P. HISTORY OF ' AATHEMATICAL - School of Mathematics

268 A HISTORY OF MATHEMATICAL NOTATIONS
function, for example, that ~ be a function of x. If there are many
functions of that same x, they may be distinguished by numbers.
Occasionally I mark the sign of relationship in this manner x1!J, X]1J,
etc., that is, any expression in x whatever; again, if one is formed from
many, as from x and y, I write X7Yl!j, ~. And when the form is
rational, I add r, as in X1.!:..JJ and X]r.21 or ~, ~. If the
form is rational integral, I write X]ri .11, X]ri. 21. But in the case of
only one function, or only a few of them, the Greek letters suffice, or
some such, as you are using." This notation to indicate functions
Leibniz never had printed, nor did he refer to it again in his letters to
Johann Bernoulli or to other correspondents. On the other hand, the
notation proposed by Johann Bernoulli was used by his brother Jakob
Bernoullil in 1701, who without use of the word "function" defined
B, F, G, C as functions, respectively of b, I, g, C by equations such as
adF=hdl or F=v(aa+ff), where a is a constant. This mode of
representation was employed extensively during the eighteenth century
and not infrequently has been found convenient also in more
recent times.
643. The use of a special symbol to stand for a function of a given
variable is found again in a memoir- of Johann Bernoulli of 1718, where
the Greek l{J is employed for this purpose. He stated: "en prenant
aussi l{J pour la caracteristique de ces Ionctions." He also writes (p. 246)
the differential equation dy:dz= (l{Jx±c) :a, and further on (p, 250),
dy:dx= (l{Jz±c) :a. It is to be observed that neither Johann Bernoulli
nor his successors Clairaut and D'Alembert in their earlier writings
inclose the variable or argument in parentheses." The use of parentheses
for this purpose occurs in Euler' in 1734, who says, "Si I(~ +C)
denotet functionem quamcunque ipsius ~ -l-c," This is the first apa
pearance of I for "function." About the same time Clairaut designates
a function of x by IIx, x, or ~x, without using parentheses," In
1 Jakob Bernoulli in Analysin magni problematis isoperimetrici (Basel, 1701);
Jacobi Bernoulli, Opera, Vol. II (Geneva, 1744), p. 907, 909.
2 Memoires d. l'acad. d. sciences de Paris (1718), p. 108; Opera (Lausanne et
Geneve, 1742), Vol. II, p. 243, 246, 250.
3 G. Ensstrom, BibliotheCfL mathematica (3d ser.), Vol. VI (1905), p. 111.
• L. Euler in Comment. Petropol. ad annos 1734-1735, Vol. VII (1840), p.
18o, 187, second paging; reference taken from J. Tropfke, op. cit., Vol. II (2d ed.,
1921), p. 35.
5 A. C. Clairaut in Histoire d. l'acad. d. sciences, annee 1734 (Paris, 1736),
su«, p. 197.