314 



G. H. KNIBBS. 



overthrow them. Riemann's argument therefore can logically lead 

 to nothing more than that we may, for sufficient reasons, regard 

 the objective universe under the form of a specialised region of 

 space, and if evidence were accumulated shewing that that view 

 offered any advantages, a change of the ordinary scheme of inter- 

 pretation might be desirable. The foundations of geometry how- 

 ever, would remain impregnable, and straight lines, planes, and 

 the definitions of deviation would be no less necessary than they 

 are now. 



The special reason adduced for doubting the validity of our 

 fundamental geometrical notions was that researches in respect of 

 the quanta of definite portions of a "manifold" 1 of which tri- 

 dimensional space is assumed to be a somewhat simple form — 

 constituted merely a division of the science of magnitude, in 

 which magnitudes are to be treated as regions in a manifold, not 

 independent of their position. The absence of such researches 

 was alleged to be the reason why the achievements of Lagrange, 

 Pfaff, Jacobi, and Abel for the theory of differential equations 

 remained unfruitful, but outside this, the researches were essential 

 for the adequate discussion of multiform analytic functions (on the 

 manner indicated in the preceding section). The Riemann sur- 

 faces have shewn that every system of points represented by a 

 function, may, with an n-ply extended magnitude, be represented 

 continuously, the manifold passing over continuously into one 

 another, and hence the determination of position in a given 

 manifold is reduced to the determination of quantity, and of 

 position in a manifold of less dimensions, i.e. n - 1, when the 

 original manifold is w-ply extended. 



Riemann's doctrine affirms that to regard a straight line as 

 by the equation 



8s= viSx^ + Syt + Sz*) (37) 



is to constitute the simplest but not the essential type of possible 



1 " Mannigfaltigkeit." 



