ON THE PARTITIONS OF A CLOSE. 
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it, or mere contour, is not an edge. It may be added that an edge does not cut or 
meet itself or any other edge except at the summit or summits of the edge itself. 
Consider now a close bounded by /3+1 mere contours: if for any partitioned 
close we have P the number of parts, S the number of summits, E the number of 
edges, B the number of breaks of contour; then, for the unpartitioned close, we have 
P = 1, S — 0, E = 0, B = f3, and therefore 
P + $ + /3 = A + l + P; 
and it is to be shown that this equation holds good in whatever manner the close is 
partitioned. The partitionment is effected by the addition, in any manner, of summits 
and mere contours, and by drawing edges, any edge from a summit to itself or to 
another summit. The effect of adding a summit is first to increase S by unity: if 
the summit added be on a contour, E will be thereby increased by unity; for if the 
contour is a mere contour, it is not an edge, but becomes so by the addition of the 
summit; if it is not a mere contour, but has upon it a summit or summits, the 
addition of the summit will increase by unity the number of edges of the contour. 
If, on the other hand, the summit added be an isolated one, then the addition of 
such summit causes a break of contour, or B is increased by unity. Hence the 
addition of a summit increases by unity S; and it also increases by unity E or else B, 
that is, it leaves the equation undisturbed. The effect of the addition of a mere con 
tour is to increase P by unity, and also to increase B by unity: it is easy to see 
that this is the case, whether the new mere contour does or does not contain within 
it any contour or contours. Hence the addition of a mere contour leaves the equation 
undisturbed. The effect of drawing an edge is first to increase E by unity; if the 
edge is drawn from a summit to itself, or from a summit on a contour to another 
summit on the same contour*, then the effect is also to increase P by unity; if, 
however, the edge is drawn from a summit on a contour to a summit on a different 
contour, then P remains unaltered, but B is diminished by unity. There are a few 
special cases, which, although apparently different, are really included in the two 
preceding ones: thus, if the edge be drawn to connect two isolated summits, these are 
in fact to be considered as summits belonging to two distinct contours, and the like 
when a summit on a contour is joined to an isolated summit. And so if there be 
two or more summits connected together in order, and a new edge is drawn connecting 
the first and last of them, this is the same as when the edge is drawn through two 
summits of the same contour. The effect of drawing a new edge is thus to increase 
E by unity, and also to increase P by unity, or else to diminish B by unity; that 
is, it leaves the equation undisturbed. Hence the equation P+S + /3 = E+1 +P, 
which subsists for the unpartitioned close, continues to subsist in whatever manner the 
close is partitioned, or it is always true. 
In particular, if /3 = 0, that is, if the original close be bounded by a mere contour, 
P + S = E +1 + P; and if, besides, B= 0, then P-\-S = E+ 1, which is the ordinary 
equation in the theory of the partitions of a polygon.