PRINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 249 



represent elements of that world in geometrical form, that 

 geometry becomes applicable, and even then not to the thing, but 

 merely to its representation. 1 The notion that had we a larger 

 opportunity of experiment, the question as to the nature of space 

 itself, might (supposititiously) be determined, fails to recognise 

 that unoccupied space must 2 physically be regarded as perfectly 

 homaloidal and homogeneous, and that observed physical facts 

 must necessarily be interpreted ultimately on that basis, as being 

 essentially the most simple; for an explanation which assumed 

 changes in the curvature and intensity of space would be essentially, 

 though not formally, identical with one that assumed these to be 

 due to variations in the sether in space. Negative parallax for 

 distant stars would involve a modification of our conception of the 

 content of stellar space; i.e. it would involve a physical and not 

 a geometrical explanation, 3 although an analytical geometry might 

 be developed which could formally represent the essential features 

 of the sethereal modifications. 



Among conceptions that have been regarded as throwing new 

 light on the theory of space Niemann's notion of an n-ply extended 

 magnitude 4 has been assigned a prominent place. 5 But Riemann 



1 Hence essentially, geometry is independent of the physical point of 

 view, and of physical interpretations, though of course both the original 

 representation and final interpretation may be false. For example, we 

 may elect to regard the sun as a fixed point, and a comet as a point mov- 

 ing in say a parabolic path about the sun as focus. With the very same 

 system of celestial coordinates, the sun would appear to the comet (the 

 latter being regarded as the fixed point) to move about it in the same 

 curve and in the same direction, the coordinates however differing 180°. 

 Hence the curve-is parabolic in either sense, but may, in reference to some 

 other fixed point, be quite different. 



2 For to suppose otherwise would require the support of some con- 

 siderations, justifying the substitution of the complex for the simple idea. 



3 Lobatchewsky's deduction that the negative parallax of some stars 

 could possibly throw light on the nature of abstract space is therefore, I 

 think wrongly founded. 



4 [1826-1866]. Ueber die Hypothesen welche der Geometrie zu Grunde 

 legen. Abh. d. k. Ges. a. Wiss. Gottingen, Bd. xin., 1868. Annali di 

 Math, in., pp. 309, 326, 1869-70. Nature, vin., pp. 14-17, 36-37, 1873. 



5 By Helmholtz, Clifford, Chrystal, and others. I need hardly say that 

 anyone who takes the trouble to acquaint himself with the part played 

 by the Kiemann's surfaces, as developed for example in the very fine 

 treatises of Whitehead, Forsyth, and Harkness and Morley (Universal 

 Algebra, and Theory of Functions and Analytic Functions) is not likely 

 to lightly esteem the significance of Biemann's work. 



