GEOMETEICAL AXIOMS. 45 



down instead that the sum of the angles of any 

 triangle is equal to two right angles. These, also, 

 are determinations of quantity. 



Now we may start with this view of space, accord- 

 ing to which the position of a point may be deter- 

 mined by measurements in relation to any given 

 figure (system of co-ordinates), taken as fixed, and 

 then inquire what are the special characteristics of our 

 space as manifested in the measurements that have 

 to be made, and how it differs from other extended 

 quantities of like variety. This path was first entered 

 by one too early lost to science, B. Eiemann of Gfott- 

 ingen. 1 It has the peculiar advantage that all its 

 operations consist in pure calculation of quantities, 

 which quite obviates the danger of habitual percep- 

 tions being taken for necessities of thought. 



The number of measurements necessary to give the 

 position of a point, is equal to the number of dimensions 

 of the space in question. In a line the distance from one 

 fixed point is sufficient, that is to say, one quantity ; 

 in a surface the distances from two fixed points must 

 be given ; in space, the distances from three ; or we 

 require, as on the earth, longitude, latitude, and height 

 above the sea, or, as is usual in analytical geometry, 

 the distances from three co-ordinate planes. Eiemann 



1 Ueber die Hypothesen welche der Geometrie zu Grunde liegen, 

 Habilitationsschrift vom 10 Juni 1854. (Abhandl. der TtonigL 

 GeselhcJi. zu Gottingen, Bd. XIII.) 



