P. HISTORY OF ' AATHEMATICAL - School of Mathematics

266 A HISTORY OF MATHEMATICAL NOTATIONS
terms vary in two or more different ways, and introduces symbols
To, 0 TI,o T 2, 0 .•••
To, I T I, I T2, I ••••
To, 2 T I, 2 T2, 2· •••
and, correspondingly, Do, 0, DI,o, .... , Do, h DI, h •••• , Do, 2, DI,2,
.... , or generally D m , n for the successive differences.
Laplace' lets the variables t, t l , t 2 , •••• increase respectively by
a, ai, a2, .... , and lets u' be the value of a function of these variables
after they have received their increment. He writes u'-u=
Llu, and uses also the symbols .l2u , LliU , 2:, and 2: i .
641. Later notations.-The symbols d, D, Ll, J', 2: were used by
Arbogast," Francais," and others in the development of functions in
series and in treatment of recurrent series. Kramp's substitution of
the German letters ~ and ® for the Greek Ll and 2: did not meet with
wide acceptance.' These symbols were used sometimes as symbols of
operation (d and D in differentiation, Ll for finite differences, f for
integration, 2: for finite summation), and sometimes as if they were
veritable algebraic quantities. Servois- studied the laws of the calculus
with symbols and showed that Ll, D, 2:, f possessed the propriele
commutative, propriete distributive, expressions which have secured
permanent adoption. Later authors have used a great variety of
symbolism in the calculus of operations, of which the more important
are due to Boole," Carmichael,' and Casorati."
Euler's notation found entrance into England at the time of the
1 (EUVTes de Laplace, Vol. IX (Paris, 1893), p, 316, 318; Memoires de l'acad. r, d.
1ICienc. de Paris, annee 1777 (Paris, 1780).
2 L. F. A. Arbogast, Du oalcul des derivations (Strasbourg, an VIII [1800]),
p. 1-159, 230. See A. Cayley in Philosophical Transactions (London), Vol. CLI
(1861), p. 37. See also Encyclopblie des scien. math., Tom. II, Vol. V, p. 4, 5 (1912).
I J.F. Francais in Annales math. pures appl. (Gergonne), Vol. III (1812-13),
p. 244-72. He uses the rounded letter a in place of d.
4 C. Kramp, RMmens d'arithmetique 'Universelle (Cologne, 1808), "Notations."
IF. J. Servois, Annales math. puree appl. (Gergonne), Vol. V (1814-15), p. 93.
a G. Boole, Philosophical Tra1l8actions, Vol. CXXXIV (London, 1844), p.
225; Math. Analysis oj Logic (Cambridge, 1847), p. 15-19. 7'reatise on Differential
Equations (Cambridge, 1859), p. 371-401.
7 R. Carmichael, TretitilIJ on the Calculm oj Operatio1l8 (London, 1855).
• F. Casorati in Annali di mat. pura edappl. (2d ser.), Vol. X (1880-82), p. 10,
ete.