"'(7). The foregoing view, although not precisely the same with that adopted by Euclid himself, in his exposition of the elements of geometry, is at least consistent therewith; and has been made the basis of an important and modern method of calculation, respecting directed lines in one plane, which seems to have been first introduced about the commencement of the present century, by Argand in France, and for which Professor De Morgan... has lately proposed the name of Double Algebra: because it recognises and employs two numerical elements (such as the numbers 1 and 60 in the foregoing example), as required for the joint determination of the length and direction of a straight line. And it is now to be shown what is the nature of the passage that has been made, by the author of the Lectures on Quaternions, from such a double system of algebraic geometry, to what may be called, by analogy and contrast, a quadruple system of calculations respecting directed lines, or a system of QUADRUPLE ALGEBRA."