Listing's theorem, (established in his Memoir, Die Census räumlicher Gestalten), is a generalisation of Euler's theorem S + F = E + 2, which connects the number of summits, faces, and edges in a polyhedron; viz. in Listing's theorem we have for a figure of any sort whateve

or, what is the same thing,

where

in which theorem a relates to the points; b, k′ relate to the lines; c, k″, π to the surfaces; d, k‴ to the spaces; and prelates to the detached parts of the figure, as will be explained.

ais the number of points; there is no question of multiplicity, but a point is always a single point. A point is either detached or situate on a line or surface.

bis the number of lines (straight or curved). A line is always finite, and if not reentrant there must be at each extremity a point: no attention is paid to cusps, inflexions, &c., and if the line cut itself there must be at each intersection a point; and in general a point placed on a line constitutes a termination or boundary of the line. Thus a line is either an oval (that is, a non–intersecting closed curve of any form whatever), a punctate oval (oval with a single point upon it), or a biterminal (line terminated by two distinct points). For instance, a figure of eight is taken to be two punctate ovals; an oval, placing upon it two points, is thereby changed into two biterminals.