Jl-NK. 1011 



KNOWLEDGE. 



215 



The image produced by the eye being invariably a 

 plane figure, we should be unable to determine 

 whether a bodv were two or three dimensioned if 

 neither \\e nor the body sur\'eyed could move 

 relativeh- to one another. 



As it is we are enabled to survey a sphere, a cube, 

 a tree, a house, or an\- other object from innumerable 

 points of view w ithin our three-dimensioned space 

 and thus to inspect its surface or surfaces and satisf\- 

 ourseh'es that it is indeed a solid ; our opinion is 

 formed from the multitudinous images impressed on 

 the retina of the eye as we move about from point to 

 point or awail <uirselves of the motion of the body 

 which serves to displa\- its various parts before our- 

 selves, the observers. 



Now just as we mistake solid figures for plane 

 ones, so in a like manner when we survey a four, 

 five or infinitelv dimensioned figure we should 

 imagine it to be three-dimensioned ; for it would 

 completely satisfv the investigations we could make, 

 bv looking at it from all [jossible points of view 

 within our three-dimensioned space : of course, its 

 shape as regards the fourth and higher dimensions 

 would remain inscrutable to us, so that we could not 

 possibK- foretell their existence or non-existence : Init 

 thev might be there all the same, for we could 

 prove nothing to the contrary 



Xow proceeding briefl\" to summarise the con- 

 clusions attained we ma\' state that : — 



(1) Algebraical equations of one. two and three 

 variables can be graphically represented : and certain 

 of these equations with two or three variables 

 are the laws of certain plane and solid figures. 

 In particular x'^ -(- y ■^ = r'^ represents a circle and 

 \'-{-\'~-{-z^^=r'^ represents a sphere. If in the 

 last-mentioned equation we put z = o. we obtain the 

 first equation which can be shown graphicalh' to 

 represent a section of the sphere. 



There are an infinite number of equations in 

 this series, and it we take the next higlier one 

 x''^-|-y^ + z"+u''^ = r" and put u = o, we obtain the 

 equation of the sphere: in other words the sphere 

 is a section of this four-dimensioned figure : we do 

 not, however, mean to impK' that it cannot also be 

 the section of other four-dimensioned figures, since 

 obviouslv it can be, preciseK' in the same wa\' that a 

 circle might be the section of a c\linder. cone, or 

 other solid. 



The four-dimensioned figure is the section of a 

 five-dimensioned one, and this progression is 

 continued to infinitx', the infinitely dimensioneif 

 figure being the onlv one that is not a section of a 

 higher dimensioned figure. In spite, however, of 

 the necessary existence of these complex figures we 

 are not acquainted with anv of them from the four- 

 dimensioned one upwards. 



(2) All our ideas of figures cannot be more than 

 three-dimensioned because of the limitations in our 

 powers of observation. 



(3) There are no one or two-dimensioned bodies : 

 all bodies appear to be three-dimensioned, and two- 

 dimensioned figures are onl\- found in realitv as 



sections of these bodies. This is in accordance with 

 the algebraical equations from which it can be 

 deduced that the straight line is the section of a 

 circle or other two-dimensioned figure, and the 

 latter a section of a three-dimensioned figure such as 

 the sphere. 



(4) That the sphere is apparent!}- not a section of 

 a four-dimensioned figure, but appears to belong to 

 that highest t}-pe of figure which extends in all 

 possible dimensions (supposed to be three in 

 number*, is a fact that is not in accordance with 

 what we should expect from the algebraical 

 etiuations : but it can be brought into agreement 

 therewith by allowing for our undoubted inability 

 to appreciate more than three dimensions, together 

 with the certainty that any four or higher 

 dimensioned figures would appear three-dimensioned 

 to ourselves. 



(5) Paragraphs 3 and 4 explain wh}- all visible 

 bodies appear to possess three dimensions and 

 neither more nor less, and since there are no one 

 or two dimensioned bodies, there is no justifiable 

 reason wh\' there should be an\' three. foLn\ or 

 finitely dimensioned ones ; on the other hand, all 

 bodies must be infinitelv dimensioned, although we, 

 by reason of our personal limiiations cannot perceive 

 more than three of these dimensions. 



(6l Olniously space itself, the ether and other 

 bodies, must also be infinitely dimensioned. 



Now from any body such as one w liose algebraical 

 equation is x" + \'-+ . . . (=o terms in alll:=r-, we can 

 take a section. ol>taining thus a figure whose equation 

 is x-+y-4- . . . (oo— 1 terms in all)=r-, and then a 

 section of this, and so on until we obtain the four- 

 dimensioned figure \-+\-^ + z-+u- = r-, the sphere 

 x^+y-+z- = r- and the circle x-+y- = r-. Con- 

 versely, starting with the circle, we can sa\- it is the 

 section of a sphere, a cone, a cvlinder, or some 

 other solid figure : and similarh- the sphere is the 

 section of a four-dimensioned figure which mav 

 belong to the spherical t\"pe, the conical t\pe, the 

 cylindrical type, or some other. 



Our purpose now is to take the sjihere. cone, and 

 c}'linder and show how they would appear if we 

 were onl\- two-dimensioned, and how we could 

 distinguish one from the other : and then, arguing 

 bv analogy to show how the four-dimensioned 

 figures of the spherical, conical, and cylindrical 

 t\-pes would similarh- appear to us with our three- 

 dimensioned faculties, and how the effects produced 

 bv them would be characteristic and distinguishable 

 from one another. 



The simplest way to proceed is to hypothecate 

 the existence of a two-dimensioned being, one who 

 can onlv percei\-e length and breadth and is 

 absoluteh- ignorant of the existence of thickness : 

 his observations must be confined to w-hat takes 

 place on a plane surface which, consisting of infinite 

 length and breadth, will be to him what space 

 is to us. 



The onl\- figures of whose existence he can 

 possibly be aware must be two-dimensioned ones, 



