214 



KNOWLEDGE. 



JrxK. l')ll. 



that there is no cause \vh\- four and other dimen- 

 sioned figures should not exist : from this it follows 

 that the only figures which are not the sections 

 of higher dimensioned ones must be those possess- 

 ing an infinite number of dimensions : in other 

 words, these last mentioned figures are the only ones 

 that can have "an individual existence. 



The first point for us to determine is the 

 possession bv ever\' visible body of at least three 

 dimensions : we shall then proceed to shov\- that 

 the\' must be inferred to possess an infinite number 

 of such dimensions: and finalh" to examine the 

 effects produced by certain four-dimensioned figures, 

 firstly when mo\ing in the direction of the axis of 

 the fourth dimension and secondlv when expanding 

 and contracting in the three-dimensions with which 

 we are familiar. 



Now all visible bodies must necessaril\- have 

 dimension of some kind. 



The impressions of things external to our mind 

 are received and transmitted to it by means of the 

 senses and their organs : and we must consequenth- 

 remain utterly ignorant cif anything that does not 

 appeal to at least one of our senses. 



We find that all sense impressions conveved to us 

 have their origin in objects that jiossess extension in 

 space. Colour, scent, light, sound, taste, heat and 

 cold, all are in\"ariabl\' found to proceed from such a 

 body : and conversely there is no visible bod\- that 

 has not extension in space, a possession which is 

 indeed its primar\- attribute, all others such as 

 colour, taste, and so forth lieing of serondar\- 

 importance. 



When we proceed to consider those things that 

 ha\e dimension, we find that some onl\- of them 

 have an independent existence and therefore comph- 

 with our definition of a body. Others, which are 

 figures only, and nothing more, do not exist apart 

 but are only attributes of a bod\'. Thus we find 

 that a solid such as a cube is an actual object, but a 

 surface such as the square merelv forms part of the 

 solid : we do not find one existing alone, absoluteh' 

 dispossessed of thickness. 



The straight line is nothing more than part of the 

 boundary of a plane figure : and so far is it from 

 possessing an independent objective existence that it 

 is even unimaginable : for geometrical purposes we 

 have to allow it some degree of thickness, although 

 we thereb}' convert it into a plane surface. 



The solid, then, is a bod\-, because in addition to 

 its length and breadth it possesses thickness : whilst 

 lines, having merelv one dimension, can only serve 

 for the construction of a plane figure, and the plane 

 figure by itself, having only two dimensions, is no 

 more an object than the line : it is an abstraction that 

 serves as an attribute of a three-dimensioned hod\". 



Now all visible bodies are found to have three 

 dimensions; there is no l)ody, either artificial or 

 natural, that has not length, breadth and thickness. 

 Our next step is to enquire wh\- this is so. On the 

 one hand it ma\' be because there are no one or two 



dimensioned bodies; and, on the other, because we 

 ourselves, on account of some inherent limitations in 

 our power of sight, are unable to perceive them. 

 This last hypothesis, however, is obviouslv untrue, 

 for not only do we see that part of a bod\- which 

 possesses only length and breadth, that is to say its 

 surface, but that is all we can see. 



It is. therefore, obvious that we could detect the 

 presence of any plane surface, such as a square, if 

 such a figure ever had an independent existence. 



Let us imagine a square of this nature to come 

 within the sphere of our observation. What charac- 

 teristic impression would it produce on us? 



Well, suppose that we first of all observed it when 

 looking in a direction at right angles to the plane in 

 which it lies: it would then appear as a square. 

 Now let us mo\'e as though we were about to pass 

 b\- its upstanding edge on one side, in order to attain 

 a position at its rear. With our eyes directed on it 

 the square would apparently' diminish in width, but 

 not in height, becoming narrower and narrower, 

 until nothing but a small strip remained; and even 

 this would finally disappear when one reached the 

 edge of the figure: the square would be completeh' 

 invisible, since it has no thickness. Then, as we 

 passed be\-ond its edge and towards the rear, the 

 surface would reappear, and increase in apparent 

 width, until it regained its full size and shape. 



Now such a phenomenon as this has never been 

 seen, and it therefore follows that there are no 

 bodies with onl\' two dimensions : and further that 

 there cannot be an\' with simpK' one. for if there 

 were, a two-duuensioned bod\' could be constructed 

 therefrom. 



In the course n\ the foregoing arguments it was 

 remarked that we could only percei\e the surface of 

 a bodv : and that, therefore, if we looked at a plane 

 surface, we only observed its length and breadth, 

 the thickness remaining unperceived and unde- 

 termined. Our next point of inquiry will endeavour 

 to explain why we only see the plane, and how it 

 is that we can ascertain the existence of the third 

 dimension. 



The e\'e is. of course, the organ of sight ; it 

 comprises a lens and a screen called the retina. 

 This screen consists of a surface, and is analogous 

 to the sheet on which magic lantern pictures are 

 displa\'ed, the eye lens being equivalent to the 

 compound lens of the lantern. 



Now. it is impossible to produce anything but a 

 flat picture on the lantern sheet ; the image of a 

 globe, for instance, cannot be projected so as to 

 possess length, breadth, and thickness ; the length 

 and breadth can be displayed, because the screen 

 itself is a plane surface, but the thickness can only 

 be simulated b\' a skilful distribution of light and 

 shade. 



Similarh' the image impressed on the retina is 

 essentialK' two-dimensioned ; and for this reason we 

 are frequenth" led to mistake a round body for a flat 

 one ; the moon, for instance, alwa\s appears to be 

 fiat, 



