Earliest uses of symbols of operation

Earliest uses of symbols of operation

by Jeff Miller

Logarithm function. Log. (with a period, capital "L") was used by Johannes Kepler (1571-1630) in 1624 in Chilias logarithmorum (Cajori vol. 2, page 105)

log. (with a period, lower case "l") was used by Bonaventura Cavalieri (1598-1647) in Directorium generale Vranometricum in 1632 (Cajori vol. 2, page 106).

log (without a period, lower case "l") appears in the 1647 edition of Clavis mathematicae by William Oughtred (1574-1660) (Cajori vol. 1, page 193).

Kline (page 378) says Leibniz introduced the notation log x (showing no period), but he does not give a source.

log a was introduced by Edmund Gunter (1581-1626) according to an Internet source. [I do not see a reference for this in Cajori.]

ln (for natural logarithm) was used in 1893 by Irving Stringham (1847-1909) in Uniplanar Algebra (Cajori vol. 2, page 107).

William Oughtred (1574-1660) used a minus sign over the characteristic of a logarithm in the Clavis Mathematicae (Key to Mathematics), "except in the 1631 edition which does not consider logarithms" (Cajori vol. 2, page 110). The Clavis Mathematicae was composed around 1628 and published in 1631 (Smith 1958, page 393). Cajori shows a use from the 1652 edition.

Greatest integer function. Although [x] is commonly used for this function, the notation an x inside
brackets from which the top bars have been removed was introduced by Kenneth E. Iverson in 1962, according to the website of the University of Tennessee at Martin.

The function is also called the floor function.

According to Grinstein (1970), "The use of the bracket notation, which has led some authors to term this the bracket function, stems back to the work of Gauss (1808) in number theory. The function is also referred to by Legendre who used the now obsolete notation E(x)."

Use of arrows. Saunders Mc Lane, in Categories for the working mathematician (Springer-Verlag, 1971, p. 29), says: "The fundamental idea of representing a function by an arrow first appeared in topology about 1940, probably in papers or lectures by W. Hurewicz on relative homotopy groups. (Hurewicz, W.: "On duality theorems," Bull. Am. Math. Soc. 47, 562-563) His initiative immediately attracted the attention of R. H. Fox and N. E. Steenrod, whose ... paper used arrows and (implicitly) functors... The arrow f: : X (arrow) Y rapidly displaced the occasional notation f(X) (subset of ) Y for a function. It expressed well a central interest of topology. Thus a notation (the arrow) led to a concept (category)". [Arturo Mena]

Sign, or signum, function. The symbol [a], to represent 0, 1, or -1, according to whether a is 0, positive, or negative, was introduced by Leopold Kronecker (1823-1891). He wrote:

Bezeichnet man naemlich mit [a] den Werth Null oder +1 oder -1, je nachdem die reelle Groesse a selbst gleich Null oder positiv oder negativ ist ... [February 14, 1878]

This citation was provided by Julio González Cabillón


Lettering of geometric figures. The designation of points, lines, and planes by a letter or letters was in vogue among the ancient Greeks and has been traced back to Hippocrates of Chios (about 440 B. C.) (Cajori vol. 1, page 420, attributed to Moritz Cantor).

Lettering of triangles. Richard Rawlinson in a pamphlet prepared at Oxford sometime between 1655 and 1668 used A, B, C for the sides of a triangle and a, b, c for the opposite angles. In his notation, A was the largest side and C the smallest (Cajori vol. 2, page 162).

Leonhard Euler and Thomas Simpson reintroduced this scheme many years later, Euler using it in 1753 in Histoire de l'académie de Berlin (Cajori vol 2., page 162). Euler used capital letters for the angles.

In 1866, Karl Theodor Reye (1838-1919) proposed the plan of using capital letters for points, lower case letters for lines, and lower case Greek letters for planes in a remarkable two-volume work on geometry, Die Geometrie der Lage (Cajori vol. 1, page 423).

As early as 1618, an anonymous writer of the "Appendix" in the 1618 edition of Edward Wright's translation of John Napier's "Mirifici logarithmorum canonis descriptio" labeled the right angle of a triangle with the letter A:

It will bee conuenient in euery calculation, to haue in your view a triangle, described according to the present occasion: and if it bee a right angled triangle, to note it with Letters A.B.C: so that A may bee alwayes the right angle; B the angle at the Base B.A and C the angle at the Cathetus CA [sic].

Cf. page 3 of "An Appendix to the Logarithmes of the Calculation of Triangles, and also a new and ready way for the exact finding out of such lines and logarithmes as are not precisely to be found in the Canons", in John Napier's "A description of the Admirable Table of Logarithmes ...", London, Printed for Simon Waterson, 1618

James W. L. Glaisher (1848-1928) has remarked that the letter A is taken to be the right angle in the right-angled triangle ABC in order that BA may represent the BAse, and CA the CAthetus, the first two initials indicating the words. The fact that this lettering was also employed by William Oughtred (1574-1660) in his books is one of the many arguments in support that Oughtred might be the author of the "Appendix" (Cajori vol. 2, p. 154).

Angle. Pierre Hérigone (1580-1643) used both the current angle symbol and < in Cursus mathematicus. This work was published in 1634 and in a second edition in 1644. Cajori lists the symbols from the 1644 edition, which shows both angle symbols (Cajori vol. 1, page 202).

Arc. The arc symbol appears in the middle of the twelfth century in Plato of Tivoli's translation of the Liber embadorum by Savasorda (Cajori vol. 1, page 402).

Circle. Heron used a modified circle with a dot in the center to represent a circle around A. D. 150 (Cajori vol. 1, page 401).

Pappus used a circle with and without a dot in the center to represent a circle in the fourth century A. D. (Cajori vol. 1, page 401).

Triangle. Heron about A. D. 150 used a triangle as a symbol for triangle (Cajori).

Congruence. Gottfried Wilhelm Leibniz (1646-1716) introduced a symbol using one horizontal bar under the
tilde for congruence in an unpublished manuscript of 1679 (Cajori vol. 1, page 414).

The first appearance in print of Leibniz' sign for congruence was in 1710 in the Miscellanea Berolinensia in the anonymous article "Monitum," which is attributed to Leibniz (Cajori vol. 2, page 195).

In 1777, Johann Friedrich Häseler (1732-1797) used a congruent symbol with two horizontal bars (with the tilde reversed) in Anfangsgründe der Arith., Alg., Geom. und Trig. (Lemgo), Elementar-Geometrie (Cajori vol. 1, page 415).

In 1824 Carl Brandan Mollweide (1774-1825) used the modern congruent symbol with two horizontal bars under a
normal tilde in Euklid's Elemente (Cajori vol. 1, page 415).

Radius. Leonhard Euler introduced the use of R for the radius of the circumscribed circle and r for the radius of the inscribed circle (Boyer, page 495).

Degrees. The symbols for degrees, minutes, and seconds were used by Claudius Ptolemy (c. 85-c. 165) in the Almagest. However, the notation differed somewhat from the modern notation, and according to Cajori (vol. 2, page 143), "it is difficult to uphold" the view that our signs for degrees, minutes, and seconds are of Greek origin.

The first modern appearance of the degree symbol ° Cajori found is in the revised 1569 edition of Gemma Frisius, Arithmeticae practicae moethodus facilis by Gemma Frisius (1508-1555), although the symbol appears in the Appendix on astronomical fractions due to Jacques Peletier (1517-1582) and dated 1558. Cajori writes:

This is the first modern appearance that I have found of ° for integra or "degrees." It is explained that the denomination of the product of two such denominate numbers is obtained by combining the denominations of the factors; minutes times seconds give thirds, because 1+2=3. The denomination ° for integers or degrees is necessary to impart generality to this mode or procedure. "Integers when multiplied by seconds make seconds, when multiplied by thirds make thirds" (fol. 62, 76). It is possible that Peletier is the originator of the ° for degrees. But nowhere in this book have I been able to find the modern angular notation ° ' " used in writing angles. The ° is used only in multiplication.

Erasmus Reinhold (1511-1553) used ° ' " in Prvtenicae tabulae coelestium motuum published in 1571 (Cajori).

Line segment. A bar above AB to indicate line segment AB was used in 1647 by Bonaventura Cavalieri (1598-1647) in Geometria indivisibilibae and Exercitationes geometriae sex, according to Cajori.

Slope. The earliest known use of m for slope appears in Vincenzo Riccati’s memoir De methodo Hermanni ad locos geometricos resolvendos, which is chapter XII of the first part of his book Vincentii Riccati Opusculorum ad res Physica, & Mathematicas pertinentium (1757):

Propositio prima. Aequationes primi gradus construere. Ut Hermanni methodo utamor, danda est aequationi hujusmodi forma y = mx + n, quod semper fieri posse certum est. (p. 151)

The reference is to the Swiss mathematician Jacob Hermann (1678-1733). This use of m was found by Dr. Sandro Caparrini of the Department of Mathematics at the University of Torino.

In 1830, Traite Elementaire D'Arithmetique, Al'Usage De L'Ecole Centrale des Quatre-Nations: Par S.F. LaCroix, Dix-Huitieme Edition has y = ax + b [Karen Dee Michalowicz].

Another use of m occurs in 1842 in An Elementary Treatise on the Differential Calculus by Rev. Matthew O'Brien, from the bottom of page 1: "Thus in the general equation to a right line, namely y = mx + c, if we suppose the line..." [Dave Cohen].

O'Brien used m for slope again in 1844 in A Treatise on Plane Co-Ordinate Geometry [V. Frederick Rickey].

George Salmon (1819-1904), an Irish mathematician, used y = mx + b in his A Treatise on Conic Sections, which was published in several editions beginning in 1848. Salmon referred in several places to O'Brien's Conic Sections and it may be that he adopted O'Brien's notation. Salmon used a to denote the x-intercept, and gave the equation (x/a) + (y/b) = 1 [David Wilkins].

Karen Dee Michalowicz has found an 1848 British analytic geometry text which has y = mx + h.

The 1855 edition of Isaac Todhunter's Treatise on Plane Co-Ordinate Geometry has y = mx + c [Dave Cohen].

In 1891, Differential and Integral Calculus by George A. Osborne has y - y' = m(x - x').

In Webster's New International Dictionary (1909), the "slope form" is y = sx + b.

In 1921, in An Introduction to Mathematical Analysis by Frank Loxley Griffin, the equation is written y = lx + k.

In Analytic Geometry (1924) by Arthur M. Harding and George W. Mullins, the "slope-intercept form" is y = mx + b.

In A Brief Course in Advanced Algebra by Buchanan and others (1937), the "slope form" is y = mx + k.

According to Erland Gadde, in Swedish textbooks the equation is usually written as y = kx + m. He writes that the technical Swedish word for "slope" is "riktningskoefficient", which literally means "direction coefficient," and he supposes k comes from "koefficient."

According to Dick Klingens, in the Netherlands the equation is usually written as y = ax + b or px + q or mx + n. He writes that the Dutch word for slope is "richtingscoëfficiënt", which literally means "direction coefficient."

In Austria k is used for the slope, and d for the y-intercept.

According to Julio González Cabillón, in Uruguay the equation is usually written as y = ax + b or y = mx + n, and slope is called "pendiente," "coeficiente angular," or "parametro de direccion."

According to George Zeliger, "in Russian textbooks the equation was frequently written as y = kx + b, especially when plotting was involved. Since in Russian the slope is called 'the angle coefficient' and the word coefficient is spelled with k in the Cyrillic alphabet, usually nobody questioned the use of k. The use of b is less clear."

It is not known why the letter m was chosen for slope; the choice may have been arbitrary. John Conway has suggested m could stand for "modulus of slope." One high school algebra textbook says the reason for m is unknown, but remarks that it is interesting that the French word for "to climb" is monter. However, there is no evidence to make any such connection. Descartes, who was French, did not use m. In Mathematical Circles Revisited (1971) mathematics historian Howard W. Eves suggests "it just happened."

Parallelism. Two vertical bars, written horizontally and resembling the modern equal sign, were used by Heron about A. D. 150 and by Pappus (Cajori).

Thee parallel symbol written vertically was first used by William Oughtred (1574-1660) in Opuscula Mathematica Hactenus Inedita, which was published posthumously in 1677 (Cajori vol. 1, page 193).

John Kersey (1616-1677) also used the vertical parallel symbol. He used it after Oughtred, but in a work which was published before Oughtred. He used the symbol in Algebra, which was published in 1673. Kersey switched the lines from horizontal to vertical because of the adoption of the equal symbol (Cajori vol. 1, page 411).

Perpendicularity. was first used by Pierre Hérigone (1580-1643) in 1634 in Cursus mathematicus, which was published in five volumes from 1634 to 1637 (Cajori vol. 1, page 408). Johnson (page 149) says, "Herigone introduced so many new symbols in this six-volume work that some suggest that the introduction of these symbols, rather than an effective mathematics text, was his goal."

Right angle. was used by Pappus (Cajori vol. 1, page 401).

Semi-perimeter. A capital S was first used by Leonhard Euler (1707-1783) in 1750 (Cajori 1919, page 235).

Similarity. ~ was introduced by Gottfried Wilhelm Leibniz (1646-1716) in a manuscripts of 1679 which were not published by him. The symbol was an S for similis, written sideways. The original manuscripts do not survive and it is uncertain whether the symbol Leibniz first used resembled the tilde or the tilde inverted (Cajori vol. 1, page 414).

In the manuscript of his Characteristica Geometrica he wrote: "similitudinem ita notabimus: a ~ b" (Cajor vol. 1, page 414).

The first appearance in print of Leibniz' sign for similarity was in 1710 in the Miscellanea Berolinensia in the anonymous article "Monitum," which is attributed to Leibniz (Cajori vol. 2, page 195).

S.S.S., S.W.S., and W.S.W. for the triangle congruence theorems and axioms were invented by Julius Worpitzky (1835-1895), professor at the Friedrich Werder Gymnasium in Berlin (Cajori vol. 1, page 424). (W for Winkel=angle)

An article in The Mathematics Teacher in March 1938 uses a.s.a. = a.s.a. and s.s.s. = s.s.s. and s.a.a. = s.a.a. as reasons in a proof. An article in the same journal in 1940 uses C.p.c.t.e., which is written out as "Corresponding parts of congruent triangles are equal."

An article in The Mathematics Teacher in April 1948 has: "The three common theorems on congruence of triangles (SAS = SAS; ASA = ASA; SSS = SSS) are 'proved' by superposition."


Sine. In 1583, Thomas Fincke (or Finck) (1561-1656) used sin. (with a period) in Book 14 of his Geometria rotundi. Cajori writes that "perhaps the first use of abbreviations for the trigonometric lines goes back to ... Finck" (Cajori vol. 2, page 150).

In 1624, Edmund Gunter (1581-1626) used sin (without a period) in a drawing representing Gunter's scale (Cajori vol. 2, page 156). However, the symbol does not appear in Gunter's work published the same year.

In 1626, Girard designated the sine of A by A, and the cosine of A by a (Smith vol. 2, page 618).

In a trigonometry published by Richard Norwood in London in 1631, the author states that "in these examples s stands for sine: t for tangent: sc for sine complement: tc for tangent complement: sc for sine complement: tc for tangent complement : sec for secant" (Smith vol. 2, page 618).

In 1632, William Oughtred (1574-1660) used sin (without a period) in Addition vnto the Vse of the Instrvment called the Circles of Proportion (Cajori vol. 1, page 193, and vol. 2, page 158).

According to Smith (vol. 2, page 618), "the first writer to use the symbol sin for sine in a book seems to have been the French mathematician Hérigone (1634)." However, the use by Oughtred would seem to predate that of Hérigone.

Cosine. In his Geometria rotundi (1588) Thomas Fincke used sin. com. for the cosine.

In 1674, Sir Jonas Moore (1617-1679) used Cos. in Mathematical Compendium (Cajori vol. 2, page 163).

Samuel Jeake (1623-1690) used cos. in Arithmetick, published in 1696 (Cajori vol. 2, page 163).

The earliest use of cos Cajori shows apparently is by Leonhard Euler in 1729 in Commentarii Academiae Scient. Petropollitanae, ad annum 1729 (Cajori vol. 2, page 166).

Ball and Asimov say cos was first used by William Oughtred (1574-1660). Ball gives the date 1657; Asimov gives 1631. However, Cajori indicates Oughtred used only sco for cosine (Cajori vol. 1, page 193) and Cajori reports Glaisher says Oughtred did not even use the word cosine.

Tangent. In 1583, Thomas Fincke (1561-1656) used tan. in Book 14 of his Geometria rotundi (Cajori vol. 2, page 150). Fincke also used tang.

In 1632, William Oughtred (1574-1660) used tan in The Circles of Proportion (Cajori vol. 1, page 193).

Secant. In 1583, Thomas Fincke (1561-1656) used sec. in Book 14 of his Geometria rotundi (Cajori vol. 2, page 150).

In 1632, William Oughtred (1574-1660) used sec in The Circles of Proportion (Cajori vol. 1, page 193).

Cosecant. In his Geometria rotundi (1588), Thomas Fincke used sec. com. for the cosecant (Cajori vol. 2, page 150).

Samuel Jeake in his Arithmetick (1696) used cosec. for cosecant (Cajori vol. 2, page 163).

Simon Klügel in Analytische Trigonometrie (1770) used cosec for cosecant.

It appears that the earliest use Cajori shows for csc is in 1881 in Treatise on Trigonometry, by Oliver, Wait, and Jones (Cajori vol. 2, page 171).

Cotangent. In his Geometria rotundi (1588), Thomas Fincke used tan. com. for the cotangent (Cajori vol. 2, page 150).

In 1674, Sir Jonas Moore (1617-1679) used Cot. in Mathematical Compendium (Cajori vol. 2, page 163).

Samuel Jeake in his Arithmetick (1696) used cot. for cotangent (Cajori vol. 2, page 163).

A. G. Kästner in Anfangsgründe der Arithmetik, Geometrie ... Trigonometrie (1758) used cot for cotangent (Cajori vol. 2, page 166).

Cis notation. Irving Stringham used cis (beta) for cos (beta) + i sin (beta) in 1893 in Uniplanar Algebra (Cajori vol. 2, page 133).

Inverse trigonometric function symbols. Except where noted otherwise, the following citations are from Cajori vol. 2, page 175-76.

Daniel Bernoulli was the first to use symbols for the inverse trigonometric functions. In 1729 he used A S. for arcsine in Comment. acad. sc. Petrop., Vol. II.

In 1736 Leonhard Euler (1707-1783) used A t for arctangent in Mechanica sive motus scientia. Later in the same publication he used simply A

In 1737 Euler used A sin for arcsine in Commentarii academiae Petropolitanae ad annum 1737, Vol. IX.

In 1769 Antoine-Nicolas Caritat Marquis de Condorcet (1743-1794) used arc (sin. = x) (Katz, page 42).

In 1772 Carl Scherffer used arc. tang. in Institutionum analyticarum pars secunda.

In 1772 Joseph Louis Lagrange (1736-1813) used arc. sin in Nouveaux memoires de l'academie r. d. sciences et belles-lettres.

According to Cajori (vol. 2, page 176) the inverse trigonometric function notation utilizing the exponent -1 was introduced by John Frederick William Herschel in 1813 in the Philosophical Transactions of London. A full-page footnote explained his choice of notation for the inverse trigonometric functions, such as cos.-1 e, which he used in the body of the article (Cajori vol. 2, page 176).

However, according to Differential and Integral Calculus (1908) by Daniel A. Murray, "this notation was explained in England first by J. F. W. Herschell in 1813, and at an earlier date in Germany by an analyst named Burmann. See Herschell, A Collection of Examples of the Application of the Calculus of Finite Differences (Cambridge, 1820), page 5, note."

According to Cajori, in France, Jules Houël (1823-1886) used arcsin.

In 1914, Plane Trigonometry by George Wentworth and David Eugene Smith has:

In American and English books the symbol sin-1 y is generally used; on the continent of Europe the symbol arc sin y is the one that is met. The symbol sin-1 y is read "the inverse sine of y," "the antisine of y," or "the angle whose sine is y." The symbol arc sin y is read "the arc whose sine is y," or "the angle whose sine is y."

In 1922 in Introduction to the Calculus, William F. Osgood wrote, "The usual notation on the Continent for sin-1, x, tan-1 x, etc., is arc sin x, arc tan x, etc. It is clumsy, and is followed for a purely academic reason; namely, that sin-1 x might be misunderstood as meaning the minus first power of sin x. It is seldom that one has occasion to write the reciprocal of sin x in terms of a negative exponent. When one wishes to do so, all ambiguity can be avoided by writing (sin x)-1."

Powers of trigonometric functions. The practice of placing the exponent beside the symbol for the trigonometric function to indicate, for example, the square of the sine of x, was used by William Jones in 1710. He wrote cs2 (for cosine) and s2 (for sine) (Cajori vol. 2, page 179).

Degrees, minutes, seconds. See the geometry page.

Radians. G. B. Halstead in Mensuration (1881) suggested using the Greek letter rho to indicate radians.

G. N. Bauer and W. E. Brooke in Plane and Spherical Trigonometry (1907) suggested the use of r (the lower case r in a raised position) to indicate radians.

A. G. Hall and F. G. Frink in Plane Trigonometry (1909) suggested the use of R (the capital R in a raised position).

P. R. Rider and A. Davis in Plane Trigonometry (1923) suggested the use of (r) (the lower case r in parentheses) to indicate radians.

Hyperbolic functions. Vincenzo Riccati (1707-1775), who introduced the hyperbolic functions, used Sh. and Ch. for hyperbolic sine and cosine (Cajori vol. 2, page 172). He used Sc. and Cc. for the circular functions.

Johann Heinrich Lambert (1728-1777) further developed the theory of hyperbolic functions in Histoire de l'académie Royale des sciences et des belles-lettres de Berlin, vol. XXIV, p. 327 (1768). According to Cajori (vol. 2, page 172), Lambert used sin h and cos h.

According to Scott (page 190), Lambert began using sinh and cosh in 1771.

According to Webster's New International Dictionary, 2nd. ed., sinh is an abbreviation for sinus hyperbolus.

In 1902, George M. Minchin proposed using hysin, hycos, hytan, etc.: "If the prefix hy were put to each of the trigonometric functions, all the names would be pronounceable and not too long." The proposal appeared in Nature, vol. 65 (April 10, 1902).


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 inThéorie 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 psi prime for d psi over dx 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 (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.

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 & curly d 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, & curly d z une difference partielle. [Throughout this paper, both dz & curly d 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 curly d z will be employed as symbols of total differential, and of partial difference, respectively.]

However, the "curly d" was first used in the form du/dx 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 du/dx, with the curly d's le coefficient de x dans la différence de u, & par du/dx 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" [appeared in Crelle's Journal, Band 22, pp. 319-352, 1841].

Sed quia uncorum accumulatio et legenti et scribenti molestior fieri solet, praetuli characteristica


differentialia vulgaria, differentialia autem partialia characteristica

the curly d symbol


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. Several weeks later, on Nov. 21, he first placed dx after the integral symbol (Burton, page 359). Later in 1675, he proposed the use of the symbol in a letter to Henry Oldenburg, secretary of the Royal Society: "Utile erit scribi [symbol] pro omnia, ut [symbol] l = omn. l, id est summa ipsorum l" [It will be useful to write [symbol] for omn. so that [symbol] 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 (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 the integral symbol with
a and b as the limits of integration 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 226 (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

lim. of delta P over delta x ,

la limite du rapport des changements simultanes de P & de x, favoir par

dP over dx ;

en sorte que

lim of delta P over delta x


dP over dx ;

designent la même chose

lim (without a period) was written in 1841 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 for the limit as x approaches x-sub-0 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 (Julio González Cabillón and Cajori vol. 2, page 257).

Delta to indicate a small quantity. In 1706, Johann Bernoulli used the Greek letter delta 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 epsilon in 1821 in Cours d'analyse, and sometimes used delta instead (Cajori vol. 2, page 256). According to Finney and Thomas (page 113), "[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 (Oeuvres II.3, p. 54) 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 An upside-down delta (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 (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.

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 An upside-down
delta ; 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 (vol. 2, page 135) 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.

Gradient. Maxwell and Riemann-Weber used grad as an abbreviation or symbol for gradient (Cajori vol. 2, page 135).

Divergence. William Kingdon Clifford (1845-1879) used the term divergence and wrote div u or dv u (Cajori vol. 2, page 135).

Laplacian operator. The capital delta for An upside-down
delta 2 was introduced by Robert Murphy in 1883 (Kline, page 786).

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/ [symbol] , sive alicuota pars infinite parva; (esto enim [symbol] 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 [symbol] 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 [the infinity symbol] 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 [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].


Intersection and union. Giuseppe Peano (1858-1932) introduced the symbol for intersection and the symbol for union in 1888 in Calcolo geometrico secondo l'Ausdehnungslehre di H. Grassmann (Cajori vol. 2, page 298).

According to Schwartzman (p. 118) the intersection symbol above dates back to Leibniz "who also used it to indicate regular multiplication." Cajori says Leibniz used the symbol for multiplication, but seems not to confirm that he used it for intersection.

Existence. Peano used the symbol
for existence in volume II, number 1, of his Formulaire de mathematiqués, which was published in 1897 (Cajori vol. 2, page 300).

Membership. Peano used the symbol
for membership in the introduction to volume I of his Formulaire de mathematiqués, which was published in Turin in 1895, although the introduction itself is dated 1894 (Cajori vol. 2, page 300).

The website at the University of St. Andrews states that Peano introduced the symbol in 1889 and that it comes from the first letter if the Greek word meaning "is."

Peano's symbol for membership was an ordinary epsilon ; the stylized epsilon now used was adopted by Bertrand Russell in Principles of Mathematics in 1903 (Julio González Cabillón).

Such that. According to Julio González Cabillón, Peano introduced the backwards lower-case epsilon for "such that" in "Formulaire de Mathematiques vol. II, #2" (p. iv, 1898).

Peano introduced the backwards lower-case epsilon for "such that" in his 1889 "Principles of arithmetic, presented by a new method," according van Heijenoort's From Frege to Gödel: A Source Book in Mathematical Logic, 1879--1931 [Judy Green].

For all. According to M. J. Cresswell and Irving H. Anellis, the upside-down A originated in Gerhard Gentzen, "Untersuchungen ueber das logische Schliessen," Math. Z. 39 (1934), p, 178. In footnote 4 on that page, Gentzen explains how he came to use the sign. It is the "All-Zeichen," an analogy with the symbol
for existence for the existential quantifier which Gentzen says that he borrowed from Russell.

Cajori, however, shows that Peano used the
symbol for existence before Russell and Whitehead (whose backwards E had serifs, unlike Peano's).

Braces enclosing the elements of a set. This symbolism was introduced in 1895 by Georg Cantor (1845-1918). Cantor sets about his famous essay [p. 481] as follows:

Unter einer 'Menge' verstehen wir jede Zusammenfassung M von bestimmten wohlunterschiedenen Objecten m unsrer Anschauung oder unseres Denkens (welche die 'Elemente' von M genannt werden) zu einem Ganzen.

In Zeichen druecken wir dies so aus:

M = {m}.

The citation above is from "Beiträge zur Begründung der transfiniten Mengelehre" [Contributions to the founding of the theory of transfinite numbers], Mathematische Annalen, Band XLVI [vol. 46], pp. 481-512, B. G. Teubner, Leipzig, 1895.

Please recall that Cantor's "Contributions to the founding of the theory of transfinite numbers" [first published by The Open Court publishing Company, Chicago-London, 1915] is a translation of the two memoirs which had appeared in Mathematische Annalen for 1895 and 1897 under the title: "Beiträge zur Begründung der transfiniten Mengelehre" -- translation from the German, introduction, and notes by Philip Edward Bertrand Jourdain (1879-1919). An unabridged and unaltered republication of the English translation mentioned was edited also by Dover Publications, Inc., New York, 1955 [ISBN: 0486600459].

M stands for the German term "Menge." Cantor may have used this notation earlier in his correspondence with the mathematicians of his day. (This entry was contributed by Julio González Cabillón.)

p, q, and r were used as "propositional letters" in 1908 in the article "Mathematical logic as based on the theory of types" by Bertrand Russell [Denis Roegel].

These three letters were also used by Alfred North Whitehead and Bertrand Russell in the first volume of Principia mathematica, which was published in 1910 (Cajori vol. 2, page 307). The OED2 shows several uses of these letters in the Principia, with the date 1903, although the three volumes were published in 1910, 1912 and 1913.

~p for "the negation of p" appears in 1908 in the article "Mathematical logic as based on the theory of types" by Bertrand Russell [Denis Roegel].

The symbolism was also used in 1910 by Alfred North Whitehead and Bertrand Russell in the first volume of Principia mathematica (Cajori vol. 2, page 307).

p \/ q for "p or q" appears in 1908 in the article "Mathematical logic as based on the theory of types" by Bertrand Russell [Denis Roegel].

The symbolism was also used in 1910 by Alfred North Whitehead and Bertrand Russell in the first volume of Principia mathematica. (These authors used p.q for "p and q.") (Cajori vol. 2, page 307)

The notation (x) for "for all x" appears in 1908 in the article "Mathematical logic as based on the theory of types" by Bertrand Russell [Denis Roegel].

The null set symbol (Ø). André Weil (1906-1998) says in his autobiography that he introduced the symbol:

Wisely, we had decided to publish an installment establishing the system of notation for set theory, rather than wait for the detailed treatment that was to follow: it was high time to fix these notations once and for all, and indeed the ones we proposed, which introduced a number of modifications to the notations previously in use, met with general approval. Much later, my own part in these discussions earned me the respect of my daughter Nicolette, when she learned the symbol Ø for the empty set at school and I told her that I had been personally responsible for its adoption. The symbol came from the Norwegian alphabet, with which I alone among the Bourbaki group was familiar.

The citation above is from page 114 of André Weil's The Apprenticeship of a Mathematician, Birkhaeuser Verlag, Basel-Boston-Berlin, 1992. Translated from the French by Jennifer Gage. The citation was provided by Julio González Cabillón.

This letter is used in the Norwegian, Danish and Faroese alphabets.

The therefore symbol ( consisting of three dots )was first published in 1659 in the original German edition of Teusche Algebra by Johann Rahn (1622-1676) (Cajori vol. 1, page 212, and vol 2., page 282).

The halmos (a box indicating the end of a proof). On the last page of his autobiography, Paul R. Halmos (1916- ) writes:

My most nearly immortal contributions are an abbreviation and a typographical symbol. I invented "iff", for "if and only if" -- but I could never believe that I was really its first inventor. I am quite prepared to beieve that it existed before me, but I don't know that it did, and my invention (re-invention?) of it is what spread it thorugh the mathematical world. The symbol is definitely not my invention -- it appeared in popular magazines (not mathematical ones) before I adopted it, but, once again, I seem to have introduced it into mathematics. It is the symbol that sometimes looks like [an empty square], and is used to indicate an end, usually the end of a proof. It is most frequently called the "tombstone", but at least one generous author referred to it as the "halmos".

This quote is from I Want to Be a Mathematician: An Automathography, by Paul R. Halmos, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1985, page 403.

The aleph null symbol was conceived by Georg Cantor (1845-1918) around 1893, and became widely known after "Beiträge zur Begründung der transfiniten Mengelehre" [Contributions to the Foundation of Transfinite Set Theory] saw the light in Mathematische Annalen, vol. 46, B. G. Teubner, Leipzig, 1895.

On page 492 of this prestigious journal we find the paragraph Die kleinste transfinite Cardinalzahl Alef-null [The minimum transfinite cardinal number Aleph null], and the following:

...wir nennen die ihr zukommende Cardinalzahl, in Zeichen, *Alef-null* ... [We call the cardinal number related to that (set); in symbol, *Alef-null* ]

P. S: Cantor's "Contributions to the founding of the theory of transfinite numbers" was translated from the German by Philip E. B. Jourdain, and published in 1915 by The Open Court Publishing Company, Chicago-London. (This entry was contributed by Julio González Cabillón.)

In Georg Cantor, Dauben (page 179) says that Cantor did not want to use Roman or Greek alphabets, because they were already widely used, and "His new numbers deserved something unique. ... Not wishing to invent a new symbol himself, he chose the aleph, the first letter of the Hebrew alphabet...the aleph could be taken to represent new beginnings...." Avinoam Mann points out that aleph is also the first letter of the Hebrew word "Einsof," which means infinity and that the Kabbalists use "einsof" for the Godhead. Mann also notes that Coleridge, in Kubla Khan, refers to the sacred river Alph, and it is thought that this name is related to Aleph. In a letter dated April 30, 1895, Cantor wrote, "it seemed to me that for this purpose, other alphabets were [already] over-used" (translation by Martin Davis). Although his father was a Lutheran and his mother was a Roman Catholic, he had at least some Jewish ancestry.

A reader of this page writes, "Please do note that the probable derivation of this is the fact that this letter alep literally is the same as elep, meaning "thousand," the largest number whereto a name was applied in ancient Hebrew. The term "Aleph null" could hardly have been taken from (as your article is claiming) the mystical term eyn-sowp, which is a compound meaning, literally, "nothingness-consuming" (which is actually referring to a mystic vision of nothingness, i.e. with the notion that there can exist nothing other than the Godhead). A mathematician, after all, can hardly expect (certainly Georg Cantor did not!) that any infinity can be derived from nothingness, by such a process as division by zero; which is, in effect, what thy published article asserteth as actual."


Congruence of numbers. The congruent symbol used in number theory three
horizontal bars was introduced in print in 1801 by Carl Friedrich Gauss (1777-1855) in Disquisitions arithmeticae:

Numerorum congruentiam hoc signo, three horizontal bars , in posterum denotabimus, modulum ubi opus erit in clausulis adiungentes, -16 three horizontal bars 9 (mod. 5), -7 three
horizontal bars 15 (modo 11).

The citation above is from Disquisitiones arithmeticae (Leipzig, 1801), art. 2; Werke, Vol. I (Gottingen, 1863), p. 10 (Cajori vol. 2, page 35).

However, Gauss had used the symbol much earlier in his personal writings (Francis, page 82).

The number of primes less than x. Edmund Landau used pi (x) for the number of primes less than or equal to x in 1909 in Handbuch der Lehre von der Verteilung der Primzahlen (Cajori vol. 2, page 36).

Letters for the sets of rational and real numbers. The authors of classical textbooks such as Weber and Fricke did not denote particular domains of computation with letters.

Richard Dedekind (1831-1916) denoted the rationals by R and the reals by gothic R in Continuity and irrational numbers (1872). Dedekind also used K for the integers and J for complex numbers.

In 1895 in his Formulaire de mathématiques, Giuseppe Peano (1858-1932) used N for the positive integers, n for integers, N0 for the positive integers and zero, R for positive rational numbers, r for rational numbers, Q for positive real numbers, q for real numbers, and Q0 for positive real numbers and zero [Cajori vol. 2, page 299].

Helmut Hasse (1898-1979) used [capital gamma] for the integers and [capital rho] for the rationals in Höhere Algebra I and II, Berlin 1926. He kept to this notation in his later books on number theory. Hasse's choice of gamma and rho may have been determined by the initial letters of the German terms "ganze Zahl" (integer) and "rationale Zahl" (rational).

Otto Haupt used G0 for the integers and [capital rho]0 for the rationals in Einführung in die Algebra I and II, Leipzig 1929.

Bartel Leendert van der Waerden (1903-1996) used C for the integers and [capital gamma] for the rationals in Moderne Algebra I, Berlin 1930, but in editions during the sixties, he changed to Z and Q.

Edmund Landau (1877-1938) denoted the set of integers by a fraktur Z with a bar over it in Grundlagen der Analysis (1930, p. 64). He does not seem to introduce symbols for the sets of rationals, reals, or complex numbers.

Q for the set of rational numbers and Z for the set of integers are apparently due to N. Bourbaki. (N. Bourbaki was a group of mostly French mathematicians which began meeting in the 1930s, aiming to write a thorough unified account of all mathematics.) The letters stand for the German Quotient and Zahlen. These notations occur in Bourbaki's Algébre, Chapter 1.

Julio González Cabillón writes that he believes Bourbaki was responsible for both of the above symbols, quoting Weil, who wrote, "...it was high time to fix these notations once and for all, and indeed the ones we proposed, which introduced a number of modifications to the notations previously in use, met with general approval."

[Walter Felscher, Stacy Langton, Peter Flor, and A. J. Franco de Oliveira contributed to this entry.]

C for the set of complex numbers. William C. Waterhouse wrote to a history of mathematics mailing list in 2001:

Checking things I have available, I found C used for the complex numbers in an early paper by Nathan Jacobson:

Structure and Automorphisms of Semi-Simple Lie Groups in the Large, Annals of Math. 40 (1939), 755-763.

The second edition of Birkhoff and MacLane, Survey of Modern Algebra (1953), also uses C (but is not using the Bourbaki system: it has J for integers, R for rationals, R^# for reals). I have't seen the first edition (1941), but I would expect to find C used there too. I'm sure I remember C used in this sense in a number of other American books published around 1950.

I think the first Bourbaki volume published was the results summary on set theory, in 1939, and it does not contain any symbol for the complex numbers. Of course Bourbaki had probably chosen the symbols by that time, but I think in fact the first appearance of (bold-face) C in Bourbaki was in the formal introduction of complex numbers in Chapter 8 of the topology book (first published in 1947).

Euler's phi function. phi (m) was introduced by Carl Friedrich Gauss (1777-1855) in 1801 in his Disquisitiones Arithmeticae, articles 38, 39 (Cajori vol. 2, page 35, and Dickson, page 113-115).

The article "Number Theory" in the Encyclopaedia Britannica claims this symbol was introduced by Leonhard Euler (1707-1783). However Dickson (page 113) and Cajori (vol. 2, p. 35) say that Euler did not use a functional notation in Novi Comm. Ac. petrop., 8, 1760-1, 74, and Comm. Arith., 1, 274, and that Euler used pi N in Acta Ac. Petrop., 4 II (or 8), 1780 (1755), 18, and Comm. Arith., 2, 127-133. Shapiro agrees, writing: "He did not employ any symbol for the function until 1780, when he used the notation pi n."

Sylvester, who used tau for this function, also believed that Euler used phi . He writes (in vol. IV p. 589 of his Collected Mathematical Papers) "I am in the habit of representing the totient of n by the symbol (tau) n, (tau) (taken from the initial of the word it denotes) being a less hackneyed letter than Euler's phi , which has no claim to preference over any other letter of the Greek alphabet, but rather the reverse." This information was taken from a post in sci.math by Robert Israel.

Quadratic reciprocity. Adrien-Marie Legendre introduced the notation that (D/p) = 1 if D is a quadratic residue of p, and (D/p) = -1 if D is a quadratic non-residue of p. According to Hardy & Wright's An Introduction to the Theory of Numbers: "Legendre introduced 'Legendre's symbol' in his Essai sur la theorie des nombres, first published in 1798. See, for example, §135 of the second edition (1808)." [Paul Pollack]

Mersenne numbers. Mersenne numbers are marked Mn by Allan Cunningham in 1911 in Mathematical Questions and Solutions from the Educational Times (Cajori vol. 2, page 41).

Fermat numbers. Fermat numbers are marked Fn in 1919 in L. E. Dickson's History of the Theory of Numbers (Cajori vol. 2, page 42).

The norm of a + bi. Dirichlet used N(a+bi) for the norm a2+b2 of the complex number a+bi in Crelle's Journal Vol. XXIV (1842) (Cajori vol. 2, page 33).

Galois field. Eliakim Hastings Moore used the symbol GF[qn] to represent the Galois field of order qn in 1893. The modern notation is "Galois-field of order qn" (Julio González Cabillón and Cajori vol. 2, page 41).

Sum of the divisors of n. Euler introduced the symbol the integral
symbol n in a paper published in 1750 (DSB, article: "Euler").

In 1888, James Joseph Sylvester continued the use of Euler's notation the integral
symbol n (Shapiro).

Allan Cunningham used [lower case sigma](N) to represent the sum of the proper divisors of N in Proceedings of the London Mathematical Society 35 (1902-03) (Cajori vol. 2, page 29).

[According to Shapiro, Cunningham used s(n) in the above paper.]

In 1927 Landau chose the notation S(n) (Shapiro).

L. E. Dickson used s(n) for the sum of the divisors of n (Cajori vol. 2, page 29).

The Möbius function. Möbius' work appeared in 1832 but the µ symbol was not used.

The notation µ(n) was introduced by Franz Mertens (1840-1927) in 1874 in "Über einige asymptotische Gesetze der Zahlentheorie," Crelle's Journal (Shapiro).

Big-O notation was introduced by Paul Bachmann (1837-1920) in his Analytische Zahlentheorie in 1894. The actual O symbol is sometimes called a Landau symbol after Edmund Landau (1877-1938), who used this notation throughout his work.

According to Wladyslaw Narkiewicz in The Development of Prime Number Theory:

The symbols O(·) and o(·) are usually called the Landau symbols. This name is only partially correct, since it seems that the first of them appeared first in the second volume of P. Bachmann's treatise on number theory (Bachmann, 1894). In any case Landau (1909a, p. 883) states that he had seen it for the first time in Bachmann's book. The symbol o(·) appears first in Landau (1909a). Earlier this relation has been usually denoted by {·}.

[Paul Pollack contributed to this entry.]

Little-oh notation was first used by Edmund Landau (1877-1938) in 1909, according to the website of the University of Tennessee at Martin. The symbol appears in 1909 in his Handbuch der Lehre von der Verteilung der Primzahlen.



Factorial. An early factorial symbol, the argument is placed inside an L-shaped
symbol , was suggested by Rev. Thomas Jarrett (1805-1882) in 1827. It occurs in a paper "On Algebraic Notation" that was printed in 1830 in the Transactions of the Cambridge Philosophical Society and it appears in 1831 in An Essay on Algebraic Development containing the Principal Expansions in Common Algebra, in the Differential and Integral Calculus and in the Calculus of Finite Differences (Cajori vol. 2, pages 69, 75).

The notation n! was introduced by Christian Kramp (1760-1826) in 1808 as a convenience to the printer. In his Élémens d'arithmétique universelle (1808), Kramp wrote:

Je me sers de la notation trés simple n! pour désigner le produit de nombres décroissans depuis n jusqu'à l'unité, savoir n(n - 1)(n - 2) ... 3.2.1. L'emploi continuel de l'analyse combinatoire que je fais dans la plupart de mes démonstrations, a rendu cette notation indispensable.

In "Mémoire sur les facultés numériques," published in J. D. Gergonne's Annales de Mathématiques [vol. III, 1812 and 1813], Kramp writes:

1. [...] Je donne le nom de Facultés aux produits dont les facteurs constituent une progression arithmétique, tels que

a(a + r)(a + 2r)...[a + (m-1)r];

et, pour désigner un pareil produit, j'ai proposé la notation


Les facultés forment une classe de fontions très-élementaires, tant que leur exposant est un nombre entier, soit positif soit négatif; mais, dans tous les autres cas, ces mêmes fonctions deviennent absolument transcendantes. [page 1]

2. J'observe que toute faculté numérique quelconque est constamment réductible ô la forme trés-simple

1m|1 = 1 . 2 . 3 ... m

ou à cette autre forme plus simple [page 2]


si l'on veut adopter la notation dont j'ai fait usage dans mes Éléments d'arithmétique universelle, no. 289. [page 3]

[Julio González Cabillón; Cajori vol. 2, p. 72]

In The Elliptic Functions As They Should Be (1958), Albert Eagle advocated writing !n rather than n!, so that the operator would precede the argument, as it does in most cases [Daren Scot Wilson].

In his article "Symbols" in the Penny Cyclopaedia (1842) De Morgan complained: "Among the worst of barabarisms is that of introducing symbols which are quite new in mathematical, but perfectly understood in common, language. Writers have borrowed from the Germans the abbreviation n! to signify 1.2.3.(n - 1).n, which gives their pages the appearance of expressing surprise and admiration that 2, 3, 4, &c. should be found in mathematical results" [Cajori vol. 2, p. 328].

Combinations and permutations. Leonhard Euler (1707-1783) designated the binomial coefficients by n over r within parentheses and using a horizontal fraction bar in a paper written in 1778 but not published until 1806. He used used the same device except with brackets in a paper written in 1781 and published in 1784 (Cajori vol. 2, page 62).

The modern notation, using parentheses and no fraction bar, appears in 1826 in Die Combinatorische Analyse by Andreas von Ettingshausen [Henry W. Gould]. According to Cajori (vol. 2, page 63) this notation was introduced in 1827 by Andreas von Ettingshausen in Vorlesungen über höhere Mathematik, Vol. I.

Harvey Goodwin used nPr for the number of permutations of n things taken r at a time in 1869 and earlier. The notation appears in his Elementary Course of Mathematics, 3rd ed. (Cajori vol. 2, page 79).

G. Chrystal used nCr for the number of combinations of n things taken r at a time in Algebra, Part II (1899) (Cajori vol. 2, page 80).


Probability. Symbols for the probability of an event A on the pattern of P(A) or Pr(A) are a relatively recent development given that probability has been studied for centuries. A. N. Kolmogorov's Grundbegriffe der Wahrscheinlichkeitsrechnung (1933) used the symbol P(A). The use of upper-case letters for events was taken from set theory. H. Cramér's Random Variables and Probability Distributions (1937), "the first modern book on probability in English," used P(A). In the same year J. V. Uspensky (Introduction to Mathematical Probability) wrote simply (A). W. Feller's influential An Introduction to Probability Theory and its Applications volume 1 (1950) uses Pr{A} and P{A}in later editions.

Conditional probability. Kolmogorov's (1933) symbol for conditional probability ("die bedingte Wahrscheinlichkeit") was P<SMALLB (A). Cramér (1937) referred to the "relative probability" and wrote PB (A). Uspensky (1937) used the term "conditional probability" with the symbol (A,B). The vertical stroke notation Pr{A | B} was made popular by Feller (1950), though it was used earlier by H. Jeffreys. In his Scientific Inference (1931) P(p | q) stands for "the probability of the proposition p on the data q." Jeffreys mentions that Keynes and Johnson, earlier Cambridge writers, had used p/q; Jeffreys himself had used P(p : q). The symbols p and q came from Whitehead and Russell's Principia Mathematica.

Expectation. A large script E was used for the expectation in W. A. Whitworth's well-known textbook Choice and Chance (fifth edition) of 1901 but neither the symbol nor the calculus of expectations became established in the English literature until much later. For example, Rietz Mathematical Statistics (1927) used the symbol E and commented that "the expected value of the variable is a concept that has been much used by various continental European writers..." For the continental European writers E signified "Erwartung" or "'éspérance."

Random variable. The use of upper and lower case letters to distinguish a random variable from the value it takes, as in Pr{X = xj }, became popular around 1950. The convention is used in Feller's Introduction to Probability Theory.


The individual with the greatest influence on present day statistical terminology and notation remains R. A. Fisher (1890-1962). Many of Fisher's papers are available on the University of Adelaide Library website. The first edition of Fisher's tremendously influential textbook, Statistical Methods for Research Workers (1925) is available on the Classics in the History of Psychology website.

Notation for Parameters and Estimates. Today there are two conventions for representing a parameter and the corresponding estimate. One is to write the estimate by adding a hat (or other accent) to the character representing the parameter (often a Greek character). The other is to use corresponding Greek and Latin characters for parameter and estimate. Both conventions owe most to R. A. Fisher who insisted on clearly distinguishing parameters (see words) and estimates. He used the hat device mostly in conjunction with theta in general discussions of estimation as in Phil. Trans. R. Soc. 1922 - see below. The most familiar example of the Graeco-Latin convention, s, for an estimate of sigma , dates from 1908 but the convention only became established in the 1930s. The growth of the system can be seen through the entries on specific symbols.

Correlation coefficient. When Galton introduced correlation in "Co-Relations and Their Measurement," Proc. R. Soc., 45, 135-145, 1888 he chose the symbol r for the index of co-relation, perhaps for its affinity with regression. The use of rho for the population linear correlation coefficient is found in 1892 in F. Y Edgeworth, "Correlated Averages," Philosophical Magazine, 5th Series, 34, 190-204. The symbol appears on page 190 (David, 1995).

Karl Pearson, who dominated correlation research from the mid-1890s, favoured the use of r (for both parameter and estimate); thus in 1896 he was writing, "Let r0 be the coefficient of correlation between parent and offspring" in Proc. R. Soc. LIX 302 (OED2). Student (W. S. Gosset) in "The Probable Error of the Correlation Coefficient" (Biometrika, 6, 302-310 1908) wrote r for the estimate and R for the parameter value. H. E. Soper (Biometrika, 9, 91-115, 1913) introduced r and rho . Soper's scheme was adopted by Fisher in his work on correlation.

G. Udny Yule introduced the notation r12.3 for the partial correlation between x1 and x2 holding x3 fixed in his 1907 "On the Theory of Correlation for any Number of Variables, Treated by a New System of Notation," Proc. R. Soc. Series A, 79, pp. 182-193. The Greek form followed in M. S. Bartlett's 1933 "On the theory of statistical regression," Proc. Royal Soc. Edinburgh, 53, 260-283.

R has been used for the double, triple, ..., n-fold or multiple correlation coefficient, at least since Yule in 1896. R is now generally used for the sample coefficient which is awkward for the population value because the upper-case rho is the unappealing P.

Moments. Pearson introduced the basic symbol mu to which numerical subscripts would be added to indicate the order and a prime could be added to indicate about which value the moment is taken. Fisher applied the Graeco-Latin convention and twinned the mu 's with m's in his paper on cumulants (1929).

Standard deviation and variance. The use of sigma for standard deviation first occurs in Karl Pearson's 1894 paper, "Contributions to the Mathematical Theory of Evolution," Philosophical Transactions of the Royal Society of London, Ser. A, 185, 71-110. On page 80, he wrote, " Then sigma will be termed its standard-deviation (error of mean square)" (David, 1995). When Fisher introduced variance (see Words) he did not introduce a new symbol but instead used sigma 2.

Pearson's notation did not distinguish between parameter and estimate. Student (W. S. Gosset) in "The Probable Error of a Mean," Biometrika, 6, 1-25, 1908 used s for an estimate of sigma , though contrary to modern practice his divisor was n, not (n - 1). Fisher eventually adopted Student's s2 (with adjusted n) as an estimate of sigma 2 beginning with his 1922 paper, "The goodness of fit of regression formulae, and the distribution of regression coefficients" (J. Royal Statist. Soc., 85, 597-612).

Regression notation. Regression analysis has its roots in Gauss's work (1809/-23) on the combination of observations and Pearson's work (1896) on correlation but the modern notation essentially dates from the 1920s when R. A. Fisher drew the Gauss and Pearson lines together. In his Statistical Methods for Research Workers (1925) Fisher presents regression using y and x and the terms "dependent variable" and "independent variable." For the population values of the intercept and slope Fisher uses alpha and beta and for the estimates a and b. This textbook exposition was based on a 1922 paper, "The goodness of fit of regression formulae, and the distribution of regression coefficients" (J. Royal Statist. Soc., 85, 597-612). (The sheer variety of early regresssion notation can be seen from the examples in Aldrich (1998).)

theta as the generic "unknown" parameter. R. A. Fisher established the role and theta in it in "On the Mathematical Foundations of Theoretical Statistics" (Phil. Trans. R. Soc. 1922) and the papers that followed. Fisher had already used the notation in his first publication, a paper he wrote as a third year undergraduate, "On an absolute criterion for fitting frequency curves" (Messenger of Mathematics, 1912).

kappa for cumulants (cumulative moment functions) and the corresponding k-statistics. Fisher introduced this notation in his 1929 paper "Moments and Product Moments of Sampling Distributions," Proceedings of the London Mathematical Society, Series 2, 30, 199-238. He introduced the cumulant notation into the 1932 (fourth) edition of the Statistical Methods for Research Workers.

Mean of the normal distribution. mu , as the symbol for the mean of the normal distribution, was surprisingly late in becoming established. Fisher adopted it in the 1936 (sixth) edition of the Statistical Methods for Research Workers. He had been using m since 1912. He used x-bar for the sample mean throughout.

Sample mean. I do not have the earliest use of x-bar for the sample mean. However, John Harper notes that R. A. Fisher used the notation in "On an absolute criterion for fitting frequency curves," Messenger of Mathematics, v. 41: 155-160 (1912) on p. 157. He does not know whether that represents the first use.

Symbols for test statistics. There are no conventions here like those governing parameters and estimates.

F distribution. Please see the entry on the mathematical words page.

Chi-squared. Please see the entry on the mathematical words page.

The letter t. Christian Kramp was apparently the first to use the symbol t (Walker, 1929). He used it to stand for (in modern notation) x/ sigma (sqrt 2) where x is a single observation.

Mansfield Merriman used t to stand for the ratio of the limiting error to the probable error. This ratio has some similarities to the statistic that Gossett was to invent under the name of x, but that later became known as t (Tankard, page 94).

For the history of the symbol t for Student's distribution, please see the entry on the math words page.

Part 2 of 2