Hamilton published a paper entitled
On Geometrical Nets in Space.which appeared in the Proceedings of the Royal Irish Academy, volume 7 (1862), pp. 532-582.
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In this paper, Hamilton introduces a quinary calculus, in which the position of a point in (three-dimensional) space relative to five chosen points in general position is described by quintuples of real numbers. Hamilton uses this quinary calculus to study the geometric net of points, lines and planes, obtained from those five points by successively intersecting planes passing through three of the given or subsequently constructed points with lines passing through two such points.