639], and later by James Gregory [1668] and by Thomas Simpson [1743]". German still calls it the Keplersche Fassregel, Kepler s barrel rule | Source 462, OConnor and Robertson; Source 453, Miller and Aldrich |
| Amos Dettonville | 1658 | Blaise Pascal | French | anagram of Louis de Montalte, his Provincial Letters pseudonym | Pascal's pen-name for the cycloid contest and the Traite des sinus | Source 143, [author unverified] |
| Newtons method | 1669 Newton (algebraic, no derivative); 1690 Raphson; 1740 Simpson (the modern iteration) | Thomas Simpson for the form taught | - | - | MacTutor: "the Newton-Raphson method for solving f(x) = 0 is, in its present form, due to Simpson ... Newton described an algebraic process for solving polynomial equations which Raphson later improved. The method of approximating the roots did not use the differential calculus. The modern iterative form x_{n+1} = x_n - f(x_n)/f (x_n) is due to Simpson, who published it in 1740." Antecedents in Sharaf al-Din al-Tusi (12th c.) and in Chinese root-extraction (Jia Xian, Qin Jiushao) are logged by other agents | Source 462, OConnor and Robertson; Source 53, O'Connor |
| Gregory-Newton interpolation formula | 1670 (Gregory, unpublished); 1687 (Newton, in print) | James Gregory | - | - | Whittaker and Robinson (1924): "The formula is often referred to as Newton s formula of interpolation, although it was discovered by James Gregory in 1670." They coined the double-barrelled name to fix it, and it partly stuck | Source 453, Miller and Aldrich |
| Rolles theorem | 1691 | Michel Rolle stated it, without proof, in a book on solving polynomial equations | - | - | It was a theorem about polynomials, produced by Rolle s "method of cascades". AND: Rolle led the campaign in the Paris Academy from 1700 to 1705 against the infinitesimal calculus, which he "described as a collection of ingenious fallacies". The theorem underpinning the differential calculus is named after one of its loudest opponents | Source 459, OConnor and Robertson; Source 453, Miller and Aldrich |
| Mean Value Theorem | 1691 (Rolle, special case); 1806/1823 (Lagrange and Cauchy); Bolzano 1817 | Cauchy for the proof now taught; Lagrange for the statement; Bolzano for the tools | - | - | The name in English is late: "theorem of mean value" 1889, "mean value theorem" 1894. Cauchy s 1823 Lecon 7 proof, the first epsilon-delta proof in history, is a proof of essentially this theorem | Source 453, Miller and Aldrich; Source 452, Miller; Source 360, Grabiner |
| LHopitals rule | 1696 (published); 1694 (bought) | Johann Bernoulli discovered it; Guillaume de lHopital published it | - | - | The 0/0 rule. LHopital paid Bernoulli a retainer for his mathematical discoveries and the right to use them; the rule appears in LHopitals Analyse des infiniment petits (1696), the first calculus textbook, which does credit Bernoulli and Leibniz in general terms in its preface | Source 315, Truesdell; Source 316, L'Hospital; Source 242, Leibniz |
| Taylor series | 1712-1715 (Taylor); earlier in Gregory and Johann Bernoulli | Brook Taylor stated it in print (Methodus incrementorum, 1715, Prop. VII Cor. II, p. 23) after communicating it to Machin on 26 July 1712 | - | - | Kline: the result was known to Gregory and Leibniz, and Johann Bernoulli published "practically the same result in 1694". Gibson (1921) concludes Taylor s Prop. XI is essentially Bernoulli s theorem and that "one can not be surprised that Bernoulli felt aggrieved at Taylor s omission of any reference to him" | Source 312, Gibson; Source 453, Miller and Aldrich |
| Maclaurin series | 1717 (Stirling); 1742 (Maclaurin) | James Stirling published it first; Maclaurin himself attributed the general theorem to Taylor | - | - | Stirling, Lineae Tertii Ordinis Neutonianae (1717), p. 32, establishes "the form of Taylor s Theorem now usually called Maclaurin s Theorem", by undetermined coefficients, with worked examples including (a+x)^n and cos x - twenty-five years before Maclaurin s Treatise of Fluxions (1742) art. 751, which uses the same method. Stirling in 1730 wrote that "the first to discover this Theorem was Mr Taylor" | Source 312, Gibson; Source 453, Miller and Aldrich; Source 461, Maclaurin |
| Stirlings approximation | 1730 (Stirling); 1730 and earlier (De Moivre) | Abraham de Moivre found the asymptotic form; Stirling supplied the constant sqrt(2 pi) | - | - | Stirling s formula appears at Methodus Differentialis (1730), Prop. 28, Example 2, p. 136. De Moivre had the form n! ~ C n^(n+1/2) e^(-n) without identifying C; Stirling identified C = sqrt(2 pi). Reasonable people split the credit; the name does not | Source 453, Miller and Aldrich |
| witch of Agnesi | c. 1760 (Colson s mistranslation; printed 1801); the curve was named versoria by Grandi in 1718 | John Colson, translating Agnesi | English, from a misreading of Italian la versiera as l'avversiera | she-devil, adversary, from Latin adversarius | Nothing. It is a mistranslation of a word for a nautical rope, and it has stuck for 265 years. See T-E18013: la versiera is a nautical rope, l avversiera is a she-devil, and Colson read the wrong one. The curve is in Fermat and Grandi before Agnesi | Source 319, Agnesi; Source 320, Agnesi; Source 327, O'Connor |
| Bolzano-Weierstrass theorem | 1817 (Bolzano); 1860s-70s (Weierstrass lectures) | Bernard Bolzano | - | - | Bolzano proved it in the 1817 Rein analytischer Beweis, as a lemma; the paper was almost unread for fifty years, and Weierstrass reproved it for his Berlin lectures. This is the rare case where BOTH names are on it, and the first one still had to be rediscovered to get there | Source 363, Bolzano; Source 370, Morscher; Source 371, O'Connor |
| Cauchy-Schwarz inequality | 1821 Cauchy (sums); 1836 Liouville (implicit); 1859 Bunyakovsky (integrals); 1862 Grassmann; 1885 Schwarz | Viktor Bunyakovsky for the integral form | - | - | Bunyakovsky, "Sur quelques inegalites concernant les integrales ordinaires et les integrales aux differences finies" (Mem. Acad. imp. sci. St-Petersbourg, 1859), inequality (C) on p. 4, WITH the equality case - twenty-six years before Schwarz s 1885 paper (S 15, p. 344), which is the source of the proof now taught. Russian and much European usage correctly says Cauchy-Bunyakovsky-Schwarz | Source 457, OConnor and Robertson; Source 458, Kichenassamy |
| Riemann integral | 1854 (Riemann, Habilitationsschrift); 1875 (Darboux) | Riemann for the definition; Gaston Darboux for the upper-and-lower-sum machinery taught today | - | - | The construction in every modern textbook - upper sums, lower sums, and the criterion that they meet - is Darboux s, from "Memoire sur les fonctions discontinues", Ann. Sci. Ecole Norm. Sup. (2) 4 (1875), 57-112. The name "Riemann integral" is in print by 1907 (Hobson) | Source 453, Miller and Aldrich; Source 391, Riemann |
| Holders inequality | 1888 | Leonard James Rogers, "An extension of a certain theorem in inequalities", Messenger of Math. 17, 145-150 | - | - | Named after Otto Holder, whose paper is 1889. A free bonus correction from the Cauchy-Schwarz literature | Source 458, Kichenassamy |
| Jensens inequality | 1889 | Otto Holder, Nachr. Gottingen, 38-47 | - | - | Named after Jensen, who pointed out in his own paper that Holder had it first, and was ignored | Source 458, Kichenassamy |
| Fundamental Theorem of Calculus (the name) | 1942 in English under that exact name | Sherwood and Taylor, Calculus (1942), per Miller | - | - | The RESULT belongs to Gregory (Geometriae pars universalis, 1668) and Barrow before Newton and Leibniz, and its modern statement to Cauchy; the NAME is a twentieth-century textbook coinage, about 270 years younger than the theorem | Source 453, Miller and Aldrich; Source 138, Nauenberg; Source 140, O'Connor |
| Stiglers law of eponymy | 1980 | Stephen M. Stigler, Transactions of the New York Academy of Sciences | - | "No scientific discovery is named after its original discoverer" | The law this whole section documents. The frequently repeated claim that Stigler credited the law s discovery to the sociologist Robert K. Merton - making the law an instance of itself - COULD NOT BE VERIFIED: the 1980 paper is paywalled (HTTP 403) and no authorised full text was reachable. Bibliographic reference confirmed; the Merton attribution is logged as unverified | Source 460, Stigler |
| Gabriel's horn | 0 | unknown | English | the trumpet of the archangel Gabriel | Torricelli's acute hyperbolic solid; the origin of the name is genuinely unknown | Source 157, Wijeratne |
| Vandermonde determinant | never discussed by Vandermonde | - | - | - | A bonus case: Lebesgue believed the attribution arose from a MISREADING of Vandermonde s unfamiliar notation. Nobody knows who coined the name; it is in print by 1888 | Source 453, Miller and Aldrich |
Phrases and quotations that travel
| Item | First attested | Introduced by | Origin | Literal meaning | What it means now | Sources |
|---|---|---|---|---|---|---|
| theorein vs. apodeiknynai | c. 250 BCE | Archimedes | Greek (θεωρεῖν / ἀποδεικνύναι) | to investigate / to prove | Archimedes' own distinction between a heuristic that suggests a theorem and a demonstration that establishes it | Source 2, Heath |
| ge yuan shu | 263 | Liu Hui | Chinese (割圓術) | circle-cutting technique | Liu Hui's method of doubling the sides of an inscribed polygon to approach the circle | Source 20, Guo |
| mouhe fanggai | 263 | Liu Hui | Chinese (牟合方蓋) | box-lid / joined covers | The solid common to two cylinders inscribed at right angles in a cube; Liu Hui saw the sphere is to it as pi is to 4, but could not compute its volume | Source 23, Wagner |
| yangma / bienao / qiandu | 263 | Liu Hui | Chinese (陽馬 / 鱉臑 / 塹堵) | male horse / turtle's foreleg / moat-wall | A pyramid, a tetrahedron and a right prism; Liu Hui proved the yangma is 2/3 and the bienao 1/3 of the qiandu | Source 20, Guo |
| Zu Geng principle | c. 480 | Zu Gengzhi | Chinese (緣幂勢既同，則積不容異) | if the corresponding areas are the same, the volumes cannot be different | Cavalieri's principle: solids with equal cross-sections at every height have equal volume, stated about 1100 years before Cavalieri | Source 23, Wagner |
| tatkalika gati | 1150 | Bhaskara II | Sanskrit | motion belonging to this instant | Instantaneous velocity | Source 50, Ramasubramanian |
| uniformiter difformis | 1335 | William Heytesbury | Latin | uniformly non-uniform | uniformly accelerated motion (constant acceleration) | Source 101, Jung; Source 107, Oresme |
| antya-samskara | c. 1400 | Madhava of Sangamagrama | Sanskrit | end-correction | A remainder term added where an infinite series is truncated, with a derived criterion for exactness | Source 50, Ramasubramanian |
| method of exhaustion | 1600s | European mathematicians, not the Greeks | Latin/English coinage describing a Greek method | using up | A double reductio proving a magnitude is neither greater nor less than a target; note the Greek method does not exhaust anything, it brackets | Source 1, Heath; Source 8, O'Connor |
| omnes lineae ('all the lines') | 1635 | Bonaventura Cavalieri | Latin | all the lines | the totality of parallel sections of a plane figure, used collectively as a magnitude | Source 156, Andersen |
| ductus plani in planum | 1647 | Gregoire de Saint-Vincent | Latin | the leading of a plane into a plane | Saint-Vincent's construction for generating solids and comparing volumes | Source 141, O'Connor |
| new analysis | 1660s-1670s | Newton's own name for his early method | English rendering of Latin usage | the modern analysis, as against ancient geometry | Wallisian induction from particular cases, free use of infinitesimal moments, and Cartesian representation of curves by equations; Newton decided by the late 1670s that it was not a language fit for print | Source 204, Guicciardini |
| method of fluxions | 1671 | Isaac Newton | English/Latin hybrid | the method of rates of flow | Newton's calculus, as opposed to Leibniz's 'differential calculus'. The two names split the mathematical world for a century | Source 185, Newton; Source 187, Newton |
| calculus differentialis | 1684 | Gottfried Wilhelm Leibniz, Acta Eruditorum | Latin | "the calculus of differences", literally "the reckoning of differences" | Differential calculus. Before coining it Leibniz called his procedure methodus tangentium directa, "the direct method of tangents" | Source 231, Leibniz; Source 232, Leibniz (trans. Walker); Source 453, Miller and Aldrich |
| calculus summatorius | 1684 and 1686 | Gottfried Wilhelm Leibniz | Latin | summing calculus | Superseded name for the integral calculus: Leibniz lost the naming argument to Johann Bernoulli but kept his integral sign, the long s of summa, so his defeated word survives inside the winner s notation | Source 324, Miller; Source 451, Cajori; Source 453, Miller and Aldrich |
| first and last ratios (rationes primae et ultimae) | 1687 | Isaac Newton | Latin | the first and the last ratios | Newton's name for his limit method: the ratio with which quantities begin to be, or with which they vanish. Book I Section I of the Principia carries this title. NOTE ON RENDERING: Motte's 1729 English gives 'first and last ratios', and that phrasing is everywhere; the standard modern translation (Cohen and Whitman, 1999) renders rationes primae et ultimae as 'first and ULTIMATE ratios'. Use whichever suits, but say which edition you are quoting | Source 186, Newton; Source 187, Newton; Source 72, Newton |
| spira mirabilis / eadem mutata resurgo | 1690s; on the tomb from 1705 | Jacob Bernoulli | Latin | the marvellous spiral / though changed, I rise again the same | The logarithmic (equiangular) spiral, which reproduces itself under scaling - the property the motto celebrates and the tomb's carving fails to show | Source 328, O'Connor; Source 329, Ashworth |
| calculus integralis / integral calculus | 1696 | Johann Bernoulli proposed it; Leibniz adopted it in 1696 in place of his own calculus summatorius | Latin | calculus of the whole | The branch dealing with integration | Source 324, Miller |
| ghosts of departed quantities | 1734 | Berkeley, The Analyst S.35 | English | - | Berkeley's name for evanescent increments: quantities that are neither finite, nor infinitely small, nor zero | Source 310, Berkeley |
| vis viva | 1740s debate | Leibniz's term, defended by du Chatelet against the Newtonian mv | Latin | living force | mv^2, twice what we now call kinetic energy; the Newtonians measured force by mv, which is momentum. Both sides were right about different quantities | Source 332, Project Vox |
| calculus of variations | 1744 (subject founded); the name is later | Euler, Methodus inveniendi (E65) | Latin variatio | a change, a turning | The calculus of functionals: finding the curve or function that maximises or minimises an integral | Source 323, Bogart |
| why Legendre dropped the curly d | 1787 | (Cajori s explanation) | - | Legendre abandoned his own notation the year after inventing it, and in the 1780s d, D and the rounded d were all being fought over by three different branches of mathematics at once | Cajori S 596: "It looked indeed as if the different mathematical architects ... found themselves confronted with the curse of having their sign language confounded" | Source 451, Cajori |
| primitive function (fonction primitive) | 1797 | Lagrange, Theorie des fonctions analytiques art. 17 | French/Latin primitivus | first, original | The function from which a derivative is derived; hence 'primitive' for an antiderivative | Source 317, Lagrange |
| fonction prime / seconde / tierce | 1797 | Joseph-Louis Lagrange, Theorie des fonctions analytiques p. 14 | French | "prime function", "second function", "third function" | Lagranges own names for f prime, f double prime, f triple prime - the origin of saying "f prime" out loud | Source 451, Cajori; Source 317, Lagrange |
| tanquam ex ungue leonem | 1831 in this Latin form | David Brewster, Life of Sir Isaac Newton p. 194, attributing it to Johann Bernoulli | Latin proverb | as by the claw, the lion | Used for recognising a master by a small sample of work. Bernoulli's 1697 print version appears to have been French with the Latin tag embedded ('comme ex ungue Leonem') | Source 321, Brewster; Source 337, Bernoulli |
| Jacobis claim to the partial-derivative sign | 1841 | Carl Gustav Jacob Jacobi, De determinantibus functionalibus, Crelle XXII p. 319 | Latin | Jacobi says he preferred the rounded d simply because "an accumulation of parentheses for reading and writing is rather onerous" | Jacobi made no historical search and reinvented Legendre s symbol; Cajori: "the mathematical public, uncritical on matters of priority, is crediting Jacobi with devices first proposed by Legendre and Hamilton". Adoption still took ~50 years: Cayley ignored it in 1857 | Source 451, Cajori; Source 452, Miller |
| matrix, the objection to | 1867 | Charles L. Dodgson (Lewis Carroll), An Elementary Treatise on Determinants | English | Dodgson wanted the word BLOCK instead: "surely the former word means rather the mould, or form, into which algebraical quantities may be introduced, than an actual assemblage of such quantities" | A lost terminological argument, by the author of Alice in Wonderland | Source 453, Miller and Aldrich |
| why Hardy insisted on the arrow | 1908 | G. H. Hardy, A Course of Pure Mathematics, Preface | English | "to write n = infinity, x = infinity (as if anything ever were equal to infinity) ... is in the early stages of mathematical training to go out of one s way to encourage incoherence and confusion of thought" | The pedagogical argument that replaced x = infinity with x -> infinity. It is a teaching argument, made by a textbook writer, and it won | Source 451, Cajori |
| scribal publication | 1993 | Harold Love (the historian's category, applied to Newton by Guicciardini) | English | publishing by handwritten copy | Circulating a text in manuscript copies instead of print, in a culture with established ways of generating, transmitting and selling such copies; its point is that the power in a text depends on possession of it being denied to others | Source 204, Guicciardini |
| geometric backlash | 1997 | Helena Pycior (quoted by Guicciardini) | English | a swing back toward geometry | The reaction in English seventeenth-century mathematics against reliance on algebraic symbolism, whose main voices were Barrow and Hobbes and whose influence shows in Newton's turn to synthetic methods after 1671 | Source 204, Guicciardini |
| bhuta-sankhya | - | Indian mathematical tradition | Sanskrit | object-numbers | A system encoding digits as words (eyes = 2, Vedas = 4, lunar mansions = 27), so numbers could be written as memorable verse; Madhava's 11-decimal pi is recorded this way | Source 50, Ramasubramanian |
| anni mirabiles / annus mirabilis | 20th century (historians) | not Newton | Latin | wonderful years / wonderful year | The historians' label for Newton's 1665-1667. NOT Newton's own phrase. Newton's own words were 'in those days I was in the prime of my age for invention' (CUL Add. 3968.41, f. 85r) | Source 190, Whiteside |
| calculos subducere | classical Latin | Cicero, De finibus 2.19.60 | Latin | "to draw the pebbles up/away" | "To do the sums, to cast up accounts." The Romans already had the metaphor; Latin also has ad calculos vocare, "to call to the pebbles" = to hold someone to strict account (Cicero, Laelius 16.58) | Source 455, Lewis and Short |