P. HISTORY OF ' AATHEMATICAL - School of Mathematics

232 A HISTORY OF MATHEMATICAL NOTATIONS
The association of x, y, z, etc., with 8, d, D, etc., respectively, makes
unnecessarily heavy demands upon the memory of the reader.
606. A. L. Cauchy.-Cauchy used a variety of different notations
for partial differentiation at different times of his long career. In
1823, in his lessons on the infinitesimal calculus,' he represents partial
differentials by d",u, d llu, d.u, and the partial derivatives by d;:, ~:'
~:'
but remarks that the latter are usually written, for brevity,
du du du .
dx' dy' ds' He writes also the more general partial derivative
tl:=:' f ..· u, but adds that the letters z, v. z at the base of the
y z .
dl+m +...... U
d's are ordinarily omitted, and the notation did ." d n is used.
x y z ....
The same notation, and similar remarks on partial derivatives and
partial differentials, are found in his lessons on the differential calculus'
of 1829.
In England, John Hinds gives the general form of partial derivatives
which we last quoted for Cauchy, and then remarks: "Another
kind of notation attended with some conveniences has of late been
partially adopted. In this, the differential coefficients are no longer
expressed in a fractional form, but are denoted by the letter d with
the principal variables suffixed: thus, ~; and ~; are equivalent
to d",u and d llu," and so that the foregoing partial derivative of the
order l+m+n+etc. is written d~d:d; etc. u. This notation which
Hind describes is that of Cauchy for partial differentials.
De Morgan, when considering partial processes, lets ~; stand for
a partial derivative, and dd~u for a total derivative, the two being
"totally distinct."!
1 Cauchy, Resume des le~ons donnees d l'ecole royale polytechnique sur le calcul
infinitesimal (Paris, 1823); (Euoree (2d ser.), Vol. IV, p. 50, 79.
2 Cauchy, Leeone sur le calcul differentiel (Paris, 1829); (Euoree (2d ser.),
Vol. IV, p. 513, 527.
3 John Hind, Principles of the Differential Calculus (2d ed.; Cambridge, 1831),
p.372.
• Augustus De Morgan, Differential and Integral Calculus (London, 1842),
p.88-91.