4. Note on Plane Algebra

By Plane Algebra I mean what De Morgan called Double Algebra. While ordinary algebra deals with quantities which are represented on a straight line, and Quaternions with quantities which are represented in space, Double Algebra deals with those which are represented on a plane. The object of this paper is to show some applications of this intermediate method.

XXXII. On the differential coefficients and determinants of lines, and their application to analytical mechanics

I propose in these pages to prove the principal theorems of dynamics in a manner which appears to me both simpler and more methodical than that in which they are generally proved; and I believe that I shall be able, by applying a few conceptions which spring naturally from the principles of higher algebra and statics, to give a clear interpretation to most of the more complicated formulæ in dynamics, as well as to the several analytical steps which lead to those formulæ. 2. There are many reasons why the diagonal AD, which is constructed on the straight lines AB, AC, should be considered as the sum of those two lines. Those reasons may be found developed in De Morgan’s 'Double Algebra,’ in Warren 'On Imaginary Quantities,’ and in the Tract of Benjamin Gompertz 'On Imaginary Quantities.'