P. HISTORY OF ' AATHEMATICAL - School of Mathematics

140 A HISTORY OF MATHEMATICAL NOTATIONS
must be omitted here were proposed in 1914 by J. A. Schouten.'
For example, he introduces ...., and l.- as signs of multiplication of
vectors with each other, and J and L as signs of multiplication of
affinors (including tensors) with each other, and uses fifty or more
multiples of these signs for multiplications, their inverses and derivations.
In the multiplication of geometric magnitudes of still higher
order (Septoren) he employs the sign --- and combinations of it with
others. These symbols have not been adopted by other writers.
A tensor calculus was elaborated in 1901 chiefly by G. Ricci and
T. Levi-Civita, and later by A. Einstein, H. Weyl, and others. There
are various notations. The distance between two world-points very
near to each other, in the theory of relativity, as expressed by Einstein
in 1914 2 is ds 2 = L!7I" dx; dy" where !71" is a symmetric tensor of second
1"
rank (having two subscripts ~ and v) which embraces sixteen products
AI'B, of two covariant vectors (AI') and (B,), where X=Xl, x..=y,
Xs=z, x4=idt. He refers to Minkowski and Laue and represents, as
Laue 3 had done, a symmetric tensor by the black-faced letter P and
its components by P""" Prl/' Prz, PI/r, PI/I/' PI/Z, Pzr, POI/' pzz. A worldtensor
has sixteen components. Laue denotes the symmetric tensor by
the black-faced letter t. Representing vectors by German letters,
either capital or small, Laue marks the vector products of a vector
and tensor by [qp] or [~t].
In writing the expression for ds 2 Einstein in other places dropped
the 2;, for the sake of brevity, the summation being understood in
such a ease.' Pauli uses the symbolism x=xl, y=x 2 , z=z3, u=x4,
writing briefly ,xi, and expressed the formula for the square of the distance
thus, ds 2 = !7ikdxidxk, where gik = gki, it being understood that i
and k, independently of each other, assume the values 1, 2, 3, 4.
Pauli" takes generally the magnitudes aiklm ....rst •••• , in which the indices
assume independently the values 1, 2, 3, 4, and calls them "tensor
components," if they satisfy certain conditions of co-ordinate
1J. A. Schouten, Grundlagen der Vektor- und Ajfinoranalysis (Leipzig, 1914),
p.64, 87, 93, 95,105,111.
2 A. Einstein, Sitzungsberichte der Konigl. P. Akademie der Wissen8ch., Vol.
XXVIII (1914), p. 1033,1036.
• M. Laue, Das RelativiUilsprinzip (Braunschweig, 1911), p. 192, 193.
4 Einstein explains the dropping of 2": in Annalen der Physik, Vol. XLIX (1916),
p. 781. See H. A. Lorentz and Others, The Principle oj Relativity, a Collection oj
Original Memoirs (London, 1923), p. 122.
6 W. Pauli, Jr., in Encyclopddie (Leipzig, 1921), Band V2, Heft 4, Art. V 19,
p.569.