P. HISTORY OF ' AATHEMATICAL - School of Mathematics

134 A HISTORY OF MATHEMATICAL NOTATIONS
toni in 1845 wrote, ".... the symbolic equation, D-C=B-A,
may denote that the point D is ordinarily related (in space) to the
point C as B is to A, and may in that view be also expressed by writing
the ordinal analogy, D . .C: :B..A; which admits of inversion and
alternation." Peano! employed the symbolism a-b=c-d.
506. Produds of vectors.-Proceeding to the different products of
vectors, one observes that there has been and is great variety of notations.
W. R. Hamilton assigned to two vectors only one product pp',
which is a quaternion and is the sum of two parts, the scalar of the
quaternion Spp', and the vector part of the quaternion Vpp'.
H. G. Grassmann developed several products, of which the "internal
product" or scalar product (the same as Hamilton's Spp')
and the "external product" or vector product (the same as Hamilton's
Vpp') occupy a central place in vector analysis. Grassmannt represented
in 1846 the scalar product by aXb; in 1862 4 the scalar product
by [ulv] and in 1844 and 1862 the vector product by [uv].
Resa1 6 wrote for the scalar product aXb, as did also Peano" in
1899, and later Burali-Forti and Marcolongo,? At an earlier date,
1888, Peano had written the scalar product ulv. Burali-Forti and
Marcolongo give the vector product in the form u /\ v. The simple
form uv for the scalar product is employed by Somov," Heaviside,?
Foppl," Ferraris." Heaviside at one time followed Hamilton in writing
the vector product, Vuv. Gibbs" represented the scalar product,
called also "dot product," by u.v, and the vector product by uXv;
Gibbs's U.v = - Suv of Hamilton.
1 W. R. Hamilton, Cambro and Dublin Math. Jour., Vol. I (1846), p. 47.
Z G. Peano, Formulaire maihemaiique, Vol. IV (1903), p. 253, 254.
3 H. G. Grassmann. Geometrische Analyse (Leipzig, 1847); Werke. Vol. P
(Leipzig, 1894), p. 345.
• H. Grassmann, Werke, Vol P, p. 345; Vol. 12, p. 56, 112.
6 H. Resal, Traits de cin~matique pure (Paris, 1862).
eG. Peano, Formulaire de mathematiquee, Vol. II (Turin, 1899), p. 156.
7 C. Burali-Forti et R. Marcolongo, Elementi di calcolo vettoriale (Bologna,
1909), p. 31. S~e Encuclopedie des scien. math., Tom. IV, Vol. II, p. 14,22.
8 P. Somov, Vector Analysis and Its Applications (Russian) (St. Petersburg,
1907).
9 O. Heaviside, Electrical Papers, Vol. II (London, 1892), p. 5.
10 Foppl, Geometric der Wirbelfelder (1897).
11 G. Ferraris, Lezioni di elettratecnica,Kap. I (1899).
U J. W. Gibbs's Vector Analysis, by E. B. Wilson (New York, 1902), p, 50.