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F. B. Wiley and G. A. Bliss 
which D has an odd number of intersection points, since otherwise 
both end points of D would be on the same side of the line /. 
If p'p'' is such a segment, then it forms with C two simply 
closed rectifiable curves Ci and Co one of which encloses p\ 
and the other p'2, for after its last intersection with p'p'Ahe 
polygon D, and hence p^o, is entirely exterior to the curve Ci. 
If yi — ^2 is greater than e, then the point p'l can be chosen so 
near to pi that its vertical distance to the line I is greater than 
e/2. The altitude of the curve Ci must then be greater than e/2. 
The effective length must at least equal twice the altitude. Thus 
the effective length of Ci is greater than e. Co is handled in like 
manner. 
We now take up the main theorem of this section: 
The interior of a simply closed rectifiable curve. 
X = ,p{t), y = (f ^ t ^ t") 
can be divided by a finite number of segments of straight lines into 
regions each of ivhich has a 'maximum diameter less than an arbi- 
trarily assigned positive constant e. 
If the altitude of any closed curve is greater than e, the effec- 
tive length of either of its two parts after subdivision by a hori- 
zontal line segment, as described in the auxiliary theorem, will 
be less than L-e where L is the length of the curve. 
If the altitude yi—yo of C is greater than e, then the effective 
arc of either Ci or Co will be greater in length than e and the 
effective arc of each will also be less than L-e, where L is the 
perimeter of C as above. 
The curves Ci, C2 may next be subdivided as in the auxiliary 
theorem. If the curve Ci for example, has still an altitude greater 
than e, the two curves into which it is subdivided will have effective 
lengths less than L-2e. By a continuation of this process of 
simultaneous subdivision the interior of C will be subdivided after 
n steps into regions bounded by simply closed rectifiable curves 
whose altitudes are < e or else whose effective lengths are less 
than L-ne. If n ^ L/e — 2 then each subdivision will be in 
altitude, or else have effective length, < 2e. But in the latter 
case also its altitude must be less than e. 
In a similar manner the regions so formed may be subdivided by 
vertical segments into others whose breadths are less than e. 
