I70 THE GRAMMAR OF SCIENCE 



ing this in mind we may now return to the geometrical 

 conceptions of space. 



S 7. — Conceptual Space. Gcojuetrical Boundaries 



It has been remarked (p. 165) that we conceive groups 

 or sense-impressions to be limited by surfaces and lines. 

 We speak of the surface of the table ; the fly-leaf of this 

 book appears to be separated from the air above it by a 

 plane surface and that plane to be bounded at its upper 

 edge by a portion of a straight line. In the first place 

 we have to ask whether our geometrical notions of line 

 and plane correspond to the limits of anything we actually 

 find in perception or whether they are purely ideal limits 

 to processes begun in perception, but which it is impossible 

 to carry to a limit in perception. The answer to these 

 questions lies in the conceptions of sameness and continuity. 

 The geometrical ideas of line and plane involve absolute 

 sameness in all their elements and absolute continuity. 

 Every element of a straight line can in conception be 

 made to fit every other element, and this however it be 

 turned about its terminal points. Every element of a 

 plane can be made to fit every other element, and this 

 without regard to side. Further, every element of a 

 straight line or a plane, however often divided up, is in 

 conception, when magnified up, still an element of straight 

 line or plane. 



The geometrical ideas correspond to absolute sameness 

 and continuity, but do we experience anything like these 

 in our perceptions .'' The fly-leaf of this book appears at 

 first sight a plane surface bounded by a straight line, but 

 a very slight inspection with a magnifying lens shows that 

 the surface has hollows and elevations in it, which quite 

 defy all geometrical definition and scientific treatment. 

 The straight line which seems to bound its edge becomes, 

 under a powerful glass, so torn and jagged that its ups 

 and downs are more like a saw-edge than a straight line. 

 The sameness and continuity are seen to be wanting on 

 more careful investigation. We take a glass cube skil- 



