P. HISTORY OF ' AATHEMATICAL - School of Mathematics

270 A HISTORY OF MATHEMATICAL NOTATIONS
tp, and took' 4> as a function tp, z, y, z, .... In 1772 Lagrange- took u
as a function of z. Another time he 3 wrote Clairaut's equation, y-px
+f(p) = 0, "f(P) denotant une fonction quelconque de p seul," and
he gave f'(p) = d~~) . A curious symbol, (J)x, for our f(x) was used by
W. Bolyai.!
645. During the early part of the nineteenth century functional
notations found their way into elementary textbooks; for instance,
into Legendre's Elements of Geometry,5 where a function of p and q
was marked 4>: (p, q).
J. F. W. Herschel 6 uses the signf(x) and says, "In generalf(f(x))
orffx maybe writtenj2(x) .... andf1l'fn(x) =f"'+n(x) , .... andf-l(x)
must denote that quantity whose function f is z."
646. G. Peano! writes Y=f(x) and x =!(Y) , where Jmeans the inverse
function of f. This notation is free from the objection raised
to that of Herschel and Burmann (§533, 645), and to gd-lcJ>, used as
the inverse Gudermannian by. some writers, for instance, by J. M.
Peirce in his Mathematical Tables (1881, p. 42) and D. A. Murray in
his Differential and Integral Calculus (1908, p. 422), but by others
more appropriately marked },,(cJ».
B. SYMBOLS FOR SOME SPECIAL FUNCTIONS
647. Symmetric functions.-Attention was directed in § 558 to
Leibniz' dot notation for symmetric functions. Many writers used no
special symbol for symmetric functions; the elementary functions
were placed equal to the respective coefficients of the given equation,
due attention being paid to algebraic signs. This course was followed
I Loc. cit.; (Euvres, Vol. II (Paris, 1868), p. 40.
2 N. Memoires d. l'acad. r. d. scienc. et bell.-Iett. de Berlin, annee 1772; (Euvres,
Vol. III (Paris, 1869), p. 442.
8 N. Memoires d. l'acad. r. d. scienc. et bell.-lett. de Berlin, annee 1774; (Euore«,
Vol. IV (Paris. 1869), p. 30. See also p. 591.
• Wolfgang Bolyai, Az arithmetioo eleje (Maras.Vasirhelyt, 1830). See B. Boncompagni,
Bullettino, Vol. I (1868), p. 287.
• A. M. Legendre, :2Lements de geometrie (ed, par J. B. Balleroy ... avec notes
... par M. A. L. Marchand; Bruxelles, 1845), p. 188.
a J. F. W. Herschel, Calculus of Finite Differences (Cambridge, 1820), p. 5.
Herschel says that he first explained his notation for inverse functions in the Philosophical
Transactions of London in 1813 in his paper "On a Remarkable Application
of Cotes's Theorem," but that he was anticipated in this notation by Burmann.
7 G. Peano, Lezioni di analiBi injinitesimale, Vol. I (Torino. 1893), p. 8.