Structure of the complex unit i:
i2 = -1
(e1e2)2 = -1
i = e1e2
Wenn ich das Werk, dessen ersten Theil ich hiermit dem Publikum übergebe, als Bearbeitung einer neuen mathematischen Disciplin bezeichne, so kann die Rechtfertigung einer solchen Behauptung nur durch das Werk selbst gegeben werden.
Hermann Grassmann - "Die Ausdehnungslehre" - Vorrede
William Kingdon Clifford
I propose to communicate in a brief form some applications of Grassmann's theory which it seems unlikely that I shall find time to set forth at proper length, though I have waited long for it. Until recently I was unacquainted with the Ausdehnungslehre, and knew only so much of it as is contained in the author's geometrical papers in Crelle's Journal and in Hankel's Lectures on Complex Numbers. I may, perhaps, therefore be permitted to express my profound admiration of that extraordinary work, and my conviction that its principles will exercise a vast influence upon the future of mathematical science.
...Grassmann looked for rules for combining vectors which would fully describe the geometrical properties of directed line segments. He noticed that two directed line segments connected end to end determined a third, which may be regarded as their sum...
...Though several people might be credited with conceiving the idea of "directed number", Hermann Grassmann, in his book of 1844, developed the idea with precision and completeness that far surpassed the work of anyone else at the time...