On May 3, 2006, Robert Eldon Taylor <email@example.com> wrote:
> In a paper "On Quaternions" > presented by W. R. Hamilton at a meeting of the Royal Irish Academy on > November 11th, 1844, and included in Vol. 3 of the Proceedings of the > Royal Irish Academy, he mentions a desire to form "a distinct > conception, and to find a manageable algebraical expression, of a fourth > proportional to three rectangular lines, when the directions of those > lines were taken into account; as Mr. Warren and Mr. Peacock had shewn > how to conceive and express the fourth proportional to any three lines > having direction, but situated in one common plane." The paper by > Warren referred to is the 1828 "Treatise on the Geometrical > Representations of the Square Roots of Negative Quantities". I cannot > find anything in Peacock's 1847 Algebra about this (but it is seven > hundred pages). > > I am trying to trace the references to Warren and Peacock...
M. J. Crowe's book _A History of Vector Analysis_ mentions these in the first chapter. In addition to the short book, Warren published two papers the next year in Phil. Trans. Royal Soc. 119 (1829):
"Considerations of the objections raised against the geometerical representation of the square roots of negative numbers" (pp. 241-254)
"On the geometerical representations of the power of quantities whose indices involve the square roots of negative numbers" (pp. 334-359)
I could read those on JSTOR. Here is the basic description, as stated on p. 242 of the first paper:
The fundamental principles and definitions which I arrived at were these: that all straight lines drawn in a given plane from a given point, in any direction whatever, are capable of being algebraically represented, both in length and direction; that the addition of such lines (when estimated both in length and direction) must be performed in the same manner as composition of motion in dynamics; and that four such lines are proportionals, both in length and direction, when they are proportionals in length, and the fourth is inclined to the third at the same angle as the second is to the first.
The Peacock reference is apparently to his "Report on the Recent Progress and Present State of Certain Branches of Analysis," in _Report of the British Association for the Advancement of Science_ (1834), 135-352. Crowe says (pp. 14-15) that Hamilton attended the meeting where this paper was presented, and that "Peacock briefly discussed (ibid., p. 228) Argand's book and the papers from Gergonne's _Annales_. Crowe mentions all of those in his Chapter I.