MATHEMATICS: O. VEBLEN 
655 
The theorem is easy in the one-dimensional case. It has been proved 
in the two-dimensional case by H. Tietze {Palermo, Rend., Circ. Mat., 
38, 1914, p. 247) and more simply, by H. L. Smith (in an article soon to 
appear in the Annals of Mathematics). No proof has been published 
so far as I am aware, for the higher cases. 
The proofs by Tietze and Smith establish a stronger theorem than 
that-stated above, for they show the existence of a family of transforma- 
tions Fx each of which carries the square into itself and leaves all points 
of the boundary invariant. This restriction on the transformations 
Fx, that each of them shall carry the square into itself, is not needed in 
some of the important applications of the theorem; and without this 
restriction the theorem can be proved very easily. 
The proof below is stated for the two-dimensional, but applies without 
change to the w-dimensional, case. 
. Let Si be a square, A BCD, whose sides are of length unity, it being 
understood that a square, unlike a cell, includes its boundary. Let T2 
be a translation parallel to the side AB which carries the side AD into 
the side BC, T3 a translation parallel to the side BC which carries the 
side AB into the side DC, and Ti the resultant of T2 and T3. Let ^2, 
S3, 5*4 be the squares into which Si is carried by T2, Ts, respectively. 
Thus ^i, S%, S3, and ^4 together constitute a square whose sides are of 
length 2. 
Let be a (1 — 1) continuous transformation of Si into itself which 
leaves all points of the boundary of Si invariant. The transformation 
T2F1 (the resultant of Fi followed by ^2) carries Si into S2. I shall first 
show that T2F1 is a deformation and it then follows easily that Fi is 
also a deformation. 
The rectangle composed of Si and 6*2 can be carried into the rectangle 
composed of S3 and S^ by a transformation A which for points of Si, is 
the same as T3 and for points of S2, is the same as TJ^ 1-^X2'^. Since 
T3 and TaFi~^T2~^ have the same effect on the common points of the 
boundaries of Si and 6*2, the transformation A is uniquely defined, (1 — 1), 
and continuous. 
The transformation K.TiFi.K^^, as applied to ^3 is the same as 
TaFi~^T2~^T2FiT3~^, which is the translation T^Ts'^ = T2, carrying S3 
into 54; denote this translation T2 by Ti. Let Tx (0 ^ x ^1) denote the 
translation carrying S3 a distance x in the direction of translation of Ti. 
The existence of the family of translations Tx (0 ^ x ^1) shows that 
Ti is a deformation. But since A.T2F1.A-' = Ti, A'^TiA = T2F1. Hence 
the existence of the set of transformations A^^r^A (0 ^ x ^ 1) shows 
that T2F1 is a deformation. 
