SECTION 23. GENERALISATION. 347 



group to which the peacock belongs." (Frank Cramer, op. cit., 

 pp. 58-59.) "The first thing to be established in proof of the 

 derivation of climbing plants from non-climbers was the exis- 

 tence of gradations in the power of climbing, and the inter- 

 mediate stages between the different methods of climbing 

 by twining of the stem, by leaf-stalks, and by tendrils." (Ibid., 

 p. 166.) 



175. Lastly. The dimensional theories appear to diminish 

 in persuasiveness when the measure of a high degree of exacti- 

 tude in analysis is applied. A point, a line, and a plane, as 

 limiting notions, are permissible concepts; but from the stand- 

 point of objective reality we are bound to assume that they 

 convey no distinct meaning. For instance, since, by definition, 

 a point does not, and a line does, occupy space, no number of 

 successive points however great, could form a line however 

 small. Similarly, since a line, by definition, is said to have no 

 width, an infinite number of juxtaposed lines could never form 

 a plane however limited; and since, by definition, a plane has 

 no depth, me superposition of any number of planes however 

 multitudinous, could not form a solid however thin. The three 

 sets of dimensions, as conceived separately, have therefore, it 

 seems, no relation whatever one to another. 



From a different standpoint, the same criticism applies if a 

 .sufficiently high degree of exactitude is employed. Any actual 

 point must have three dimensions, and so must any actual line 

 and plane. A one or two-dimensioned being is apparently a 

 mere logical or verbal figment. So-called plane beings resolve 

 themselves into three-dimensional beings whom we, for theoreti- 

 cal purposes or owing to an insufficient degree of clarity of 

 thought, regard as two-dimensional. .For the same reason, when 

 we speak of four-dimensional beings, we are almost certainly 

 carried away by the verbal decomposition of solids into three 

 parts. As a matter of fact, every conceivable object is a solid, 

 no more and no less. It is as if, misled by a useful division, 

 men reasoned that we could imagine the existence of a single 

 horizontal direction east, or of horizontal directions, additional 

 to east, west, north, and south, and their intermediates. 



Or to illustrate the matter differently. Conscious lines floating 

 in space and coming into contact or collision with other lines, 

 would become aware of the existence of those other lines. If 

 these lines collided with planes or cubes, they would be con- 

 scious of them, but only in so far as lines. So with planes, 

 colliding with lines and cubes. The planes would recognise the 

 lines as lines, and the cubes in so far as planes. Accordingly, 

 a four-dimensional being coming into collision with a three- 

 dimensional being would be recognised by the latter so far as 

 three out of the four dimensions are in question. The four- 

 dimensional being cannot consequently live more than one 

 fourth outside the three-dimensional world. 



