Hamilton's Papers on Geometry

Much of Sir William Rowan Hamilton's research in the final two decades of his life was devoted to applying quaternion and vector methods to the study of problems in mechanics and in geometry. He published a number of short papers on the application of quaternions to the study of geometrical problems.

One of the more substantial papers dating from this period is On Symbolical Geometry, which appeared in installments in The Cambridge and Dublin Mathematical Journal in the years 1846-9. This paper presents what is essentially a `coordinate-free' approach to the theory of quaternions, as applied to geometrical problems. Hamilton begins with a discussion of addition and subtraction of directed line segments (via the usual `parallelogram law of addition'). He then introduces `geometrical quotients', which are represented as quotients of directed lines in space, and derives the algebraic laws that such quotients would have to satisfy, to be consistent with certain natural requirements set out in the paper. (Geometrical quotients can be decomposed into their scalar and vector parts, and the algebra of geometrical quotients is isomorphic to the algebra of quaternions.) Hamilton then applies his algebra of geometrical quotients to the study of certain geometrical problems. In particular, there is a detailed study of the property of cyclic cones, and the paper concludes with a geometrical construction of ellipsoids in space. At no point in the paper does Hamilton employ either Cartesian coordinates or trigometry.

Hamilton's last substantial mathematical paper is On Geometrical Nets in Space, which appeared in volume~7 of the Proceedings of the Royal Irish Academy in 1862. Hamilton begins by describing a `quinary calculus' for specifying the location of points of three-dimensional Euclidean space relative to five given points in general position. He then applies this quinary calculus to the study of `geometrical nets', which had earlier been investigated by Möbius in Der Calcul Barycentrische, published in 1827. Starting from five points in general position in three-dimensional space, one can construct a `geometrical net' of points, lines, and planes by successively choosing five points in general position from those already given or constructed, and intersecting the plane through three of these points with the line through the remaining two in order to obtain new points of the net. Hamilton, in his paper On Geometrical Nets in Space, investigates in detail the properties of such a net determined as far as the points, lines and planes of `second construction'. He enumerates and classifies the points obtained, and investigates the geometrical relations of collinear points, where at least four constructed points lie on any constructed line.