P. HISTORY OF ' AATHEMATICAL - School of Mathematics

THEORY OF FUNCTIONS 267
introduction of the Leibnizian notation of the calculus, in the early
part of the nineteenth century. Euler's symbols are used by J. F. W.
HerscheP and later by George Boole. 2 Boole says (p. 16, 17): "In
addition to the symbol A, we shall introduce a symbol E to denote the
operation of giving to x in a proposed subject function the increment
unity;-its definition being E ux = Ux+l." "The two symbols are con­
nected by the equation E=1+A."
SYMBOLS I~ THE THEORY OF FUNCTIONS
A. SYMBOLS FOR FUNCTIONS IN GENERAL
642. In 1694 Johann Bernoulli" represented by the letter n any
function whatever of z ("posito n esse quantitatem quomodocunque
formatam ex indeterminatis et constantibus") and then proceeded
to consider the integral of ndz. In an article! of the following year,
Jakob (James) Bernoulli proposed a problem in which he lets p and q
be any two functions whatever of x ("... pet q quantitates utcunque
datas per x denotant"). In 1697 Johann Bernoulli' let the capital
letter X and also the corresponding Greek letter ~ represent functions
of x ("per ~ intelligo quantitatem datam per x et constantes"; "X
quantitati itidem ex x et constantibus compositae"). He writes also
"fX=quantitati pure dependenti ab x et constantibus." In a letter
to Leibniz he said:" "For'denoting any function of a variable quantity
x, I rather prefer to use the capital letter having the same name X
or the Greek ~, for it appears at once of what variable it is a function;
this relieves the memory." To this Leibniz replied:" "You do excellently
in letting the mark of a function indicate of what letter it is the
I J. F. W. Herschel, Calculus of Finite Differences (Cambridge, 1820), p. 1,4.
2 George Boole, Calculus of Finite Differences (3d ed.; London, 1880), p. 2, 4, 5.
[Preface to 1st ed. is dated April 18, 1860.]
I Johann Bernoulli in Acta eruditorum (Leipzig, 1694), p. 437. I am indebted
for this reference and a few others to G. Enestrom's article in Bibliotheca mathematica
(2d ser.), Vol. V (1891), p. 89.
4 Jakob Bernoulli in Acta eruditorum (1695), p, 553.
• Johann Bernoulli, in Acta eruditorum (1697), p, 115.
f Leibnizens MathematiBche Schriften, Vol. III (1856), p. 531. Letter dated
Aug. 16/26, 1698.
7 Letter of Leibniz to Johann Bernoulli, n.d., in Leibnizens Mathern. Schri!ten,
Vol. III (1856), p. 537. It should be stated that Leibniz' letter, as printed in the
Commercium Philosophicum et Mathematicum of G. W. Leibnis and Johann Bernoulli,
Vol. I (1745), p. 399, has some slight variations in the symbolism; thus,
instead of X1.!:JJ we find X]r:11. Here, and in the other cases given in the letter,
a colon is printed where in Gerhardt's edition there is simply a dot.