376 
F. B. Wiley and G. A. Bliss 
applying his method of proof to the triangle to which Jordaid 
reduces the problem. 
Our purpose in this paper is to accomplish the same result for a 
simply closed curve by applying a subdivision theorem due to 
Bliss" which takes the place of the Jordon reduction to a triangle. 
We start with a simply closed rectifiable curve C which is 
entirely interior to a region in which the integrand function /( 2 ) 
is continuous and has at each point a unique derivative. Our 
method consists in showing in the first place, in §1, that the region 
enclosed by the curve can always be subdivided into a finite 
number of regions^ each of which can be surrounded by a rectangle 
in which f(z) satisfies the above hypothesis. Then in §2 it is 
shown that Cauchy’s theorem holds for each of these rectangles 
and consequently ( §3) the Cauchy theorem holds for any simply 
closed rectifiable curve in each rectangle. It readily follows that 
(§4) the Cauchy theorem holds for our original curve C. 
§1. A Method of Subdividing the Intekior of a Simply 
Closed Rectifiable Curve 
In this section we show, after stating certain preliminary 
theorems, first, by means of an auxilliary theorem and then a 
main theorem, how the interior of a simply closed rectifiable 
curve C may be subdivided into regions each of which has its 
maximum diameter less than an arbitrarily assigned constant e ; 
and second, for a curve C lying in a simply connected continuum, 
that the number of subdivisions necessary to permit each sub- 
region to be surrounded by a rectangle lying wholly in the con- 
tinuum is finite — a formula for the maximum number being exhib- 
ited. It is understood that by a rectifiable curve is meant a 
continuous curve with length. 
We are presupposing the following theorems.^ Any simply 
closed rectifiable curve C in an xy-plane divides the plane into two 
continua, an exterior and a finite interior. Any two interior 
^ Loc. cit. 
^ Bliss, Princeton Colloquium, Part I, p. 29. 
® Osgood, Lehrhuch der Funktionen Theorie, chapt. V.; Bliss, A Proof of the 
Fundamental Theorem of Analysis Situs, Bulletin of the American Mathematical 
Society, Vol. 12 (1905-06), p. 336; also Brouwer, Beweis des Jordanschen Kur- 
vensatzes, Mathematische Annalen, vol. 69 (1910), p. 169. 
