P. HISTORY OF ' AATHEMATICAL - School of Mathematics

IMAGINARIES AND VECTOR ANALYSIS
133
pendicular to one another, such that i.j= -j.i=k, j.k= -k.j=i,
k.i= -ik=j.
Stringham' denoted cos l3+i sin 13 by "cis 13," a notation used also
by Harkness and Morley," Study'! represented vectorial quantities
by a biplane, or two non-perpendicular planes, the initial plane q"
and the final plane q,'. The biplane is represented by Stf Similarly,
Study defines a "motor" by two non-perpendicular straight lines (~
the initial line, and ID the final), and represents the motor by the
symbol ID1~.
504. Length of vector.-The length of a vector was marked by
Bellavitis AB (the same designation as for vector), by H. G. Grassmann'
VR2, by W. R. Hamilton- TR (i.e., tensor of the vector R),
by R. Gans" \R!, the Weierstrassian symbol for absolute value. Later
Gans discarded IRI because IR[ is a function of R and functional symbols
should precede or follow the entity affected, but IRI does both.
505. Equality of vectors.-L. N. M. Carnot? employed =l= as
signe d'equipollence, practically a sign of identity. "Si les droites AlJ,
CD concurrent au point E, j'ecrirai AB'CD*E," where AB'VJ)
means the point of intersection of the two lines.
To express the equality of.vectors the sign = has been used extensively.
Mobius" had a few followers in employing ==; that symbol
was used by H. G. Grassmann? in 1851. Bellavitis'? adopted the astronomical
sign libra "c:", H. G. Grassmann'! in 1844 wrote #, which was
also chosen by Voigt l 2 for the expression of complete equality. Hamil­
1 Irving Stringham, Uniplanar Algebra (San Francisco, 1893), p. xiii, p. 101.
'J. Harkness and F. Morley, Theory of Analytic Functions (London, 1898),
p. 18, 22, 48, 52, 170.
3 E. Study, Geometric der Dynamen (Leipzig, 1903), p. 30, 51; Encyclop&lie
des scien. math., Tom. IV, Vol. II (1912), p. 55, 59.
4 H. Grassmann, Werke, Vol. .II (Leipzig, 1894), p. 345; Vol. 12 (1896), p.1l8.
6W. R. Hamilton, Cambro and Dublin Math. Jour., Vol. I (1846), p. 2.
6R. Gans, Einjuhrung in die Vector Analysis (Leipzig, 1905), p. 5.
7 L. N. M. Carnot, Geometrie de position (Paris, 1803), p. 83, 84.
8 A. F. Mobius, Der barye. Calcul (1827), § 15; Werke, Vol. I, p. 39.
·Crelle's Journal, Vol. XLII, p. 193-203; Hermann Grassmann's Gesam'TMll6
math. 'Und physik. Werke, Band II (ed. F. Engel; Leipzig, 1904), p. 90.
10 G. Bellavitis, op. cit., p. 243-61.
U Hermann Grassmann, Gesammelte • . • . Werke, Band I; Ausdehnungslehra
(von 1844), p. 67.
12 W. Voigt, N achricht. K. GeseUsch. d. Wissen8ch. s: Giittingen (1904), p. 495­
513.