216 



KXOWLEDCxE. 



Junk. 1911. 



and these only if they lie in his plane of observation. 

 Such figures would be presented when a solid was 

 intersected by the plane surface. Now we can 

 imagine a solid to move downwards towards the 

 surface from above and to pass completel\- through 

 it. Before and after the passage of the solid, no 

 part of it would be seen b\- the being, but during the 

 event, he would observe a series of sections, each 

 consisting of a plane figure. 



Now these sections, of course. W(.)uld be definiteh- 

 correlated to one another, since one and all belong 

 to the same bodv. and from their change or 

 constancy in size and shape, it would be possible for 

 the being to determine the nature of the solid figure. 

 Thus from the fact that he saw a circle of unvarying 

 diameter, he could hvpothecate the existence of a 

 cylinder : and similarly, if the circle were one that 

 appeared to increase or decrease in diameter, it could 

 be ascribed to the existence of a right circular 

 cone, sphere, or some other solid figure whose cross 

 section was a circle. 



Furthermore, the particular kind of solid could be 

 deduced, for the\" wciuld each produce a different and 

 characteristic change in the size of the circle. A 

 sphere, for instance, would produce a circle at first no 

 larger than a point, but which would increase 

 gradually until it became a great circle of the sphere 

 and then decrease again to a point. A right circular 

 cone, assuming it to approach apex first, and parallel 

 with the axis of the third dimension, would also 

 first produce a circle no larger than a point : this 

 ^\■ould also gradually grow, but it would not subse- 

 quentJN' decrease like that produced b\- the sphere. 

 Furthermore, provided that both cone and sphere 

 moved downwards with uniform velocitw there would 

 also be this difference — that any point on the circular 

 section of the cone would mo\-e radialh- outwards 

 with a uniform velocit\', whilst an\' point on the 

 similar section of the sphere would move outwards 

 with variable velocity : and its acceleration w ould be 

 just as characteristic of the sphere as the uniform 

 velocity was of the cone. Any other figure would 

 produce a characteristic acceleration. 



Now just as the sphere and cone moving down- 

 wards along the axis of the third dimension pre- 

 sented the phenomenon of a plane: figure (the circle) 

 growing equally in the two dimensions of the plane, 

 so would the spherical figure of tlie fourth dimension 

 (equation x''^+y-+z'^+u"=r^), and the cone-like figure 

 of the same degree, each moving downwards 

 along the axis of the fourth dimension with a uniform 

 velocity, present to us the phenomenon of a solid 

 figure (the sphere) growing equally in the three 

 dimensions of space : in the first instance with 

 characteristic acceleration, and in the second with 

 uniform velocity; and just as it is possible to 

 account for any imaginable change in the size and 

 shape of a plane figure b_\' the hypothesis that it 

 forms part of a solid figure moving downwards along 

 the axis of the third dimension, so similarh- is it 

 possible to explain any increase or decrease in the 

 length, breadth and thickness of a solid figure h\' the 



hvpothesis that it forms part of a four-dimensioned 

 figure moving downwards along the axis of the 

 fourth dimension. 



It is likewise [)ossible to explain the propagation 

 of ether and other vibrations, originating from a 

 point, in the same way ; for we know that such 

 vibrations, starting from the point, advance at a 

 uniform rate in all directions, so that the wave front 

 of each \'ihration consists of a sphere constantl}- 

 growing outwards at a uniform velocitv, the precise 

 phenomena that would be produced bv a downward 

 movement along the fourth dimension of some four- 

 dimensioned cone-like figure. 



There is this objection, however, to such a theor\-, 

 that if ether is infinitely dimensioned, there is no 

 reason wh\- the \'ihrations should not extend equally 

 in all of them, so that the shape of each wave front 

 would be represented by the formula x-+y^+ . . . (oo 

 terms in all) = r"-^, and it is impossible to presuppose 

 this figure to be the section of any other figure or 

 for it to be caused bv motion along a dimension not 

 included in the equation, for all the dimensions are 

 included therein. 



There is. however, another point of view from 

 which to regard the propagation of these vibrations. 



It is known that before the discharge of an electric 

 spark, the ether in the neighbourhood is in a state of 

 tension; it might be regarded as being attracted 

 towards the point where the electrical charge is 

 collected. 



Now suppose for the moment that we regard the 

 ether as only three-dimensioned, and ourselves to be 

 only capable of appreciating two dimensions; and let 

 the charged point lie in the plane of our obser\-ation. 

 The ether will Ix' attracted to the point from all 

 directions, and wc may imagine the space about the 

 point divided up int(.) an infinite number of con- 

 tiguous cones of ether, all with their apices at the 

 charged point; in the normal state of affairs, with 

 no electrical charge in the neighbourhood, these 

 cones would become cylindrical tubes, and the conical 

 shajje is therefore always an unstable one, tending to 

 re\ert to the cvlindrical form as soon as the point is 

 discharged. It is obvious that in undergoing this 

 change of form on the discharge taking jjlace, all we 

 should appreciate would be an ether disturbance 

 travelling outwards from the point in the form of an 

 e\er-increasing circle; for we could only observe the 

 lateral expansion of the cone whose axis was perpen- 

 dicular to our plane of observation; all the other cones 

 in their expansion would immediately extend above 

 or below our plane, and thus become unobservable. 



Now similarly, since we observe an electrical dis- 

 turbance to travel outwards in an ever-growing 

 sphere, we may imagine the ether in the neighbour- 

 hood of the charged point to be composed of a series 

 of four-dimensioned, cone-like figures ; or assuming 

 that the disturbance travels outwards in as many 

 dimensions as possible, to be composed of infinitel\' 

 dimensioned cone-like figures, each of which has for 

 cross section a figure whose equation is x-+y-+ . . . 

 (oo — 1 terms in all)=r-, 



