"If I wished to attract the student of any of these sciences to an algebra for vectors, I should tell him that the fundamental notions of this algebra were exactly those with which he was daily conversant. ...I should call his attention to the fact that Lagrange and Gauss used the notation (αβγ) to denote precisely the same as Hamilton by his S(αβγ) except that Lagrange limited the expression to s, and Gauss to vectors of which the length is the secant of the latitude, and I should show him that we have only to give up these limitations, and the expression (in connection with the notion of geometrical addition) is endowed with an immense wealth of transformations. I should call his attention to the fact that the notation [r_1r_2], universal in the theory of orbits, is identical with Hamilton's V(\rho_1\rho_2) except that Hamilton takes the area as a vector... I confess that one of my objects was to show that a system of vector analysis does not require any support from the notion of the quaternion, or... of the imaginary in algebra."