P. HISTORY OF ' AATHEMATICAL - School of Mathematics

DIFFERENTIAL CALCULUS
209
1797 edition of Lagrange's Theorie des fonctions analytique as well as
in the first volume of S. F. Lacroix's Traite du Calcul dijJerentiel
et du Calcul integral. Arbogast expresses himself in his Preface as
follows:
"To form the algorithm of derivations, it became necessary to
introduce new signs; I have given this subject particular attention,
being persuaded that the secret of the power of analysis consists in
the happy choice and use of signs, simple and characteristic of the
things which they are to represent. In this regard I have set myself
the following rules: (1) To make the notations as much as possible
analogous to the received notations; (2) Not to introduce notations
which are not needed and which I can replace without confusion by
those already in use; (3) To select very simple ones, yet such that
will exhibit ail the varieties which the different operations require."
His principal symbol is the D, as the sign for "derivation." This
symbol had been previously used by Johann Bernoulli (§§ 528, 560).
During the latter part of the eighteenth century the D had been used
by several authors to represent a finite difference. Arbogast lets
F(a+x) be any function (une fonction quelconque) of the binomial
a+x; one knows, he says, that one can develop that function in a series
proceeding according to the powers of x, viz., a+bx+ 1 ~ 2 z2+ .... ,
where a=Fa. He designates by D the operation upon Fa that yields
b, so that b=DFa, c=DDFa, etc. While Arbogast's symbol D for
our derivative has maintained its place in many books to the present
time, a large variety of satellites to it, which Arbogast introduced,
are now only of antiquarian interest. Placing a dot (p. 2) after the D
gives him D • Fa. to represent DFa • D • a for cases where D • a. is not 1.
He writes (p. 33): "lim au lieu de 1 • 2 •~~ ... m'" On page 308 he
states: "Nous designerons meme a l'avenir les coeficiens differentiels
dq,x d}q,x d ~t t i . bl 2
dx ' 1 • 2 . dz2 ' .... , X e an mvana e, par (Jq,x, ~ q,x, •••• , ce
qui est la meme chose que les derivees Dea, D2q,X, ..•." Some of his
•
signs are exhibited in Figure 123.
On page 330 Arbogast explains d-l, d- 2 , •••• , and D-l, D-2,
.... , as meaning, respectively, dijJerentielles inverses and derivees
inverses. As an incidental rather than a systematic notation, the d-: 1
and Ir:' have maintained themselves to the present day. This nota­