*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)*".

In 1770 Joseph Louis Lagrange (1736-1813) wrote
for
in his memoir *Nouvelle méthode pour résoudre les équations
littérales par le moyen des séries.* The notation also occurs
in a memoir by François Daviet de Foncenex in 1759 believed actually to
have been written by Lagrange.

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.*

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. Bernoulli
used the symbol in a non-operational sense.

**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,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,"dz&zdésigneront ou deux differences partielles dez,, dont une par rapport ax,l'autre par rapport ay,ou biendzsera une différentielle totale, &zune difference partielle. [Throughout this paper, bothdz&zwill either denote two partial differences ofz,where one of them is with respect tox,and the other, with respect toy,ordzandzwill be employed as symbols of total differential, and of partial difference, respectively.]

Pour éviter toute ambiguité, je répresentarie par le coefficient deLegendre 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" [appeared in Crelle's Journal, Band 22, pp. 319-352, 1841].xdans la différence deu,& par la différence complète deudivisée pardx.

Sed quia uncorum accumulatio et legenti et scribenti molestior fieri solet, praetuli characteristicaThe "curly d" symbol is sometimes called the "rounded d" or "curved d" or

ddifferentialia vulgaria, differentialia autem partialia characteristica

denotare.

**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. Several weeks later, on Nov.
21, he first placed *dx* after the integral symbol.
Later in 1675, he proposed the use of the symbol in a letter to Henry Oldenburg,
secretary of the Royal Society: "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.] 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.

**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*.

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 àThe citation above is from "Théorie analytique de la chaleur" [The Analytical Theory of Heat], Firmin Didot, Paris, 1822, page 226 (paragraph 231.a,et qui est complète lorsque la variable équivaut àb. . .

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.

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.

**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 parlim (without a period) was written in 1841 Karl Weierstrass (1815-1897) in one of his papers published in 1894 in,

la limite du rapport des changements simultanes de

P& dex,favoir par;

en sorte que

ou

;

designent la même chose

The arrow notation for limits was introduced by Godfrey Harold Hardy (1877-1947) in his remarkable "A Course of Pure Mathematics," Cambridge: At the University Press, xv, pp. 428, 1908. Check the preface of this first edition.

**Delta and epsilon.** Augustin-Louis Cauchy (1789-1857) used epsilon in
1821 in *Cours d'analyse,* and sometimes used delta instead. According to Finney and Thomas, "[delta] meant
"différence" (French for *difference* and [epsilon] meant
"erreur" (French for *error*).

The first theorem on limits that Cauchy sets out to prove in the *Cours
d'Analyse* has as hypothesis that

for increasing values of x, the difference f(x+1) - f(x) converges to a certain limit k.The proof then begins by saying

denote by [epsilon] 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 - [epsilon], k + [epsilon].[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)]

**Nabla.** The vector differential operator
(also called *del* or *atled*) was introduced by William Rowan
Hamilton (1805-1865).

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. This
information is taken from the OED2 entry on *nabla.*

Hamilton 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 nabla, rotated 90 degrees, 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.

According to Stein and Barcellos, 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.

Hamilton also used the nabla as the vector differential operator, rotated 90
degrees, in "On Quaternions; or on a new System of Imaginaries in
Algebra"; which he published in installments in the *Philosophical
Magazine* between 1844 and 1850. The relevant portion of the paper consists
of articles 49-50, in the installment 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 through 90 degrees to avoid confusion with other uses of the symbol.

Cajori writes that Hamilton introduced the operator, and a
footnote references *Lectures on Quaternions* (1853), page 610. The OED2
indicates that the nabla appears, rotated 90 degrees in *Lect. Quaternions*
vii. 610.

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.

**Gradient.** Maxwell and Riemann-Weber used grad as an abbreviation or
symbol for *gradient* .

**Divergence.** William Kingdon Clifford (1845-1879) used the term
divergence and wrote *div u* or *dv u*.

**Laplacian operator.** The capital delta for ^{2} was introduced by
Robert Murphy in 1883.

**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 CavalleriiWallis also used the infinity symbol in various passages of hisGeometriam 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.

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

Cajori 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.]