P. HISTORY OF ' AATHEMATICAL - School of Mathematics

128 A HISTORY OF MATHEMATICAL NOTATIONS
Kastner! was among the first to represent a pure imaginary by a
letter; for V -1 he wrote 'If" where a is an even integer. He speaks of
"den unmoglichen Factor b+fv-1 fur den ich b+f'lf" schreiben will."
Before this he had used 'If" for va, where awas positive."
Caspar Wessel'' in an Essay presented to the Danish Academy
in 1797 designates "by + 1 positive rectilinear unity, by +E another
unity, perpendicular to the first and having the same origin," and
then writes "v=1=E"; "cos V+E sin v."
498. It was Euler who first used the letter i for v=1. He gave
it in a memoir presented in 1777 to the Academy at St. Petersburg,
and entitled "De formulis differentialibus etc.," but it was not published
until 1794 after the death of Euler.'
As far as is now known, the symbol i for V -1 did not again
appear in print for seven years, until 1801. In that year Gauss" began
to make systematic use of it; the example of Gauss was followed in 1808
by Kramp,"
499. Argand,? in his Essai (Paris, 1806), proposes the suppression
of V -1 and the substitution for it of particular signs similar to the
signs of operation + and -. He would write '" for +-v=1 and
for -v-1, both indicating a rotation through 90°, the former
1 A. G. Kastner, AnjangsgrUnde der Analysis endliche:r Grossen (Gottiugen,
1760), p. 133.
sA. G. Kastner, op. cii., p. 117.
a Caspar Wessel, Essai sur la representation analytique de la direction (Copenhague,
1897), p. 9, 10, 12. This edition is a translation from the Danish; the
original publication appeared in 1799 in Vol. V of the Nye Samling aj det Kongelige
Danske Videnskabernes Selskabs Skrifter.
• The article was published in his Institutiones calculi integralis (2d ed.),
Vol. IV (St. Petersburg, 1794), p. 184. See W. W. Beman in Bull. Amer. Math.
Soc., Vol. IV (1897-98), p. 274. See also Encyclopedie des scien. math., Tom. I,
Vol. I (1904), p. 343, n. 60.
I K. F. GaU8S, Disquisitiones arithmeticae (Leipzig, 1801), No. 337; translated
by A. Ch. M. Poullet-Delisle, Reche:rches arithmttiques (Paris, 1807); GaU8S,
We:rke, Vol. I (Gottingen, 1870), p. 414.
I C. Kramp, RUmens d'arithmttique universeUe (Cologne, 1808), "Notations."
7 J. R. Argand, Essai sur une manib-e de representer lea quanliUs imaginaires
dans lea constructions gWm~triques (Paris, 1806; published without the author's
name); a second edition was brought out by G. J. Hoiiel (Paris, 1874); an English
translation by A. S. Hardy appeared in New York in 1881. We quote from Hardy's
edition, p. 35, 45.