Hamilton published a paper entitled
On Symbolical Geometry.which appeared in installments in the first four volumes of the Cambridge and Dublin Mathematical Journal for the years 1846-1849.
This paper is available in the following formats:
In this paper, Hamilton develops an algebra of directed lines in space, avoiding completely the use of Cartesian coordinates or trigonometry. The `geometrical quotients' introduced in this paper correspond to the quaternions. Hamilton applies the principles of his `symbolical geometry' to obtain coordinate-free proofs of certain geometrical theorems relating to cones and surfaces of the second degree.