568 GEOMETRY 



formazioni geometriche delle figure piane, published in 1863. Within 

 a few years, Clifford, Noether, and Rosanes, working independently, 

 established the remarkable result that every Cremona transforma- 

 tion in a plane can be decomposed into a succession of quadratic 

 transformations, thus bringing to light the fact that there are at 

 bottom only two types of algebraic one-to-one correspondence, the 

 homographic and the quadratic. 1 



The development of a corresponding theory in space has been one 

 of the chief aims of the geometers of Italy, Germany, and England 

 for the last thirty years, but the essential question of decomposition 

 still remains unanswered. Is it possible to reduce the general Cremona 

 transformation of space to a finite number of fundamental types ? 



In its application to the study of the properties of algebraic 

 curves and surfaces, the theory of the Cremona transformation 

 is usually merged in the more general theory of the birational trans- 

 formation. By means of the latter, a correspondence is established 

 which is one-to-one for the points of the particular figure considered 

 and the transformed figure, but not for all the points of space. In 

 the plane theory an important result is that a curve with the most 

 complicated singularities can, by means of Cremona transformations. 

 be converted into a curve whose only singularities are multiple 

 points with distinct tangents (Noether); furthermore, by means of 

 birational transformations, the singularities may be reduced to the 

 very simplest type, ordinary double points (Bertini). The known 

 theory of space curves is also, in this aspect, quite complete. The 

 analogous problem of the reduction of higher singularities of a sur- 

 face has been considered by Noether, Del Pezzo, Segre, Kobb, and 

 others, but no ultimate conclusion has yet been obtained. 



One principal source of difficulty is that, while in case of two 

 birationally equivalent curves the correspondence is one-to-one 

 without exception, on the other hand, in the case of two surfaces, 

 there may be isolated points which correspond to curves, and just 

 such irregular phenomena escape the ordinary methods. Again, 

 not only singular points require consideration, as is the case in the 

 plane theory, but also singular lines, and the points may be isolated 

 or superimposed on the lines. Most success is to be expected from 

 further application of the method of projection from a higher space 

 due to Clifford and Veronese. In this direction the most important 

 result hitherto obtained is the theorem, of Picard and Simart, that 

 any algebraic surface (in ordinary space) can be regarded as the 

 projection of a surface free from singularities situated in five-dimen- 

 sional space. 



1 Segre recently called attention to a case where the usual methods of discus- 

 sion fail to apply; the proof has been completed by Castelnuovo. Cf. Atti di 

 Torino, vol. xxxvi (1901). 



