PROFESSOR AT HEIDELBERG 263 



angular sum from those of small superficies ; but the results 

 of geometrical and astronomical measurements which give 

 the sum of the angles of a triangle only approximately, and 

 never exactly, as two right angles, only justify us in concluding 

 that the measure of our spatial curvature is exceedingly 

 small; that it actually vanishes is not to be proven, it is 

 an axiom. 



In an interesting passage in the lecture Helmholtz describes 

 how we can picture to ourselves the appearance of a pseudo- 

 spherical world extending in all directions, and hence the 

 axioms of our geometry can in no wise be founded on the 

 given form of our capacity of intuition. Beltrami had con- 

 structed a pseudo-spherical space within a sphere of Euclidean 

 space so that every straightest line and planest surface of the 

 former was represented by a straight line and plane surface 

 in the latter: Helmholtz makes it probable by similar con- 

 siderations that if our eyes were provided with suitable convex 

 glasses, pseudo-spherical space would no longer appear very 

 singular to us, and it would only be at the outset that we should 

 be deceived in our estimation of the size and distance of 

 remote objects. 



In April, 1869, Beltrami contributed an interesting letter to 

 the discussion, pointing out Helmholtz's error as above, and 

 Helmholtz lost no time in correcting his statement in a com- 

 munication to the Scientific Society at Heidelberg. 



Helmholtz, like all philosophers and scientific men at the 

 beginning of the Nineteenth Century, was profoundly exercised 

 by epistemological questions : ' What is true in our ideas and 

 conceptions ? How far do our notions correspond with reality ? ' 

 These and kindred problems relating to the theory of know- 

 ledge were the logical outcome of such work as the preceding, 

 although not explicitly developed till a later period. 



A note found among Helmholtz's papers gives us in this 

 connexion a slight but highly interesting 'Analysis of Knowledge 

 as we actually have it ' : 



' The content, (i) Sensations are the only direct and pure 

 perceptions. (2) Our conceptual images of external individual 

 objects are the aggregate of a large number of different 

 conceptions. (3) The concept of an object present expressly 

 includes the assurance that with suitable conditions of observa- 



