**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.

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

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 *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

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 *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 *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.** *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.**

**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 cs^{2} (for cosine) and s^{2} (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 in*Théorie des fonctions analytiques* the symbols *f'x*
and *f''x* are found; in the *Oeuvres,* Vol. X, "which purports
to be a reprint of the 1806 edition, on p. 15, 17, one finds the corresponding
parts given as *f(x), f'(x), f''(x), f'''(x)*" (Cajori vol. 2, page
207).

In 1770 Joseph Louis Lagrange
(1736-1813) wrote *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.*

*History
of Mathematics* that Arbogast introduced this symbol, but it seems he does
not show this symbol in *A History of Mathematical Notations.*)

*D* was used by Arbogast in the same
work, although this symbol had previously been used by Johann Bernoulli (Cajori
vol. 2, page 209). Bernoulli used the symbol in a non-operational sense (Maor,
page 97).

**Partial derivative.** The "curly d" was used in
1770 by Antoine-Nicolas Caritat, Marquis de Condorcet (1743-1794) in
"Memoire sur les Equations aux différence partielles," which was
published in *Histoire de L'Academie Royale des Sciences,* pp. 151-178,
Annee M. DCCLXXIII (1773). On
page 152, Condorcet says:

Dans toute la suite de ce Memoire, *dz* & *z* désigneront ou deux differences partielles de *z,*, dont une par
rapport a *x,* l'autre par rapport a *y,* ou bien *dz* sera une
différentielle totale, & *z* une difference partielle. [Throughout this paper, both *dz* & *z* will either denote two partial differences of *z,* where one of
them is with respect to *x,* and the other, with respect to *y,* or *dz*
and *z* will be employed as symbols of total differential, and of partial
difference, respectively.]

However, the
"curly d" was first used in the form *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 *x* dans la
différence de *u,* & par *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

*d*

differentialia vulgaria, differentialia autem partialia characteristica

denotare.

The
"curly d" symbol is sometimes called the "rounded d" or
"curved d" or *Jacobi's delta.* It corresponds to the cursive
"dey" (equivalent to our *d*) in the Cyrillic alphabet.

**Integral.** Before introducing the integral symbol,
Leibniz wrote omn. for "omnia" in front of the term to be integrated.

The integral symbol was first used by Gottfried Wilhelm Leibniz
(1646-1716) on October 29, 1675, in an unpublished manuscript. 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 *l* = omn. *l*, id est summa ipsorum *l*" [It will be useful
to write *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 *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

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

en sorte que

ou

designent la même chose

lim
(without a period) was written in 1841 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

**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 *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

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 ^{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/

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

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

Cajori (vol. 2, p 44) says the conjecture has been made that Wallis
adopted this symbol from the late Roman symbol *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 *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 *Formulaire
de mathematiqués,* which was published in 1897 (Cajori vol. 2, page 300).

**Membership.** Peano used *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 *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

Cajori, however, shows that
Peano used

**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

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 "

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

**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** ( *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,

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 *Disquisitions arithmeticae*:

Numerorum congruentiam hoc signo,

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

**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, *N _{0}*
for the positive integers and zero,

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 *G*^{0} 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.** *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 *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 *n.*"

Sylvester, who used tau for this function, also believed that Euler used

**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 *M _{n}*
by Allan Cunningham in 1911 in

**Fermat numbers.** Fermat numbers are marked *F _{n}*
in 1919 in L. E. Dickson's

**The norm of a + bi.** Dirichlet used

**Galois field.** Eliakim Hastings Moore used the symbol
*GF*[*q ^{n}*] to represent the Galois field of order

**Sum of the divisors of n.** Euler introduced the symbol

In 1888, James Joseph Sylvester continued the use of Euler's notation *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, *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* + 2*r*)...[*a*
+ (*m*-1)*r*];

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

*a*^{m}^{|r}*.*

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

1^{m}^{|1} = 1 . 2 . 3 ... *m*

ou à cette autre forme plus simple [page 2]

*m*!,

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 * _{n}P_{r}* for the number of
permutations of

G. Chrystal used * _{n}C_{r}* for the number of
combinations of

**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 P* _{B}* (

**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*
= *x** _{j}* },
became popular around 1950. The convention is used in Feller's

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 *Phil. Trans. R. Soc.*
1922 - see below. The most familiar example of the Graeco-Latin convention, *s,*
for an estimate of

**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 *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 *r*_{0} 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

G. Udny Yule introduced the notation *r*_{12.3} for the
partial correlation between *x*_{1} and *x*_{2}
holding *x*_{3} 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 *P.*

**Moments.** Pearson introduced the basic symbol *m*'s in his paper on cumulants (1929).

**Standard deviation and variance.** The use of *Philosophical
Transactions of the Royal Society of London, Ser. A,* 185, 71-110. On page
80, he wrote, " Then ^{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 *n*, not (*n*
- 1). Fisher eventually adopted Student's *s*^{2} (with adjusted *n*)
as an estimate of ^{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 *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).)

**
**

**
**

**Mean of the normal distribution.** *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

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.