March 24, 1S99.] 



SCIENCE. 



447 



disposition for later reactions. To be sure, 

 many of these discharges lead finallj' to 

 muscle contractions which bring with them 

 centi-ipetal sensations from the joints, the 

 muscles, the tendons, and these muscle and 

 joint sensations themselves then become a 

 part in the idea, for instance, of time, of 

 space, of feeling. But the new part only 

 reinforces the general tone which is given 

 in the general discharge, and gives to it 

 only the exact detail which gets its charac- 

 ter just through the blending of these sen- 

 sations of completed reactions with the 

 accompaniments of the central discharge. 



A consistent psychology thus may start 

 with the follo-ndng principles : It considers 

 all variations of mental life as variations of 

 the content of consciousness, and this con- 

 tent as a complex object, including in this 

 first presupposition a complicated trans- 

 formation of the real inner life, a transfor- 

 mation by which the subjectifying view of 

 real life is denied for the causal psychological 

 system. Everj^ content of consciovTsness is 

 further considered as a complex of sensa- 

 tions, that is, of possible elements of per- 

 ceptive ideas. Every sensation is con- 

 sidered as having a fourfold manifoldness, 

 varying in kind, in streng-th, in vividness 

 and in value. The physiological basis of 

 every sensation, and thus of everj^ psychical 

 element, is the physical process by which a 

 centripetal stimulation becomes transformed 

 into a centrifugal impulse, the kind depend- 

 ing upon the locality of the centripetal 

 channel, the strength irpon the quantity of 

 the stimulus, the value upon the locality of 

 the centrifugal channel, and the vividness 

 vifton the quantity of the discharge. 



Hugo Mijnsterbeeg. 



Haevaed Univeksity. 



SOPHUS LIE. 

 On the eighteenth of February, 1899, the 

 greatest mathematician in the world, 

 Sophus Lie, died at Christiania in Norway. 



He was essentially a geometer, though 

 applying his splendid powers of space cre- 

 ation to questions of analysis. From Lie 

 comes the idea that every system of geom- 

 etry is characterized by its group. In or- 

 dinary geometry a surface is a locus of 

 points ; in Lie's Kugel-rjeometrie it is the ag- 

 gregate of spheres touching this surface. 

 By a simple correlation of this sphere- 

 geometry with Pluecker's line-geometry, Lie 

 reached results as unexpected as elegant. 

 The transition from this line-geometry to 

 this sphere-geometry was an example of 

 contact-transformations. 



Now contact-transformations find appli- 

 cation in the theory of partial differential 

 equations, whereby this theory is vastly 

 clarified. Old problems were settled as 

 sweepingly as new problems were created 

 and solved. 



Again, with his Tlieorie der Tratisforma- 

 tionsgriqjpen, Lie changed the very face and 

 fashion of modern mathematics. 



A magnificent application of his theory 

 of continuous groups is to the general prob- 

 lem of non-Euclidean geometry as formu- 

 lated by Helmholtz. To this was awarded 

 the great Lobachevski Prize. Not even 

 this award could sufficiently emphasize the 

 epoch-making importance of Lie's work in 

 the evolution of geometry. 



Moreover, the foundations of all philoso- 

 phy are involved. To know the non-Eu- 

 clidean geometry involves abandonment of 

 the position that axioms as to their concrete 

 content are necessities of the inner intui- 

 tion ; likewise abandonment of the position 

 that axioms are derivable from experience 

 alone. 



Lie said that in the whole of modern 

 mathematics the weightiest part is the 

 theory of diff'erential equations, and, true 

 to this conviction, it has always been his 

 aim to deepen and advance this theory. 

 Now it may justly be maintained that in 

 his theory of transformation groups Lie has 



