56 ORIGIN AND SIGNIFICANCE OF 



Meanwhile, we must not forget that all geometrical 

 measurements rest ultimately upon the principle of 

 congruence. We measure the distance between points 

 by applying to them the compass, rule, or chain. We 

 measure angles by bringing the divided circle or theo- 

 dolite to the vertex of the angle. We also determine 

 straight lines by the path of rays of light which in 

 our experience is rectilinear ; but that light travels in 

 shortest lines as long as it continues in a medium of 

 constant refraction would be equally true in space of a 

 different measure of curvature. Thus all our geo- 

 metrical measurements depend on our instruments 

 being really, as we consider them, invariable in form, 

 or at least on their undergoing no other than the small 

 changes we know of, as arising from variation of tem- 

 perature, or from gravity acting differently at different 

 places. 



In measuring, we only employ the best and surest 

 means we know of to determine, what we otherwise are 

 in the habit of making out by sight and touch or by 

 pacing. Here our own body with its organs is the 

 instrument we carry about in space. Now it is the 

 hand, now the leg, that serves for a compass, or the eye 

 turning in all directions is our theodolite for measur- 

 ing arcs and angles in the visual field. 



Every comparative estimate of magnitudes or mea- 

 surement of their spatial relations proceeds therefore 



