360 



SCIENCE. 



[N. S. Vol. XX. No. 507. 



ing a continuous curve and without going 

 out from the region. 



"2. The points of this new plane are 

 susceptible of transformations called move- 

 ment and which form a group. 



"3. Among these movements, there are 

 an infinity which leave fixed a certain point 

 M and which are called rotations about M. 



"The ensemble of the transformes of the 

 same point A by all these rotations is called 

 a circle. Every circle has an infinity of 

 points. 



"4. The group of movements forms a 

 closed system; which means this: if there 

 are an infinity of movements which change 

 two points Aa and B^ the first into A.^ and 



B., the second into J., and B„ 



the 7ith 



into A and B ; and if the point A„ tends 



n n 



towards A and the point 5„ towards B 

 when n increases indefinitely, there will 

 also be in the group a movement which 

 will exactly change A^ into A and B^ into 

 B; and the same holds if in place of two 

 points we consider three of them or only 

 one. 



' ' I have slightly abridged the statements, 

 making them lose, it is true, a little of their 

 precision, but without taking away any- 

 thing essential. About these enunciations 

 I have certain observations to make. 



"The question in brief is to find all the 

 groups of transformations of the plane into 

 itself, or of a part of the plane into itself, 

 these groups being subjected only to con- 

 ditions in appearance very slightly restrict- 

 ive. How, therefore, can one arrive at 

 conclusions so precise? 



"This results from the definition which 

 Hilbert gives of movement. To be a move- 

 ment, a transformation must satisfy many 

 conditions; first it must be continuous and 

 transform two points infinitely near into 

 two points infinitely near; then it must be 

 biuniform, that is to say, that every point 

 of the plane must have one transforme and 



only one, and be the transforme of one 

 point and of only one. 



' ' By that are found to be excluded a very 

 great number of groups; for example, the 

 group of the projective transformations of 

 the plane and the group of homothetics, 

 that is to say, transformations which 

 change every plane figure into a homothetic 

 figure [a figure similar and similarly 

 placed]. 



"Why? If we take, for example, the 

 homothetic group we see that it contains 

 degenereseent transformations, those where, 

 the center of homothety moreover being 

 any whatever, the ratio of homothety is 

 nul or infinite. In these transformations 

 the center of homothety has an infinity of 

 transformes or is the transforme of ah in- 

 finity of points. These degenereseent trans- 

 formations, without which the group would 

 not be a closed system, can not be excluded, 

 nor any more can they be retained, since 

 they do not satisfy the definition of move- 

 ment. 



"One may see in the same manner that 

 a circle can not contain all the points of 

 the plane, otherwise, among the rotations 

 about the center of this circle, there would 

 be one which would bring to the center a 

 point of the plane, other than the center, 

 so that the center would be the transforme 

 of two points, of this point and of itself. 



"That implies the existence of an in- 

 variant analogous to distance. 



"One sees that the conditions are more 

 restrictive than they seemed. In relation 

 to the ideas of Lie, the progress realized 

 is considerable. Lie supposed that his 

 groups were defined by analytic equations. 



"The hypotheses of Hilbert are much 

 more general. 



"Without doubt this is not yet entirely 

 satisfactory, since if the form of the group 

 is supposed any whatsoever, its matter, that 

 is to say, the plane which undergoes the 

 transformations, is still obliged to be a 



