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Seehere for a listof calculus and analysis entrries on theWords pages.
Derivative. The symbolsdx, dy, anddx/dy were introduced byGottfried 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 byJoseph Louis Lagrange (1736-1813). In 1797 inThéorie des fonctions analytiquesthe symbolsf'x andf''x are found; in theOeuvres, Vol. X,"which purports to be a reprint of the 1806 edition, on p. 15, 17, one finds the corresponding partsgiven asf(x), f'(x), f''(x), f'''(x)" (Cajori vol. 2, page 207).
In 1770 Joseph Louis Lagrange (1736-1813) wrote
for
in his memoirNouvelle 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 1759believed actually to have been written by Lagrange (Cajori 1919, page 256).
In 1772 Lagrange wroteu' =du/dx anddu =u'dx in "Sur une nouvelle espèce de calcul relatif àla 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 byLouis François Antoine Arbogast(1759-1803) in "De Calcul des dérivations et ses usages dans la théorie des suiteset 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 hisHistory of Mathematics that Arbogast introducedthis symbol, but it seems he does not show this symbol inA History of MathematicalNotations.)
D was used by Arbogast in the same work, although this symbolhad previously been used by Johann Bernoulli (Cajori vol. 2, page209). Bernoulli used the symbol in a non-operational sense (Maor,page 97).
Partial derivative. The "curly d" was used in 1770 byAntoine-Nicolas Caritat, Marquis de Condorcet(1743-1794) in "Memoire sur les Equations aux différence partielles,"which was published inHistoire 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 &However, the "curly d" was first used in the formz désigneront ou deuxdifferences partielles dez,, dont une par rapport ax,l'autre par rapport ay, ou biendz sera unedifférentielle totale, &
byAdrien Marie Legendrein 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, jerépresentarie parLegendre abandoned the symbol and it was re-introduced byCarl Gustav Jacob Jacobiin 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 theCollected Works).la différence complète deu divisée pardx.
Sed quia uncorum accumulatio et legenti et scribentimolestior fieri solet, praetuli characteristicaddifferentialia vulgaria, differentialia autem partialia characteristicaThe "curly d" symbol is sometimes called the "rounded d" or "curved d"orJacobi’s delta. It corresponds to the cursive "dey"(equivalent to ourd) in the Cyrillic alphabet.
Integral. Before introducing the integral symbol, Leibnizwrote 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 scribipro omnia, ut
Two weeks later, on Nov. 11, inMethodi tangentium inversae exempla, he first placeddx 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:
http://www.gwlb.de/Leibniz/Leibnizarchiv/Veroeffentlichungen/VII5A.pdf
http://www.gwlb.de/Leibniz/Leibnizarchiv/Veroeffentlichungen/VII5B.pdf
The first appearance of the integral symbol in printwas in a paper by Leibniz in theActa Eruditorum. The integralsymbol was actually a long letter S for “summa.”
In hisQuadratura curvarum of 1704, Newton wrote a smallvertical bar above x to indicate the integral of x. He wrote twoside-by-side vertical bars over x to indicate the integral of (x witha single bar over it). Another notation he used was to enclose theterm in a rectangle to indicate its integral. Cajori writes thatNewton’s symbolism for integration was defective because the x with abar could be misinterpreted as x-prime and the placement of arectangle about the term was difficult for the printer, and thattherefore Newton’s symbolism was never popular, even in England.[Siegmund Probst contributed to this entry.]
Limits of integration. Limits of integration werefirst indicated only in words. Euler was the first to use a symbolinInstitutiones calculi integralis, where he wrote thelimits in brackets and used the Latin wordsab andad(Cajori vol. 2, page 249).
The modern definite integral symbol was originated by JeanBaptiste Joseph Fourier (1768-1830). In 1822 in his famousThe AnalyticalTheory of Heat he wrote:
Nous désignons en général par le signeThe citation above is from"Théorie analytique de la chaleur"[The Analytical Theory of Heat], Firmin Didot, Paris, 1822, page 252 (paragraph 231).l'intégrale qui commence lorsque la variable équivaut àa, et qui est complète lorsquela variable équivaut àb. . .
Fourier had used this notation somewhat earlier in theMémoires of the French Academy for 1819-20, in anarticle of which the early part of his book of 1822 is a reprint(Cajori vol. 2 page 250).
Thebar notation to indicate evaluation of an antiderivativeat the two limits of integration was first used by Pierre FredericSarrus (1798-1861) in 1823 in Gergonne’sAnnales, Vol. XIV.The notation was used later by Moigno and Cauchy (Cajori vol. 2, page250).
Integration around a closed path. Dan Ruttle, a reader of this page, has found a use of the integralsymbol with a circle in the middle byArnold Sommerfeld (1868-1951)in 1917 inAnnalen derPhysik, "Die Drudesche Dispersionstheorie vom Standpunkte desBohrschen Modelles und die Konstitution von H2, O2 und N2." This useis earlier than the 1923 use shown by Cajori. Ruttle reports that J.W. Gibbs used only the standard integral sign in hisElements ofVector Analysis (1881-1884), and that and E. B. Wilson used asmall circle below the standard integral symbol to denote integrationaround a closed curve in hisVector Analysis (1901, 1909) andinAdvanced Calculus (1911, 1912).
Limit. lim. (with a period) was used first bySimon-Antoine-Jean L'Huilier (1750-1840).In 1786, L'Huilier gainedmuch popularity by winning the prize offered by *l'Academie royaledes Sciences et Belles-Lettres de Berlin*. His essay, "Expositionélémentaire des principes des calculs superieurs,"accepted the challenge thrown by the Academy -- a clear and precisetheory on the nature of infinity. On page 31 of this remarkable paper,L'Huilier states:
Pour abreger & pour faciliter le calcul par une notationplus commode, on est convenu de désigner autrement que par,la limite du rapport des changements simultanes deP & dex, favoir par
;en sorte que
lim (without a period) was written in 1841 byKarl Weierstrass(1815-1897) in one of his papers published in 1894 inMathematische 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 haveoriginated with the English mathematician John Gaston Leathem in his 1905 bookVolume and Surface Integrals Used in Physics.
Leathem wrote the following in hisundated Preface (p. v, paragraph 3) toElementsof the Mathematical Theory of Limits, G. Belland Sons, 1925:
The arrow symbol for tendencyto limit was introduced in my tract onSurfaceand Volume Integrals published by the CambridgeUniversity Press in 1905. It has been erroneouslyattributed to another writer owing to its use,with inadvertent omission of acknowledgment, inan important book published three years later.It is now a well-established notation, and I havethought it desirable to supplement it in thepresent work by using sloped arrows to distinguishupward 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 andA 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 byJohn Wallis (1616-1703)in 1655 in hisDe sectionibus conicis (On Conic Sections) as follows:
Suppono in limine (juxta^ Bonaventurae CavalleriiGeometriam Indivisibilium) Planum quodlibet quasi ex infinitislineis parallelis conflari: Vel potiu\s (quod ego mallem) exinfinitis Prallelogrammis [sic] aeque\ altis; quorum quidemsingulorum altitudo sit totius altitudinis 1/Wallis also used the infinity symbol in various passages of hisArithmetica infinitorum (Arithmetic of Infinites) (1655 or1656). For instance, he wrote (p. 70):
Cum enim primus terminus in serie Primanorum sit 0,primus terminus in serie reciproca eritInZero to Lazy Eight, Alexander Humez, Nicholas Humez, andJoseph Maguire write: "Wallis was a classical scholar and it ispossible 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 fromthe lowercase omega, omega being the last letter of the Greek alphabetand thus a metaphor of long standing for the upper limit, theend."Cajori (vol. 2, p 44) says the conjecture has been made that Wallisadopted this symbol from the late Roman symbol for 1,000. He attributes theconjecture 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 Romanhand abacus stored at the Bibliothèque Nationale in Paris. Aplaster cast of this abacus is shown in a photo on page 305 of theEnglish translation of Karl Menninger’sNumber Words and NumberSymbols; at the time, the cast was held in the Cabinet desMédailles in Paris. The photo reveals that the column devotedto 1000 on this abacus is inscribed with a symbol quite close inshape to the lemniscate symbol, and which Menninger shows wouldeasily have evolved into the symbol M, the eventual Roman symbol for1000 [Randy K. Schwartz].
[Julio González Cabillón contributed to this entry.]
Delta to indicate a small quantity. In 1706,Johann Bernoulliused δ to denote the difference of functions. Julio González Cabillón believes thisis 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 inCours d'analyse(Oeuvres II.3),and sometimes used δ instead (Cajori vol. 2, page 256). According to Finney and Thomas (page113), "δ meant 'différence' (French fordifferenceand ε meant 'erreur' (French forerror)."
The first theorem on limits that Cauchy sets out to prove in theCours 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 ofx make the differencef(x + 1) -f(x)converge to the limitk, one can assign a sufficiently substantialvalue to a numberh so that, forx bigger than or equal toh,the difference in question is always between the boundsk -ε,k + ε.[William C. Waterhouse]
The first delta-epsilon proof is Cauchy’s proof of what isessentially the mean-value theorem for derivatives. It comes from his lectureson the Calcul infinitesimal, 1823, Leçon 7, inOeuvres, Ser. 2, vol. 4, pp. 44-45.The proof translates Cauchy’s verbal definition of the derivative as the limit (when it exists) ofthe quotient of the differences into the language of algebraicinequalities using both delta and epsilon. In the 1820s Cauchy didnot specify on what, given an epsilon, his delta or n depended, soone can read his proofs as holding for all values of the variable.Thus he does not make the distinction between converging to a limitpointwise and convering to it uniformly.
[Judith V. Grabiner, author ofThe Origins of Cauchy’s Rigorous Calculus (MIT, 1981)]
For vector analysis entries on theWords pages,seehere for a list.
The vector differential operator, now written
and callednabla ordel, was introduced by William Rowan Hamilton(1805-1865). Hamilton wrote the operator as
and it was P. G. Tait who establishedas the conventional symbol--see hisAn Elementary Treatise on Quaternions(1867). Tait was also responsible for establishing the termnabla. SeeNABLA on theEarliest Uses of Words page.
David Wilkins suggests that Hamilton may have used the nablaas a general purpose symbol or abbreviation for whatever operator he wantedto introduce at any time. In 1837 Hamilton used the nabla, in its modern orientation,as a symbol for any arbitrary function inTrans. R. Irish Acad. XVII.236. (OED.) He used the nabla to signify a permutation operator in "Onthe Argument of Abel, respecting the Impossibility of expressing a Root of anyGeneral Equation above the Fourth Degree, by any finite Combination of Radicalsand 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 meetingof July 20, 1846. This paper appeared in volume 3 (1847), pp. 273-292. Hamiltonalso used the rotated nabla as the vector differential operator in "OnQuaternions; or on a new System of Imaginaries in Algebra"; which he publishedin instalments in thePhilosophical Magazine between 1844 and 1850. Therelevant portion of the paper consists of articles 49-50, in the instalmentwhich appeared in October 1847 in volume 31 (3rd series, 1847) of the PhilosophicalMagazine, pp. 278-283. A footnote in vol. 31, page 291, reads:
In that paper designed for Southampton the characteristic was writtenWilkins writes that "that paper"refers to an unpublished paper that Hamilton had prepared for a meeting of theBritish Association for the Advancement of Science, but which had been forwardedby mistake to Sir John Herschel’s home address, not to the meeting itself inSouthampton, and which therefore was not communicated at that meeting. The footnoteindicates that Hamilton had originally intended to use the nabla symbol thatis used today but then decided to rotate it to avoid confusion with other usesof the symbol.
The rotated form appears in Hamilton’s magnum opus, theLectures on Quaternions (1853, p. 610).
Cajori (vol. 2, page 135) and the OED give this reference.
According to Stein and Barcellos (page 836), Hamilton denotedthe gradient with an ordinary capital delta in 1846. However, this informationmay be incorrect, as David Wilkins writes that he has never seen the gradientdenoted by an ordinary capital delta in any paper of Hamilton published in hislifetime.
David Wilkins of the School of Mathematics at Trinity College in Dublin has madeavailable texts of the mathematical papers published by Hamilton in his lifetime at hisHistory of Mathematics website.
grad as a symbol for gradient appears in H. Weber’sDie partiellen differential-gleichungender mathematischen physik nach Riemanns Vorlesungen of 1900 (Cajori vol. 2, page 135). SeeGRADIENT on theEarliest Uses of Words page.
William Kingdon Clifford (1845-1879) useddiv u ordv u as symbols for divergence (Cajori vol. 2, page135).
The symbol Δ for the Laplacian operator (also represented by2)was introduced byRobert Murphyin 1833 inElementary Principles of the Theories of Electricity. (Kline, page 786).SeeLAPLACE’s OPERATOR on theEarliest Uses of Words page.