312] 
ON THE PARTITIONS OF A CLOSE. 
63 
Several definitions and explanations are required. The words line and curve are 
used indifferently to denote any path which can be described currente calcuno without 
lifting the pen from the paper. A closed curve, not cutting or meeting itself (*), is 
called a contour. An enclosed space, such that no part of it is shut out from any 
other part of it, or, what is the same thing, such that any part can be joined with 
any other part by a line not cutting the boundary, is termed a close. The boundary 
of a close may be considered as the limit of a single contour, or of two or more 
contours lying wholly within the close. The reason for speaking of a limit will appear 
by an example. Consider a circle, and within it, but wholly detached from it, a 
figure of eight; the space interior to the circle but exterior to the figure of eight is 
a close: its boundary may be considered as the limit of two contours,—the first of 
them interior to the close, and indefinitely near the circle (in this case we might say 
the circle itself); the second of them an hour-glass-shaped curve, interior to the close 
(that is, exterior to the figure of eight) and indefinitely near to the figure of eight. 
The figure of eight, as being a curve which cuts itself, is not a contour; and in the 
case in question we could not have said that the boundary of the close consisted of 
two contours. A similar instance is afforded by a circle having within it two circles 
exterior to each other, but connected by a line not cutting or meeting itself; or even 
two points, or, as they may be called, summits, connected by a line not cutting or 
meeting itself; or, again, a single summit: in each of these cases the boundary of 
the close may be considered as the limit of two contours. But this explanation once 
given, we may for shortness speak of the close as bounded by a single contour, or by 
two or more contours ; and I shall throughout do so, instead of using the more precise 
expression of the boundary being the limit of a contour, or of two or more contours. 
The excess above unity of the number of the contours which form the boundary of 
a close is the break of contour for such close; in the case of a close bounded by a 
single contour, the break of contour is zero. 
Any point whatever on a curve may be considered as the point of meeting of 
two curves, or, in the case of a closed curve, as the point where the curve meets 
itself, but it is not of necessity so considered. A point where a curve cuts or meets 
itself or any other curve, is a summit; each point of termination of an unclosed 
curve is also a summit; any isolated point may be taken to be a summit. It follows 
that, in the case of a closed curve not cutting or meeting itself (that is, a contour), 
any point or points on the curve may be taken to be summits ; but the contour need 
not have upon it any summit: it is in this case termed a mere contour. The curve 
which is the path from a summit to itself, or to any other summit, is an edge: the 
former case is that of a contour having upon it a single summit, the latter that of 
an edge having, that is terminated by, two summits, and no more. It is hardly 
necessary to remark that a contour having upon it two or more summits consists of 
the same number of edges, and, by what precedes, a contour having upon it a single 
summit is an edge; but it is to be noted that a contour without any summit upon 
1 It is hardly necessary to add, except in so far as any point whatever of the curve may be considered 
as a point where the curve meets itself.