654 
MATHEMATICS: 0. VEBLEN 
tions, and hence less than the inferred great thickness of many atoll 
masses, but it would presumably be sufficient to cause a moderate pre- 
ponderance of submergence on continental coasts which themselves suf- 
fer many diverse movements of upheaval and depression. It is not, 
however, to be supposed that general warpings and deformations of the 
ocean floor, upward and downward, should be left out of consideration; 
such movements have surely taken place to a less or greater^ degree, 
particularly in the western Pacific, where coral reefs border continental 
islands. The integrated effect of all these causes of change in the level 
of the ocean surface cannot now be determined, because so little is known 
regarding the various factors of the problem: but nothing in the little 
that is known and in the much more that may be fairly inferred should 
be regarded as discountenancing the theory of upgrowing reefs on sub- 
siding foundations, essentially as Darwin supposed. His primary theory 
of coral reefs holds good, although his supplementary theory of broad 
ocean-floor subsidence needs modification. 
1 Guppy, H. B., Scot. Geogr. Mag., 14, 1888, (121-137); see p. 135, 136. 
2 Hickson, S. J., A naturalist in Celebes, London, 1889; see p. 42. 
3 Murray, J., Proc. Roy. Sac. Edinh., 10, 1880, (505-518); see p. 516. 
^ Geikie, Sir A., The ancient volcanoes of Great Britain, London, 1897; see vol. 2, p. 470. 
6 Branner, J. C., Amer. J. Sci., 16, 1903, (301-316); see p. 301-303. 
6 Molengraaf, G. A. F.,Proc. k. Akad. Wet. Amsterdam, 19, 1916, (610-627); see p. 619-620. 
^ This statement depends on the fact, certified by chemists, that the withdrawal of 
limestone from solutions in water diminishes the water volume by only a small portion of 
the volume of the withdrawn limestone. 
ON THE DEFORMATION OF AN N-CELL 
By Oswald Veblen 
DEPARTMENT OF MATHEMATICS. PRINCETON UNIVERSITY 
Communicated by E. H. Moore. October 8. 1917 
I propose to prove that any {1 — 1) continuous transformation of an 
n-cell and its boundary into themselves, which leaves all points of the bound- 
ary invariant, is a deformation. 
For the purposes of this proof the w-cell may be taken to be the interior 
of an ;z-dimensional cube. A deformation is a (1 — 1) continuous trans- 
formation Fi which is a member (corresponding to x = 1) of a one-para- 
meter continuous family of (1 — 1) continuous transformations Fx 
(0 ^ X ^1) such that Fo is the identity. It is understood that each 
Fx is a transformation of the ^^-dimensional cube into a set of points of 
an w-dimensional Euclidean space in which the w-dimensional cube is 
situated. 
