Last revision: June 23, 2017
See here for a list of calculus and analysis entrries on the Words pages.
Derivative. The symbols dx, dy, and dx/dy were introduced by Gottfried Wilhelm Leibniz (1646-1716) in a manuscript of November 11, 1675 (Cajori vol. 2, page 204).
f'(x) for the first derivative, f''(x) for the second derivative, etc., were introduced by Joseph Louis Lagrange (1736-1813). In 1797 in Théorie des fonctions analytiques the symbols f'x and f''x are found; in the Oeuvres, Vol. X, "which purports to be a reprint of the 1806 edition, on p. 15, 17, one finds the corresponding parts given as f(x), f'(x), f''(x), f'''(x)" (Cajori vol. 2, page 207).
In 1770 Joseph Louis Lagrange (1736-1813) wrote for in his memoir Nouvelle méthode pour résoudre les équations littérales par le moyen des séries (Oeuvres, Vol. III, pp. 5-76). The notation also occurs in a memoir by François Daviet de Foncenex in 1759 believed actually to have been written by Lagrange (Cajori 1919, page 256).
In 1772 Lagrange wrote u' = du/dx and du = u'dx in "Sur une nouvelle espèce de calcul relatif à la différentiation et à l'integration des quantités variables," Nouveaux Memoires de l'Academie royale des Sciences et Belles-Lettres de Berlin (Oeuvres, Vol. III, pp. 451-478).
Dx y was introduced by Louis François Antoine Arbogast (1759-1803) in "De Calcul des dérivations et ses usages dans la théorie des suites et dans le calcul différentiel," Strasbourg, xxii, pp. 404, Impr. de Levrault, fréres, an VIII (1800). (This information comes from Julio González Cabillón; Cajori indicates in his History of Mathematics that Arbogast introduced this symbol, but it seems he does not show this symbol in A History of Mathematical Notations.)
D was used by Arbogast in the same work, although this symbol had previously been used by Johann Bernoulli (Cajori vol. 2, page 209). Bernoulli used the symbol in a non-operational sense (Maor, page 97).
Partial derivative. The "curly d" was used in 1770 by Antoine-Nicolas Caritat, Marquis de Condorcet (1743-1794) in "Memoire sur les Equations aux différence partielles," which was published in Histoire de L'Academie Royale des Sciences, pp. 151-178, Annee M. DCCLXXIII (1773). On page 152, Condorcet says:
Dans toute la suite de ce Memoire, dz & z désigneront ou deux differences partielles de z,, dont une par rapport a x, l'autre par rapport a y, ou bien dz sera une différentielle totale, & z une difference partielle. [Throughout this paper, both dz & z will either denote two partial differences of z, where one of them is with respect to x, and the other, with respect to y, or dz and z will be employed as symbols of total differential, and of partial difference, respectively.]However, the "curly d" was first used in the form by Adrien Marie Legendre in 1786 in his "Memoire sur la manière de distinguer les maxima des minima dans le Calcul des Variations," Histoire de l'Academie Royale des Sciences, Annee M. DCCLXXXVI (1786), pp. 7-37, Paris, M. DCCXXXVIII (1788). On page 8, it reads:
Pour éviter toute ambiguité, je répresentarie par le coefficient de x dans la différence de u, & par la différence complète de u divisée par dx.Legendre abandoned the symbol and it was re-introduced by Carl Gustav Jacob Jacobi in 1841. Jacobi used it extensively in his remarkable paper "De determinantibus Functionalibus" Crelle’s Journal, Band 22, pp. 319-352, 1841 ( pp. 393-438 of vol. 1 of the Collected Works).
Sed quia uncorum accumulatio et legenti et scribenti molestior fieri solet, praetuli characteristica d differentialia vulgaria, differentialia autem partialia characteristica denotare.The "curly d" symbol is sometimes called the "rounded d" or "curved d" or Jacobi’s delta. It corresponds to the cursive "dey" (equivalent to our d) in the Cyrillic alphabet.
Integral. Before introducing the integral symbol, Leibniz wrote omn. for “omnia” in front of the term to be integrated.
The integral symbol was first used by Gottfried Wilhelm Leibniz (1646-1716) on October 29, 1675, in an unpublished manuscript, Analyseos tetragonisticae pars secunda:
Utile erit scribi pro omnia, ut l = omn. l, id est summa ipsorum l. [It will be useful to write for omn. so that l = omn. l, or the sum of all the l’s.]
Two weeks later, on Nov. 11, in Methodi tangentium inversae exempla, he first placed dx after the integral symbol, replacing
Both manuscripts were first published by Gerhardt. There is now a critical edition (Gottfried Wilhelm Leibniz, Sämtliche Schriften und Briefe, Reihe VII: Mathematische Schriften, vol 5: Infinitesimalmathematik 1674-1676, Berlin : Akademie Verlag, 2008, pp 288-295 and 321-331), online:
The first appearance of the integral symbol in print was in a paper by Leibniz in the Acta Eruditorum. The integral symbol was actually a long letter S for “summa.”
In his Quadratura curvarum of 1704, Newton wrote a small vertical bar above x to indicate the integral of x. He wrote two side-by-side vertical bars over x to indicate the integral of (x with a single bar over it). Another notation he used was to enclose the term in a rectangle to indicate its integral. Cajori writes that Newton’s symbolism for integration was defective because the x with a bar could be misinterpreted as x-prime and the placement of a rectangle about the term was difficult for the printer, and that therefore Newton’s symbolism was never popular, even in England. [Siegmund Probst contributed to this entry.]
Limits of integration. Limits of integration were first indicated only in words. Euler was the first to use a symbol in Institutiones calculi integralis, where he wrote the limits in brackets and used the Latin words ab and ad (Cajori vol. 2, page 249).
The modern definite integral symbol was originated by Jean Baptiste Joseph Fourier (1768-1830). In 1822 in his famous The Analytical Theory of Heat he wrote:
Nous désignons en général par le signe l'intégrale qui commence lorsque la variable équivaut à a, et qui est complète lorsque la variable équivaut à b. . .The citation above is from "Théorie analytique de la chaleur" [The Analytical Theory of Heat], Firmin Didot, Paris, 1822, page 252 (paragraph 231).
Fourier had used this notation somewhat earlier in the Mémoires of the French Academy for 1819-20, in an article of which the early part of his book of 1822 is a reprint (Cajori vol. 2 page 250).
The bar notation to indicate evaluation of an antiderivative at the two limits of integration was first used by Pierre Frederic Sarrus (1798-1861) in 1823 in Gergonne’s Annales, Vol. XIV. The notation was used later by Moigno and Cauchy (Cajori vol. 2, page 250).
Integration around a closed path. Dan Ruttle, a reader of this page, has found a use of the integral symbol with a circle in the middle by Arnold Sommerfeld (1868-1951) in 1917 in Annalen der Physik, "Die Drudesche Dispersionstheorie vom Standpunkte des Bohrschen Modelles und die Konstitution von H2, O2 und N2." This use is earlier than the 1923 use shown by Cajori. Ruttle reports that J. W. Gibbs used only the standard integral sign in his Elements of Vector Analysis (1881-1884), and that and E. B. Wilson used a small circle below the standard integral symbol to denote integration around a closed curve in his Vector Analysis (1901, 1909) and in Advanced Calculus (1911, 1912).
Limit. lim. (with a period) was used first by Simon-Antoine-Jean L'Huilier (1750-1840). In 1786, L'Huilier gained much popularity by winning the prize offered by *l'Academie royale des Sciences et Belles-Lettres de Berlin*. His essay, "Exposition élémentaire des principes des calculs superieurs," accepted the challenge thrown by the Academy -- a clear and precise theory on the nature of infinity. On page 31 of this remarkable paper, L'Huilier states:
Pour abreger & pour faciliter le calcul par une notation plus commode, on est convenu de désigner autrement que par , la limite du rapport des changements simultanes de P & de x, favoir par ; en sorte que ou ; designent la même chose
lim (without a period) was written in 1841 by Karl Weierstrass (1815-1897) in one of his papers published in 1894 in Mathematische Werke, Band I, page 60 (Cajori vol. 2, page 255).
The arrow notation for limits. In the 1850s, Weierstrass began to use .
Our present day expression seems to have originated with the English mathematician John Gaston Leathem in his 1905 book Volume and Surface Integrals Used in Physics.
Leathem wrote the following in his undated Preface (p. v, paragraph 3) to Elements of the Mathematical Theory of Limits, G. Bell and Sons, 1925:
The arrow symbol for tendency to limit was introduced in my tract on Surface and Volume Integrals published by the Cambridge University Press in 1905. It has been erroneously attributed to another writer owing to its use, with inadvertent omission of acknowledgment, in an important book published three years later. It is now a well-established notation, and I have thought it desirable to supplement it in the present work by using sloped arrows to distinguish upward and downward tendencies.
The widespread use of the arrow notation can probably be attributed to its appearance in two books in 1908: An Introduction to the Theory of Infinite Series (1st ed.) by Thomas John I'Anson Bromwich and A Course of Pure Mathematics by Godfrey Harold Hardy. [See pp. 116-118 and 163-164 in Hardy.]
The following comments appear in the Preface of Hardy’s 1908 book (p. ix):
[This entry was contributed by Dave L. Renfro.]
Infinity. The infinity symbol was introduced by John Wallis (1616-1703) in 1655 in his De sectionibus conicis (On Conic Sections) as follows:
Suppono in limine (juxta^ Bonaventurae Cavallerii Geometriam Indivisibilium) Planum quodlibet quasi ex infinitis lineis parallelis conflari: Vel potiu\s (quod ego mallem) ex infinitis Prallelogrammis [sic] aeque\ altis; quorum quidem singulorum altitudo sit totius altitudinis 1/, sive alicuota pars infinite parva; (esto enim nota numeri infiniti;) adeo/q; omnium simul altitude aequalis altitudini figurae.Wallis also used the infinity symbol in various passages of his Arithmetica infinitorum (Arithmetic of Infinites) (1655 or 1656). For instance, he wrote (p. 70):
Cum enim primus terminus in serie Primanorum sit 0, primus terminus in serie reciproca erit vel infinitus: (sicut, in divisione, si diviso sit 0, quotiens erit infinitus)In Zero to Lazy Eight, Alexander Humez, Nicholas Humez, and Joseph Maguire write: "Wallis was a classical scholar and it is possible that he derived from the old Roman sign for 1,000, CD, also written M--though it is also possible that he got the idea from the lowercase omega, omega being the last letter of the Greek alphabet and thus a metaphor of long standing for the upper limit, the end."
Cajori (vol. 2, p 44) says the conjecture has been made that Wallis adopted this symbol from the late Roman symbol for 1,000. He attributes the conjecture to Wilhelm Wattenbach (1819-1897), Anleitung zur lateinischen Paläographie 2. Aufl., Leipzig: S. Hirzel, 1872. Appendix: p. 41.
This conjecture is lent credence by the labels inscribed on a Roman hand abacus stored at the Bibliothèque Nationale in Paris. A plaster cast of this abacus is shown in a photo on page 305 of the English translation of Karl Menninger’s Number Words and Number Symbols; at the time, the cast was held in the Cabinet des Médailles in Paris. The photo reveals that the column devoted to 1000 on this abacus is inscribed with a symbol quite close in shape to the lemniscate symbol, and which Menninger shows would easily have evolved into the symbol M, the eventual Roman symbol for 1000 [Randy K. Schwartz].
[Julio González Cabillón contributed to this entry.]
Delta to indicate a small quantity. In 1706, Johann Bernoulli used δ to denote the difference of functions. Julio González Cabillón believes this is probably one of the first if not the first use of delta in this sense.
Delta and epsilon. Augustin-Louis Cauchy (1789-1857) used ε in 1821 in Cours d'analyse (Oeuvres II.3), and sometimes used δ instead (Cajori vol. 2, page 256). According to Finney and Thomas (page 113), "δ meant 'différence' (French for difference and ε meant 'erreur' (French for error)."
The first theorem on limits that Cauchy sets out to prove in the Cours d'Analyse (Oeuvres II.3, p. 54) has as hypothesis that
The proof then begins by saying
denote by ε a number as small as one may wish. Since the increasing values of x make the difference f(x + 1) - f(x) converge to the limit k, one can assign a sufficiently substantial value to a number h so that, for x bigger than or equal to h, the difference in question is always between the bounds k - ε, k + ε.[William C. Waterhouse]
The first delta-epsilon proof is Cauchy’s proof of what is essentially the mean-value theorem for derivatives. It comes from his lectures on the Calcul infinitesimal, 1823, Leçon 7, in Oeuvres, Ser. 2, vol. 4, pp. 44-45. The proof translates Cauchy’s verbal definition of the derivative as the limit (when it exists) of the quotient of the differences into the language of algebraic inequalities using both delta and epsilon. In the 1820s Cauchy did not specify on what, given an epsilon, his delta or n depended, so one can read his proofs as holding for all values of the variable. Thus he does not make the distinction between converging to a limit pointwise and convering to it uniformly.
[Judith V. Grabiner, author of The Origins of Cauchy’s Rigorous Calculus (MIT, 1981)]
For vector analysis entries on the Words pages, see here for a list.
The vector differential operator, now written and called nabla or del, was introduced by William Rowan Hamilton (1805-1865). Hamilton wrote the operator as and it was P. G. Tait who established as the conventional symbol--see his An Elementary Treatise on Quaternions (1867). Tait was also responsible for establishing the term nabla. See NABLA on the Earliest Uses of Words page.
David Wilkins suggests that Hamilton may have used the nabla as a general purpose symbol or abbreviation for whatever operator he wanted to introduce at any time. In 1837 Hamilton used the nabla, in its modern orientation, as a symbol for any arbitrary function in Trans. R. Irish Acad. XVII. 236. (OED.) He used the nabla to signify a permutation operator in "On the Argument of Abel, respecting the Impossibility of expressing a Root of any General Equation above the Fourth Degree, by any finite Combination of Radicals and Rational Functions," Transactions of the Royal Irish Academy, 18 (1839), pp. 171-259.
Hamilton used the rotated nabla, i.e. , for the vector differential operator in the "Proceedings of the Royal Irish Academy" for the meeting of July 20, 1846. This paper appeared in volume 3 (1847), pp. 273-292. Hamilton also used the rotated nabla as the vector differential operator in "On Quaternions; or on a new System of Imaginaries in Algebra"; which he published in instalments in the Philosophical Magazine between 1844 and 1850. The relevant portion of the paper consists of articles 49-50, in the instalment which appeared in October 1847 in volume 31 (3rd series, 1847) of the Philosophical Magazine, pp. 278-283. A footnote in vol. 31, page 291, reads:
In that paper designed for Southampton the characteristic was written ; but this more common sign has been so often used with other meanings, that it seems desirable to abstain from appropriating it to the new signification here proposed.Wilkins writes that "that paper" refers to an unpublished paper that Hamilton had prepared for a meeting of the British Association for the Advancement of Science, but which had been forwarded by mistake to Sir John Herschel’s home address, not to the meeting itself in Southampton, and which therefore was not communicated at that meeting. The footnote indicates that Hamilton had originally intended to use the nabla symbol that is used today but then decided to rotate it to avoid confusion with other uses of the symbol.
The rotated form appears in Hamilton’s magnum opus, the Lectures on Quaternions (1853, p. 610).
Cajori (vol. 2, page 135) and the OED give this reference.
According to Stein and Barcellos (page 836), Hamilton denoted the gradient with an ordinary capital delta in 1846. However, this information may be incorrect, as David Wilkins writes that he has never seen the gradient denoted by an ordinary capital delta in any paper of Hamilton published in his lifetime.
David Wilkins of the School of Mathematics at Trinity College in Dublin has made available texts of the mathematical papers published by Hamilton in his lifetime at his History of Mathematics website.
grad as a symbol for gradient appears in H. Weber’s Die partiellen differential-gleichungen der mathematischen physik nach Riemanns Vorlesungen of 1900 (Cajori vol. 2, page 135). See GRADIENT on the Earliest Uses of Words page.
William Kingdon Clifford (1845-1879) used div u or dv u as symbols for divergence (Cajori vol. 2, page 135).
The symbol Δ for the Laplacian operator (also represented by 2) was introduced by Robert Murphy in 1833 in Elementary Principles of the Theories of Electricity. (Kline, page 786). See LAPLACE’s OPERATOR on the Earliest Uses of Words page.