50 OKIGIN AND SIGNIFICANCE OF 



conclusions, 1 partly from seeking to represent in space 

 the system of colours, involving the comparison of one 

 threefold extended aggregate with another, and partly 

 from inquiries on the origin of our ocular measure for 

 distances in the field of vision. Biemann starts by 

 assuming the above-mentioned algebraical expression 

 which represents in the most general form the distance 

 between two infinitely near points, and deduces there- 

 from, the conditions of mobility of rigid figures. I, on 

 the other hand, starting from the observed fact that 

 the movement of rigid figures is possible in our space, 

 with the degree of freedom that we know, deduce the 

 necessity of the algebraic expression taken by Riemann 

 as an axiom. The assumptions that I had to make as 

 the basis of the calculation were the following. 



First, to make algebraical treatment at all possible, 

 it must be assumed that the position of any point A 

 can be determined, in relation to certain given figures 

 taken as fixed bases, by measurement of some kind of 

 magnitudes, as lines, angles between lines, angles 

 between surfaces, and so forth. The measurements 

 necessary for determining the position of A are known 

 as its co-ordinates. In general, the number of co- 

 ordinates necessary for the complete determination of 

 the position of a point, marks the number of the dirnen- 



1 Ueber die Thatsachen die der Geometric zum Grunde liegen 

 (Nachrichtenvonder konigl. Ges. d. Wiss.zu Gottingen, Juni 3, 1868). 



