A paper by Hamilton entitled
On Quaternions; or on a new System of Imaginaries in Algebraappeared in installments in the Philosophical Magazine between 1844 and 1850.
This paper is available in the following formats:
The paper commences by giving the definition of the quaternions. (This is the first published account of the theory). Hamilton discusses the relationship between quaternions and spherical trigonometry. He then discusses the decomposition of quaternions into their scalar and vector parts, and proves an identity which is essentially the same as that for the vector triple product in vector algebra. He uses quaternions to give a proof of a theorem equivalent to the famous theorem of Pascal which states that if a hexagon is inscribed within a conic section, then the points of intersection of the continuations of opposite sides of the hexagon are collinear. He also shows how this theorem of Pascal is related to a theorem of Chasles. The paper continues with a lengthy discussion of the geometry of ellipsoids in three-dimensional space. Hamilton's investigations using quaternions lead to several geometric constructions of such ellipsoids. The `nabla operator' (acting on scalar and vector fields) is later introduced, and the `Laplacian' operator is expressed as the square of the `nabla operator'. The paper concludes with the statement of various theorems concerning `gauche' polygons in surfaces of the second order.