Subdividing the Interior of a Rectifiable Curve 379 
If in the above discussion we replace ehy e/V 2, the theorem 
follows at once since each sub-region lies in a square of side eV 2 
and hence has a diameter less than e. 
The number of line segments necessary for the subdivision of the 
region in the interior of C into regions of diameters less than e is not 
greater than 
To develop- this formula we note, by continuing the process 
indicated in the main theorem, that 2'" — 1 line segments give 2'' 
regions each with altitude less than e or else with an effective 
arc less than L — ne. By taking n greater than L/e we see that 
any arc with altitude still greater than e would necessarily have 
effective length less than zero, which is impossible. This gives 
2^-^ as a maximum for the number of regions necessary for the 
subdivision of C into sub-regions with altitudes less than e. In 
a like manner we show that 2^^^ is a maximum for the number 
of sub-regions into which each of the regions just obtained needs 
to be divided to insure that the breadth of each will be less than 
e. It follows that 4^^^ is a maximum for the number of sub- 
regions into which it is necessary to divide the interior of C so 
that the length and breadth of each region will be less than e, 
while jg ^ maximum for the number of sub-regions into 
which it is necessary to divide the interior of C to insure that 
the maximum diameter of each region be less than e, and this 
can be accomplished by drawing not more than 4 ^b 2/6 ggg, 
ments. 
The continuation of the subdividing process until the maximum 
diameter of each region is at most e, where e is taken less than the 
minimum distance from C to the boundary of a continuum in the 
interior of which the curve is supposed to lie, insures that each 
region may be surrounded by a rectangle lying wholly in the 
continuum and establishes the proof of the corollary we now state. 
If our given curve C lies in a simply connected continuum and if e 
is taken less than the minimum distance from C to the boundary of 
the continuum, then 4^kve maximum for the number of line 
segments necessary to divide the region in the interior of C into siib- 
regions each of which may he surrounded by a rectangle lying ivholly 
in the continuum. 
