,,the      ,,mystery ,,of ,,space       
,the domain of the senses1 in nature1 is
 almost      infinitely small in        
comparison with the vast      region    
accessible to thought which lies beyond 
them4--,,tyndall1 ,on ,radiant ,heat4   
,,the      ,,mystery ,,of ,,space       
,a ,study of the ,hyperspace ,movement  
in the ,light of the ,evolution of      
,new ,psychic ,faculties      and       
,an ,inquiry into the ,genesis and      
,essential ,nature of ,space        ,,by
      ,,robert ,t4 ,,browne             
,,new ,,york      ,e4 ,p4 ,,dutton      
,,company      #fha ,,fifth ,,avenue    
,,copyright #aiai1 ,,by      ,e4 ,p4    
,,dutton  ,,company        ,all ,rights 
,reserved          ,printed in the      
,united ,states of ,america             
,,to      ,,the ,,cherished ,,memory    
,,of        ,mylie ,de ,pr              
,,whose ,,wifely ,,devotion1            
,,sympathetic ,,assistance ,,and        
,,encouragement      ,,during ,,the     
,,early ,,labors ,,on ,,the ,,text      

,,were ,a ,,constant ,,source ,,of      
,,inspiration ,,and ,,fortitude ,,to    
,,the ,,author1      ,,this ,,book ,,is 
,,dedicated      ,,affectionately       
,,preface      ,mathematics is the      
biometer of intellectual evolution4     
,hence1 the  determination of the status
 quo of the intellect at any time can be
  accomplished most satisfactorily by   
applying to it the rigorous measure  of 
the mathematical method4 ,the intellect 
has but one true divining  rod and that 
is mathematics4 ,by day and by night it 
points the way  unerringly1 so long as  
it leads through materiality2 but1      
falteringly1  blindly1 fatally1 when    
that way veers into the territory of    
vitality  and spirituality4             
,wherefore1 when we have wished to      
ascertain the real status of the        
intellect1 as well as its limitations1  
tendencies1 possibilities1 we  have     
turned to its conduct in the field of   
mathesis where it is least  trammeled in
 its ingressive and egressive           

motivations because of the  natural and 
easy accommodation which is offered for 
intellectual  movement4 ,whether there  
are signs of moribundity or symptomatic 
evidences of marked growth or of a      
termination of intellectual  regnancy1  
or whatever may be the occasion for the 
examination1 no surer  index than the   
mathematical may be found for the       
purpose4 ,full logical  justification   
is1 therefore1 claimed for the choice of
 mathematical  evidences to test the    
assumption that a new era of conscious  
mental  conquest is opening for the     
vanguard of humanity and sequently for  
the  human family4    ,the treatment of 
these evidences has fallen logically    
into two  divisions1 namely1 the first1 
a brief and elementary review of the    
principles of the non-,euclidean        
geometry and their bearings upon the    
question of space as the subject of     
mathematical study2 and second1  the    
consideration of space as a             
psychological1 vital and dynamic or     

creative phenomenon4    ,in ,part ,i an 
effort has been made to trace the growth
 of the notion  of hyperspace and to    
show that it is a symbol of a new epoch 
of  intellectual expansion1 an actual   
seizure of a new domain of awareness  by
 the mind4 ,and for this purpose a      
critical examination of the  fundamental
 question of dimensionality is entered  
upon from which it  develops that the   
status of this primary norm of          
mathematical thought  exhibits a        
relatively inchoate character because of
 its insufficiency  as a definitive     
quantity1 and further1 because of its   
rather  superfoetated aspect when       
utilized as a panacea for mathetic      
symptoms4  ,also1 it has been found     
necessary to survey the field of the    
four-space  which has been accorded such
 a prominent place in the mathematical  
thought of the day4 ,the reader should  
find in the chapter devoted to  it      
adequate material for thought and       
sufficient comprehension of its         

meanings as a mathematical contrivance4 
,in ,part ,,ii an endeavor is made to   
interpret the evidences offered by  high
 mathematic considerations in the light 
of the larger psychogenetic  movement4  
,for this undertaking the quality of    
awareness has been  studied with the    
view to establishing its priority as a  
determining  agency in the consideration
 of space in aspects broader and less   
restricted than those embraced in the   
mathematical premises4 ,wherefore1  it  
appears that there are massive          
implications arising out of the         
hyperspace propaganda which have        
hitherto been neglected in the          
preliminaries incident to the           
fabrication of its structure4 ,a very   
brief1 and more or less symbolic1 sketch
 of the genesis of space has  served to 
demonstrate its essential nature as a   
consubstantive of  materiality1 vitality
 and intellectuality1 the three major   
objective  processes4 ,consequently it  
becomes imperatively necessary that any 

view of space which neglects its kosmic 
intent and purpose should be  regarded  
as gravely fragmentary and insufficient4
 ,it is only by linking  up the two     
aspects of space1 the mathematical and  
the psychological1 in  such a manner    
that the one shall supplement the other1
 that we shall be  able to arrive at a  
truly satisfactory understanding of its 
nature4    ,in ,chapter ,,ix attention  
is invited to some of the extremities of
  mathematical laws wherein it is shown 
that1 because mathematical goods  are   
strictly of intellectual texture and    
fabric1 vain is the hope of  reaching   
any reliable certainty with respect to  
many vital questions1  even regarding   
space itself1 by means of the           
mathematical method4 ,the  intellect1   
and1 therefore1 mathematics encounter   
the most formidable  stricture when     
effort is made to maneuver in the field 
of vitality  or realism4 ,in addition1  
it is shown that1 when pushed to the    
utmost  logical limits1 metageometry    

proves not only futile1 but emphasizes  
the  need for a sharp turning of the    
path of search from the intellectual  or
 material to the spiritual or           
intuitional4 ,indeed1 it becomes        
painfully certain that the ,golden      
,fleece of profounder knowledge will  be
 discovered never by an expedition whose
 bark has its sails set for  the winds  
of mathematical seas4 ,but1 contrarily1 
a new bark1 moored at  the furthermost  
shore of the sea of intellectuality with
 sails set for  the winds which come    
from the realm of intuitional           
perception1 must be  seized4 ,whereupon1
 by the straightest line1 we shall1 at  
the last1 land  upon the shore of       
realism1 of truth all inclusive4        
,mathematical evidences have been used  
in these discussions because  they1 of  
all lines of knowledge1 afford a more   
just exemplification  of intellectual   
evolution4 ,the science of mathematics  
is the measure  of the quality of       
intellectual growth and1 therefore1 its 

data1 its  postulates1 hypotheses and   
advances clearly mark the stages of the 
intellectual movement4    ,chapter ,x is
 the natural and logical sequence of the
 inquiry into the  question of          
spatiality4 ,the conclusions reached    
therein and the obvious  inferences     
which should be drawn from the arguments
 presented flow  inevitably not only    
from the evidences of mathematical data 
but of the  common observations of life4
 ,and while we disclaim any intention of
  demanding acceptance of them as final1
 authoritative declarations1 we  shall  
be satisfied if the readers of this     
volume be incited to solve  for         
themselves the problems which these     
queries naturally suggest4  ,happy      
indeed shall be the outcome if there be 
any who1 following the  path sketched   
herein1 shall find the solution of the  
,mystery of  ,space and apply its       
meanings to the enhancement of the      
values of the  intuitive life4    ,in   
conclusion1 the author esteems it a     

special obligation of gratitude  that he
 should here acknowledge the debt which 
he owes to all of  his friends who have 
in any way assisted or encouraged him in
 the  completion of this work4 ,among   
those whom he is permitted to thank in  
this way is ,mr4 ,,james ,,rindfleisch  
who1 having very kindly prepared the    
illustrations for the photo-engravers1  
is deserving of special mention4        
,,robert ,t4 ,,browne4     ,new ,york   
,city1 #aiai4            ,,contents     
,,page      ,,preface                   
vii      ,,introduction3 ,explanatory   
,notes                           x      
,,part ,,one          ,,chapter ,i      
,,the ,,prologue      ,on the           
,variability of ,psychic ,powers--,the  
,discovery of    the ,fourth ,dimension 
,marks a ,distinct ,stage in            
,psychogenesis--,the non-,methodical    
,character of    ,discoveries--,the     
,three ,periods of ,psychogenetic       
,development--,the ,scope and           
,permissibility of    ,mathetic         

,license--,kosmic ,unitariness          
,underlying    ,diversity               
#bc          ,,chapter ,,ii             
,,historical ,,sketch ,,of ,,the        
,,hyperspace ,,movement      ,egypt the 
,birthplace of ,geometry--,precursors3  
,nasir-,eddin1    ,christoph ,clavius1  
,saccheri1 ,lambert1 ,la ,grange1       
,kant--,influence of the ,mecanique     
,analytique--,the ,parallel             
,postulate the ,root and ,substance of  
the ,non-,euclidean    ,geometry--,the  
,three ,great ,periods3 ,the ,formative1
    ,determinative and                  
,elaborative--,riemann and the          
,properties    of ,analytic ,spaces     
#dd          ,,chapter ,,iii            
,,essentials ,,of ,,the                 
,,non-,,euclidean ,,geometry      ,the  
,non-,euclidean ,geometry ,concerned    
with ,conceptual    ,space              
,entirely--,outcome of ,failures at     
,solving    the                         
,parallel-,postulate--,the ,basis of the
 ,non-,euclidean    ,geometry--,space   

,curvature and ,manifoldness--,some     
,elements of the ,non-,euclidean        
,geometry--,certainty1    ,necessity and
 ,universality as ,bulwarks of          
,geometry--,some ,consequences of       
,efforts at ,solving    the             
,parallel-,postulate--,the ,final ,issue
 of the    ,non-,euclidean              
,geometry--,extended ,consciousness     
#fi          ,,chapter ,,iv             
,,dimensionality      ,arbitrary        
,character of ,dimensionality--,various 
,definitions    of ,dimension--,real    
,space and ,geometric ,space            
,differentiated--,the ,finity of        
,space--,difference ,between    the     
,purely ,formal and the ,actual--,space 
as ,dynamic    ,appearance--,the ,a     
,priori and the ,a ,posteriori as       
,defined by ,paul ,carus                
#ib          ,,chapter ,v          ,,the
 ,,fourth ,,dimension      ,the ,ideal  
and the ,representative ,nature of      
,objects in    the ,sensible            
,world--,the ,fluxional1 the ,basis of  

,mental ,differences--,natural ,symbols 
and ,artificial    ,symbols--,use of    
,analogies to ,prove the ,existence of  
a ,fourth ,dimension--,the ,generation  
of a ,hypercube    or                   
,tesseract--,possibilities in the ,world
 of the    ,fourth ,dimension--,some    
,logical ,difficulties ,inhering    in  
the ,four-,space ,conception--,the      
,fallacy of the    ,plane-,rotation     
,hypothesis--,c4 ,h4 ,hinton and        
,major ,ellis on the ,fourth ,dimension 
#aah            ,,part ,,two            
,,spatiality3 ,,an ,,inquiry ,,into     
,,the ,,essential ,,nature        ,,of  
,,space ,,as ,,distinguished ,,from     
,,the ,,mathematical                    
,,interpretation          ,,chapter ,,vi
          ,,consciousness ,,the ,,norm  
,,of ,,space ,,determinations           
,realism ,is ,determined by             
,awareness--,succession of ,degrees     
of ,realism--,sufficiency of            
,tridimensionality--,the                
,insufficiency of ,self-consistency as a

 ,norm of    ,truth--,general ,forward  
,movement in the ,evolution    of       
,consciousness ,implied in the          
,hyperspace ,concept--,the              
,hypothetical ,nature of ,our           
,knowledge--,hyperspace    the ,symbol  
of a ,more ,extensive ,realm of         
,awareness--,variations in the ,methods 
of ,interpreting    ,intellectual       
,notions--,the ,tuitional and the       
,intuitional ,faculties--,the           
,illusionary ,character of    the       
,phenomenal--,consciousness and the     
,degrees of    ,realism                 
#afa          ,,chapter ,,vii           
,,the ,,genesis ,,and ,,nature ,,of     
,,space      ,symbology of ,mathematical
 ,knowledge--,manifestation    and      
,non-manifestation ,defined--,the       
,pyknon and    ,pyknosis--,the ,kosmic  
,engenderment of ,space--,on    the     
,consubstantiality of ,spatiality1      
,intellectuality1    ,materiality1      
,vitality and ,kosmic                   
,geometrism--,chaos-,theos-,kosmos--,cha

ogeny and    ,chaomorphogeny--,n4       
,malebranche on ,god and the            
,world--,the ,space-,mind--,space and   
,mind ,are    ,one--,the ,kosmic        
,pentoglyph                             
#bjc          ,,chapter ,,viii          
,,the ,,mystery ,,of ,,space      ,the  
,thinker and the ,ego--,increscent      
,automatism of    the ,intellect--,the  
,egopsyche and the ,omnipsyche--,kosmic 
,order or ,geometrism--,life as         
,engendering    ,element--,the ,mystery 
of ,space ,stated--,kathekos    and     
,kathekotic ,consciousness--,function of
 the ,ideal--,the    ,path of ,search   
for an ,understanding of the ,nature and
    ,extent of ,space ,must ,proceed in 
an ,inverse ,direction    #bdb          
,,chapter ,,ix                          
,,metageometrical ,,near-,,truths       
,realism ,is ,psychological and         
,vital--,the ,impermanence of           
,facts--,on the ,tendency of the        
,intellect to ,fragmentate--,the        
,intellect and ,logic--,the ,passage of 

,space--,kosmometer    and ,zoometer1   
,instruments for the ,measurement of the
    ,passage of ,space and the ,flow of 
,life--,the ,disposal of    ,life and   
the ,power to ,create--,space1 a        
,dynamic1 ,creative                     
,process--,numbers and                  
,kosmogony--,kosmic ,significance of    
the ,circle and the                     
,pi-proportion--,mechanical ,tendence   
of the ,intellect and ,its ,inaptitude  
for the ,understanding    of ,life--,the
 ,criterion of ,truth                   
#bhd          ,,chapter ,x              
,,media ,,of ,,new ,,perceptive         
,,faculties      ,the ,spiritualization 
of ,matter ,is the ,end of              
,evolution--,sequence    and ,design in 
the ,evolution of ,human                
,faculties--,the    ,upspringing        
,intuition--,evidences of ,supernormal  
,powers    of ,perception and the       
,possibility of ,attainment--,the       
,influence and ,place of the ,pituitary 
,body and the ,pineal    ,gland in the  

,evolution of ,additional               
,faculties--,the    ,skeptical ,attitude
 of ,empirical ,science and the ,need   
for a ,more ,liberal ,posture--,the     
,general ,results of    ,pituitarial    
,awakening ,upon ,man and the ,theory of
    ,knowledge                          
#cbg      ,,bibliography                
#cei      ,,index                       
#cfg            ,,the ,,mystery ,,of    
,,space            ,,introduction       
,,explanatory ,,notes      ,the         
following interpretation of words1      
phrases and notions occurring in  the   
text1 and also biographical sketches    
which the scope and purpose of  the book
 itself make it impracticable to        
elaborate1 are appended with  the view  
to facilitating its perusal4            
,,at-,,one-,,ment 7state of unity1      
unitariness72 denotes the ultimate state
  of oneness towards which all          
evolutionary movement tends2 applied to 
consciousness1 indicates the final      
expansion of consciousness wherein  it  

coincides with the universal            
consciousness in extent and quality of  
comprehension4 ,as applied to things1   
denotes the unification of all          
movements1 tendencies1 and evolutions as
 a singularity2 the end of all          
evolutionary activity 7vide p4 #bgj74   
,,beltrami1 ,,eugenio1 was born at      
,cremona1 ,november #af1 #ahce2 there   
he attended the elementary schools1 the 
gymnasium and the lyceum1  excepting the
 scholastic year #ahdh-di when he was at
 the ,gymnasium of  ,venice1 now known  
as ,marco ,polo4 ,he finished his lyceal
 studies in the  summer of #ahec1 and in
 the following autumn 7,november7 became
 a student  in the ,mathematical        
,faculty of the ,university of ,pavia1  
after having  obtained a scholarship    
there on the ,castiglioni ,foundation in
 the  ,collegio ,ghisleri4    ,in #ahed1
 the succeeding year1 he was expelled   
from this college in  company with five 
of his colleagues who were accused of   
promoting  disorders against the ,abbot 

,,leonardi1 rector of the college4 ,the 
expulsion brought him many hardships and
 disappointments1 and for two  years he 
drifted along merely existing as his    
family was too poor to  have him        
matriculated at another university4 ,but
 in #ahef1 he went to  ,verona where he 
succeeded in securing employment as     
secretary to the  engineer1 ,,diday1 in 
the ,government service of              
,lombardy-,venice4 ,on  ,january #aj1   
#aheg1 he was dismissed from this       
position for political  reasons2 but as 
the annexation of ,lombardy to ,piedmont
 occurred soon  thereafter1 he became   
again attached to the office of ,,diday1
 his  former employer1 when it was      
transferred to ,milan as a consequence  
of  political changes4    ,at ,milan    
,,beltrami took up his mathematical     
education in real earnest  as he now had
 access to ,professor ,,brioschi1 his   
former tutor1 and also  ,,luigi         
,,cremona4 ,through the influence of    
these two men he was designated         

7,october #ah1 #ahfb7 ,professore       
straordinario in the ,university of     
,bologna4 ,his work on ,surfaces of     
,constant ,negative ,curvature1 as  the 
pseudosphere1 and his application of the
 expression given by  ,,lobachevski     
7q4v47 for the angle of parallelism1    
very definitely  secure for him a place 
among the foremost workers in the field 
of the  non-,euclidean geometry4 ,he    
postulated a theorem1 known as          
,beltrami's  ,theorem1 which he stated  
as follows3 ,the center of a circle     
circumscribing a triangle is the center 
of gravity of the centers of  its       
inscribed and escribed circles4 ,he died
 in the year #aijj4 7,vide  ,amer4      
,math4 ,mo41 ,vol4 ,,ix1 p4 #ei47       
,,bolyai1 ,,janos 7#ahjb-ahfj71 was born
 at ,kleansenburg1 ,hungary4 ,he is     
said to have inherited his mathematical 
genius from his father1 ,,bolyai        
,,farkas 7#agge-ahef71 who was born at  
,bolya1 ,hungary4 ,being a very         
spirited youth1 his progress in his     

studies was most remarkable4 ,he        
completed the curriculum at the ,latin  
school when only twelve years of  age4  
,was graduated from the ,philosophical  
,curriculum as a result of two  years of
 study and then entered the ,viennese   
,academy of ,engineers4 ,was  appointed 
lieutenant at ,temesvrlin1 #ahbc1 whence
 on ,november #c1 #ahbc1  he wrote his  
father3 ,i have discovered such         
magnificent things that ,i  am myself   
astonished at them4 ,it would be damage 
eternal if they were  lost4 ,when you   
see them1 my father1 you yourself will  
acknowledge it4  ,now1 ,i cannot say    
more1 only so much3 that from nothing ,i
 have created  another wholly new world4
 ,this letter was written in the ,magyar
  language and has been preserved at the
 ,marcos ,vsrhely1 ,hungary4 ,the       
mathematical conceptions formulated by  
him became the appendix of  the         
,tentamen1 a book which his father had  
written on the ,theory of  ,parallels4  
,his ,science ,absolute of ,space was   

translated into the ,french in #ahfh  by
 the ,french mathematician1 ,j4         
,,ho,,el1 to whom belongs the credit of 
popularizing the works both of ,,bolyai 
and ,,lobachevski4 7,vide ,science1  n4 
s41 ,vol4 #ce1 ,no4 #ijf1 #aiab47       
,,cayley1 ,,arthur1 born at ,richmond1  
,surrey1 ,england1 ,august #af1 #ahba2  
studied at ,king's ,college school2     
entered ,trinity ,college1 ,cambridge1  
already a well equipped mathematician at
 the age of seventeen4 ,when  but       
twenty-one years of age he took two of  
the highest honors in the  ,university  
of ,cambridge4 ,he was ,senior ,wrangler
 and ,first ,smith's  ,prizeman4 ,he    
published his first paper in #ahda and  
this was followed by  eight hundred     
memoirs4    ,for fourteen years he      
practiced as ,conveyancer4 ,in #ahfc    
,lady ,,sadler',s  various trusts were  
consolidated1 and a new ,sadlerian      
professorship of  ,pure ,mathematics was
 created for the express purpose of     
affording a  place for ,cayley4         

,meanwhile1 as early as #aheb he was a  
fellow of the  ,royal ,society2 in #aheh
 he joined ,,sylvester and ,,stokes in  
publishing the  ,quarterly ,journal of  
,pure and ,applied ,mathematics4    ,he 
was for a considerable time principal   
adviser as to the merits of  all        
mathematical papers which were presented
 for publication to the  ,royal         
,society1 the ,astronomical ,society1   
the ,mathematical ,society and  the     
,cambridge ,philosophical ,society4 ,he 
is said to have been the most  learned  
and erudite of mathematicians1 and much 
of the material1  therefore1 which now  
constitutes the basis of the            
non-,euclidean  geometry is due to his  
laborious efforts and comprehensive     
knowledge  of mathematics4 7,vide       
,review of ,reviews1 ,vol4 ,,ii1 #ahie1 
,sketch1  reprinted from ,monist47      
,,chaogeny 7,gr4 ,chaos1 disorder1      
geny--generating1 evolution72 the       
evolution of chaos into order4 ,a kosmic
 process involving the  elaboration of  

the original1 formless world-plasm into 
the first faint  signs of orderliness2  
the beginnings of the movement of life  
or the  ,creative ,logos in preparation 
of the field of evolution4              
,,chaomorphogeny 7,gr4 ,chaos1          
disorder--,morphe1 form--geny1 becoming1
  generating72 evolution of the         
space-form1 the universe2 the           
establishment of the metes and bounds of
 the universe2 also1 the  origination   
and characterization of all forms as to 
tendence1 purpose  and limitations4     
,,conceptualization--,the act of        
conceptualizing1 the formulation of     
concepts2 the process by which the      
,thinker arrives at concepts2 the       
logical procedure by which the          
consistency of a scheme of thought is   
established4    ,,construction1         
,,ideal--,a purely formal conception2 a 
theory1 hypothesis2  a logical          
determination not necessarily based upon
 facts1 but  possessing virtue because  
of consistency2 a self-consistent scheme

 of  thought4    ,,cosmos--,whenever the
 term cosmos appears in the text spelled
 as here  shown1 it refers to phenomena 
pertaining to the earth or the solar    
system2 when spelled kosmos reference is
 made to the universe as a  whole4      
,,criterion ,,of ,,truth--,defined in   
the text as a four-fold standard of     
reference1 embracing the following      
elements1 namely1 the causal1  the      
sustentative1 relational and            
developmental4 ,lacking any one of      
these1 no view of truth is more than    
fragmentary4 ,applied to space1  it     
contemplates an inquiry into the genesis
 or causal aspect1 an  accounting for   
the duration aspect1 a recognition of   
its relation  to the totality of        
objects1 and lastly1 a prophecy of its  
telestic  or perfective culmination4    
,this test has been applied to the study
  of space as sketched in the text and  
the conclusions reached are an  outcome 
of the inquiry directed along these     
lines4    ,,curvature ,,of ,,space--,a  

doctrine formulated by ,,riemann and    
which  maintains that space is curved1  
and consequently1 all lines drawn       
therein are curved lines4 ,professor    
,,pickering aptly describes the  results
 of movements in a curved space by      
pointing out that if we go  far enough  
east we arrive at the west2 north1 we   
arrive at the south2  towards the       
zenith1 we arrive at the nadir1 and vice
 versa4    ,,deiform--,the basic idea   
indicated is that the universe is the   
form or  body of the supreme deity1     
since ,he is not only immanent in the   
kosmos1  but sustains it by ,his life2  
that in order to create a manifested    
universe1 it was necessary to limit1 or 
sacrifice1 in a measure1 ,his  own      
illimitability4 ,viewed in this light1  
the kosmos assumes an added             
significance4    ,,dimension--7,l4      
,dimensio1 to measure71 measurement2 a  
system of space  measurement4 ,the      
,euclidean geometry recognizes three    
dimensions or  cordinates as being      

necessary to establish a point position2
 witness1  the corners of a cube to form
 which three of the edges come together 
at a point4 ,these edges represent      
cordinates4 ,for the purposes of        
metageometry1 the term dimension has    
been variously defined1 as1  direction1 
extent1 a system of space measurement1  
or a system of  cordinates4 ,regarded as
 a series of cordinates1 it became      
possible  to postulate a system which   
required four cordinates to establish   
the position of a point1 as in the      
hypercube4 ,there may be five1 six1     
seven1 eight1 or any number of such     
cordinate systems according to  the kind
 of space involved in the calculations4 
,determinations based  upon the logical 
necessities of the various cordinate    
systems have  been found to be          
self-consistent throughout and1         
therefore1 valid for  metageometrical   
purposes4 ,much depends upon the        
definition2 for1 after  the definition  
has been once determined it remains then

 merely to make  inferences and         
conclusions conform to the intent of the
 definition4                            
,,diversity--,philosophically1 the idea 
indicated has reference to all          
dissimilarities1 differences1           
inequalities1 divergent tendencies1     
movements and characteristics to be     
noted in the universum of life2  the    
antithesis of kosmic unity2 the natural 
outcome of life in seeking  expression2 
the result of the fragmentative tendency
 of life4    ,,duodim 7duo1 two2 dim1   
abbreviation of dimension7--,a          
hypothetical  being supposed to be      
possessed with a consciousness adapted  
only to  two dimensions2 a dweller in   
,flatland or two-space whose scope of   
motility is limited to two directions1  
as on the face of a plane2 a  term      
invented by hyperspace advocates for the
 purpose of establishing  by analogy    
some of the characteristics of the four 
space and also its  rationality4        
,,duopyknon 7duo1 two--pyknon1 primary  

unit in the process of kosmic           
involution1 a condensation7--,secondary 
phase in the elaboration of  chaos into 
kosmic order4 ,,duopyknosis 7duo1 two1  
secondary--pyknosis1  process of        
condensation and origination7--,the     
second period of  the involutionary     
movement of life during which the duadic
 plane  of the kosmos is being          
established2 the second1 in the series  
of  seven distinct phases1 of           
space-genesis2 dual differentiation of  
kosmic plasm4 ,duopyknosis contemplates 
that1 in the passage of the             
kosmos-to-be from the plane of          
non-manifestation to the plane of       
manifestation1 there are seven distinct1
 though interdependent and              
interrelational1 stages through which   
life passes1 and that1 of these1  it is 
the second4 ,it relates to the plane of 
non-manifestation and is1  therefore1   
beyond the ken of the intellectuality1  
being a symbol4    ,,egopsyche 7,ego1   
the self-conscious ,i--psyche1          

soul7--,the mental1  emotional and      
physical mechanisms of man1 the         
,thinker4 ,these include  the purely    
mental system1 the emotional or         
affective mechanism1 the  nerve-systems 
7cerebro-spinal and sympathetic7 and the
 brain2 the  objective or sense-derived 
consciousness of the ,thinker which is  
elaborated from the total mass of       
perceptions transmitted through  the    
senses2 the medium of                   
self-consciousness2 the intellectual    
consciousness as distinguished from the 
intuitional or omnipsychic              
consciousness 7q4v474    ,the notion of 
the egopsychic consciousness is based   
upon data  already empirically          
determined from the mass of evidences   
everywhere  observable4 ,it seems to be 
apparent that there is a consciousness1 
a seat of knowledge1 in man the content 
of which is unknown to the              
sense-consciousness4 ,dreams1           
premonitions1 intuitions1 impressions   
and  the totality of all such phenomena 

substantiate this view4 ,furthermore1   
it is agreed that the source of the     
intuition is not identical with  that of
 the intellect4 ,the egopsychic         
consciousness1 accordingly1 is  purely  
intellectual4    ,,fluxion1             
,,psychic--,the difference between a    
mental image and an  object2 an image is
 the representation of certain salient  
or cardinal  characteristics of an      
object1 sufficient for identification2  
but an  image is not congruent1 in every
 respect1 with the object4 ,thus when we
  perceive an object1 although as       
,,bergson contends we perceive it in the
  place where it is and not in the      
brain1 it is the image of the object    
which takes its place in memory and not 
the object itself4 ,there is1 of        
course1 a marked disparity between this 
memory-image and the object4  ,even if  
the image possessed one of the          
properties of the object1 as1  size1 it 
could not take its place in memory1 and 
neither could it do so  if it possessed 

any of the properties of the real       
object4 ,consciousness  is such that all
 due allowances are made for these      
conditions and the  mind is able to     
retain more or less exact knowledge of  
these properties  in the image2 but     
there is a difference1 small though it  
may be4 ,this  difference is the psychic
 fluxional4    ,,fohat 7,skt7--,a term  
applied to the ,creative ,logos who is  
said to be  the generating element in   
the differentiation of chaos into kosmic
  orderliness2 the supreme deity in the 
rle of ,creator4    ,,form1 ,,pure--,an 
abstraction arrived at by subtracting   
the last vestige  of materiality or     
substantiality from an idea and viewing 
the remains  as a pure unsubstantial    
form or idealization2 the shell or      
frame-work  of a material object or     
condition2 existing in idea or thought  
only2  a mental conception regarded as a
 type or norm2 a purely hypothetical    
construction4    ,,four ,,space--,often 
referred to as the fourth dimension     

7vide ,chapter ,v72  a space in which   
four cordinates 7four lines drawn       
perpendicular to  each other7 are       
necessary and sufficient to establish   
the position of a  point1 as1 a         
hypercube4    ,,gauss1 ,,charles        
,,frederick1 born at ,brunswick1 ,april 
#cj1 #aggg4 ,his  father1 being a       
bricklayer1 had intended that he should 
follow the same  occupation4 ,so1 in    
#aghd1 ,charles was sent to the ,btner  
,public ,school  in ,brunswick1 in order
 that he might be taught the ordinary   
elements  of education4 ,but during his 
attendance at this school1 his unusual  
intelligence and aptitude attracted the 
attention and friendship of  ,professor 
,,bartels who later became the          
,professor of ,mathematics at  ,dorpat4 
,in #agib1 through the kindly           
representations of ,professor  ,,bartels
 to the ,duke of ,brunswick1 young      
,gauss was sent to the  ,collegium      
,carolinum4 ,this greatly displeased his
 father as he saw in  this move the     

frustration of his plans for ,charles4  
,in #agid1 however1  ,,gauss entered the
 ,university of ,gttingen still         
undecided whether he  should make       
mathematics or philology his life work4 
,while residing at  ,gttingen1 he made  
his celebrated discoveries in analysis  
and these  turned his attention         
definitely to the field of mathematics4 
,he completed his studies at ,gttingen  
and returned to ,brunswick in  #agih1   
residing at ,helmstadt where he had     
access to the ,library in the           
preparation of his ,disquisitiones      
,arithmeticae which was published  in   
#ahja4 ,he received his doctorate degree
 7,ph4,d47 on ,july #af1 #agii4  ,his   
next notable work was the invention of a
 method by which he  calculated the     
elements of the orbit of the planet     
,ceres which had  been discovered by    
,,piazzi1 ,january #a1 #ahjj1 and who   
had left no record  of his calculations 
by which other astronomers could locate 
the planet4  ,,gauss also calculated an 

ephemeris of ,ceres' motion by means of 
which  ,,de ,,zach rediscovered the     
planet ,december #ca1 #ahjj4    ,his    
,theoria ,motus ,corporum ,coelestium in
 ,conicis ,sectionibus ,solem           
,ambientium1 in which the author gives a
 complete system of formul  and         
processes for computing the movements of
 a heavenly body revolving  in a conic  
section is an outgrowth of his early    
researches and brought  him lasting     
fame4    ,through the influence of his  
friend ,,olbers1 he was appointed1 ,july
 #i1  #ahjg1 first director of the new  
,gttingen ,observatory1 and ,professor  
of ,astronomy in the ,university1 a     
position which he held until the end  of
 his life4 ,he died ,february #bc1      
#ahee4 7,vide ,astronomical ,society    
,notices1 ,vol4 #af1 p4 #hj1 #ahef2 also
 ,nature1 ,vol4 ,,xv1 pp4 #ecc-ecg1     
#ahgg47    ,,geometrism--,of geometrical
 quality2 a notion derived from         
,,plato',s  declaration3 ,god           
geometrizes4 ,it was his belief that the

 creative  acts of the deity are        
executed in accordance with geometric   
design and  laws2 that in the totality  
of such acts there necessarily inheres a
  latent geometric quality4 ,,kant      
closely adhered to this notion in his   
discussions of space as an aspect of    
divine intelligence4 ,he believed  that 
the intellect merely rediscovers this   
latent geometrism when  it turns to the 
study of materiality1 and this belief is
 shared by  ,,bergson1 the foremost     
metaphysician of the present time4      
,,hyperspace 7hyper1 above1 beyond1     
transcending--space7--,that species of  
space constructed by the intellect for  
convenience of measurement2 an          
idealized construction2 a purely        
arbitrary1 conventional mathematical    
determination2 the fourth dimension2 any
 space that requires more than  three   
cordinates to fix a point position in   
it1 as1 a five space1 an  n-space4      
,,intuitograph--,the means by which the 
omnipsychic consciousness  transmits    

intuitional impressions to the          
egopsychic or intellectual              
consciousness4 ,an intuitogram is a     
direct cognition1 an intuition2  a      
primary truth projected into the        
egopsychic consciousness by the         
,thinker4 ,it is recognized that1 under 
the necessities of the present          
schematism of things1 it is exceedingly 
difficult to propagate an  intuition1   
especially with the same degree of ease 
as concepts are  propagable2 yet1 this  
is believed to be a condition which will
 be  overcome as the evolution of the   
higher faculties proceeds4              
,,involution--,process of enfolding1    
involving2 antithesis of evolution2     
philosophically1 the doctrine of        
involution maintains that1 during  the  
process of kosmic pyknosis              
7space-genesis71 all that is to be      
expressed1 developed and perfected as a 
result of the evolutionary  movement was
 first involved1 enfolded or deposited  
as latent archetypal  tendencies and    

radicles in the original world-plasm2   
that1 as the  involutionary movement    
proceeded through the various phases of 
space-genesis1 these became more and    
more phenomenal until at last  they     
terminated in the elaboration of a      
manifest universe3 each  stage1         
accordingly1 of the involutionary       
procedure became the basic  substructure
 of a plane of specialized substance or 
materiality  and consciousness4 ,thus it
 appears that evolution really begins   
where involution ends 7vide ,fig4 #ah71 
and the two opposing processes          
constitute the dualism of life as       
generating element4 ,this notion has    
been symbolized in the ,lingam yoni of  
,hellenistic philosophies1 also  in     
,yang and ,yin of ,chinese philosophy1  
which represent the original  pair of   
opposites4    ,,kathekos--,a purely     
arbitrary term devised for the express  
purpose  of providing a convenient      
symbol to convey the idea embodied in   
the  triglyph1 ,chaos-,theos-,kosmos1   

and is composed of the first three      
letters in each one of the terms of the 
triglyph2 hence1 symbolizes  the        
triunity and interaction involved in the
 resolution of chaos into  an orderly   
kosmos by the will of the ,creative     
,logos4 ,thus1 kathekos  embodies a     
quadruplicate notion1 namely1 chaos1    
,creative ,logos1  manifested kosmos1   
and the creational activity of the      
,logos in the  transmutation of disorder
 into order4 ,the justification for this
 term1  therefore1 resides in its       
convenience1 brevity and                
comprehensiveness4    ,by referring to  
figure #ag1 it will be seen that        
,kathekos divides into  two             
kinds--involutionary1 or that which     
pertains to involution1 and             
evolutionary1 or that that pertains to  
evolution4 ,it thus comprises  the      
beginning and the end of the world age  
or cycle and pertains to                
non-manifestation4 ,the raison d'tre of 
this differentiation is  embodied in the

 notion that1 on the involutionary arc  
of the cycle1 the  chaogenic period     
represents a phase of the world age when
 space-genesis  is in an archetypal     
state wherein are involved all          
possibilities that  are to become       
manifest in the kosmos1 and on the      
evolutionary arc1 the  kathekotic period
 which is parallel to the chaogenic and 
represents a  phase of the world age    
when the kosmos has reached ultimate    
perfection1  embodying the perfected    
results of the possibilities which      
inhered in  the chaogenic period or in  
involutionary kathekos4 ,thus1 kathekos 
is  dual in nature1 on the one hand     
representing kosmic potency1 and on the 
other1 kosmic perfection of these       
potencies4 ,it is ,alpha1 as related  to
 involution1 and ,omega1 as related to  
evolution4    ,,kathekosity--,a         
derivative1 signifying creative activity
 and all that it  implies2 the state of 
consciousness or cognition corresponding
 thereto4    ,,klein1 ,,felix 7#ahdi--71

 born at ,dusseldorf2 studied at ,bonn1 
and when  only seventeen years of age   
was made assistant to the noted         
,,pl,,cker  in the ,physical ,institute4
 ,he took his doctorate degree in #ahfh1
 then  went to ,berlin1 and later to    
,gttingen where he assisted in editing  
,,pl,,cker',s works4 ,he entered the    
,gttingen faculty in #ahga2 became      
,professor of ,mathematics at ,erlangen 
in #ahgb2 and subsequently held         
professorships at ,munich1 #ahge2       
,leipzig1 #ahhj1 and ,gttingen1 #ahhf4  
,no one else in ,germany has exerted so 
great influence upon ,american          
mathematics as he4    ,,kosmos--,see    
,cosmos4    ,,la ,,grange1 ,,joseph     
,,louis1 born at ,turin1 ,january #be1  
#agcf2 died at  ,paris1 ,april #aj1     
#ahac2 regarded as the greatest         
mathematician since the  time of        
,newton4 ,it may be interesting to note 
that ,,la ,,grange remarked  that       
mechanics is really a branch of pure    
mathematics analogous to a  geometry of 

four dimensions1 namely1 time1 and the  
three cordinates  of the point in space4
 7,vide ,ball's ,account of the ,history
 of  ,mathematics47    ,,lie1 ,,sophus1 
a noted mathematician1 referred to as   
the great  comparative anatomist of     
geometric theories1 creator of the      
doctrines  of ,contact ,transformations 
and ,infinite ,continuous ,groups1 and  
revolutionizer of the ,theory of        
,differential ,equations4               
,,logos--,the supreme deity of the      
phenomenal universe2 ,creator2 ,fohat2 a
  planetary god2 the deity of a solar   
system4    ,,manvantara 7,skt47--,a     
world age2 the periods of involution and
 evolution  combined2 the stage during  
which the universe is in manifestation2 
a  ,day of ,brahma4    ,,mathesis 7,gr4 
mathein1 to learn7--,erudition2 profound
 learning2 the  realm of metaphysical   
conceptions2 the field of higher        
mathematics2 the  sphere of             
conceivability2 the theoretical4        
,,mentograph--,a cognitive factor       

consisting of a complete perception     
fused or in coalescence with a          
memory-image4 ,pure memory1 of itself1  
is without utility as an aid to         
cognition2 but1 when nourished or       
supplemented by the substance of        
perception it becomes the basis of      
intellectual consciousness4             
,,metageometry 7,gr4 ,meta1 beyond1     
transcending--geometry7--,commonly1 any 
kind of geometry that differs from the  
,euclidean1 as the non-,euclidean2  a   
geometry based upon the assumption that 
the angular sum of a triangle  is       
greater or less than two right angles2  
the highest form of geometry2  a system 
of idealized mathematical constructions4
 ,sometimes called  pangeometry2        
designated by ,,gauss as ,astral        
,geometry2 the geometry  of hyperspace4 
,it consists of results arrived at by   
geometers in  seeking a proof of the    
parallel-postulate4                     
,,meta-,,self--,the higher self in man2 
the universal self2 the one self  of    

which all individual selves are but     
fragments or parts4 ,in man1 it  is     
cordinate with the omnipsyche 7q4v47 and
 as such is the medium of  kosmic       
consciousness4    ,,morphogeny 7,gr4    
,morphe1 form1 vehicle1 body--geny1     
evolution7--,the  evolution of forms1   
the production of individual bodies or  
vehicles  for life1 including organs and
 faculties4 ,morphogenic--a derivative2 
pertaining to morphogeny2 a kosmic      
process 7vide figs4 #ag and #ah74       
,n-,,dimensionality--,quality of        
conceptual space by virtue of which it  
may be regarded as possessing an        
indefinite number of dimensions4        
,,near-,,truth--,any statement or view  
which is based upon partial  knowledge2 
predicates concerning a class or genus  
derived from limited  acquaintance with 
particulars of the class or genus2      
statements based  upon logical          
determinations inhering in idealized    
constructions and  applied to concrete  
or objective conditions2 an abstraction 

viewed as a  reality2 the application of
 the qualities of abstractions to       
realities4                              
,,neurogram--,psychologically1 a        
movement received by the afferent       
nerves in the form of a stimulation and 
transmitted through the brain  and      
efferent nerves as either a reflex or   
voluntary action2 a nerve  impulse2 a   
perception2 a primary unit of           
intellectual consciousness2  cf4        
,intuitogram4    ,,newcomb1 ,,simon     
7#ahce-aiji71 born at ,wallace1 ,nova   
,scotia2 educated  in his father's      
school and came to the ,united ,states  
in #ahec4 ,began1  in #ahed1 teaching in
 ,maryland2 was appointed computer on   
,nautical  ,almanac at ,cambridge in    
#aheg2 was graduated at ,lawrence       
,scientific  ,school in #aheh2 appointed
 ,professor of ,mathematics in the ,u4  
,s4 ,navy  in #ahfa4 ,he supervised the 
construction of the #bf-;inch equatorial
  telescope at the ,naval ,observatory1 
and was secretary of the ,transit  of   

,venus ,commission2 was a member of     
nearly all of the ,imperial and  ,royal 
societies of ,europe and of the various 
societies in the ,united  ,states1      
receiving the ,copley ,medal in #ahgd2  
the ,huygens1 #ahgh2 the  ,royal        
,society1 #ahij1 and the ,bruce ,medal  
in #ahih2 held the presidency  of the   
following learned societies1 viz3       
,american ,association for  the         
,advancement of ,science1 #ahgg2        
,society for ,psychical ,research1      
#ahhe-ahhf2 ,american ,mathematical     
,society1 #ahig-ahih2 the ,astronomical 
and ,astrophysical ,society of ,america 
from its foundation in #ahii4 ,he       
rendered notable service in popularizing
 the doctrine of hyperspace4            
,,norm--,an authoritative standard2     
model or type2 standard of reference4   
,the choice of a norm for spatial       
determinations cannot abide in  any     
premise except that which naturally1 and
 not artificially and  conventionally1  
conforms to what is actually perceived2 

if so1 there  should be justification   
for challenging the wisdom and utility  
of  the present schematism of things4   
,there is an inherent conformity of     
space with intellect and intellect with 
space1 and because of this  natural     
complementarity of part with part and   
whole with whole1 space  cannot be      
otherwise than the intellectuality      
normally conceives it  to be1 provided1 
of course1 that the cognitive movement  
is free and  untrammeled by arbitrary   
hindrances4 ,consciousness1 therefore1  
is  the norm or standard of reference   
for all questions arising out of a      
consideration of spatiality4            
,,omnipsyche--,a term used to denote the
 ,thinker's cognitive apparatus2  the   
universal soul manifesting in           
individuals2 the consciousness  of the  
,thinker in virtue of which he is at-one
 with the universal  consciousness2 the 
medium of kosmic consciousness2 the     
source of the  intuition1 cf4           
,egopsyche4 ,the divinity in man 7which 

is taken for  granted71 or his highest  
self can in no way be said justly to    
take its  rise from sense-experience or 
from any bodily process4 ,if divine1    
then  eternal1 and therefore1           
persistent4 ,broadly1 the doctrine of   
evolution  recognizes the passage of    
life from form to form1 adding a little 
to  each successive form and inevitably 
pushing each to a higher degree  of     
perfection4 ,now1 what is it that passes
 from form to form8 ,is it              
undifferentiated life or is it a        
specialized form of life8 ,from every   
evidence1 it would be judged that the   
life that ensouls an individual  form is
 a specialized principle1 i4e41 limited 
to the execution of a  given purpose4   
,if life as a specialized principle1    
limited to the  execution of a given    
purpose in each form1 passes on1 it must
 preserve1  at least1 the sublimated    
results obtained during its residence in
 each  individual form4 ,it would thus  
become a sort of reservoir containing   

all these transmuted results4 ,the      
omnipsyche1 within the meaning of the   
text1 is precisely this specialized life
 principle4                             
,,parallel-,,postulate--,variously      
referred to as the ,x,ith1 ,x,i,ith and 
,x,i,i,ith  axiom of the ,elements of   
,euclid2 stated by ,,manning1 in        
,non-,euclidean  ,geometry1 p4 #ia1 in  
the following form3 ,if two lines are   
cut by a  third1 and the sum of the     
interior angles on the same side of the 
cutting line is less than two right     
angles1 the lines will meet on that     
side when sufficiently produced4 ,this  
celebrated postulate has proven  to be  
the most fruitful ever devised2 for it  
embodies in itself the  possibility of  
three geometries based respectively upon
 the following  assumptions1 namely3 ,i4
 ,that there exists a triangle1 the sum 
of whose  angles is congruent to a      
straight angle1 the ,euclidean2 ,,ii4   
,that there  exists a triangle the sum  
of whose angles is less than a straight 

angle1 the ,,lobachevskian2 ,,iii4 ,that
 there exists a triangle the sum  of    
whose angles is greater than a straight 
angle1 the ,,cayley-,,klein4  ,speaking 
of the content of the last two named1   
,,edward ,,moffat ,,weyer#a  says3      
,hypothetical realms1 wherein the       
dimensions of space are assumed  to be  
greater in number than three1 yield     
strange geometries1 which  are only card
 castles1 products of a sort of         
intellectual play in  the construction  
of which the laws of logic supply the   
rules of the  game4 ,the character of   
each is determined by whatsoever        
assumption its  builder lays down at the
 start4      #a ,vide ,popular ,science 
,monthly1 vol4 #gh1 p4 #eed1 #aiaa4     
,,passage ,,of ,,space--,a phrase       
connoting the movement of space from    
chaos  to perfect order1 a process      
believed to be infinite4 ,the genesis of
  space necessarily implies an          
elaboration1 a procedure1 by which the  
metamorphosis from disorder to kosmic   

order is made1 and this movement  is    
referred to as the passage of space1 a  
phenomenon thought to be  measurable by 
means of a suitable instrumentality4    
,,perisophism--,see ,near-,truth4       
,,pseudosphere--,a surface of constant  
negative curvature2 basis of            
,,beltrami',s metageometrical           
calculations2 surface resembling a      
champagne  glass or common spool4 ,the  
assumption that space is pseudospherical
  has given rise to the notion of       
space-curvature and various other       
conceptions4    ,,psychogeny 7,gr4      
,psyche1 ,soul--geny7--,history of the  
evolution of the  soul or the           
development of the senso-mechanism in   
organisms4 ,,ernst  ,,haeckel has traced
 the psychogeny of man through          
twenty-two different  stages from the   
moneron to the anthropoid apes1 and man4
    ,,pralaya 7,skt47--,kosmic quietude2
 the period during which the universe   
is not in manifestation2 gestatory      
period2 kosmic inactivity2 opposed  to  

manvantara 7q4v472 figuratively1 the    
kosmic womb2 world egg4    ,,pyknon     
7,gr4 pyknon1 hard7--,the principle of  
kosmic condensation2 the  primary basis 
of space-genesis2 the initiation of the 
process by  virtue of which chaos is    
elaborated into kosmic order4           
,,pyknosis--,the  process of spatial    
engenderment4 ,there are seven of these 
processes1  each indicating a phase of  
duration1 namely3 ,,monopyknosis1 the   
primary  phase2 ,,duopyknosis1          
secondary2 ,,tripyknosis1 tertiary4     
,these three  pertain to the plane of   
non-manifestation1 the pralayic or      
gestatory  duration-phase4 ,the results 
arrived at during these duration-phases 
are concentrated in the ,quartopyknotic 
which corresponds to the causal  plane  
of manifestation or pure kosmic         
spirituality4 ,,quintopyknosis1 a       
process concerned in the genesis of     
mentality2 ,,sextopyknosis1 kosmic      
sensibility2 ,,septopyknosis1 kosmic    
materiality4 ,these seven phases  of    

duration constitute the scope of space  
genesis or kosmogenesis1  and           
incidentally depose the substructure of 
kosmic materiality1  sensibility1       
intellectuality and spirituality1 as    
well as the higher  trinity of kosmic   
modes4 ,the ramifications of these      
principles are  innumerable and         
omnipresent4 7,see ,chapter ,,vii47     
,,quartodim--,a hypothetical being      
assumed to have a consciousness adapted 
to hyperspace or the fourth dimension1  
and whose scope of action is            
encompassed within a space which        
requires four cordinates1 as the        
four-space4    ,,reality                
7,realism7--,life2 the harmony existing 
among the parts to  maintain their      
equilibrium in the whole2 the principle 
of integrity  subsisting among parts2   
kosmic vitality4    ,,riemann1 ,,george 
,,frederich ,,bernhard1 was born        
,september #ag1 #ahbf1 in the  village  
of ,breselenz1 near ,dannenburg1 in     
,hanover4 ,until he was eight  years of 

age his father was his sole tutor1 but  
even at this age he  exhibited great    
powers of arithmetical calculation4 ,in 
the ,spring of  #ahdj young ,,riemann   
was sent to the ,hanover ,lyceum where  
he remained for  two years1 leaving in  
#ahdb for the ,gymnasium at ,luneburg4  
,here1 under  the direction of          
,professor ,,schmalfuss1 he learned very
 rapidly1 and is  said to have required 
only one week thoroughly to familiarize 
himself  with ,,legendre',s ,theory of  
,numbers4    ,on ,april #ab1 #ahdf      
7,easter71 he entered the ,university of
 ,gttingen as a  student of ,theology in
 accordance with his father's wishes4   
,his passion  for mathematics1 however1 
was so aroused by the lectures of       
,,gauss that  ,he begged his father to  
be allowed to devote himself entirely to
 the  studies of his choice4 ,for two   
years he studied under ,,jacobi at      
,berlin4  ,he then returned to          
,gttingen1 and was graduated1 his thesis
 being a  dissertation on the           

foundations of a general theory of      
functions of  a variable complex        
magnitude4 ,in #ahed he qualified as a  
teacher by  giving a lecture on the     
,hypothesis on which ,geometry is       
,founded4 ,in  #aheg he became          
,professor ,extraordinarius1 and in     
#ahei was elected  ,corresponding       
,member of the ,academy of ,sciences of 
,berlin and in #ahfj a  member of the   
,academy of ,sciences of ,gttingen4     
,after four years of failing health1    
during which he visited ,messina1       
,palermo1 ,naples1 ,rome1 ,florence1    
,pisa and ,milan1 he died at ,lago      
,maggiore1 ,july #bj1 #ahff1 in full    
possession of his faculties and         
conscious of his approaching end4       
,,schweikart1 ,,ferdinand ,,karl        
7#aghj-aheg71 studied from #agif to     
#agih in  ,marburg1 attending the       
mathematical lectures of ,j4 ,k4 ,f4    
,,hauff4 ,in  #ahab he became professor 
in ,charkov1 a position which he held   
for  four years4 ,in #ahaf he became a  

tutor in the ,city of ,marburg where  he
 remained until #ahbj when he           
transferred his labors to ,knigsberg4   
,it was during his tutorship at         
,charkov1 ,marburg and ,knigsberg that  
he1 entirely alone and without the      
slightest suggestion from any man1      
developed and taught a non-,euclidean   
geometry to the students under  his     
care4 ,for copy of his treatise on      
non-,euclidean geometry1 see            
,historical ,sketch of the ,hyperspace  
,movement1 ,chapter ,,ii4               
,,scopographic ,,impressions--,sight    
perceptions fused with an associated    
memory-image1 and forming the basis of  
action on external phenomena4           
,,sensographic                          
,,impressions--,perceptions or impulses 
transmitted through  the nerves of a    
sense-organ2 any impression acting      
through the media of  the senses4       
,,sensible ,,world--,the world of the   
senses2 that which responds to the      
senses2 the domain of perception2 the   

phenomenal world2 world of  perceptual  
space4    ,,space-,,curvature 7see      
,curvature of ,space74                  
,,space-,,genesis--,the process of      
spatial engenderment2 the movement of   
life as engendering agent in bringing   
into manifestation the kosmos2  the     
story of the appearance of the organized
 kosmos4 ,the genesis of  space can only
 be symbolized1 as has been done in the 
text1 for the  limitations of human     
consciousness do not otherwise admit of 
the  empirical establishment of the     
notion of its detailed procedure4       
,,spatiality--,space as a dynamic1      
creative movement2 kosmic order1 as     
opposed to disorder2 the path of the    
engendering movement of life2  the place
 of life4 ,spatiality1 materiality1     
intellectuality and  geometricity or the
 latent geometrism of the kosmos are    
thought of as  being consubstantial and 
interdependent2 but1 of these1          
spatiality is  regarded as the substance
 out of which the latter three are      

elaborated4    ,,superconceptual--,the  
purely intuitional2 an act of cognition 
performed  without the detailed work of 
conception derived from sense-data2     
conception of intuitions and their      
inter-relations2 the ,thinker's         
consciousness freed from intellectual   
characterization4                       
,,superperception--,perception of       
conceptual relations2 a state of        
cognition wherein1 instead of receiving 
percepts or images from the  external   
world1 then elaborating them into       
concepts1 the ,thinker  apprehends      
composite images or concepts at first   
hand4 ,it is a power  which the         
liberated mind of the future will       
possess owing to the  growing automatism
 of the intellect and the more facile   
expression of  the intuitional          
consciousness4    ,,tesseract 7,gr4     
,tessera1 four1 cube1 tessella7--,a     
hypercube 7see ,chapter  ,v47           
,,thinker 7,skt4 ,manu1 thinker7--,the  
real1 spiritual man1 as  differentiated 

from his perceptive vehicles--mind1     
emotions and  physical body2 the        
omnipsychic intelligence who receives1  
classifies1  interprets and preserves   
percepts2 the manipulator of concepts2  
in  fine1 the higher1 spiritual man4    
,the ,thinker uses the various          
perceptive instrumentalities as so  many
 tentacles or antennae by which he      
contacts the sensible world  and makes  
the necessary adaptations to            
environment4 ,he is the pure            
intelligence which is the source of all 
cognitive motivation2 opposed  to ego1  
because the egopsychic instrumentality  
is essentially an  individualizing1     
separative agency2 while the ,thinker's 
omnipsychic  intelligence is the basis  
of his unity with the universal         
intelligence4  ,this conception of the  
,thinker implies that1 as a spiritual   
intelligence1 he is within and without  
the body1 filling it as the  ocean fills
 the sponge1 encompassing1 enveloping it
 and1 at the same  time1 originating the

 totality of activities which manifest  
in and  through the body4 ,he is        
limited1 therefore1 in his              
manifestations in  the sensible world   
only by the pliability of his vehicles4 
,,transfinity--,a state or condition    
that is incomprehensible to finite      
intelligence2 that which transcends the 
finite1 yet is not infinite2  less than 
infinity and greater than finity4 ,space
 is referred to as  being transfinite   
rather than infinite in extent4 ,but    
space transfinite  should be            
distinguished from space finite though  
unbounded4 ,for1  there would seem to be
 little worthy of choice between a      
finite1  unbounded space and an infinite
 one4 ,the absence of boundary would    
naturally suggest an infinite extent4   
,and although ,,riemann who is the      
author of the unbounded space           
arbitrarily determined that such a      
space should be a manifold possessing a 
measure of curvature which  could be    
determined either by counting or actual 

measurement1 he  undoubtedly knew1      
nevertheless1 that while each manifold  
might be an  unbounded space the        
totality of such manifolds1 infinite in 
number1  must also be infinite in       
extent4 ,it would seem to do violence to
 common  sense1 if not to logical       
necessity1 to view space both as        
unbounded  and finite in extent1 yet    
there would be no such difficulty in the
  recognition of space as being both    
transfinite and finite2 because it  is  
conceivable that the extent and         
character of space finite should        
transcend a finite intellectuality1 and 
yet not be infinite4    ,,tridim--,a    
being whose scope of consciousness is   
limited to a space of  three dimensions1
 as ordinary human beings4              
,,tridimensionality--,that  quality     
possessed by perceptual space by virtue 
of which it is  necessary and sufficient
 to have three cordinates1 and only     
three1 to  establish the position of a  
point4    ,,unodim--,a hypothetical     

being assumed to have a consciousness   
limited to  linear or one-space4        
,,zones ,,of ,,affinity--,regions in the
 domain of intellectuality wherein      
minds1 possessing a common differential1
 rate of vibration or quality1  adhere  
to certain tenets from choice4 ,schools 
of philosophy1 religions1  and all those
 major divisions of intellectual effort 
which divide and  subdivide intellectual
 allegiance are believed to take their  
rise in  this property of               
intellectuality in virtue of which all  
minds having a  similar coefficient     
gravitate towards a common agreement1   
especially  where the movement is       
voluntary and untrammeled4        ,,part
 ,,one      ,,the ,,essentials ,,of     
,,the ,,geometry ,,of ,,hyperspace ,,and
 ,,their ,,significations               
,,chapter ,i      ,,the ,,prologue      
,on the ,variability of ,psychic        
,powers--,the ,discovery of the ,fourth 
,dimension ,marks a ,distinct ,stage in 
,psychogenesis--,the ,non-,methodical   

,character of ,discoveries--,the ,three 
,periods of ,psychogenetic              
,development--,the ,scope and           
,permissibility of ,mathetic            
,license--,kosmic    ,unitariness       
,underlying ,diversity4    ,in          
presenting this volume to the public    
profound apologies are made  to the     
professional mathematician for the      
temerity which is shown  thereby4 ,all  
technical discussion of the problems    
pertinent to the  geometry of           
hyperspace1 however1 has been carefully 
avoided4 ,the reader  is1 therefore1    
referred to the bibliography published  
at the end of  this volume for matter   
relating to this aspect of the subject4 
,the aim  rather has been to outline    
briefly the progress of mathematical    
thought  which has led up to the idea of
 the multiple dimensionality of space2  
to state the cardinal principles of the 
,non-,euclidean geometry and to  offer  
an interpretation of the metageometrical
 concept in the light of  the           

evolutionary nature of human faculties  
and material characteristics  and       
properties4    ,the onus of this        
treatise is1 therefore1 to distinguish  
between what is  commonly known as      
sensible space and that other species of
 space known  as geometric spaces4 ,also
 to show that the notion which has been 
styled  hyperspace is nothing more nor  
less than an evidence of the faint1     
early outcroppings in the human mind of 
a faculty which1 in the course  of time1
 will become the normal possession of   
the entire human race4  ,thus the weight
 of all presentations will be to give   
currency to the  belief1 very strongly  
held1 that humanity1 now in its infancy1
 is yet  to evolve faculties and        
capabilities1 both mental and spiritual1
 to a  degree hitherto viewed as        
inconceivable4    ,on this view it must 
appear that the faculty of thought      
including the  powers of imagination and
 conceptualization are not psychological
  invariants1 but1 on the other hand1   

are true variants4 ,they are1           
consequently1 answerable to the         
principle of evolution just as all      
vital phenomena are4 ,some have thought 
that no matter what idea may  come into 
the mind of the human race or at what   
time the idea may  be born the mind     
always has been able to conceive it4    
,that is1 many  believe that the nature 
of mind is such that no matter how      
complex an  idea may be there has always
 been in the mind the power of          
conceiving  it4 ,but this view cannot be
 said to have the support of any        
trustworthy  testimony4 ,if so1 then the
 mind must at once be recognized as     
fully  matured and capable during every 
epoch of human evolution1 no less  in   
the first than in the latest1 which1 of 
course1 is absurd4 ,it is  undoubtedly  
more reasonable and correct to believe  
that the powers of  conceptualization   
are matters of evolutionary concern4    
,for instance1  the assertion that the  
mind was incapable of conceiving1 in the

 realm  of theology1 a                  
non-anthropomorphic god1 or1 in the     
field of biology1  the doctrine of      
evolution1 or1 in the domain of         
invention1 the wireless  telegraph1 or1 
in mathematics1 the concept of          
hyperspace before the  actual time of   
these conceptions1 cannot be            
successfully controverted4    ,in fact1 
it may be laid down as one of the first 
principles of  psychogenesis that the   
mind rarely1 if ever1 conceives an idea 
until it  has previously developed the  
power of conceptualizing it and giving  
it  expression in the terms of prior    
experience4 ,as in the growth of the    
body there are certain processes which  
require the full development  of the    
organ of expression before they can be  
safely executed so  in the phyletic     
development of faculties there are      
certain ideas1  conceptions and scopes  
of mental vision which cannot be        
visualized  or conceptualized until the 
basis for such mentation has been laid  

by the appearance of previously         
developed faculties of expression4  ,and
 especially is this true of the         
intellect4 ,inasmuch as the entire      
content of the intellect is constituted 
of sense-derived knowledge1  with the   
exception of intuitions which are not of
 intellectual origin  though dependent  
upon the intellect for interpretation1  
there can  be no doubt as to the        
necessity of there being first deposed  
in the  intellect a sense-derived basis 
for intellection before it can become   
manifest4 ,the ,sensationalists1 led by 
,,leibnitz1 propounded as their         
fundamental premise this dictum3 ,there 
is nothing in the intellect  which has  
not first been in the senses except the 
intellect itself1  and this has never   
been gainsaid by any school that could  
disprove  it4 ,the intuitionalist does  
not deny it3 he merely claims that we   
are  the recipients of another form of  
knowledge1 the intuitional1 which1      
instead of being derived from           

sense-experience1 is projected into the 
intellectual consciousness from another 
source which we designate the  ,thinker4
 ,thus1 from the two forms of           
consciousness1 come into the area  of   
awareness truths that spring from       
entirely different sources4 ,from  the  
one source a steady stream of           
impressions flow constituting the       
substance of intellectual consciousness2
 from the other only a drop1  every now 
and then1 falls into the great inrushing
 mass so as to add  a dim               
phosphorescence to an otherwise         
unilluminated pool4 ,obviously1  when   
there is a lack of sensuous data from   
which a certain concept may  be         
elaborated there can be no conception   
based upon them1 and as the  variety and
 quality of concepts are in exact       
proportion to the variety  and quality  
of sense-experience there can be no     
demand for a particular  species of     
notions such as might be elaborated out 
of the absent  or non-existent          

perception4 ,hence1 the power of        
conceiving springs  forth from          
sense-experience4 ,sense-experience is  
essentially a mass  of perceptions3     
these1 creating a demand for additional 
adaptations1  conspire1 as if1 to evoke 
the power or faculty to meet the demand1
 and  consequently1 an added            
conceptualization is made4    ,progress 
in human thought is made in a manner    
similar to that which  prevails in the  
development of other natural processes1 
such as1 the  power of speech in the    
child4 ,in the development of this      
faculty there  are certain definite     
stages which appear in due sequence4    
,the child is  not gifted with the power
 of speech at once4 ,it comes1 by       
gradual and  sometimes painful growth1  
into a full use of this faculty4 ,now1  
much  the same principle holds true in  
the evolution of the mind in the human  
species4 ,it is an established biologic 
principle that the ontogenetic          
processes manifested in the individual  

are but a recapitulation of the         
phylogenetic processes which are        
observable in the progress of the       
entire species4 ,the view becomes even  
more cogent when note is taken  of the  
fact that the foetus1 during            
embryogenesis1 passes successively      
through stages of growth which have been
 shown to be analogous1 if  not         
identical1 with those stages through    
which the human species has  developed1 
namely1 the mineral1 vegetal and animal4
    ,wherefore it may be said that the  
fourth dimensional concept marks a      
distinct stage in psychogenesis or      
evolution of mind4 ,it required1  as    
will be shown in ,chapter ,,ii1 nearly  
two thousand years for it to  germinate1
 take root and come to full fruition4   
,for it was not until  the early years  
of the nineteenth century that          
mathematicians1 taking  inspiration from
 ,,riemann 7#ahbf-ahff7 fully recognized
 the concept as  a metaphysical         
possibility1 or even the idea was       

conceived at all4  ,serious doubt is    
entertained as to the possibility of its
 conception by  any human mind before   
this date1 that is1 the time when it was
 actually  born4 ,prior to that time1   
mathematical thought was taking upon    
itself  that shape and tendence which   
would eventually lead to the discovery  
of hyperspace2 but it could not have    
reached the zenith of its upward        
strivings at one bound4 ,that would have
 been unnatural4    ,such is the        
constitution of the mind that although  
it is the quantity  which bridges the   
chasm between the two stages of man's   
evolution when  he merely thinks and    
when he really knows it is entirely     
under the  domain of law and must       
observe the times and seasons1 as it    
were1 in  the performance of its        
functions4 ,the scope of psychogenesis  
is very  broad1 perhaps unlimited2 but  
its various stages are very clearly     
defined notwithstanding the breadth of  
its scope of motility4 ,and while  the  

distance from moneron to man1 or from   
feeling to thinking is  vast1 the gulf  
which separates man1 the ,thinker1 from 
man1 the knower1  is vaster still4 ,who1
 therefore1 can say what are the        
delights yet in  store for the mind as  
it approaches1 by slow paces1 the goal  
whereat  it will not need to struggle   
through the devious paths of perceiving1
  conceiving1 analyzing1 comparing1     
generalizing1 inferring and judging2    
but will be able to know definitely1    
absolutely and instantaneously8  ,that  
some such consummation as this shall    
crown the labors of mental  evolution   
seems only natural and logical4    ,it  
may be thought by some that the         
character and content of  revelational  
impressions constitute a variation from 
the requirements  of the law above      
referred to1 but a little thought will  
expose the  fallacy of this view4 ,the  
nature of a revealed message is such as 
to make it thoroughly amenable to the   
restrictions imposed by the             

evolutionary aspects of mind in general4
 ,that this is true becomes  apparent   
upon an examination of the four cardinal
 characteristics of  such impressions4  
,first1 we have to consider the         
indefinite character  of an apocalyptic 
ideograph which is due to its symbolic  
nature4 ,this  is a feature which       
relieves the impression of any pragmatic
 value  whatsoever1 especially for the  
period embracing its promulgation4      
,then1  such cryptic messages may or may
 not be understood by the recipient  in 
which latter case it is nonpropagable4  
,second1 the necessity of  previous     
experience in the mind of the recipient 
in order that he may  be able to        
interpret to his own mind the psychic   
impingement4 ,the basis  which such     
experience affords must necessarily be  
present in order  that there may be an  
adequate medium of mental qualities and 
powers in  which the ideogram may be    
preserved4 ,a third characteristic is   
that  revelations quite invariably      

presuppose a contemplative attitude of  
mind which1 in the very nature of the   
case1 superinduces a state of           
preparedness in the mind for the proper 
entertainment of the concept  involved4 
,this fact proves quite conclusively    
that revelational  impressions are not  
exceptions to the general rule4 ,lastly1
 a  dissatisfaction with the conditions 
with which the symbolism deals or  to   
which it pertains is also a             
prerequisite4 ,this condition is really 
that which calls forth the cryptic      
annunciation1 and yet1 preceding  it is 
a long series of causes which have      
produced both the conditions  and the   
revolt which the revelator feels at     
their presence4 ,in view of  the        
foregoing1 it would appear that         
objections based upon the alleged       
nonconformity of the revealed or        
inspired cannot be entertained as it    
must be manifest that it1 too1 falls    
within the scope of the laws of  mental 
growth4    ,discoveries1 whether of     

philosophical or mechanical nature1 or  
whether  of ethical or purely           
mathematical tendence1 are never the    
results of  a deliberate1 methodical or 
purposive reflection4 ,for instance1 let
  us take ,,lie',s transformation       
groups1 mathematic contrivances used  in
 the solution of certain theorems4 ,now1
 it ought to be obvious that  these     
mathetic machinations were not          
discovered by ,,sophus ,,lie as a       
consequence of any methodic or          
purposeful intention on his part4  ,that
 is1 he did not set out deliberately to 
discover transformation  groups4 ,for   
back of the groups lay the entire range 
of analytic  investigations2 the        
mathematical thought of more than a     
thousand years  furnished the           
substructure upon which ,lie built the  
conception of his  groups4 ,similarly1  
it may be said with equal assurance that
 no matter  how great the intensity of  
thought1 nor how purposeful1 nor of how 
long  duration the series of            

concentrated abstractions which led up  
to the  invention of the printing press1
 the linotype or multiplex printing     
press of our day could not have been    
produced abruptly1 nor by use of  the   
mental dynamics of the human mind of    
remoter days4 ,its production  had to   
follow the path outlaid by the laws of  
psychogenesis and await  the development
 of those powers which alone could give 
it birth4  ,the whole question resolves 
itself1 therefore1 into the idea of the 
complete subserviency of the mind1 in   
all matters of special moment1 to  the  
laws aforementioned4 ,the supersession  
of the law of its own life by  the mind 
is well-nigh unthinkable1 if not quite  
so4    ,if we now view the history of   
the mind as manifested in the human     
species1 three great epochs which divide
 the scope of mental evolution  into    
more or less well-defined stages present
 themselves4 ,these are3  first1 the    
formative stage2 second1 the            
determinative stage2 third1  the stage  

of freedom1 or the elaborative stage4   
,in all of the early races of men1      
through every step which even  preceded 
the genus homo1 the generic mind was    
being formulated4 ,it  was being given  
shape1 outline and direction4 ,all of   
the first stage1  the formative1 was    
devoted to organization and direction4  
,those  elementary sensations which     
constituted the basis of mind in the    
primitive man were accordingly strongly 
determinative of what the mind  should  
be in these latter days4 ,to this       
general result were contributed  the    
effects of the activity of cells1       
nerves1 bones1 fibers1 muscles  and the 
blood4    ,the formative period         
naturally covered a very extensive area 
in the  history of mind or psychogenetic
 development4 ,it was followed closely1 
but almost insensibly1 by the           
determinative period during which all   
the latent powers1 capacities and       
faculties which were the direct         
products of the formative period were   

being utilized in meeting the  demands  
of the law of necessity4 ,the making of 
provisions against  domestic want1      
against the attacks of external foes2   
the combating  of diseases1 physical    
inefficiency1 the weather1 wild beasts1 
the  asperities of tribal enmities2 as  
well as furthering the production of    
art1 music1 sculpture1 the various      
branches of handiwork1 literature1      
philosophies1 religions and the         
effectuation of all those things which  
now appear as the result of the mental  
activity of the present-day man  make up
 the essence and purpose of the         
determinative period4    ,signs of the  
dawn of the elaborative stage1 also     
called the stage of  freedom1 have been 
manifest now for upwards of three       
centuries and  it is1 therefore1 in its 
beginnings4 ,it is not fully upon us4   
,not yet  can we fully realize what it  
may mean1 nor can we unerringly forecast
  its ultimate outcome2 but we feel that
 it is even now here in all the  glories

 of its matutinal freshness4 ,and the   
mind is beginning to be  free from the  
grinding necessities of the constructive
 period having  already freed itself    
from the restrictive handicaps of the   
primeval  formulation period4 ,already  
the upgrowing rejuvenescences so common 
at the beginning of a new period are    
commencing to show themselves  in every 
department of human activity in the     
almost universal desire  for greater    
freedom4 ,and this is particularly      
noticeable in the many  political       
upheavals which1 from time to time1 are 
coming to the  surface as well as in the
 countless other aspects of the         
wide-spread  renaissance4 ,perhaps the  
time may come1 never quite fully1 when  
there  will be no longer any necessity  
to provide against the external         
exigencies of life2 perhaps1 the time   
will never be when the mind  shall no   
more be bound by the law of             
self-preservation1 not even when  it has
 attained unto the immortality of       

absolute knowledge2 yet1 it  is         
intuitively felt that it must come to   
pass that the mind shall be  vastly     
freer than it is to-day4 ,and with this 
new freedom must come  liberation from  
the necessities of the elementary       
problems of mere  physical existence4   
,the inference is1 therefore1 drawn that
 the fourth dimensional concept1  and   
all that it connotes of hyperspace or   
spaces of n-dimensionality  are some of 
the evidences that this stage of freedom
 is dawning4 ,and  the mind1 joyous at  
the prospect of unbounded liberty which 
these  concepts offer1 cannot restrain  
itself but has already begun to revel   
in the sunlit glories of a newer day4   
,what the end shall be2 what  effect    
this new liberty will have on man's     
spiritual and economic life2  and what  
it may mean in the upward strivings of  
the ,thinker for that  sublime          
perpetuity which is always the property 
of immediate knowledge  no one can hope1
 at the present time1 to fathom4 ,it is1

 however1  believed with ,,keyser that  
it is by the creation of hyperspaces    
that  the rational spirit secures       
release from limitation2 for1 as he     
says1  in them it lives ever joyously1  
sustained by an unfailing sense of      
infinite freedom4    ,the elevating     
influence of abstract thinking1 such as 
excogitation upon  problems dealing with
 entities inhabiting the domain of      
mathesis is1  without doubt1            
incalculable in view of the fact that it
 is only through  this kind of thought  
that the spirit is enabled to reach its 
highest  possibilities4 ,this is        
undoubtedly the philosophy of those     
religious  and occult exercises known as
 meditations1 and this perhaps was  the 
main idea in the mind of the ,hebrew    
poet when he exclaimed3 ,let  the words 
of my mouth and the meditation of my    
heart be acceptable  in thy sight1 ,o   
,lord1 my strength and my ,redeemer4    
,the principal1  if not the only1 value 
possessed by the summitless hierarchies 

of  hyperspaces which the mathematician 
constructs in the world of pure  thought
 is the enrichening and ennobling       
influence which they exert  upon the    
mind4 ,but admittedly this unbounded    
domain of mathetic  territory which he  
explores and which he finds peopled with
 ideas1  ensembles1 propositions1       
relations and implications in endless   
variety  and multiplicity is quite real 
to him and subsists under a reign of    
law the penalties of which1 while not as
 austere and unreasonable  as some which
 we find in our tridimensional world1   
are nevertheless  quite as palpable and 
as much to be feared4 ,for the orthodoxy
 of  mathematics is as cold and         
intolerant as ever the religious fanatic
  can be4 ,but the reality and even the 
actuality which may be imputed to  the  
domain of mathesis is of an entirely    
different quality from that  which we   
experience in our world of triune       
dimensionality and it is a  regrettable 
error of judgment to identify them4 ,it 

ought1 therefore1  never be expected1   
nor is it logically reasonable to assume
 that the  entities which inhabit the   
mathetic realm of the analyst should be 
submissive to the laws of sensible      
space2 nor that the conditions which    
may be found therein can ever be made   
conformable to the conditions  which    
exist in perceptual space4    ,it was   
,,plato',s belief that ideas alone      
possessed reality and what  we regard as
 actual and real is on account of its   
ephemerality and  evanescence not real  
but illusionary4 ,this view has been    
shared by a  number of eminent thinkers 
who followed1 with some ostentation1 the
  lead established by ,,plato4 ,for a   
considerable period of time this  school
 of thinkers had many adherents2 but the
 principles at length  fell into        
disrepute owing to the absurdities      
indulged in by some of the  less careful
 followers4 ,the realism1 or for that   
matter1 the actuality  of ideas cannot  
be denied2 yet it is a realism which is 

neither to  be compared with the        
physical reality of sense-impressions   
nor its  phenomena4 ,the character and  
peculiarity of ideas are in a class     
apart  from similar notions of          
perceptual space content4 ,it is as if  
we were  considering the potentialities 
of the spirit world and the entities    
therein in connection with incarnate    
entities which in the very nature  of   
the case is not allowable4 ,furthermore1
 it is unreasonable to  suppose that the
 conditions on a higher plane than the  
physical can be  made responsible to a  
similar set of conditions on the        
physical plane4    ,there are certain   
astronomers who base their speculations 
as to the  habitability of other planets
 upon the absurd hypothesis that the    
conditions of life upon all planets must
 be the same as those on the  earth1    
forgetting that the extent of the       
universe and the scope of  motility of  
life itself are of such a nature as to  
admit of endless  variations and        

adaptations4 ,there is a realism of     
ideas and a realism  of perceptual      
space4 ,yet this is no reason why the   
two should be  identified4 ,on the other
 hand1 owing to the diversity in the    
universe1  every consideration would    
naturally lead to the assumption that   
they  are dissimilar4 ,to invest ideas1 
notions1 implications and inferences    
with a reality need not logically or    
otherwise affect the reality of a       
stone1 a fig1 or even of a              
sense-impression4    ,to a being on the 
spirit levels our grossest realities    
must appear  as non-existent4 ,they are 
neither palpable nor contactable in any 
manner within the ordinary range of     
physical possibilities4 ,for us his     
gravest experiences can have no reality 
whatsoever2 for no matter how  real an  
experience may be to him it is          
altogether beyond our powers of         
perception1 and therefore1 to us        
non-existent also4 ,it should1 however1 
be stated that the state of our         

knowledge about a given condition can   
in no way affect its existence4 ,it     
merely establishes the fact that  two or
 more realities may exist independent of
 one another and further  that the gamut
 of realism in the universe is infinite 
and approaches a  final state when its  
occlusion into absolute being follows as
 a logical  sequence4    ,recurring to  
the consideration of the reality of     
spirit-realms  as compared with that of 
sensible space1 it comes to view that   
our idealism1 that is1 the idealism     
which is a quality of                   
conceptualization1 may be regarded as   
identical with their realism1 at  least 
as being on the same plane as it4       
,stated differently1 the things  that   
are ideal to us and which constitute the
 data of our consciousness  may be as   
real to them as the commonest object of 
sense-knowledge is to  us4 ,what1       
therefore1 appears to us as the most    
ethereal and idealistic  may have quite 
a realistic character for them4         

,ultimately1 however1 and in the final  
deeps of analysis it will  be found     
undoubtedly that both our realism and   
our idealism as  well as similar        
qualities of the spirit world are in all
 essential  considerations quite        
illusionary4 ,all knowledge gained in a 
condition  short of divinity itself is  
sadly relative4 ,even mathematical      
knowledge  falls far short of the       
absolute1 the fondest claims of the     
orthodox  mathematician to the contrary 
notwithstanding4 ,it has been said      
frequently that a mathematical fact is  
an absolute fact and that its  verity1  
necessity and certainty cannot be       
questioned anywhere in the  universe    
whether on ,jupiter1 ,neptune1          
,fomalhaut1 ,canopus or ,spica4  ,but   
having so declared1 the fact of the     
sheer relativity of our  knowledge is   
not disturbed thereby nor controverted4 
,happily1 neither  distance nor a lack  
of distance can in any way affect the   
quality of  human knowledge1            

mathematical knowledge not excepted4    
,that can only  be affected by          
conditions which cause it to approach   
perfection and  nothing but evolution   
can do that4    ,in the light of results
 obtained in analytic investigations the
  question of the flexibility of        
mathematical applications becomes       
evident and one instead of being        
convinced of the vaunted invariability  
of the laws obtaining in the world of   
mathesis is1 on the other  hand1 made   
aware of the remarkable and seemingly   
unrestrained  facility with which these 
laws may be made to apply to any        
conditions  or set of assumptions within
 the range of the mind's powers of      
conception4 ,mathematicians have deified
 the definition and endowed  it with    
omnific powers imputing unto it all the 
attributes of  divinity--immutability1  
invariance1 and sempiternity4 ,in this  
they have  erred grievously although1   
perhaps1 necessarily4 ,mathetic         
conclusions  are entirely conditional   

and depend for their certainty upon the 
imputed certitude of other propositions 
which in turn are dependent1  in ever   
increasing and endlessly complex        
relations1 upon previously  assumed     
postulates4 ,these facts make it        
exceedingly difficult to  understand the
 attitude of mind which has obscured the
 utter mutability  and consequent       
ultimate unreliability of the fine-spun 
theories of  analytic machinations4     
,the apriority of all mathematical      
knowledge is open to serious            
questioning4 ,and although there is no  
hesitancy in admitting the basic        
agreement of the most primary facts of  
mathematical knowledge with the         
essential character of the intellect the
 existence of well-defined  limits for  
such congruence cannot be gainsaid4 ,the
 subjunctive  quality of geometric and  
analytical propositions is made apparent
  by an examination of the possibilities
 falling within the scope of            
permissibility offered by mathetic      

license4 ,for instance1 privileged to   
proceed according to the analytic method
 it is allowable to reconstruct  the    
sequence of values in our ordinary      
system of enumeration so as to  admit of
 the specification of a new value for   
say1 the entire series  of odd numbers4 
,this value might be assumed to be a    
plus-or-minus one1  dependent upon its  
posture in the series4 ,that is1 all odd
 numbers in  the series beginning with  
the digit #c1 and continuing1 #e1 #g1   
#i1 #aa1 #ac1  #ae1 #ag1 #ai1 444 n1    
could be assumed to have only a place   
value which  might be regarded as a     
constant-variable4 ,the series of even  
numbers1  #b1 #d1 #f1 #h1 #aj1 #ab1 #ad1
 #af1 444 n1 may be assumed to retain   
their  present sequence values4 ,under  
this system the digit #a would have an  
absolute value2 all other odd numbers   
would have a constant-variable  value2  
constant1 because always no more nor    
less than #a dependent  upon their place
 in the operations and whether their    

values were to  be applied by addition  
or subtraction to or from one of the    
values  in the even number series2      
variable1 because their values would be 
determinable by their application and   
algebraic use4    ,there would1 of      
course1 be utilitarian objection to a   
system of this  kind2 but under the     
conditions of a suppositionary          
hypothesis1 it  would be self-consistent
 throughout1 and if given universal     
assent  would suit our purposes equally 
as well as our present system4 ,but  the
 fact that this can be done under the   
mathematic method verily  proves the    
violability of mathematical laws and    
completely negatives  the assumption    
that the sum of any two digits1 as say  
#b plus #b equals  #d1 is necessarily   
and unavoidably immutable4 ,for it can  
be seen that  the sum-value of all      
numbers may be made dependent upon the  
assumed  value which may be assigned to 
them or to any collection thereof4      
,furthermore1 it is a matter of         

historical knowledge that it was the    
custom of ancient races of men to       
account for values by an entirely       
different method from what we use       
to-day4 ,the latter is a result of      
evolution and while experience teaches  
that it is by far the most  convenient1 
it is nevertheless true that earlier men
 managed at least  fairly well on a     
different basis4 ,then1 too1 the fact of
 the utility  and universal             
applicability of our present system1    
based upon universal  assent1 does not  
obviate the conclusion that any other   
system1  consistent in itself1 might be 
made to serve our purposes as well4     
,it ought to be said1 however1 in       
justice to the rather utilitarian       
results obtained by ,,la ,,grange1      
,,helmholtz1 ,,fechner1 and others who  
strove to make use of their discoveries 
in analysis in solving  mechanical1     
physiological and other problems of more
 or less pragmatic  import that1 in so  
far as this is true1 mathematical       

knowledge  must be recognized as being  
consistent with the necessities of a    
priori requirements4 ,but even these    
results may not be regarded  as         
transcending the scope of the most      
fundamental principles of               
sense-experience4 ,it will be discovered
 finally1 perhaps1 that the  energy     
spent in elaborating complicate series  
of analytic curiosities  has been       
misappropriated4 ,it will then be       
necessary to turn the  attention        
definitely to the study of that which   
lies not at the  terminus of the        
intellect's modus vivendi1 but which is 
both the  origin of the intellect and   
its eternal sustainer--the intuition1   
or life itself4 ,this can result in     
nothing less than the complete          
spiritualization of man's mental outlook
 and the consequent inevitable          
recognition of the underlying and       
ever-sustaining one-ness of all  vital  
manifestations4    ,one of the          
curiosities of the tendency in man's    

mind to specialize in  analytics1       
whether in the field of pure mathematics
 or metaphysics1  is the fact that it   
almost invariably leads to an attempt to
 account  for cosmic origins on the     
basis of paralogic theories4 ,this in   
times  past has given rise to the theory
 of the purely mechanical origin  of the
 universe as well as many other         
fantastic fallacies the chief  error of 
which lay in the failure to distinguish 
between the realism  of mental concepts 
and that of the sensible world4 ,in     
spite of this1  however1 one is bound to
 appreciate the beneficial effects of   
analytic  operations because they serve 
as invigorants to mental growth4 ,it    
could  not1 therefore1 be wished that   
there were no such thing as analytics2  
for the equilibria-restoring property of
 the mind may at all times be  relied   
upon to minimize the danger of excesses 
in either direction4  ,just as the tide 
flowing in flows out again1 thereby     
restoring  the ocean's equilibrium1 so  

the mind ascending in one generation    
beyond the safety mark has its          
equilibrium restored in the next by a   
relinquishment of the follies of the    
former4    ,the four-space is one of the
 curiosities of analytics2 yet it need  
not be a menace to the sane             
contemplation of the variegated products
  of analysis4 ,safety here abides in   
the restraint which should  characterize
 all discussion and application of the  
concept4 ,if  enthusiasts would be      
content not to transport the so-called  
fourth  dimensional space out of the    
sphere of hyperspace and cease trying   
to speculate upon the results of its    
interposal into three space  conditions1
 which is in every way a constructual   
impossibility1 there  could not be any  
possible objection to its due           
consideration4 ,this  would obviate the 
danger of calling into question either  
the sincerity  or perspicacity of those 
whose enthusiasm tempts them to         
transgress the  limits of propriety in  

their behavior towards the inquiry4     
,there is but one life1 one mind1 one   
extension1 one quantity1 one  quality1  
one being1 one state1 one condition1 one
 mood1 one affection1  one desire1 one  
feeling1 one consciousness4 ,there is   
also but one  number and that is unity4 
,all so-called integers are but         
fractional  parts of this kosmic unity4 
,the idea represented by the word two   
really connotates two parts of unity and
 the same is true of a  decillion1 or   
any number of parts4 ,these are merely  
the infinitesimals  of unity and they   
grow less in size and consequence as the
 divisions  increase in number4 ,the    
analysis of unity into an infinity of   
parts is  purely an a posteriori        
procedure4 ,that it is an inherent      
mind-process  is a fallacy4 ,all our    
common quantities1 as the mile1         
kilometer1  yard1 foot1 inch1 gallon1   
quart1 are conventional and arbitrary   
and  susceptible of wide variations4 ,as
 the basis of all physical phenomena  is

 unity2 it is only in the ephemeral     
manifestations of sensuous  objects that
 they appear as separate and distinct   
quantities4    ,we see on a tree many   
leaves1 many apples or cherries2 on a   
cob many  grains of corn4 ,we have      
learned to assign to each of these      
quantities  in their summation a        
sequence value4 ,but this is an         
empirical notion  and cannot be said to 
inhere in the mind itself4 ,let us take1
 for  instance1 the mustard seed4 ,if it
 were true that in one of these seeds   
there existed all the subsequent seeds  
which appear in the mustard  plant as   
separate and identifiable quantities1   
and not in essence1 then  there would   
perhaps be warrant for the notion that  
diversity1 as the  calculable element1  
is an a priori conception4 ,but1 as this
 is not  the case and since diversity is
 purely empirical and pertains only to  
the efflorescence of the one life it is 
manifestly absurd to take that  view4   
,under the most charitable allowances1  

therefore1 there can be but two         
quantities--unity and diversity2 yet not
 two1 for these are one4 ,unity  is the 
one quantity and diversity is the       
division of unity into a  transfinity of
 parts4 ,unity is infinite1 absolute and
 all-inclusive4  ,diversity is finite   
although it may be admitted to be       
transfinite1 or  greater than any       
assignable value4 ,unity alone is       
incomprehensible4  ,in order to         
understand something of its nature we   
divide it into a  diversity of parts2   
and because we fail to understand the   
transfinity  of the multitude of parts  
we mistakenly call them infinite4       
,when analysis shall have proceeded far 
enough into the abysmal  mysteries of   
diversity2 when the mathematical mind   
shall have been  overcome by the        
overwhelming perplexity of the maze of  
diverse parts1  it shall then fall      
asleep and upon awaking shall find that 
wonderfully  simple thing--unity4 ,it is
 the one quantity that is endowed with  

a magnitude which is both inconceivable 
and irresolvable4 ,the one  ineluctable 
fact in the universe is the             
incomprehensibility and  all-inclusivity
 of one-ness4 ,it is incomprehensible1  
inconceivable  and infinite at the      
present stage of mind development4 ,but 
the goal  of mind is to understand the  
essential character of unity1 of life4  
,its evolution will then stop1 for it   
will have reached the prize of  divinity
 itself whereupon the intellect exalted 
by and united with the  intuition shall 
also become one with the divine         
consciousness4            ,,chapter ,,ii
      ,,historical ,,sketch ,,of ,,the  
,,hyperspace ,,movement        ,egypt   
the ,birthplace of                      
,geometry--,precursors3 ,,nasir-,,eddin1
 ,,christoph    ,,clavius1 ,,saccheri1  
,,lambert1 ,,la ,,grange1               
,,kant--,influence of the    ,mecanique 
,analytique--,the ,parallel-,postulate  
the ,root and ,substance    of the      
,non-,euclidean ,geometry--,the ,three  

,great ,periods3 ,the ,formative1       
,determinative and                      
,elaborative--,,riemann and the         
,properties of ,analytic    ,spaces4    
,the evolution of the idea of a fourth  
dimension of space covers a long  period
 of years4 ,the earliest known record of
 the beginnings of the  study of space  
is found in a hieratic papyrus which    
forms a part of the  ,rhind ,collection 
in the ,british ,museum and which has   
been deciphered by  ,,eisenlohr4 ,it is 
believed to be a copy of an older       
manuscript of date  #cdjj ,b4 ,c41 and  
is entitled ,directions for ,knowing    
,all ,dark ,things  ,the copy is said to
 have been made by ,,ahmes1 an ,egyptian
 priest  between #agjj and #aajj ,b4 ,c4
 ,it begins by giving the dimensions of 
barns2 then follows the consideration of
 various rectilineal figures1  circles1 
pyramids1 and the value of pi 7,greek3  
p74 ,although many of  the solutions    
given in the manuscript have been found 
to be incorrect  in minor particulars1  

the fact remains that ,egypt is really  
the  birth-place of geometry4 ,and this 
fact is buttressed by the knowledge     
that ,,thales1 long before he founded   
the ,ionian ,school which was the       
beginning of ,greek influence in the    
study of mathematics1 is found  studying
 geometry and astronomy in ,egypt4      
,the concept of hyperspace began to     
germinate in the latter part of the     
first century1 ,b4 ,c4 ,for it was at   
this date that ,,geminos of ,,rhodes    
7,b4 ,c4 #gj7 began to think seriously  
of the mathematical labyrinth  into     
which ,,euclid',s parallel-postulate    
most certainly would lead if  an attempt
 at demonstrating its certitude were    
made4 ,he recognized  the difficulties  
which would engage the attention of     
those who might  venture to delve into  
the mysterious possibilities of the     
problem4  ,there is no doubt1 too1 but  
that ,,euclid himself was aware1 in some
  measure at least1 of these            
difficulties2 for his own attitude      

towards  this postulate seems to have   
been one of noncommittance4 ,it is1     
therefore1 not strange that the         
astronomer1 ,,ptolemy 7,a4 ,d4 #hg-afe71
  should be found seeking to prove the  
postulate by a consideration of  the    
possibilities of interstellar triangles4
 ,his researches1 however1  brought him 
no relief from the general              
dissatisfaction which he felt  with     
respect to the validity of the problem  
itself4    ,for nearly one thousand     
years after the attempts at solving the 
postulate by ,,geminos and ,,ptolemy1   
the field of mathematics lay            
undisturbed4 ,for it was at this time   
that there arose a strange  phenomenon1 
more commonly known as the ,dark ,ages1 
which put an  effectual check to further
 research or independent investigations4
  ,mathematicians throughout this long  
lapse of time were content to  accept   
,,euclid as the one incontrovertible1   
unimpeachable authority1  and even such 
investigations as were made did not have

 a rebellious  tendence1 but were mainly
 endeavors to substantiate his claims4  
,accordingly1 it was not until about the
 first half of the thirteenth  century  
that any real advance was made4 ,at this
 time there appeared  an ,arab1         
,,nasir-,,eddin 7#abja-abgd7 who        
attempted to make an improvement  on the
 problem of parallelism4 ,his work on   
,,euclid was printed in  ,rome in #aeid 
,a4 ,d41 about three hundred and twenty 
years after his  demise and was         
communicated in #afea by ,,john ,,wallis
 7#afaf-agjc7 to the  mathematicians of 
,oxford ,university4 ,although his      
calculations and  conclusions were      
respectfully received by the ,oxford    
authorities no  definite results were   
regarded as accomplished by what he had 
done4  ,it is believed1 however1 that   
his work reopened speculation upon  the 
problem and served as a basis1 however  
slight1 for the greater  work that was  
to be done by those who followed him    
during the next  succeeding eight       

hundred years4    ,about twenty years   
before the printing of the work of      
,,nasir-,,eddin1  ,,christoph ,,clavius 
7#aegd7 deduced the axiom of parallels  
from the  assumption that a line whose  
points are all equidistant from a       
straight  line is itself straight4 ,in  
his consideration of the                
parallel-postulate  he is said to have  
regarded it as ,,euclid',s ,x,i,i,ith   
axiom4 ,later ,,bolyai  spoke of it as  
the ,x,ith and later still1 ,,todhunter 
treated it as the  ,x,i,ith4 ,hence1    
there does not seem to have been any    
general unanimity  of opinion as to the 
exact status of the parallel-postulate1 
and  especially is this true in view of 
the uncertainty now known to have       
existed in ,,euclid',s mind concerning  
it4    ,,girolamo ,,saccheri            
7#affg-agcc71 a learned ,jesuit1 born at
 ,san ,remo1  came next upon the stage4 
,and so important was his work that it  
will  perpetuate the memory of his name 
in the history of mathematics4 ,he was  

a teacher of grammar in the ,jesuit     
,collegio di ,brera where ,,tommaso     
,,ceva1 a brother of ,,giovanni1 the    
well-known mathematician1 was teacher   
of mathematics4 ,his association with   
the ,,ceva brothers was especially      
beneficial to him4 ,he made use of      
,,ceva',s very ingenious methods in his 
first published book1 #afic1 entitled   
,solutions of ,six ,geometrical         
,problems ,proposed by ,count ,roger    
,ventimiglia4        ,a                 
-------------------------------- ,b     
,c -------------------------------- ,d  
,,fig4 #a4    ,,saccheri attacked the   
problem of parallels in quite a new way4
  ,examining a quadrilateral1 ,,abcd1 in
 which the angles ,a and ,b are  right  
angles and the sides ,,ac and ,,bd are  
equal1 he determined to  show that the  
angles ,c and ,d are equal4 ,he also    
sought to prove  that they are either   
right angles1 obtus acute4 ,he undertook
 to prove  the falsity of the latter two
 propositions 7that they are either     

obtuse or acute71 leaving as the only   
possibility that they must be  right    
angles4 ,in doing so1 he found that his 
assumptions led him into  contradictions
 which he experienced difficulty in     
explaining4    ,his labors in connection
 with the solution of the problems      
proposed by  ,,count ,,ventimiglia1     
including his work on the question of   
parallels1  led directly into the field 
of metageometrical researches1 and      
perhaps  to him as to no other who had  
preceded him1 or at least to him in a   
larger degree1 belongs the credit for a 
continued renewal of interest  in that  
series of investigations which resulted 
in the formulation of  the              
non-,euclidean geometry4    ,the last   
published work of ,,saccheri was a      
recital of his endeavors at             
demonstrating the parallel-postulate4   
,this received the ,imprimatur  of the  
,inquisition1 ,july #ac1 #agcc2 the     
,provincial ,company of ,jesus  took    
possession of the book for perusal on   

,august #af1 #agcc2 but  unfortunately  
within two months after it had been     
reviewed by these  authorities1         
,,saccheri passed away4    ,all efforts 
which had been made prior to the work of
 ,,saccheri were  based upon the        
assumption that there must be an        
equivalent postulate  which1 if it could
 be demonstrated1 would lead to a       
direct1 positive  proof of ,,euclid',s  
proposition4 ,although these and all    
other attempts  at reaching such a proof
 have signally failed and although it   
may  correctly be said that the entire  
history of demonstrations aiming at  the
 solution of the famous postulate has   
been one long series of utter  failures1
 it can be asserted with equal certitude
 that it has proven  to be one of the   
most fruitful problems in the history of
 mathematical  thought4 ,for out of     
these failures has been built a         
superstructure  of analytical           
investigations which surpasses the most 
sanguine  expectations of those who had 

labored and failed4    ,in #agff ,,john 
,,lambert 7#agbh-aggg7 wrote a paper    
upon the ,theory  of ,parallels dated   
,sept4 #e1 #agff1 first published in    
#aghf1 from  the papers left by ,f4     
,,bernoulli1 which contained the        
following  assertions3#b    #a4 ,the    
parallel-axiom needs proof1 since it    
does not hold for geometry  on the      
surface of the sphere4    #b4 ,in order 
to make intuitive a geometry in which   
the triangle's sum  is less than two    
right angles1 we need an imaginary      
sphere 7the  pseudosphere74    #c4 ,in a
 space in which the triangle's sum is   
different from two right  angles there  
is an absolute measure 7a natural unit  
for length74    ,at this time ,,immanuel
 ,,kant 7#agbd-ahjd71 the noted ,german 
metaphysician1  was in the midst of his 
philosophical labors4 ,and it is        
believed that  it was he who first      
suggested the idea of different spaces4 
,below is  given a statement taken from 
his ,prolegomena#c which corroborates   

this view4      ,that complete space    
7which is itself no longer the boundary 
of    another space7 has three          
dimensions1 and that space in general   
cannot have more1 is based on the       
proposition that not more than    three 
lines can intersect at right angles in  
one point4444 ,that we    can require a 
line to be drawn to infinity1 a series  
of changes to    be continued 7for      
example1 spaces passed through by       
motion7 in    indefinitum1 presupposes a
 representation of space and time which 
can only attach to intuition4      #b   
,vide ,new ,york ,mathematical ,society 
,bulletin1 ,vol4 ,,iii1    #ahic-d1 p4  
#gi1 ,g4 ,b4 ,,halstead on ,lambert's   
,non-,euclidean ,geometry4      #c      
,prolegomena1 ,,kant1 p4 #cg1 ,trans4 by
 ,j4 ,p4 ,,mahaffy and ,j4    ,h4       
,,bernard4    ,his differentiation      
between space in general and space which
 may be  considered as the boundary of  
another space shows1 in the light of    
the subsequent developments of the      

mathematical idea of space that  he very
 fully appreciated the marvelous scope  
of analytic spaces4 ,his  conception of 
space1 therefore1 must have had a       
profound influence  upon the mathematic 
thought of the day causing it to undergo
 a rapid  reconstruction at the hands of
 geometers who came after him4    ,under
 the masterly influence of ,,la ,,grange
 7#agcf-ahac7 the idea of  different    
spaces began to take definite shape and 
direction2 the  geometry of hyperspace  
began to crystallize2 and the field of  
mathesis  prepared for the growth of a  
conception the comprehension of which   
was destined to be the profoundest      
undertaking ever attempted by the  human
 mind4 ,unlike most great men whom the  
world learns tardily to  admire1 ,,la   
,,grange lived to see his talents and   
genius fully recognized  by his         
compeers2 for he was the recipient of   
many honors both from his  countrymen   
and his admirers in foreign lands4 ,he  
spent twenty years in  ,prussia where he

 went upon the invitation of ,,frederick
 the ,great who in  the ,royal summons  
referred to himself as the greatest king
 in ,europe  and to ,,la ,,grange as the
 greatest mathematician in ,europe4 ,in 
,prussia  the ,mecanique ,analytique and
 a long series of memoirs which were    
published in the ,berlin and ,turin     
,transactions were produced4 ,,la       
,,grange  did not exhibit any marked    
taste for mathematics until he was #ag  
years  of age4 ,soon thereafter he came 
into possession of a memoir by ,,halley 
quite by accident and this so aroused   
his latent genius that within one  year 
after he had reviewed ,,halley',s memoir
 he became an accomplished              
mathematician4    ,he created the       
calculus of variations1 solved most of  
the problems  proposed by ,fermat1      
adding a number of theorems of his own  
contrivance2  raised the theory of      
differential equations to the position  
of a  science rather than a series of   
ingenious methods for the solution  of  

special problems and furnished a        
solution for the famous  isoperimetrical
 problem which had baffled the skill of 
the foremost  mathematicians for nearly 
half a century4 ,all these stupendous   
tasks he  performed by the time he      
reached the age of nineteen4    ,the    
,mecanique ,analytique is his greatest  
and most comprehensive  work4 ,in this  
he established the law of virtual work  
from which1  by the aid of his calculus 
of variations1 he deduced the whole of  
mechanics1 including both solids and    
liquids4 ,it was his object in the      
,analytique to show that the whole      
subject of mechanics is implicitly      
embraced in a single principle1 and to  
lay down certain formulae from  which   
any particular result can be obtained4  
,he frequently made the  assertion that 
he had1 in the ,mecanique ,analytique1  
transformed  mechanics which he         
persistently defined as a geometry of   
four  dimensions#d into a branch of     
analytics and had shown the so-called   

mechanical principles to be the simple  
results of the calculus4  ,hence1 there 
can be no doubt but that ,,la ,,grange  
not only completed  the foundation1 but 
provided most of the material in his    
analyses and  other abstract results of 
great generality which he obtained in   
his  numerous calculations1 for the     
superstructure subsequently known as the
  geometry of hyperspace1 and in which  
the fourth dimensional concept  occupies
 a very fundamental place4      #d ,in  
#aged ,d',,alembert 7#agag-aghc7        
published an article in the    famous   
old ,encyclopedia edited by ,,diderot   
and himself on ,dimension4    ,in this  
article the idea of the fourth dimension
 is dwelt upon at    length4 ,the view  
which he expressed in this article1 of  
course1 served    greatly to popularize 
the conception among the learned men of 
the day1    and owing to the close      
relationship existing between           
,d',,alembert and ,,la    ,,grange1 it  
is not surprising that the latter should

 have been very much    enamored of the 
idea4    ,it is as if for nearly        
seventeen hundred years workmen1 such as
  ,,geminos1 of ,,rhodes1 ,,ptolemy1    
,,saccheri1 ,,nasir-,,eddin1 ,,lambert1 
,,clavius1  and hundred of others who   
struggled with the problem of parallels1
  had made more or less sporadic        
attempts at the excavation of the land  
whereon a marvelously intricate building
 was to be constructed4 ,there  is no   
historical evidence to show that any of 
them ever dreamed that  the results of  
their labors would be utilized in the   
manner in which  they have been used4   
,then came ,,kant with the wonderfully  
penetrating  searchlight of his         
masterful intellect who from the        
elevation which  he occupied saw that   
the site had great possibilities1 but he
 had not  the mathematical talent to    
undertake the work of actual1 methodical
  construction4 ,indeed his task was of 
a different sort4 ,however1 he          
succeeded in opening the way for ,,la   

,,grange and others who followed  him4  
,,la ,,grange immediately seized upon   
the idea which for more than  a thousand
 years had been impinging upon the minds
 of mathematicians  vainly seeking      
lodgment and began the elaboration of a 
plan in  accordance with which minds    
better skilled in the pragmatic         
application  of abstract principles than
 his could complete the work begun4     
,unfortunately1 on account of his       
intense devotion and loyalty to the     
study of pure mathematics1 and when he  
had reached the summit of  his greatness
 where he stood without a rival as the  
foremost living  mathematician1 his     
health became seriously affected1       
causing him to  suffer constant attacks 
of profound melancholia from which he   
died on  ,april #aj1 #ahac4    ,we come 
now to one of the most remarkable       
periods in the history of  mental       
development4 ,during the six hundred    
years between the birth of              
,,nasir-,,eddin and the death of ,,la   

,,grange the entire world of mathesis   
was being reconstituted4 ,since there   
had been gradually going on an  internal
 process which1 when completed1 forever 
would liberate the mind  from the narrow
 confines of consciousness limited to   
the three-space1  it is not surprising  
that we should find1 in the mathematical
 thought  of the time1 an absolutely    
epoch-making departure4 ,the innumerable
  attempts at the solution of the       
parallel-postulate1 all failures in the 
sense that they did not prove1 have     
intensified greatly the esteem in  which
 the never-dying elements of ,,euclid   
are held to-day4 ,and despite  the fact 
that there may come a time when his     
axioms and conclusions  may be found to 
be incongruent with the facts of        
sensuous reality2 and  though all of his
 fundamental conceptions of space in    
general1 his  theorems1 propositions and
 postulates may have to give way before 
the  searching glare of a deeper        
knowledge because of some revealed      

fault1  the perfection of his work in   
the realm of pure mathematics will      
remain  forever a master piece demanding
 the undiminished admiration of mankind4
    ,the parallel-postulate1 as stated  
by ,,euclid in his ,elements of         
,geometry1 reads as follows3      ,if a 
straight line meet two straight lines so
 as to make the two    interior angles  
on the same side of it taken together   
less than    two right angles1 these    
straight lines being continually        
produced1    shall at length meet upon  
that side on which are the angles which 
are less than two right angles4    ,on  
this postulate hang all the law and the 
prophets of the  non-,euclidean         
,geometry4 ,in it are the virtual       
elements of three  possible geometries4 
,furthermore1 it is both the warp and   
the woof  of the loom of present-day    
metageometrical researches4 ,it is the  
golden egg laid by the god ,,seb at the 
beginning of a new life cycle in        
psychogenesis4 ,its progeny are         

numerous--hyperspaces1 sects1 straights1
  digons1 equidistantials1 polars1      
planars1 coplanars1 invariants1         
quaternions1 complex variables1 groups  
and many others4 ,a wonderfully         
interesting breed1 full of meaning and  
pregnant with the power of final        
emancipations for the human intellect6  
,when the conclusions which were        
systematically formulated as a result   
of the investigations along the lines of
 hypotheses which controverted  the     
parallel-postulate were examined it was 
found that they fell into  three main   
divisions1 namely3 the synthetic or     
hyperbolic2 the analytic  or            
,,riemannian and the elliptic or        
,,cayley-,,klein4 ,these divisions or   
groups are based upon the three         
possibilities which inhere in the       
conception taken of the sum of the      
angles referred to in the above         
postulate as to whether it is equal to1 
greater or less than two right  angles4 
,the assumption that the angular sum is 

congruent to a straight angle is  called
 the ,euclidean or parabolic hypothesis 
and is to be distinguished  from the    
synthetic or hyperbolic hypothesis      
established by ,,gauss1  ,,lobachevski  
and ,,bolyai and which assumes that the 
angular sum is less  than a straight    
angle4 ,the elliptic or ,,cayley-,,klein
 hypothesis assumes  that the angular   
sum is greater than a straight angle4   
,,lobachevski1  however1 not satisfied  
with the statement of the               
parallel-postulate  as given by ,,euclid
 and which had caused the age-long      
controversy1  substituted for it the    
following3      ,all straight lines     
which1 in a plane1 radiate from a given 
point1 can1 with respect to any other   
straight line1 in the same    plane1 be 
divided into two classes--the           
intersecting and the                    
non-intersecting4 ,the boundary line of 
the one and the other class    is called
 parallel to the given line4    ,this is
 but another way of saying about the    

same thing that ,,euclid had  declared  
before1 and yet1 curiously enough it    
afforded just the liberty  that         
,,lobachevski needed to enable him to   
elaborate his theory4    ,for the       
purposes of this sketch the field of the
 development of  non-,euclidean geometry
 is divided into three periods to be    
known as3  7#a7 the formative period in 
which mathematical thought was being    
formulated for the new departure2 7#b7  
the determinative period during  which  
the mathematical ideas were given       
direction1 purpose and a  general       
tendence2 7#c7 the elaborative period   
during which the results  of the former 
periods were elaborated into definite   
kinds of geometries  and attempts made  
at popularizing the hypotheses4         
,the ,formative ,period    ,,charles    
,,frederich ,,gauss 7#aggg-ahee7 by some
 has been regarded as the  most         
influential mathematician that figured  
in the formulation of  the              
non-,euclidean geometry2 but closer     

examination into his efforts  at        
investigating the properties of a       
triangle shows that while his           
researches led to the establishment of  
the theorem that a regular  polygon of  
seventeen sides 7or of any number which 
is prime1 and also  one more than a     
power of two7 can be inscribed1 under   
the ,euclidean  restrictions as to      
means1 in a circle1 and also that the   
common  spherical angle on the surface  
of a sphere is closely connected with   
the constitution of the area inclosed   
thereby1 he cannot justly be  designated
 as the leader of those who formulated  
the synthetic school4  ,and this1 too1  
for the simple reason that1 as he       
himself admits in  one of his letters to
 ,,taurinus1 he had not published       
anything on the  subject4 ,in this same 
letter he informs ,,taurinus that he had
 pondered  the subject for more than    
thirty years and expressed the belief   
that  there could not be any one who had
 concerned himself more exhaustively    

with this second part 7that the sum of  
the angles of a triangle cannot  be more
 than #ahj degrees7 than he had4        
,writing from ,gttingen to ,,taurinus1  
,november #h1 #ahbd1 and commenting     
upon the geometric value of the sum of  
the angles of a triangle1 he  says3     
,your presentation of the demonstration 
that the sum of the angles    of a plane
 triangle cannot be greater than #ahj   
degrees does1    indeed1 leave something
 to be desired in point of geometrical  
precision4 ,but this could be supplied1 
and there is no doubt    that the       
impossibility in question admits of the 
most rigorous    demonstration4 ,but the
 case is quite different with the second
    part1 namely1 that the sum of the   
angles cannot be smaller than    #ahj   
degrees2 this is the real difficulty1   
the rock upon which    all endeavors are
 wrecked4444 ,the assumption that the   
sum of the    three angles is smaller   
than #ahj degrees leads to a new        
geometry    entirely different from our 

,euclidean--a geometry which is         
throughout consistent with itself1 and  
which ,i have elaborated    in a manner 
entirely satisfactory to myself1 so that
 ,i can solve    every problem in it    
with the exception of the determining of
 a    constant which is not a priori    
obtainable4    ,it appears from this    
correspondence that ,,gauss had in the  
privacy  of his own study elaborated a  
complete non-,euclidean geometry1 and   
had so thoroughly familiarized himself  
with its characteristics and            
possibilities that the solution of every
 problem embraced within it was  very   
clear to him except that of the         
determination of a constant4 ,he        
concluded the above letter by saying3   
,all my endeavors to discover           
contradiction or inconsistencies in     
this non-,euclidean geometry have been  
in vain1 and the only thing     in it   
that conflicts with our reason is the   
fact that if it were     true there     
would necessarily exist in space a      

linear magnitude     quite determinate  
in itself2 yet unknown to us4           
,judging from the correspondence between
 ,,gauss and ,,gerling 7#aghh-aheg71    
,,bessel 7#aghd-ahdf71 ,,schumacher and 
,,taurinus1 the nephew of ,,schweikart1 
and that between ,,schweikart and       
,,gerling1 there had grown up a general 
dissatisfaction in the minds of         
mathematicians of this period with      
,euclidean geometry and especially the  
parallel-postulate and its              
connotations4 ,,bessel expresses this   
general discontent in one of his        
letters to ,,gauss1 dated ,february #aj1
 #ahbi1 in which he says3      ,through 
that which ,,lambert said and what      
,,schweikart disclosed    orally1 it has
 become clear to me that our geometry is
 incomplete1    and should receive a    
correction1 which is hypothetical1 and  
if the    sum of the angles of the plane
 triangle is equal to #ahj degrees1     
vanishes4    ,the opinion of leading    
mathematicians at this time seems to    

have been  crystallizing very rapidly4  
,unconsciously the men of this formative
  period were adducing evidence which   
would give form and tendence to the     
developments in the field of mathesis at
 a later date4 ,they appear to  have    
been reaching out for that which1 ignis 
fatuus-like1 was always  within easy    
reach1 but not quite apprehensible4     
,a bolder student than ,,gauss was      
,,ferdinand ,,carl ,,schweikart         
7#aghj-aheg7  who also has been credited
 with the founding of the non-,euclidean
  geometry4 ,in fact1 if judged by the  
same standards as ,,gauss1 he would  be 
called the father of the geometry of    
hyperspace2 for he really  published the
 first treatise on the subject4 ,this   
was in the nature of  an inclosure which
 he inserted between the leaves of a    
book he loaned  to ,,gerling4 ,he also  
asked that it be shown to ,,gauss that  
he might give  his judgment as to its   
merits4    ,,schweikart',s treatise1    
dated ,marburg1 ,december1 #ahah1 is    

here quoted in  full3      ,there is a  
two-fold geometry--a geometry in the    
narrower sense1    the ,euclidean1 and  
an astral science of magnitude4         
,the triangles of the latter have the   
peculiarity that the sum of    the three
 angles is not equal to two right       
angles4      ,this presumed1 it can be  
most rigorously proven3 7a7 ,that the   
sum of the three angles in the triangle 
is less than two right    angles4       
7b7 ,that this sum becomes ever smaller1
 the more content    the angle incloses4
 7c7 ,that the altitude of an isosceles 
right-angled triangle indeed ever       
increases1 the more one    lengthens the
 side2 that it1 however1 cannot surpass 
a certain    line which ,i call the     
constant4      ,squares have            
consequently the following form3        
,illustration3 ,,fig4 #b4      ,if this 
constant were for us the radius of the  
earth 7so that    every line drawn in   
the universe1 from one fixed star to    
another1    distant #ij from the first1 

would be a tangent to the surface of    
the earth7 it would be infinitely great 
in comparison with the    spaces which  
occur in daily life4    ,the above1     
being the first published1 not printed1 
treatise on the new  geometry occupies a
 unique place in the history of higher  
mathematics4  ,it gave additional       
strength to the formative tendencies    
which  characterized this period and    
marked ,,schweikart as a constructive   
and  original thinker4    ,the nascent  
aspects of this stage received a        
fruitful contribution  when ,,nicolai   
,,lobachevski 7#agic-ahdg7 created his  
,imaginary ,geometry  and ,,janos       
,,bolyai 7#ahjb-ahfj7 published as an   
appendix to his father's  ,tentamen1 his
 ,science ,absolute of ,space4          
,,lobachevski and ,,bolyai  have been   
called the ,creators of the             
,non-,euclidean ,geometry4 ,and this    
appellation seems richly to be deserved 
by these pioneers4 ,their work  gave    
just the impetus most needed to fix the 

status of the new line  of researches   
which led to such remarkable discoveries
 in the more  recent years4 ,the        
,imaginary ,geometry and the ,science   
,absolute of  ,space were translated by 
the ,french mathematician1 ,j4 ,,ho,,el 
in #ahfh  and by him elevated out of    
their forty-five years of obscurity and 
non-effectiveness to a position where   
they became available for the           
mathematical public4 ,to ,,bolyai and   
,,lobachevski1 consequently1 belong  the
 honor of starting the movement which   
resulted in the development of          
metageometry and hence that which has   
proved to be the gateway of a new       
mathematical freedom4    ,,gauss1       
,,schweikart1 ,,lobachevski1 ,,wolfgang 
and ,,janos ,,bolyai were the  principal
 figures of the formative period and the
 value of their  work with respect to   
the formulation of principles upon which
 was  constructed the ,temple of        
,metageometry cannot be overestimated4  
,the ,determinative ,period    ,this    

period is characterized chiefly by its  
close relationship to the  theory of    
surfaces4 ,,riemann',s ,habilitation    
,lecture on ,the ,hypotheses  ,which    
,constitute the ,bases of ,geometry     
marks the beginning of this  epoch4 ,in 
this dissertation1 ,,riemann not only   
promulgated the system  upon which      
,,gauss had spent more than thirty years
 of his life in  elaborating1 for he was
 a disciple of ,,gauss2 but he disclosed
 his own  views with respect to space   
which he regarded as a particular case  
of manifold4 ,his work contains two     
fundamental concepts1 namely1 the       
manifold and the measure of curvature of
 a continuous manifold1  possessed of   
what he called flatness in the smallest 
parts4 ,the  conception of the measure  
of curvature is extended by ,,riemann   
from  surfaces to spaces and a new kind 
of space1 finite1 but unbounded1 is     
shown to be possible4 ,he showed that   
the dimensions of any space are         
determined by the number of measurements

 necessary to establish the  position of
 a point in that space4 ,conceiving1    
therefore1 that space  is a manifold of 
finite1 but unbounded1 extension1 he    
established the  fact that the passage  
from one element of a manifold to       
another may  be either discrete or      
continuous and that the manifold is     
discrete or  continuous according to the
 manner of passage4 ,where the manifold 
is regarded as discrete two portions of 
it can be compared1 as to  magnitude1 by
 counting2 where continuous1 by         
measurement4 ,if the whole  manifold be 
caused to pass over into another        
manifold each of its  elements passing  
through a one-dimensional manifold1 a   
two-dimensional  manifold is thus       
generated4 ,in this way1 a manifold of  
n-dimensions  can be generated4 ,on the 
other hand1 a manifold of n-dimensions  
can  be analyzed into one of one        
dimension and one of 7n-#a7 dimensions4 
,to ,,riemann1 then1 is due the credit  
for first promulgating the  idea that   

space being a special case of manifold  
is generable1 and  therefore1 finite4   
,he laid the foundation for the         
establishment of  a special kind of     
geometry known as the elliptic4 ,space1 
as viewed  by him1 possessed the        
following properties1 viz43             
generability1  divisibility1            
measurability1 ponderability1 finity and
 flexity4    ,these are the six pillars 
upon which rests the structure of       
hyperspace  analyses4#e      #e ,vide   
,nature1 ,vol4 ,,viii1 pp4 #ad-ag2 #cf1 
#cg 7#ahgc72 also    ,mathematical      
,papers1 pp4 #fe-ga4    ,generability is
 that property of geometric space by    
virtue of which  it may be generated1 or
 constructed1 by the movement of a line1
 plane1  surface or solid in a direction
 without itself4 ,divisibility is that  
property of geometric space by virtue of
 which it may be segmented  or divided  
into separate parts and superposed1 or  
inserted1 upon  or between each other4  
,measurability is that property by      

virtue  of which geometric space is     
determined to be a manifold of either  a
 positive or negative curvature1 also by
 which its extent may be  measured4     
,ponderability is that property of      
geometric space by virtue  of which it  
may be regarded as a quantity which can 
be manipulated1  assorted1 shelved or   
otherwise disposed of4 ,finity is that  
property  by virtue of which geometric  
space is limited to the scope of the    
individual consciousness of a unodim1 a 
duodim or a tridim and by  virtue of    
which it is finite in extent4 ,flexity  
is that property by  virtue of which    
geometric space is regarded as          
possessing curvature1  and in           
consequence of which progress through it
 is made in a curved1  rather than a    
geodetic line1 also by virtue of which  
it may be flexed  without disruption or 
dilatation4    ,,riemann who thus       
prepared the way for entrance into a    
veritable  labyrinth of hyperspaces is1 
therefore1 correctly styled ,the father 

of metageometry1 and the fourth         
dimension is his eldest born4 ,he  died 
while but forty years of age and never  
lived long enough fully  to elaborate   
his theory with respect to its          
application to the  measure of curvature
 of space4 ,this was left for his very  
energetic  disciple1 ,,eugenio          
,,beltrami 7#ahce-aijj7 who was born    
nine years after  ,,riemann and lived   
thirty-four years longer than he4 ,his  
labors mark  the characteristic         
standpoint of the determinative period4 
,,beltrami',s  mathematical             
investigations were devoted mainly to   
the non-,euclidean  geometry4 ,these led
 him to the rather remarkable conclusion
 that the  propositions embodied therein
 relate to figures lying upon surfaces  
of  constant negative curvature4        
,,beltrami sought to show that such     
surfaces partake of the nature of the   
pseudosphere1 and in doing so1 made use 
of the following illustration3          
,illustration3 ,,fig4 #c4               

,illustration3 ,,fig4 #d4    ,if the    
plane figure aabb is made to revolve    
upon its axis of symmetry  ,,ab the two 
arcs1 ab and ab will describe a         
pseudospherical  concave-convex surface 
like that of a solid anchor ring4 ,above
 and  below1 toward aa and bb1 the      
surface will turn outward with          
ever-increasing flexure till it becomes 
perpendicular to the axis  and ends at  
the edge with one curvature infinite4   
,or1 the half of a  pseudospherical     
surface may be rolled up into the shape 
of a champagne  glass1 as in ,fig4 #d4  
,in this way1 the two straightest lines 
of the  pseudospherical surface may be  
indefinitely produced1 giving a kind of 
space 7pseudospherical7 in which the    
axiom of parallels does not hold  true4 
,the determinative period marks the most
 important stage in the  development of 
non-,euclidean geometry and certainly   
the most  significant in the evolution  
of the idea of hyperspaces and multiple 
dimensionality4 ,,riemann and ,,beltrami

 are chief among those whose labors     
characterize the scope of this period4  
,their work gave direction and  general 
outline for later developments and all  
subsequent researches  along these lines
 have been conducted in strict          
conformity with the  principles laid    
down by these pioneer constructionists4 
,they laid out  the field and designated
 its confines beyond which no adventurer
 has  since dared to pass4    ,the great
 importance of the work of ,,riemann at 
this time may be seen  further in the   
fact that it not only marked the        
beginning of a new  epoch in geometry2  
but his pronouncement of the hypothesis 
that space  is unbounded1 though finite1
 is really the first time in the history
  of human thought that expression was  
ever given to the idea that space  may  
yet be only of limited extent4 ,before  
that time the minds of all  men seemed  
to have been unanimous in the           
consideration of space as an            
illimitable and infinite quantity4      

,the ,elaborative ,period    ,the       
elaborative stage includes the work of  
all those who1 working upon  the bases  
laid down by ,,lobachevski1 ,,bolyai1   
,,schweikart and ,,riemann1  have sought
 to amplify the conclusions reached by  
them4 ,among those  whose investigations
 have greatly multiplied the            
applications of  hyperspace conceptions 
are ,,ho,,el 7#ahff7 and ,,flye ,,st4   
,,marie 7#ahga7  of ,france2 ,,helmholtz
 7#ahfh71 ,,frischauf 7#ahgb71 ,,klein  
7#ahdi71 and  ,,baltzer 7#ahgg7 of      
,germany2 ,,beltrami 7#ahgb7 of ,italy2 
,,de ,,tilly 7#ahgi7  of ,belgium2      
,,clifford and ,,cayley 7#ahba7 of      
,england2 ,,newcomb 7#ahce7 and         
,,halstead of ,america4    ,these have  
been most active in popularizing the    
subject of  non-,euclidean geometry and 
incidentally the idea of the fourth     
dimension4 ,the great mass of           
non-professional mathematical readers1  
therefore1 owe these men an immeasurable
 debt of gratitude for the work  that   

they have done in the matter of         
rendering the conceptions which         
constitute the fabric of metageometry   
understandable and thinkable4 ,a  glance
 at the bibliography appended at the end
 of this volume will give  some idea of 
the enormous amount of labor that has   
been expended in an  effort to translate
 the most abstract mathematical         
principles into a  language that could  
easily be comprehended by the average   
intelligent  person4    ,the            
characteristic standpoint of this period
 is the popular  comprehension of the   
hyperspace concept and the consequent   
mental  liberation which follows4 ,for  
there is no doubt but that unheard of   
possibilities of thought have been      
revealed by investigations into  the    
nature of space4 ,an entirely new world 
has been opened to view and  only a     
beginning has been made at the          
exploration of its extent and           
resources4    ,one of the notable       
incidents of the early years of this    

period is the  position taken by ,,felix
 ,,klein who stands in about the same   
relation  to ,,cayley as ,,beltrami does
 to ,,riemann1 in that he assumed the   
task of  completing the work of his     
predecessor4 ,,klein held that there are
 only  two kinds of ,,riemannian        
space--the elliptical and the spherical4
 ,or  in other words1 that there are    
only two possible kinds of space in     
which the propositions announced by     
,,riemann could apply4 ,,sophus ,,lie1  
called the great comparative anatomist  
of geometric theories1 carried  his     
classifications to a final conclusion in
 connection with spaces of  all kinds   
and decided that there are possible only
 four kinds of three  dimensional       
spaces4    ,but whether men with        
peering1 microscopic1 histological      
vision shall  establish the existence of
 one or many spaces1 and regardless of  
the  mathematic rigor with which they   
shall demonstrate the self-consistency  
of the doctrines which they hold1 the   

fact remains that the hypotheses  thus  
maintained1 while they may be regarded  
as true descriptions of  the spaces     
concerned1 are1 nevertheless1           
incompatible4 ,all of them  cannot be   
valid4 ,it will perhaps be found that   
none of them are  valid1 especially     
objectively so4 ,the only true view1    
therefore1 of  these systems of         
hyperspaces is that which assigns them  
to their  rightful place in the         
infinitely vast world of pure mathesis  
where  their validity may go            
unchallenged and their existence        
unquestioned2  for in that domain of    
unconfined mentation1 in that realm of  
divine  intuitability1 the marvelous    
wonderland of ideas and notions1 one is 
not only disinclined to doubt their     
logical actuality1 but is quite  willing
 to accede their claims4                
,,chapter ,,iii      ,,essentials ,,of  
,,the ,,non-,,euclidean ,,geometry      
,the ,non-,euclidean ,geometry          
,concerned with ,conceptual    ,space   

,entirely--,outcome of ,failures at     
,solving the                            
,parallel-,postulate--,the ,basis of the
 ,non-,euclidean ,geometry--,space      
,curvature and ,manifoldness--,some     
,elements of the ,non-,euclidean        
,geometry--,certainty1 ,necessity and   
,universality as ,bulwarks    of        
,geometry--,some ,consequences of       
,efforts at ,solving the                
,parallel-,postulate--,the ,final ,issue
 of the ,non-,euclidean                 
,geometry--,extended ,consciousness4    
,the term non-,euclidean is used to     
designate any system of geometry  which 
is not strictly ,euclidean in content4  
,it is interesting to note how the term 
came to be used4 ,it appears to  have   
been employed first by ,,gauss4 ,he did 
not strike upon it suddenly1  however1  
as in the correspondence between him and
 ,,wachter in #ahaf  he used the        
designation anti-,euclidean and then1   
later1 following  ,,schweikart1 he      
adopted the latter's terminology and    

called it ,astral  ,geometry4 ,this he  
found in ,,schweikart',s first published
 treatise  known by that name and which 
made its appearance at ,marburg in      
,december1 #ahah4 ,finally1 in his      
correspondence with ,,taurinus in #ahbd1
  ,,gauss first used the expression     
non-,euclidean to designate the system  
which he had elaborated and continued to
 use it in his correspondence  with     
,,schumacher in #ahca4                  
,non-,legendrean1 semi-,euclidean and   
non-,archimedean are titles  used by ,m4
 ,,dehn to denote all kinds of          
geometries which represented  variations
 from the hypotheses laid down by       
,,legendre1 ,,euclid and  ,,archimedes4 
,the semi-,euclidean is a system of     
geometry in which the sum of the  angles
 of a triangle is said to be equal to   
two right angles1 but  in which one may 
draw an infinity of parallels to a      
straight line  through a given point4   
,the non-,euclidean geometry embraces   
all the  results obtained as a          

consequence of efforts made at finding a
  satisfactory proof of the             
parallel-postulate and is1 therefore1   
based  upon a conception of space which 
is at variance with that held by        
,,euclid4 ,according to the ,ionian     
school space is an infinite continuum   
possessing uniformity throughout its    
entire extent4 ,the non-,euclideans     
maintain that space is not an infinite  
extension2 but a finite though          
unbounded manifold capable of being     
generated by the movement of  a point1  
line or plane in a direction without    
itself4 ,it is also  held that space is 
curved and exists in the shape of a     
sphere or  pseudosphere and is          
consequently elliptical4    ,the        
inapplicability of ,,euclid's           
parallel-postulate to lines drawn  upon 
the surface of a sphere suggested the   
possibility of a space  in which the    
postulate could apply to all possible   
surfaces or that  space itself may be   
spherical in which case the postulate   

would be  invalidated altogether4       
,hence1 it is quite natural that        
mathematicians  finding themselves      
unable to prove the postulate with due  
mathetic  precision should turn their   
attention to the conceptually possible4 
,in this virtual abandonment of the     
perceptual for the conceptual lies  the 
fundamental difference between the      
,euclidean and the non-,euclidean       
geometries4 ,it may be said to the      
credit of the ,euclideans that they     
have sought to make their geometric     
conceptions conform as closely as       
possible to the actual nature of things 
in the sensuous world while  at the same
 time they must have perceived that at  
best their spatial  notions were only   
approximations to the sensuous actuality
 of objects  in space4    ,on the other 
hand1 non-,euclideans make no pretense  
at discovering any  congruency between  
their notions and things as they        
actually are4  ,the attitude of the     
metageometricians in this respect is    

very aptly  described by ,,cassius      
,,jackson ,,keyser who says3      ,he   
constructs in thought a summitless      
hierarchy of hyperspaces1    an endless 
series of orderly worlds1 worlds that   
are possible and    logically actual1   
and he is content not to know if any of 
them be    otherwise actual or          
actualized4#f      #f ,mathematics1 by  
,c4 ,j4 ,,keyser1 ,adrian ,professor of 
,mathematics1 ,columbia ,university4    
,the non-,euclidean is1 therefore1 not  
concerned about the applicability  of   
ensembles1 notions and propositions to  
real1 perceptual space  conditions4 ,it 
is sufficient for him to know that his  
creations  are thinkable4 ,as soon as he
 can resolve the nebulosity of his      
consciousness into the conceptual       
star-forms of definite ideas  and       
notions1 he sits down to the feast which
 he finds provided by  superfoetated    
hypotheses fabricated in the deeps of   
mind and logical  actualities           
imperturbed and unmindful of the weal of

 perceptual space  in its homogeneity of
 form and dimensionality4               
,fundamentally1 the non-,euclidean      
geometry is constructed upon the  basis 
of conceptual space almost entirely4    
,knowledge of its content is            
accordingly derived from a              
superperceptual representation of       
relations  and interrelations subsisting
 between and among notions1 ideas1      
propositions and magnitudes arising out 
of a conceptual consideration  thereof4 
,in other words1 representations of the 
non-,euclidean  magnitudes1 cannot be   
said to be strictly perceptual in the   
same sense  that three-space magnitudes 
are perceived2 for three-space          
magnitudes  are really sense objects    
while hyperspace magnitudes are not     
sense  objects4 ,they are far removed   
from the sensuous world and in order to 
conceive them one must raise his        
consciousness from the sensuous plane   
to the conceptual plane and become aware
 of a class of perceptions  which are   

not perceptions in the strict sense of  
the word1 but  superperceptions2 because
 they are representations of concepts   
rather  than precepts4    ,notions of   
perceptual space are constituted of the 
triple presentations  arising out of the
 visual1 tactual and motor sensations   
which are fused  together in their final
 delivery to the consciousness4 ,the    
synthesis  of these three               
sense-deliveries is accomplished by     
equilibrating  their respective         
differences and by correcting the       
perceptions of one  sense by those of   
another in such a way as to obtain a    
completely  reliable perception of the  
object4 ,this is the manner in which the
  characteristics of ,euclidean space   
are established4    ,the characteristics
 of non-,euclidean space are not arrived
 at exactly  in this way4 ,being beyond 
the scope of the visual1 tactile and    
motor  sense apprehensions1 it cannot be
 said to represent judgments derived    
from any consideration or elaboration of

 the deliveries presented  through these
 media4 ,yet1 the substance of          
metageometry1 or the science  of the    
measurement of hyperspaces1 may not be  
regarded as an a priori  substructure   
upon which the system is founded4 ,that 
is1 the conceptual  space of            
non-,euclidean geometry is not presented
 to the consciousness  as an a priori   
notion4 ,on the other hand1 the a       
posterioristic  quality of metageometric
 spaces marks the entire scope of       
motility of  the notions appertaining   
thereto4    ,the notions1 therefore1 of 
conceptual space are derivable only from
 the  perception of concepts1 or1       
otherwise consist of judgments          
concerning  interconceptual relations4  
,the process of apperception involved in
 the  recognition of relations which may
 be methodically determined is much     
removed from the primary procedure of   
perceiving sense-impressions  and fusing
 them into final deliveries to the      
consciousness for  conceptualization or 

the elaboration into concepts or general
  notions4 ,it is a procedure which is  
in every way superconceptual and        
extra-sensuous4 ,the metageometrician or
 analyst in no way relies upon          
sense-deliveries for the data of his    
constructions2 for1 if he did1 he       
should1 then1 be reduced to the         
necessity of confining his conclusions  
to the sphere of motility imposed by the
 sensible world with the result  that we
 should be able to verify empirically   
all his postulations4 ,but1  contrarily1
 he goes to the extra-sensuous1 and     
there in the realm of  pure             
conceptuality1 he finds the requisite   
freedom for his theories2  thus1        
environed by a sort of intellectual     
anarchism1 he pursues  analytical       
pleasures quite unrestrainedly4 ,the    
difference between  the two mental      
processes--that which leads from the    
sensible world to  conception and that  
which veers into the fields beyond--is  
so great  that it is hardly permissible 

to view the results arrived at in the   
outcome of the separate processes as    
being identical4    ,to illustrate this 
difference1 let us draw an analogy4 ,the
 miner  digs the iron ore out of the    
ground4 ,the iron is separated from the 
extraneous material and delivered to the
 furnaces where the metal is  melted and
 turned out as pig iron4 ,it is further 
treated1 and steel1  of various grades1 
cast iron and other kinds of iron are   
produced4 ,the  treatment of the iron   
ore up to this stage is similar to the  
treatment  of sense-impressions by the  
,thinker4 ,steel1 cast iron1 et cetera1 
are  similar to mental concepts4 ,later1
 the steel and other products are       
converted into instruments and numerous 
articles4 ,this represents  the         
superperceptual process4 ,trafficking in
 iron ore products1 such  as instruments
 of precision1 watch springs1 and the   
like1 represents  a stage still farther 
removed from the primary treatment of   
the  ore and is similar to that to which

 concepts are treated when the          
metageometrician manipulates them in the
 construction of conceptual             
space-forms4 ,perception is the dealing 
with raw iron ore while  conception is  
analogous to the production of the      
finished product4    ,superperception   
would be analogous to the trafficking in
 the  finished product as such and      
without any reference to the source  or 
the preceding processes4 ,thus the      
notions and judgments of the            
non-,euclidean geometry are arrived at  
as a result of a triple process  of     
perception1 conception and              
superperception the latter being merely 
superconceived as formal space-notions4 
,but it is obvious that the  more       
complex the processes by which judgments
 purporting to relate to  perceptual    
things are derived the more likely are  
those judgments to be  at variance with 
the nature of the things themselves4    
,in view of the foregoing1 the dangers  
resulting from identifying  the products

 of the two processes are very obvious  
indeed4 ,but the  difference between the
 two procedures is the difference       
between  ,euclidean and non-,euclidean  
geometries or the difference between    
perceptual space notions and conceptual 
space notions4 ,hence1 it is  not       
understood just how or why it has       
occurred to anyone that the two  notions
 could be made congruent4 ,magnitudes in
 perceptual1 sensible  space are things 
apart from those that may be said to    
exist in  mathematical space or that    
space whose qualities and properties    
have no  existence outside of the mind  
which has conceived them4 ,it is        
believed  to be quite impossible to     
approach the study of metageometrical   
propositions with a clear1 open mind    
without previously understanding  the   
fundamental distinctions which exist    
between them4    ,it follows1 therefore1
 as a logical conclusion that geometric 
space of  whatsoever nature is a purely 
formal construction of the intellect1   

and  for this reason is completely under
 the sovereignty of the intellect       
however whimsical its demands may be4   
,being thus the creature of the         
intellect1 its possibilities are limited
 only by the limitations of the         
intellect itself4 ,perceptual space1    
being neither the creature of the       
intellect nor necessarily an a priori   
notion resident in the mental           
substructure1 but existing entirely     
independent of the intellect or  its    
apprehension thereof1 cannot be expected
 to conform to the purely  formal       
restrictions imposed by the mind except 
in so far as those  restrictions may be 
determined by the nature of perceptual  
space4 ,and  for that matter1 it should 
not be forgotten that1 as yet1 we have  
no  means of determining whether or not 
the testimony of the intellect is       
thoroughly credible simply because there
 is no other standard by which  we may  
prove its testimony4 ,it is possible to 
justify the deliveries  of the eye by   

the sense of touch1 or vice versa4 ,it  
is also possible  to prove all our      
sense-deliveries by one or the other of 
the senses4  ,but we have no such good  
fortune with the deliveries of the      
intellect4  ,we have simply to accept   
its testimony as final2 because we      
cannot  do any better4 ,but if it were  
possible to correct the testimony of    
the intellect by some other faculty or  
power which by nature might be  more    
accurate than the intellect it should be
 found that the intellect  itself is    
sadly limited4    ,the possible         
curvature of space is a notion which    
also characterizes  the content of the  
non-,euclidean geometry4 ,it is upon    
this notion  that the question of the   
finity and unboundedness of space1 in   
the  mathematical sense1 rests4 ,in the 
curved space1 the straightest line  is a
 curved line which returns upon itself4 
,progression eastward  brings one to the
 west2 progression northward brings one 
to the south1  et cetera4 ,on this view 

space is finite1 but may not be regarded
 as  possessing boundaries4             
,space-curvature1 reinforced by the idea
 that space is also a manifold  is the  
enabling clause of metageometry and     
without them the analyst  dares not     
proceed4 ,here again1 we are led to the 
confession that  however fantastic these
 two notions may seem and evidently are1
 there  is nevertheless to be recognized
 in them a dim glimpse of a veritable   
reality--a slight foreshadowing of the  
revelation of some great kosmic         
mystery4    ,the manifoldness of space  
is the fiat of analysis4 ,it is the     
inevitable  outcome of the analyst's    
method of procedure4 ,his education1    
training  and view of things in general 
inhibit his arriving at any other       
result and he may be pardoned with good 
grace for his manufacture of  the       
space-manifold4 ,for by it perhaps a    
better appreciation of that  wonderful  
extension of consciousness in the nature
 of which is involved  the explanation  

of the perplexing problems which the    
manifold and other  metageometrical     
expedients faintly adumbrate may be     
gained4    ,it is pertinent1 in the     
light of the above1 to examine into some
 of the  relative merits of the three   
formal bulwarks of geometrical          
knowledge4  ,these are certainty1       
necessity and universality4             
,geometric certainty is derived solely  
from the nature of the premises  upon   
which it is based4 ,if the premises be  
contradictory1 it is1  of course1       
defective4 ,but if the premises are     
non-contradictory  or self-evident1 then
 the certainty of geometric notions and 
conclusions is valid4 ,another          
consideration of prime importance in    
this connection is the definition4 ,from
 it all premises proceed4  ,hence1 the  
definition is even more important than  
the premise2 for  it is the persisting  
determinant of all geometric conclusions
 while  the premise is dependent upon   
the limitations of the definition4 ,the 

determinative character of the          
definition has led to its apotheosis2   
but this1 admittedly1 has been necessary
 in order to give stability  and        
permanency to the conclusions which     
followed4 ,but in spite of this  it     
would appear that the certainty of      
geometric conclusions is not a  quality 
to be reckoned as absolute or final4    
,with the same certainty that it can be 
said the sum of the angles of  the      
triangle is equal to two right angles it
 may be asserted that  that sum is also 
greater or less than two right angles4  
,certainty  which is based upon the     
inherent congruity of definitions1      
premises  and propositions is an        
entirely different matter from that     
certainty  which arises out of the real1
 abiding validity of a scheme of        
thought4  ,but this difference is not   
lessened by the fact that the latter is 
dependent1 in a measure1 upon the       
correct systematization of our  spatial 
experiences by means of methodical      

processes4 ,euclidean  geometry1        
accordingly1 is not so certain in its   
applications as it is  utilitarian2 but 
non-,euclidean geometry is even less    
certain than the  former and            
consequently more lacking in its        
utilitarian possibilities4    ,the      
necessity of geometrical determinations 
is merely the necessity  which inheres  
in logical inferences or deductions4    
,these may or may  not be valid4        
,inasmuch as the necessariness of       
deductions is primarily  based upon the 
conditional certainty of premises and   
definitions it  appears that this       
quality is in no way peculiar to        
geometry whether  ,euclidean or         
non-,euclidean4 ,in like manner1 the    
universality of  geometric judgments may
 not properly be regarded as a          
peculiarity of  geometry2 but is        
explicable upon the basis of the formal 
character  of the assumptions which     
underlie it4 ,the chief value1 then1 of 
non-,euclidean geometry seems to abide  

in the fact that it clarifies our       
understanding as to the complex         
processes by which it is possible to    
organize and systematize our spatial    
experiences for assimilation and  use in
 other branches of knowledge4    ,with  
the above statement of the case of the  
non-,euclidean geometry  it is now      
thought permissible to state briefly    
some of the elements  thereof4#g      #g
 ,the science of pure mathematics is    
perhaps indebted to no one in so        
great a degree as to ,,george ,,bruce   
,,halstead1 formerly of the ,university 
of ,texas1 whose labors in connection   
with the popular exposition of    the   
non-,euclidean geometry have been most  
untiring and effectual4 ,vide           
,popular ,astronomy1 ,vol4 ,,vii and    
,,viii1 #aijj1 ,dr4 ,g4 ,b4 ,,halstead4 
,below will be found some of the        
elements obtained as a consequence of   
efforts made both at proving and        
disproving the parallel-postulate of    
,euclid3    ,if two points determine a  

line it is called a straight4    ,if two
 straights make with a transversal equal
 alternate angles they  have a common   
perpendicular4    ,a piece of a straight
 is called a sect4    ,if two equal     
coplanar sects are erected perpendicular
 to a straight1  if they do not meet1   
then the sect joining their extremities 
makes  equal angles with them and is    
bisected by a perpendicular erected     
midway between their feet4    ,the sum  
of the angles of a rectilineal triangle 
is a straight angle1  in the hypothesis 
of the right 7angle72 is greater than a 
straight  angle in the hypothesis of the
 obtuse 7angle72 is less than a straight
  angle in the hypothesis of the acute  
7angle74    ,the hypothesis of right is 
,euclidean2 the hypothesis of the acute 
is  ,,bolyai-,,lobachevskian2 the       
hypothesis of obtuse is ,,riemannian4   
,if one straight is parallel to a second
 the second is parallel to the  first4  
,parallels continually approach each    
other4    ,the perpendiculars erected at

 the middle point of the sides of a     
triangle are all parallel1 if two are   
parallel4    ,if the foot of a          
perpendicular slides on a straight its  
extremity  describes a curve called an  
equidistant curve1 or an equidistantial4
    ,an equidistantial will slide on its
 trace4    ,in the hypothesis of the    
obtuse a straight is of finite size and 
returns into itself4    ,two straights  
always intersect4    ,two straights     
perpendicular to a third straight       
intersect at a point  half a straight   
from the third either way4    ,a pole is
 half a straight from its polar4    ,a  
polar is the locus of coplanar points   
half a straight from its pole4          
,therefore1 if the pole of one straight 
lies on another straight the  pole of   
this second straight is on the first    
straight4    ,the cross of two straights
 is the pole of the join of their poles4
    ,any two straights inclose a plane  
figure1 a digon4    ,two digons are     
congruent if their angles are equal4    

,the equidistantial is a circle with    
center at the poles of its basal        
straight4    ,a typical postulate based 
upon the ,,bolyai hypothesis of the     
acute angle  is the following3    ,from 
any point ,p drop ,,pc1 a perpendicular 
to any given straight  line ,,ab4 ,if ,d
 move off indefinitely on the ray ,,cb1 
the sect will  approach as limit ,,pf   
copunctal with ,,ab at infinity4        
,illustration3 ,,fig4 #e4    ,,pd is    
said to be at ,p the parallel to ,,ab   
toward ,b4 ,,pf makes  with ,,pc an     
angle ,,cpf which is called the angle of
 parallelism for  the perpendicular     
,,pc4 ,it is less than a right angle by 
an amount  which is the limit of the    
deficiency of the triangle ,,pcd4 ,on   
the  other side of ,,pc1 an equal angle 
of parallelism gives the parallel  ,p to
 ,,ba towards ,,am4#h ,thus at any point
 there are two parallels  to a straight4
 ,a straight has1 therefore1 two        
separate points at  infinity4           
,straights through ,p which make with   

,,pc an angle greater than the  angle of
 parallelism and less than its          
supplement do not meet the  straight    
,,ab at all not even at infinity4       
#h ,,note4--,m may be any point on the  
line ,,ba indefinitely    produced4     
,the parallel-postulate is stated in the
 non-,euclidean geometry as  follows3   
,if a straight line meeting two straight
 lines make those angles which  are     
inward and upon the same side of it less
 than two right angles the  two straight
 lines being produced indefinitely will 
meet each other on  this side where the 
angles are less than two right angles4  
,it is stated by ,,manning#i in the     
following language3    ,if two lines are
 cut by a third and the sum of the      
interior angles on  the same side of the
 cutting line is less than two right    
angles the  line will meet on that side 
when sufficiently produced4    ,it is   
rather significant that in this         
postulate which is really  a definition 
of space should be found grounds for    

such diverse  interpretations as to its 
nature4 ,of course1 the moment the mind 
seeks to understand the infinite by     
interpreting it in the unmodified  terms
 of the apparently unchangeable finite  
it entangles itself into  insurmountable
 difficulties4 ,as a drowning man grasps
 after straws so  the mind1 immersed in 
endless abysses of infinity1 fails to   
conduct  itself in a seemly manner2 but 
gasps1 struggles and flounders and  is  
happy if it can1 in the depths of its   
perplexity1 discover a way  of logical  
escape4 ,the pure mathematician has a   
hankering after the  logically          
consistent in all his pursuits2 to him  
it is the ,holy  ,grail of his highest  
aspirations4 ,he seeks it as the devotee
 seeks  immortality4 ,it is to him a    
philosopher's stone1 the elixir of      
perpetual youth1 the eternal criterion  
of all knowledge4      #i ,vide         
,non-,euclidean ,geometry1 p4 #ia4      
,failures to demonstrate the celebrated 
postulate of ,,euclid led1 as a  matter 

of course1 to the substitution of       
various other postulates more  or less  
equivalent to it in that each of them   
may be deduced from the  other without  
the aid of any new hypothesis4    ,among
 those who sought proof by a restatement
 of the problem are the  following3     
#a4 ,,ptolemy3 ,the internal angles     
which two parallels make with a         
transversal on the same side are        
supplementary4    #b4 ,,clavius3 ,two   
parallel straight lines are equidistant4
    #c4 ,,proclus3 ,if a straight line  
intersects one of two parallels it also 
intersects the other4    #d4 ,,wallis3  
,a triangle being given another triangle
 can be constructed  similar to the     
given one and of any size whatever4     
#e4 ,,bolyai 7,w473 ,through three      
points not lying on a straight line a   
sphere can always be drawn4    #f4      
,,lorenz3 ,through a point between the  
lines bounding an angle a  straight line
 can always be drawn which will         
intersect these two lines4    #g4       

,,saccheri3 ,the sum of the angles of a 
triangle is equal to two right  angles4 
,there were1 of course1 many other      
statements and substitutions used by    
mathematicians in their endeavors       
satisfactorily to establish the truth   
of the parallel-postulate4 ,that their  
labors should have terminated1  first1  
by doubting it1 then by denying1 and    
finally1 by building up a  system of    
geometries which altogether ignores the 
postulate is just  what might naturally 
be expected of these men who have given 
to the  world the non-,euclidean        
geometry4 ,in doing what they did many1 
if  not all of them1 were not aware in  
any measure of the proportions  of the  
imposing superstructure that would be   
built upon their  apparent failures4    
,all of them undoubtedly must have      
sensed the vague  adumbrations forecast 
by the unfolding mysteries which they   
sought to  lay bare2 all of them must   
have felt as they executed the early    
tasks  of those crepuscular days of pure

 mathematics that the way which they    
were traveling would lead to the inner  
shrine of a higher knowledge  and a     
wider freedom2 they may have been led by
 divine intuition to  strike out on this
 new path and yet they could not have   
known how fully  their dreams would be  
realized by the mathematicians of the   
twentieth  century4 ,if so1 they were   
truly gods and mathesis is their        
kingdom4    ,the analyst proceeds upon a
 basis entirely at variance with that   
which  guides the ordinary investigator 
in the formulation of his conclusions4  
,the empirical scientist in arriving at 
his theories or hypotheses is  governed 
at all times by the degree of conformity
 which his postulates  exhibit to the   
actual phenomena of nature4 ,he         
endeavors to ascertain  just how far or 
in what degree his hypothesis is        
congruent with  things found in nature4 
,if the dissidence is found to          
predominate he  abandons his theory and 
makes another statement and again sets  

out to  determine the degree of         
conformity4 ,if he then finds that the  
natural  phenomena agree with his theory
 he accepts it as for the time being    
finally settling the question4 ,in all  
things he is limited by the  answer     
which nature gives to his queries4 ,not 
so with the exponent of  pure           
mathematics4 ,for him the truth of      
hypotheses and postulates is  not       
dependent upon the fact that physical   
nature contains phenomena  which answer 
to them4 ,the sole determining factor   
for him is whether  or not he is able to
 state with rational consistency the    
assumed  first principles and then      
logically develop their consequences4   
,if he  can do this1 that is1 if he can 
state his hypotheses with consistency   
and develop their consequences into a   
logical system of thought1 he is  quite 
satisfied and well pleased with his     
performances4 ,but the fact  that this  
is true is of vital significance for all
 who seek clearly to  understand the    

essential character of hyperspatiality4 
,it appears1 therefore1 that the science
 of consequences is the radical  essence
 of pure geometry4 ,the metageometrician
 enjoys unlimited freedom  in the choice
 of his postulates and suffers          
curtailment only when it  comes to the  
question of consistency4 ,he is at      
liberty to formulate as  many systems of
 geometry as the barriers of consistency
 will permit  and these are practically 
innumerable4 ,so long then as the laws  
of  compatibility remain inviolate his  
multiplication of postulate-systems  may
 proceed indefinitely4 ,is it strange   
then that under conditions  where an    
investigator has such unbridled liberty 
he should be found  indulging in        
mathetic excesses8    ,,kant held that  
the axioms of geometry are synthetic    
judgments a  priori2 but it appears that
 in the strictest sense this is not the 
case4 ,it depends upon the type of mind 
which is taken as a standard of         
reference4 ,if it be the uncultivated   

mind1 it is certain that to it  the     
relations expressed by an axiom would   
never appear spontaneously4  ,if on the 
other hand1 the standard be that of a   
cultivated mind it is  also equally     
certain that to it these relations would
 be discovered  only after methodical   
operations4 ,all judgments arrived at as
 a result  of logical processes should1 
it seems1 be regarded as judgments a    
posteriori1 i4e41 the results of        
empirical operations4 ,confessedly1  the
 facts adduced in course of             
experimentation serve as guides in      
choosing among all of the many possible 
logical conventions2 but our  choice    
remains untrammeled except by the       
compulsion arising out of  a fear of    
inconsistency4 ,the real criterion then 
of all geometries  is neither truth1    
conformability nor necessity1 but       
consistency and  convenience4    ,the   
difficulty with the non-,euclideans     
resolved itself into the  question as to
 whether it is more consistent1 as well 

as convenient1 to  establish a proof of 
the postulate by taking advantage of the
 support  to be found in other          
postulates or whether1 by seeking a     
demonstration  based upon the deliveries
 of sense-experience as to the nature of
  space and its properties1 a still more
 consistent conclusion might  be        
reached4 ,they had further perplexity1  
however1 when it came to a  decision as 
to whether the organic world is produced
 and maintained  in ,euclidean space or 
in a purely conceptual space which alone
 can  be apprehended by the mind's      
powers of representation4 ,unwilling to 
admit the existence of the world in     
,euclidean space1 they turned their     
attention to the examination of the     
properties of another kind of  space    
so-called which unlike the space of the 
,ionian school could be  made to answer 
not only all the purposes of plane and  
solid figures1  but of spherics as well4
 ,and so1 the manifold space was        
invented by  ,,riemann and later        

underwent some remarkable improvements  
at the hands  of his disciple1          
,,beltrami4 ,but it may be said here1   
parenthetically1  that the truth of the 
whole matter is that our world is       
neither in  ,euclidean nor              
non-,euclidean space1 both of which1 in 
the last analysis1  are conceptual      
abstractions4 ,although it may not be   
denied that the  ,euclidean space is the
 more compatible4    ,the problem of    
devising a space1 if only a very limited
 portion1 in  which could be            
demonstrated the assumed alternative    
hypothesis and  its consequences        
logically developed1 occasioned no      
inconsiderable  concern for the         
non-,euclidean investigators2 but       
neither ,,lobachevski1  ,,bolyai nor    
,,riemann were to be baffled by the     
difficulties which they  met4 ,these    
only cited them to more laborious toil4 
,having succeeded in  mentally          
constructing the particular kind of     
space which was adaptable  to their     

rigorous mathetic requirements it       
immediately occurred to  them that all  
the qualities of the limited space thus 
devised might  logically be amplified   
and extended to the entire world of     
space and  that what is true of figures 
constructed in the segmented portion  of
 space which they used for experimental 
purposes is also true of  figures drawn 
anywhere in the universe of this space  
as all lines  drawn in the finite1      
bounded portion could be extended       
indefinitely  and all magnitudes        
similarly treated4 ,from these results1 
it was  but a single step to the        
conclusion which followed--that either  
an  entirely new world of space had been
 discovered or that our notion of  the  
space in which the organic world was    
produced is wholly wrong and  needs     
revision4 ,but notwithstanding the      
insurmountable obstacles which  stood in
 the way of the investigators who made  
the attempt to discover  the homology   
which might exist between the           

characteristics of the  newly fabricated
 space and the phenomenal world1        
investigations were  carried forward    
with almost amazing recklessness and    
loyalty to the  mathetic spirit until it
 was discovered that all efforts to     
trace out  any definite lines of        
correspondence were futile4 ,then the   
policy of  ignoring the question of     
conformability was adopted and has since
 been  pursued with unchangeable        
regularity by the analytical            
investigator4    ,among the results     
obtained by the non-,euclideans in their
 profound  researches into the nature of
 hyperspace are these3 #a4 ,it was found
  that the angular sum of a triangle1   
being ordinarily assumed to be a        
variable quantity1 is either less or    
greater than two right angles  so that a
 strictly ,euclidean rectangle could not
 be constructed4 #b4  ,the angle sums of
 two triangles of equal area are equal4 
#c4 ,no two  triangles not equal can    
have the same angles so that similar    

triangles  are impossible unless they   
are of the same size4 #d4 ,if two equal 
perpendiculars are erected to the same  
line1 their distance apart  increases   
with their length4 #e4 ,a line every    
point of which is equally  distant from 
a given straight line is a curved line4 
#f4 ,any two lines  which do not meet1  
even at infinity1 have one common       
perpendicular  which measures their     
minimum distance4 #g4 ,lines which meet 
at infinity  are parallel4 ,but it is   
apparent that these results have not    
followed  upon any mathematical         
consequence of other supporting         
postulates or  axioms such as would     
place them on a cordinate basis with    
those used  as a support for the        
parallel-postulate2 for they are based  
upon the  envisagement of an entirely   
new principle of space-perception and   
belong to a wholly different set of     
space qualities4    ,the final issue    
then of the non-,euclidean geometry is  
neither in  the utility of its processes

 and conclusions nor in the increscent  
inclination towards a new outlook upon  
the world of mathesis2 but  resides     
solely in the possibilities yet to be   
developed in that vast  domain of       
analytical thought which it has         
discovered and opened to  view4 ,to say 
that it sheds any light upon the nature 
of the universe  is perhaps to take the 
radical view2 yet it cannot be doubted  
that the  researches incident to the    
formulation of the non-,euclidean       
geometry  have greatly extended the     
scope of consciousness4 ,whether the    
extension  is valid and normal or simply
 a hypertrophic excrescence of mental   
feverishness2 whether by virtue of it we
 shall more closely approach  an        
understanding of the true nature of the 
mind of the ,infinite1 or  shall all    
fall into insanity1 are certainly       
debatable questions4 ,it  nevertheless  
appears evident that humanity has gained
 something of  real1 abiding permanence 
by this new departure4 ,if that         

something be  merely an extended        
consciousness or an awakening to the    
fact that  there are stages of awareness
 beyond the strictly sensuous1 and every
  observable evidence points to this1   
then there has only begun the  process  
by which the faculty of conscious       
functioning in this new world  shall    
become the normal possession of the     
human species4 ,but this new  world     
cannot be said to be of mathematical    
import2 for it is doubtful  if          
mathematical laws such as have been     
devised up to the present time1  would  
obtain therein4 ,so that if anything1 it
 must be psychological and  vital4      
,on this view the worlds of hyperspace  
inlaid with analytic  manifoldnesses and
 constant curvatures are but the primal 
excitants  which will finally awaken in 
the mind the faculty of awareness in the
  new domain of psychological content4  
,then will come the blooming of  the    
diurnal flower of the mind's immortality
 and the outputting of the  organ of    

consciousness wherewith the infinite    
stretches of hyperspaces1  the low-lying
 valleys of reals and imaginaries and   
the uplifting hills  of finites and     
infinites shall be divested of their    
mysteries and  stand out in their       
unitariness no longer draped in the veil
 of the  inscrutable and the            
incomprehensible4    ,the fourth        
dimension1 regarded by some as a new    
scope of motion for  objects in space1  
by others as a new and strange direction
 of spatial  extent and by others still 
as the doorway of the temple of exegesis
  wherein an explanation may be found   
for the entire congeries of  mysteries  
and supermysteries which now perplex the
 human mind1 may also  be said to be the
 key to the non-,euclidean geometry4    
,but it really  complicates the         
situation2 for one has to be capable of 
prolonged  abstract thought even to     
envisage is as a conceptual possibility4
  ,,poincar#aj says3 ,any one who should
 dedicate his life to it could1         

perhaps1 eventually imagine the fourth  
dimension1 implying thereby  that a     
lifetime of prolonged abstract thought  
is necessary to bring the  mind to that 
point of ecstasy where it could even so 
much as imagine  this additional        
dimension4 ,nevertheless by it 7the     
fourth dimension7  was the              
non-,euclidean geometry made and without
 it was not any of the  hyperspaces made
 that were made4 ,it is the view which  
geometers have  taken of space in       
general that has made the fourth        
dimension possible1  and not only the   
fourth1 but dimensions of all degrees4  
,the basis of  the non-,euclidean       
geometry may be found then in the notion
 of space  which has been predominant in
 the minds of the investigators4        
#aj ,vide ,nature1 ,vol4 ,,xlv1 #ahib4  
,finally1 it should be pointed out that 
the non-,euclidean geometry1  though a  
consistent system of postulates1 has    
been constructed upon a  misconception  
based upon the identification of real1  

perceptual space  with systems of       
space-measurements4 ,hyperspaces which  
are not spaces  at all should not be    
confounded with real space4 ,but they   
constitute  the substance of            
non-,euclidean geometry2 they are its   
blood and sinews4  ,their study is      
interesting1 because of the             
possibilities of speculation  which it  
offers4 ,no mind that has thought deeply
 upon the intricacies  of the fourth    
dimension1 or hyperspace1 remains the   
same after the  process4 ,it is bound to
 experience a certain sense of humility1
 and  yet some pride born of a knowledge
 that it has been in the presence of  a 
great mystery and has delved into the   
fearful deeps of kosmic mind4  ,to the  
mind that has thus been anointed by the 
sacred chrism of the  inner mysteries of
 creative mentality there always come   
that stillness  and calm such as        
characterize the aftermath of reflection
 upon the  incomprehensible and the     
transfinite4            ,,chapter ,,iv  

,,dimensionality      ,arbitrary        
,character of ,dimensionality--,various 
,definitions of    ,dimension--,real    
,space and ,geometric ,space            
,differentiated--,the    ,finity of     
,space--,difference ,between the ,purely
 ,formal and the    ,actual--,space as  
,dynamic ,appearance--,the ,a ,priori   
and the ,a    ,posteriori as ,defined by
 ,,paul ,,carus4      ,in previous      
chapters we have traced the growth and  
development of the  non-,euclidean      
geometry showing that the so-called     
fourth dimension is  an aspect thereof4 
,it is now deemed fitting that we should
 enter into  a more detailed study of   
the question of dimensionality with a   
view to  examining some of the          
difficulties which encompass it4    ,the
 question of dimension is as old as     
geometry itself4 ,without it  geometric 
conclusions are void and meaningless4   
,yet the conception  of dimensionality  
itself is purely conventional4 ,in its  
application  to space there is involved 

a great deal of confusion because of the
  inferential character of its          
definition4 ,for instance1 commonly we  
measure a body in space and arbitrarily 
assign three elements to  determine its 
position4 ,the simplest standard for    
this purpose is the  cube having three  
of its edges terminating at one of its  
corners4                 ,d             
,a                                      
-------------------------             ,b
                        ,c              
,,fig4 #f4    ,thus because it is found 
that the entire volume of a cube is     
actually  comprehended within the       
directions indicated by the lines ab1 bc
  and db it is determined that the three
 cordinates of the point b  are         
necessary and sufficient to establish   
the dimensions of the cube  and         
consequently of the space in which it   
rests4 ,the conception may be  stated in
 this way3 ,if a collection of elements1
 say points or lines1  be of such a     
nature or order that it is sufficient to

 know a certain  definite number of     
facts about it in order to be able to   
distinguish  every one of the elements  
from all the others1 then the assemblage
 or  collection of elements is said to  
be of the same number of dimensions  as 
there are elements necessary to its     
determination4 ,in the above  figure    
there are three elements1 namely1 the   
lines ab1 bc1 and  db1 which are        
necessary and sufficient for the        
determination of the  position of the   
point b4 ,in this way geometers have    
determined that  our space is           
tridimensional2 but it is obvious that  
this conclusion is  based not upon any  
examination of space itself but upon the
 measurement  of bodies in space4 ,upon 
this view it is seen that conclusions   
based  upon such a procedure render our 
notion of the extension of bodies  in   
space identical with the notion of      
spatial extensity4 ,in other  words1 we 
take bodies in space and by examining   
their characteristics  and properties   

arrive at an alleged apodeictic judgment
 of space4 ,it  is by means of this     
conventional norm of geometric knowledge
 that  various other spaces1 notably the
 one-1 two-1 four-and n-space1 have     
been devised4 ,it would appear that if  
some more absolute standard of          
measurement or definition of space were 
adopted the confusion which  now clings 
to the conception of dimension could be 
obviated4 ,for if it  be true that three
 and only three elements are necessary  
to determine  a point-position in our   
space and that in this determination we 
also  find the number of dimensions of  
space1 then it may also be true that    
n-cordinates would just as truly        
determine the dimensionality of  an     
n-space1 which is granted4 ,but then the
 n-space would be just  as legitimate as
 the three-space2 for it is determined  
by exactly  the same standards4 ,it is  
both quantitatively and qualitatively   
the  same4 ,if1 however1 on account of  
the exigencies that might arise1  we are

 forced to seek solace in the notion of 
an n-space whither  shall we turn for   
it8 ,it cannot be found2 for it is      
imperceptible1  uninhabitable1          
non-existent1 and therefore1 absolutely 
and purely an  abstraction4             
,consequently1 there must be something  
radically wrong with  the definition of 
space or with its determinants4    ,the 
purely arbitrary character of           
dimensionality is very aptly  described 
by ,,cassius ,,jackson ,,keyser1 who    
says3      444 ,the dimensionality of a 
given space is not unique1 but          
depends upon the choice of the geometric
 entity for primary or    generating    
element4 ,a space being given1 its      
dimensionality    is not therewith      
determined1 but depends upon the will of
 the    investigator who by a proper    
choice of generating element endows     
the space with any dimensionality he    
pleases4 ,that fact is of    cardinal   
significance for science and            
philosophy4#aa      #aa ,vide ,monist1  

,vol4 ,,xvi1 #ahif1 ,mathematical       
,emancipations4    ,it is a fact of     
cardinal significance for science2      
because it  emphasizes the necessity for
 some more rational procedure than that 
of the geometrician in arriving at an   
absolutely unique method of  determining
 the dimension and essential nature of  
real space4 ,its  significance for      
philosophy lies in the need of a        
logical1 rigidly  exclusive and         
absolutely peculiar standard of space   
definition4 ,the  definition of         
perceptual space should be such as      
rigorously inhibits  its inclusion as a 
particular in any general class4 ,the   
necessity for  this is warranted by its 
universality and uniqueness4    ,the    
lines of demarkation between what is    
recognized as perceptual  space and what
 has been called geometric or conceptual
 space should  be very sharply drawn4   
,so that when reference is made to      
either there  will be no doubt as to    
which is meant4 ,and then1 too1         

conceptual  space is no space at all1   
properly speaking4 ,it is merely a      
system of  space-measurement4 ,and as   
such has no logical right to be put in  
the  same category as perceptual space4 
,real space is unique4 ,geometric space 
belongs to a class whose members  are   
capable of indefinite multiplication4   
,it is certainly most  illogical to     
identify them4 ,perceptual space1       
figuratively speaking1  is a quantity2  
analytic space is the foot-rule1 the    
yard-stick1 the  kilometer1 by which it 
is measured and apportioned4 ,it is     
logically  impossible to predicate the  
same conclusion for both of them4 ,that 
is1 to do so causes a profound fracture 
of the fundamental norms of  logic4     
,such conclusions being thus            
illegitimate it is rather surprising    
that an error of this nature should have
 been made4 ,it is perhaps  accountable 
for on the grounds of the geometer's    
complete insouciance  as to how his     
postulates shall stand in their relation

 to things in the  phenomenal world4    
,it is agreed that as convenient as is  
,euclid's system of  space-measurement  
it is not by any means congruent with   
the extension  of real space objects4   
,it does1 however1 approximate congruity
 with  these objects as nearly as       
possible4 ,how then could it be expected
  that a system of space-measurement so 
far removed from this primary           
congruence as the non-,euclidean system 
is should exhibit more obvious  signs of
 correspondence8 ,but the advocates of  
the n-dimensionality  of space have     
illatively asserted the identity of     
space and its  dimensions4 ,accordingly1
 there is not recognized any distinction
  between their conception of space     
itself and its qualitative              
peculiarities4 ,they use the terms      
interchangeably4 ,so that dimension     
means space and vice versa4 ,in this    
lack of discrimination may  be found the
 source of much of the confusion which  
attaches to the  conception of space4   

,if it were arguable that the relation  
between space and its dimensions  is the
 same as that between matter and its    
properties then the  restriction of this
 relation to three and only three       
directions of  extent would be          
disallowed2 for the reason that if1 as  
is commonly  done1 dimension be made to 
mean direction of extent1 there would   
be an unlimited number of directions of 
extent and they would  all be           
perceptible4 ,but this is really another
 fundamental fault4  ,non-,euclideans   
have stretched the meaning of the term  
dimension so that  it not only includes 
the idea of direction but an entirely   
new class of  qualities--the fourth     
dimension4 ,and despite this reformation
 of the  original conception1 they      
demand that it shall be called space4   
,we have just shown that the generic    
concept of dimensionality is that  three
 and only three cordinates are necessary
 and sufficient for  its determination4 
,granting that this is true1 are we not 

compelled  consequently to see that we  
have1 by adding a fourth or             
n-dimensions1  involved ourselves into a
 more complex situation than before8    
,for by  postulating a fourth dimension 
either we have created a new world whose
  dimensions are four in number or we   
have explicitly admitted that the  three
 dimensions have a fourth4 ,aside from  
the logical difficulties  which beset   
these conclusions there is also set up a
 condition which is  at variance with   
the most elementary requirements of     
common sense4    ,thus far mathematical 
thought has not served to clarify our   
notions  of space nor to shed any new   
light upon the vital processes which are
  alleged to have their explanation in  
the new discovery4 ,simply stated1      
metageometricians have brought us to the
 place where we must either  recognize  
that the fourth dimension is another    
sphere lying dangerously  near the earth
 in which space extends in four primary 
directions and  in which four cordinates

 are necessary for its determination or 
we  are driven to the other horn of the 
dilemma where we are brought face  to   
face with the conclusion that the three 
perceptual space dimensions  have in    
common a hitherto unknown property or   
extension in virtue of  which it may be 
viewed as having an unlimited number of 
dimensions4  ,to accept the latter view 
is equivalent to saying that1 in the    
above  figure1 the three lines ab1 bc   
and db have formed a triple  entente by 
which they have mutually and severally  
acquired a new  domain1 hyperspace1 and 
in which1 because of the vast resources 
of the  region1 they are able to perform
 wondrous things4    ,let us examine    
briefly the various current definitions 
of dimension4  ,it is assumed by not a  
few that dimension is the same as       
direction4  ,but can we grant this      
wholly to be true8 ,if so1 then a mere  
child  may see that there are and must  
necessarily be as many dimensions as    
there are directions4 ,primarily1 there 

are six directions of space  and an     
unlimited number of subsidiary          
directions4 ,on this view it  is not    
necessary to invent a new domain of     
space if the object be  merely to       
discover and utilize a greater number of
 dimensions than has  heretofore been   
allowed4 ,for the identification of the 
term dimension  with direction already  
makes available an almost infinite      
number of  dimensions4 ,but this view is
 objected to by the advocates1 for it is
  contrary to the hypothesis of         
n-dimensionality4    ,dimension also    
means extent4 ,this is partially true4  
,it cannot be  wholly true4 ,for1 if it 
were1 then1 space would have only one   
dimension  which is also not allowable  
under the hypothesis4 ,then the         
definition  leaves out of account the   
idea that space is at the same time a   
direction or collection of directions4  
,the term extension is generic  and when
 applied to space means extension in all
 possible directions  and not in any one

 direction4 ,so that it is not          
permissible to say  that space extends  
in this direction or that because it    
extends in all  directions              
simultaneously and equally4             
,geometers claim that space is a system 
of cordinates necessary for  the        
establishment of a point-position in it4
 ,this view1 however1  identifies space 
with a system of space-measurement and  
is therefore  faulty4 ,according to this
 view there may be as many spaces as    
there are  systems of space-measurement 
and the latter may be limitless4 ,but if
  the totality of spaces are to be      
viewed as one space then we shall have  
one space with an indefinite number of  
dimensions2 also an indefinite  number  
of space measurements which would be    
confusing4 ,much1 if not  all1 of such a
 system's utility and convenience would 
be unavailable  or useless4 ,that1 too1 
would be in violation of the avowed     
purpose of  these investigations which  
is to enhance the utility and           

convenience of  mathematic operations4  
,now it is evident that space is neither
 direction1 extension1 a system  of     
space-measurement nor a system of       
manifolds whose dimensions are          
generable4 ,and this is so for the same 
reason that a piece of cloth  is not the
 elements of measurement--inches1 feet1 
yards--by which it  is apportioned4 ,and
 because we find that the fabric of     
space lends  itself accommodatingly to  
our conventional norms of measurement is
 not  sufficient reason for identifying 
it with these norms4 ,here we have the  
source of all error in mathematical     
conclusions about the nature of  space2 
because all such conclusions are based  
not upon the intrinsic  nature of space1
 but upon artificial forms which we     
choose to impose  upon it for our own   
convenience4 ,but it should be          
remembered that the  irregularities     
which we note are not in space itself   
but inhere in  the forms which we use4  
,for these purposes space is extremely  

elastic  and accommodates itself to the 
shape and scope of any construction  we 
may decide to try upon it4 ,in this     
respect it is like water which  has no  
regard for the shape1 size or kind of   
vessel into which it may  be posited4   
,there is one thing certain that judging
 from the above  considerations there   
has been not yet any absolute1          
all-satisfying  definition devised for  
space by mathematicians4    ,the best   
definitions hitherto constructed are    
purely artificial and  arbitrary        
determinations4 ,it is rather anomalous 
that there should be  so little         
unanimity about what is the most        
fundamental consideration  of           
mathematical conclusions which are      
supposed to be so certain1 so  necessary
 and universal as to be                 
incontrovertible4 ,confessedly1 it  is a
 condition which raises again the       
question as to just what are  the limits
 of mathematical certainty and necessity
 and just how far  we shall depend upon 

the validity of mathematics to determine
 for us  absolutely certain conclusions 
about the nature of space4 ,in view  of 
the uncertainty noted1 are we justified 
in following too closely  the mathematic
 lead even in matters of logic1 to say  
nothing of our  conception of space8 ,it
 seems that we shall have necessarily1  
on  account of the recognized           
limitations of mathematics in this      
matter1  to turn to some more tenable   
source for the norms of our knowledge   
concerning space4 ,for in the light of  
the rather indefensible position  which 
metageometricians have involved         
themselves there appears to be no  hope 
in this direction4    ,it is undoubtedly
 safer not to rely altogether upon the  
purely  abstract1 even in the world of  
mathesis1 for any absolute criterion  of
 knowledge4 ,it is perhaps well that we 
should expunge the word  absolute from  
our vocabularies4 ,it is really a       
misnomer and has no  meaning in the     
lexicon of nature4 ,there is in reality 

no absolute in  the sense of final      
absolution from all conditions or       
restrictions4    ,in the ultimate       
analysis there is unquestionably no hue1
 tone1 quality1  condition nor any      
imaginable posture of life1 being or    
manifestation  that is absolved from    
every other one of its class or from the
  totality4 ,all these are relational   
and interdependent4 ,there is no room   
for the absolute4 ,in fact1 it is a     
quality which cannot in any way  be     
ascribed to any aspect of kosmic        
manifestation4 ,it has existence  only  
in the mind and has been devised for the
 purpose of marking the  limits of its  
scope4 ,all being is relative2 all life 
is relative and is  destined to change  
its qualities as it evolves4 ,all       
knowledge is also  relative and what is 
true of one state may not be true of    
another2 what  is true of one life may  
not be true of another life2 the        
limitations  of one degree of knowledge 
may not have any bearings upon another  

degree4 ,the norms of one will not      
satisfy the conditions of another  stage
 of manifestation4 ,it is always within 
limits that the criterion  of knowledge 
will be found to satisfy a given set of 
conditions4 ,hence  within certain      
limits mathematical conclusions will    
maintain their  validity4 ,error is     
committed by pushing the validity of    
these limits  to a position without the 
sphere of limitations4 ,this seems to be
 the  crux of the whole matter4         
,mathematicians1 notably                
non-,euclideans1  have sought to extend 
the comparatively small sphere of limits
 of  congruence between mathematic and  
perceptual space to such an extent  as  
to cause it to encroach upon forbidden  
territory4 ,in doing this they  have    
erred grievously1 causing serious       
offense to the more sensitive  spirit of
 the high-caste mathematicians among    
whom are none more truly  conservative  
than ,,paul ,,carus1#ab who says3       
,metageometricians are a hot-headed race

 and display sometimes all    the       
characteristics of sectarian fanatics4  
,to them it is quite    clear there may 
be two straight lines through one and   
the same    point which do not coincide 
and yet are both parallel to a third    
line4      #ab ,vide ,monist1 ,vol4     
,,xix1 p4 #djb 7#aiji74    ,to the      
student who has carefully followed the  
development of the  non-,euclidean      
geometry and the notion of hyperspace   
the above  characterization is none too 
severe nor ill-deserved4 ,nothing could 
more vividly yet correctly portray the  
impious tactics of the  metageometrician
 and establish his perceptual obliquity 
more surely  than the mere fact1        
mentioned by ,,carus1 that he can with  
evident lack  of mental perturbation    
proclaim that two straight lines1       
noncoincident  with each other1 may pass
 through a point and yet be parallel to 
a third line4 ,but this is a mere       
trifle1 a bagatelle1 to the many  other 
infractions of which he is guilty4 ,the 

wonder is that he is  able to secure    
such obsequious acceptance of his       
offerings as many  of the most serious  
minded mathematicians are inclined to   
give4 ,is  it to be wondered at that1   
despite the profuse protestations of the
  advocates1 many who take up the study 
of the question of hyperspace  should   
experience a deep revulsion from the    
posture assumed by  metageometricians   
with respect to these queries8          
,linked with the idea of dimensionality 
is the notion that space is  infinite4  
,this is a conception which has its     
roots imbedded in the  depths of        
antiquity4 ,primitive man1 looking up   
into the heavens at  what appeared to   
him as a never ending extension1 was    
awed by its  vastness2 but the minds of 
the most learned of the present-day men 
are not free from this innate dread of  
infinity4 ,it permeates the  thought    
life of all alike and none seems to be  
able to rise above it4  ,mathematicians1
 philosophers1 scientists all share in  

the general  belief that space is       
without limit1 unending in extent and   
eternally  existent4 ,,riemann1 whose   
thought life found its most convenient  
mode of  expression by means of pure    
mathematics1 was the first in the       
history  of human thought to surmise    
that space is not infinite but limited  
even though unbounded4 ,but his         
conception has been much vitiated on    
account of its entanglement with an     
idealized construction by which  space  
is regarded as a thing to be manipulated
 and generated by act  of thought4 ,were
 it not for this his conception would   
indeed mark  the beginning of a new era 
in psychogenesis4 ,as it is1 when all   
the  nonsensical effusions have been    
cleared away from our space conceptions 
and men come really to understand       
something of the essential nature  of   
space this new era will find its true   
beginnings in the mind of  ,,riemann4   
,although it must be said1 as is the    
case with all progressive  movements1   

the later development of a rationale for
 this conclusion  will vary greatly from
 his original conception4 ,for he had in
 mind a  space that is generable and    
therefore a logical construction while  
ultimately the mind will swing back to a
 consideration of real space4           
,already men are beginning to see a new 
light4 ,already they are  beginning to  
take a new view of space in general4    
,the departure is  especially noticeable
 in the attitude assumed by ,,hiram ,m4 
,,stanley4#ac  ,he says3      ,if we    
seek the most satisfactory understanding
 of space we shall    look neither to   
mathematics nor ,psychology but to      
,physics4 ,the    trend of ,physics1 say
 with such a representative as          
,,ostwald1 is    to make things the     
expression of force2 the constitution   
and    appearance of things are         
determined by dynamism2 and we may best 
interpret space as a mode of this       
dynamic appearance4    ,space1 as a mode
 of dynamic appearance is a slight      

improvement upon the  old idea of a pure
 vacuity2 for in the light of what we   
now know about  space content much of   
the dignity of that view has been lost4 
,men now  know that space is not an     
empty void4 ,they know that the         
atmosphere  fills a great deal of space4
 ,they also have extended their         
conception  in this direction to include
 the ether and occultism goes further   
and postulates four kinds of ether--the 
chemical1 life1 light and  psychographic
 ethers4 ,but it does not stop here4 ,it
 postulates a series  of grades of finer
 matter than the physical which fills   
space and  permeates its entire extent  
even to identification with its         
essential  nature4      #ac             
,philosophical ,review1 ,vol4 ,,vii     
7#ahih74    ,,stanley continues3        
,everything does not1 as commonly       
conceived1 fall into some               
pre-existent space convenient for it2   
but everything makes its    own         
spaciousness by its own defensive and   

offensive force1 and the    totality of 
all appearance is space in general4     
,according to ,,stanley1 not only do    
physical1 perceptual objects1 by  their 
offensive and defensive force make their
 own space but the  appearance of that  
in which no physical object is makes    
room for itself  by its own dynamic     
force4 ,in other words1 that which we   
call pure  extensity is by virtue of its
 dynamism the cause of its own          
existence4    ,at first hand there      
appears to be little worthy of serious  
consideration in this view of ,,stanley2
 yet1 if carried to its logical         
conclusion1 the merit of the hypothesis 
becomes apparent4 ,accordingly1         
interstellar distances which are        
commonly said to be even without air  or
 life of any kind are really an         
appearance possessed of a dynamism      
peculiar to itself4 ,and this very      
force-appearance1 constituting  space1  
is that which makes it perceivable4 ,for
 instance1 let us say  the space that   

exists between the earth and the moon1  
is not really  empty nor does it have an
 existence prior to itself1 but is a    
mode of  dynamic appearance which is the
 cause of its own existence4 ,its       
dynamic  character makes it to appear   
perceptible to our senses4 ,logically1  
if  the dynamism were removed there     
would remain neither space nor the      
appearance of space4 ,if this were true1
 and it is worthy of serious  thought1  
then space is certainly finite1 as in   
its totality1 according  to ,,stanley',s
 view1 it would have to be regarded as a
 phenomenon of the  inner and finite    
life of the infinite4    ,it is believed
 that we may go a step further and      
unqualifiedly assert  that space is     
finite1 even denying its infinity as a  
general mode  of the activity of the    
whole4 ,yet it is transfinite in the    
sense  that it transcends the           
comprehension of finite minds or        
processes4 ,it  is finite because it is 
in manifestation4 ,everything that is in

  manifestation is finite4 ,the infinite
 is not in manifestation4 ,infinity  has
 to be limited always to become         
manifest4 ,the ,deity has limited ,his  
being in order that there may be a      
manifested universe4 ,all things1  all  
appearances are finite2 because they are
 phenomena connected with               
manifestation4    ,this question may be 
viewed from another standpoint4 ,all    
things in  manifestation or existence   
are polar in their constitution4 ,for   
instance3 there cannot be a here without
 a there4 ,there cannot  be an upper    
without a lower4 ,right is copolar with 
wrong2  good is copolar with evil2 night
 with day2 manifestation with           
non-manifestation2 truth with falsity2  
infinity with finity and so  on1        
throughout the whole gamut of the pairs 
of opposites4 ,what is the  logical     
inference8 ,space is paired with a lack 
of space4 ,there cannot  be what we call
 space without there being at the same  
time the  possibility1 at least1 of the 

lack of space or spacelessness4 ,this is
  a conclusion that is rigorously       
logical and incontrovertible4    ,but it
 has been urged that it is impossible   
for the mind to imagine  a condition    
where there is no space4 ,it even has   
been asserted that  it is contrary to   
the constitution of the mind itself to  
imagine no  space4 ,but whether         
imaginable or not has no effect whatever
 upon the  validity of the conception4  
,neither1 it is said1 can we imagine a  
fourth dimension but the mind has come  
dangerously near to imagining  it4 ,the 
distance from excogitating upon1        
discussing and describing  the          
properties of four-space to imagining it
 is not so great after  all4 ,truly it  
is difficult indeed1 it seems1 to be    
able to describe a  thing yet not be    
able to imagine or make a mental image  
of it4 ,there  is an evident fallacy    
here4 ,either the description of        
four-space is  no description at all or 
it is a true delineation of an idealized

  construction which is well within the 
mind's powers of imagination4  ,indeed  
the question of imaginability is not    
determinative in itself2  for what the  
mind may now be unable to imagine1      
because of its more or  less nebulous   
character1 and owing to its infancy may 
in the course of  time be easily        
accomplished4    ,the universe is a     
compacted plenum4 ,it is chock-full of  
mind1 of  life1 of energy and matter4   
,these four are basically one4 ,they    
exist1  of course1 in varying degrees of
 tenuity and intensity and answer to a  
wide range of vibrations4 ,together1 in 
their manifestation of action  and      
interaction1 in their dynamic           
appearance1 if you please1 they         
constitute space4 ,if these were removed
 with all that their existence  implies 
there would result a condition of       
spacelessness in which no  one of the   
appearances which we now perceive would 
be possible4 ,even  sheer extensity     
would be non-existent4 ,all scope of    

motility would be  lacking4 ,dimension1 
cordinates1 direction1                  
space-relations--all would  be          
impossible4    ,a straight line is an   
ideal construction of the mind4 ,it does
 not exist  in nature4 ,it can never be 
actualized in the phenomenal universe4  
,between the ideal and the real1 or     
actual1 there is a kosmic chasm4 ,it    
broadens or narrows according as the    
phenomenal appearance approaches  or    
recedes from the ideal4 ,what1          
therefore1 can be postulated of  the one
 will not apply with equal force to the 
other4 ,they are not  congruent and can 
never be in the actualized universe4    
,the moment the  actual becomes         
identified with the ideal it ceases to  
be the actual4  ,the universe does not  
exist as pure form1 neither does space4 
,as purely formal constructions of the  
intellect these can have no  perceptible
 existence4 ,the phenomenal or sensible 
may not be judged by  exactly the same  
standard as the formal4 ,the phenomenal 

or sensible  represents things as they  
appear to the senses1 or1 so far as the 
actualized universe is concerned1 as    
they really are4 ,the formal  represents
 things as they are made to appear by   
the mind4 ,it cannot  be actualized4 ,it
 may be said that the purely formal is  
the limit of  evolution4 ,the phenomenal
 may approach the ideal as a limit1 but 
can  never become fully congruent with  
it4 ,the difference between the ideal   
and the actual is a dynamic one2 it is  
by virtue of this difference  that the  
universe is held in manifestation4      
,evolution is the decrement  of this    
difference between the purely formal and
 the actual4 ,so long  then as a kosmic 
differential is maintained the          
phenomenal continues  to be manifest3   
when it is finally reduced to nothing it
 goes out of  manifestation4 ,the       
phenomenal is finite2 the ideal         
infinite4    ,wherefore1 it is          
undoubtedly improper to refer to space  
as being  infinite4 ,the term really is 

inapplicable4 ,transfinity is much      
better  and more accurate4 ,space is    
transfinite because its scope is greater
  than any finite scope of motility can 
encompass1 because it exceeds  finite   
comprehensibility4    ,,riemann',s      
notion that space is limited gains      
weight in the light of the  foregoing   
considerations4 ,but he could not       
conceive of the limitability  and       
unboundedness of space as such in its   
pure essence2 but was  compelled1 by his
 own limitations1 to make an idealized  
construction in  which he could         
actualize his conception4 ,and for real1
 dynamic space1  he substituted his     
ideal construction and proceeded upon   
that basis4  ,and of course1 his view   
while it had no reference to perceptual 
space  nevertheless possessed an        
illative relation thereto and should be 
recognized as construable in that light4
    ,the process of squaring the circle 
recognized as a geometric  impossibility
 is significant of the fluxional nature 

of the universal  residuum perpetually  
maintained between the archetypal and   
the  manifested kosmos4 ,it seems that  
there is a profound truth embodied in   
this problem4 ,there is a lesson that   
may be learned by mathematicians1       
philosophers1 scientists and thinkers in
 general4 ,there is an element  of      
eternal necessity and universality about
 it which is truly symbolic  of the     
finity of the universe and the infinity 
of the archetypal4 ,just  as a square or
 a series of polygonal figures inscribed
 in a circle  cannot be made to coincide
 exactly with the circle so cannot the  
actual  be made to coincide with the    
ideal4 ,the circumference of the circle 
is the unapproachable limit of inscribed
 squares4 ,if it were possible  so to   
multiply squares thus inscribed that a  
figure coincident with  the             
circumference of a circle might be      
constructed1 such a figure  would not be
 a square but a circle4 ,the manifested 
universe is like  that--the process of  

inscribing squares within a circle4 ,it 
is ever  becoming1 evolving1 developing1
 but never quite attains4 ,infinity  is 
a process4 ,but no single stage in that 
process is infinite4 ,each  is finite   
and their totality makes the infinity of
 the process4 ,the  universe manifested 
to the senses or the intellect is       
finite4    ,space1 says ,,paul ,,carus1 
is the possibility of motion in all     
directions4#ad ,to be sure1 it is       
admitted that space offers  opportunity 
for motion in all directions4 ,but is   
space this opportunity  of motility8 ,or
 is possibility of motion space8 ,the   
possibility of  motion must rest in the 
thing that moves4 ,it implies a potency 
in the  moving entity1 not in space4 ,if
 it is meant that space is the potency  
that resides in the moving element it is
 still more difficult to  understand the
 connotation4 ,but even granting this   
view1 are we not  compelled to recognize
 the dynamism of space as a necessary   
inference8  ,another definition which   

,,carus gives is that space is a pure   
form of  extension4 ,if it be granted   
that space is a pure form of extension  
we should have to conclude that it has  
no actual existence2 for pure  form does
 not exist except as an idealized       
construction4 ,it cannot  be found in   
nature4 ,pure form is ideal4 ,impure or 
natural form is  actual4 ,therefore the 
space in which we live and in which the 
universe  exists cannot be a pure form  
because life cannot exist in the purely 
formal4 ,it is useless to talk about    
space as mere form so long as  it       
maintains life4 ,the difficulty which   
this phase of the question  presents is 
another evidence of the inadequacy of   
our definitions4      #ad ,vide         
,foundations of ,mathematics1 p4 #ajg4  
,it is also found to be impossible to   
concur in ,,carus' conception of        
knowledge a priori4 ,his notion of the a
 priori varies somewhat from  the       
,kantian view4 ,he defines it as an     
idealized construction1 the  mind made1 

abstract thought1 and places it in the  
same category as  a concept4 ,this is   
undoubtedly born of his desire to get   
rid of ,,kant',s  innate ideas which    
seem to be distasteful to him4 ,but in  
doing so it  appears that the real a    
priori has been overlooked4 ,let us     
examine  for a moment this important    
question4 ,the a posteriori connotates  
all  knowledge gained through the       
senses1 or sense experience4 ,all       
knowledge  therefore whose origin can be
 traced to the senses is knowledge a    
posteriori4 ,now1 knowledge a priori    
should be just the opposite of  this4   
,it should indicate such knowledge as   
that which does not have  its origin in 
the senses1 or which is not dependent   
upon the ordinary  avenues of           
sense-experience4 ,abstract thought is  
as truly experience as  smelling1 seeing
 or hearing4 ,it is by traversing its   
scope of motility  that the mind finds  
out what the norms of logic are4 ,it    
could not  remain quiescent and discover

 them4 ,it has to be active1 examining1 
comparing and judging4 ,almost the      
entire range of thought1 its entire     
scope1 is characterized by the a        
posterioristic method4 ,in fact1  all   
thought is a posterioristic4 ,despite   
the fact that1 in thinking  in the      
abstract1 it is necessary mentally to   
remove all elements of  concreteness1   
all materiality and all actuality1 the  
conclusions  reached have to be referred
 to the standards maintained by the     
actual1  the concrete and the material4 
,then we do not start with the abstract 
in our thinking4 ,we begin with the     
concrete and by mentally removing  all  
physical qualities arrive at the        
abstract4    ,the mind has a            
constitution4 ,it acts in a given way   
because it is its  nature so to act4    
,not because it has learned to act in   
that manner4 ,it  performs certain      
functions intuitively without previous  
instruction  or experience for the same 
reason that water dampens or heat warms4

  ,it is natural for it to do so4 ,this 
naturalness1 this performance of        
function without being taught or without
 experience constitute the  principle of
 apriority in the mind4 ,aprioriness is 
a principle  of mind partaking of the   
very nature and essence of mind4 ,it is 
the  very mainspring of mentality4      
,perception and conception are processes
  which the mind performs intuitively4  
,the mind perceives and conceives       
because it is impossible for the normal 
mind to do otherwise4 ,we take  a view  
upon a given question2 we assume certain
 mental attitudes of  affirmation1      
negation or indifference because we have
 learned to do so  by virtue of the     
tuitional capability of mind4 ,these    
describe the a  posteriori4 ,that is1   
all knowledge obtained as a result of   
voluntary  mental processes constitutes 
the mass of knowledge a posteriori4 ,the
  a priori is what the mind is by       
nature3 the a posteriori is what  the   
mind becomes4 ,it is the mind-content4  

,the a priori is not a mental           
construction2 it is an essential        
principle of mind4 ,it should not be    
identified with the purely  formal1 as  
is done by ,,paul ,,carus3#ae      #ae  
,vide ,foundations of ,mathematics1 p4  
#db4    ,he says3      ,the a priori is 
identical with the purely formal which  
originates in our mind by abstraction4  
,when we limit our attention    to the  
purely relational1 dropping all other   
features out of    sight1 we produce a  
field of abstraction in which we can    
construct    purely formal combination1 
such as numbers1 or the ideas of types  
and species4 ,thus we create a world of 
pure thought which has the    advantage 
of being applicable to any purely formal
 consideration    and we work out       
systems of numbers which1 when counting1
 we can    use as standards of reference
 for our experience in practical        
life4    ,thus ,,carus definitely links 
up the a priori to a factor which is    
nothing more nor less than a mental     

by-product4 ,for such is the  category  
in which would be placed both the       
process of abstraction and  its results4
 ,it is therefore exceedingly difficult 
to understand why  so cursory a         
consideration should have been given to 
the principle of  apriority than which  
no other element of mind is more        
essentially a  part of the mind itself4 
,the formal is symbolic4 ,it signifies  
an informing quantity4 ,pure form       
itself is but a negation of that which  
formerly filled it4 ,then1 too1  the    
formal is purely artificial because it  
is a mental construction4  ,essentially 
there is as much difference between the 
purely formal and  the a priori as      
between creator and creature1 as between
 potter and  clay4 ,the one is the      
builder1 the other is the material2 the 
one the  knower and the other the known4
 ,thus1 the only reason that the formal 
is found to be answerable to the a      
priori at all is due to the fact  that  
it is construable only upon the basis of

 the a priori4 ,but  being so is not    
sufficient warrant for its              
identification with the a  priori4 ,the 
formal merely represents the totality of
 possibilities in  the universe as      
viewed by the mind2 but as the number of
 possibilities  open to the mind is1 on 
account of its nature and purpose       
limited1 it  is not to be supposed that 
it 7the mind7 shall measure up to all   
the  possibilities offered by the       
formal4 ,moreover1 it is certain that no
  sane mind cherishes the hope that     
there shall ever be found in the        
universe of life and form a congruence  
for all of the possibilities  held out  
by the purely formal4    ,as an eternal 
principle of mind1 the a priori is in   
agreement with  the divine mind of the  
kosmos4 ,in its aposteriority the mind  
is of  diverse tendences1 qualities and 
characteristics4 ,apriorily1 it acts in 
unison with the eternal purpose of life 
and the universal mind4 ,in its         
aposteriority1 it often goes awry4 ,in  

its apriority it can never be  insane2  
insanity is a symptom of the morbid a   
posteriori4    ,the mind in man acts the
 same as mind in the vegetal and lower  
animal  kingdoms4 ,metabolism and       
katabolism1 indeed all cell-activity1   
are  a priori performances of the mind4 
,growth and all its phenomena1  the     
cyclicism of natural processes1 and     
every activity connected  therewith     
belong to the category of the a priori4 
,cells multiply1  divide1 build up and  
tear down tissues and they do it        
intuitively4 ,most  certainly these     
functions are performed without any     
assistance from the  intellect4 ,all the
 myriad activities in nature with which 
the intellect  in man has not the       
slightest concern1 truly acting in      
accord with some  primordial impetus1   
are activities a priori4    ,now what is
 the attitude of the intellect1 in the  
light of the a  priori1 towards space   
and the question of dimensionality8 ,it 
is  evident that no matter what this    

attitude may be it is in agreement  with
 the constitution of things and of the  
universe4 ,and if so1 it is  right and  
without illusion4 ,it is also evident   
that whatever notion  a posteriori the  
intellect may entertain with respect to 
these  questions is unavoidably liable  
to the illusionary drawbacks common to  
conclusions based upon limited          
experience4 ,the geometric view of space
  belongs to the category of the a      
posteriori4 ,hence it is subject to  the
 usual imposition of error4    ,tersely 
stated1 ,,kant's view of space is that  
it is a form of intuition1  a form a    
priori1 a transcendental form4 ,as such 
he considered it  to be a native form of
 perception not belonging to the        
category of  sense-deliveries4          
,accordingly1 space is a form of        
intuition arising  out of and inhering  
in the constitution of mind4 ,it is a   
notion  which constitutes the universal 
and eternal prerequisite of mind and    
is1 therefore1 intrinsically necessary  

to all phases of mentation4  ,now1 this 
being true just what may be said to be  
the relation of  dimensionality to this 
a priori form of space which is found to
  exist in the mind as an eternal aspect
 of its nature8 ,does the mind          
intuitively measure its contents or its 
operations by the empirical  standard of
 space-measurement known as dimension8  
,is the attitude of  the mind towards   
the objectively real one of             
discrimination a priori  as to the      
direction or dimension in which a       
percept may originate8 ,in  other words1
 does the mind habitually and           
intuitively refer its data  to a system 
of cordinates for final determination8  
,there is no other  answer but that the 
mind makes no such reference and is     
dependent upon  no kind of cordinate    
system in any of its operations a       
priori4 ,as a  form of intuition1 the   
space notion is present in the mind as a
 scope  of existence1 of motility1 of   
being and of sheer roominess4 ,the      

notion  of direction or dimension1 being
 an artificial construction1 does not   
enter into this form of intuition at    
all4 ,it is only when the mind  comes to
 elaborate upon its perceptive          
performances and possibilities  that the
 questions of relations1 positions and  
directions arise4  ,but this latter is a
 matter separate and distinct from the  
state of  awareness which embodies the  
notion of space4    ,dimension is an    
arbitrary norm constructed by the mind  
for the  determination of various       
positions in space4 ,it is an accident  
or  by-product of the process of        
elaborative cognition1 a convenient  and
 appropriate means of measurement for   
objects in space and their              
space-relations4 ,but it is no more a   
priori than a foot rule or a  square4   
,but being purely an empirical product  
it may be said to be  an aspect of      
psychogenesis because it relates to the 
evolutionary  aspect of mind4 ,the      
assumption may therefore be allowed that

 the  mind may1 in the course of its    
evolution1 find it convenient and       
appropriate to devise an additional     
ordinate or dimension to satisfy  the   
necessities of its more complex         
ramifications into the nature of  things
 and to determine their greatly         
increased space-relations4 ,it  may be  
even possible for the mind to function  
normally in a space of  four dimensions4
 ,but this would simply be a new        
adjustment1 not a  change in the        
essential nature of mind4 ,it would be  
like the series of  adjustments to      
environments which man has made in the  
onward movement  of civilization4 ,there
 has been no serious change in the      
manhood  per se of man4 ,that has       
remained the same2 there has been merely
 a  complication of environmental       
influences4 ,similarly1 in the          
acquisition  of four-dimensional powers1
 granting that such an acquisition is   
possible1 there is nothing to be added  
to the aprioriness of mind  itself4 ,is 

it not1 therefore1 logical to assume    
that the discovery  of a fourth         
cordinate and the consequent            
conceptualization of the  same1 point to
 the development in the mind of a       
greatly extended  faculty1 more keenly  
penetrative powers of cognition and a   
further  diversification of its         
environments than it has hitherto       
enjoyed8  ,indeed1 it seems so4         
,,chapter ,v      ,,the ,,fourth        
,,dimension      ,the ,ideal and the    
,representative ,nature of ,objects in  
the    ,sensible ,world--,the ,psychic  
,fluxional the ,basis of ,mental        
,differences--,natural and ,artificial  
,symbols--,use of ,analogies to         
,prove the ,existence of a ,fourth      
,dimension--,the ,generation of a       
,hypercube or ,tesseract--,possibilities
 in the ,world of the ,fourth           
,dimension--,some ,logical ,difficulties
 ,inhering in the ,four-,space          
,conception--,the ,fallacy of the       
,plane-,rotation ,hypothesis--,c4 ,h4   

,,hinton and ,major ,,ellis on the      
,fourth ,dimension4      ,the world of  
mathesis is truly a marvelous domain4   
,vast are its  possibilities and vaster 
still its sweep of conceivability4 ,it  
is the  kingdom of the mind where1 in   
regal freedom1 it may perform feats     
which it is impossible to actualize in  
the phenomenal universe4 ,in  fact1     
there is no necessity to consider the   
limitations imposed by  the actualities 
of the sensuous world4 ,logic is the    
architect of  this region1 and for it   
there is no limit to the admissibility  
of  hypotheses4 ,these may be multiplied
 at will1 and legitimately so4 ,the     
chief error lies in the attempt to make 
them appear as actual facts of  the     
physical world4    ,mathematicians1     
speculating upon the possibilities of   
mathetic  constructions and forgetting  
the necessary distinctions which should 
be recognized as differentiating the two
 worlds1 in their enthusiasm  have been 
led into the error of postulating as    

qualities of the  phenomenal world the  
characteristics of the conceptual4      
,accordingly1  a great deal of confusion
 as to the proper limits and            
restrictions of  these conceptions has  
arisen and there still may be found     
those who are  enthusiastically         
endeavoring to push the actualities of  
the physical  over into the conceptual4 
,but in assuming any attitude towards   
mathetic  propositions1 especially with 
a view to demonstrating their actuality1
  very careful discrimination as to the 
essential qualities and their           
connotations should be made4 ,hence1    
before taking up a brief study of  the  
fourth dimension proper1 it is deemed   
fitting to indicate some of  the        
fundamental distinctions which every    
student of these questions  should be   
able to make with reference to the data 
which he meets4    ,all objects of the  
sensible world have both an essential or
 ideal  nature and a representative or  
sensuous nature4 ,that is1 they may be  

studied from the standpoint of the ideal
 as well as the sensuous4  ,the         
representative nature is that which we  
recognize as the mode of  appearance to 
our senses which1 as ,,kant held1 is not
 the essential  or ideal character of   
the thing itself4 ,for there is quite as
 much  difference between the sensuous  
percept and the real thing itself as    
between an object and its shadow4 ,in   
fact1 a concept viewed in this  light1  
may be seen to have all the             
characteristics of an ordinary  shadow2 
for instance1 the shadow of a tree4     
,view it as the sun is  rising2 it will 
then be seen to appear very much        
elongated1 becoming  less in length and 
more distinct in outline as the sun     
rises to a  position directly overhead4 
,the elongation may again be seen when  
the  sun is setting4 ,throughout the day
 as the sun assumes different angles    
with reference to the tree the          
proportions and definiteness of the     
shadow vary accordingly4 ,thus the      

angularity of the sun1 the intensity    
and fullness of the light and the shape 
and size of the tree operate to         
determine the character of the shadow4  
,much the same thing is true of a       
sensuous representation4 ,if we examine 
carefully our ideas of geometric        
quantities and magnitudes1 it will be   
found that the concepts themselves are  
not identical with the objects  of the  
physical world1 but mere mental shadows 
of them4 ,the angularity  of            
consciousness1 or the distinctness of   
one's state of awareness1  being        
analogous to similar attitudes in the   
solar influence are the  main           
determinants of the character of the    
mental shadow or concept4  ,wherefore   
mathematical spaces or magnitudes are   
not sensuous things  and have therefore 
no more real existence than a shadow1   
and strictly  speaking not as much2 for 
a shadow may be seen1 while such        
magnitudes  can only be conceived4 ,it  
may be urged that since we can conceive 

of  such things they must have existence
 of some kind4 ,and so they have1  but  
it is an existence of a different kind  
from that which we recognize  as        
belonging to things in the sensible     
world4 ,they have a conceptual          
existence1 but not a sensuous one4      
,therein lies the great difference4     
,to be sure1 a shadow is a more or less 
true representation of the thing  to    
which it pertains4 ,that this is true   
can be established empirically4         
,similarly1 the degree of congruity     
between objects and concepts  likewise  
may be determined4 ,if this were not    
true we should be very  much            
disappointed with what we find in the   
phenomenal world and could  never be    
quite sure that the mentograph existing 
in our minds was a  faithful            
representation of the thing which we    
might be examining4 ,but  really the    
foundation for such a disappointment is 
present in every  concept1 every percept
 with which the mind deals4 ,this       

disappointment1  although in actual     
experience is reduced to an almost      
negligible  quantity1 is due to the     
failure of sensuous objects to conform  
wholly  to the specific details of the  
mental shadow or mentograph4 ,this  lack
 of congruence between the mental       
picture and the object itself  is       
necessary for obvious reasons4 ,it is   
markedly observable in the  early       
efforts of a child in learning          
distances1 weights1 resistances1        
temperatures and the like4 ,no          
inconsiderable time is required for the 
child to be able correctly to harmonize 
his sense-deliveries with  actual       
conditions4 ,otherwise1 the child would 
never make any of the  ludicrous        
mistakes of judgment of which it is     
guilty when trying to  get its bearings 
in the world of the senses4 ,in the     
course of time  the child gradually     
learns by experience that certain things
 are  true of objects1 distances1       
temperatures1 resistances1 etc41 and    

that certain things are not true of     
them4 ,he learns these things by        
actually contacting various objects4 ,he
 is then competent to render  correct   
judgments1 within certain limits1 as to 
the conditions which  he finds in the   
sensible world4 ,and the allowances1    
equations and  corrections which his    
motor1 sensory and psychic mechanisms   
learn to  make in childhood serve for   
all subsequent time4 ,and this is       
important  to remember2 for the mature  
mind is apt to forget or overlook the   
adaptations which the child-mind has    
made in its growth4    ,if there were no
 such differences between the concept   
and the thing  itself1 actual physical  
contact would not be necessary4 ,for one
 could  rely wholly upon the            
sense-deliveries and each sense might   
operate  entirely independently of all  
the others as there would be no         
necessity  to correct the delivery of   
one by those of the others4 ,this1 of   
course1  raises the question as to the  

necessity of sense-experience at all    
under conditions where there would be no
 disparity between the thing  itself and
 the ideal representation of it in the  
mind4 ,the absence of  this variable    
quantity would open to the mind the     
possibility of really  knowing the      
essential nature of objects in the      
phenomenal world1 a  condition of       
affairs which is admittedly now without 
the range of the  powers of the mind4   
,at any rate1 the essential thingness of
 objects can never be  comprehended by  
the mind until the diminution of this   
disparity between  the object of sense  
and the mental picture of it which      
exists in the  consciousness has        
proceeded to such a limit as either     
completely to  have obliterated it or to
 such an extent that the psychic fluxion
 is so  slight as not to matter4    ,it 
is believed that the results of mental  
evolution1 as the mind  approaches the  
transfinite as a limit1 will operate to 
minimize the  fluxional quantity which  

subsists between all objects of sense   
and  their ideal representation as data 
of consciousness4 ,the conclusion  that 
the mind of early men who lived hundreds
 of thousands and perhaps  millions of  
years ago on this planet consumed a much
 longer time in  learning the           
adjustments between the objects which it
 contacted in the  sensuous world and   
the elementary representations which    
were registered  in its youthful        
consciousness than is to-day required   
for similar  processes seems to be      
demanded1 and substantiated as well1 by 
what is  known of the phyletic          
development of the mind in the human    
race4    ,in view of the above1 it is   
thought that the duration of such simple
  mental processes served not only to   
prolong the physical life of  the man of
 those early days1 but may also account 
for the puerility  and incapacity of the
 mind at that stage4 ,not that the slow 
mental  processes were active causative 
agencies in lengthening the life  of    

man1 but that they together with the    
crass physicality of man  necessitated a
 longer physical life4 ,this1 perhaps in
 a larger  sense than any other         
consideration1 accounts for the         
fundamental  discrepancies in the mind  
of the primitive man in comparison with 
the  efficiency of the mind of the      
present-day man4 ,in view of the        
potential  character of mind and in the 
light of the well graduated scale of its
  accomplishments1 it is undoubtedly    
safe to conclude that the quality of    
mental capacities is proportional to the
 psychic fluxional which may  exist at  
any time between the ideal and the      
essential or real4 ,mental  differences 
and potentialities in general may be due
 to the magnitude  of the psychic       
fluxional or differential that exists   
between the  conceptual and the         
perceptual universe4 ,in some minds it  
may be greater  than in others4 ,the    
chasm between things-in-themselves and  
the mental  notion pertaining thereto   

may vary in a direct ratio to the       
individual  mind's place in             
psychogenesis1 and therefore1 be the key
 to all mental  differences in this     
respect4    ,most certain it is that    
there may be marked fluctuations in the 
judicial approach of minds towards any  
psychic end4 ,in other words1  there is 
not only a fluxional or differential    
between the object and  its             
representation1 but also a differential 
between the approach  of one mind and   
another in the judicial determination of
 notions  concerning ideas4 ,in this    
way1 differences of opinions as to the  
right and wrong of judgments arise4     
,indeed1 there seem to be zones  of     
affinity for minds of similar           
characteristics1 or minds that have  the
 same degree of differential2 so that1  
in choosing among the many  possible    
judgments predicable upon a species of  
data1 all those minds  having the same  
degree of psychic differential discover 
a special  affinity or agreement among  

themselves4 ,hence1 we have cults1      
schools  of thought1 and various other  
sectional bodies that find a basis  of  
agreement for their operations in this  
way4 ,the outcome of this  remarkable   
intellectual phenomenon is that there   
are as many different  kinds of         
judgments as there are zones of affinity
 among minds4 ,various  systems of      
philosophy owe their existence to these 
considerations1 and  the considerations 
themselves flow from the fact that all  
intellectual  operations are essentially
 superficial2 because there is no means 
by  which they may penetrate to the     
steady flowing stream of reality which  
pervades and sustains objects in the    
sensible world4    ,in view1 therefore1 
of the foregoing and with special       
reference to  geometric constructions1  
it is necessary in approaching a study  
of  the four-space that it be understood
 at the outset that the fourth          
dimension can neither be actualized nor 
made objectively possible even  in the  

slightest degree in the perceptual      
world2 because it belongs  to the world 
of pure thought and exists there as an  
extra personal  affair1 separate and    
distinct from the world of the senses4  
,as says ,,simon ,,newcomb3#af      ,the
 experience of the race and all the     
refinements of modern    science may be 
regarded as showing quite conclusively  
that1 within    the limits of our       
experience1 there is no motion of       
material    masses1 in the direction of 
a fourth dimension1 no physical agency  
which we can assume to have its origin  
in regions to which matter    cannot    
move1 when it has three degrees of      
freedom4    ,there is1 however1 no      
logical objection to the study of the   
fourth  dimension as a purely           
hypothetical question1 if by pursuit of 
the same  an improvement of methods of  
research and of the outlook upon the    
field  of the actual may be gained4     
,hence1 it is with this attitude of mind
  that we approach the consideration of 

the fourth dimension4      #af ,vide    
,science1 ,vol4 ,,vii1 p4 #b1 ,no4 #aeh1
 #ahih4    ,various efforts have been   
made to render the conception of a      
fourth  dimension of space thinkable4   
,the student of space has reasoned3 ,we 
say that there are three dimensions of  
space4 ,why should we stop here8  ,may  
there not be spaces of four dimensions  
and more8 ,or he has said3  ,if ',a' may
 represent the side of a square1 ,a#b   
its area1 and ,a#c the  volume of a cube
 with edge equal to ,a2 what may ,a#d1  
,a#e or ,anth  represent in our space8  
,the conclusion1 with respect to the    
quantity  ,a#d1 has been that it should 
represent a space of four dimensions4   
,algebraic quantities1 however1         
represent neither objects in space nor  
space qualities except in a purely      
conventional manner4 ,all efforts to    
justify the objective existence of a    
fourth dimension based upon such        
reasoning will1 therefore1 fail2 because
 the basis of such arguments  is itself 

faulty4 ,in the sentence3 ,the man loves
 his bottle1 the  thing meant is not the
 bottle1 but what the bottle contains4  
,for the  purpose of the figure the     
bottle signifies its contents4 ,there is
 no  more real connection between the   
bottle and what it contains than        
between any word and the object for     
which it stands4 ,words are said  to be 
symbols of ideas4 ,but they are not     
natural symbols2 they are  conventional 
symbols1 made for the purpose of        
cataloguing1 indexing  and systematizing
 our knowledge4 ,words can be divorced  
from ideas  and objects1 or rather have 
never had any real connection with them4
  ,there are two classes of natural     
symbols1 namely2 objects and  ideas4    
,these1 objects and ideas1 symbolize    
realities4 ,realities  are imperceptible
 and incomprehensible to the intellect  
which has  aptitude only for a slight   
comprehension of the symbols of         
realities4  ,for instance1 a tree is a  
natural symbol4 ,it represents an       

actuality  which is imperceptible to the
 intellect4 ,the intellect can deal only
  with the sensible symbol4 ,it is a    
natural symbol2 because it is  possible 
directly to trace a living connection   
between the tree and  the tree-reality4 
,that is1 it would be possible so to    
trace out the  vital connection between 
the tree and its reality if the         
intellect  had aptitude for such        
tracery4 ,but1 in reality1 since it has 
no such  aptitude1 it remains for the   
work of that higher faculty than the    
intellect which recognizes both the     
connection and the intellect's          
inability to trace it4 ,further1 an     
object is called a natural symbol       
because it is the bridge between        
sensuous representation and reality4    
,it is as if one could begin at the     
surface of an object and by a  subtle   
process of elimination and excortication
 arrive at the heart  of the universum  
of reality4 ,no such consummation may be
 reached by  dealing with words which   

have merely an artificial relationship  
with  the objects which they signify4   
,again1 ideas1 that is1 ideas that are  
universal in application and have their 
roots in the great ocean of  reality1   
are natural symbols2 because if it were 
possible to handle  an idea with the    
physical hands it would be possible to  
arrive at the  heart of that which it   
symbolized without ever losing our      
connection  with the idea itself4 ,in   
other words1 ideas and objects1 unlike  
words1  can never be divorced from that 
which they symbolize4 ,both1 being of   
the same class1 are the opposite poles  
of realities4 ,this then is  the        
difference between natural symbols and  
artificial symbols--that  a natural     
symbol1 such as objects and ideas1 is   
copolar with reality  whereas an        
artificial symbol1 such as words1       
geometric constructions  and the like   
not only lacks this copolarity but is   
itself a symbol of  natural symbols4    
,it is1 therefore1 inconceivable that   

because the algebraic quantity  ,a#c has
 been arbitrarily decreed to be a       
representation of the volume  of a cube1
 every such quantity in the algebraic   
series shall actually  represent some   
object or set of objects in the physical
 world4 ,even  if it be granted that    
such may be the case1 is it not certain 
that  there is a limit to things in the 
objective universe8 ,yet there may  not 
be any limit to algebraic or            
mathematical determinations4 ,the       
material universe is limited and        
conditioned2 the world of mathesis is   
unlimited and unconditioned save by its 
own limitations and conditions4  ,it is 
irrational to expect that physical      
phenomena shall justify all             
mathematical predicates4    ,there is   
perhaps no single mathematical          
desideratum or consideration  which may 
be said to be the natural symbolism of  
realities2 for the  whole of            
mathematical conclusions is a mass of   
artificial and arbitrary  but concordant

 symbols of the crasser or nether pole  
of the antipodes  of realism4 ,it is    
exceedingly dangerous1 therefore1 to    
predicate  upon such a far-fetched      
symbolism as mathematics furnishes      
anything  purporting to deal with       
ultimate realities4 ,and those who      
insist upon  doing so are either blind  
themselves to these limitations or are  
madly  endeavoring to befog the minds of
 others who are dependent upon them  for
 leadership in questions of mathematical
 import4    ,analogies have been        
unsparingly used in efforts to          
popularize the  four-space conception   
and much of the violence which has been 
done  to the notion is due to this      
vagary4 ,the mathematical publicist1 in 
trying to give a mental picture of the  
fourth dimension1 examines the          
appearances of three dimensional beings 
as they might appear to a two           
dimensional being or duodim4 ,he        
imagines a race of beings endowed  with 
all the human faculties except that they

 live in a land of but  two             
dimensions--length and breadth4 ,he     
thinks of them as shadows of  three     
dimensional beings to whom there are no 
such conceptions as up  and down4 ,they 
can see nothing nor sense anything in   
any way that  is without their plane4   
,they can move in any direction within  
the  plane in which they live1 but can  
have no idea of any movement that  might
 carry them without that plane4 ,a house
 for such beings might be  simply a     
series of rectangles4 ,one of them might
 be as safe behind  a line as a tridim  
or three dimensional being would be     
behind a  stone wall4 ,a bank safe for  
the unodim would be a mere circle4 ,a   
duodim in any two dimensional prison    
might be rescued by a tridim  without   
the opening of doors or the breaking of 
walls4 ,an action of  a tridim performed
 so as to contact their plane would be  
to them a  miracle1 absolutely          
unaccountable upon the basis of any     
known fact to  the unodim or duodim4 ,a 

tridim might go into a house where lived
  a family of duodims1 appear and       
disappear without being detected  or its
 ever being discovered how he           
accomplished such a marvelous  feat4    
,our miracles1 after the same fashion1  
are said to be the antics  of some four 
dimensional being who has similar access
 to our three  dimensional world and    
whose actions are similarly inexplicable
 to us4  ,so the analogies have been    
multiplied4 ,but the temptation to apply
 the  consequences of such reasoning to 
actual three-space conditions has  been 
so great that many have yielded to it   
and have consequently sought  actually  
to explain physical phenomena upon the  
basis of the fourth  dimension4    ,the 
utilitarian side of the question of     
hyperspace has not been  neglected      
either4 ,and so1 early in the           
development of the hypothesis  and its  
various connotations1 the attention of  
investigators was turned  to this aspect
 of the inquiry4 ,strange possibilities 

were revealed as  a result4 ,for        
instance1 it was found that an expert   
fourth dimensional  operator is         
possessed of extraordinary advantages   
over ordinary  tridimensional beings4   
,operating from his mysterious hiding   
place in  hyperspace1 he could easily   
appear and disappear in so mysterious a 
manner that even the most strongly      
sealed chests of treasures would  be    
easily and entirely at his disposal4 ,no
 city police1 ,scotland ,yard  detective
 nor gendarme could have any terrors for
 him4 ,,drs4 ,,jekyll  and ,,messrs4    
,,hyde might abound everywhere without  
fear of detection4  ,objects as well as 
persons might be made to pass into or   
out of  closed rooms without penetrating
 the walls1 thus making escape easy  for
 the imprisoned4 ,no tridimensional     
state1 condition or system of           
arrangements1 accordingly1 would be safe
 from the ravages of evilly  inclined   
four dimensional entities4 ,objects that
 now are limited to a  point or line    

rotation could in the fourth dimension  
rotate about a  plane and thus further  
increase the perplexities of our        
engineering  and mechanical problems2   
four lines could be erected             
perpendicular  to each other whereas in 
three space only three such lines can be
  erected2 the right hand could be      
maneuvered into the fourth dimension    
and be recovered as a left hand2 the    
mysteries of growth1 decay and  death   
would find a satisfactory explanation on
 the basis of the  fourth dimensional   
hypothesis and many1 if not all1 of the 
perplexing  problems of physiology1     
chemistry1 physics1 astronomy1          
anthropology  and psychology would yield
 up their mysteries to the skill of the 
fourth dimensional operator4 ,marvelous 
possibilities these and much  to be     
desired6 ,but the most remarkable thing 
about these so-called  possibilities is 
their impossibility4 ,it is this kind of
 erratic  reasoning that has brought the
 conception of a fourth dimension into  

general disrepute with the popular mind4
 ,it is to be regretted1 too1  for the  
notion is a perfectly legitimate one in 
the domain of mathesis  where it        
originated and rightly belongs4    ,it  
is not to be wondered at that           
metageometricians and others should  at 
first surmise that1 in the four-space1  
they had found the key to  the deep     
mysteries of nature in all branches of  
inquiry4 ,for so vast  was the domain   
and so marvelous were the possibilities 
which the new  movement revealed that it
 was to be expected that those who were 
privileged to get the first glimpses    
thereof would not be able to  realize   
fully their significance4 ,but the      
stound of their minds and  the attendant
 magnification of the elements which    
they discovered  were but incidents in  
the larger and more comprehensive       
process of  adjustment to the great     
outstanding facts of psychogenesis which
 is  only faintly foreshadowed in the   
so-called hyperdimensional4 ,the whole  

scope of inquiry connected with         
hyperspace is not an end in itself4  ,it
 is merely a means to an end4 ,and that 
is the preparation of the  human mind   
for the inborning of a new faculty and  
consequently more  largely extended     
powers of cognition4 ,metageometrical   
discoveries are  therefore the          
excrescences of a deeper1 more          
significant world process  of mental    
unfoldment4 ,they belong to the         
matutinal phenomena incident  to this   
new stage of mental evolution4 ,all such
 investigations are but  the preliminary
 exercises which give birth to new      
tendencies which are  destined to flower
 forth into additional faculties and    
capacities4 ,so  that it is well that   
the evolutionary aspect of the question 
be not  overlooked2 for there is danger 
of this on account of the magnitude and 
kosmic importance of its scope of       
motility4    ,a geometric line is said  
to be a space of one dimension4 ,a plane
 is a  space of two dimensions and a    

cube1 a space of three dimensions4 ,in  
figure #g below1 the line ab is said to 
be one dimensional2 because  only one   
cordinate is necessary to locate a      
point-position in it4  ,the plane1 abcd1
 figure #h1 is said to be two           
dimensional because  two cordinates1 ab 
and db are required to locate a point1  
as the  point b4 ,the cube abcdefgh1    
figure #i1 is said to be tridimensional1
  because1 in order to locate the point 
b1 for instance1 it is  necessary to    
have three cordinates1 ab1 bc and gb4   
,the tesseract  is said to be four      
dimensional1 because1 in order to locate
 the point  b1 in the tesseract1 it is  
necessary to have four cordinates1 ab1  
bc1 bb' and h'b1 figure #aj4        ,a  
-------------------------------- ,b     
,,fig4 #g4          ,a                  
------------------------------ ,b       
,c ------------------------------ ,d    
,,fig4 #h4    ,it will be noted that in 
figures #h1 #i and #aj1 the element of  
perpendicularity enters as a necessary  

determination4 ,in figure #h1 the  lines
 ab and bd are perpendicular to each    
other4 ,similarly1 in ,fig4  #aj1 lines 
ab1 bc1 bb' and h'b are perpendicular to
 one another4  ,that is1 at their       
intersections1 they make right angles4  
,similarly1  figures representing any   
number of dimensions may be constructed4
    ,illustration3 ,,fig4 #i4           
,illustration3 ,,fig4 #aj4--,the        
,tesseract4    ,the line ab represents a
 one-space4 ,an entity living in a one  
space  is called a unodim4 ,the plane1  
abcd1 represents a two-space1  and      
entities living in such a space are     
called duodims4 ,the cube1  abcdefgh1   
represents a three-space and entities   
inhabiting such a  space are called     
tridims4 ,figure #aj represents a       
four-space1 and its  inhabitants are    
called quartodims4 ,each of the         
above-mentioned spaces  is said to have 
certain limitations peculiar to itself4 
,the fourth dimension is said to lie in 
a direction at right angles to  each of 

our three-space directions4 ,this1 of   
course1 gives rise to the  possibility  
of generating a new kind of volume1 the 
hypervolume4 ,the  hypercube or         
tesseract is described by moving the    
generating cube in  the direction in    
which the fourth dimension extends4 ,for
 instance1 if  the cube1 ,fig4 #i1 were 
moved in a direction at right angles to 
each  of its sides a distance equal to  
one of its sides1 a figure of four      
dimensions1 the tesseract1 would result4
    ,the initial cube1 abcc'e'fhh'1 when
 moved in a direction at right  angles  
to each of its faces1 generates the     
hypercube1 ,fig4 #aj4 ,the  lines1 aa'1 
bb'1 cc'1 dd'1 ee'1 ff'1 gg'1 hh'1 are  
assumed to be perpendicular to the lines
 meeting at the points1 a1  b1 c1 d1 e1 
f1 g1 h4 ,hence a'b'1 b'd1 dd'1 d'a'1   
ef1 fg1 gg'1 g'e1 represent the final   
cube resulting from  the hyperspace     
movement4 ,counting the number of cubes 
that compose  the hypercube we find that
 there are eight4 ,the generating cube1 

abcc'e'f'hh'1 and the final cube1 a'b'1 
b'd1 dd'1 d'a'1 ef1  fg1 gg'1 g'e1 make 
two cubes2 and each face generates a    
cube  making eight in all4 ,a tesseract1
 therefore1 is a figure bounded by      
eight cubes4    ,to find the different  
elements of a tesseract1 the following  
rules will  apply3    #a4 ,to find the  
number of lines3 ,multiply the number of
 lines in the  generating cube by two1  
and add a line for each point or corner 
in it4  ,e4g41 #b  #ab  #bd  #h  #cb4   
#b4 ,to find the number of planes1 faces
 or squares3 ,multiply the  number of   
planes in the generating cube by #b and 
add a plane for each  line in it4 ,e4g41
 #b  #f  #ab  #bd4    #c4 ,to find the  
number of cubes in a hypercube3         
,multiply the number of  cubes in the   
generating cube1 one1 by two and add a  
cube for each plane  in it4 ,e4g41 #b   
#a  #f  #h4    #d4 ,to find the number  
of points or corners3 ,multiply the     
number of  corners in the generating    
cube by #b4 ,e4g41 #b  #h  #af4         

99       99       99       99       99  
,in a plane there may be three points   
each equally distant from one  another4 
,these may be joined1 forming an        
equilateral triangle in which  there are
 three vertices or points1 three lines  
or sides and one  surface4    ,in       
three-space there may be four points    
each equidistant from the  others4 ,at  
the vertices of a regular tetrahedron   
may be found such  points4 ,the         
tetrahedron has four points1 one at each
 vertex1 #f lines  and #d equilateral   
triangles1 as in ,fig4 #aa4    ,in      
four-space1 we have #e points each      
equidistant from all the rest1  giving  
the hypertetrahedron4 ,this four        
dimensional figure may be  generated by 
moving the tetrahedron in the direction 
of the fourth  dimension1 as in ,fig4   
#ab4 ,if a plane be passed through each 
of the  six edges of the tetrahedron and
 the new vertex there will be six new   
planes or faces1 making #aj in all1     
counting the original four4 ,from  the  

new vertex there is also a tetrahedron  
resting upon each base of  the original 
tetrahedron so that there are five      
tetrahedra in all4  ,a hypertetrahedron 
is a four-dimensional figure consisting 
of five  tetrahedra1 ten faces1 #aj     
lines and #e points4    ,illustration3  
,,fig4 #aa4--,tetrahedron4              
,illustration3 ,,fig4                   
#ab4--;,hypertetrahedron4    ,,paul     
,,carus#ag suggests the use of mirrors  
so arranged that they  give eight       
representations of a cube when placed at
 their point of  intersection4 ,he says3
      ,if we build up three mirrors at  
right angles and place any object    in 
the intersecting corner we shall see the
 object not once1 but    eight times4   
,the body is reflected below and the    
object thus    doubled is mirrored not  
only on both upright sides but in       
addition    in the corner beyond1       
appearing in either of the upright      
mirrors    coincidingly in the same     
place4 ,thus the total multiplication of

    our tridimensional boundaries of a  
four dimensional complex is    rendered 
eight-fold4      ,we must now bear in   
mind that this representation of a      
fourth    dimension suffers from all the
 faults of the analogous figure    of a 
cube in two dimensional space4 ,the     
several figures are not    eight        
independent bodies but are mere         
boundaries and the four    dimensional  
space is conditioned by their           
interrelation4 ,it is    that           
unrepresentable something which they    
inclose1 or in other    words1 of which 
they are assumed to be boundaries4 ,if  
we were    four dimensional beings we   
could naturally and easily enter into   
the mirrored space and transfer         
tridimensional bodies or parts    of    
them into those other objects reflected 
here in the mirrors    representing the 
boundaries of the four dimensional      
object4 ,while    thus on the one hand  
the mirrored pictures would be as real  
as the    original object1 they would   

not take up the space of our three      
dimensions1 and in this respect1 our    
method of representing the    fourth    
dimension by mirrors would be quite     
analogous to the cube    pictured on a  
plane surface1 for the space to which we
 7being    limited to our tridimensional
 space-conception71 would naturally     
relegate the seven additional mirrored  
images is unoccupied and if    we should
 make the trial1 we would find it empty4
      #ag ,vide ,foundations of         
,mathematics1 pp4 #ic-id4    ,the       
utility of such a representation as that
 which ,,carus outlines in  the above is
 granted1 i4e41 so far as the purpose   
which it serves in  giving a general    
idea of what a four-space object might  
be imagined  to be like1 but the        
illustration does not demonstrate the   
existence  of a fourth dimension4 ,it   
only shows what might be if there were a
  four-space in which objects could     
exist and be examined4 ,we1 of course1  
have no right to assume that because it 

can be shown by analogous  reasoning    
that certain characteristics of the     
fourth dimensional object  can be       
represented in three-space the possible 
existence of such an  object is thereby 
established4 ,not at all4 ,for there is 
no imaginable  condition of             
tridimensional mechanics in which an    
object may be said to  have an objective
 existence similar to that represented  
by the mirrored  cube4    ,but there are
 discrepancies in this representation   
which well might be  considered4 ,they  
have virtually the force of invalidating
 somewhat the  conception which the     
analogy is designed to illustrate4 ,for 
instance1  in the case of the mirrored  
object placed at the point of           
intersection  of the three mirrors built
 up at right angles to each other4 ,upon
  examination of such a construction it 
is found that the reflection of  the    
object in the mirrors has not any       
perceptible connection with the  object 
itself4 ,and this1 too1 despite the fact

 that they are regarded  as boundaries  
of the hypercube2 especially is this    
true when it is  noted that these       
reflections are called upon to play the 
part of real1  palpable boundaries4 ,if 
a fourth dimensional object were really 
like  the mirror-representation it would
 be open to serious objections from  all
 viewpoints4 ,the replacement of any of 
the boundaries required in  the analogy 
would necessarily mean the replacement  
of the hypercube  itself4 ,in other     
words1 if the real cube be removed from 
its position  at the intersection of the
 mirrors no reflection will be seen1 and
  hence no boundaries and no hypercube4 
,the analogy while admittedly           
possessing some slight value in the     
direction meant1 is nevertheless        
valueless so far as a detailed          
representation is concerned4 ,so the    
analogy falls down2 but once again is   
the question raised as to whether  the  
so-called fourth dimension can be       
established or proven at all upon       

purely mathematical grounds4 ,it also   
emphasizes the necessity for a  clearer 
conception of the meaning of dimension  
and space4    ,the logical difficulties 
which beset the hyperspace conception   
are  dwelt upon at length by ,,james ,h4
 ,,hyslop4 ,he says3#ah      ,the       
supposition that there are three        
dimensions instead of one1    or that   
there are only three dimensions is      
purely arbitrary1    though convenient  
for certain practical purposes4 ,here   
the    supposition expresses only       
differences of directions from an       
assumed point4 ,thus what would be said 
to lie in a plane in one    relation    
would lie in the third dimension in     
another4 ,there    is nothing to        
determine absolutely what is the first1 
second1    or third dimension4 ,if the  
plane horizontal to the sensorium    be 
called plane dimension1 the plane       
vertical to it will be    called solid1 
or the third dimension1 but a change of 
position    will change the names of    

these dimensions without involving the  
slightest qualitative change or         
difference in meaning4      ,moreover1  
we usually select three lines or planes 
terminating    vertically at the same   
point1 the lines connecting the three   
surfaces of a cube with the same point1 
as the representative of    what is     
meant by three dimensions1 and reduce   
all other lines    and planes to these4 
,but interesting facts are observable   
here4    #a4 ,if the vertical relation  
between two lines be necessary for      
defining a dimension1 then all other    
lines than the specified ones    are    
either not in any dimension at all1 or  
they are outside the    three given     
dimensions4 ,this is denied by all      
parties1 which only    shows that a     
vertical relation to other lines is not 
necessary to    the determination of a  
dimension4 #b4 ,if lines outside the    
three    vertically intersecting lines  
still lie in dimension or are           
reducible to the other dimensions they  

may lie in more than one    dimension at
 the same time which after all is a     
fact4 ,this only    shows that          
qualitatively all three dimensions are  
the same and    that any line outside of
 another can only represent a dimension 
in    the sense of direction from a     
given point or line1 and we are         
entitled to assume as many dimensions as
 we please1 all within    three         
dimensions4      ,this mode of treatment
 shows the source of the illusion about 
the 'fourth dimension4' ,the term in its
 generic import denotes    commensurable
 quality and denotes only one such      
quality1 so that    the property        
supposed to determine non-,euclidean    
geometry must    be qualitatively       
different from this1 if its figures     
involve the    necessary qualitative    
differentiation from ,euclidean         
mathematics4    ,but this would shut out
 the idea of 'dimension' as its basis   
which is contrary to the supposition4   
,on the other hand1 the term    has a   

specific meaning which as different     
qualitatively from the    generic       
includes a right to use the generic term
 to describe them    differentially1 but
 if used only quantitatively1 that is1  
to    express direction as it1 in fact1 
does in these cases1 involves    the    
admission of the actual1 not a          
supposititious1 existence of    a fourth
 dimension which again is contrary to   
the supposition    of the non-,euclidean
 geometry4 ,stated briefly1 dimension as
    commensurable quality makes the     
existence of the fourth dimension    a  
transcendental problem1 but as mere     
direction1 an empirical    problem4 ,and
 the last conception satisfies all the  
requirements    of the case because it  
conforms to the purely quantitative     
differences which exist between         
,euclidean and non-,euclidean           
geometry as the very language about     
'surfaces1' 'triangles1' etc41    in    
spite of the prefix 'pseudo1'           
necessarily implies4      #ah ,vide     

,philosophical ,review1 ,vol4 ,v1 #ahif1
 p4 #ceb1 et4    seq4    ,thus it would 
seem that those who have been most      
diligent in  constructing the hyperspace
 conception have been the least careful 
of the logical difficulties which beset 
the elaboration of their  assumptions4  
,yet it sometimes requires the          
illogical1 the absurd and  the aberrant 
to bring us to a right conception of the
 truth1 and when  we come to a          
comparison of the two1 truth and        
absurdity1 we are the  more surprised   
that error could have gained so great   
foothold in face  of so overwhelming    
evidences to the contrary4    ,the      
entire situation is1 accordingly1 aptly 
set forth by ,,hyslop when he  says1    
continuing3      ,there are either a    
confusion of the abstract with the      
concrete    or of quantitative with     
qualitative logic1 444 so that all      
discussion about a fourth dimension is  
simply an extended mass    of           
equivocations turning upon the various  

meanings of the    term 'dimension4'    
,this when once discovered1 either makes
 the    controversy ridiculous or the   
claim for non-,euclidean properties a   
mere truism1 but effectually explodes   
the logical claims for a new            
dimensional quality of space as a piece 
of mere jugglery in which    the juggler
 is as badly deceived as his spectators4
 ,it simply    forces mathematics to    
transcend its own functions as defined  
by    its own advocates and to assume   
the prerogatives of metaphysics4        
,shall we1 therefore1 assent to the     
imperialistic policy of  mathematicians 
who would fain usurp the preserves of   
the metaphysician  in order that they   
may exploit a superfoetated hypothesis8 
,it is not  believed that the harshness 
of ,,hyslop',s judgment in this respect 
is undeserved4 ,it is1 however1         
regretted that the notions of           
mathematicians have been so inchoate as 
to justify this rather caustic1  though 
appropriate criticism4 ,for it does     

appear that the moment the              
mathematician deserts the province of   
his restricted sphere of motility  and  
enters the realm of the transcendental1 
that moment he loses his  way and       
becomes an inexperienced mariner on an  
uncharted sea4    ,it is interesting to 
note that ,,cassius ,,jackson           
,,keyser1#ai while  recognizing the     
purely arbitrary character of the       
so-called  dimensionality of space1     
nevertheless lends himself to the view  
that  if we think of the line as        
generating element we shall find that   
our  space has four dimensions4 ,that   
fact may be seen in various ways1 as    
follows3      ,a line is determined by  
any two of its points4 ,every line      
pierces    every plane4 ,by joining the 
points of one plane to all the points   
of another1 all the lines of space are  
obtained4 ,to determine    a line1 it   
is1 then1 enough to determine two of its
 points1    one in the one plane and one
 in the other4 ,for each of these       

determinations two data1 as before      
explained1 are necessary and            
sufficient4 ,the position of the line is
 thus seen to depend upon    four       
independent variables1 and the four     
dimensionality of our    space in lines 
is obvious4      #ai3 ,vide ,monist1    
,vol4 ,,xvi1 #ahif1 ,mathematical       
,emancipations4    ,similarly he argues 
for the four dimensionality of space in 
spheres3      ,we may view our space as 
an assemblage of its spheres4 ,to       
distinguish a sphere from all other     
spheres1 we need to know four    and but
 four independent facts about it1 as    
say1 three that shall    determine its  
center and one its size4 ,hence our     
space is four    dimensional also in    
spheres4 ,in circles1 its dimensionality
 is    six2 in surfaces of second order 
7those that are pierced by a    straight
 line in two points71 nine2 and so on ad
 infinitum4    ,the view taken by       
,,keyser is a typical one4 ,it is the   
mathematical  view and is characterized 

by a certain lack of restraint which is 
found to be peculiar to the whole scheme
 of thought relating to  hyperspace4 ,it
 is clear that the kind of space that   
will permit  of such radical changes in 
its nature as to be at one time three   
dimensional1 at another time four       
dimensional1 then six1 nine and  even   
n-dimensional is not the kind of space  
in which the objective  world is known  
to exist4 ,indeed1 it is not the kind of
 space that  really exists at all4 ,in  
the first place1 a line cannot generate 
perceptual space4 ,neither can a circle1
 nor a sphere nor any other  geometrical
 construction4 ,it is1 therefore1 not   
permissible1 except  mathematically1 to 
view our space either as an assemblage  
of its  spheres1 its circles or its     
surfaces2 for obviously perceptual space
  is not a geometrical construction even
 though the intellect naturally  finds  
inhering in it a sort of latent         
geometrism which is kosmical4  ,for     
there is a wide difference between that 

kosmic order which is  space and the    
finely elaborated abstraction which the 
geometer  deceives himself into         
identifying with space4 ,there is       
absolutely  neither perceptible nor     
imperceptible means by which perceptual 
space  in anywise can be affected by an 
act of will1 ideation or movement4      
,just why mathematicians persist in     
vagarizing upon the generability  of    
space by movement of lines1 circles1    
planes1 etc41 is confessedly  not easily
 understood especially when the natural 
outcome of such  procedure is           
self-stultification4 ,it is far better  
to recognize1 as  a guiding principle in
 all mathematical disquisitions         
respecting the  nature of space that the
 possibilities found to inhere in an    
idealized  construction cannot be       
objectified in kosmic1 sensible space4  
,the line  of demarkation should be     
drawn once for all1 and all             
metageometrical  calculations and       
theories should be prefaced by the      

remark that3 if  objective space were   
amenable to the peculiarities of an     
idealized  construction such and such a 
result would be possible1 or words to   
that effect4 ,this mode of procedure    
would serve to clarify many if  not all 
of the hyperspace conceptions for the   
non-mathematician as  well as for the   
metageometricians themselves1 especially
 those who are  unwilling to recognize  
the utter impossibility of their        
constructions  as applied to perceptual 
space4 ,we should then cease to have the
  spectacle of otherwise well-demeanored
 men committing the error of  trying to 
realize abstractions or abstractionizing
 realities4 ,herein  is the crux of the 
whole matter1 that mathematicians1      
rather than be  content with realities  
as they find them in the kosmos1 should 
seek  to reduce them to abstractions1   
or1 on the other hand1 make their       
abstractions appear to be realities4    
,,keyser proceeds to show how the       
concept of the generability of          

hyperspace may be conceived by beginning
 with the point1 moving it in  a        
direction without itself and generating 
a line2 beginning with the  line1       
treating it similarly1 and generating a 
plane2 taking the plane1  moving it in a
 direction at right angles to itself and
 generating a  cube2 finally1 using the 
cube as generating element and          
constructing  a four-space figure1 the  
tesseract4 ,now1 as a matter of fact1 a 
point being intangible cannot be moved  
in any direction neither can  a         
point-portion of sensible space be      
removed4 ,nevertheless1 we quite  agree 
with him when he asserts3               
,certainly there is naught of absurdity 
in supposing that under    suitable     
stimulation the human mind may1 in the  
course of time1    speedily develop a   
spatial intuition of four or more       
dimensions4    7,the italics in the     
above quotation are ours47    ,here we  
have a tacit implication that the notion
 which geometers  have heretofore       

designated as dimension really is a     
matter of  consciousness1 of intuition1 
and therefore1 determinable only by the 
limitations of consciousness and the    
deliveries of our intuitive  cognitions4
 ,as a more detailed discussion of this 
phase of the subject  shall be entered  
into when we come to a consideration of 
,chapter ,,vi on  ,consciousness as the 
,norm of ,space ,determinations further 
comment is  deferred until then4        
,now1 as it appears certain that what   
geometers are accustomed to call        
dimension is both relative and          
interchangeable in meaning--the one     
becoming the other according as it is   
viewed--the conclusion very  naturally  
follows that neither constructive nor   
symbolic geometry  is based upon        
dimension as commensurable quality4 ,the
 real basis of  the non-,euclidean      
geometry is dimension as direction4 ,for
 whatever  else may be said of the      
fourth dimension so-called it is        
certainly  unthinkable1 even to the     

metageometricians1 when it is absolved  
from  direction although no specific    
direction can be assigned to it4 ,it  is
 agreed perhaps among all non-,euclidean
 publicists that the fourth  dimension  
must lie in a direction which is at     
right angles to all the  three          
dimensions4 ,but if they are asked how  
this direction may be  ascertained or   
even imagined they are nonplused because
 they simply do  not know4 ,the         
difficulty in this connection seems to  
hinge about the  question of identifying
 the conditions of the world of phantasy
 with  those of the world of sense4     
,there are distortions1 ramifications1  
submersibles1 duplex convolutions and   
other mathetic acrobatics which  can be 
performed in the realm of the conceptual
 the execution of which  could never be 
actualized in the objective world4      
,because these antics  are possible in  
the premises of the mathematical        
imagination is scarce  justification for
 the attempts at reproduction in an     

actualized and  phenomenal universe4    
,one of the proudest boasts of the      
fourth dimensionist is that  hyperspace 
offers the possibility of a new species 
of rotation1 namely1  rotation about a  
plane4 ,he refers to the fact that in   
the so-called  one-space1 rotation can  
take place only about a point4 ,for     
instance  in ,figure #g1 the line ab    
represents a one-space in which rotation
  can take place only about one of the  
two points a and b4 ,in  ,figure #h     
which represents a two-space1 rotation  
may take place about  the line ab or the
 line cd1 etc41 or1 in other words1 the 
plane  abcd can be rotated on the axial 
line ab in the direction of the  third  
dimension4 ,in tridimensional space only
 two kinds of rotation  are possible1   
namely1 rotation about a point and about
 a line4 ,in the  fourth dimension it is
 claimed that rotation can take place   
about a  plane4 ,for example1 the cube  
in ,figure #i1 by manipulation in the   
direction of the fourth dimension1 can  

be made to rotate about the side  abgf4 
,a very ingenious argument is used to   
show how rotation about a plane  is     
thinkable and possible in hyperspace4   
,but with this1 as with the  entire     
fabric of hyperspace speculations1      
dependence is placed almost  entirely   
upon analogous and symbolic conceptions 
for evidence as to the  consistency and 
rationality of the conclusions arrived  
at4          ,d             ,c          
,d#a        -------------4444444444444  
3                                 3     
3                                 3     
3        -------------4444444444444     
,a             ,b            ,a#a       
,,fig4 #ac4    ,it is urged that        
inasmuch as the rotation about the line 
bc in ,figure  #ac would be             
incomprehensible or unimaginable to a   
plane being for the  reason that such a 
rotation involves a movement of the     
plane into the  third dimension1 a      
dimension of which the plane being has  
no knowledge1  in like manner rotation  

about a plane is also unimaginable or   
incomprehensible to a tridim or a three 
dimensional being4 ,it is shown1        
however1 that the plane being1 by making
 use of the possibilities of an  assumed
 tridimension1 could arrive at a        
rational explanation of line  rotation4 
,illustration3 ,,fig4 #ad4    ,figure   
#ad offers an illustration by means of  
which a two dimensional  mathematician  
could demonstrate the possibility of    
line rotation4  ,he is already          
acquainted with rotation about a point2 
for it is the  only possible rotation   
that is observable in his two           
dimensional  world4 ,by conceiving of a 
line as an infinity or succession of    
points  extending in the same direction2
 by imagining the movement of his  plane
 in the direction of the third dimension
 thereby generating a  cube and at the  
same time assuming that the lines thus  
generated were  merely successions of   
points extending in the same direction1 
he could  demonstrate that the entire   

cube ,figure #ad1 could be rotated about
  the line ,,bhx used as an axis4 ,for  
upon this hypothesis it would be        
arguable that a cube is a succession of 
planes piled one upon the other  and    
limited only by the length of the cube  
which would be extending  in the1 to    
him1 unknown direction of the third     
dimension4 ,he could  very logically    
conclude that as a plane can rotate     
about a point1 a  succession of planes  
constituting a tridimensional cube1     
could also  be conceived as rotating    
about a line which would be a succession
  of points under the condition of the  
hypothesis4 ,his demonstration1         
therefore1 that the cube1 ,figure #ad1  
can be made to rotate around  the line  
,,bhx would be thoroughly rational4 ,he 
could thus prove  line-rotation without 
even being able to actualize in his     
experience  such a rotation4            
,analogously1 it is sought by           
metageometricians to prove in like      
manner  the possibility of rotation     

about a plane4 ,thus in ,figure #af is  
shown a  cube which has been rotated    
about one of its faces and changed from 
its  initial position to the position it
 would occupy when the rotation had     
been completed or its final position    
attained4    ,illustration3 ,,fig4 #ae4 
,illustration3 ,,fig4 #af4--,plane      
,rotation    ,the gist of the arguments 
put forward as a basis for              
plane-rotation  is briefly stated thus3 
,the face cefg is conceived as          
consisting of  an infinity of lines4 ,a 
cube1 as in ,figure #ae1 is imagined or 
assumed  to be sected into an infinity  
of such lines1 each line being the      
terminus of one of the planes which make
 up the cube4 ,each one of the          
constituting planes is thought of as    
rotating about its line-boundary  which 
intersects the side of the cube4 ,the   
process is continued  indefinitely until
 the entire series of planes is rotated1
 one by  one1 around the series of lines
 which constitute the axial plane4      

,hence1 in order that the cube1 ,figure 
#af1 may change from its initial        
position to its final position each one 
of the infinitesimal planes  of which   
the cube is assumed to be composed must 
be made to rotate  about each one of the
 infinitesimal lines of which the plane 
used as  an axis is composed4 ,in this  
way1 it is shown that the entire cube   
has been made to rotate about its face1 
cefg4 ,this concisely1 is  the ,,quod   
,,erat ,,demonstrandum of the           
metageometrician who sets out to  prove 
rotation about a plane4 ,thus it is made
 to appear that in order  that          
tridimensional beings may be enabled to 
conceive of four-space  rotation1 as in 
,figures #ae and #af1 in which the      
rotation must also be  thought of as    
taking place in the direction of the    
fourth dimension1  they must adopt the  
same tactics that a two dimensional     
being would use  to understand some of  
the possibilities of the tridimensional 
world4    ,it is1 of course1 unwise to  

assume that because a thing can be      
shown to be possible by analogical      
reasoning its actuality is  thereby     
established4 ,this consideration cannot 
be too emphatically  insisted upon2 for 
many have been led into the error by    
relying too  confidentially upon results
 based upon this line of argumentation4 
,there is a vast difference between     
mentally doing what may be assumed  to  
be possible1 the hypothetical1 and the  
doing of what is actually  possible1 the
 practical4    ,in the first place1     
plane-rotation in the actual universe is
 a  structural impossibility4 ,the very 
nature and constitution of material     
bodies will not admit of such contortion
 as that required by the  rotation of a 
body1 say a cube1 about one of its      
faces4 ,let us examine  some of the     
results of plane rotation4 #a4 ,the     
rotation must take  place in the        
direction of the fourth dimension4 ,now1
 as it is utterly  impossible for any   
one1 whether layman or metageometrician1

 even to  imagine or conceive1 in any   
way that is practical1 the direction of 
the fourth dimension it is also         
impossible for one to move or rotate  a 
plane1 surface1 line or any other body  
in that direction4 ,we are  in the very 
beginning of the process of             
plane-rotation so-called  confronted    
with a physical impossibility4 #b4      
,plane rotation necessarily  involves   
the orbital diversion of every particle 
in the cube4 ,this  alone is sufficient 
to prohibit such a rotation2 for it is  
obvious  that the moment a particle or  
any series of particles is diverted from
  its established orbital path          
disruption of that portion of the cube  
must necessarily follow4 ,this upon the 
assumption that the particles  of matter
 are in motion and revolving in their   
corpuscular orbits4 #c4  ,plane-rotation
 necessitates a radical change in the   
absolute motion  of each individual     
particle1 electron1 atom or molecule of 
matter in  the cube and a consequent    

retardation or acceleration of this     
motion4  ,this upon the hypothesis that 
the particles of matter are vibrating   
at the rate of absolute motion4 #d4 ,it 
presupposes a reconstitution  of each   
atom1 molecule or particle in the cube1 
changing the path of  intra-corpuscular 
rotation either from a right to left    
direction or  from a left to right      
direction1 as the case may be4 ,the     
particles of  matter in the cube will be
 acted upon in much the same manner as  
the  particles in a glove when it is    
maneuvered in the fourth dimension4 ,in 
describing this phenomenon1 ,,manning   
says3#bj      ,every part by itself1 in 
its own place is turned over with only a
    slight possible stretching and      
slight changing of positions of the     
different particles of matter which go  
to make up the glove4      #bj ,vide    
,fourth ,dimension1 ,simply ,explained1 
edited by    ,h4 ,p4 ,,manning1 p4 #bh4 
,the slight stretching and slight       
changing of the positions of the        

particles referred to would be of small 
consequence if applied to  ponderable   
bodies4 ,but when used in connection    
with particles of matter  which are     
themselves of very infinitesimal size   
means far more--enough1  as we have     
said1 to militate severely against the  
integrity of the  cube4 ,it is not      
deemed necessary to go further into the 
physical  aspects of plane-rotation as  
it is believed sufficient has been said 
to  negative the assumption from a      
purely structural viewpoint4    ,among  
the vagaries of hyperspace publicists   
none is perhaps more  notable than the  
view taken by ,c4 ,h4 ,,hinton3#ba      
,if it could be shown that the electric 
current in the negative    direction    
were exactly alike the electric current 
in the positive    direction1 except for
 a reversal of the components of the    
motion in three dimensional space1 then 
the dissimilarity of    the discharge   
from the positive and negative poles    
would be an    indication of the        

one-sidedness of our space4 ,the only   
cause of    difference in the two       
discharges would be due to a component  
in    the fourth dimension1 which       
directed in one direction transverse    
to our space1 met with a different      
resistance to that which it met    when 
directed in the opposite direction4     
#ba ,vide ,fourth ,dimension1 p4 #ge1   
,c4 ,h4 ,,hinton4    ,to be sure4 ,and  
with equal certainty it might be said   
that if the  moon were made of green    
cheese it might well be the ambition of 
the  world's chefs to be able at some   
time to flavor macaroni with it1  thus  
serving a rare dish4 ,even so1 if there 
were an actual1 objective  fourth       
dimension to our space we might be able 
to shove into it all the  perplexing    
problems of life and let it solve them  
for us4 ,but the fact  that the fourth  
dimensional hypothesis is itself a mere 
supposition  seems to have been         
overlooked or rather completely ignored 
by ,,hinton4  ,or else1 ought it not be 

an obvious folly to hope to construct a 
rational explanation of perplexing      
physical conditions upon the basis  of a
 purely suppositionary1 and therefore   
unproven1 hypothesis8    ,the recognized
 domain of the four-space1              
mathematically considered1  is according
 to the most generous allowance very    
small1 so small1 in  fact1 that the     
disposition of some to crowd into it the
 essential  content of the manifested   
universe is a matter of profound        
amazement4  ,then1 too1 it cannot be    
denied that there is no appreciable     
urgency or  necessity for having        
recourse to a purely hypothetical       
construction  for explicatory data      
regarding a phenomenon which has not    
been shown  to be without the scope of  
ordinary scientific methods of procedure
 to  unravel4    ,the claim of certain  
spiritualists1 notably ,,zollner of     
,leipsig1 that  the phenomena of        
spiritism is accountable for on the     
grounds that the  fourth dimension      

affords a residential area for          
discarnate beings  whence spiritistic   
forayers may impose their presence upon 
unprotected  three dimensional beings is
 no less fatuous than the original      
supposition itself4 ,for upon this      
latter is built the entire fabric of    
meaningless speculations so gleefully   
indulged in by those who glibly         
proclaim the reality of the four-space4 
,indeed1 clearer second thought  will   
reveal that1 when the pendulum of       
erratic thinking and trafficking  in    
mental constructions swings back1       
hyperspaces1 after all1 are but  the    
ignes fatuii of mathetic obscurantism4  
,then1 why should it be deemed necessary
 to discover some more  mysterious realm
 of four dimensional proportions in     
which the spirits  of the dead may find 
a habitation8 ,are the spiritualists1   
too1 reduced  to the necessity of       
further mystifying their already        
adequately  mysterious phenomena8 ,if   
there were not quite enough of          

physicality  upon the basis of which all
 the antics of these entities can be    
explained1 and that satisfactorily1 one 
would1 as a matter of course1  be       
inclined to lend some credence to these 
claims2 but as it is clear  that all    
organized beings have some power1 if no 
more than that which  maintains their   
organization1 and as it ought also be an
 acceptable  fact that such a being is  
directed by mind2 and further1 that     
owing to  the nature of a spirit body it
 can penetrate solid matter or matter of
  any other degree of density below the 
coefficient of spirit matter1 it  ought 
likewise be unnecessary to go without   
the province of strictly  tridimensional
 mechanics for an explanation of        
spiritistic phenomena4    ,equally      
unnecessary and uncalled for is the     
attempt of certain others  who lean     
toward the view of speculative chemists 
to account for the  none too securely   
established hypothesis that eight       
different alcohols1  each having the    

formula ,c#e;,h#ab,o may be produced    
without variation4  ,this is said to be 
due to the fact that certain of the     
component atoms1  notably the carbon    
atoms1 take a fourth dimensional        
position in the  compound and thus      
produce the unusual spectacle of eight  
alcohols from  one formula4 ,have       
chemists actually exhausted all purely  
physical means  of reaching an          
understanding of the carbon compounds   
and are therefore  compelled to resort  
to questionable means in order to make  
additional  progress in their field8 ,it
 is incredible4 ,hence the more         
facetious  appears the mathematical     
extravaganza in which originates the    
tendence  among the more sanguine       
advocates to make of the fourth         
dimension a  sort of jack of all trades1
 a veritable ,aladdin's lamp wherewith  
all kosmic profundities may be          
illuminated and made plain4 ,not until  
the perfection of instruments of        
precision has been reached1 and not     

until human ingenuity has been exhausted
 in its efforts to produce more  refined
 methods of research should it be       
permissible even to venture  into       
untried and more or less debatable      
fields in search of a relief  which     
after all is unobtainable4              
,notwithstanding the fact that all      
attempts at accounting for physical     
phenomena on the basis of               
n-dimensionality 7which is itself by all
  the standards of objective reference a
 non-existent quantity and  therefore   
irreconcilable with perceptual space    
requirements7 are to be  characterized  
simply as a senseless dalliance with    
otherwise deeply  profound questions1   
many have fallen into a complete        
forgetfulness of  the logical barriers  
inhering in and hedging about the query 
and have  committed other and less      
excusable errors in the premises4 ,take1
 for  instance1 the suggestion that the 
action of a tartrate upon a beam  of    
polarized light is due to the assumption

 of a fourth dimensional  direction by  
some component in the acid4 ,this for   
the reason that  experimentation has    
shown that tartaric acid1 in one form1  
will turn  the plane of polarized light 
to the right while in another form will 
turn it to the left4 ,it is not         
believed1 however1 that there is any    
warrant for such an assumption4 ,there  
is also another kind of tartrate  which 
seems to be neutral in that it has no   
effect whatever upon the  beam of light1
 turning it neither to the right nor to 
the left nor  having other visible or   
determinable effect upon it4 ,indeed1 it
 is  not clear how it is hoped to prove 
such a case by constituting as a  norm a
 hypothesis which is essentially        
indemonstrable4 ,a more logical         
procedure would be first to establish   
the objective1 discoverable  posture of 
four-space2 show the actual movement of 
matter and entities  therein2 locate it 
by empirical methods of research1 and   
then1 basing  our assertions upon       

apodeictic evidences1 assume a new      
attitude toward  these phenomena because
 of the support found in established and
  verifiable facts4 ,some hope of       
gaining a respectful hearing might then 
be entertained2 but at least to do so   
now appears to be quite untimely4       
,,major ,,wilmot ,e4 ,,ellis1 ,coast    
,artillery ,corps1 ,united ,states      
,army1 in  ,the ,fourth ,dimension      
,simply ,explained1#bb remarks3      444
 in the ether1 if anywhere1 we should   
expect to find some    fourth           
dimensional characteristics4            
,gravitation1 electricity1    magnetism 
and light are known to be due to        
stresses in1 or    motions of1 the      
infinitesimal particles of the ether4   
,the real    nature of these phenomena  
has never been fully explained by       
three dimensional mathematical analysis4
 ,indeed1 the unexplained    residuum   
would seem to indicate that so far we   
have merely been    considering the     
three dimensional aspects of four       

dimensional    processes4 ,as one       
illustration of many1 it has been shown 
both    mathematically and              
experimentally that no more than five   
corpuscles may have an independent      
grouping in an atom4      #bb ,q4 v41 p4
 #bdb1 edited by ,h4 ,p4 ,manning4      
,the weakness of this view may be due to
 the fact that at that  time ,,major    
,,ellis was emphasizing in his own mind 
the necessity of  simplifying the       
conception so as to make it of easy     
comprehension  rather than the          
establishment of any fealty to truth or 
the spirit  of mathesis in his          
examination of the problem4 ,what       
therefore of  reality the student fails 
to find in his view may be attributed to
 the  sacrifice which the writer        
7,,major ,,ellis7 felt himself called   
upon to  make for the sake of           
simplicity4 ,hence a certain expressed  
connivance  at his position is          
allowable4 ,but1 on the other hand1 if  
such were not  the conscious intent of  

,,major ,,ellis it is not understood how
 it should  appear that the unexplained 
residuum would seem to indicate that so 
far we have merely been considering the 
three dimensional aspects of  four      
dimensional processes4 ,contrarily1 it  
has yet to be proved that  three        
dimensional space does not afford ample 
scope of motility for  all observable or
 recognizable physical processes and    
that there is  no necessity for         
reference to hyperspace phenomena for an
 explanation  of the unexplained        
residuum4 ,it is1 of course1 understood 
that many  of the possibilities         
predicated for hyperspace are purely    
nonsensical  so far as their actual     
realization is concerned4 ,our concern  
is1  therefore1 not with that class of  
predicates1 but with those wherein      
reside some slight show of probability  
of their response to the  conditions of 
n-dimensionality either as a system of  
space-measurement  or a so-called space 
or series of spaces4    ,,major ,,ellis 

concludes his simple study of four-space
 by proposing the  following query3     
,may not birth be an unfolding through  
the ether into the    symmetrical       
life-cell1 and death1 the reverse       
process of a    folding-up into four    
dimensional unity8    ,it is confessed  
that there seems to be nothing to       
warrant the giving  of an affirmative   
reply to this query4 ,it is1 perhaps1   
sentimentally  speaking a very beautiful
 thing to contemplate death as a        
painless1  unconscious involvement into 
a glorious one-ness with all life1 and  
birth1 as the reverse of all this4 ,but 
where is the utility of such a  dream if
 it be merely a dream and impossible of 
realization8    ,,simon ,,newcomb1#bc at
 one time one of the outstanding figures
 in  the early development of the fourth
 dimensional hypothesis1 openly         
declared that there is no proof that the
 molecule may not vibrate in a  fourth  
dimension4 ,there are facts which seem  
to indicate at least the  possibility of

 molecular motion or change of some sort
 not expressible  in terms of time and  
the three cordinates in space4    ,of   
course1 there is no proof that a        
molecule may not at times be  ensconced 
in a four-space neither is there proof  
nor probability that  it is so hidden4  
,indeed1 there is no proof that there is
 such a thing  as a molecule for that   
matter4      #bc ,vide ,science1 ,vol4  
,,vii1 #aeh1 #ahih1 p4 #d4    ,in all of
 the foregoing proposals it is assumed  
that the fourth  dimension really exists
 and that it lies just beneath the      
surface of  the visible1 palpable limits
 of the material universe2 that lying   
in close juxtaposition to all that we   
are able to see1 to hear or  sense in   
any way is this mysterious1 eternally   
prolific1 all-powerful  something1      
hyperspace1 ever-ready to nourish and   
sustain the forms  which have the nether
 parts firmly encysted in one or the    
other of her  n-dimensional berths4     
,thus it would seem that while yet      

functioning  in a strictly              
tridimensional atmosphere1 some one1    
more reckless  than the rest1 should at 
last stumble upon some up-lying portion 
of  it and be instantly transformed into
 a mathetic fay of etherealized         
four-dimensional stuff4        ,,part   
,,two      ,,spatiality      ,,an       
,,inquiry ,,into ,,the ,,essential      
,,nature ,,of ,,space ,,as              
,,distinguished    ,,from ,,the         
,,mathematical ,,interpretation         
,,chapter ,,vi      ,,consciousness     
,,the ,,norm ,,of ,,space               
,,determinations      ,realism ,is      
,determined by ,awareness--,succession  
of ,degrees of    ,realism--,sufficiency
 of ,tridimensionality--,the            
,insufficiency of    ,self-,consistency 
as a ,norm of ,truth--,general ,forward 
,movement    in the ,evolution of       
,consciousness ,implied in the          
,hyperspace    ,concept--,the           
,hypothetical ,nature of ,our           
,knowledge--,hyperspace    the ,symbol  

of a ,more ,extensive ,realm of         
,awareness--,variations in    the       
,method of ,interpreting ,intellectual  
,notions--,the ,tuitional    and the    
,intuitional ,faculties--,the           
,illusionary ,character of the          
,phenomenal--,consciousness and the     
,degrees of ,realism4      ,things have 
value for us only to the extent to which
 we can become  aware of their being4   
,the appraisement of all objects1       
conditions1  states or qualities is     
determined directly by the degree or    
quality  of awareness with which we     
apprehend them4 ,those elements which   
are  without the intellect's scope of   
awareness have no interest and hence  no
 value so far as the individual         
intellect is concerned4 ,and this is    
true of all degrees and states of       
consciousness from the lowest to the    
highest1 from the human to the divine4  
,there enter into all conscious         
determinations three factors1 namely3   
7a7 the scope1 or totality1 of          

adaptations which an organism can  make 
in the sensible world1 7b7 the power of 
consciousness to make  adaptations and  
7c7 environment4 ,these three are       
interdependent4  ,the totality of       
adaptations depends primarily1 of       
course1 upon the  quality of conscious  
powers or faculties1 and also1 in a     
lesser  degree1 upon opportunities      
afforded by environment4 ,faculties of  
consciousness are derived directly from 
the influences exerted upon  the        
organism by his environment and the     
results of the struggle to  overcome    
them4 ,environment is of two kinds1     
artificial and natural4 ,the  artificial
 environment is such as has been        
modified by our conscious  action upon  
external phenomena4 ,the residue is     
natural4 ,and thus the  scope of        
adaptability becomes an unvarying       
witness to the quality of  consciousness
 manifesting through a given organism4  
,the universe is so constructed that the
 essential character of its  various    

states and qualities is a fixed quantity
 for a given scope of  consciousness and
 varies only as the sphere of           
consciousness varies4  ,states of       
existence or scopes of adaptation which 
are registering upon  a higher plane or 
in a more subtle sphere of existence    
than that in  which we may at any time  
be functioning can only appear          
evidential  to us when the mechanism of 
our consciousness becomes congruently   
adjusted therewith4 ,so that the focus  
of consciousness must always be  a      
variable quantity adaptable1 under      
proper conditions1 to any plane  in the 
kosmos4 ,consciousness1 then1 becomes   
the sphere of limits both  of knowledge 
and adaptability4 ,but lest we seem to  
admit implicitly  part of the           
contentions which mathematical          
publicists have made in  postulating the
 unodim and duodim consciousness1 it is 
necessary  carefully to differentiate   
between the results arrived at as a     
result  of the two procedures4 ,in the  

first place1 analysts assume the        
existence of a unodim and duodim plane  
of consciousness and proceed  to        
construct thereon an analogy designed to
 show the feasibility  of another       
assumption1 the fourth dimension4       
,while1 in laying the  foundation of    
consciousness upon a tridimensional     
plane we do not start  with an          
assumption1 but with a fact4 ,therein   
lies the difference4  ,enormous         
advantages inhere in a procedure based  
upon facts1 but in  a series of planes  
built upon assumptions no such          
advantages are  discovered4 ,for however
 much the series of hypothetical planes 
may be  extended or elaborated there    
must inhere necessarily throughout the  
series an assumptional value which      
vitiates the conclusions no less  than  
the premises4 ,the sanity and integrity 
of intellectual operations  depend      
almost entirely upon the differentiation
 which we make between  the necessities 
arising out of assumptions and those    

which spring  up empirically from       
established facts4 ,no procedure is     
necessary to  establish the value of    
such a differentiation1 nevertheless it 
may  be suggested that it is allowable1 
under the rules of logic1 to make  any  
assumption whatsoever so long as care is
 taken to see that the  conclusions     
embody in themselves the characteristics
 of the original  premise4 ,for         
instance1 it is permissible to assume   
that space is  curved4 ,under such an   
assumption1 it is only necessary that   
the  constructions which follow shall be
 self-consistent4 ,but the case is      
different when we come to deal with     
spatiality and vitality4 ,these are     
quantities which cannot1 in the last    
analysis1 be made to conform to  the    
rules of the game of logic4    ,thus1   
when it is intimated that realism lends 
itself to an apparent  division into    
degrees1 and that each degree has a     
corresponding state  of consciousness1  
it is by no means to be inferred that   

such apparent  divisions are of         
mathematical import4 ,for1 in reality1  
i4e41 when the  consciousness has       
expanded so as to become congruent with 
the limits  of even the space mind 7vide
 ,fig4 #bj71 there appear to be no      
divisions  in realism4 ,it is only      
because of the fragmentariness of our   
outlook  upon the kosmos that realism   
appears to be divided into various      
planes2  for all of these planes are    
one4 ,the divisions exist for relative  
knowledge1 but not for complete         
knowledge2 they exist for a finite      
intelligence1 but not for a transfinite 
intelligence4 ,that is why we  view     
realism as a series of planes4 ,it is   
because we discover that1 as  we        
proceed1 as our consciousness expands   
and we take in more and more  of the    
vital activities of the kosmos and      
understand better the causes  underlying
 that which we contact1 we have passed  
from a state of lesser  knowledge to one
 of greater knowledge4 ,and so we say we

 have passed  from one degree of realism
 to another1 whereas1 really we have not
  passed from one degree of realism to  
another degree4 ,instead1 it is  our    
consciousness that has expanded4    ,if 
now1 we conceive reality to be a scale  
extending from one  extremity to another
 7that is1 from supreme consciousness to
 entire  unconsciousness1 from final    
knowledge to total ignorance71 and the  
intellectual consciousness as the       
indicator which traverses the scale     
denoting at all times the precise degree
 of our comprehension of  reality1 and  
hence the degree of expansion of        
consciousness1 we shall  constitute a   
similitude closely approximating the    
real status quo of  humanity with       
respect to the sensible and             
supersensible worlds4 ,the  quantity or 
force which causes the indicator to move
 along the scale  is the quality of     
awareness4 ,and this varies directly as 
the scope of  adaptability varies4      
,realism is homogeneous throughout its  

extent2  but the scale marked upon it   
registers from naught to unity4 ,and    
between these every conceivable degree  
of awareness may be registered4  ,the   
indicator moves only as the scope       
widens1 and thus is shown a  change in  
the quality of awareness4 ,for1 however 
paradoxical it may  seem1 the wider the 
scope of knowledge the better its       
quality3 the more  one knows1 the more  
complete and of higher quality becomes  
that which  he knows4    ,the intellect 
is of scientific tendence1 studiously   
rejecting all  phenomena which do not   
yield to its senso-mechanisms4 ,even    
intuitions  suffer the humility of      
rejection and do not escape the         
limitations  which the intellect imposes
 upon them4 ,this is so1 because1 as    
yet1  there is no adequate perceptive   
and conceptive apparatus for the        
propagation and classification of       
intuitions1 as apart from concepts4     
,the outcome of these proscriptions is  
that intuitions--free1 mobile1  and more

 or less formless in themselves1 must   
first be rehabilitated  and vestured in 
garments a la intellect to conform to   
the prevailing  mode4 ,but intuitions   
thus treated are no longer intuitions1  
but  empirical concepts4 ,true          
intuitions are like aqueous             
vapor--amorphous1  permeating1          
diffusive3 axioms or empirical concepts 
are like cakes of  ice--formal1         
inflexible and conforming to the shape  
of the mold into  which they are poured4
 ,because of this--the scientific       
tendence of the  intellect and the      
consequent necessity of reforming so    
much of the  data which constitute its  
substructure1 of pressing1 condensing   
and  reshaping it to suit its own       
ready-made patterns--it can be perceived
  how profound is the influence of the  
intellectual consciousness in           
determining the character of the        
totality of data which the sensible     
world1 and for that matter1 the         
supersensible1 offer us4 ,the intellect 

is the only means at hand for the       
interpretation of the meaning and       
significance of the world of phenomena4 
,consequently1 whatever meaning  or     
significance we are led to attach to    
that part of the universe which  we     
contact1 in any way1 is dictated by the 
intellectual consciousness4  ,there is  
no escape from the decisions of the     
intellect so long as the  present scheme
 of things endures4    ,thus1 by        
whatever standard of reference the      
matter may be determined1  it remains   
indisputably established that the       
intellectual consciousness  is the sole 
determinant of the phenomenal value of  
everything within  our scope of         
awareness or adaptability4 ,and whatever
 the fault1  incongruity or discrepancy 
that may be revealed by a more intimate 
knowledge of the genesis and character  
of the appearance of the  sensible      
world1 it will be found to be due to the
 peculiar cut and mode  of the intellect
 and not to things themselves4 ,the     

value1 qualitative  or existential1     
which the intellect irrevocably assigns 
to objects  and conditions in the world 
of the senses is the exclusive norm not 
only by which these are judged1 but     
also1 by which our action upon  them and
 their action upon us are determined4   
,images or objects which  do not act    
upon us and upon which we cannot act    
have no interest for  us4 ,but as an    
integral part of the totality of images 
or objects in  the sensible world1 we   
must inevitably act upon all that is    
outside of  ourselves1 and these1 in    
turn1 must react upon us4 ,on the other 
hand1  there must be objects and images 
in the universe of life and form upon   
which1 because of their inherent nature 
and on account of the lack of  our      
interest in them and their interest in  
us1 we can neither act nor  become the  
object of their action4    ,but herein  
is a mystery4 ,for1 either we act upon  
and are recipients of  the action of the
 totality of images or objects in both  

the sensible  and supersensible worlds1 
or we are so placed in the grand scheme 
of  things that both ourselves and the  
sphere of our interests and possible    
actions are closed circuits1            
hermetically sealed and                 
non-communicative  with the other like  
spheres1 which do not and cannot act    
upon us4 ,there  is yet a third         
possibility--that we are so fashioned1  
in the entirety  of our being1 that some
 part of us is exactly congruent with   
some part  of every sphere of possible  
actions and interests in the kosmos1 and
  therefore1 each of us has being or    
consciousness of a kind that is  keyed  
to and registering in the totality of   
such spheres2 and that1  at present1    
because our interests and possible      
actions are limited to  the plane of    
sensibility1 we are conscious only      
there4 ,and further1  that although     
those spheres of our consciousness which
 are fixed to  register in other planes 
do not answer to the lowest on which we 

now  operate1 having a character of     
which we are unaware1 they nevertheless 
cannot be said not to exist1 because of 
the lack of communication  between them4
 ,among these three possible choices1 we
 have no hesitancy  in expressing a     
decided preference for the last         
mentioned--that the  range of our being 
is co-extensive with the range of       
reality1 and  like a pendulum1 we       
oscillate1 at long intervals1 between   
two kosmic  extremities--nescience and  
omniscience4    ,the intellectual       
consciousness is the touch-stone of     
realism4 ,it  is like a spreading light 
which1 as it expands1 reveals previously
  darkened corners and conditions1 only 
it has power both to reveal and  to     
bring into manifestation4 ,in its       
present state1 man's consciousness  is  
like a searchlight4 ,it illumines and   
takes cognizance of everything  that    
falls within its scope of motility and  
is consequently able to  study in detail
 that which it reveals4 ,but that which 

is beyond its  scope is as if it never  
existed so far as the individual        
consciousness  is concerned4 ,it is not 
reasonable to predict that the same     
characteristics that are observable in  
any given state shall persist           
throughout all the various scopes       
through which the consciousness must    
proceed in its evolutionary expansion4  
,for the scale of kosmic realism  is one
 grand panorama extending from the      
grossest to the most subtle  and        
refined4 ,while in general the thread of
 realism may pervade the  entire scale  
it is nevertheless marked by many and   
diverse changes  in its characteristics 
as it is followed from one stage to     
another4  ,so that the realistic        
character of one stage may vary greatly 
from  that which next preceded it or    
from that which will succeed it4 ,it    
would appear1 therefore1 that in passing
 from one stage of realism to  another  
there need not remain anything but the  
mere fact of reality in  its connection 

with ultimate reality2 for it is obvious
 that in every  condition of realism    
which may be encountered in the kosmos  
there  must be a basic thread of        
ultimate reality running through the    
whole4  ,the entire gamut of realism may
 accordingly be traversed without the   
danger of being diverted from the golden
 thread of realism which thus  permeates
 all4 ,it is always the phenomena of    
realism with which we are  concerned and
 which we are trying to understand      
rather than realism  itself4 ,it is this
 that confounds us4 ,if it were not for 
the phenomena1  which is the way realism
 or life presents itself to our         
consciousness1  we should experience no 
trouble in discovering the reality1 all 
other  things being equal4 ,for the     
former ever obscures the latter4 ,it is 
the supreme task of mental evolution to 
break through the clouds of  phenomena  
in the search for the eternal substratum
 of reality which  runs through the     
sensible universe of things4    ,the    

first view of conditions that the mind  
takes upon awakening to  consciousness  
in any new sphere of cognition is       
necessarily hazy  and inchoate4 ,there  
is more or less of astonishment1 wonder 
and  bewilderment upon first becoming   
aware of a new scope of realism4 ,in    
this state it is natural that the mind  
should overlook or ignore much  that is 
essential and perhaps all that is so    
even escaping the true  import of the   
phenomena which it senses4 ,it is       
reasonable1 too1 that in  such a state  
the main outlines of what is really seen
 may be greatly  distorted and          
exaggerated so that it is well-nigh     
impossible to secure  a correct         
comprehension of the character of a new 
scope of realism  from any early survey4
 ,it is not until later years1 after    
much study  and circumspection that the 
mind1 becoming used to the new          
conditions1  begins to get correct      
impressions and to make valid judgments 
as to  that which it discerns4 ,and even

 then1 it not infrequently happens  that
 the resultant view of things in general
 is found to be in need of  revision and
 correction4 ,hence1 after everything is
 sifted down to the  ultimate allowance 
for the illusion incident to too much   
enthusiasm  and wonder we have only a   
very small residuum of truth upon which 
to  build and this latter we often find 
to be the single thread of reality      
which runs through all the phenomena and
 which is1 therefore1 the only  quantity
 that remains worthy of much            
consideration4    ,thus it is with      
religion4 ,the path of progress over    
which our religious  conceptions have   
come need not be outlined here1 but to  
any one at  all acquainted with the     
history of religious thought and ideals 
it  at once must be patent that it has  
been one continuous surrender of  the   
old for the new1 of one degree of       
realism for another newer and  higher   
degree2 that always it has been the     
phenomena1 the flora of  the ideals     

which have had to give way1 while       
nothing was left but the  roots of      
realism from which they have sprung4 ,it
 has been the same  with scientific     
knowledge4 ,facts have been collected   
and hypotheses  proposed to synthesize  
them and yet these have had to give way 
for  others1 and others still1 until the
 data of scientific knowledge to-day    
are quite different from what they were 
in earlier days4 ,and yet  permeating   
the scientific knowledge of all times   
has been the golden  thread of reality1 
and of all facts and systems of facts   
which man  has successively assumed and 
surrendered nothing has remained but the
  reality2 indeed1 nothing could so     
remain1 but reality4 ,so it is with  air
 phenomena with which consciousness has 
to deal4 ,this perhaps is due  to the   
fact that the mind interprets phenomena 
in accordance with the  quality of its  
awareness1 and as consciousness is a    
variable quantity1  its standards of    
interpretation will likewise vary4 ,each

 new scope of  awareness1 after this    
manner1 yields higher and more exact    
standards of  interpretation4 ,and then1
 progressing in awareness from the      
segment to  the whole a fuller view of  
the phenomena as well as of reality     
itself  is gained and also a more       
comprehensive judgment of the relations 
which  exist between the segment and the
 whole4 ,in other words1 as the scope   
of consciousness widens it becomes more 
and more apparent that what was  first  
thought to be a separate segment is in  
reality identified with  the whole in an
 indissoluble manner4 ,for the ,thinker 
is then not only  aware of the segment  
as such1 but he is also conscious of the
 fact that  it has definite relations   
with the entirety and that what he needs
 is  merely a more extended             
consciousness4    ,in denying the       
existence of the four-space or spaces of
  n-dimensionality as described and     
defined by geometricians1 we do not     
thereby deny the existence of a plane of

 consciousness which is as  much unlike 
the conditions of the tridimensional    
world as it is said  to be unlike the   
four-dimensional world2 but what we do  
deny is that  such a higher plane of    
existence has necessarily to be         
conditioned  by such characteristics as 
the metageometricians have proposed4 ,it
  is maintained that there is no basis  
in consciousness for a world of  four   
dimensions2 that the consciousness has  
no tendency for action in  four-space4  
,neither has matter nor life any        
inclination or potency  to behave in a  
four-dimensional manner4 ,it is indeed  
more rational  to suppose that there is 
a higher plane1 in fact1 a series of    
higher  planes1 in which the thread of  
realism is continuous1 not broken as it 
necessarily would have to be in         
extending to hyperspace1 nor curved  as 
in a manifold2 that this series of      
subtler and finer planes of             
consciousness are merely an elongation  
of our three dimensional scope  of      

realism4 ,it1 therefore1 remains only to
 master the phenomena of  each in just  
the same manner as we have1 in a        
measure1 mastered the  phenomena of     
tridimensionality4 ,for it is easily    
conceivable that the  quality of        
consciousness is such that it may adapt 
itself to a far  wider range of         
possibilities than may be discovered in 
hyperspace and  still be a tri-space    
quantity4    ,it is believed1 however1  
that in all the new and higher planes of
  consciousness tridimensionality is the
 norm both of the phenomena  and of the 
reality peculiar to them4 ,and that1    
being such1 does not  change or vary1   
but is a fixed quantity regardless of   
the plane of  consciousness4            
,furthermore1 it is believed that the   
highest state of  consciousness in the  
entire kosmos could easily exist1 and   
does so  exist1 upon the basis of       
three-space as the norm of its extent4  
,a sharp line of demarkation should be  
drawn between the reality which  is life

 and consciousness and that which       
belongs to the realm of  phantasy4 ,for 
it is the prerogative of the intellect  
to create1 out  of the remains and      
deposits which it finds in the pathway  
of life1  whatsoever it wills4 ,this it 
does continuously2 but it scarcely can  
be  expected that such creations shall  
be endowed with the same dynamic        
character as that which life bestows    
upon its creations4 ,the creations  of  
the one are merely dead carcasses while 
those of the other are vital  and real4 
,between them the same marked difference
 exists as between the  growing tree and
 the lumber which the builder converts  
into a house4  ,the organization which  
we witness when we look upon a building 
made of  the dead body of a tree is not 
the same kind of organization as that   
which we see when we view the living1   
growing1 vital tree4 ,the dead  tree is 
a deposit of life cast off by it when it
 passed on4 ,whatever  the intellect can
 do in disposing of the remains of the  

tree-life is  conventional and          
artificial4 ,if it convert it into an   
edifice it will  then bestow upon it a  
sort of consistency which is quite      
sufficient for  all purposes4 ,but the  
consistency which holds the organization
 of an  edifice together is not the kind
 of consistency which holds a living    
tree together4 ,in fact1 there is a     
consistency that is not consistent4     
,such is the consistency of             
metageometry4 ,it is self-consistent and
 yet  inconsistent with the consistency 
of the kosmos and its norm of being     
which is consciousness4                 
,self-consistency is one thing and      
kosmic consistency is quite  another4   
,it does not necessarily follow that    
because a given scheme  of thought is   
consistent in all its parts that it is  
also consistent  with universal truth or
 with life4 ,this very vital fact was   
overlooked  by ,,gauss and all those who
 followed in his wake when he discovered
  that his ,astral ,geometry was        

consistent throughout in all its parts4 
,there is only one norm of truth and    
that is kosmic consistency4 ,it  matters
 little that a thing shall be           
self-consistent2 it matters much        
whether it is consistent with the       
universal standard4 ,it has been shown  
to be logically possible to elaborate at
 least two different systems  of        
geometry1 namely1 the geometry of the   
acute angle and that of the  obtuse1    
which1 while each of them is            
self-consistent throughout1 are         
nevertheless inconsistent with each     
other and with the geometry of the      
right angle 7,euclidean74 ,this1 it     
would seem1 appears to be sufficient    
for the invalidation of either one or   
both of the non-,euclidean systems  of  
geometric thought4 ,indeed1 if it can be
 shown that the ,euclidean  geometry is 
more representative of the true approach
 to the norm of  space-genesis and of   
creation so far as its mode of          
manifestation is  concerned1 and        

consequently true of the norm set up by 
consciousness1  the rejection of both   
systems of non-,euclidean geometry seems
 to be  thoroughly warranted4 ,but this 
is obvious and requires no demonstration
  nor comment to make it clear4 ,we have
 only to ask ourselves whether  it has  
ever occurred to us that consciousness  
has either a tendency  to or            
adaptability for action in a curvilinear
 manner2 or1 if when  we contemplate    
ideas or idea-relations we have the     
impression of  perceiving a curvilinear 
or manifold tendence in them either of a
  positive or negative nature1 and also 
whether it has been observed that  our  
thought processes naturally assume      
four-dimensional attitudes4  ,if we find
 that such a query must be answered     
negatively1 and indeed  we must so find1
 then1 we have no basis for the         
assumption that any  one of the systems 
of non-,euclidean geometry is valid     
either for the  present status of       
consciousness or for a future existence1

 since it is  true that the future is   
but an elongation of the present4       
,evolution is  to bring no radical      
changes in the norms of reality2 it has 
merely to  deepen and widen and make    
more intense1 efficient and             
comprehensive  the present scope of our 
consciousness and thereby1 while the    
,thinker  is passing from one degree of 
realism to another1 to bring him into a 
clearer conception of what his own      
limited scope of motility means to  the 
whole4    ,the four-space is a mathetic 
divertisement4 ,that is1 it cannot be   
said to lie in the direction of a       
straight line which proceeds either  in 
a forward or lateral direction4 ,neither
 does it lie in a plane  which is       
vertical or horizontal to the sensorium4
 ,it is1 therefore1  a fractural        
departure from any conceivable          
tridimensional direction1  a geometric  
anomaly4 ,evolution1 despite the minor  
aspects of its  movement1 undoubtedly   
proceeds in a straight line and not by a

 zigzag  nor discontinuous line and     
hence it is irrational to assume that it
  will1 when it passes on to the next   
advanced stage1 emerge into the  realm  
of the four-space4 ,for the so-called   
hyperspace of geometry  cannot by any   
standards of reference be said to lie in
 the plane of  any straight line which  
can be described in three-space4 ,if    
life is  to evolve more efficient forms 
and if the forms are to evolve into     
more perfect organizations and mind and 
consciousness to become more  intense   
and comprehensive expressions of the    
divine mind of the kosmos  it is        
certainly not in the domain of          
hyperspace that these shall find  the   
substructure of their higher            
development2 but1 if at all1 it shall   
be found1 as in all times past1 in the  
realm of perceptual space where  bodies 
are said to have three and only three   
possibilities of motion4    ,what then  
is the significance of the more than a  
thousand years of  mathematical labors2 

of all that has been said and done in an
 endeavor  to bring into the popular    
consciousness a conception of           
hyperspace8 ,is  it a question of       
,love's ,labour's ,lost8 ,or is it a    
mere prostitution  of mathematical      
talent8 ,to answer these queries is the 
burden of this  treatise and it is hoped
 that as the text continues the reader  
may be  able to arrive at his own       
conclusions as to the relative value of 
the  work of the mathematicians in this 
respect and be able to judge for        
himself the true significance of it all4
    ,the specific value of consciousness
 as a determinative factor in           
space-measurement has been recognized by
 all who have sought to arrive  at a    
logical justification for the conception
 of four-dimensionality by  analogous   
reasoning4 ,the existence of the unodim 
with consciousness  limited to a line or
 point has been assumed and it has been 
shown  how greatly such a being would be
 handicapped by his limited area of     

consciousness1 it having been proposed  
to confine his consciousness to  one    
dimension4 ,an unodim would1 of course1 
be entirely unaware of any  other       
dimension than that in which he could   
consciously function4 ,so  that with    
respect to his own consciousness no     
other dimension would be  necessary for 
the continuance of his life processes4  
,he might live his  life without any    
knowledge even of any limitations or    
barriers to other  things higher than   
those of his plane4 ,he would be content
 to exist in  the one-space and enjoy   
the benefits which it offered4 ,he could
 have no  notion of the two-space1 but  
it has been allowed that a super-unodim1
  an unodim metageometrician1 if you    
please1 could reason out a mental       
conception of what the two-space might  
be4 ,passing on to a space of  two      
dimensions1 the domain of the duodim1 a 
greater freedom of  movement is allowed 
and instead of being able to function in
 only one  dimension the inhabitants of 

this plane would find themselves able to
  move about in at least two directions4
 ,consciousness would accordingly  enjoy
 a more comprehensive scope4 ,but in a  
manner similar to that used  by the     
unodim metageometrician it is held that 
the duodim could  get a conception of   
the three-space by analogous reasoning  
and so  gradually become conscious of a 
higher degree of spatiality than his    
own4 ,in the conscious reasoning of     
both1 however1 is the condition of      
perpendicularity4 ,that is1 it must be  
assumed by both the unodim and  the     
duodim that the new dimension must lie  
in a plane perpendicular  to their      
space4 ,so1 the unodim would postulate  
that the two-space  must lie in a       
direction at right angles to his space1 
and yet he would  not be able to        
indicate the direction owing to his     
ignorance of any  experience that would 
acquaint him with the new space as well 
as the  want of possibility of motion   
therein4 ,similarly1 the duodim would   

arrive at a conception of three-space4  
,thus1 it has been argued that  tridims1
 or people living in our tridimensional 
world1 could1 by  using a like line of  
argument or reasoning1 arrive at a      
conception or  understanding of the     
four-space1 which1 of course1 must also 
lie in a  direction at right angles to  
three-space4    ,the implications of    
this mode of thought show how thoroughly
 the  mathematician recognizes the      
limitations which consciousness imposes 
upon our knowledge of the world and the 
subtler conditions about us4  ,but1     
moreover1 it is even obvious to all who 
stop to think about  it2 for it can     
readily be seen how little those things 
which do not  enter our scope of        
awareness affect us either physically1  
mentally  or spiritually4 ,but no one   
can be so bold as to deny utterly that  
anything exists but what is found in our
 consciousnesses4 ,it is even  true that
 in the great centers of population     
where people are compelled  to live many

 families in the same house1 it is the  
usual thing for  these individual       
families to live in complete            
forgetfulness of all  the others in the 
house and live their lives so completely
 that it  would be exceedingly difficult
 to measure the effect the one has upon 
the other4 ,the mathematician1 as is    
shown by the hyperspace movement1       
recognizes that there are planes of     
superconsciousness the nature and       
character of which persons confined to  
limited areas of consciousness  can have
 no knowledge and may only arrive at    
that knowledge by serious  thought and  
contemplation4 ,in other words1 they    
tacitly admit the  existence of higher  
planes of consciousness as well as the  
necessity  of elevating the personal    
consciousness in order to comprehend    
them4  ,although it was not expressly   
allowable in the analogy of the unodim1 
it is nevertheless one of the strongest 
implications of the process  of         
reasoning that the unodim could have    

easily raised the plane of  his         
consciousness by continuing his         
researches until he1 too1 became        
conscious of the three-space1           
mathematically1 as well as the          
two-space4  ,for it was not necessary   
for him to raise the plane of           
consciousness in  order to contact the  
two-space4 ,he had need only to widen   
it4 ,but in  order to comprehend the    
mathematical three-space it would be    
necessary  for him to elevate his       
consciousness4    ,the fundamental error
 in the foregoing line of thought rests 
in the  fact that awareness in the human
 family has not developed in the manner 
outlined4 ,the human species has not    
come into conscious relations  with the 
three-space by outgrowing the one-space 
and the two-space in  succession4 ,the  
fact of the matter is that when         
consciousness first  dawned it must have
 encompassed all three dimensions       
simultaneously  and equally and there is
 nothing to indicate that its rise was  

otherwise4 ,then1 specifically there is 
no evidence that the evolution  of      
consciousness has proceeded in a        
rectangular manner4 ,indeed1 there  is  
undoubtedly no warrant for the          
assumption that it has progressed  in   
ways that are mathematically            
determinable at all4 ,the question  very
 naturally rises in view of the above as
 to the relative value of  mathematical 
knowledge in the scheme of              
psychogenesis4 ,can mathematical        
knowledge or laws be said actually and  
finally to settle once for all  time any
 question in which consciousness or life
 enters as a factor8  ,upon the response
 to this question hinges unanswerably   
the decision as  to the category which  
mathematical knowledge should by right  
occupy in  the entire schematism of     
life4 ,if it can be successfully        
maintained  that final judicative power 
abides in mathematics in the            
determination  of these questions1 then 
it would be useless to struggle against 

the  fiat of mathematics and            
mathematicians2 verily1 we should be    
compelled  to accept nolens volens all  
that mathematicians have devised about  
hyperspace and its connotations4 ,if1 on
 the other hand1 it can be shown  that  
no such judicative power inheres in     
mathematical knowledge we  shall then be
 able to establish for mathematics a    
true category and to  dispose of the    
hyperspace movement in a manner that    
shall at once be  logical and necessary4
    ,that the discovery of hyperspace by
 the mathematician is merely an  aspect 
of a general forward movement in the    
evolution of consciousness  can be shown
 by a brief correlative study4          
,hyperspace is the artificial  symbol of
 a higher and more extensive realm of   
awareness4 ,for it cannot  be doubted   
that to be able to think in the terms of
 hyperspace1 to  study the various      
relations and interrelations upspringing
 from the  original premises1 actually  
to become conscious in the hyperspatial 

realm thus constructed1 requires a      
different species or quality of         
consciousness than that required for    
ordinary thinking4 ,the period  covering
 the rise of artificial spatiality is   
contemporaneous with the  rise of the   
phenomena identified with the spiritual 
life of ,,swedenborg2  for during the   
same time he began a series of visions  
which revealed to  him great knowledge  
of the unseen and supersensuous         
realities of life  and existence4 ,his  
consciousness was being flooded with the
 light from  so-called celestial spheres
 and he was gradually becoming conscious
  of a new dimension1 a new space1 a    
higher world that is altogether  unlike 
the world of the senses4 ,during this   
period1 too1 ,,dante1 the  great kosmic 
seer began to jot down the results of   
his hyperspace  experiences1 after which
 he wrote his ,divina ,commedia in which
 he  describes more or less minutely    
some of the characteristics of the      
hyperspace domain which was revealed to 

his consciousness as he saw  and        
interpreted it4 ,both ,,swedenborg and  
,,dante being deeply religious  and     
pious-minded had their reports of the   
new world colored by their  own mental  
experiences and proclivities4 ,,plato   
had at an earlier day  set down what he 
conceived to be the ethical and civic   
characteristics  of the new age1 the    
,utopia of mankind living on a higher   
plane4  ,it was during these days that  
the doctrine of evolution was born1     
although it remained for ,,darwin to    
formulate and buttress it with a        
stupendous congeries of facts4 ,,martin 
,,luther1 the great religious  reformer1
 likewise contacted the radiating       
light-glow of a higher  consciousness   
into which the race was coming but of   
which only the  foremost were able to   
get advance glimpses4 ,,kant1 one of the
 peerless  leaders of the vanguard of   
humanity1 at this time also1 conceived  
and  wrote his ,critique of ,pure       
,reason which is likewise an evidence of

  the upliftment of his consciousness on
 the side of pure intellectuality  and  
the commencement of a general period of 
illumination4 ,and then1  later1 but    
embraced within the same period1 artists
 began to get  glimpses of this higher  
consciousness which showed itself in a  
new and  strange departure in art4 ,in  
rapid succession new schools sprang up  
and  came to be known as the cubist1    
post-impressionist1 futurist1           
orphimist1 the synchromist and the      
vorticist4 ,art really began  the search
 for the plastic essence of the world   
trying to portray  its conception of the
 real image of the spirit of the world4 
,color  acquired a new kind of splendor 
and painting gave birth to a new        
intrinsic beauty of material and sheer  
loveliness of texture4 ,all of  which   
were evidences of an intellectual       
up-reaching in response to the  sharp   
appulsions from above4 ,,darwin's mind1 
being of scientific bent1  saw and      
interpreted everything in the terms of  

materialistic science2  but there is no 
doubt but that the expansion of the area
 of awareness  which his mind           
experienced in his great conception of  
evolution as a  continuous process and  
all that it implies thereby was a result
 of the  universal appulsion of the     
human intellect against the new domain  
of  consciousness4 ,and ,,kant's        
conception of space in general perhaps  
may be  said to have been the           
seed-thought for the metageometrician4  
,but thus it will be noted that in all  
the cases mentioned in the  foregoing   
there is always present the personal    
element of the  investigator1 and that  
the reports of each of these have been  
colored  and characterized by their     
individual consciousness and            
experiences4  ,that all reports would   
agree with respect to details connected 
with  the new domain of consciousness   
could scarcely be expected owing to  the
 wonder and bewilderment which seize the
 intellect under such  circumstances4   

,no implication that the mathematicians 
have been unduly  excited by what they  
have discovered after years of patient  
research  in this direction is indicated
 by the foregoing observations2 but it  
cannot be denied that the enthusiasm of 
the moment and the consequent           
minimization of all other phenomena but 
the special line being  investigated    
serve very effectively to obscure the   
mental vision of  the more partisan4 ,it
 perhaps is sufficient that the         
investigator  should set down in as     
orderly manner as possible the things   
which he  conceives1 and that he should 
interpret them according to the         
standards  of his own intellect4 ,more  
than this cannot be expected4 ,moreover1
 it  usually suffices that the future   
investigator1 far removed from the      
beclouding influences of partisanship1  
who successfully raises his             
consciousness to that higher plane shall
 be able to synthesize the  findings of 
all and thereby with the aid which comes

 to him from a  more advantageous       
position arrive at sounder views and a  
more reliable  judgment4    ,it will    
thus be seen that the metageometrician's
 method of  interpretation is no more   
entitled to final credence and general  
acceptance than that of the             
spiritualist1 ,,swedenborg1 or the      
occult  seer1 ,,dante4 ,for in their    
best moods and at their highest points  
of  mental efficiency these have only   
succeeded in vaguely symbolizing what   
they have conceived of the realities of 
the supersensuous realm in  terms of    
their own experiences4 ,is there any    
more cogent reason1 then1  for accepting
 the analyst's conception of a world of 
hyperspace peopled  with ensembles1     
propositions1 spaces of n-dimensionality
 and other  mathetic contrivances than  
the ,inferno of ,,dante1 inhabited by   
hideous  shapes and repellent denizens1 
the remains of ill-spent earth-lives  or
 ,,swedenborg',s ,celestial ,realm1     
wherein dwell numerous beings of        

celestial character performing various  
tasks in the work of the world8  ,these 
observations should not lead the reader 
to come to the conclusion  that the     
visions of ,,dante and ,,swedenborg are 
deemed to be more worthy  of credence   
than mathematical knowledge when that   
knowledge is limited  to the sphere     
where it rightfully belongs2 but the    
proper view is that  which would make it
 appear that it is the way these widely 
differing  workers interpret what they  
have seen2 that it is the adaptation of 
the  unseen realms to the peculiarities 
of the mentalities which observe  them4 
,the mathematicians have simply         
portrayed as well as they could  their  
conception of the new stage of          
consciousness and its contents1  and    
following the modus vivendi of all      
intellects have interpreted  these      
things in the terms of mathematics1     
merely because mathematics  constituted 
the best available symbology at hand for
 the purpose4  ,similarly1 the painter  

sees a new world of color2 the          
politician1 a new  era of political     
freedom2 the religious enthusiast1 a new
 religious  conception2 the scientist1 a
 new condition of matter and energy1 and
  so on1 to the most ordinary mind1 each
 sees something new while at the  same  
time is necessarily limited to the      
confines of his own mentality  when he  
comes to interpret what he sees and     
conceives4 ,hence1 there  would appear  
to be only one way to regard all these  
advances and  that is by synthesizing   
them1 by correlating1 and by tracing    
them to  a common source1 and finally by
 seeing them as one general forward     
movement of intellectual evolution4     
,man1 the ,thinker1 who in essence is a 
pure intelligence1 has two  mental      
mechanisms or organs of consciousness4  
,one of these is the                    
brain-consciousness or the egoic4 ,it is
 so called because the brain is  its    
organ of expression and impression4 ,it 
manifests through the brain  and uses it

 to further the various objective       
cognitive processes4 ,the               
brain-consciousness is a child of the   
physical body and its life is           
intimately identified with the life of  
the body4 ,this consciousness may  be   
called the a posterioristic mechanism or
 organ of the ,thinker and  is therefore
 his means of interpreting the phenomena
 of the objective  world4               
,cell-consciousness is a phase of the   
ergonic functions of the  a             
posterioristic mechanism4 ,the other    
organ of consciousness is an  aspect of 
the intelligence of the ,thinker himself
 and perhaps may be  said to be the     
active1 organized portion of that       
intelligence4 ,it is  separate and      
distinct from the a posterioristic      
consciousness yet  sustaining a         
substructural relationship with it1     
being the source of  the egoic or       
brain-consciousness4 ,it may be called  
the a priori  consciousness4 ,its roots 
are buried deep within the heart of the 

space-mind and it is therefrom nourished
 and developed by what it  receives in  
the way of intelligence4 ,it is the     
intuitional faculty2  knows without     
being taught2 conceives without reason2 
interprets  according to the norms of   
the space-mind or the divine mind of the
  kosmos4 ,it always resides on a higher
 plane than that of the brain-mind  or  
consciousness1 only at rare intervals   
being able to contact it with  flashes  
of its own intelligence as intuitions4  
,the a priori consciousness being the   
intuitive faculty of the ,thinker  is1  
therefore1 a phase of his mental life on
 a higher plane than the  sensuous4 ,all
 its conceptions constitute the a priori
 knowledge  of the brain mind so-called4
 ,the a priori faculty of man's higher  
consciousness gives the character       
possessed by that form of knowledge     
known to philosophy as the a priori4 ,so
 that the a priori has a  more          
substantial basis than has hitherto been
 surmised4 ,it is not  only that which  

may be said to transcend experience but 
that which is  the organ of contact with
 the supersensuous realities as well as 
of  expression through the              
brain-consciousness4    ,the mind's     
method of apprehending objective        
phenomena is not a  direct process but  
an indirect process by virtue of which  
neurograms  or nerve-impacts registered 
in the brain are interpreted4 ,external 
sense-impressions are1 of course1       
conveyed to the cortical area by  means 
of appropriate vibrations which traverse
 the lines of the neural  mechanism4    
,these are recorded in the brain areas  
just as a telegraphic  communication is 
registered in the apparatus of the      
receiving end1  and in being so1 they   
make terminal registrations which man1  
the  ,thinker1 interprets after a       
psychic code which has been built up    
empirically4 ,that is1 he comes to know 
that certain rates of vibration  and    
certain peculiarities therein mean      
certain things when referred  to the    

sensorium4 ,he then interprets according
 to this experience the  symbolism of   
all neurographical impressions4 ,but it 
is obvious that  under such             
circumstances1 where the interpreter is 
far removed from the  thing itself and  
finds it necessary to interpret rates of
 vibration or  symbols in order to      
arrive at a knowledge of the            
intelligence which is  conveyed to him1 
the chances of inadequate conception are
 very great  indeed4 ,in fact1 it is not
 possible through the use of            
neurographical  symbols alone to get any
 complete notion of the phenomena       
considered4  ,and thus there stands     
between the ,thinker and absolute       
knowledge a  barrier which prevents his 
arriving at a state of certitude in his 
knowledge of the world of sensible      
objects4 ,it is1 however1 a barrier     
which will always remain1 checking ever 
his approach to finality in his         
understanding of the universum of       
appearances4    ,a markedly different   

condition obtains in the realm of the a 
priori  or intuitional for the reason   
that the barriers which inhere in  the  
neurographical or a posterioristic      
method are absent and the  ,thinker has 
a more direct approach to the objects of
 cognition4  ,hence the chance for error
 is very small indeed4 ,this will       
account1  therefore1 for the superiority
 of the intuitional over the rational   
or the perceptual4 ,indeed1 it is       
doubtful whether the purely rational    
possesses any value whatsoever when its 
modus vivendi is unsanctioned  by the   
intuitional4    ,else why can we not be 
certain that the results of our rational
  processes are correct at all times8   
,is it not because we lack the power  to
 perceive whether our premises are      
correct in the first place8 ,quite      
truly4 ,for if the ,thinker can intuit  
the necessity and certitude of  any     
given premise it follows that the       
consequences of that premise  are true4 
,it would1 therefore1 appear that the   

more the intuitional  faculty is        
developed the clearer will be our       
perceptions not alone  of abstract      
values but of objective things          
themselves4 ,further1 it  is doubtlessly
 true that the more the space-mind is   
developed in the  human race the deeper 
will become our perceptions of the      
essential  be-ness of things so that    
whatever may be the presentations of the
  space-mind to the brain-mind they will
 be by far more accurate than the       
impressions we receive through the      
latter as a medium of apprehension4  ,it
 is but natural1 however1 that in the   
present more or less chaotic  condition 
in which the faculty of the intuitional 
is found it should  be difficult even to
 interpret its presentations accurately4
 ,it is  perhaps due to the fact that we
 are unused to its deliveries and mode  
of registration as well as to the fact  
that it has been overshadowed  by the   
intellectual or rational faculty4 ,but  
the mere fact that it is  present and   

functioning1 even if but rudimentarily1 
is evidence of its  potentiality and the
 possibility of its future development  
to a still  higher degree of efficiency4
    ,there is no doubt but that the     
original impulse which resulted in the  
creation of the faculty of              
perceptibility in the ,thinker also     
marked  out the metes and bounds of our 
entire range of perceivability which    
includes not only the intuitional but   
something higher still4 ,there is  no   
doubting either the obvious fact that   
these metes and bounds cannot  have been
 other than rudimentary or general lines
 of denotation1 and  that the work of   
their further elaboration and refinement
 is a matter  of evolutionary detail4   
,for if we assume that the general      
principles  of evolution are true we    
immediately recognize the cogency of    
this  view4 ,that which we now call the 
hand has not always been the perfect    
instrument that it is nor has the ear   
always been so keenly adjusted  as at   

present4 ,it has required undoubtedly   
many million of years for  the eye to   
reach its present degree of complexity  
and adaptability4  ,yet in all these    
cases the different organs existed in   
potentiality  from the beginning2 the   
metes and bounds of the hand1 the ear   
and the  eye were laid out primordially4
 ,evolution has specialized and adjusted
  them to environments and needs4 ,thus 
it will be seen that while the          
intuitional faculty was designed for    
manifestation from the beginning  it has
 nevertheless required ages for its     
appearance even in the most  rudimentary
 fashion4    ,almost the entire content 
of human knowledge is based upon        
assumptions  or hypotheses2 in fact1 is 
but a mass of these1 and especially is  
this true of mathematics1 science and   
philosophy4 ,of course1 there  are      
certain minor observable facts which by 
reason of the seeming  permanence of    
their existence have been eliminated    
from the category of  assumptions4 ,but 

even this elimination when it is traced 
to its depths  may be found to be       
erroneous1 and perhaps after all1 when  
we have  really begun to know something 
of the reality of things1 may have again
  to be placed in this category4 ,and   
then1 too1 the hypothetical nature  of  
our knowledge is due largely to the     
,thinker's method of contacting  the    
objective world which is the subject of 
his knowledge4 ,it is  because it is    
necessary for him to interpret the      
neurographical symbols  which           
sense-impressions make in the brain     
matter according to a psychic  code that
 renders his knowledge of things in     
general hypothetical4 ,his              
interpretations are based upon an       
assumed value which experience has      
taught him to give to each neurogram4   
,but even when his interpretations      
leave nothing to be desired in respect  
to their accuracy of  apprehension of   
what the neurographical vibration       
implies there is that  further barrier  

to his cognition of reality which is due
 to his remote  removal from the object 
itself and the consequent extreme       
difficulty1  if not present             
impossibility1 of identifying his       
consciousness with the  essence of the  
objects which he contemplates4    ,when 
the ,thinker's consciousness is         
presented with a neurograph  of say1 a  
cube1 it is not the cube itself which he
 contemplates  or observes2 it is the   
neurograph or psychic symbol which the  
sense-impressions make in the brain4    
,his consciousness deals not with  the  
object but with the symbols4 ,it is true
 that when he verifies one  neurograph  
by another1 as the scopographic or sight
 impressions by  the tactographic or    
touch impressions it is found that the  
delivery  thus determined is a true     
enough representation4 ,it is also true 
that  the ,thinker1 as a rule1 does not 
accept a neurograph as valid until it   
has been verified by at least one or    
more presentations through his  outer   

sense organs4 ,it occurs1 therefore1    
that all such deliveries  are verified  
and corrected by one or more sense      
witnesses before  final acceptance by   
the ,thinker2 but even then it cannot be
 said that  his notions thus gained are 
in all respects correct and true to the 
standards set up by the                 
brain-consciousness not to mention      
higher forms  of consciousness4 ,and    
then1 when we consider that in addition 
to the  numerous chances for error which
 naturally inhere in this method of     
cognition it must also be apparent that 
the ,thinker's approach to the  reality 
of things is much impeded by his        
separation therefrom1 the  unreliability
 of our ordinary methods of cognition is
 much emphasized4    ,but aside from the
 egoic or brain-consciousness there is  
the higher  consciousness of the        
,thinker himself4 ,for the              
brain-consciousness is  merely his      
method of regarding and comprehending   
the neurographical  deliveries1 the     

psychic code by which he systematizes   
and organizes  his cognitions or        
impressions of the sensible world4 ,this
 higher  consciousness constitutes the  
faculty a priori for the ,thinker on  a 
higher plane of existence1 and because  
it deals with elements  altogether      
unlike those which make up the data of  
brain-consciousness  is1 accordingly1   
less liable to error in its judgments of
 the  supersensuous presentations than  
is the objective or brain-mind4 ,in     
fact1 it is difficult to conceive of a  
state or conditions wherein1  supported 
as this view contemplates1 the intuition
 should err in  judgment4 ,viewed from  
the standpoint of external impedimenta1 
this  condition may be said to be due to
 the absence of sensuous barriers  which
 would otherwise prevent the near       
approach of the ,thinker's              
consciousness to the essence of things  
which is the object of  his             
consciousness on this higher plane4     
,directly1 however1 it is  undoubtedly  

due to the fact that1 following the lead
 of life itself1  yea1 as the veritable 
handmaid of life1 it cannot err where   
life  is concerned4 ,when dealing with  
notions a priori or intuitograms  the   
,thinker is relieved of the onerous     
necessities and limitations  incident to
 the examination and determination of   
neurographic symbols  registered in the 
brain cortex and so is free to study1 to
 examine  and judge at first hand the   
impressions which are received from his 
own plane of intuition4 ,the difference 
is about the same as that  which should 
exist between the methods of            
communication between two  telegraphic  
operators when in one instance they     
would have to depend  upon the          
deliveries conveyed over the wires1     
while in the other1 when  they stood    
face to face with each other1 they could
 communicate by  direct conversation4   
,in the one case the method of          
communication is  direct and simple1    
while in the other it is indirect1      

circuitous and  complex4 ,it can1       
therefore1 be readily seen that in all  
cases where  the approach is made in a  
direct1 simple manner the probability of
  error is much less than in cases where
 the intellectual approach is  less     
direct and more complicated4 ,hence in  
drawing conclusions as to  the relative 
importance of the two mechanisms of     
consciousness1 the a  posterioristic and
 the a priori1 it is necessary to bear  
in mind  the comparative superiority of 
the one over the other as a means of    
cognition4 ,it matters little that the  
intuitional faculty is not so  well     
developed as the tuitional because it is
 but natural that inasmuch  as the      
,thinker's needs are subserved in the   
sensuous realm by the  tuitional        
consciousness it should1 from more      
active use1 gain somewhat  over the     
intuitional in facility of expression   
and general utility4 ,and  the more so1 
because the two faculties serve         
different purposes2 one is  attuned to  

receive impressions from a subtler plane
 while the other is  fitted for contact 
with the phenomenal universe2 one is    
related to the  conceptual while the    
other is related to the perceptual4     
,they differ  not only in function1 but 
in nature as well4 ,there is1 of course1
 a  natural barrier consisting of the   
inherent limitations of each faculty    
which prevents the full mergence or     
unification of the two states of        
consciousness so that there exists a    
state of consciousness the data of      
which the brain-mind is unaware1 it     
being able only at rare intervals  even 
to receive so much as slight impressions
 from it in the nature  of intuitional  
flashes or inspirations and the like4   
,viewed in this  light it would appear  
that the cognitions which are most      
truthworthy  are those which are        
presented by the intuitional faculty    
because  they are nearer to the         
essential reality of things2 they have  
to do  more specifically with the nature

 of that which appears while the        
tuitional mind can only regard that     
which is the appearance4 ,herein  lies  
the whole difference4    ,the natural   
outcome of this division of labor       
between the tuitional and  the          
intuitional is the establishment of the 
fact of man's relationship  both to the 
phenomenal and the real2 that in his    
psychic nature must  reside the faculty 
of apprehending the real and that he    
shall one day  awaken into activity this
 now latent faculty whereby he may make 
a  direct and naked contact with        
reality4    ,if it be true that1 as     
,,plato said1 ,god does geometrize1 and 
that  the divine geometry1 as will      
appear1 is based upon a system1 an      
alphabet which taken together are the   
point 4 1 the line ----1 the  triangle  
,illustration1 the square ,illustration 
and the circle  ,illustration1 then1 we 
have in this geometric alphabet the very
  secret of the divine geometry4 ,with  
these1 and in the kosmic laboratory  of 

chaogeny1 the ,creative ,logos has      
measured off the limits and  confines of
 space2 with them ,he has traced out its
 dimensions the  archeological evidences
 of which we may view in the space-mind 
itself2  and with them he has           
established the manner of its appearance
 to the  ,thinker4 ,in dimensions1      
three1 and yet not three1 but one1      
,space1  the eternal progenitor of all  
forms and energies1 having received  the
 divine fiat in the beginning that thus 
far it should extend  and no further1   
persists in faithful obedience to the   
law of its  being--tridimensionality4   
,it must be so because it is thus       
sanctioned  by the highest faculty in   
man that can render judgment thereon4   
,if  tridimensionality inhere in the    
space-mind1 as the law of its being and 
in the intuitional consciousness as the 
norm of its essential nature  and as the
 easiest and simplest expression of the 
tuitional mind1 how  can it be gainsaid 
that these considerations obviate the   

necessity of  the mathetical hyperspace8
    ,if the reality of things is hidden 
from us and if we are1 therefore1       
unable to perceive their real essences  
it is because our mode of  thought and  
our consciousness have obscured our     
vision and limited us  to this state of 
paucity of perception4 ,it is not       
because reality is  itself a hidden1    
inscrutable quantity nor that its modus 
vivendi  is unknowable2 but because we  
being multiformly limited1 cabined1     
cribbed and confined are resultantly    
lacking in the power to discern  that   
which otherwise would be most obvious to
 us4 ,it may well be set  down as       
axiomatic that when1 in the process of  
our thinking1 we arrive  at the         
inscrutable1 the unknowable and the     
infinite1 it is evident  that our       
thought processes are dealing with a    
form of realism which is  higher and    
beyond the possibilities of our loftiest
 thought-reaches4  ,and in order to     
symbolize to itself this condition the  

intellect  poses such terms as          
inscrutable1 unknowable and the infinite
  simply because that is the best it can
 do4 ,hence when it is said that  space 
is infinite it is apparent that the mind
 recognizes that when  it contemplates  
space it is dealing with something whose
 degree of  realism transcends its      
powers of comprehension4 ,infinity is a 
relative  term1 and in fact1 decreases  
in extensity in the proportion that the 
consciousness expands and comprehends4  
,it is not unlikely that should  the    
intellect one day discover that it had  
awakened into union with  the space-mind
 it would immediately reject its        
preconceived notion of  the infinity of 
space4 ,but we need not wait until the  
coming of this  far off event in the    
path of psychogenesis2 for we can here  
and now  perceive with what must be a   
higher faculty than the intellect the   
verity of this conclusion4    ,but      
certain it is that the intellect1 in the
 pride and arrogance of  its traditional

 heritage1 will not without a great     
struggle yield the  ground and prestige 
it has held for an aeon of time2 and in 
vain does  the intuition serve notice of
 dispossessal in these premises2 but    
however stubbornly fought the battle1   
however tenaciously held the  position  
time will discover the weakening of the 
intellect's hand4  ,death for the       
intellect may ensue as a result of the  
conflict but it  will be a death        
wherefrom it will arise1 quickened1     
revivified and  uplifted by its         
disposer1 the intuition1 upon the       
remains of its dead  self to a higher   
and grander state than it has ever      
enjoyed before4    ,space is not static4
 ,it is dynamic1 potential and kinetic4 
,it is a  process1 a becoming4 ,its     
duration as a process is never ending4  
,its  extensity is limited and finite4  
,the so-called infinity of space is one 
of the capital illusions of the         
intellect which can only be removed  by 
an expansion of the consciousness1 by a 

mergence of the individual              
consciousness with the                  
space-consciousness4 ,in the            
ever-widening circle  of the individual 
consciousness lesser realities give way 
to greater as  the darkness recedes from
 the light--the lesser appearing in     
comparison  with the greater1 as the    
consciousness broadens1 as matter to    
spirit1  as night to day or as          
limitation to non-limitation4 ,thus the 
most solid  facts and conditions of our 
limited life are but the shadows of the 
deeper realities which shall be revealed
 to the ,thinker in the days of  his    
larger and more glorious life of freedom
 from limitations4    ,and now it will  
appear that the whole fabric of our     
knowledge shall  have to be reduced to  
the bare warp and woof2 for nothing is  
real but  these4 ,it is as if the       
,thinker1 using the tuitional mind1 had 
been in  all times past studying the    
design woven in the surface of a very   
thick  plush carpet4 ,there are the warp

 and woof1 the long vertical threads    
which make the plush and the intricate  
design appearing on the surface4  ,our  
knowledge may be likened to the design4 
,it represents the contents  of our     
knowledge4 ,we have not even so much as 
begun the study of the  nature of the   
vertical threads as they appear beneath 
the surface to  say nothing of beginning
 the study of the warp and woof4 ,the   
warp and  the woof are the realism of   
the kosmos2 the vertical threads are the
  roots and stem of the phenomenal      
world2 the design is our sensible  world
 as it appears to the intellect4 ,the   
life of the intellect has  been spent in
 contemplating this design2 while of the
 hands which wove  the carpet1 of the   
mind which directed the hands and of the
 spirit  which vitalized all1 it knows  
nothing nor indeed can it know anything4
  ,where shall we say are those hands1  
that mind and that spirit which  made   
the carpet possible and an actuality8   
,in vain do we search among  the        

remains1 among the soft1 glistening     
threads of the carpet or among  the     
intricacies of the design4 ,for they are
 not there4 ,they have passed  on4 ,the 
intellect looks at the design or at the 
vertical threads and  because it is     
unable to follow them to their source1  
it decides that  they are infinite1     
inscrutable and unknowable4 ,but not so4
 ,all that is  required are eyes to see 
and a mind 7or shall we say a mind      
vitalized  by the intuition7 trained to 
discern the threads as they point upward
  with their termini firmly rooted in   
the warp and woof of the fabric4  ,but  
we must first master the design1 and    
then turning to the threads1  master    
them4 ,then shall the doors of kosmic   
reality swing wide and the  ,thinker    
shall be ushered into the eternal palace
 of kosmic realism  wherein he shall    
find the great secret1 the heart1 the   
purpose1 the  beginning and the end1 the
 very nature of things-in-themselves4   
,the nature of every degree or condition

 of realism is so constituted  that its 
qualities1 characteristics and          
limitations are exactly  adequate for   
the satisfaction and fulfillment of all 
the requirements  and needs of every    
possible state of normal consciousness4 
,so that each  degree of reality and    
each state of normal consciousness is   
sufficient  and complete in itself and  
mutually satisfies the necessities of   
each  other4 ,the substratum of reality 
or life which extends from the heart  of
 the kosmos to the extreme limits of the
 phenomenal universe exists  in degrees1
 not discrete1 but continuous4 ,and     
these merge into one  another by        
insensible stages4 ,such is the         
imperceptible continuity of  the whole  
as each degree is gradually immerged    
into the other that  only the           
limitations of consciousness itself make
 it to appear as if  it were            
discontinuous4 ,for every stage of      
realism there is a state  of            
consciousness which answers to it       

completely and sufficiently4  ,so both  
the state of consciousness and that of  
reality1 manifesting  at any given      
stage1 seem to be complete and final for
 that stage4  ,realism or life and      
consciousness possess only a relative   
finality  fashioned upon the necessities
 and requirements for any given state   
of being4 ,consciousness alone fixes the
 apparent limits of life2 it  also      
determines the state of our knowledge of
 life4 ,and thus when the  ,thinker is  
confined to any stage of reality and    
congruent degree of  consciousness it   
appears that what he there finds is     
ample for all his  purposes4            
,accordingly he is convinced that that  
stage is the final  consideration of his
 scope of motility4 ,it is only when he 
is able  to raise his consciousness to a
 point where he can contact higher      
realities that he becomes aware that    
there are higher stages in which  his   
consciousness may manifest4 ,this       
peculiarity of the ,thinker's           

consciousness is accentuated when he    
allows himself to become wholly         
engrossed with a study of the phenomena 
of that stage in which he can           
consciously function4 ,hence it         
constantly occurs that men exhausting   
the study of the phenomenal find        
themselves floundering upon the beach   
of the outlying shores of consciousness 
where in sheer desperation  they fall   
into the illusion that they have indeed 
reached the limits  of manifested life  
and that beyond those limits there is no
 organized  being4 ,unconscious are they
 that in ever widening circles the      
fertile  lands extend and await the     
awakening of their consciousness when   
they  may till the fallow ground of this
 new domain and begin again the  search 
for the ultimately real4    ,with       
respect to the present powers of        
consciousness1 it cannot be             
successfully controverted that the      
concept of tridimensionality of  space  
is sufficient for all purposes4 ,it must

 be so for it is not only  an aspect of 
the phenomena of space but of reality as
 well4 ,this fact  is attested by the   
nature of mind that answers to the      
nature of space4  ,tridimensionality    
characterizes the entire extent of      
consciousness and  life1 and therefore1 
of space itself4 ,this characterization 
may be  traced to the very doors of the 
heart of space where the three become   
one4 ,nor would this conception be in   
the least vitiated if it were  allowed  
that the mass of the phenomena of the   
supersensuous world1  lying in close    
proximity to the sensuous world1 does   
present itself to  the consciousness in 
a four-dimensional manner and that the  
phenomena  of a still higher plane      
present themselves in a five or         
n-dimensional  manner to that state of  
consciousness which may be congruent    
with  them2 because then we should be   
making allowances for the changes in    
phenomena and their mode of presentation
 to the consciousness which  by no means

 implies a corresponding change in      
reality or life4 ,all  phenomena are    
fashioned by the intellect4 ,the        
phenomenal world is  just what the      
intellect interprets it to be4 ,it is   
that and nothing  more4 ,its qualities1 
attributes and characteristics are such 
as the  consciousness gives to it4 ,it  
exists only for the purposes of the     
evolving consciousness4 ,and1 as an     
instrument of consciousness1 its        
existence is strictly subject to the    
evolutionary needs thereof4 ,in  that   
moment that the immediate needs of the  
consciousness shall no  longer be able  
to find satisfaction in the phenomena of
 any plane of  nature1 in that moment   
the phenomena of that plane disappear1  
recede  and are swallowed up in the     
maelstrom of eternal reality4    ,in the
 gradual expansion of consciousness as  
it passes through the  infinite series  
of grades of awareness meantime becoming
 deeper1  broader and more comprehensive
 as it proceeds1 there may be observed  

running through all these planes and    
orders that which is neither the        
phenomena of the various planes nor the 
consciousness2 but which must  be the   
substructural basis of both1 remaining  
the same1 unchanged and  unchangeable4  
,that is the thread of reality1 the     
passage of life itself  which is the    
eternal basis of all4 ,now it is to this
 reality1 life1  that the space-mind is 
related and in which its roots1 its     
heart and  the very center of its being 
are at one with the divine mind of the  
kosmos4    ,the question of             
dimensionality is solely a concern of   
the objective  or brain-mind which is   
the intellect4 ,it is one of the ways in
 which  the intellect endeavors to      
understand phenomena4 ,it is an         
arbitrary  contrivance devised by the   
intellect for its convenience in        
studying  the world of things4 ,without 
it1 as obviously appears1 the intellect 
would not be able to go very far in its 
consideration of the minor  problems    

which inhere in material things4 ,the   
fourth dimension is  but another        
attitude1 another contrivance1 which the
 intellect has  devised in order that it
 may study from another angle the       
evanescent  phenomena of the world of   
appearances4 ,having apparently         
exhausted  the possibilities of motion  
in three dimensions1 and being driven on
  to the acquirement of more picturesque
 views by the very necessity  of its    
continued growth1 it has betaken itself 
to another higher  mountain peak1 called
 hyperspace where with larger lenses and
 higher  powered instruments it is      
beginning to scan the landscapes of a   
new  intellectual realm of              
consciousness4 ,yet the celestial       
wonders of this  new-found realm of     
consciousness remain in undisturbed     
forgetfulness  or neglect4 ,but it is   
not by a scrutiny of mathetic landscapes
 nor  by a study of the celestial       
wonders that the ,thinker shall one day 
realize the object of his eagerly pushed

 quest after the real2 for he  shall    
find it1 if at all1 in the temple of the
 kosmic mind which is not  made by the  
intellect nor meted and bounded by      
geometric systems of  space-measurement4
    ,in all the learned pother incident 
to the mastery of the phenomenal1  the  
furniture of the world of the senses1 it
 is as if the self in man1  the         
,thinker1 sat secluded in a six-walled  
tenement1 and hence six times  removed  
from the subject of his study1 and      
endeavored to interpret that  which     
appeared to his vision4 ,and thus1      
thinking that what he sees is  the only 
reality1 he remains in inglorious       
nescience of the reality of  that upon  
which he himself stands1 unconscious    
that the tunnel-shaped  aperture through
 which he peers leads not outward1 but  
backward and  within to the habitation  
of the real of which he himself is a    
part4  ,men are deeply and well-nigh    
hopelessly concerned with appearances1  
with static views of life1 with         

instantaneous exposures4 ,life1 reality 
and all the eternal verities pass on and
 assume countless postures1  attitudes1 
moods1 tenses and nuances4 ,the         
intellect is content to  occupy itself  
with a single tense or mood4 ,indeed1 it
 has no aptitude  or power to consider  
more than one at a single time4 ,it thus
 misses  the continuity1 the            
ceaselessness1 of life4 ,what is more1  
every  singularity1 every attitude1 mood
 or tense which the intellect grasps    
for consideration is immediately remade 
so as to fit its own moods  and tenses4 
,and upon each and every nuance the     
intellect immediately  imposes its own  
form--actually and literally            
rehabilitates them with  its own        
habiliments4 ,unfortunately1 this       
peculiarity occludes the  intellect from
 any approach to the true nature of that
 phase which it  can grasp4             
,hyperspace is one of the illusions of  
the phenomenal2 it is the dress  which  
the intellect has superimposed upon a   

single nuance2 it is a mask  which is an
 exact replica of the mood of the       
intellect4 ,yet through  this mask the  
intellect grandly hopes to approach     
reality4 ,the period  through which the 
mind is now passing is a repetition of  
the evil  days of scholasticism when men
 set out to determine the exact number  
of celestial beings that could be       
perched upon the extremity of a  needle 
point4 ,it is a time when men's minds   
easily assume grotesque  and hideous    
shapes and their thoughts become the    
embodiment of  fantastic entities4 ,the 
exclusive occupation of such minds      
becomes the  fabrication of mathetic    
monstrosities which rapidly deliquesce  
upon  the first approach of the real or 
the appearance of the first ray of      
intuition which may escape through the  
dim and misty condition of the          
intellectual over-hangings4 ,it will not
 be ever thus2 for the ,thinker  will   
one day pass from a study of the        
arrangement of phenomena in space  and  

by well-ordered steps come once again to
 himself4 ,and then through  the maze of
 it all set out upon the true           
path----the tridimensionality  of space 
following which he will inevitably      
approach the citadel of the  real1 the  
kosmic space-mind4            ,,chapter 
,,vii      ,,the ,,genesis ,,and        
,,nature ,,of ,,space      ,symbology of
 ,mathematical                          
,knowledge--,manifestation and          
,non-manifestation ,defined--,the       
,pyknon and ,pyknosis--,the    ,kosmic  
,engenderment of ,space--,on the        
,consubstantiality of    ,spatiality1   
,intellectuality1 ,materiality1         
,vitality and ,kosmic                   
,geometrism--,chaos-,theos-,kosmos--,cha
ogeny and ,chaomorphogeny--,n4          
,,malebranche ,on ,god and the          
,world--,the ,space-,mind--,space and   
,mind    ,are ,one--,the ,kosmic        
,pentoglyph4      ,geometry is concerned
 primarily with a study of the          
measurement of  magnitudes in space4    

,three cordinates are necessary and     
sufficient for  all of its              
determinations4 ,metageometry           
comprehends the study of the            
measurement of magnitudes in conceptual 
space4 ,for its purposes four  or       
n-cordinates are necessary and          
sufficient4 ,perceptual space is  that  
form of extension in which the physical 
universe is recognized  to have been    
created and in which it now exists4     
,conceptual space is  an idealized      
conception belonging to the domain of   
mathesis and has  no actual1 physical   
existence outside of the mind4          
,mathematical space  represents the     
idealism of perceptual space4           
,geometrical magnitudes may be defined  
as symbols of physical objects  and     
geometry as a treatise on the symbology 
of forms in space4 ,in  fact1 all       
cognitive processes are simply efforts  
at interpreting  the symbolism of       
sense-deliveries2 and the difference    
between mere  knowledge and wisdom1     

which is the essence of all knowledge1  
is the  difference between the          
understanding of a symbol and the       
comprehension  of the essential nature  
of the thing symbolized4 ,so long as    
knowledge  of space is limited to the   
understanding of a symbol or symbols by 
which it is presented to the            
consciousness so long will it fall short
  of the comprehension of the essential 
nature of space4 ,in vain have  we      
sought in times past to understand space
 by studying relations1  positions and  
the characteristics of forms in space2  
in vain have  we based our conclusions  
as to its real nature upon the          
fragmentary  evidences which our senses 
present to our consciousnesses4 ,it is  
as if one had busied himself with one of
 the meshes in a great net  and confined
 his entire attention to what he found  
there1 meanwhile  remaining in complete 
ignorance of the nature of the net1 how 
it came  to be there1 of what it is made
 and how great its extent may be4       

,there is ever a marked difference      
between a symbol and the thing which  it
 symbolizes4 ,words are the symbols of  
ideas2 ideas1 as they exist  in the     
mind1 are the symbols of eternal        
verities as they exist in  the          
consciousness of the ,logos of the      
universe4 ,there may be a wide          
diversity of symbolic forms which       
represent one single idea2 as1 for      
instance1 the variety of word forms     
which represent the idea of deity  in   
the various languages4 ,likewise there  
may be a multiplicity of ideas  which   
represent a single verity4 ,but neither 
is the idea nor the word  the real thing
 in itself4 ,that quality of a          
life-aspect which we call  its thingness
 has an essential nature which cannot   
become the object  of consciousness     
except by virtue of its representation  
through ideas  and their symbolisms1 and
 even then1 the thing which we conceive 
is  not the nature of a quality of the  
life-aspect but an idea of it--a  symbol

 which stands for that idea4 ,in order1 
therefore1 for the mind  to arrive at an
 understanding of an eternal verity1    
such as space1  it must first be able to
 synthesize all of the representative   
ideas  and then abstract from their     
compositeness a notion of its essential 
nature4 ,but this can be done only by   
identifying the consciousness  with the 
essential being of the object           
considered4 ,in other words1 the        
consciousness and the intrinsic being of
 forms1 principles1 forces  and         
processes must embrace each other in the
 intimacies of direct  cognition2 the   
life which is consciousness and that    
life which is  essential being1 being   
coeval1 cordinate and mutually          
responsive1 must  in so close a contact 
as here intimated reach an understanding
 of the  realism shared by both4 ,that  
is1 the human consciousness1 following  
in  the wake of life and consisting of a
 specialized aspect of life itself1     
will1 by such an intimate approach to   

the life-principle of forms1  readily   
understand2 for it has only to recognize
 a replica of itself  in rendering its  
judgment4 ,but it is not claimed that   
such a state of  recognition by the     
consciousness of life itself can be     
attained at all  by ordinary means1     
neither is it believed that it is the   
next stage  in conscious evolution4     
,however1 it is not doubted but that    
such an  exaltation of the consciousness
 is possible1 yea practical2 but the    
difficulties which beset the path of    
attainment in this direction are  so    
great that it may as well be considered 
unattainable4 ,the mere fact  of these  
difficulties1 however1 only             
re-emphasizes the insufficiency  of the 
intellectual method4 ,the identification
 of consciousness with  essential being 
is a procedure which cannot be          
accomplished by an act  of will directly
 and immediately4 ,because it is a      
process1 a series of  unfoldments1 an   
adjustment of the focus of consciousness

 to the kosmic  essentialities which    
constitute the substructure of the      
manifested  universe4 ,in the very      
nature of things1 a kosmic essentiality 
cannot  be viewed as being in           
manifestation especially in the same    
degree  as ordinary physical objects are
 manifest4 ,the former is a state1 a    
potentiality1 a dynamic force1 an       
existence which should be thought  of as
 an extra-kosmic affair dwelling on the 
plane of unity or kosmic  origins2 while
 the latter are the exact opposite of   
this4 ,the one can  be seen1 felt and   
sensed while the other is the roots     
which are not  seen but lie buried      
deeply in the heart of the universal    
plasm of being  and beyond the ken of   
sensuous apperception4 ,the term        
manifestation  is both relative and     
flexible in its use4 ,it is relative    
because  it will apply equally to all   
stages of cognition4 ,a thing is in     
manifestation when it is presentable to 
the ordinary means of cognition         

belonging to any stage of conscious     
functioning2 it is not in  manifestation
 when it is beyond the scope of the     
,thinker's schematism  of cognitive     
powers4 ,its flexibility is seen in its 
ready yieldance to  the entire range of 
implications inhering in the process of 
cognition1  fitting the simplest as well
 as the highest and most complex4       
,great is the gulf which is interposed  
between manifestation and               
non-manifestation2 and yet the two1 in  
essence1 are one4 ,they are  linked     
together as the stem of a flower is     
joined to its roots4  ,likewise one is  
visible1 palpable while the other is    
invisible1  impalpable though no less   
real and abiding4 ,as the thin crust of 
earth separates the stem1 leaves and    
flowers from the roots so the           
limitations of man's consciousness      
separate the manifest from the          
unmanifest4 ,similarly1 as when the     
surrounding earth is removed from  the  
roots and they are laid bare revealing  

their continuity and unity  with the    
outputting stem and flowers1 so1 when   
the limitations of  consciousness are   
removed by the subtle process of        
expansion to which  the consciousness is
 amenable so that it can encompass the  
erstwhile  unmanifest1 it1 too1 will    
reveal the eternal unity of the kosmic  
polars--manifestation and               
unmanifestation4    ,there is but one   
barrier to ultimate knowledge and that  
is the human  consciousness however     
paradoxical this may seem4 ,the         
unutterable  darkness which shuts out   
the so-called unknowable from our       
cognition is the limitation of man's    
upreaching consciousness4 ,these        
limitations constitute the difference   
between the human intellect and  the    
mind of the ,logos4 ,nevertheless the   
outlying frontiers of man's             
consciousness gradually are being pushed
 farther and farther without4  ,every   
new idea gained1 each new emotion       
indulged1 each new conception           

conquered1 and every mental foray which 
the ,thinker makes into the  realm of   
the conceptual1 every exploration into  
the abysmal labyrinth  of man's inner   
nature are the self's expeditionary     
forces which are  gradually annihilating
 the frontier barriers of consciousness 
and thus  approaching more closely upon 
the ,ultima ,thule of man's spiritual   
possibilities4    ,space is in          
manifestation4 ,it exists and has being 
whether it is  viewed as an object1 an  
entity or the mere possibility of       
motion4 ,that  it offers an opportunity 
of motion and renders it possible for   
objects  to move freely from point to   
point cannot be denied and yet this fact
  has no bearing whatever upon the      
essential nature of space4 ,the very    
fact of its appearance1 its             
manifestation1 makes it obvious that it 
is  the nether pole of that eternal pair
 of opposites--manifestation and        
non-manifestation1 being and non-being1 
which are essentially one4    ,it will  

be seen from figure #ag that the period 
of involution embraces  seven separate  
stages1 the monopyknon1 duopyknon1      
tripyknon1  etc41 being the unit        
principle or engendering elements of the
  respective stages4 ,involution        
comprises all creative activity from the
  first faint stirrings of the void and 
formless chaos until the universe  has  
actually become manifest and dense      
physical matter has appeared4  ,it is   
divided into two cardinal periods1      
namely1 the chaogenic period  during    
which primordial chaos is given its     
character and directive  tendencies for 
the world age4 ,it is a phase of        
duration wherein the  fiat of kosmic    
order is promulgated1 and consist of    
three stages1  monopyknosis1            
duopyknosis1 tripyknosis#bd gradually1  
insensibly1  gradating into the fourth  
or quartopyknotic2 the chaomorphogenic  
period is likewise divided into three   
stages--quintopkynosis1  sextopyknosis  
and septopyknosis1 developing out of the

 fourth  gradually4 ,the quartopyknotic 
stage is the stage of metamorphosis  or 
transmutation wherein the transition    
from non-manifestation to  manifestation
 is completed2 it is also the stage of  
kosmic causation1  because from it      
spring the matured causes or vital      
impetus which  engender all that follow4
      #bd ,see ,fig4 #ah4               
,illustration3 ,,fig4 #ag4--;,involution
 and ,evolution    ,the close of the    
involutionary phase of the world age is 
marked by the  final deposition of dense
 physical matter and this is closely    
followed  by the beginnings of the      
evolutionary movement which1 like the   
involutionary movement1 is divided into 
two cardinal periods1 namely1  the      
morphogenic 7during which are produced1 
in turn1 insensible forms1  sensible    
forms and spiritual forms7 and the      
kathekotic period which  marks the      
perfection1 the consummation of the     
evolutionary movement4  ,these two      
cardinal periods of the evolutionary    

phase of duration and  the two cardinal 
periods of the involutionary phase      
complete the kosmic  age1 the ,great    
,day of ,brahma4 ,the concentric        
circles1 beginning with  the dot and    
ending with the seven concentric        
circles1 and designated  as a1 b1 c1 d1 
e1 f1 g1 are representations of the     
constitution of the  respective units   
corresponding to each of the seven      
subdivisions4 ,they  symbolize the seven
 degrees of condensation or pyknosis    
which comprise  the genesis of space1 on
 the one hand1 and on the other1 the    
stages  of unfoldment4 ,because1 during 
involution all potencies1 powers and    
characters were being infolded1         
involved2 but during evolution1 these   
are being unfolded1 expressed1 evolved4 
,the figure #ah is another view of these
 two major movements1 involution  and   
evolution4 ,the genesis of space is here
 shown symbolized by the  ,kosmic ,egg4 
,the seven stages of involution are     
referred to as1 the  monopyknotic1      

duopyknotic1 tripyknotic1               
quartopyknotic1 quintopyknotic1         
sextopyknotic and the septopyknotic2    
while the corresponding stages  of      
evolution are referred to as1 the       
physical1 the sentient1 mental1         
causative or spiritual1 the triadic1    
duadic and monadic1 indicating  that the
 principle of physicality is succeeded  
by the principles of  sentience1        
mentality1 spirituality1 and the three  
forms of kathekotic  being4 ,this       
symbolism1 it should be stated1 is      
designed with respect  to the universe  
and man and has no reference to other   
possible  evolutions than the human and 
contemporaneous animal1 plant and       
mineral  evolutions4    ,illustration3  
,,fig4 #ah4--,the ,genesis of ,space    
,to follow the ramifications of the     
symbolism above would involve a  survey 
of all branches of knowledge1 and       
indeed1 would be out of place  in this  
book4 ,only the widest general outlines 
can be suggested here  and it is        

believed that this is sufficient to     
enable the reader to  grasp the         
magnitude of the symbol and to          
understand its purpose and  intent4     
,it would appear1 therefore1 that if it 
is possible for the intellect to        
traverse by means of a study of kosmic  
symbolism1 used as a standard of        
reference1 the entire length of the     
bridge which engages the antipodes  into
 an eternal unity1 something may be     
gained in the way of a more  definite   
and clearer understanding of the        
essential nature of space in  its       
relation to kosmogenesis4    ,in the    
diagram1 figure #ag1 is shown a table   
which represents the  stages of         
space-genesis4 ,it will be noted that   
the whole scheme is  divided into seven 
stages4 ,it is not an arbitrary division
 simply but  a symbolic one and         
represents fullness1 completeness1      
entirety4 ,the  names given1 namely1    
monopyknotic1 duopyknotic1 etc41        
represent the  symbolic characteristics 

of each stage in its relation to the    
universe  in the process of becoming4   
,the terms monadic1 duadic1 triadic1    
etc41 are representative of the seven   
planes of matter in the universe4    ,a 
pyknon is a kosmic principle and        
represents the typal aspect  of         
kosmogenesis4 ,it is a generic term and 
may be identified in its  relation to   
the various stages by the prefix4 ,the  
monopyknon belongs to  the ulterior pole
 of the antipodes on the side of        
non-manifestation4  ,so do the duopyknon
 and the tripyknon4 ,pyknosis is a      
process of  kosmic condensation1 or     
limitation for purposes of              
manifestation4 ,it  is a stage in the   
descent of the kosmic ,spirit-,life1 a  
degradation of  non-manifestation into  
manifestation1 and is1 therefore1 the   
cardinal  causative principle of        
creation4 ,the term pyknon being generic
  is applicable alike to a particle of  
matter1 a state of being1 a  condition  
of existence1 a process or a principle4 

,the monopyknon is1  accordingly1 the   
primary aspect of the process of kosmic 
pyknosis4 ,it  is the archetype and     
therefore all inclusive and             
omnipotential4 ,but  whether regarded as
 a singularity or as a whole it should  
never be  divorced in thought from the  
primal act of creation4 ,it represents  
the  first act of material              
differentiation in the being of the     
creative  ,logos on the plane of        
non-manifestation4 ,it is the beginning 
of every  great planetary or kosmic     
manvantara or period of manifestation4  
,during either a planetary1 solar or    
kosmic pralaya1 or gestatory  period1   
the kosmic plasm is in a quiescent1     
undifferentiated condition4  ,this      
undifferentiated plasm when acted upon  
by the will of the ,creative  ,logos1   
,fohat1 as ,he is sometimes called in   
,eastern ,philosophies1  begins to      
become conditioned1 begins to           
differentiate4 ,the primal  act or stage
 in such a process is the formation or  

appearance of a  monopyknon4 ,it then   
becomes the characteristic aspect of    
that stage4    ,monopyknons are the     
quiescent1 unawakened1 though potential 
and  archetypal principles peculiar to  
the monopyknotic period of              
space-genesis which are ultimately to   
become1 on the physical plane1          
singularities of life of whatsoever     
kind4 ,thus the lineage of every  single
 life-form or principle in the universe 
runs unbroken back from  the present of 
the ,now to the present of monopyknosis4
 ,so great is the  design of the kosmos 
that the entity which is now man or the 
atom was  started on its journey to this
 culmination at the break of the ,great 
,kosmic ,day when the omni-pregnant     
wheels of monopyknosis first began to   
turn4 ,duopyknons and tripyknons        
constitute the two remaining stages  on 
the plane of non-manifestation4 ,and    
their correspondences in every  stage of
 involution or the descent of spirit    
into matter are eternal  and kosmic4    

,likewise the lineage of the dual and   
triple aspects of  all life forms on the
 path of evolution may be traced        
rearward to the  duopyknotic and        
tripyknotic stages of kosmogenesis4     
,the metamorphosis by which the         
monopyknon becomes a duopyknon contrives
  the differentiation of the pristine   
plasm of kosmic being so that the  first
 becomes the ensouling or vitalizing    
principle of the second2  and1 in turn1 
the second becomes the vitalizing or    
inner principle  of the third2 the third
 of the fourth and the fourth of the    
fifth  and so on throughout the series  
until the last is reached which is  the 
septopyknon4 ,the septopyknon is1       
therefore1 a seven-principled  form4 ,it
 is both unitary and septenary--unitary 
in the sense that  the seven are really 
one and septenary for the reason that   
each of  the seven principles1 in the   
course of evolution1 becomes a separate 
process specially adapted to functioning
 upon its peculiar plane of  matter4    

,thus it is seen that the utmost        
significance attaches to this  septenary
 pyknosis of the kosmic plasm of life4  
,the implications of  this conception   
are1 of course1 too vast and            
multifarious to be set  down here4 ,we  
shall have to dismiss it with one       
observation only1 and  that is3 every   
single appearance of life and form in   
the totality of  such appearances is    
rooted in kosmic pyknosis where it has  
received its  inner vitalizing force1   
its form and the law of its mode and    
manner of  appearance together with the 
metes and bounds of its existence4      
,these processes1 monopyknosis and      
duopyknosis1 are to be regarded as      
taking place1 each in its own period1   
everywhere throughout the ,body  of     
,being of the ,logos but on the plane of
 non-manifestation4 ,they are  states of
 preparation for manifestation analogous
 to the germinative  period of the seed 
or the egg4 ,they represent the first   
stirrings of  the kosmic plasm and      

contain the promise and potency of all  
that is  to succeed them4 ,there is one 
other stage1 cordinate with these two1  
and that is the tripyknotic which       
completes the unmanifest trinity  and   
constitutes the archetypal vehicle      
whence proceeds the manifested          
universe4 ,the ensouling principle of   
the tripyknon is the duopyknotic        
principle4 ,but when the descent has    
reached the tripyknotic stage the  three
 have been merged into one and the      
characteristics which were  peculiar to 
each as a separate pyknon are then fused
 into a single  quality having three    
aspects which are mutually              
interdependent and  cordinate4    ,the  
unmanifest trinity1 now complete as a   
result of the triple  pyknotic process1 
is the imperishable and ever sustaining 
radix1 the  all-mother of the manifested
 universe4 ,it is the ,golden ,egg laid 
by  the god ,seb at the beginning of a  
great life-cycle4 ,it is also           
,chaos-,theos-,kosmos1 i4e41 kosmic     

disorder1 divine will or generating     
element1 and kosmic order or space4 ,in 
it1 as in an egg1 resides  in kosmic    
potency all that the universe is to     
become in any ,great  ,day of ,brahma or
 any ,great ,kosmic ,life ,cycle4 ,its  
eldest born is  ,,space1 physiological  
and perceptual1 and the latter is the   
eternal  father-mother of the universe4 
,space is1 therefore1 the male-female   
principle of manifestation2 in its      
kosmic womb all forms are created1      
developed1 evolved and sustained4 ,into 
it again1 at the close of  the ,great   
,day1 all existences1 forms and all     
principles matured and  ripened by the  
vicissitudes of kosmic evolution will be
 inhaled with  the return of the ,great 
,breath of ,life4 ,the unmanifest       
trinity is  the archetype1 and          
therefore1 the pattern or model for the 
manifest1  embodying in potency all that
 the manifest may ever become4 ,forth   
from  the unmanifest proceeds space as a
 dynamic process endued with the        

potentiality for generating all that the
 manifested universe contains4    ,the  
terms unmanifest1 unconditioned and     
unlimited have a special  meaning here  
and are used in the same sense as the   
mathematical term  transfinite1 and     
therefore1 imply a transcending of any  
finite or  assignable degree or quality 
of manifestation4 ,hence they should  be
 distinguished from the term absolute   
which has a different  implication4 ,so 
that although the triple process        
outlined above  may not be viewed except
 as a characterization of the plane of  
non-manifestation1 and hence of the     
primordial activity of the ,creative    
,logos1 there is nothing in the         
symbolism to warrant the identification 
of this process in any way with the     
,logos in absolution4 ,for on this  view
 ,absolute ,being is1 in a large        
measure1 sacrificed when the            
monopyknotic process is begun and the   
monopyknon 7kosmic principle7  begins to
 appear4 ,absolute ,being1 while it may 

not be defined1  delineated or described
 may be symbolized by the ideograph3    
action in  inaction2 being in non-being2
 manifestation in non-manifestation2    
for these are symbols merely and do not 
describe or delineate4    ,we have      
observed the subtle connection which    
exists between  manifestation and       
non-manifestation and have seen how     
that1 as the  roots of a plant sustain  
the outer growth of stem and flowers1   
the  former being the matrix out of     
which proceeds the latter1 and that  in 
like manner does the unmanifest sustain 
the manifest2 it should1  therefore1 be 
clear that the ,body of ,being of the   
,unmanifest ,logos1 in  a similar       
manner1 is the basis and cause of all   
that is manifest or  in existence2 and  
yet it is more1 it is essentially all   
that is and  all that the all is to be  
in manifestation at any time throughout 
the  immeasurable process of            
kosmogenesis4 ,it is the ,self of the   
,universe1  the meta-self of the great  

diversity of selves4 ,the ,self1        
however1  although it cannot be said to 
exist except as a simple1 homogeneous   
quantity is nevertheless1 in the very   
nature of things1 a triunity1  and in   
essence1 not only the basal element of  
the ,all-,space and the  concrete forms 
which exist therein1 but is also        
identical in essence  and substance with
 space4    ,a stage has now been reached
 in the description of our symbolism    
when  it may be assumed1 upon the basis 
of the foregoing1 that the meta-self    
has become manifest1 i4e41 in potential 
and dynamic appearance1 as a  result of 
the triple process of pyknosis          
hereinbefore outlined4 ,the  meta-self  
may then be identified with the ,supreme
 ,manifest ,deity2  for there is ever a 
subtle identification of the manifest   
with the  unmanifest4 ,out of their     
action and interaction the universe is  
made  manifest and phenomenal and is    
thereby sustained4 ,the reciprocity  of 
action between these two kosmic polars1 

is the metamorphotic key  to creation2  
it is the symbol of the procedure of    
,creative ,will in  the act of creating4
 ,it is the transitional process whereby
 the  passage is made from non-being to 
being2 from the unconditioned to  the   
conditioned2 from undifferentiation to  
differentiation and is  reflected and   
symbolized in every natural process     
wherein matter is  transformed from one 
state to another1 or life and mind and  
spirit  diffused1 centralized and       
organized into ever new and higher forms
 and  expressions4    ,another important
 notion to be gained in this connection 
is the fact  that it appears as a       
logical sequence to the foregoing that  
the being  of the manifested ,logos must
 necessarily fill all space1 yea more1  
is that space in every conceivable      
essentiality4 ,his limitations are  the 
limitations of space4 ,his qualities1   
properties and attributes  are the      
qualities1 properties and attributes of 
space and are only  different from the  

original spatial character when         
manifested through  a diversity of forms
 by whose very inner constitution and   
exterior  form the modifications are    
accomplished4 ,all matter in the        
universe1  all energy1 and indeed1 all  
manifestations or emanations of         
whatsoever  character1 hue1 tone or     
quality are1 in reality1 ,his being and 
nothing  but ,his being4 ,there is1     
therefore1 no form nor ensouling        
principle  whether of life1 mind or     
sheer dynamism which can exist outside  
of  ,his being and be1 even in the      
slightest degree1 absolved from an      
eternal identity therewith4 ,once this  
idea is grasped and its varied          
implications noted it then is no longer 
conceivable that any other  order or    
schematism can be possible in our       
universe1 and that1 too1  despite the   
multiform conceptions peculiar to the   
varied systems of  philosophy4    ,the  
matutinal dawn of creation came at the  
close of the tripyknotic  movement which

 resulted in the elaboration of         
materials1 the initiation  of           
principles1 processes and types1 and the
 final preparation of the  field of     
evolution4 ,the three processes or      
aspects of non-manifestation  projected 
in preparation for manifestation1       
namely1 monopyknosis1  duopyknosis and  
tripyknosis represent the earliest      
stages of germinal  development4 ,when  
these had closed1 the ,great ,kosmic    
,egg began to  germinate2 the first     
faint1 indescribable signs of manifested
 life  began to appear4 ,involution set 
in4 ,the fourth or quartopyknotic stage1
  though only slightly differentiated1  
or rather representing that period  of  
kosmic involution when that which is to 
become the manifested  universe first   
begins to fall under the sway of kosmic 
order1 is  nevertheless the basis of all
 great world processes4 ,it is just     
midway  between the poles--manifestation
 and non-manifestation4 ,during this    
stage the life elements are receiving   

the imprints of character1 being        
endowed with directive tendencies and   
stored with such dynamism as will       
persist throughout the ,great ,life     
,cycle in which they are to manifest4   
,it is here that begins the movement of 
involution1 the storing away  of those  
elements and factors1 no more and no    
less1 that are to show  forth on the    
upward path of evolution2 it is here    
that matter begins to  assume form2     
electrons1 ions and atoms created1 or1  
that those minute  processes which on   
the evolutionary side are to culminate  
in these  are originated4 ,this is the  
metamorphotic stage4 ,it is the         
laboratory  of the universe wherein     
,fohat1 the ,creative ,logos1 prepares  
the  materials out of which and in which
 the vast diversity of morphons  or     
forms is created4 ,quartopyknosis1      
accordingly1 is the first active  step1 
on the plane of manifestation1 which    
results in the appearance  of perceptual
 space and consequently of physical     

matter itself as well  as all the other 
grades of matter in the kosmos4 ,space1 
brought into  existence by the act of   
the ,creative ,logos in imposing        
limitation upon  ,his being1 is in its  
primordial form composed of             
quartopyknons or  quadruplicate         
principles and tendencies which act in  
unison and to the  accomplishment of a  
single end or purpose4 ,on this plane or
 during the  continuance of this period 
of space-creation1 the roots of         
universal  law and order are produced4  
,in it are planted the principles of    
good  and evil and a sharp line of      
demarkation established between all the 
conceivable pairs of opposites which    
exist4 ,it accounts for the duality  of 
life and form4 ,male-female2            
father-mother2 positive-negative2       
,rajah-,tamas 7action-inaction7--all    
these find in this process their        
eternal origination4 ,it is the stage of
 harmony1 bliss1 ideality1  perfection1 
perfect equilibrium and balance4 ,here1 

innumerable ages  before they actually  
appear1 the glow-worm and the daisy1 the
 amoeba  and the dynosaur1 man and the  
planetary gods alike abode their time   
awaiting the toppling of the scales of  
kosmic potency when all would be        
plunged headlong into the endless       
labyrinth of becoming4    ,the          
quartopyknotic process is similar in all
 details to the three  preceding        
processes1 these latter being prototypes
 of all succeeding  stages of           
involution4 ,the quintopyknon1          
accordingly1 symbolizes the             
quintuplicative action of life in its   
descending movement toward the  creation
 of matter in its densest form4 ,that   
is1 it is a five-fold  principle acting 
in unison and kosmic consistency1       
infolding in the  universal wherewithal 
that which is to become mental matter on
 the  side of evolution4 ,just1 as may  
be seen in the diagram1 ,figure #ag1    
the quartopyknotic process symbolizes1  
on the involutionary side of  the life  

current that which is to become on the  
evolutionary side1  spiritual essence1  
so the quintopyknotic deposits the seeds
 of that  which is to become mental     
matter4 ,during both the quartopyknotic 
and  the quintopyknotic processes all   
the potencies and promises1 residing  on
 the plane of non-manifestation and     
destined to show forth as  spirit and   
mind are brought into a fuller and more 
marked degree of  manifestation and     
become the seeds of spirituality and    
mentality which  are to ripen and be    
ladened with the fruitage thereof many  
ages hence4    ,the reader should bear  
in mind that the processes here         
described are  thought of as taking     
place at the beginning and as having    
their roots  planted in eternal         
duration2 that they refer to a period   
long before  the universe even resembled
 anything like its present aspect2 at a 
time even before there were individual  
minds to perceive it1 before  even the  
gods--solar1 planetary1 super-solar and 

super-planetary were  in existence to   
take part in the matutinal ceremonials  
of creation's  vast hour of stillness4  
,the mind must accustom itself to go    
back  of appearances1 back of time1 back
 of space itself and discern the        
foundations of time1 space and          
appearances being laid and to perceive  
that which is no less than the action of
 the ,supreme ,deity ,himself in        
brooding over the primordial            
formlessness which is ,himself1 and from
  which will gradually evolve the       
universe of qualities1 conditions and   
appearances4    ,the quintopyknon is1   
therefore1 the base of the              
mind-principle in the  kosmos4 ,all the 
qualities of mind whether in man or in  
the planetary  gods1 whether in the     
moneron or the tyrannosaur1 in the      
mountain or  the oak1 reside in kosmic  
potency in the quintopyknon4            
,,nicholas ,,malebranche1#be in one of  
his very lucid moments1 beheld  the     
essential character of the symbology of 

space and was led to the  conclusion    
that ,god is space itself4 ,to him it   
was equally certain that  all our ideas 
of space1 geometrical or purely         
physiological1 as well  as our notions  
of the great suprasensual domain of     
ideas1 exist in  the kosmic deiform1 or 
body of the ,logos of ,being4 ,he saw   
all things  in ,god4 ,god did not create
 ideas2 they are a part of ,god         
,himself2  ,god did not create mind2 it 
is a part of ,himself2 no kind of matter
  did ,he create2 it is a veritable part
 of ,himself and indissoluble  from     
,himself4 ,the great outstanding        
implication of this philosophy is  that 
our consciousness of ,god is but a part 
of ,god's consciousness  of ,himself2   
our consciousness of self and the       
not-self is but ,god's  consciousness of
 these things4 ,there is no existence of
 anything1  either of the self or the   
not-self1 except in this consciousness4 
,it is  refreshing1 therefore1 to note  
that although the approach is made from 

another and entirely different          
direction1 almost the same conclusion as
  to the ultimate resolution of all     
chaogenetic elements into what is the   
very systasis or consistence of the     
great kosmic deiform1 is reached4       
#be ,vide ,recherche1 ,chap4 ,,vii1 also
 ,philosophical ,review1    ,v4 #ae1 p4 
#dja4--,,malebranche4    ,but a         
marvelous vision comes with the dawn of 
this truth upon the lower  mind4 ,it    
establishes clearly the truth of        
,,kant',s notion when he said3  ,since  
everything we conceive is conceived as  
being in space1 there is  nothing which 
comes before our mind from which the    
idea of space can  be derived2 it is    
equally present in the most rudimentary 
perception  and the most complete4 ,the 
mind cannot get away from the conception
  of space1 because1 out of the very    
essence of space1 as a result of  the   
quintopyknotic process1 it was produced1
 created and organized4  ,the idea of   
space is1 therefore1 not derived from   

things in space nor  from their         
relations in space4 ,it sprang up with  
self-consciousness4  ,as soon as the    
,thinker became conscious of himself he 
became aware of  space4 ,the very state 
of self-consciousness implies space4    
,the self in  man is a specialized      
aspect of space4 ,indeed1 it is a       
projection of  space4 ,the moment the   
self can say3 ,i am1 it also can        
complete the  declaration by saying3 ,i 
am ,space4 ,when the self looks out from
  his six-walled cabin of imprisonment  
into the immensity of what we  call     
space he looks out into that which is   
himself and his immensity2  he perceives
 the source and the ever-present        
sustenance of his being  and recognizes 
his identity therewith1 provided he does
 not allow  himself to become entangled 
in the philosophical difficulties which 
the  intellect is prone to throw around 
the simple1 yet marvelously complex1    
notion of self-consciousness4    ,this  
should settle1 once for all1 the        

question of apriority4 ,the  a priori   
inheres in quintopyknosis or kosmic     
psychogenesis4 ,it is  the essential    
nature of mind2 it is the mind's lines  
of organization2  it is the law of the  
mind's being and action4 ,all mental    
perception  originates from things in   
space4 ,no thought of any detail1 of any
 state  or condition1 whether limited or
 unlimited1 related or isolated can be  
conceived except it be of things in     
space4 ,and this is so1 because  mind   
and space are one4 ,it is not so with   
our conception of time4  ,time is merely
 an aspect of consciousness in its      
limitation and does  not inhere in the  
mind in the same manner that does space4
 ,in fact1  it is not a part of the     
mind's nature as it has been shown that 
space  is4 ,it would1 therefore1 seem to
 be a grave mistake so to cordinate  the
 two notions4 ,space is the progenitor  
of mind and is continually  identified  
with mind4 ,time is the child of        
consciousness4 ,that is1 it  is one of  

the illusions of consciousness which the
 ego will shed as his  consciousness    
expands4 ,duration alone is a cordinate 
of space4    ,the mind now recognizes   
space as something apart and separate   
from  itself only because of its        
unconsciousness of the identity existing
  between it and space4 ,just so1 it is 
not by mind alone that the  at-one state
 of consciousness shall be attained2 for
 although in  one form or another it is 
able to gain some knowledge of the      
apparent  oneness of all life it cannot 
directly realize this oneness4 ,in order
  to do this fully it must be able      
consciously to identify itself with  the
 life1 feel what it feels and experience
 what it experiences and  otherwise come
 into a conscious relationship with the 
root and source  of life4               
,space-consciousness is a simple1 direct
 cognitive process2  while              
time-consciousness is a complex1 and    
therefore1 indirect process4  ,the      
former cannot be analyzed4 ,that is1 no 

analysis is necessary to  its sufficient
 comprehension2 the latter must always  
be analyzed and  categorized for its    
sufficient apprehension4 ,every moment  
of time  whether past1 present or       
future1 when presented to the           
consciousness1  is determined by its    
relationship to some other moment of    
time4 ,space  is indivisible2 time is   
divisible4 ,space is an intuitional     
concept2  time is an intellectual       
concept4 ,time belongs to phenomena4    
,space is  the root and source of       
phenomena4 ,time is the leaves of a tree
 while  space is the life of the tree4  
,space-consciousness1 in its relation to
 the present status of                  
mind-development1 is itself an illusion2
 for despite the fact  that the         
,thinker's apperception of it as a state
 is simple1  direct and fundamental1 it 
is only so because of the inability to  
realize to itself the unity of the      
seeming two4 ,the attainment  of        
space-consciousness or the space mind1  

which contrives the  understanding of   
the identity of mind and space also     
annihilates the  consciousness of space 
as a separate notion from the mind4     
,once the  ,thinker's consciousness has 
arisen to that state where it perceives 
its unity with space all sense of       
separateness is lost4 ,just as when  two
 molecules of hydrogen uniting with one 
molecule of oxygen to form  a new       
compound lose their identity in the new 
realization of unity1 so  does the      
consciousness when by the alchemy of    
psychogenesis it becomes  identified    
with space1 not only lose its identity  
as such1 but also  any consciousness    
whatever that space exists as something 
separate  and distinct from itself4     
,imagine the whole of the duration      
aspect of  kosmogenesis crowded into an 
infinitesimal instant and the bulk of   
all  matter1 suns1 stars1 worlds and    
planets1 condensed into a space less    
than the magnitude of an hydrogen ion   
and in this way a symbol of what  it may

 mean to attain unto absolute knowledge 
or unto the space-mind1  may be         
obtained4    ,recurring to the process  
of quintopyknosis1 it may be noted that 
the  quintopyknon or five-fold kosmic   
principle of life which we have seen  to
 be identical to the seeds of mental    
matter brought into existence  by the   
reaction of ,fohatic energy or the ,will
 of the ,creative ,logos  upon the      
substance of the quartopyknotic stage1  
is1 symbolically  speaking1 more dense  
and compact than the pyknons of the     
preceding  stages4 ,it is ensouled by   
the quartopyknon4 ,it is a rather       
complex  state of ensoulment consisting 
of four condensations or pyknoses4      
,the next stage in the process of kosmic
 involution which is also  concerned    
with the preparation of the evolutionary
 field is that of  sextopyknosis and    
implies the senary condensation of the  
original  world-stuff with the view to  
the formation of emotional or astral    
matter4 ,identically the same process of

 ensoulment or involution  obtains upon 
the plane of sextopyknosis as have been 
observed to obtain  upon the preceding  
planes of involution4 ,involution must  
necessarily  precede evolution4 ,that   
which has not been involved1 enfolded or
  ensouled cannot be evolved or         
unfolded4 ,whatever potencies1 powers   
and  capabilities or qualities and      
characteristics that may appear at any  
time in the universe of life and form   
must have first been involved  or       
enfolded or else they could not have    
been evolved4 ,space itself is  an      
evolution4 ,it is a process of becoming1
 of unfolding1 of flowering  forth4 ,as 
it evolves more and more there will     
appear new and added  characteristics   
and qualities of life and form4 ,new    
possibilities will  arise and in the end
 a supernal vision of a glorified       
universe will  burst into view4    ,the 
scheme of space-genesis is completed    
during the septopyknotic  process       
wherein the basal elements of dense     

physical matter and  its various        
gradations are produced and given       
character1 form and  direction4 ,but    
this completion means merely a temporary
 estopment in  the process of           
kosmogenesis which actually results in  
the formation  of physical matter in its
 crassest state4 ,it does not mean a    
final  arrest of the entire process     
which is conceived of as continuing only
  in a regressive manner back to a      
kathekotic#bf condition wherein  it     
embodies the fruitage of the entire     
scheme4 ,the septopyknon1  accordingly1 
is a seven-ply pyknon in which are      
embodied1 in varying  degrees of        
manifestation and phanerobiogenic       
7life-exhibiting7  quality1 the         
essentialities of all that has preceded 
on all planes and  during all stages of 
space-genesis4 ,that is to say--in the  
physical  life of the universe is       
confined the essence of all the series  
of  grades of life in the kosmos4 ,in   
man's physical body are wrapped up  all 

the glories attainable in his long1     
almost unending pilgrimage  of          
evolution2 in it are stored all the     
possibilities of the spirit2  all       
powers1 all qualities1 all              
characteristics1 ever intended for      
man's attainment are in the physical4   
,but they must be evolved1 they  must be
 unfolded and expressed4 ,the physical  
must be glorified1  spiritualized1      
deified4 ,for by the way of the         
glorification and  spiritualization of  
the flesh man may attain unto oneness   
with the  divinity in himself and       
consequently with the divine life of the
 world4      #bf ,kathekosis 7from      
,chaos-,theos-,kosmos7 is to evolution  
what chaogeny is to involution4 ,it is  
the end of evolution1 but also    the   
beginning of involution1 and in the     
latter function is known as    chaogeny4
 ,see diagram ,no4 #ag4    ,to          
summarize3 ,the genesis of space        
embraces seven stages1 namely1 the      
monopyknotic1 the duopyknotic and the   

tripyknotic which belong to the  plane  
of non-manifestation and are the        
primordial world-stuff and  together    
make up the unmanifest body of the      
,logos of ,being4 ,these  become the    
seed-germ of the universe of spatiality4
 ,the quartopyknotic  is the fourth     
stage in the process of space-genesis   
and is the  metamorphotic or crucial    
stage during which non-manifestation is 
metamorphosed into manifestation4 ,in it
 the unmanifest becomes the  manifest4  
,it corresponds to the plane of pure    
spirit1 and indeed1  embodies within    
itself all the qualities which          
spirituality is to show  forth during   
the life of the kosmos4 ,the            
quintopyknotic is the fifth  stage and  
corresponds to the mental plane1        
embodying in itself all  qualities of   
mentality in the universe and furnishing
 the basis and  essence of that which is
 to become the kosmic mind in           
manifestation4  ,the sextopyknotic is   
the sixth process and symbolizes the    

sixth stage  which embodies all the     
characteristics and properties of       
emotional  matter in the universe and is
 the basal element of the plastic       
essence  of sentient existence in the   
kosmos4 ,the septopyknotic is the       
seventh  and final stage corresponding  
to the physical plane of the kosmos and 
contains in its seven-fold constitution 
the seeds and potencies of all  the     
preceding stages1 as well as all the    
characteristics and properties  which   
physical matter is destined to show     
forth during the manvantara  or world   
age4 ,these seven processes result in   
the dynamic appearance  of space1 the   
mother and container of all things1 and 
complete the  involutionary aspects of  
kosmogenesis4 ,evolution began where    
involution  ceased and will end for this
 manvantara when the last vestige of    
those powers1 capabilities and potencies
 which were involved shall have  been   
evolved unto kosmic perfection4    ,the 
measure of the ,great ,kosmic           

,space-form was sealed at the close of  
the involutionary movement of the ,great
 ,life ,wave4 ,then its metes and       
bounds were fixed by the fringe of      
kathekosity which circumscribed it4     
,if it be true that the reader found it 
extremely difficult to grant  the       
connotations of the symbolism when the  
mental or quintopyknotic  stage was     
reached when illative cognizance was    
given to the fact that  space is also   
composed of mental matter1 it may be    
still more difficult  to grant the claim
 that physical matter is also           
essentially a part of  space4 ,but this 
is the implication4 ,and1 therefore1 it 
follows that  all matter1 all energy1   
all life and all mind wherever it may be
 found  in the ,great ,space-,form is   
space itself and nothing but space4     
,hence1  it appears that space is indeed
 the dynamism of the universe4 ,in its  
kosmic womb the great world egg was     
formed of its own substance solely  and 
in it still the universe of form        

persists and evolves withal4    ,if it  
be suitable for the physicist to talk of
 gravitation1  electricity1 magnetism   
and force let him do so1 for these terms
 serve  the present category of human   
knowledge2 but the human mind will not  
lament the day when it comes to         
recognize that these things1 these      
forces1 these aspects are nothing more  
than space-activities and               
space-phenomena4 ,if a planet's place be
 preserved in space it is  because      
space1 vital1 dynamic1 creative space1  
sustains it and from  its gentle1 yet   
eternally firm grasp there is no escape4
 ,all that  the planets1 suns and worlds
 are and all that they may ever become  
in this manvantara or world age have    
been derived from space1 yea1  are of   
the very essence of spatiality4 ,if the 
chemist choose to  talk of chemism1     
negative and positive1 of combining     
properties and  dissociative phenomena  
let him also become aware that these    
phenomena  are but the external aspects 

of the inner and ephemeral              
life-processes  of space-forms and that 
ultimately these1 too1 may be traced    
back into  an eternal originality within
 the bosom of the all-mother1           
spatiality4    ,dense physical matter1  
such as constitutes the physicality of  
celestial bodies in its ultimate        
dissociation would1 accordingly1 be     
resolved into the original chaogenetic  
formlessness which marked the  chaogeny 
of non-manifestation although it would  
naturally be orderly  and progressive   
passing through the seven stages1       
septopyknosis1  sextopyknosis1          
quintopyknosis1 etc41 until the end had 
been reached1  meanwhile exhibiting in  
each plane the phenomena peculiar to the
  dissociative processes thereof4 ,on   
this view space is a plenum of  matter  
of varying degrees of intensity1 ranging
 from the densest  physical to the most 
tenuous and formless matter of the      
highest levels  of the manifested       
universe4 ,but as neither the dense     

material forms  nor the other grades of 
matter have an eternally enduring       
quality1  being alike subject to        
mutation1 space likewise falls under the
 law of  becoming whereby it1 too1 must 
yield to the edict of kosmic disorder4  
,some may be inclined to argue that     
since space and mind are one and  the   
same thing it must necessarily follow   
that whatever possibilities  of         
measurement may be found to exist in the
 mind would logically be  found to exist
 in space2 and that since all the       
necessary conditions  of hyperspatial   
operations are proved to be existent in 
the mind the  case of the               
hyperspatiality of perceptual space is  
proved thereby4 ,in  other words1 if the
 fourth dimension can be proved to be   
mentally  construable it is also        
possible in perceptual space4 ,but these
  hypotheses are not granted1 and       
neither will they be acceptable to      
those minds who choose to take that view
 when it is known that there  is a      

marked difference between the mind that 
is purely intellectual  and mind that is
 purely intuitional or mind a priori4   
,the intellect  is fashioned for matter 
only2 it is so constructed as to fit    
squarely  into every nook and cranny1   
every groove and interstice in matter2  
yet for the generating element1 life1 it
 has no aptitude nor suitable           
congruence4    ,the attainment of the   
space-mind or kosmic consciousness would
 then  imply a mastery of all           
fundamental possibilities pertaining to 
all  degrees of matter4 ,thus by        
becoming conscious in the matter of all 
the planes one makes a certain definite 
approach to this ultimate  state of     
consciousness until all the barriers    
between ordinary  self-consciousness and
 the consciousness of the space-mind    
have been  entirely obliterated4        
,pyknosis1 in all of its septenary      
aspects1 is concerned primarily  with   
involution or the preparation of the    
chaogenetic elements for  the work of   

kosmic evolution4 ,it may be thought of 
as being divided  into two great        
divisions1 namely3 chaogeny and         
chaomorphogeny4  ,it is concerned with  
the organization of chaos1 the          
establishment  of kosmic geometrism in  
the formless1 void1 arupic substance and
  preparation for evolution4 ,chaogeny1 
of course1 is that kosmic process  by   
virtue of which space itself becomes    
manifest and in which there  is no      
established order4 ,chaomorphogeny 7from
 ,chaos  ,morphe   ,geny7 signifies the 
activities of the creative ,logos in    
laying the  foundations in primordial   
space-matter of the various star-forms1 
including nebul1 worlds1 planets1 suns1 
etc41 of which ,canopus1  ,jupiter1     
,fomalhaut and ,sirius and our own sun  
are examples1 giving  direction and     
general tendence to their varied        
life-processes4 ,both  these processes  
are concerned with the preparation of   
the field and  its consequent           
fertilization in anticipation of its    

cultivation and  harvest4 ,these two    
constitute kosmic involution or the     
great life  wave's passage on the       
downward arc of the ,great ,world ,egg  
or ,circle4  ,it is during the          
chaomorphogenic cycle that the          
constitution of the  universe of        
manifestation is promulgated2 when laws 
for its government  during that         
manvantara are sketched out in the world
 of nascent  spatiality2 when the       
archetype of every imaginable or        
possible form  is projected upon the    
impregnated screen of creation1 then    
folded in1  pushed toward the center1   
involved1 awaiting that time when the   
life  wave begins its passage upon the  
upward arc and evolution ensues1        
calling forth all that has been enfolded
 in the bosom of the pyknotic  centers  
of manifestation4 ,it is easily         
conceivable that here during the        
troublous times of the chaomorphogenic  
enfoldment the now known six  directions
 of space were among the eternal edicts 

of space-genesis and  that that law     
which now makes it appear that three    
cordinates1 and  only three1 are        
sufficient for the determination of a   
point position  in space was imprinted  
in the very nature of that which was to 
become  space4    ,the kosmic field     
having been prepared as a result of the 
chaomorphogenic activities1 lowly and   
scarcely organized forms begin  to      
appear and the ascent upon the upward   
arc of the ,great ,cycle  commences4    
,evolution begins4 ,its scope is        
likewise divided into two  great stages1
 namely3 7a7 ,morphogeny1 the purpose of
 which is the  development of life-forms
 or pyknons which are to appear on the  
various  planets1 stars1 worlds and suns
 of the universe4 ,it embraces the whole
  span of the life-aspect on the        
evolutionary side of manifestation4  ,in
 this aspect is included also every     
conceivable adaptation of the  universal
 principle of life from the beginning of
 its movement to  the end4 ,the universe

 is now functioning in and progressing  
through  the vicissitudes of this stage4
 ,that is1 all the present observable   
adaptations which the life-pyknon or    
principle is making1 has made or  will  
make1 are embraced within the scope of  
what is here designated  as the         
morphogenetic aspect of evolution4 7b7  
,the second and  last division of the   
arc of evolution is called ,kathekos1   
thus  symbolizing the syncretism of the 
trinitarian aspects of kosmogenesis1    
,chaos-,theos-,kosmos1 perfected and    
united as a result of the labors  of    
manifestation4 ,in this final summation 
of the labors of the  life-wave as it   
has progressed from involution through  
all the devious  manifestations of      
evolution are embodied the perfection   
and ultimate  elaboration and expression
 of all the pyknotic tendencies which   
were  established during the entire     
scope of space-genesis4    ,thus it will
 be seen that the first three stages of 
space-genesis1  called chaogeny1        

encompass the first three pyknotic      
processes or  are analogous thereto     
while the latter called chaomorphogeny1 
the  organization and ensoulment of     
space-forms1 embraces the latter four1  
quartopyknosis1 quintopyknosis1         
sextopyknosis and septopyknosis4 ,this  
division obtains on the involutionary   
side of the great life-cycle4  ,the     
upward arc of the ,great ,kosmic ,egg or
 ,cycle is also divided into  two great 
stages1 namely1 ,morphogeny             
7manifestation of life through  the     
various forms which it assumes7 and     
,kathekos1 or the kathekotic  plane of  
perfected triunity which is represented 
by the evolutionary  union of           
,chaos-,theos-,kosmos4 ,kathekos would1 
therefore1 symbolize  the ultimate      
elaboration of ,chaos into a            
well-ordered kosmos wherein  are        
expressed all the possibilities which   
inhered in the archetypal  plan of the  
,creative ,logos or ,theos and in which 
all had reached  the ultimate perfection

 in the body of being of the ,logos     
,himself4  ,but the kathekotic plane is 
to be distinguished from the original   
,chaos-,theos-,kosmos represented as    
functionating upon the plane of         
non-manifestation during chaogeny4      
,kathekos symbolizes the perfected      
manifestations of the triune aspects of 
the ,creative ,logos through the        
perfected forms resulting from the      
labors of kosmic evolution1 while       
,chaos-,theos-,kosmos symbolized1 as a  
triune glyph1 the ,unmanifest  ,trinity 
in the primordial beginnings of         
space-genesis4 ,one is the seed2  the   
other the fully matured plant2 one the  
egg2 while the other is the  full grown 
bird2 one the root2 the other the       
fruitage2 one ,alpha1 the  other ,omega2
 one the beginning1 the other the end4  
,the end1 however1  is reached only     
that1 in due time1 the entire scheme may
 be commenced  again1 once more         
utilizing the results of the preceding  
scheme of  evolution as the basis of the

 ensuing one4 ,thus after every ,kosmic 
,day1  commences the ,kosmic ,night4    
,the succession of kosmic days and      
nights is  infinite4 ,this infinity of  
becomings in the life of the kosmos is a
  necessary outcome of eternal duration4
    ,the above1 thus briefly set down1  
is the symbolism of space-genesis4  ,it 
is commended to the reader as a basis   
for the conception that the  real1      
essential1 perceptual space is something
 far more wonderful1  more fundamental  
than either the geometrician or the     
metageometrician  has ever dreamed of1  
and yet the latter's consciousness is   
undoubtedly  being appulsed by the      
fingers of a new species of             
conceptualizations  which1 one day in   
the not too distant future1 will arouse 
in it the  faint hungerings after the   
realization of the real space-nature4   
,these  mathetic appetites thus brought 
into being will finally lead the  human 
mind into the ,elysian fields of kosmic 
consciousness where for  another million

 years1 perhaps1 it may feed upon the   
mysteries and  hypermysteries to be     
found in the granaries of the           
,space-,mind4    ,the study of space in 
its wider and deeper meanings is        
necessary in  order that a clearer      
understanding of its true significance1 
as the  subject of geometric researches1
 may be gained4 ,it is confessed1       
however1 that there is neither direct   
evidence nor implicative  authority for 
any assumption that the view herein     
outlined affords any  justification for 
the notion of the n-dimensionality of   
space4 ,for1  although the line of      
reasoning indulged in must lead         
inevitably to the  conclusion that the  
worlds of spatiality1 materiality1      
intellectuality  and spirituality1      
essentially and fundamentally one so far
 as origins  and qualities are          
concerned1 were alike engendered by the 
same  generating element1 life2 and that
 spatiality being the primal basis  of  
the others is1 nevertheless1 under the  

exigencies of this aspect of  the       
kosmos1 highly susceptible to the       
mensurative requirements of the         
grossest1 there appears to be no        
necessity for calling upon extraneous   
considerations for assistance in our    
efforts to comprehend the various       
connotations of the symbolism4 ,then1   
too1 it is easily conceivable that      
under conditions where these elements1  
spatiality1 intellectuality and         
materiality1 are not only co-extensive  
but interpenetrative1 there is  no      
justification for the assumption that   
they must exist in layers or            
manifoldnesses or in discrete degrees1  
separated from one another as if  they  
were constituted of different substances
 and occupied different  spheres4 ,for  
every single point in perceptual space  
is a focus for  lines drawn through     
every conceivable grade of materiality1 
spatiality  or intellectuality in the   
kosmos4 ,and the same system of         
cordinates  which is necessary and      

sufficient for the localization of a    
point in  our space is also sufficient  
for the location of a point anywhere in 
the entire world of spatiality1         
intellectuality or spirituality4 ,in    
fact1 the external1 visible worlds of   
materiality and spatiality are  nothing 
more than the mass-termini of lines     
extending from divinity  to physicality2
 from primordial originality to kosmic  
modernity and it  is intellectually     
conceivable that progression back over  
the grooves  made by these mass-termini 
of lines would lead directly and        
unerringly  to originality itself4 ,in  
spite of the manifold pyknoses which we 
have  shown to characterize the         
symbolism of space-genesis it is a very 
simple matter2 for the entire scheme    
could and must have proceeded  along    
strictly tridimensional lines4          
,tridimensionality must have  inhered in
 the primeval archetype of space or else
 it could not appear  as an outstanding 
fact of perceptual space now2 for all   

that we can now  observe in space as    
characteristics must have first been    
included1  enfolded1 involved1 before it
 could have been evolved4 ,hence1 it is 
to be remembered that we are to-day     
dealing with the expressions of         
tendencies and principles which inhered 
in the manifested universe as           
potentialities in the very beginning4   
,the alphabet of space-genesis consists 
of five characters1 namely1  the point1 
the line1 the triangle1 the square and  
the circle4 ,these  are the             
pentagrammaton of space1 of             
intellectuality1 materiality and  of    
spirituality4 ,they constitute the basis
 of kosmic geometrism4  ,with these all 
geometrical figures may be constructed2 
with them  all magnitudes may be        
delineated and projected4 ,they describe
 every  conceivable activity of the     
,creative ,logos and designate the      
bounds of  the entire scope of motility 
of kosmogenesis4    ,in figure #ai1 are 
shown the dot1 the line1 triangle1      

square and the  circle which together   
form the kosmic pentoglyph4 ,the point  
symbolizes  kosmic inertia1 inactivity  
or the beginning of motion2 the line is 
the  first aspect of motion1 the        
beginning of creation2 the triple aspect
 of  kosmogenesis is symbolized by the  
triangle1 ,chaos-,theos-,kosmos1 the    
,unmanifest ,trinity2 the square        
emblematizes kosmic being in evolution2 
while the circle is the syncretism of   
all these and stands for the  perfected 
kosmos1 or the kosmos in process of     
perfection4 ,very truly  did ,,plato    
remark3 ,god geometrizes2 for the       
pentagrammaton--the                     
point-line-triangle-square-circle--is   
the deity's way of manifesting          
,himself4 ,but there is here no need for
 space-curvature nor for  triangles     
whose value is greater or less than #ahj
 degrees2 there is no  need even for the
 mathematical fourth dimension4         
,illustration3 ,,fig4 #ai4--,kosmic     
,pentoglyph    ,it cannot be believed1  

however1 that metageometricians are     
really in  earnest in what they suggest 
of hyperspace and n-dimensionality2  it 
cannot be believed that they are        
entirely satisfied with what  they have 
found of the so-called hyperspatial1 and
 yet1 some of  them are fanatically     
patriotic over the new-found domain2    
some are  even intolerant4 ,but there   
are others who look upon the fabric of  
metageometry as a stepping stone to     
space-realities1 a mile-post on the     
path to the realization of a higher     
consciousness1 the consciousness of  the
 space-mind or kosmic consciousness4    
,and may this not1 after all1  be the   
goal of the human intellect1 now        
slightly distraught by the  exuberances 
of youth and the joys of a new mental   
freedom8 ,the work  of the future       
mathematicians will be the destruction  
of the tumorous  inconsistences to be   
found in the various non-,euclidean     
systems  of geometrical thought1 the    
elimination of the novelties and the    

nonsensicals1 the synthesizing of those 
elements which are sanctioned  by the   
space-mind and the building thereon a   
sane interpretation of  space-phenomena 
in the light of illuminations received  
from greatly  extended faculties and a  
participation in that larger            
consciousness  into which the human race
 seems slowly to be immerging4    ,the  
domain of hyperspace is but the         
fairy-land of mathesis1 peopled  with   
goblins1 gnomes1 kobolds1 elves and fays
 which are the  spaces1 dimensions1     
propositions1 ensembles and theorems of 
the  metageometrician4 ,but like the    
fairies and nature spirits of the unseen
  about us1 they have their bases in the
 real1 objective world of facts  however
 difficult it may be to establish their 
direct connection with  it4 ,as the gay1
 invisible sprites of phantom-land      
represent intelligent  natural forces at
 work in the furtherance of the         
evolution of forms1 so  the impalpable  
things of mathesis are emblems of kosmic

 forces at work  in the upbuilding of   
structures of higher consciousness which
 shall be  towers of vision for the     
human soul whence it may view the       
hill-crests  of infinite knowledge and  
the low-lying plains of kosmic          
mysteries4    ,finally1 it has been     
noted that space is the very consistence
 of the  kosmos2 it is the life1 the    
form and both the outer and the inner   
manifestation of the combined life and  
form2 it is reality1 also  illusion2 it 
is concrete1 also ideal4 ,we have noted 
also that mind is  consubstantial with  
space and that space gives it its inner 
life and  nature as well as nourishes   
its outer growth and development4 ,in   
fact1  we have seen that space and mind 
are one essentially and that they  exist
 as aspects of the same thing1 life4 ,in
 whatever way1 then1 that  the mind     
normally views space that is the natural
 way4 ,all attempts to  deviate from the
 natural way are1 therefore1            
unsanctioned by the nature  of things4  

,so long as geometry remains true to the
 nature of mind and  space so long will 
it be valid universally and possessed of
 kosmic  necessity and invariance4 ,it  
behaves most unseemly when it departs   
from  its fealty to the nature of things
 per se4    ,both the outlook of the    
mind upon the objective world as well as
 its  inlook upon its own states of     
consciousness or the subjective world   
are tridimensional4 ,its growth is      
tridimensional1 its nature is  likewise 
tridimensional1 and there is not even   
the slightest tendence  either to       
perceive1 conceive or perform in a      
four-dimensional manner1  mathematically
 speaking4 ,trace out the biologic      
development of each  mental faculty1    
from the mind of the moneron to the mind
 of the most  highly developed man and  
it will be found that everywhere and    
always1  without variation or exception1
 the nature of each of these has  been  
to express itself tridimensionally and  
naturally4 ,there is not  even the      

slightest sign of so much as a germinal 
appetence for the  four-space2 it would1
 therefore1 seem almost a prostitution  
of mental  faculty to divert mental     
energy into the seemingly useless       
channel of  present-day metageometrical 
researches2 yet1 it must be admitted    
that  even though the end sought cannot 
be attained1 the final results of the   
intellectual delvings into the dread    
homogeneity of kosmic origins and  the  
consequent realization of the awesome   
coevalism of mind and space  whence     
shall arise the recognition of the      
wondrous unitariness of all  existences1
 will be that we shall come upon that   
thrice mysterious  contrivance--the     
,heart of ,divinity1 the ,kosmic        
,space-,center in which  abide the roots
 of the ,great ,all in a marvelously    
indescribable unity  and infinite       
originality4    ,we may conclude1 then1 
that hyperspatiality and all its        
appurtenances  are but the toys of the  
childhood of humanity4 ,but1 as the     

years  pass and the days of maturity    
come on apace1 it1 toy-like1 will also  
be discarded4 ,and the mind will seize  
then upon the seriousness of  reality   
just as the matured youth responds to   
the stern realities of  life and manhood
 responsibilities4 ,but no one can say  
that the toys of  childhood are wholly  
useless2 no one can say that the joys   
which they  bring are entirely fatuous  
and unreal nor shall we attempt to      
intimate  that mathetic contrivances are
 without utility1 without purpose and   
significance in the life of the growing 
mind of humanity4 ,but they1  too1 will 
pass away4            ,,chapter ,,viii  
,,the ,,mystery ,,of ,,space      ,the  
,thinker and the ,ego--,increscent      
,automatism of the    ,intellect--,the  
,egopsyche and the ,omnipsyche--,kosmic 
,order or    ,geometrism--,life as      
,engendering ,element--,the ,mystery of 
,space    ,stated--,kathekos and        
,kathekotic ,consciousness--,function of
 the    ,ideal--,the ,path of ,search   

for an ,understanding of the ,nature and
    ,extent of ,space ,must ,proceed in 
an ,inverse ,direction4      ,the       
fragmentariness of the ,thinker's       
outlook upon the universe of  spatiality
 is due to the inhibitive action set up 
by the constrictive  bonds which his    
complicate mechanism of intellectuality 
interposes  between himself and reality4
 ,the ,thinker1 who stands back of and  
uses  the various media of objective    
consciousness1 such as the neural       
mechanisms1 brain1 emotions1 his        
individualized life-force and the  mind 
which together make up the instruments  
with which he contacts the  sensuous    
domain1 by adapting his consciousness to
 these means1 as the  artisan utilizes  
his tools1 constitutes his own          
intellectuality4 ,the  intellectuality1 
then1 is the totality of media by which 
consciousness  effects its entrance into
 the sensuous world and by which it     
receives  impressions therefrom4 ,in    
other words1 it is the sum of all those 

qualities1 operations1 processes and    
mechanisms which are recognized as      
constituting the modus vivendi of man's 
intellectuality1 and these  are1 in     
reality1 nothing more than the ego      
himself4    ,many have been inclined to 
regard that which has been called the   
ego as the highest sovereign power in   
the state of manhood4 ,he has  been     
looked upon as the final consideration  
in the constitution of  the human being4
 ,but the ego is an evolutionary product
 and the  concomitant of                
self-consciousness which is the         
,i-making faculty in  man's psychic     
life4 ,it is that quality of            
consciousness which makes  man conceive 
of himself as a separate1 detached and  
independent being4  ,it is a purely     
intellectual or tuitional product1 and1 
as such1 is to  be differentiated from  
the intuitional or life-quality which is
 the  essence of man's real selfhood4   
,with respect to the ,thinker1 the ego  
occupies precisely the same status as   

the agent to his principal4 ,as  the    
agent is the representative of the      
principal in all matters which  come    
within the scope of his prescribed      
jurisdiction so is the ego the  agent of
 the ,thinker who is a spiritual        
intelligence4 ,accordingly1  from an    
ethical viewpoint the ,thinker is       
responsible for the acts of  the agent  
and can in no wise escape the penalties 
accruing as a result  of the agent's    
violations4 ,just as a commercial firm  
sends out a  representative for the     
collection of data concerning certain   
phases of  its business or it may be of 
any business or the entire world market 
so the ,thinker projects his own        
consciousness into the mechanisms  which
 are in their totality the egoic life4  
,that is1 he sends out his  agent1 the  
ego1 into life and into the objective   
world of facts and  demands that he     
shall convey to him1 from all points of 
the territory  which he is expected to  
cover1 reports of his findings4 ,of     

course1  these reports which are        
transmitted by the ego 7the intellectual
  mechanism of the ,thinker7 are more or
 less well prepared summations  of his  
individual observations and deductions4 
,these are the percepts  which the ego  
presents to the ,thinker's              
consciousness4 ,concepts are  formed by 
the ,thinker in his treatment of these  
sense-presentations4  ,it very          
frequently happens that the ego         
transmits reports which1 for  one reason
 or another1 give very imperfect        
knowledge of the matter  which his      
reports are designed to cover4 ,often it
 is necessary that  additional and      
supplemental reports be made about the  
same thing1 and  even then1 it is       
well-nigh impossible1 if not quite so1  
for him fully  to cover every detail of 
the matter under consideration and in no
 case  is it possible for him to do more
 than report on the superficialities  of
 the question under scrutiny4 ,if the   
ego1 in his operations1 be  imagined to 

be hampered by similar circumstances and
 difficulties as  those which would     
ordinarily beset a commercial attach it 
will then  be clear that his reports    
must ever be fragmentary because of the 
inaccessibility of much of the data     
which would be necessary for a full     
report1 and further1 because of the     
inadequacy of his methods and means  of 
gathering data due to the inherent      
limitations of his capabilities1        
endurance and perspicacity and          
innumerable other limitations and       
difficulties which must be faced in all 
search for the real4 ,so that1  while   
the sufficiency of the means which the  
ego enjoys at this stage  for all       
practical purposes is granted no        
hesitancy is entertained when  it comes 
to a discovery of the reals of knowledge
 in declaring their  insufficiency4     
,then1 too1 when it is remembered that  
these egoic reports are in  the nature  
of neurographical communications which  
are similar to  telegraphic despatches  

and must pass through several stations1 
as  ganglia1 etc41 often being relayed  
from one to another1 it will be  quite  
apparent that much1 even of the original
 quality of the missives  forwarded1    
will have been lost or radically changed
 in some way before  it is finally      
delivered for the inspection of the     
,thinker himself4 ,it  not infrequently 
happens1 even in perfectly normal       
beings1 that the  ego in filing1        
recording1 transcribing1 interpreting1  
translating and  otherwise preparing    
these data for the ,thinker's use1 lets 
a cog slip1  misplaces some of the data1
 loses or destroys fragments of it and  
so is  unable to maintain a complete    
portfolio of his materials4    ,as the  
,thinker is entirely dependent upon his 
agent1 the ego1 for  the trustworthiness
 of his information covering the matter 
of the  sensuous world it is obvious    
that at best his information is very    
fragmentary indeed1 and necessarily so  
when it is considered that the  modus   

operandi of his agent and the difficulty
 of his operations are  so complicate as
 to magnify the obstructions in the way 
to complete  freedom in this regard4    
,to continue the similitude of principal
 and agent it may be asserted  that it  
is also true that the commercial house  
that sends out its  attaches frequently 
will send letters containing directions 
as to  procedure1 sometimes censuring   
for past delinquencies and sometimes    
commending for praiseworthy deeds2 and  
this1 too1 in addition to the  original 
instructions which were given at the    
outset4 ,it even comes to  pass that the
 home office1 because of some           
meretricious accomplishment1  as the    
marked increase of efficiency shown by  
the agent's close  application to his   
duties and the consequent success of his
 operations1  confers certain favors    
upon the agent or removes some of the   
restrictions which were originally      
imposed1 gives an increase in salary  or
 promotes the agent to a higher and more

 lucrative office with larger  powers   
and greater authority4 ,this is         
analogous to what the ,thinker  does for
 the ego4 ,for he not only receives     
reports from the ego1 but  often1 in the
 shape of intuitions1 gives additional  
information as to  the proper manner of 
doing things1 sheds more light upon some
 obscure  operation1 commends for duty  
well performed1 condemns for failures or
  for wrong-doing1 rewards arduous toil 
with greater powers of vision1  keener  
insight1 greater capabilities2 in fact1 
promotes the ego in the  sight of other 
egos by marking him out as an           
exceptional ego4 ,but the  curious      
aspect of this procedure is that1 in    
time and after the ego  has been        
repeatedly commended and promoted and   
otherwise favored by  the ,thinker1 he  
begins to think that he owns the firm1  
that he is the  life and main support of
 the whole corporation4 ,he becomes     
arrogant1  self-willed and finally falls
 into the illusion that he alone is     

responsible for the phenomenal success  
of the firm4 ,this is the source  of    
that illusion of the intellect which    
makes itself think that it1 the  ego1 is
 all there is to man1 that his          
instruments of operation in the         
objective world are the only kind of    
instruments that may be used2 that  his 
method of gathering data about things is
 the only safe and sure  method2 and so 
it develops that the intellectuality is 
the source of  man's separateness1 his  
individuality and his apparent aloofness
 from  other men and things4 ,it is1 of 
course1 needless to point out that in   
this way the intellect comes to be the  
tyrant of man1 ruling with a  rigid     
monopoly and as an all-exclusive        
autocracy4    ,from the above           
implications it would appear that the   
intellect and the  intuitive faculty are
 two separate and distinct processes1   
and so they  are4 ,one is the inverse of
 the other4 ,the tendence of the egoic  
life or  the intellect is for the       

external while the intuition is an      
internal  process4 ,the intellect acts  
from without towards the interior while 
the  intuition acts from within outward4
 ,the intellect is the product of the   
intuition which is another term for the 
consciousness of the ,thinker on  his   
own plane4 ,just as the child lives a   
separate and distinct1 though           
dependent1 life from the parents so the 
intellect has a modus vivendi  which is 
distinct and separate from that of the  
,thinker1 and yet it is  in all points  
dependent upon the life of the ,thinker4
 ,here again1 we  find an analogy in the
 relation of the child to the parent4   
,as some  children are more amenable to 
the will of the parent than others1  so1
 in some persons1 the intellect is more 
amenable to the action of  the intuition
 than in others4 ,yet it is a certain   
fact that the more  the outward life is 
governed by the intuition1 i4e41 the    
more the  intellect responds to the     
intuitive faculty of the ,thinker1 the  

higher  the order of the life of the ego
 and the more accurate his decisions    
and judgments4 ,in fact1 it assuredly   
may be asserted that the place  of every
 individual in the scale of evolution is
 determined in a very  large measure by 
the degree of agreement between the     
intuition and the  intellect or by the  
ease with which the intuition may       
operate through  the intellect as a     
medium4 ,at least1 the quality of one's 
life may be  determined directly by     
these considerations4    ,the ,thinker  
being himself a pure spiritual          
intelligence1 living upon  the plane of 
spirit and therefore unhampered by the  
difficulties  which the ego meets in his
 operations in the objective sensorium1 
and  possessed of far greater knowledge1
 is correspondingly free from the       
limitations of the ego and very         
naturally closer to kosmic realities4   
,hence1 he is better situated for the   
procurement of correct notions  of      
relations1 essentialities and the like4 

,it is believed1 therefore1  that in the
 proportion that these two processes1   
the intellectual and  the intuitional1  
are brought1 in the course of evolution1
 to a closer  and more rigid agreement1 
in the proportion that the ,thinker is  
able  to transmit the intuitograms in   
the shape of concepts or that the       
intuition is made more and more         
conceptual1 in just that proportion is  
humanity becoming perfect and its       
evolution complete4 ,the difficulty     
found to inhere in the conceptualization
 of intuitions so that they may  be     
propagated from man to man seems not to 
lie in the ,thinker himself1  but more  
essentially in the ego1 in the          
intellectuality and its  complicate     
schematism or plan of action4 ,it would 
appear1 therefore1  that the only way of
 escaping or transcending this          
difficulty is for  the ego so to refine 
his vehicles or so facilitate his plan  
of action  by eliminating the numerous  
relays or sub-stations intervening      

between  the consciousness of the       
,thinker and that which may be said to  
be his  own that the transmission of    
intuitograms may be accomplished with   
the greatest ease and clearness4 ,while 
no attempt will be made to  indicate the
 probable line of action which the ego  
or objective man  will adopt for this   
purpose1 it is believed that it may be  
said without  pedanticism that the only 
true method of attaining unto this much 
desired state of things is1 first of    
all1 by assuming a sympathetic  attitude
 not only towards the question of the   
intuition itself but  to all phenomena  
which are an outgrowth of1 or incident  
to1 the  manifestations of the intuitive
 faculty through the intellectuality1   
and second1 by the practice of prolonged
 abstract thought1 this latter          
procedure effecting a suspension of the 
intellectuality temporarily  at the same
 time allowing it to experience an      
undisturbed contact  with the           
intuitional consciousness1 thereby      

laying the basis for  future recognition
 of its nature and quality4 ,it would   
seem that  these two conditions are     
absolutely necessary in order that a    
more  congruent relationship may be     
promoted between these two cognitive    
faculties4 ,ordinarily1 it would appear 
that the philosopher who is  undoubtedly
 inured to the necessities of continuous
 abstraction or  the mathematician whose
 most common tasks naturally fall in    
this  category would be among all men   
most apt to develop to the point of     
conceptualizing intuitograms readily1   
yet it seems that this is not  the case4
 ,and there is good reason for it4 ,the 
mind of the philosopher  and the        
mathematician is intellectual rather    
than intuitional and is1  therefore1    
wedded to matter1 to the action and     
reaction of matter  against matter and  
hence operating in a direction at       
variance with the  trend of an          
intuitional mind4 ,and this condition is
 undoubtedly due to  a lack of a        

sympathetic attitude towards this       
species of consciousness4  ,at any rate1
 it is thought that a too great anxiety 
in this respect  need not be entertained
 by humanity at all1 for the reason that
 in  the case of a faculty1 the         
rudimentary outcroppings of which are so
  marked and universally observable and 
existing in greater or lesser  degrees  
in various human beings1 there is ample 
evidence for the  belief that it is     
being carefully and duly promoted by a  
well-directed  evolution of psychic     
faculties and powers1 so that at the    
proper time1  determinable by the state 
of perfection reached by the            
intellectuality  or the ego in the      
operation of his cognitive processes1   
the much  desired agreement of these two
 faculties will have been realized and  
the conceptualization of intuitograms   
into propagable conceptions an          
accomplished fact4 ,until this goal     
shall have been reached and the         
intuition shall have overshadowed the   

intellect as the intellect now          
overshadows the intuition2 or the       
consciousness of the ego1 derived  from 
the interplay of the ,thinker's         
consciousness among the various         
elements which constitute the ego       
himself1 shall have been merged with    
that of the ,thinker1 the outlook must  
remain fragmentary1 only becoming  a    
well-ordered whole as the barriers of   
dissidence are broken down in           
succession4    ,the evolution of        
consciousness1 from the simple1         
undifferentiated  moneron to the        
differentiated cell and from that to the
 cell-colony  and from the cell-colony  
to the organism1 traversing in          
successive  paces through all the stages
 of lower life--mineral1 vegetable and  
animal--to the stages of the simple1    
communal consciousness of the  higher   
animals1 to the self or individual      
consciousness of the human  being1 each 
requiring millions of years for its     
perfection before a  more advanced stage

 is entered1 has been one continuous    
relinquishment  of the lower and less   
complicate for the higher and more      
complex  expression of itself through   
the given media4 ,when a newer and      
higher  stage of consciousness is being 
entered by humanity its appearance  or  
manifestation is first made in the most 
advanced of the race and  that only in a
 dim1 vague way4 ,this rudimentary      
condition persists  for some time1      
perhaps many thousands of years1 then   
the faculty  becomes more general in    
appearance1 the number of advanced      
individuals  increases1 and             
consequently1 as in the case of the     
intuitive faculty1  it becomes          
universally prevalent in all humanity2  
becomes transmissible  as so-called     
acquired characters1 and then appears as
 the normal  faculty of the entire human
 family cropping out in each individual4
  ,thus1 in passing from the few        
advanced ones in the beginning to that  
stage where it becomes the common       

possession of all1 a faculty requires   
many thousands of years for its         
perfection1 and especially has this     
been true in the past history of the    
development of human faculties4  ,but it
 is believed that the sweep of the life 
current as it proceeds  from form to    
form1 from faculty to faculty1 gains in 
momentum as it  proceeds1 so that in    
these latter years due to the already   
highly  developed vehicular mechanisms  
at its disposal not so great a period   
of time as formerly is required for the 
out-bringing of a new  faculty4 ,it     
might well be that while in the past    
hundreds of thousands  of years were    
necessary in the perfection of organs   
and faculties1  in these latter days    
only a few thousand1 perhaps hundreds1  
may be  necessary and that in the days  
of the future not even so many years    
may be required to universalize a       
faculty4 ,and especially does this      
appear to be true in a state of affairs 
where so large a number of  persons are 

beginning consciously to take their     
evolution in hand and  by volitional    
activities are supplying greatly        
increased impetus to  their psychic     
processes which under ordinary1 natural 
methods would  be considerably slower in
 their development4 ,it is quite obvious
 that  all cultural efforts when applied
 to the betterment of a given plant1    
animal or faculty result in a           
corresponding hastening of the process  
of  growth far in excess of what that   
growth would be under normal1 natural   
conditions4 ,all the present faculties  
possessed by man are remarkably         
susceptible to cultural influences2 in  
fact1 the standing edict of  ethical and
 social law is that the human faculties 
must be cultivated  as highly as        
possible1 thereby giving the spirit a   
more perfect medium  of expression4     
,these observations1 therefore1 lead    
irresistibly and  unavoidably to the    
conclusion that the time for the        
upspringing of  the intuitional faculty 

in the human organism is even now upon  
us1  that undoubtedly in certain very   
advanced ones it has already reached  a 
notable degree of perfection and is     
rather more general than would  appear  
in the absence of careful investigation4
    ,now1 just as the intellect has made
 for individuality1 has emphasized  the 
separateness of the ,thinker's existence
 from that of other  thinkers1 has      
developed self-consciousness to a very  
high degree1 even  pushing it far over  
into the domain of the higher           
consciousness to the  temporary         
obscuration of the latter1 so the       
intuitional will make for  union1 for   
the brotherhood of man1 for co-operation
 and for the common  weal4 ,through it  
man will come gradually into the        
consciousness that  fundamentally1 in   
his inner nature1 in every respect of   
vital concern1  he is at-one with his   
fellowmen and not only with the apparent
 units  of life but with all life as    
expressed in whatsoever form throughout 

the universe4 ,then1 too1 he will be    
closer to the reality of things1  of    
actions and natural processes2 in fine1 
he will have begun the  development of  
the space-mind which will bring him to  
the knowledge  that he is one with space
 also and1 therefore1 with the divine   
life of  the world4    ,one of the      
peculiarities of the vital force which  
shows itself in  the consciousness as   
man's intellect1 is its growing         
automatism1  or that tendency which     
enables the consciousness to perform its
  functions automatically and thus allow
 opportunity for the development  of    
newer and higher faculties4 ,actions1   
oft repeated1 tend to become  automatic4
 ,this is also true of thought and      
consciousness4 ,it is one  of the       
beneficent results of abstract thought  
that it develops1 or  tends to develop1 
a kind of automatism whereby a marked   
saving in time  and energy is effected4 
,this affords opportunity for other     
things4  ,it is undoubtedly true that in

 the days of the truly primitive man    
his consciousness was more completely   
engaged in the execution of  the ergonic
 functions of cells1 organs and tissues2
 that all those  processes which are now
 said to be involuntary and reflexive   
were  at one time1 in the distant past  
of man's evolution1 the results of      
conscious volitions4 ,this is a         
condition which must have preceded  even
 the development of the intellect       
itself4 ,indeed1 there could be  no     
intellect in a state where the entire   
modicum of consciousness was  being     
utilized in the performance of cellular 
and histologic functions4    ,the rise  
of the intellect must have been in      
direct ratio to the  development of     
automatism among the cells1 tissues and 
organs1 so that  as these came gradually
 to perform their special labors        
reflexively  the intellect began to be  
formulated and to grow1 at first only   
incipiently1 then more and more         
completely until it reached its present 

state4 ,at the present stage of its     
evolution1 a great deal of the  labor of
 the intellect is beginning to fall into
 a kind of increscent  automatism1      
although only rudimentarily1 in many    
instances4 ,yet1 as a  result of this   
tendency1 quite the whole of the        
phenomena of perception  is             
characterized by a sort of automatic    
action4 ,and the mind perceives  without
 conscious volition4 ,many of the steps 
of conceptualization  are automatic1 in 
part1 if not wholly4 ,certain it is that
 impulses  once set in operation whether
 consciously or unconsciously continue  
to act along the same line until        
exhausted or until the end has  been    
attained4 ,consequently1 it is a proven 
fact that often serious  mathematical   
and philosophic problems have been      
solved by the mind long  after any      
conscious effort to solve them had      
ceased4 ,often solutions  have been     
arrived at during sleep4 ,many such     
cases might be cited1 but  the          

phenomenon is now so common that almost 
every one can cite some  experience in  
his own life that will substantiate the 
claim4    ,there is no doubt but that   
these phenomena are evidences of a      
reflexive  development in the intellect4
 ,the time will come undoubtedly1 and   
necessarily so if the intellect is to   
give way to a higher faculty1  which    
shall be as much above the intellect in 
its grasp of things  as the intellect is
 now above the simple consciousness of  
the lower  animal1 when quite the       
entirety of our intellectual processes  
will  become automatic or               
self-performing4 ,what then remains of  
the egoic  schematism1 after its        
transmutation or elevation as the organ 
of  the intuitional consciousness will  
be utilized as the organ of the         
,thinker's involuntary cognitive        
processes4 ,this will mean that all  of 
that laborious ideation which is now the
 abstract thought of the  ,thinker will 
be performed automatically1 leaving the 

higher aspect  of the egoic             
consciousness free to conceptualize or  
intuitograph  the intuitions4           
,perceptualization then will be replaced
 by  conceptualization4 ,this latter    
will occupy about the same status as the
  former does now4 ,and necessarily1    
perception will become more complex4    
,in other words1 while we now perceive  
simple percepts which are again         
arranged into concepts making a         
composite picture of the object1 we     
shall then be taking in the composite   
picture of the object at first  hand1   
thereby dispensing with the rather slow 
process of perception as  it now        
operates4 ,we shall still be perceiving1
 but what we perceive  will be concepts 
rather than percepts1 as at present4    
,the increased powers of intellection   
gained as a result of the  increscent   
automatism in the intellect1 the        
flowering forth of  the intuitive       
faculty and the general enhancement of  
the intellect  throughout all its       

processes will enable it to entertain   
concepts or  composite picture of things
 just as readily and as perfectly as it 
can  at present deal with a single      
percept4 ,concepts will be replaced by  
super-concepts or intuitographs4        
,increased perspicacity will enable the 
,thinker to manipulate the concepts and 
intuitographs with the same ease  and   
readiness and withal the mind will have 
attained unto an almost  unrealizable   
freedom in its search after truth4      
,the outcome of this new adjustment     
which1 of course1 will not spring  up at
 once1 but by insensible degrees1 will  
be the clarification and  unification of
 our knowledge4 ,it will mean also the  
simplification  of it2 the obviation of 
diversities of opinions1 the springing  
up of  a new and winnowed system of     
philosophy which shall be the true one2 
further1 it will imply the lessening of 
the probability of error in our         
judgments and conclusions2 the removal  
of illusion to a much larger  degree    

than to-day is possible and the         
realization by every one of  something  
of the essence of things1 of causes and 
effects1 of actions1  operations1       
natural forces and laws2 in fact1 a     
condition of mind  which will present to
 the consciousness the simple truth     
above every  conceivable phase of kosmic
 life which may come within the scope of
 the  ,thinker's observation4    ,the   
further implications of this view are   
that there is a difference  between the 
,thinker and the intellectuality4 ,the  
,thinker is eternal  and partakes1      
therefore1 of the very essence of       
primordial originality  while the       
mentality is an artificial process1 the 
resultant of the  adaptation of the     
,thinker's consciousness to his         
vehicular contrivances  of objective    
cognition and the interplay of his life 
among them4    ,if the appearance of a  
choppy sea disturbed by the passage of a
 brisk  breeze over its surface be      
imagined1 a similitude of the great     

ocean  of life may be envisaged4 ,the   
wavelet crests symbolize the egos2 the  
base of the wavelet which is one with   
the great sea of water represents  the  
,thinker which is one with the divine   
life and consciousness of  the kosmos4  
,just as wavelet crests are continually 
springing up and  falling back into the 
sea1 so are egos continually being cast 
forth  and reabsorbed into the          
universality of life only to be recast1 
as a  wavelet crest or ego1 upon the    
surface of the moving ocean of life4    
,and so1 in this respect1 the universum 
of life and consciousness  which are    
essentially one is in a constant state  
of ever-becoming1  un-becoming and      
re-becoming4    ,another implication is 
that1 on account of the diversity and   
complexity  of the means of contact with
 the external world1 it is not possible 
for  the ego to arrive at more than a   
fragmentary understanding of even the   
latent geometrism of life1 mind and     
materiality4 ,in our examination  of the

 sensuous world1 we are very much like  
the three blind men set  to examining an
 elephant4 ,one set to scrutinizing his 
trunk by means  of his sense of feeling4
 ,when asked for his judgment as to what
 the  elephant was he declared it was a 
snake2 a second who began with the  legs
 found it to be like huge pillars2 and a
 third who caught hold of  the          
elephant's tail and declared the        
elephant to be like a rope4 ,each  one  
of the blind men described what he was  
able to perceive4 ,to each  what he felt
 was all there was upon which he could  
render judgment4 ,and  so1 artists1     
philosophers1 mathematicians1 musicians1
 mechanicians1  religious seers1        
metaphysicians and all other types of   
mind1 are  just so many blind men set to
 the examination of an elephant1 or the 
sensuous world4 ,each one confidently   
believes his view to be correct2  each  
one is satisfied with the deliveries of 
his senses4 ,yet no one  of them is     
wholly correct1 no one of them has seen 

every phase and  aspect of the problem4 
,does it not1 therefore1 appear to be   
the more  reasonable and urgent that the
 view which synthesizes the judgments of
  all the possible examiners thereby    
constructing a composite idea of  the   
entire mass of judgments is the more    
reliable and the more correct8          
,referring again to the dual            
intelligence1 the ego and the ,thinker1 
which together constitute man1 it is    
deemed necessary1 in order to  present  
the concept of this duality to the mind 
of the reader in the  way that shall    
enable him easily to recall it1 to      
designate the egoic  intelligence as the
 egopsyche1 and the ,thinker's          
intelligence as the  omnipsyche4    ,the
 egopsyche is the ,i-making faculty1 the
 faculty of  self-consciousness and the 
synthesis of all those psychic states   
and  functions known as the intellect or
 mind and includes the ethical  aspect  
of man's nature4 ,the omnipsyche is the 
organism of kosmic  consciousness1 the  

space-mind1 or man's higher self and    
that which  connects with or allies him 
to all life2 it is the basis of human   
unity and of unity with divinity1 just  
as the egopsyche is the basis  of       
separation and individuality2 it is the 
organ of direct and  instantaneous      
cognition and the permanent essence     
which has persisted  through every form 
which the being1 man1 has ever assumed  
and through  every stage of human       
evolution4 ,in it are stored up the     
memories of  the ,thinker's past1 the   
secrets of life1 mind1 being1 reality1  
and the  history of life from the       
beginning2 in it also the plan of action
 for  the future of the life-wave as it 
passes from plane to plane1 from  stage 
to stage1 and from form to form4 ,it is 
the spark from the flame  that is never 
quite free from its source2 it is the   
continuous spark1  the prolonged ray    
which does not go out and cannot be     
extinguished4 ,it  is that in man which 
when full union therewith has been      

attained makes  him a god in full       
consciousness4    ,the omnipsyche is    
really a neglected and overlooked factor
 in the  doctrine of evolution4         
,evolutionists1 while they claim life to
 be  continuous and that man has come   
through all the kingdoms of nature in   
succession and has spent millions of    
years in the perfection of his  various 
organs1 faculties and stages of         
consciousness1 make no ample  allowance 
for what is in reality the basal element
 in evolution--a  continuous1           
persisting1 permanent life-force which  
does not lose its  identity from the    
beginning to the end of the process4    
,this fact--that  that spark of life    
which set out upon the evolutionary     
journey as a  moneron has glowed        
steadily from that stage to manhood1    
maintaining  meantime its original      
purposiveness and intent--seems to be   
the most  obvious consideration of the  
whole doctrine1 yet it has been more or 
less completely ignored4 ,the elementary

 requirements of evolution  would seem  
to establish clearly the necessity for  
some such eternally  persisting         
principle as the omnipsyche which is    
capable of such subtle  adaptations to  
every conceivable form of life and in   
which should be  gathered up the        
evolutionary results of every           
life-cycle4 ,for this  purpose the      
omnipsyche or unifying principle in man 
was designed  from the beginning and it 
is that which constitutes the basis of  
his  intellectual nature while in a far 
larger sense it is the divinity in  man 
himself4 ,it is indeed strange that so  
important a factor as the  omnipsyche   
should have been omitted by             
evolutionists4 ,yet it can be  accounted
 for upon the grounds of the purely     
mechanistic character  of all           
intellectual attempts at solving the    
problems of vital  manifestations4 ,but 
so long as men rely upon mechanical     
explanations  of such phenomena so long 
will they be prone to overlook the very 

essentialities of the problems which    
they devoutly wish to solve4  ,the      
continuity of the physical germ-plasm of
 the human species1#bg  now quite       
generally admitted1 would suggest1 it   
seems1 an analogous  condition to the   
continuity of the psychic plasm called  
the omnipsyche1  the only difference    
being that the omnipsyche is an         
intelligent  factor while the physical  
plasm is a medium of transmission though
  non-intelligent4 ,the omnipsyche is1  
therefore1 the psychic reservoir  of    
evolution into which are stored the     
transmuted psychics of moneron1  amoeba1
 jellyfish and every other form which it
 has ensouled and acts  as the storeroom
 of man's psychic operations as well as 
the source of  his intellectuality4     
,we turn now from the study of a sketch 
of the mechanism of man's  consciousness
 which gives at its best only a         
fragmentary view of the  universe of    
spatiality to a consideration of space  
itself in the light  of its             

interrelational bearings upon the       
question of intellectuality4      #bg   
,see ,the ,germ ,plasm2 ,a ,theory of   
,heredity1 by ,a4 ,,weissman4    ,in the
 chapter on the ,genesis and ,nature of 
,space we have1 in  tracing out the     
engenderment of space1 proved it to be  
basically one  with matter 7and indeed  
the progenitor of matter71 also with    
life and  consciousness4 ,further1 it   
has been shown that all the             
characteristics  of materiality are due 
to the adaptation of consciousness to it
  and that out of this adaptation grew  
the intellectuality4 ,a close           
approximation to this view was          
maintained by ,,kant when he discovered 
that our faculty of thinking or the     
intellect only finds again in  matter   
the mathematical order or properties    
which our faculty of  perceiving or     
consciousness has deposed there4 ,it    
appears1 therefore1  that when the      
intellect approaches matter or          
spatiality it finds always  a ready     

yieldance to its demands simply because 
intellectuality has  previously         
established therein the delineation or  
map of the path  over which it          
necessarily must traverse in its        
examination of the  object of its       
pursuit4 ,in other words1 the kosmic    
mind in engendering  materiality and    
spatiality has set up therein a kosmic  
order or  geometrism4 ,both motor and   
intellectual progress1 therefore1 can be
  made through the world of spatiality  
because of the immanence of this  kosmic
 geometrism which lies latent in the    
very fabric of the world of  substance  
fashioning both the character and the   
nature of the intellect  as well as of  
space itself4 ,so that there is a       
perfect congruity  subsisting between   
spatiality and intellectuality4         
,accordingly it is  impossible for      
either one or the other to transcend the
 grim grasp of  the mathematical order  
which binds them in such lasting and    
fundamental  agreement4                 

,extra-spatiality may degrade itself    
into spatiality1  and indeed in the very
 nature of the case1 does so degrade    
itself1  yet spatiality can never raise 
itself beyond the limits set by its     
engendering parent4 ,materiality may    
become more and more spatialized  and   
consciousness more and more             
intellectualized1 but they must proceed 
hand-in-hand one not superseding the    
other4    ,being the essence of the     
natural geometry which is everywhere    
immanent  in the universum of matter1   
space becomes an organized and ordered  
extension1 in fact is the totality of   
such organized and ordered  extension1  
which conforms to the latent geometrism 
the engenderment of  which it is the    
sole cause in the last analysis4 ,does  
it not appear  then that all that mass  
of artificial geometry which has sprung 
up as a result of departures made from  
this natural geometry is  utterly       
baseless and most certainly lacking in  
the kosmic agreement  which spatiality  

lends to our primary conceptions8 ,of   
course1 it is  admittedly possible to   
devise certain conventional forms of    
logic and  endow them with all the      
evidences of a rigid consistency but    
which1  because of their purely         
artificial character1 will fall far     
short  of any real conformity to the    
potential geometrism which has been     
established in spatiality4 ,and this    
fact is of utmost significance for  all 
those who seek to find justification    
either logically or naturally  for the  
existence of a multi-dimensional quality
 in space2 for1 if a  clear1            
discriminative conception as to the     
categorical relationship1  each to each1
 of the two kinds of geometry be carried
 in mind1 it will  not be easy to       
confound them neither will it be        
difficult to discern  where the one ends
 and the other begins4    ,now1 the     
fourth dimension and the entirety of    
those mathematical  speculations        
touching upon the question of           

hyperspace1 dimensionality1             
space-curvature and the manifoldness of 
space are purely conventional  and      
arbitrary contrivances and do not meet  
with any agreement or  authority in the 
native geometrism which we find inhering
 in space  and which the intellect      
recognizes there4 ,this conclusion seems
 to be  obvious for the reason that1 in 
the first place1 the non-,euclidean     
geometries have been constructed upon   
the basis of a negation of  the latent  
geometrism of space and intellectuality2
 and if so1 is it  reasonable to expect 
that either they or any of their        
conclusions  should accord with the     
nature of that form of geometry so      
admirably  delineated by ,,euclid8      
,obviously not4 ,it is a matter of      
historical  knowledge that the whole of 
the artificial non-,euclidean geometries
  consists of those purely conventional 
results which investigators  arrived at 
when they denied or controverted the    
norms supplied by the  natural geometry4

 ,when metageometricians found that they
 could neither  prove nor disprove the  
,euclidean parallel-postulate they then 
set  upon the examination of idealized  
constructions which negatived the       
postulate4 ,the results1 thus obtained1 
although self-consistent enough1  were  
compiled into systems of geometry which 
naturally were at variance  with each   
other and with this inherent geometrism 
which is found in  spatiality and       
answered to by the intellect both       
normally and logically4    ,furthermore1
 there is another consideration which to
 us seems to be  equally if not more    
forbidding1 in its objections to the    
cordination  of the two systems of      
geometry1 and that is the fact that the 
geometry  of hyperspace is denied the   
corroborative testimony of experience   
and this is true of practically the     
whole of its data4 ,indeed1 there  is   
perhaps no single element in its entire 
constitution which claims  the authority
 of experience4 ,this is undoubtedly the

 weakest point in  the structure of the 
hyperspatial geometries4 ,contrarily1   
such is not  the case with the natural  
geometry2 for1 in this1 the intellect in
  retracing its steps over the path laid
 out by that movement which has  at the 
same time created both the intellect and
 spatiality1 finds an  orderly and      
commodious arrangement into which it    
naturally and easily  falls4 ,so exact  
is this agreement of the intellect with 
the kosmic  order that if it were       
possible to remove the whole of         
spatiality and  materiality there would 
still be left the frame work which is   
this  latent geometrism of kosmogenesis4
 ,but the fact that the intellect       
naturally fills all the interstices of  
materiality and spatiality1  fitting    
snugly into all of them as if molded for
 just that purpose1 by  no means        
warrants the assumption that it would or
 does also fit the  engendering factor  
which has created these interstices4    
,the frame work1  the order or the      

geometrism of the kosmos has been       
established by life  acting consciously 
upon the universum of materiality4 ,and 
in order to  establish this geometrism  
life had to be mobile1 active1 creative4
 ,it  could not remain static1 immobile1
 and accomplish it4 ,being mobile1      
dynamic1 creative1 it passes on4 ,it is 
like a fashioning tool which  the       
cabinet makers use in cutting out       
designs upon a piece of wood4 ,it       
moves1 and keeps moving until the design
 is finished1 and then it is  ready for 
more designing4 ,life is like that4 ,it 
cuts out the designs  in materiality1   
fashions the form1 molds the material1  
and passes on  to other forms4 ,the     
intellect fits into these designs       
gracefully4 ,but  what it finds is not  
life itself1 only the design which life 
has made4  ,hence1 as there is neither  
an empirical spatiality nor materiality 
in  conformity with which the artificial
 geometry of the analyst may be  said to
 exist1 and as it may not be said to    

conform to the path which  life has made
 in passing through either of these1 it 
is absurd to  predicate it upon the same
 basis as the natural geometry4 ,and so1
 we  are forced1 in the light of these  
considerations to deny the validity  and
 hence the acceptability of the         
non-,euclidean geometries as either     
reasonable or warrantable substitutes   
for the ,euclidean1 and denying  which  
we also formally ignore the claims of   
the fourth dimension1 as  mathematically
 designed1 to any legitimate anchorage  
in either our  vital or intellectual    
movements4    ,it has been shown that   
the flow of life1 as it describes that  
movement  which we call evolution1      
engenders simultaneously and            
consubstantially  spatiality1           
materiality and intellectuality1 and    
these1 in turn1 the  natural order or   
geometrism everywhere immanent in the   
universe2 and  that automatically1 one  
out of the other and each out of the    
all1  these constitute the totality of  

kosmic fundamentals4 ,also we have      
sketched the mechanism of man's         
consciousness and discovered how1  in   
its evolutionary development it has     
divided into two aspects1  the          
egopsychic and the omnipsychic1 and     
these two factors ally him  definitely  
and adequately to the world of the      
senses and to the world  of             
supersensuous cognitions4 ,and thus we  
have cleared up some of the             
misconceptions which had to be          
confronted and made more easy the       
approach to the central idea1 thereby   
conserving the substantiating  influence
 which a general and more comprehensive 
view of the whole  would naturally give4
    ,the totality of kosmic order is    
space4 ,it is circumscribed by an       
orderless envelope of chaos just as the 
germ of an egg is surrounded  by the    
egg-plasm4 ,the organized kosmos is the 
germ1 kernel or central1  nucleated     
mass1 enduring in a state of becoming4  
,involutionary  kathekos or primordial  

chaos is the egg-plasm which nourishes  
the  germ or the kosmos and is that out 
of which the germ evolves4  ,kathekos or
 chaos is the unmanifest1 unorganized1  
unconditioned1  unlimited and           
undifferentiated plasm4 ,space is the   
manifest1  limited1 finite1 organized   
germ that1 feeding upon the enveloping  
chaos1 exists in a perpetual state of   
alternate manifestation  and            
non-manifestation--appearing1           
disappearing and reappearing            
indefinitely4    ,the appearance of the 
kosmos as an orderly elaboration of the 
involutionary phase of kosmogenesis1 in 
so far as kosmic order may  be said to  
be an accomplished fact1 marked the     
turning point in that  procedure whose  
function it was to make manifest a      
universe possessing  certain definite   
characteristics of orderliness2 but the 
kosmos1 as it  now stands1 may not be   
thought of as having attained unto a    
state of  ultimate orderliness4 ,the    
idea meant to be conveyed is that       

between  the point of becoming and the  
actually pyknosed1 or solidified stage  
in the process of creation there is a   
more or less well defined line  of      
demarkation cutting off that which is   
spatiality from that which  is          
non-spatiality4 ,beyond the limits of   
spatiality is an absence of  geometric  
order4 ,here geometry breaks down1      
becomes impotent1 because  it is an     
intellectual construction2 at least1 it 
is not so apparent  as in the manifested
 kosmos4 ,it is a state about which it  
is utterly  futile to predicate         
anything2 because no words can describe 
it4 ,the  most that may be said is that 
it is absence of geometric order as it  
inheres in space4 ,and if so1 all those 
movements comprehended under  the       
general notions of spatiality1          
materiality1 intellectuality and        
geometricity have both their extensive  
and detensive or inverse  movements     
nullified in their approach to it4      
,involutionary ,kathekos1  therefore1   

may be said to be the primordial        
wilderness of disorder  which outskirts 
the well laid-out and carefully planned 
garden of  the spatial universe4 ,we may
 excogitate upon some of the obvious    
functions of this kathekotic            
world-plasm2 but in doing so we must    
leave  off all attempts at a description
 of its appearance1 its magnitude1      
extent or other qualities1 and think    
only of its kosmic function4  ,we cannot
 say that there is back of it a         
spatiality nor can we say  that it is a 
spatiality2 for whatever may be its     
extent or volume1  it suffices that it  
may not be said to be space4 ,it is     
chaos4 ,space  is order1 organization1  
geometricity4 ,it cannot be said that   
there  is a latent geometrism in chaos2 
because geometric order is found  only  
in spatiality and is that which         
distinguishes spatiality from           
kathekosity or non-spatiality4 ,chaos is
 the lack of spatiality4 ,this1  of     
course1 implies that it is impenetrable 

to the intellectuality or to  vitality4 
,all inverse movement such as is        
discovered as taking place  in          
spatiality and which results in the     
phenomenalization of space runs  aground
 when it strikes against the rock-bound 
coast of kathekosity4  ,we can only say 
that it is both the point of origin for 
the evolving  universe of life and form 
and its terminus4 ,it is the nebulosity 
out of  which the whole came and into   
which all is ultimately occluded4    ,a 
great and far-reaching error is made in 
all our thinking with respect  to the   
kosmogonic processes when we postulate  
the complete absorption  of chaos as an 
early act of kosmogony4 ,customarily1 we
 think of kosmic  chaos as a primordial 
condition whose existence was done away 
as soon  as the universe came into      
active manifestation4 ,this because it  
has  been exceedingly difficult1 if not 
quite impossible1 for those whose       
privilege it was to determine the trend 
of philosophic thought to free          

themselves from the bondage of a dogma  
which owed its existence to  a          
traditional or legendary interpretation 
of facts that ought never  have been so 
interpreted4 ,chaos ,,is and ,,ever     
,,shall ,,be1 so long as the  universe  
itself lacks completion1 fullness or    
perfection in purpose1  extent and      
possibility4 ,it is undoubtedly being   
diminished1 however1  in proportion as  
the kosmos is approaching absolute      
perfection4 ,and  when the last vestige 
of chaos disappears from the outerskirts
 of  the maturing kosmos there shall    
appear a glorified universe of          
indescribable qualities4    ,space being
 a perception a priori cannot be        
determined wholly by  purely objective  
methods4 ,the yard-stick1 the telescope 
and the  light-year are objects which   
belong exclusively to the phenomenal and
  with them alone never can we arrive at
 a true conception of the nature  of    
space4 ,we can no more demonstrate the  
nature of space by the use  of objective

 instruments and movements than we can  
measure the spirit  in a balance4       
,certainly1 then1 it cannot be hoped    
that by taking the  measurement of      
space-distances in light-years1 or      
studying the nature  of material bodies1
 we shall be able to fathom this most   
objectively  incomprehensible and       
ineluctable thing which we call space4  
,it is such  that every ,thinker must1  
in his own inner consciousness1 come    
into the  realization of that awfully   
mysterious something which is the nature
  of space both as to existence and     
extent by his own subjective efforts    
unaided1 uncharted and alone4 ,when we  
measure1 weigh1 apportion and  otherwise
 try to determine a thing we are dealing
 with the phenomenal  which is no more  
the thing itself than a shadow is the   
object which  casts it4    ,what does it
 matter that metageometricians shall be 
able to  demonstrate that space exhibits
 itself to the senses in a four-or      
n-dimensional manner8 ,granting that    

they may be able to do this1  if merely 
for the sake of the discussion1 when    
they have finished1  it will not be     
space that they have determined1 but the
 phenomena  of space1 its arborescence1 
while space itself remain indeterminate 
and unapproachable by phenomenal        
methods4 ,if there are curvature1       
manifoldness and n-dimensionality these 
are not properties of space1  but of    
intellectuality in its cultured state   
and when it is1 therefore1  removed from
 the native state of conception4        
,scientists may be able to  weigh the   
human body1 count every cell1 name and  
describe every nerve1  muscle and fiber2
 they may even be able to know it in    
every conceivable  part and from every  
physical angle and relationship1 and yet
 know  nothing of the life which        
vitalizes that body and makes it appear 
the  phenomenal thing that it is4 ,so it
 is not by instruments which man may    
devise that we shall be able to         
determine the true nature and purpose   

of space4 ,we must adopt other methods  
and means and assume other  angles of   
approach than the purely objective in   
order to comprehend  space which1 being 
the sole inherent aspect of             
consciousness1 can be  understood best  
by applying the measures which the      
latter provides for  its understanding4 
,it would appear1 therefore1 that the   
best study of  space is the             
consciousness itself1 knowing which we  
shall know space4    ,the universum of  
space1 including the phenomenal         
universe1 and its  relation to          
consciousness may be likened to a       
conical funnel whose  base represents   
the phenomenal world of the senses and  
whose apex or  smallest point represents
 ultimate reality4    ,in ,figure #bj1  
we have endeavored to symbolize         
graphically this  conception of space4  
,the base marked ,sensorium represents  
the  sensible world4 ,that marked       
,realism symbolizes the ultimate  plane 
of reality1 the inner essence of the    

world1 the plane of                     
things-in-themselves4    ,the cone      
arising from the base sensorium         
symbolizes the objective  world as      
compared with consciousness2 the        
subverted cone1 with apex in  the       
sensorium1 represents the evolving human
 consciousness4    ,the successive bases
 have the following symbology3          
,self-consciousness1  ,communal         
,consciousness1 ,mikrocosmic            
,consciousness1 ,makrokosmic or         
,universal ,consciousness1 the ,plane of
 the ,space-,mind ,consciousness1       
,divine ,consciousness1 ,kathekotic     
,consciousness1 or the ,plane of ,final 
,union with the ,manifest ,logos4       
,self-consciousness is that form of     
consciousness which enables the  ego to 
become aware of himself as distinguished
 from other selves or  the ,not-self2   
the ,omnipsychic or ,communal           
,consciousness is that form  of         
consciousness from which arises the     
realization by the ,thinker of  his     

oneness with all other thinkers and with
 other forms of life4  ,mikrocosmic     
consciousness denotes a still higher    
form of consciousness1  as that which   
enables the ,thinker to become conscious
 of his living  identity with the life  
of the world or the planet on which he  
lives4  ,it represents a stage in the   
expansion of consciousness when it      
becomes one with the consciousness of   
the planet upon which it may  be        
functioning4 ,makrokosmic consciousness 
accomplishes the awareness  of the      
,thinker's unity with the life of the   
kosmos or universe4 ,the  space-mind and
 the consciousness which constitutes it 
enable the  ,thinker to comprehend the  
originality and the terminality of      
kosmic  processes4 ,it is archetypal so 
far as the life-cycle of the universe   
is concerned because the beginning1 the 
intermediate portion and the  ending of 
the kosmos are encompassed within it4   
,divine consciousness  is that form of  
consciousness which arises upon the     

unification of the  ,thinker's          
consciousness with that of the manifest 
deity2 it is1 in  fact1 omniscience4    
,the kathekotic consciousness belongs to
 the ultimate  plane of reality2 to     
kosmic origins and chaogeny1 and        
therefore1  pertains to the plane of    
non-manifestation4    ,illustration3    
,,fig4 #bj4--,kosmos and ,consciousness 
,the implications are that in comparison
 with the sensorium1 the  ,thinker's    
consciousness is a mere point in space4 
,it is1 in reality1  so small and       
insignificant that the extensity of the 
physical world  or universe seems       
unlimited1 unfathomable in meaning and  
infinite in  extent4 ,but as his        
consciousness expands1 as it passes1 in 
evolutionary  succession from one plane 
of reality to another and higher one1   
the illimitability1 the                 
incomprehensibility and infinity of the 
universe grow ever smaller and smaller1 
until the plane of divine  consciousness
 is reached4 ,then the previously       

incomprehensible dwindles  into         
insignificance1 lost in the real        
illimitability1 infinity and            
unfathomability of consciousness itself4
 ,kosmic psychogenesis1 as  exhibited   
and specialized for the purposes of the 
evolution of the  ,thinker1 can have no 
other destiny than the flowering forth  
as the ne  plus ultra of manifestation  
which is nothing short of unification   
with the highest form of consciousness  
existent in the kosmos4    ,it is not to
 be considered really that the scope of 
space is diminished  but that the       
growing1 expanding consciousness of the 
,thinker will  so reduce the relative   
extension of it that illimitability will
 be  swallowed up in its extensity4     
,consciousness1 in becoming infinite  in
 comprehension1 annihilates the         
imaginary infinity of space4            
,accordingly1 that which now appears to 
be beyond mental encompassment          
undergoes a corresponding diminution in 
every respect as the  consciousness     

expands and becomes more comprehensive4 
,the mystery  of space decreases as the 
scope of consciousness increases4 ,as   
the  ,thinker's consciousness expands   
the extensity of the manifested         
universe decreases4 ,thus the mystery of
 every aspect of kosmic life  lessens1  
and fades away1 as the intimacy of our  
knowledge concerning it  becomes more   
and more complete4 ,there is no mystery 
where knowledge is4  ,mysteriousness is 
a symbol of ignorance or                
unconsciousness1 and that  which we do  
not understand acts as a ,flaming ,sword
 keeping the way of  the ,temple of     
,reality lest ignorance break in and    
despoil the treasures  thereof4         
,figure #ba is a graph showing a        
sectional view of consciousness on  all 
planes represented as seven concentric  
circles4 ,this describes  the analogous 
enveilment of the consciousness when it 
ensouls a  physical body or when bound  
to the purely objective world of the    
senses4 ,the overcoming of the barriers 

of reality1 represented by  the         
circumscribing circles is the work of   
the ,thinker who is forever  seeking to 
expand and to know4 ,for only at its    
center1 as symbolized  here1 is the     
consciousness at one with the highest   
aspect of kosmic  consciousness and     
there alone is the mystery of space     
despoiled of its  habiliments4          
,illustration3 ,,fig4 #ba4--,septenary  
,enveilment of ,consciousness           
,accordingly1 as consciousness or the   
,thinker is more and more divested  of  
carnal barriers and illusions there     
develops a gradual recognition  of the  
unitariness of spatial extent and       
magnitude2 there arises the  certain    
knowledge that space has but one        
dimension and that dimension  is sheer  
extension4 ,the ,thinker's sphere of    
awareness is represented  as if it      
begins as a point in space and develops 
into a line which  divides into two     
lines1 the boundaries of the space      
cones4 ,thus it may  be perceived that  

the ancients had a similar conception in
 mind when  they symbolized kosmogenesis
 with the dot 7471 the line1 and the    
circle  with diameter inscribed1 which  
together represent the universe in      
manifestation4    ,we realize the       
impossibility of adequately depicting   
the full  significance of the inverse   
ratio existing between the extensity of 
space and the increscent inclusivity of 
consciousness by means of  graphs2 for  
neither words nor diagrams can portray  
the scope and  meaning of the conception
 in its entirety4 ,yet they aid the     
intellect  to grasp a ray of light1 an  
intimation of what the ,thinker sees and
  understands interiorly4    ,in this   
connection it is interesting to note the
 function of the  ideal in the evolution
 and expansion of consciousness4 ,the   
ideal has  no perceptual value2 it has  
no status in the world of the senses4   
,it  is unapproachable either in thought
 or action1 and therefore1 lies  beyond 
the grasp of both the intellectuality   

and the vitality4 ,it is  indescribable1
 inconceptible and searchless2 for the  
moment that we  describe1 define1 or    
approach the ideal1 either              
intellectually or  vitally1 in that     
moment it ceases to be ideal1 but       
actual4 ,it flees  from even the        
slightest approach2 it never remains the
 same2 it cannot  be attained1 at least 
its attainment causes it to lose its    
idealty4 ,it  is then no longer the     
ideal4 ,it is like an ignis fatuus2 the 
closer  we come to it the farther away  
it recedes4 ,it hangs suspended before  
the mind like the luscious grapes which 
hung before the mouth of the  hungry    
,tantalus4 ,as the grapes and the water 
receded from his reach  at every effort 
he made to seize them so the ideal      
remains eternally  unseizable and       
unattainable4 ,whatever1 therefore1 is  
in our thought  processes1 or in our    
knowledge1 that may be said to be ideal1
 does  not really exist4 ,the ideal is a
 phantom growing out of the nature and  

essence of the intellectuality4 ,its    
purpose is to lead merely the mind  on2 
to allure it1 to tantalize1 and compel  
it to grow by exertion1 by  the struggle
 to attain1 by the desire to overcome4  
,in this respect1 it  serves well its   
function in the economy of intellectual 
evolution4 ,it  is a mysterious aspect  
of the original and eternal desire to   
live which  is the kosmic urge present  
in all organized being and has its roots
  hidden in the divine purpose of       
creation4    ,idealized constructions1  
then1 are like ,arabian feasts conjured 
up by  a famishing mentality4 ,they are 
like the dreams of a starving man in    
which he actualizes in phantom-stuff the
 choicest viands in abundant  supply for
 his imaginary delectation4 ,the mind   
that is satisfied never  idealizes1     
never makes an idealized construction4  
,it is only when an  aching void is     
felt1 when a longing for the realization
 of that which  it has not arises within
 itself1 when a feeling of distinct     

lack1 a  want1 a hankering after        
something not in its reach1 takes       
possession of  the mind that it begins  
to idealize4 ,that is why some minds are
 without  ideals4 ,it is because they   
are satisfied with what they have and   
can  understand4 ,they feel no hungering
 for better and grander things2 they    
have no desire to understand the unknown
 and the mysterious2 hence they  do not 
idealize2 they make no attempt to       
represent unto themselves  a picture of 
that which is beyond them4 ,such minds  
are dormant1  hibernant1 asleep1        
unfeeling and unresponsive to the divine
 urge4    ,but the ideal is neither     
obtainable objectively nor subjectively1
  neither phenomenally nor really1 so   
that when we come upon the ideal  in our
 mode of thinking we have arrived not at
 a finality or end1  but at that which  
is designed to lead us on to something  
higher1 to  nobler accomplishments and  
more extensive conquests4 ,when we have 
devised idealized constructions1        

therefore1 we should not therewith be   
content but should scrutinize them1     
examine and study them for their        
implications2 for thereby we may        
discover the path and the guide-posts   
to a new domain1 a new ideal1 following 
which we shall1 in time1 come  to a     
point in our search for the real where  
the fluxional is at a  minimum2 we shall
 reach that something which will admit  
of no further  struggle--the last chasm 
between the phenomenal and the real--and
  standing on the bridge1 consciousness1
 which engages the twain1 shall  have a 
complete view of that ,sacred and       
,imperishable ,land of ,kosmic  ,realism
 where like a fleeting cloud of sheerest
 vapor shall be seen the  phantom-ideal 
deliquescing and disappearing in the    
cold1 thin air of  the real and the     
eternal4    ,since space is judged to be
 infinite by the intellect occluded in  
such clouds of illusion and hampered by 
such constrictive bonds of  limitations1
 as it now endures1 we have no right to 

conclude that the  concept of infinity  
would still linger before the mind's    
eyes when  the illusionary veil is      
removed2 in fact1 there is ample reason 
to  believe1 nay for the assertion1 that
 the recession of the veil will  reveal 
just the opposite of this illusion1     
namely that space is finite1  and even  
bounded by the fringe of chaogenetic    
disorderliness4 ,either  we perceive the
 real or we do not2 either the pure     
thingness of all  objects can be        
perceived or it cannot be perceived4 ,if
 not1 granting  that there is such a    
thing as the real1 it must be within the
 ultimate  range of conceivability4 ,it 
also seems reasonable that realism      
exists  somewhere1 and if so1 must be   
sought in a direction inverse to that in
  which we find the phenomenal and the  
approach thereto must necessarily  be   
gradual1 continuous and direct and not  
by abrupt breaks1 by twists  and turns4 
,the phenomenal must lie at the terminus
 of the real1 and  vice versa4 ,so that 

by retracing the path blazed out by the 
real in  coming to phenomenalization we 
shall perhaps find that which casts our 
shadowy world1 just as by tracing a     
shadow in a direction inverse to  that  
in which it extends we may find the     
object which projects it4    ,it is not 
out and beyond that we shall find the   
end of space2 it is  not by counting    
tens of thousands of light-years that   
the supposed  limits of space shall be  
attained4 ,the path of search must      
project  in an opposite direction--not  
star-ward but ,thinker-ward1 toward the 
subtle habitation of the consciousness  
itself4 ,we err greatly when  we think  
that by measuring distances we shall    
encompass space2 for  that which we     
measure and determine is but the clouds 
caused by the  vapor of reality4 ,it is1
 therefore1 not without1 but within1 in 
an  inverse direction that the search   
must proceed4 ,going back over the      
life-stream1 beginning where it strikes 
against the shores of solid             

objectivity1 deeper and deeper still1   
past the innermost mile-stone of  the   
self-consciousness1 back into the very  
heart of the imperturbable  interior of 
being where the ,thinker's castle opens 
its doors to the  ,great ,kosmic ,self1 
from that open door-way we may step out 
into that  great mystery of             
space--limited1 yet not limited1        
multi-dimensioned1  and yet having only 
one dimension1 veritably real and       
fundamental1 the  ,father-,mother of all
 phenomena4 ,here the great mystery of  
mysteries  is revealed as the citadel of
 the universal and the ultimate real4   
,in this citadel1 the plane of kosmic   
consciousness1 space loses its          
spaciousness and time its timeliness1   
diversity its multiplicity and  oneness 
alone reigns supreme4    ,but the       
movement towards the center and         
circumference of space1 after  this     
manner1 requires aid neither from the   
notion of space-curvature  nor that of  
the space-manifold1 except1 indeed1 only

 in so far as a  state of consciousness 
or a degree of realism may be said to be
 a  tridimensional manifold4 ,the       
feeling that space is single-pointed1   
and yet ubiquitously centered1 has been 
indulged by mathematicians  and others  
in a more or less modified form2 but    
they have imagined it  in the terms of  
an indefinite proceeding outward until  
in some manner  unaccountable alike to  
all we come back to the point of origin4
 ,it  has been expressed by ,,pickering 
when he says that if we go far enough   
east we shall arrive at the west2 far   
enough north we shall come to  the      
south2 far enough into the zenith we    
shall come to the nadir4  ,but this     
conception is based upon a notion of    
space which is the  exclusive result of 
mathematical determinations and subject 
to all the  restrictions of mathetic    
rigorousness4 ,it requires that we shall
 allow  space to be curved4 ,this we    
decline to do for the reason that it is 
both unnecessary and contrary to the    

most fundamental affirmations of  the a 
priori faculty of the ,thinker's        
cognitive apparatus4 ,it would  seem to 
be necessary only that we should extend 
our consciousness  backward1 revert it  
into the direction whence life came to  
find that  which we seek4 ,by extension 
of consciousness is meant the ability to
  function consciously upon the various 
superkosmic spheres or planes  just as  
we do on the physical4 ,yet it should be
 quite as easy to  devise an idealized  
construction which would imagify the    
results of  this ingressive movement of 
the consciousness as to represent the   
results of a progressive outward        
movement star-ward4 ,having done so the 
examination of them could be conducted  
along lines similar to those  followed  
in the scrutiny of objective results4   
,what would it mean to the ,thinker if  
he were able to identify his            
consciousness with the ether in all its 
varying degrees2 what would it  mean if 
he were able to identify his            

consciousness with life and with  the   
pure mind-matter of the kosmos2 and     
lastly1 with the spiritual  essence of  
the universe8 ,what if his various      
vehicles of awareness were  available   
for his purposes of cognition8 ,what1   
indeed1 if he could  traverse           
consciously the entire gamut of realism 
and consciousness  from man to the      
divine consciousness8 ,does it not      
appear reasonable  that as he assumed   
each of these various vestures of       
consciousness1  in succession1 he would 
gradually and finally1 come to a full   
understanding of reality itself8 ,it    
seems so4 ,this view is even more       
cogent when it is considered that the   
limitations1 and consequent  obscuration
 of consciousness are proportional to   
the number of vehicles  or barriers     
through which the ,thinker is required  
to act in contacting  the phenomenal    
universe4 ,common sense suggests that   
freedom of motility  is determined by   
the presence or absence 7more           

particularly the latter7  of bonds and  
barriers2 that the less the number of   
such barriers the  greater the scope of 
motility and consequently greater the   
knowledge4    ,,plato evidently had this
 in mind when he imagined the life of   
men  spent in cave-walled prisons in    
which their bodies were so fixed that   
they were compelled to sit in one       
prescribed position1 and therefore1  be 
unable to see anything except the       
shadows of persons or objects  as they  
passed by4 ,he conceived that men thus  
conditioned would1 in  time1 suffer the 
diminution of their scope of            
consciousness to such an  extent as to  
reduce it to identification with the    
shadows on the walls4  ,their           
consciousness would be mere             
shadow-consciousnesses the entire  data 
of which would be shadowgraphs4 ,so that
 for them the only reality  would be the
 shadows which they constantly saw4 ,a  
similar thing really  happens to man's  
consciousness limited to the plane of   

the objective  world4 ,things which are 
not objective do not appear as real to  
him1  if they do appear at all4 ,it is  
not that there are no other realities   
than those which appear to the          
egopsychic consciousness or that fall   
within its scope2 but that this form of 
consciousness is incompetent  to judge  
of the nature and appearance of those   
realities which do not  answer to the   
limitations under which it exists4 ,and 
so1 with men whose  data of             
consciousness or whose outlook upon the 
world of facts1 or  rather life1 are    
confined to the narrow bounds of        
mathematic rigor and  exclusiveness1    
there may appear to exist no realities  
which may not  be defined in the terms  
of mathematics4 ,similarly1 to the      
empiricist1  used to measurements of    
magnitudes1 weights1 and rates of       
motion1  there may also appear to be no 
realities which are not amenable to the 
mold of his empirical contrivances--the 
balance1 the chromatic and  the scalpel4

 ,all of these are shadow men           
constricted to the metes and  bounds of 
shadows which they observe only because 
they are ignorant of  the realities     
which lie without their plane4    ,life 
has so many ways of exhibiting its      
remains to the intellect2 and  these    
remains have so many facets or          
viewpoints from which they may  be      
studied1 that nothing short of a        
panoramic view of all the modes  of     
exhibition and of all the facets and    
angles of appearances will  suffice to  
present a trustworthy and comprehensive 
view of the whole4  ,then1 life itself  
is so illusive1 so unseizable by the    
intellect that  the testimony of all    
investigators are required to summarize 
its modes  of appearance4 ,and1         
therefore1 eventual contentment shall be
 secured  only when the mass of diverse 
testimonies is reduced to the lowest    
common divisor1 and for this purpose the
 operations of every class of           
investigators must be viewed as the work

 of specialists upon separate  phases1  
facets and angles of life's remains4    
,and so it is manifestly absurd for the 
empiricists1 by taking note of  the     
dimension1 extent1 quality and character
 of the shadows1 or one  single class of
 angles1 to hope to predicate any       
trustworthy judgments  about either the 
realities which cast the shadows or     
underlie the  angles2 because whatever  
notion or conception they may be able to
  gain must of necessity be merely      
fragmentary and entirely inadequate4    
,despite this fact1 however1 we still   
have the spectacle of men who1  studying
 the sensible universum of              
space-content1 endeavor either to  make 
it appear as a finality in itself1 or   
that the world of the real  must        
necessarily be conformable to the       
precise standards which they            
arbitrarily set up in their examination 
of the objective world4 ,it can  be said
 with assurance that we shall never be  
able even so much as to  approach a true

 understanding of the unseen1 real world
 until we shall  have changed our mental
 attitude towards it and ceased to      
expect that  it shall necessarily be    
fashioned and ordered in exactly the    
same way  as the world of our senses1 or
 that it shall be understood by applying
  the same methods of procedure as those
 which we use in our examination  of the
 phenomenal1 sensuous world4 ,it is a   
matter of logical necessity  that1 as   
there are no senses which can respond to
 the real1 as there  are no organs which
 vibrate in accord with the rates of    
vibration of  the real1 there can be no 
reasonable hope of understanding it by  
means  of sensuous contrivances and     
standards4    ,let the consciousness1   
therefore1 be turned not outward1 but   
inward  where is situated the temple of 
divine life2 let there be taken away    
the outward sheaths which enshrine the  
pure intelligence of the  ,thinker2 let 
him grow and expand his sphere of       
awareness2 let there be  an exploration 

of the abysmal deeps of mind1 of life   
and consciousness2  for buried deeply in
 man's own inner nature is the answer to
 all  queries which may vex his         
impuissant intellectuality4             
,,chapter ,,ix      ,,metageometrical   
,,near-,,truths      ,realism is        
,psychological and ,vital--,the         
,impermanence of ,facts--,on    the     
,tendency of the ,intellect to          
,fragmentate--,the ,intellect    and    
,logic--,the ,passage of ,space1 the    
,kosmometer and ,zoometer1              
,instruments for the ,measurement of the
 ,passage of ,space and the    ,flow of 
,life--,the ,disposal of ,life and the  
,power to ,create--,space    a ,dynamic1
 ,creative ,process--,numbers and       
,kosmogony--,the ,kosmic                
,significance of the ,circle and the    
,pi-,proportion--,mechanical            
,tendence of the ,intellect and its     
,inaptitude for the ,understanding    of
 ,life--,the ,criterion of ,truth4      
,kosmic truth has many facets4 ,the rays

 of light which we see darting  from its
 surface do not always come from the    
core4 ,often they are  reflections of   
rays whose light stops short at the     
superfice2 and  these1 in turn1 are     
reflections of deeper realities4 ,thus  
the reflected  light may be traced to   
its source by following the lead of     
external  reflections4 ,it is now known 
that moonlight1 and perhaps1 in many    
cases1 starlight1 are reflections of    
sunlight1 if not of our sun1  some other
 in the universe4 ,but it is only at    
certain times and under  certain        
conditions that we can see the sun which
 is the source of the  other kinds of   
light4 ,the stars which owe their light 
to suns are so  many facets of sunlight4
 ,the moon is a facet of sunlight also4 
,facts  are facets of truth4 ,they are  
so many faces of eternal truth4 ,they   
represent the many ways reality exhibits
 itself1 or rather its effects1  to the 
consciousness4 ,when we1 therefore1     
become aware of facts we have  not in   

virtue thereof become aware of the      
reality which produces the  facts4 ,we  
have come to know only something of the 
termini of realism  while the           
complexities and internal ramifications 
which lie between  realism itself and   
these termini altogether elude our      
cognition4    ,let us examine briefly an
 icosahedron1 for instance4 ,an         
icosahedron is  a figure comprehended   
under twenty equal sides4 ,these various
 sides  are so many faces by means of   
which the figure presents itself to the 
consciousness4 ,these faces1 however1   
are not the real object4 ,the  figure   
may be examined by viewing it from any  
one of its sides2 yet1  by simply       
examining a single face1 or any number  
of faces1 less the  total number1 we    
arrive at no satisfactory knowledge of  
the magnitude  or its substance4 ,we    
must first become conscious of all the  
faces1  holding them in mind as a       
composite picture1 before we can even   
begin  to have anything like a complete 

notion of the icosahedron4 ,then by     
continuing the examination we may find  
that the magnitude is composed  of wood 
fiber or stone or metal1 as the case may
 be4 ,in this way we  might carry the   
examination to indefinite limits and    
finally arrive at  a very comprehensive 
knowledge of the icosahedron and yet be 
unaware  altogether of the forces which 
have been at work in the production of  
the magnitude or of the reality which   
lies back of it4    ,realism is         
psychological and vital4 ,in essence it 
is mind1 spirit1  life4 ,yet these three
 are one4 ,mind is the outward vehicle  
of life2  spirit is the form or the     
interior vehicle which life assumes in  
order  to express itself4 ,realism1     
then1 is life4 ,is the logician dealing 
with reality when he collects and       
cordinates the various modes of         
interpretation by which we learn to     
understand the symbolism of life8       
,obviously not4 ,the data of logic are  
simply a collection of rules for        

interpreting concepts4 ,it is a         
compendium of indices for the ,book of  
,life4 ,it is no more the book itself   
than a table of contents is a  book4    
,but logic occupies about the same      
category as does an index to a  volume4 
,a book1 however1 is more than its      
conventional contents4 ,it is  the      
thought that is symbolized therein4 ,the
 book of life1 accordingly1  is the sum 
total of life's expressions2 but it is  
not life itself4 ,that  is the subtle1  
evasive something which the contents of 
the book of life  symbolize4 ,nature1   
both in her palpable and impalpable     
aspects1 may be  said to be the book of 
life wherein are recorded the movements1
 the  expressions1 and the diacritics of
 life4 ,the whole is a magnitude of     
many facets 7little faces74 ,we shall   
have to know all the faces before  we   
can say that we have a comprehensive    
knowledge of nature4 ,for so  long as we
 have only a fragmentary knowledge of   
the whole1 so long even  as we have     

merely a superficial knowledge of any   
aspect of nature1  just so long will our
 knowledge be in vain4 ,just as it      
frequently  happens that1 on account of 
the partial view of things1 we are led  
to make incorrect judgments concerning  
them1 so when we come to make           
assertions about life or nature in      
general1 we are apt to fall into the    
error of rendering judgments upon       
insufficient data4 ,and it is not at    
all likely that judgments thus arrived  
at can possess true validity  because it
 may happen1 and undoubtedly does always
 so happen under the  present           
limitations of human knowledge1 that the
 very elements which  are ignored or    
neglected in forming a judgment possess 
enough of virtue  to alter the intrinsic
 value of determinations based upon     
otherwise  insufficient data4 ,hence it 
develops not infrequently that our      
judgments repeatedly have to be changed 
in proportion as our data are  made more
 and more comprehensive4 ,men searching 

eagerly for the truth  sometimes allow  
themselves to be carried away by the    
enthusiasm of the  moment which arises  
upon the discovery of a new facet of    
truth2 but  if all searchers were to    
bear in mind the fact that reality      
presents  itself to consciousness in    
myriad ways and that there are          
innumerable  facets all leading         
eventually back to the source of all    
they not so  easily would be induced to 
jump to the conclusion that they had    
covered  the entire ground4 ,for when we
 have discovered a million facts1 or    
many  millions of them1 about nature we 
may say that we have only merely  begun 
and that what we have found is not to be
 compared with the  totality even of the
 directly observable phases of nature4  
,logic1 therefore1 deals with the       
symbolism existing between and  among   
facets of truth1 and not directly with  
truth itself1 although  the conclusions 
reached by the logicians may be true    
enough from  an intrinsic standpoint4   

,logic is not truth1 however2 it is     
merely  the consistence of relations and
 inter-relations between facts and      
among groups of facts4 ,truth is not    
established by logic2 it stands  in no  
need of the light of logic for its      
revelation2 indeed1 more  apt than not  
is logic to obscure truth4 ,truth is its
 own proof2 it  is self-evident4 ,logic 
is a mere modeler of facts2 it is       
static1  immobile1 fixed4 ,all truly    
logical processes need a starting point1
 a  foundation1 a premise1 a base4      
,truth1 being eternal1 mobile1 dynamic1 
vital1 needs no starting point2 needs no
 foundation because it is  itself       
fundamental2 it requires no premise     
because no premise is  comprehensive    
enough to encompass it4 ,there is only  
one way of arriving  at truth and that  
is not to arrive at all--just to        
recognize it without  procedure4 ,the   
fact that facts are1 and the fact of    
their relations and  inter-relations1   
their sequence and implications1 can be 

arrived at  only by logical processes4  
,life1 in its passage through the       
universum  of spatiality1 carefully     
diacriticizes between the realm of facts
 and  the domain of truth1 marking each 
off from the other by unmistakable      
signs and barriers4 ,truth is perceived 
as an axiomatic1 self-evident  principle
 and no amount of logic could prove or  
establish its verity4  ,facts are       
intellectual creatures2 truth is        
intuitional1 vital4 ,the  intellect     
conceives the consistence of facts while
 the intuition  recognizes truth--is    
truth1 and therefore1 follows in the    
wake of life  as consciousness4         
,there is no permanence in facts and the
 intellectual recognition of  their     
consistence4 ,the discovery of a single 
new fact may destroy the  consistence of
 a whole mass of previously correlated  
facts4 ,thus is  revealed the miracle   
power of logic over facts4 ,it can take 
a mass of  facts1 related or unrelated1 
mold them into hypotheses1 endow them   

with a sort of interior consistency1 and
 make these hypotheses take  the posture
 of truth4 ,hence logic is often an     
effective mask which  the intellect     
commonly imposes upon its material2 but 
it does so  instinctively and can no    
more escape the rigorous compulsion of  
this  instinctive functioning than water
 can escape its liquidity4 ,wherefore1  
we conclude that true permanence abides 
alone in truth because truth  is        
duration itself4 ,for the foundations of
 the whole structure of  facts in       
religion1 science1 art and philosophy   
which man has toilfully  built up in the
 last million years might easily be     
destroyed or  overturned by the         
discovery of some great fact or by      
appreciating the  true value of truth4  
,let us suppose it should suddenly be   
realized  by men that they are really   
and truly gods capable of creating and  
possessing all the other virtues1 powers
 and capabilities which we are          
accustomed to impute to supreme         

divinity2 and suppose that the fact  of 
their omnipotence and divine omniscience
 always had been obvious  but that men  
were so engaged upon details and the    
non-essentials of  life and matter that 
they had not noticed nor realized it    
before1  would not this realization make
 a vast difference in the character  of 
our knowledge and the attitude which we 
would necessarily assume  thereafter    
towards matter1 life and the problems   
which they present8  ,would not it      
completely revolutionize our arts1 our  
sciences and our  philosophies8 ,how    
much1 then1 of the facts of these would 
be left  when the light of omniscience  
had been turned on--when truth itself   
could be perceived and interiorly       
realized8 ,not much1 to be sure4 ,we    
should undoubtedly have to dispense with
 the entirety of our fact-mass1  for it 
should then be entirely useless and     
meaningless in the light  of the        
resplendent omniscience of truth4 ,as at
 present constituted  consciousness is  

focused upon the material plane for the 
purposes of  superficial observances4   
,but if the focus of consciousness      
should be  changed so as to reveal      
conditions upon what must be a higher   
and more  interior level1 the aspect of 
things would be entirely changed and the
  whole of our theory of knowledge would
 have to be reconstituted4 ,it  is      
conceivable1 yea obvious1 that the stern
 reality of being is far  removed into  
the ,great ,interior of that which is2  
and there is a point  in the path to the
 interiority of being where there is no 
illusion1 no  appearance1 indeed1       
nothing but the cold1 illuminating body 
of reality  itself4 ,it must appear also
 that along the journey interior-ward   
there  are many apparent levels or      
planes1 each of which requires a new    
focus4  ,it is unreasonable1 then1 to   
suppose that the conclusions arrived at 
as a result of purely logical processes1
 confined to the lowest levels  of      
reality1 are pertinent and valid for the

 entirety of realism which  is neither  
of mathematical nor logical import4 ,for
 instance1 if we  take the purely       
axiomatic assertion3 x equals x1 the    
intellect  is at once certain that this 
is so1 and cannot be otherwise1 and yet 
a proposition of this kind is purely    
conceptual1 conventional and  arbitrary4
 x may also equal #a1 #b1 #c1 #d1 or any
 other quantity4 ,then1  if each x in   
the above equation be replaced1 by say1 
a horse1 there  immediately arises a    
difficulty4 ,for it is not possible to  
find two  horses which are in all       
respects mutually equal4 ,so that as    
soon as  we pass from the conceptual    
into the actual1 whether on the side of 
objective reality or that of absolute   
reality1 the validity of the  axiom is  
immediately exposed to serious          
questioning4 ,the truth of the  matter  
is that on both sides of the conceptual 
it is always found that  there is a     
variance from the standards set up by   
the conceptual1 this  variance being    

more marked on the side nearest to      
absolute reality than  on the side of   
objectivism4 ,objectively1 the          
conformity of the sensible  with the    
conceptual is of such approximation as  
to lend trustworthy  utility to the     
conceptual in its application to the    
sensuous4 ,thus by  simply eliminating  
the vital factors from our equations we 
are enabled  to proceed in a reasonably 
safe manner with our judgments4 ,really1
  however1 no such approximate          
congruence can be found2 for on the side
  of reality we are dealing with an     
indivisible something--something  that  
is eternally and absolutely unitary in  
its constitution while  when we transfer
 the scene of our observations to the   
objective world  we discover a contrary 
situation4 ,here we are everywhere beset
 by  diversities1 multiplicities and    
dissimilarities4 ,this is so because the
  intellect naturally tends toward the  
objective where it finds a most         
comfortable atmosphere for its          

operations4 ,the conceptual is related  
to  the objective as a train of cars is 
related to the railway4 ,that is to     
say1 the constitution of the intellect  
is such that it finds its most  facile  
expression in the objective world and is
 about as comfortable in  the domain of 
realism as the same train of cars would 
be on the ocean4    ,the intellectuality
 is designed to deal with facets of     
truth2 it is  made to manipulate        
segments1 parts1 fractions1 and cannot  
chart its  way through a continuum such 
as reality4 ,being constitutionally a   
conventionality of the ,thinker's own   
contrivance1 and arising out of  the    
subtle adaptation of his vehicles to the
 environment afforded by  the sensuous  
world1 it can only find congruence in   
that conventionality  which is the      
instrumentality of a higher             
intellectuality expressed  in a         
diversity of forms1 into which reality  
divides itself for  manifestation4 ,the 
human intellect is1 therefore1 the      

bridge over  which is made the passage  
from the individual consciousness to the
  ,all-,consciousness2 simultaneously1  
the medium whereby the physics  of the  
brain are converted into the psychics of
 unconsciousness4 ,it  may be likened to
 a pair of specially constructed tongs  
which are so  formed as to fit exactly  
the objects which a higher              
intellectuality  has made4 ,it is       
without the province of the intellect to
 take note of  what intervenes between  
physics and psychics2 it is always      
oblivious  of interstices while taking  
cognizance of objects or things4 ,in    
this  respect1 the intellect is much    
like a steerage passenger on board an   
ocean liner who sees only his port of   
departure and port of arrival1  knowing 
nothing in the meantime of what happens 
during the voyage1  nothing of what the 
other passengers on the upper decks may 
experience  and taking no part in any of
 the passing show until he lands4 ,so   
that the passage of the intellect from  

fact to fact is an altogether           
uninteresting voyage2 it may as well be 
made unconsciously1 and to all  intents 
and purposes1 is so made4               
,accordingly1 the advocates of          
n-dimensionality find it quite          
impossible to predicate anything        
whatsoever of the passage1 say1 from    
tridimensionality to                    
quartodimensionality4 ,they find        
themselves at ease  in tridimensionality
 and have even contrived to find        
pleasant environs  in the four-space    
having made therein such idealized      
constructions as  will afford ample     
hospitality to the intellect4 ,but the  
questions as to  how the passage from   
the three-space to the four-space is to 
be made  and how the intellect shall    
demean itself during the passage have   
been  completely ignored and1 therefore1
 left unanswered4 ,what1 then1 shall  be
 said of an explorer who says he has    
found a new land and yet can  give no   
intimation as to how one may proceed to 

arrive at the new land1  what changes   
are to be made en route1 nor the        
slightest suggestion  as to the         
direction one should take in setting out
 for it8 ,it is not  likely that the    
report of such an explorer1 in practical
 life1 would  be taken seriously2 and   
yet1 there are those who1 relying       
utterly  upon similar reports made by   
certain enthusiastic analysts1 dare to  
place credence in their asseverations4  
,not only have they given wide  credence
 to these reports1 but have1 indeed1    
sought to rehabilitate  their own       
territory in accordance with the strange
 descriptions given  by unhappy         
analytical explorers4 ,now the question 
of greatest concern1  granting for the  
nonce that there is such a domain as    
hyperspace1  is the passage4 ,how shall 
we make the passage8 ,or1 is the passage
  possible8 ,in vain do we interrogate  
the analyst2 for he does not know1  nor 
does he confess to know4 ,evidently it  
is impossible for him to know  by means 

of the intellect alone2 for the         
intellect not being fitted to  take     
cognizance of the passage1 but only the 
starts and stops1 has no  aptitude for  
such questions4 ,hence1 what seems to be
 the most important  phase of the entire
 question will have to remain utterly   
inscrutable  until the intellect        
nourished by the intuition shall be     
aroused from  its lethargy and brought  
to a certain high point of illumination 
where  it1 too1 may take note of the    
passage4    ,space is the path which    
life makes in its downward sweep through
  all the stages of pyknosis or kosmic  
condensation by virtue of  which it     
accomplishes the engenderment of        
materiality as also the  path marked out
 by it in its upward swing whereby it   
accomplishes  the spiritualization of   
matter4 ,it is the kosmic order which   
life  establishes by means of its       
outgoings and incomings4 ,when we look  
out  into space we perceive that which  
is a dynamic appearance of life  itself1

 and not a pure form4 ,nothing that is a
 pure form can exist in  nature and in  
as much as space is not only            
indissoluble from nature  but partakes  
of its very essence it cannot be said to
 be a pure form4  ,the intellect1       
however1 prone to follow the grooves    
laid out by pure  logic1 never fails to 
seek to make everything that it contacts
 conform  to these logical necessities4 
,but1 if the analyst were to make       
careful  discrimination as to the       
respective categories--that into which  
life  falls and that in which the       
intellect is forced by its nature to    
proceed--he not so easily would be led  
into the fault of attempting to  shape  
realities upon models which being       
strictly conventional were not  meant   
for such uses4 ,but neither the logician
 nor the mathematician can  be condemned
 for such generosity if such            
condemnation were justifiable4  ,for    
they everywhere and at all times insist 
upon realizing  abstractions and        

abstractionizing realities1 and they do 
this with  an insouciance that is at    
times surprising4 ,yet it is in this    
very  vagary that is discovered the true
 nature of the intellect4 ,there is  a  
sort of dual tendence observed in the   
method of the intellect's  operation4 ,a
 polarity is maintained throughout3 the 
abstractive and the  concretional4 ,it  
vacillates continually between the      
abstract and the  concrete and no sooner
 has it found a concrete than it begins 
to set up  an abstract for it2 and vice 
versa--as soon as it is has constructed 
an abstract it immediately seeks either 
its concrete or sets out to hew  some   
other concrete into such shape as will  
fit it4 ,and between these  two extremes
 numerous excuses are found for         
exercising this peculiar                
characteristic1 and that too1 without   
regard to consequences4 ,it would  seem 
that the intellect1 in thus functioning1
 was really engaged at  a sort of       
sensuous play out of which it derived an

 intense and not  altogether unselfish  
pleasure4    ,of course1 it must be     
granted that diversity has its specific 
and  withal necessary uses in that it   
affords a field for the operation  of   
human intellectuality and represents the
 adaptation of the kosmic  intellect to 
the human for the purposes of evolution4
 ,this adaptation  while necessary for  
the intellectual development is1        
however1 not an  end in itself4 ,it is  
merely a means to a higher purpose4 ,in 
fact1 if  we regard materiality as a    
deposit of life1 carried by it as a kind
 of  impedimentum1 and consciousness1   
which is life1 as being identical  with 
the intellectuality which makes these   
adaptations1 there should be  no grounds
 for the statement that the one is      
adaptable to the other at  all4 ,and as 
this is really the view which we assume 
it would perhaps  be more strict to     
regard the adaptation as subsisting     
between the  human intellect and        
materiality both of which having been   

constructed  by kosmic intellectuality4 
,pursuant to the diversity of uses to   
which  materiality lends itself there   
arises in the intellect a supreme       
tendency to segment1 to break up into   
separate parts1 to multiply and         
diversify4 ,it is not content unless it 
is at this favorite and natural         
pastime4 ,it delights in taking a whole 
and dividing it into innumerable  parts4
 ,this it will do again and again2      
because all its muscles1 sinews  and    
nerves are molded in that mold and can  
no more cease in their  tendency to     
fragmentation than can the muscles of a 
dancing mouse  cease in their circular  
twirling of the mouse's body4 ,yet1 in  
this  it is but creating a well-nigh    
endless task for itself--which task     
must be performed to the uttermost4 ,but
 in its performance1 that is1  in the   
intellect's complete understanding of   
the diversity of parts1  in the         
knowledge of their relations and        
inter-relations and in their  synthesis1

 it may arrive at that one ineluctable  
something which is  called unity4 ,and  
so doing1 become ultimately free4    ,in
 view of the foregoing1 it is not       
surprising that the intellect  should   
have1 finally1 fallen upon the notion of
 n-dimensionality4 ,it  has come to that
 as naturally as it has performed its   
most common task4  ,left alone and      
unhampered in its movements1 it has     
simply followed  the lead of the ,great 
,highway through the domain of          
materiality4 ,and  now it has arrived at
 a stage where it thinks it has         
succeeded in  fractionalizing space4    
,time has long ago yielded to           
fragmentation1  been divided into minute
 parts and each part carefully measured4
  ,space1 not having a visible indicator
 like time to denote its passage  or    
parts1 suffered a long and tedious delay
 before it could boast  of a measurer4  
,as the sun-dial measured time in the   
past and became  the forerunner of the  
modern clock so n-dimensionality        

measures  space for the mathematician4  
,what more practical instrument for this
  purpose may yet be devised is not ours
 to prophesy2 yet it is not  to be      
despaired of that some one shall find a 
suitable means for  this purpose4       
,seriously1 however1 it is not without  
possibility that  should some subtle    
mind devise an instrument for marking   
the passage  of space as we have for    
denoting the passage of time a great    
stride  forward would be accomplished in
 the evolution of the human intellect4  
,for the general outcome of the         
intellect's attention being turned to   
the passage of space would undoubtedly  
be to recognize not only  its dynamism  
but its becoming-ness1 as a process of  
kosmogenesis4  ,because such an         
instrument would have to be so          
constructed as to take  note of the     
movement of life1 and for this reason1  
it would have to be  extremely sensitive
 necessarily and keyed to the subtleties
 of vitality  and not to materiality4   

,mathematics shall have failed utterly  
in the  utilitarian aspects of this     
phase of its latest diversion if it do  
not  justify its claims by crowning its 
work in the field of hyperspace  with a 
,kosmometer1 an instrument devised for  
the measurement of  the movement of     
space or a ,zoometer1 an instrument     
devised for  the measurement of the     
passage of life4 ,we should like to     
encourage  inventive minds to turn their
 attention life-ward and space-ward with
  the end in view of constructing such  
an instrument4 ,when once we have       
learned accurately to measure life we   
shall then be able to dispose of  it--to
 create4 ,it is not doubted that if ever
 humanity is to arrive  at that point in
 its evolution where it can understand  
life2 if ever it  is to attain unto the 
supreme mastery both of vitality and    
materiality  and to come to the ultimate
 attainment of divine consciousness 7all
  of which we confidently believe to be 
in store for humanity7 it must  be      

accomplished after this manner3 first1  
by syncretizing materiality  with       
vitality1 and then1 by intuitionally    
recognizing the truth of the            
implications of the syncretism4    ,the 
history of consciousness in the human   
family is identical with the  history of
 man's conquest over matter and physical
 forces4 ,and this  is clearly indicated
 in the incidentals contingent upon the 
toilsome  rise of the genus homo from   
the earliest caveman whose status       
denoted  a comparatively negligible     
transcendence of material forces1 to the
  present-day man who has gained a      
markedly notable conquest over these    
forces4 ,always consciousness seeks the 
means of adequately expressing  itself  
in the sensible world4 ,and to this end 
it engenders faculties1  organs and     
processes in the bodily mechanism1 and1 
in matter1 devises  instruments of      
application whereupon and wherewith it  
may test1  analyze1 combine and         
recombine the forces and materials it   

finds4  ,the unlimited range of         
expressions lying open to the           
consciousness  makes it necessary       
continually to devise higher and higher 
grades of  appliances to meet its needs 
as it expands4 ,it will not be gainsaid 
that the telescope has served actually  
to lay bare to the consciousness  an    
immeasurable realm of knowledge nor that
 the microscope1 turning  its attention 
in an opposite direction1 marvelously   
has enlarged and  enrichened our        
knowledge of the world about us4 ,and   
similar declaration  may be made anent  
almost every invention1 discovery and   
conquest which  man has made over       
natural phenomena4 ,thus1 by externally 
applying  mechanical implements to the  
subject of his consciousness1 man has   
extended actually his consciousness1 his
 sphere of knowledge2 has  greatly      
enhanced its quality1 and1 in the       
process1 has urged the  intellect to    
endeavors that have wrought its present 
unequaled mastery  of things4 ,nor have 

the spiritual aspects of our advance    
along these  lines been the least       
notable4 ,for these have enjoyed the    
essence of all  that has been gained in 
the process and have1 therefore1 kept   
pace with  the onward movement of the   
intellectual consciousness4 ,but        
heretofore  no advance has been made as 
a result of methodic or reflexive       
determinations4 ,that is1 men did not   
set out from the beginning1  equipped   
with foreknowledge of what their efforts
 would bring1 to  develop the present   
quality of human consciousness4 ,they   
simply worked  on1 their attention being
 absorbed by the problems that lay      
nearest and  demanded earliest          
consideration4 ,so the advance has come 
as a resultant  of man's close          
application to his ever-present         
needs--shelter1  clothing1 food1        
protection and other preservative       
measures--and it  has come naturally and
 inevitably and without prepense4       
,nevertheless1  if man1 knowing what to 

expect from the syncretization of matter
 and  mind1 after this fashion1 should  
set out deliberately to accelerate  the 
intensification1 expansion and growth of
 his consciousness1 there  is no doubt  
but that the consequence would be most  
far-reaching and  satisfactory4    ,but 
the path that leads to this grand       
consummation does not lie in the        
direction of diversity2 it lies in the  
opposite direction4 ,in vain1  then1    
does the intellect fractionalize in the 
hope that by doing so it  shall come to 
the solid substructure of life2 in vain 
does the analyst  segment space into any
 number of parts or orders2 in vain does
 he ask  how many and how much2 for by  
answering none of these queries will he 
find the satisfaction which he vaguely  
seeks4    ,if it be true that it is not 
by analysis but by synthesis that the   
true  norm of life1 and therefore1 of   
reality shall be found it is futile to  
entertain serious hope of finding it in 
any other way4 ,as a perisophism  or    

near-truth1 then1 n-dimensionality takes
 foremost rank4 ,and this  is so for the
 reason that when we proceed in the     
direction of multiple  dimensions1 that 
is1 one dimension piled upon another    
dimension or  inserted between two      
others we are traveling in a direction  
which1 the  more we multiply our        
dimensions1 leads us farther and farther
 away from  the truth4 ,this is a simple
 truism4 ,if we take1 for instance1 a   
wooden  ball and cut it up into four    
quarters1 and divide each one of these  
quarters into eighths1 into sixteenths1 
thirty-seconds1 sixty-fourths1  etc41   
indefinitely1 we shall have a           
multiplicity of parts1 each one  unlike 
the original ball4 ,but from no         
examination of the multipartite         
segments can we derive anything like an 
adequate conception of the  original    
ball4 ,something1 of course1 can be     
learned1 but not enough  to enable the  
rendering of a correct judgment as to   
the nature1 size1  shape and general    

appearance of the ball4 ,but this is    
precisely what  happens when the analyst
 divides space into many dimensions4 ,he
 cuts  it up into n-dimensional parts   
and the more minutely he divides it     
into parts the more remote will each    
part be in its similarity to the        
original shape and form of space1 and   
the farther away from the true          
conception of the nature of space he is 
led thereby4    ,now1 n-dimensionality  
or that phase of metageometry which     
regards  space as being divisible into  
any number of dimensions or systems of  
cordinates is a direct and inevitable   
product of that tendency of the         
intellect to individuate and to         
singularize phenomena4 ,biologically    
speaking1 it is a peculiarity which     
harks back to the time when life  was   
manifested through the cell-colony and  
when the individual cells  began1       
because of increasing consciousness1 to 
detach themselves  from the colony and  
set out for themselves1 and thus each   

intellect  recapitulates in its modus   
vivendi the salient tendencies of       
phylogenesis4 ,let it suffice1 then1 to 
point out that this universal  tendency 
to segment and fragmentate which        
rigorously characterizes  intellectual  
operations upon every phenomenon with   
which it deals is a  culmination of the 
primordial tendency among cells to      
divide1 inasmuch  as this phase of cell 
life must be the work of the kosmic     
intellect4  ,the natural inference is   
that from the extreme of                
individualization  there shall be a     
gradual turning1 whether of the         
intellect per se  or of the intellect   
joined to the intuition does not matter1
 towards  that other extreme of         
communalization4 ,and from this latter  
shall  grow up1 as one of the inevitable
 and ineluctable tendencies of the      
,thinker's consciousness a torrentious  
movement in human society towards       
coperation1 brotherhood1 mutuality and  
union in everything4 ,so that  whereas  

in the past and at the present time the 
intellect has been  developing under the
 dominant note of individuality it will 
then be  coming gradually under another 
dominant note--communality4 ,the result 
of this development will be the         
unification of all things1 and instead  
of many dimensions of space1 many       
measures of time1 and a general         
diversification of all phenomena1 we    
shall come to the only true notion  of  
these things and realize pragmatically  
the true value and extent of  unity in  
the universe4    ,it is admitted that   
the intellect1 in treating objects      
singly and  dealing only with the starts
 and stops of a movement1 is withal     
loyal to the kosmic order1 design and   
purpose which have priorly              
characterized manifested phenomena by   
segmentation4 ,and in this  loyalty it  
has been following merely a natural lead
 which1 while  admitting of the widest  
development and experience1 nevertheless
 at  the same time beneficently obscures

 the underlying reality in order  that  
in its adaptation to the sensuous world 
the intellect might  have the greatest  
freedom for the development suited to   
the given  stage of its evolution4 ,but 
in thus admitting the natural congruence
  between the intellectuality and the   
phenomenal or sensuous we do  not       
thereby unite with those who already    
believe that this kosmic  agreement is  
the ne plus ultra of psychogenesis4 ,on 
the other  hand1 it is maintained that  
this is merely a phase of psychogenesis 
which shall be outgrown in just the same
 measure as other phases  have been     
outgrown4 ,and notwithstanding the fact 
that judgments of  the intellect with   
respect to inter-factual relations or   
the ens  of facts themselves are as     
valid as its judicial determination of  
self-consciousness1 no more and no less1
 we are1 by the very rigor  and         
exclusiveness of this logical necessity 
and inherent limitation1  led to view   
the intellect's interpretation of       

phenomena as partial  and fragmentary2  
for the reason that the necessitous     
confinement of  its understanding and   
interpretative powers to fact-relations 
quite  effectively inhibits the use of  
these powers for the contemplation  of  
the deeper causative agencies which have
 operated to produce the  phenomena4    
,but it is apparent that just as the    
transmuted results of  other phases of  
psychogenesis are now being utilized as 
a basis for the  efficient operation of 
the intellect in the sensuous world1    
thereby  enabling the attainment of a   
very high mastery over matter1 so will  
the  functional dynamism acquired by it 
in the pursuit and comprehension of     
diversity serve well when1 in later     
days1 it has acquired the power to  deal
 directly with reality1 to create and   
dispose of life just as  the kosmic     
intellect has and is now using it in the
 execution of the  infinite process of  
becoming through which creation is      
proceeding4 ,it  would seem that the    

necessary prerequisite to the           
development of any  higher functional   
capability is that the intellect should 
be capable of  disposing of innumerable 
details1 indeed the totality of kosmic  
detail1  before it can come wholly into 
the power and capacity to understand    
and manipulate life4 ,furthermore1 it   
appears that the acquirement of  this   
power quite necessarily has been delayed
 awaiting that time when1  dominated by 
the intuition1 the intellect shall have 
attained the  requisite managerial      
ability for marshaling an exceedingly   
large number  of details4    ,the       
supreme tendency of life is expression4 
,and this expression1  singularly       
enough1 reaches its most perfect        
phenomenalization by means  of that     
movement which results in the           
multiplication of forms4 ,despite  the  
fact1 therefore1 that the comprehension 
of reality involves a  gradual turning  
away from the exclusive occupation of   
organizing a  multitude of separate and 

apparently unrelated facts to a monistic
  view which at once recognizes the     
unitariness and co-originality of  all  
things1 of life1 mind and form1 the     
intellect will need the  training and   
development which come from the mastery 
of diversity4 ,it  is1 then1 not        
difficult to perceive the wise          
utilitarianism of the  present          
schematism of things as shown in the    
universal tendency in the  intellect to 
devote itself exclusively to parts or   
segments of truth4    ,whenever an      
individual intellectuality1 on account  
of prolonged  thought and the consequent
 inurement of the mind to higher and    
higher  vibrations of the kosmic        
intellect1 brings itself to such a high 
point  of sensitiveness that it can     
receive so much as an intimation of some
  great truth1 it begins to sense1 in a 
more or less vague way1 something  of   
the substance and general tendence of   
the underlying reality of that  which   
foreshadows its appearance4 ,then1      

confounded by the multiformal           
characteristics of kosmic truth because 
of the fact that it presents  itself in 
such numerous ways and forms1 men often 
are induced to  attempt the reformation 
of all facts1 or a great mass of kindred
  facts1 in accordance with the         
newly-found fact or principle4 ,they    
forget evidently that no fact in the    
universe can be at variance with  any   
other fact and still be a fact4 ,so that
 in the totality of facts  every        
separate and distinct fact must be      
congruent with every other  fact forming
 a beautiful1 harmonious and symmetrical
 whole2 but often  the whole is made to 
suffer in the attempt at making it      
conform to the  substance of a mere     
intimation4 ,moreover1 it is conceivable
 that even  the totality of facts may   
lack a rigid conformity with reality in 
all  its parts and that having compassed
 the entire mass of facts one may  fall 
short of the understanding of realism4  
,this is practically what has happened  

in the mind of the  metageometrician who
 having received an intimation as to the
 real  nature of space as that whose    
center is everywhere and yet nowhere    
and whose nature is psychological and   
vital rather than mathematical  and     
logical1 misses the great outstanding   
facts and clings to the  intimations    
which he experiences as to the nature of
 space4 ,he1  therefore1 concludes that 
the form of space is that of a flexure  
or  curve4 ,there is a valid element in 
the notion of the curvature of  space   
but not enough of truth wholly to       
validate the notion4 ,since  the very   
reality of space is a matter which can  
be determined only  by the conformance  
of the consciousness with it in such a  
manner  as to render the conception of  
it entirely unintelligible to the       
intellect except in so far as it may be 
able to identify itself with  the       
space-process1 there is much room for   
the serious questioning of  the         
mathematic conclusion upon the grounds  

of its fragmentariness if  not entirely 
upon the basis of its invalidity4       
,wherefore it may be  seen that any     
search for either the center or the     
extreme outer limits  which proceeds in 
a manner conformable to the external    
indications of  the intellectual order  
is vain1 indeed4 ,although it is        
undoubtedly  true that the attainment of
 a central or frontier position in space
  does not involve any lineal           
progression whatsoever1 the same being  
attainable1 not by progression nor by   
overcoming distances1 but by a  subtle  
adjustment1 yea1 a sort of attachment of
 the consciousness to  the order of     
becoming which binds the appearance of  
space1 wherever  one may be1 it is      
nevertheless difficult and painful for  
the intellect  to grasp the totality of 
this truth at one sweep4 ,indeed1 it is 
not  possible for it1 alone and unaided 
by the intuition1 to grasp it at  all4  
,hence1 the mathematician who depends   
entirely upon the deliveries  of the    

intellect which conform1 in their       
passage from the conceptual to  the     
written or spoken word1 to all the      
rigors of mathetic requirements1  fails 
utterly in perceiving the magnitude of  
this conception and all  its            
connotations2 he fails because his      
prejudices and the woof and warp  of his
 intellectual habits prevent his        
assuming a sympathetic attitude  toward 
it and thereby precluding at the start  
any calm consideration  of it4 ,and not 
only is this true of the mathematician  
but of all those  whose endeavors are   
confined to the plane of purely sensuous
 and  logical data4 ,it would1          
therefore1 appear that our entire       
attitude  towards things spatial must be
 changed before we can even begin to    
perceive the reality which is really the
 object of all researches in  this      
domain4 ,but1 on the surface1 there is  
after all little difference  between the
 ultimate facts involved in these two   
totally different  conceptions4         

,mathematically speaking1 all           
progression eastward would  terminate at
 the west1 and vice versa2 and the same 
would be true  regardless of the point  
from which progression might originate4 
,always  the terminus would be the      
opposite of the starting point4 ,then1  
too1  it might be said that if we sought
 the space-center we should arrive  at  
the circumference4 ,the difficulty with 
this view is that there is  a very      
remote1 though important1 connection    
between it and the truth  of the matter4
 ,but the partiality of this view1 and  
the absence of  either experience or    
intuition to intimate a more reasonable 
view1  serve effectively to buttress it 
as a hypothesis acceptable to many4     
,thus it is ever more difficult to      
supplant a near-truth than it is to     
gain credence for the whole truth4 ,on  
the other hand1 according to the  view  
which we maintain here1 it is quite true
 that the seeking of the  kosmic        
space-center will reveal the            

circumference2 that the search for  the 
nadir will uncover the zenith2 the east 
effloresces as the west1  and a         
northward journey will wind up at the   
south1 etc41 but in quite a  different  
manner from that which the mathematician
 has in mind when he  postulates the    
curvature of space4 ,our view involves  
no space curvature  nor any other       
spatial distortion4 ,it deals with space
 as reality1 as  a dynamic process1 a   
flux which1 like the sea1 is continually
 casting  itself upon the shores of     
chaos and falling back upon itself only 
to  be recast against the rock-bound    
coast of its chaotic limits4 ,now1  that
 which falls back upon itself and rolls 
in a recurrent movement  upon its own   
surface is life which1 in its recession 
is the natural  and kosmic limitations  
of itself1 generates matter in all its  
varied  expressions4 ,space1 in its     
extensity1 cannot transcend life2 for it
 is  the path which life makes in its   
out-coming1 its manifestation4 ,of  the 

chaotic fringe which circumscribes the  
manifested universe it is  absurd to say
 that it is vital or psychological in   
any sense of these  terms4 ,for         
notwithstanding the fact that out of its
 very substance are  engendered life1   
intellectuality1 spatiality and         
materiality1 it is  nevertheless none of
 these in its primary essence4 ,it is   
,chaos-,kosmos2  because from its       
content the kosmos is evolved1 and it   
still remains2  it is chaos-spatiality2 
chaos-materiality2                      
chaos-intellectuality2                  
chaos-geometricity2 because these are   
engendered by the movement of  life in  
chaos while at the same time there      
remains a residuum of the  chaogenetic  
substance which constitutes the         
limitations of all these  subsequent    
processes4 ,in this sense1 the          
chaogenetic fringe becomes  the limits  
of the manifested universe so that it   
would appear that all  those major      
processes outlined above are finite     

manifestations of the  eternal chaos4   
,but none of those possibilities of     
motion which are  found in these major  
movements of the kosmos can be logically
 said to  exist in chaos4 ,it is the    
embodiment of everything that is the    
opposite  of those qualities which may  
be found in them1 that is1 in           
materiality1  vitality1 spatiality1     
intellectuality and geometricity4       
,apropos to this phase of the discussion
 let us examine briefly one  of its most
 significant implications1 both         
mathematical and kosmic1  which arises  
out of the fact that space is an        
engendered product of  life that is     
bound by the fringe of chaos which      
sustains and limits  it4 ,the chaotic   
fringe plus manifesting kosmos          
constitute the absolute  magnitude of   
the kosmos4 ,the manifestation factor is
 complemented by  the chaos factor and  
together the two define the full        
universe4  ,kosmogony is the universal  
movement of all kosmic elements or      

factors  in diminishing the chaotic     
complement and reducing it to kosmic    
order  or geometrism4 ,it is undoubtedly
 impossible to determine mathematically 
the exact volume of either complement or
 the ratio of the one to the  other2 yet
 it is conceivable that the chaotic     
fringe is greater in  extent than the   
ordered portion of the kosmic uni-circle
 or universe4  ,it is even conceivable  
that the difference1 upon the basis of  
the  meaning of the ,pythagorean        
,tetragrammaton and the view outlined in
 the  ,chapter on the ,mystery of       
,space1 is as seven to three wherefrom  
the  conclusion might be drawn that the 
universe has yet seven complete  stages 
more or less of evolution before the    
close of the ,great ,cycle of           
,manifestation when the fringe of chaos 
shall have been totally used up  in the 
work of creation4 ,but for those who may
 experience impatience at  the          
infinitude of the process when viewed in
 this light the terms may  be reversed  

and the difference may be conceived as  
the ratio of three  to seven wherefrom  
the conclusion would follow that the    
kosmogonic  process is seven-tenths     
complete1 as it will not vary the       
seeming  infinitude either way it may be
 determined4 ,the notion1 despite its   
speculative character1 offers an        
explanation of otherwise inexplicable   
conditions1 and1 on account of its      
profound connotations1 may even  be     
found to be productive of the highest   
good in its equilibrating  influence    
upon our mode of thinking4    ,in any   
event1 there does appear to be a subtle 
relation subsisting  between true       
numbers and kosmogony4 ,number is a     
phase of the kosmogonic  movement1 a    
measurer of the intellect and the       
establisher of the  geometrism of space1
 answering tentatively to the numericity
 of pure  being4 ,in fact1 being        
actually expresses number and number    
itself is an  evolution and not a thing 
posited once for all as a pure1         

invariable  form in the universe4 ,it   
is1 like the kosmos1 in a state of      
becoming  and there may yet appear to   
our cognitive powers a whole series of  
new  numbers pure in itself and         
altogether conformable to the conditions
  reigning at the time4    ,the         
symbology of the circle1 in all times   
recognized as the true symbol  of the   
kosmos in eternity1 of eternity itself1 
of the archetypal1 of  space1 duration  
and ,ultimate ,perfection1 is replete   
with profound  significations4 ,but it  
should be understood that the circle is 
a  symbol of the perfected universe and 
not the universe in a state  of         
evolution4 ,it symbolizes perfection1   
completion and the ultimate  union of   
the manifesting with the archetypal     
which results in the  crowning deed of  
,perfection4 ,the circle is1 therefore1 
not a symbol  of the universe as it now 
stands2 it does not represent a snapshot
  view of the kosmos but the universe as
 a full4 ,it cannot be a full  until it 

has attained the ne plus ultra of       
completion2 for a kosmic  full is that  
state to be attained by the manifested  
kosmos upon the  termination of all the 
fundamental processes now in operation4 
,but it  is this state that the circle  
really represents1 and by virtue of     
which  it possesses its intrinsic       
qualities and also in virtue of which   
the  intellect recognizes these         
qualities4 ,the properties of           
understanding  and recognizance in the  
intellect are veritably fixed by the    
status  quo of the universe during every
 stage4 ,that is1 the focus of the      
intellect1 like the focus of a chromatic
 lens1 is adjusted by the fiat  of the  
nature and eternal fitness of things to 
correspond exactly with  every state    
through which the kosmos itself passes4 
,this is one of the  obvious            
implications of the phanerobiogenic     
behavior of the kosmos  and is          
necessarily resident in the notion of   
the genesis of space and                

intellectuality as consubstantial and   
cordinate factors4    ,wherefore the    
more cogent is the reason for the belief
 that the  inherent qualities of the    
kosmogonic fundamentals2 as1 vitality1  
materiality1 spatiality1 intellectuality
 and geometricity1 are true  variants1  
and that their variability is           
proportional to the progress of  these  
major movements toward the ultimate     
satisfaction of the original  creative  
impulse4 ,may it not be1 therefore1 that
 the indeterminate  character of the    
ratio of the diameter to the            
circumference  7#c4adaeibf44471 is due  
to causes far more profound than the    
crudity of  our micrometers or the mere 
supposed fact of the circle's           
peculiarity8  ,may it not also be true  
that the pi proportion shall become a   
whole  number1 and in its integration1  
keep apace with the perfecting process  
of the kosmos1 diminishing1 by          
retrogression to one or increasing1 by  
progression1 to ten which1 after all1 is

 essentially unity1 being the  perfect  
numeral8 ,it is not without the utmost  
assurance that these  queries will be   
categorically questioned by the         
orthodox1 creed-loyal1  strictly        
intellectual type that we sketch these  
implications1 but it is  felt to be an  
urgent duty to remind all such that the 
most effective  barrier to realization  
in the field of philosophy is an        
intolerant  attitude towards all lines  
of thought which suggest the            
impermanence  of conditions as we find  
them in the kosmos at the present time4 
,the  fact is that our lives are so     
distressingly short that we have neither
  time nor opportunity to watch the     
changing moods of the kosmos nor        
discern the gradual reduction of mere   
appearance to the firm basis of         
reality1 and accordingly1 the intellect 
tenaciously clings to those  notions    
which it derives from the               
instant-exposure which the lens of      
intellectual conceivability allots to   

it4 ,once the view is taken it  is      
immediately invested with               
everlastingness4 ,this everlastingness  
is  then imputed to the kosmos in that  
particular pose1 attitude or state4     
,always the intellect beholds in that   
passing view1 snatched from the         
fleeting panorama of eternal duration1 a
 picture of itself which it  mistakes   
for the reality of the not-self4    ,the
 inclination of the axis of the earth   
toward the plane of its orbit  is       
approximately twenty-three and one-half 
degrees4 ,no well-informed  astronomer1 
however1 doubts now the fact that this  
ecliptic angle  is being gradually      
lessened2 because1 as a result of       
centuries of  observation1 it has been  
found to be decreasing at the rate of   
about  #df4c seconds per century4 ,yet  
no intellect is able to perceive in  any
 given lifetime the actual decrement of 
this angle4 ,it is only by  careful     
measurements after centuries of waiting 
that a difference can  be discovered at 

all4 ,thus it may even be so with the   
ratio of the  diameter to the           
circumference of a circle1 the only     
difference being  that it has not yet   
been determined whether there is a      
decrement or  an increase in the size of
 the ratio4    ,the pi proportion is1   
then1 a register or measurer of the     
slow1  measured approach of the         
manifesting kosmos to the standard of   
ultimate  perfection4 ,therefore1 and in
 view of these considerations1 we may   
not  hesitate to confirm our belief in  
the validity of the notion that it      
actually and literally expresses the key
 to the evolutionary status  of         
kosmogony4 ,the mathematical            
determination which limits it as an     
unchangeable1 inelastic quantity is1    
consequently1 only partially true  and  
leads to the inclusion of this quantity 
under the category of  mathematical     
near-truths1 for such it appears to be  
in spite of its  rigorous establishment4
    ,the formal topography into which   

the intellect spreads when seeking the  
ideal and the abstract is not a         
condition which is derivable from the   
real essence of life or matter1 but1 on 
the other hand1 is a product  of the    
intellect itself partaking of its nature
 rather than of the  nature of reality4 
,there is1 therefore1 a very important  
distinction to  be made between all     
deliveries of the intellect and the     
realism both  of the objects and        
conditions to which the intellectual    
deliveries  pertain4 ,one of the most   
marked peculiarities of the human       
intellect is  the fact that it always   
unavoidably stamps its own nature and   
features  upon every datum which passes 
through it to the consciousness4 ,the   
utmost importance attaches to this      
phenomenon1 for the reason that  it     
points to the necessity of carefully    
scrutinizing intellectual  deliveries   
and the making of allowances for those  
ever-present  characteristics which the 
intellect superimposes upon its data4   

,perhaps  the inherent colorific quality
 which it imposes upon our knowledge    
would be better understood if a         
similitude were indulged at this        
juncture4 ,the intellect may be likened 
to a color-bearing instrument  which1   
when it has once handled an object1     
leaves forever its own color  transfused
 into every cell and fiber of the object
 so that when the  same object is       
presented to the consciousness for      
purposes of cognition  it bears always  
the same peculiar marks and colorations 
which the  intellect1 in its            
manipulation of it1 places thereon4 ,in 
this respect  the intellect may also be 
said to be like a potter who has but one
 mold  and that of a peculiar formation4
 ,hence1 whatever wares it presents to  
the consciousness will invariably be    
found to be molded in conformity  with  
that particular mold4 ,if it were       
possible to view reality or the         
essential nature of things the          
difficulty which now the intellect  lays

 in the path of direct and uncolored    
cognition would be obviated2  for then  
there no longer would be any necessity  
of viewing things as  they are colored  
or molded by the intellect4 ,the        
intuition1 being a  process of pure     
consciousness1 will1 when it has arisen 
to a position  where it may dominate the
 intellect as the intellect now         
dominates  it1 so modify this tendency  
which we see so ineradicably bound up in
  the very nature of the intellect that 
the apparently insurmountable           
difficulties which it has interposed    
between mere perception and a  direct   
cognitive operation will be quite       
completely overcome4 ,thus1 in  the     
above1 is discovered another obstacle   
which posits itself between  the notion 
of space as reality and the intellectual
 determination of  it which the         
mathematician examines and to which his 
consciousness  is necessarily limited4  
,furthermore1 it may be perceived also  
how  easily the mind may be deluded into

 thinking that the intellectual  notion 
which it entertains of space is         
necessarily correct1 when  obversely1 it
 is simply examining a concept which has
 been remade by  the intellect into a   
form which is not at all unlike its own 
peculiar  nature1 and therefore1 as much
 short of reality as the intellect      
itself is4 ,similarly1 if the           
mathematical mind succeed in catching a 
glimpse of the reality of space in the  
form of an intimation1 which1  in itself
 though fragmentary1 is nevertheless    
true1 its consciousness  is finally     
deprived of the true validity thereof   
simply because of the  behavior of the  
intellect in its manipulation of it4    
,the importance  of these intellectual  
difficulties cannot be over-estimated   
for they  furnish the grounds for the   
ineptitude of intellectual              
determinations  made in a sphere of     
motility to which the intellect is a    
stranger4  ,and this fact will appear   
more evident when it is perceived that  

quite the entire content of human       
knowledge has been thoroughly  vitiated 
by them4 ,so that only in those very    
rare moments which 7in a  highly        
sensitive mentality7 enable the         
intuition to gain a momentary           
ascendancy over the intellect is it     
possible for the ,thinker to catch  hold
 of realism itself1 and project the     
truth of what he sees into the  lower1  
intellectual consciousness4 ,but so     
small is that portion of our  knowledge 
which owes its origin to the intuition  
that when compared  with the totality of
 that which we seem to understand it is 
well-nigh  negligible4 ,and then1 when  
it is considered that at present there  
is  no way of conceptualizing adequately
 the intuitograph so as to make it      
propagable the insignificance of this   
form of knowledge is even more  notable4
 ,it can now be seen in how large a     
measure the notion of the  curvature of 
space is merely an intellectual         
translation of a true  intuition into   

the terms of the intellect which1 in the
 very nature  of the case1 can only     
approximate the truth because of its    
colorific  habits4    ,a similar        
declaration may be made of that other   
datum of  metageometrical knowledge     
which postulates the ultimate           
convergence  of parallel lines4 ,in     
fact1 what has been said as to the      
perisophical  nature of the notion of   
space-curvature will apply with equal   
force  to the idea of parallel          
convergence since the latter is a       
derivative  of the former4 ,but there is
 yet another consideration1 apart from  
the  colorific influence of the         
intellect1 which1 although it partakes  
of  the nature of this quality1 is      
nevertheless a near-truth of quite  a   
different order4 ,this may be better    
understood by referring to  the graph   
showing the inverse ratio of objective  
space to the  consciousness4#bh ,let us 
suppose that the graph may also         
represent  the ,thinker's outlook into  

the world of spatiality4 ,it then       
appears  that1 because of that movement 
of consciousness in its pursuit of  life
 which1 as it expands1 makes the        
objective world to appear to be         
diminished in proportion to the extent  
of its expansion1 it is quite  natural1 
under such circumstances1 that parallel 
lines drawn anywhere  in the limits of  
the objective world should seem to come 
to a point in  the ultimate extension of
 themselves4 ,while this graph is not   
meant  to depict such a view1 it may be 
found nevertheless1 to be a true        
delineation of the topography of that   
state of mind into which the            
metageometrician brings himself when he 
visualizes space as curved2  for there  
is no doubt but that a state of         
intellectual ecstasy1 such  as that in  
which the mind of the metageometrician  
must be functioning  in order to        
perceive space in that form1 is quite   
different from the  normal and1         
therefore1 in need of a different       

topographical survey4  ,but1 if we grant
 that in the creational aspects of space
 there is  conceivable an ever-present  
tendency to convolution1 or a rolling   
back  upon itself1 it is imaginable that
 parallel lines inscribed either upon   
its surface or in its texture need not  
necessarily meet but maintain  their    
parallelism regardless of the complexity
 of the convolutions4 ,the  convergence 
of parallel lines is much like a tangent
 in the outgrowth  of the idea from the 
notion of space-curvature4 ,the more a  
tangential  line is extended the farther
 away from the circumference it becomes 
and consequently less in agreement      
therewith4 ,the more subsidiary         
propositions or corollaries are         
multiplied the more remote from the     
truth the determinations become and     
especially is this true of the          
hypothesis of space curvature4      #bh 
,figure #bj4    ,in the notion of the   
manifoldness of space1 by virtue of     
which it is  conceived as existing in a 

series of superimposable and generable  
manifolds of varying degrees of         
complexity1 are discernible traces  of  
that intuitional intimation which       
underlies the assertion that  because of
 the necessary phenomenalization of     
reality for the purpose  of             
manifestation to the intellect it       
appears to exist in a series of         
separate degrees1 each one more refined 
and subtle than the preceding  one and  
requiring a more highly developed       
species of consciousness for  its       
comprehension4 ,in other words1 that    
intuitional glimpse of the  essential   
character of reality1 as viewed by the  
human consciousness1  which impinged    
upon the minds of ,,riemann and         
,,beltrami leading them  to postulate as
 a corollary proposition to             
space-curvature1 its  manifoldness1 is  
nothing more nor less than the intuition
 that the  universum of spatiality      
cannot otherwise present itself to the  
intellect1 owing to its peculiar        

adaptation to the sensuous1 except by  a
 series of continuous degrees which are 
perceptible only in proportion  as the  
understanding is magnified to conform   
with it4 ,after all1  however1 it is not
 improbable that the very objectivism of
 the universe  in manifestation subsists
 in just the manner in which this       
intuitive  glimpse implies and that the 
wisdom and utilitariness of the         
kosmogonic  process which engendered    
spatiality are clearly demonstrated in  
that  arrangement of the contents of the
 kosmos which presents the grossest     
elements of phenomena first to the      
intellect in its most impotent  state   
while reserving the less crass for that 
time when the ,thinker  shall have      
evolved a cognitive organ adaptable to  
its presentations4  ,those              
metageometricians who cling to the idea 
of the manifoldness of  space1 based as 
we have shown upon the                  
pseudo-interpretation of a  rather vague
 hint arising out of an unquestionably  

true intuition1 have  allowed themselves
 to fall into the unconscious error of  
magnifying  the importance of the mere  
insinuation as to the space-nature to   
such  an extent as wholly to obscure in 
their own minds and in the minds of     
those who think after them whatever of  
the true vision that may have  been     
grasped by them4 ,furthermore1 it is    
indubitably true that that  same        
peculiarity of arrangement by which     
impalpable and invisible  forces really 
subtend gross matter producing that     
subtle schematism in  virtue of which   
the visible is subjoined to the         
invisible1 the sensuous  to the         
non-sensuous1 spirit to matter1 etc41   
also characterizes the  appearance of   
spatiality to the human understanding4  
,while there is a  superficial semblance
 of separate and discrete manifolds into
 which  space may be divided there are1 
in reality1 no such sharp lines of      
demarkation between the subtle and the  
gross1 between the visible  and the     

invisible or between spirit and matter1 
each of these being  capable of         
reduction1 by insensible degrees1 into  
the other regardless  as to whether the 
reductional process originates on the   
side of the  most refined or on that of 
the grossest4 ,accordingly1 there are no
  reasonable grounds upon which the     
notion of a space-manifold may be       
justified except as a metageometrical   
near-truth4    ,in addition to the      
foregoing1 there are yet other very     
fundamental  considerations which would 
seem to debar the totality of analytical
  conclusions as to the nature of space 
from any claim to ultimate  reliability 
and trustworthiness4 ,these are first3  
the fact that  analyses are absolutely  
incapable of dealing with life2 that    
being  the direct product of a sort of  
mechanical consistency which marks  the 
intellectual operations it has          
adaptability only for dealing  with     
fragments or disconnected parts1 and    
that without any reference  whatsoever  

to the current of life or the flow of   
reality which has  produced the parts4  
,this fact is clearly shown in that     
attitude of the  understanding which    
inevitably leads it to the declaration  
that a line  is an infinite series of   
points1 a plane an infinite series of   
lines1  and a cube1 an infinite series  
of planes1 and so on1 indefinitely4  ,to
 do this1 to look upon all phenomena as 
a series of parts similar  to each other
 and piled1 one upon the other1 or      
juxtaposed in the  manner which they are
 discovered in the sensible world1 is   
the natural  tendency of the intellect  
and this tendency finds its most facile 
expression in analytics4 ,inadaptability
 of this sort is especially  observable 
in all problems of arithmetical analysis
 in which the vital  element is a       
factor4 ,when these analyses are carried
 to their logical  conclusion1 as has   
been shown in the chapter on ,the       
,fourth ,dimension1  invariably they end
 in an evident absurdity4 ,but it is at 

their very  conclusion where the        
life-element is encountered1 where      
reality is  approached1 that they break 
down4 ,the failure of analysis1 then1 to
  encompass life1 to fit into its       
requirements and to satisfy its natural 
outcome seems clearly to establish the  
basis of the perisophical  nature of the
 entirety of analytical claims1         
especially that species  of analysis    
which seeks the remoter fields of the   
conceptual for its  determinations4     
,second3 the close connection which has 
been seen to  subsist between space and 
life as joint products of the same      
movement  makes it obvious that the same
 ultimate rule of interpretation must be
  applied to both in order to insure    
correct and dependable judgments        
regarding them4 ,how different would be 
the intellectual attitude  towards space
 if it were considered in the same light
 as vitality1  provided one really      
understood anything about vitality6     
,moreover1 as  it appears certain that  

the path of the intellect does not run  
in  the same direction as the path which
 life makes1 but in an inverse          
direction1 it is clear that the         
judgments of the former1 as to the      
action and essence of the latter1 must  
necessarily be ultimately  unreliable4  
,it can readily be seen1 however1 that  
should the intellect  be focused so as  
to follow the path of life1 to attach   
itself to the  very stream of life1 it  
would have necessarily to neglect       
materiality4  ,and such an adjustment   
would1 of course1 obviate the need of a 
material life at all for humanity4 ,in  
fact1 a physical life with an  intellect
 would be impossible under such         
conditions4 ,it is well to  recognize   
the suitability of the present          
schematism and not to become  unwisely  
restive because of it2 but it is also   
fitting that we should  discriminate    
between that which is possible for the  
intellect chained  to materiality and   
that which is impossible for it1 in such

 a state1  when foraying in a territory 
foreign to its nature1 and beyond its   
powers to master4    ,the predominating 
tendency in the intellect to account for
 the  universe of life1 mind and matter 
upon a strictly mechanical basis  is    
undoubtedly due to the constitution of  
the intellect which does  not admit of  
its direct consideration of the vital   
essence of things4  ,we are bound       
ineluctably to the surface of things4   
,all our knowledge  is therefore        
superficial4 ,we are even bound to the  
surface of ideas1  and cannot penetrate 
to the interior of these realities4 ,our
 art  is the reproductions of           
superfices2 our philosophies are the    
husks  of eternalities2 our religions1  
the habiliments of relations1 and  while
 it cannot be doubted that this         
arrangement is pre-eminently  the best  
possible one for the present stage of   
man's evolution1 it  is nevertheless    
worth while to note that it is this very
 restricted  activity of the intellect  

which shuts out from man's consciousness
  those very elements about which he is 
most concerned when he goes into  the   
field of philosophy in search of a      
solution to his unanswerable  queries4  
,but some progress most surely is made  
when the mind is  enabled to see its    
plight and recognize what are the       
difficulties and  limitations that lie  
in its path of ultimate attainment4     
,it is believed that the mechanistic1 or
 true1 character of the  intellect      
reached its zenith in the mind of       
,,lagrange when he succeeded  in        
reducing the entirety of physics to     
certain mechanical laws and  formul     
which he embodied in his ,mecanique     
,analytique ,this work  is undoubtedly  
the capstone of intellectual endeavors  
and stands  as a monument which marks   
the culmination of the present stage of 
intellectual development4 ,in thus      
placing the ,mecanique at the apex  of  
intellectual endeavors it is not thereby
 meant to be implied that  the intellect

 shall not make more progress nor that  
other formul1  equally as marvelous as  
those which ,,lagrange discovered1 may  
be  devised1 nor that other laws1       
heretofore undreamed of1 may be found2  
but what is maintained is the fact that 
while there will be growth and          
development these will run along other  
channels1 perhaps in the realm  of the  
intuitable1 and not any longer1         
especially so notably as now1  in an    
opposite direction against the current  
of life and reality2 and  further1 that 
there will be a gradual turning away    
from mechanics to  biogenetics1 from    
diversity to unity1 from the purely     
intellectual  to the intuitional1 and   
withal a final getting rid of the bonds 
of  illusion1 of that thralldom of      
mechanics1 whereupon will slowly arise  
the obsolescence of all those           
disparities which may now be recognized 
in our knowledge and in the applications
 of the intellect to the data  of the   
objective world4    ,because the        

intellect is unsuited to deal with      
reality1 and because of  its peculiar   
adaptation for diversity1 for           
multiplicity1 due to its  mechanistic   
modus vivendi1 there has grown up a     
voluminous catalogue  of systems of     
philosophy4 ,these embody such a        
multitudinous array of  beliefs1 ideas1 
conceptions1 theories and conjectures   
and constitute  a movement in human     
thought which oscillates between the    
empiricist  on the one hand and the     
transcendentalist on the other2 between 
the  idealist and the realist1 leaning  
sometimes towards the ,platonic1  the   
,cartesian and the ,kantian and at other
 times towards ,,spinoza1  ,,aristotle1 
,,spencer and ,,socrates1 always        
terminating by multiplying the  number  
of diverse beliefs rather than unifying 
them that the conclusion  is unavoidable
 that so marked a lack of unanimity is  
indicative of  a profound mental        
prestriction4 ,it was1 therefore1       
inevitable that  mathematics should fall

 under the same spell and brook no let  
nor  hindrance until it had succeeded in
 devising several diverse systems  of   
geometry which it has done for the mere 
joy of doing something1 of  following   
its instinctive aptitudes4 ,there is no 
other basis for the  heterogeneity of   
our philosophies1 our mathematics1      
indeed our beliefs  than this           
mechanical1 and hence1 radically        
illusionary character of the  intellect 
in consequence of which we have had to  
be satisfied with mere  glimpses1 hints1
 intimations and faint glimmerings of   
reality1 of life1  and of those kosmic  
movements which1 if we had the ability  
to trace  them from their source        
outward1 would lead us unerringly to a  
truer and  deeper knowledge of those    
things that under the present schematism
 must  remain for us a closed book4     
,the criterion of truth for us1         
constituted as we are and wedged  in    
between the stream of life and its shore
 of materiality1 must  be that which    

relates our knowledge both to the stream
 and to the  shore4 ,it must be so that 
all predicates which purport to approach
  it shall exhibit a dual reference--one
 that relates to materiality  and       
another that relates to vitality1 and   
yet a third that shall  combine these   
two relations into one4 ,all assertions1
 therefore1 which  pertain exclusively  
to either of these elements--to         
materiality or to  life--are necessarily
 partial1 fragmentary and perisophical  
in nature4  ,mathematics1 because it    
relates to matter and the mechanical    
forces set  up by matter acting against 
matter cannot be said to agree with such
  a criterion2 art1 because it relates  
to snapshots or static views of  matter 
is even more remote in its agreement2   
philosophy1 as it has been  known in the
 past and is known to-day1 because it   
seeks to deal with  a vitality fashioned
 after the image of materiality has     
failed when  posited alongside of this  
criterion2 and thus1 the intellectual   

toil of  millions of years has been in  
vain in so far as it has not succeeded  
even in raising a corner of the cover   
which hides reality from our view4    ,a
 near-truth is any variation from this  
standard1 this norm or  criterion4 ,it  
may be either logical1 cognitive1       
scientific or even  metaphysical4 ,to   
define3 a logical truth is a predicate  
based upon  and involving the coherency 
and consistency of thoughts themselves2 
a  cognitive truth is the conformity of 
knowledge with so much of reality  as is
 known2 scientific truth is the         
conformity of thoughts to things  and   
conditions4 ,all of these are obviously 
near-truths4 ,then1 too1 a  near-truth  
may be defined as an assertion based    
upon the criterion of  truth but falling
 within the category of cognitive truths
 owing to  insufficiency of data or     
vision4 ,such indeed are those          
metageometrical                         
predicates--n-dimensionality1           
space-flexure1 space-manifoldness and   

all other assertions based upon these in
 general and specifically4 ,any         
recognition of truth must clearly       
embrace both the vital and material     
aspects of its subject in order to be   
adequately inclusive1 that is1 it       
should include the causative1 the       
sustentative1 the relational and the    
developmental factors4 ,these four      
factors are considered necessary and    
sufficient to determine the conformity  
of any view to the criterion of  truth  
for when we are cognizant of the cause  
of a subject1 understand  the           
sustentative factors which keep it in   
existence1 are conversant  with its     
relations to other subjects and can     
follow its developmental  variations    
until we come to its final status1 why  
then1 our knowledge  is both sufficient 
and ultimate so far as that subject is  
concerned4  ,is it asking too much of   
mathematics or of philosophy or any     
system of  thought that it conform to   
these standards or to this criterion    

before  we shall accept it as final8 ,or
 shall we be satisfied with less than   
this8 ,let us hope not4    ,in the      
foregoing presentation stress has been  
placed upon the  fragmentary1 and       
therefore1 illusionary character of the 
intellect in  order to arrive at an     
understanding of the difficulties under 
which  real knowledge has to be acquired
 and to indicate the inanity of all     
attempts to resolve the riddle of space 
by means of mathematics though  regarded
 as the most typical exemplification of 
the mechanistic nature  of the          
intellect4 ,and further1 to show that1  
on account of the radical  incongruity  
which estranges life1 the producer of   
spatiality1 from the  intellect which   
returns again to scrutinize the passage 
of life in  its outward expressions1 no 
hope of ever gaining the true viewpoint 
by means of the intellect need be       
entertained4 ,but in doing so1 it is    
deemed fitting that a note of warning   
should be sounded against any  abortive 

attempts that may be made to obscure or 
distort the results of  such a close    
discrimination lest the true import of  
the examination be  lost for1 if we     
emphasize the vanity of the intellect in
 the pursuit of  that which it is by    
nature unsuited to attain we also       
equally stress  the wise utilitarianism 
which limits it to the performance of   
the  tasks assigned while at the same   
time reserving for the function of  more
 highly evolved powers1 and indeed1 for 
the intuition itself1 the  solution of  
the riddle of spatiality4 ,and if we    
declare the futility  of the            
mathematical method in all endeavors    
aimed at unveiling the  mysteries of    
life and mind1 and of that movement     
which has its roots  set in eternal     
duration from which it proceeds in an   
endless continuity  of purpose and      
promise1 we do also recognize that in   
the science of  mathematics the         
intellect shall1 as in no other method  
of cognition1  most fully fulfill the   

kosmic intent of its existence2 and     
moreover1 in  the pursuit thereof it    
shall push the frontiers of its         
possibilities  outward until it can be  
said almost to be able to make          
disposition of  life itself--at least to
 that point where1 when the intuition   
shall  have come into its own1 the      
passage from the mechanics of matter to 
the  dynamics of life1 shall be         
comparatively easy and natural4         
,,chapter ,x      ,,the ,,media ,,of    
,,new ,,perceptive ,,faculties      ,the
 ,spiritualization of ,matter the ,end  
of ,evolution--,sequence    and ,design 
in the ,evolution of ,human             
,faculties--,the ,upspringing           
,intuition--,evidences of ,supernormal  
,powers of ,perception and    the       
,possibility of ,attainment--,the       
,influence and ,place of the            
,pituitary ,body and the ,pineal ,gland 
in the ,evolution of ,additional        
,faculties--,the ,skeptical ,attitude of
 ,empirical ,science and    the ,need   

for a ,more ,liberal ,posture--,the     
,general ,results of    ,pituitarial    
,awakening ,upon ,man and the ,theory of
 ,knowledge4      ,evolution is a       
continuous process and the primal       
impetus back of the  great on-flowing   
ocean of life acts infinitively4 ,it is 
not terminated  when life has succeeded 
in perfecting a form for the perfection 
of  forms in themselves is not the end  
of vital activity4 ,the end of          
evolution is the complete               
spiritualization of matter4 ,so that it 
does  not matter how perfect a form may 
be either subjectively or in its        
adaptation to environments2 it does not 
matter how faultless a medium  for the  
ensouling life it may be1 there is ever 
the eternal necessity  that life must   
drive it back over the path of its      
genesis until it  shall be transmuted   
into pure spirit4 ,adaptation succeeds  
adaptation  and with each there is a    
change in the form and this process     
continues  until there is a more or less

 perfect congruence between form and    
juxtaposed environmental conditions4    
,but no sooner than agreement  has been 
attained under one set of conditions new
 conditions arise  and require a new    
setting1 new adaptative movements4 ,thus
 there is a  continuous proceeding from 
stage to stage1 going from the grossest 
to  the subtlest and most refined1      
always the form is being pushed onward  
and upward by life4 ,but adaptation is  
not undergone for the benefit  of the   
form1 but more truly for the informing  
principle4 ,it is the  progression of   
the life-element which constitutes the  
adaptation of  form to form and to their
 peculiar environs4 ,the form is a tool 
or  instrument of life which it discards
 the moment it fails to respond  to its 
requirements4 ,thus forms are constantly
 being assumed and as  constantly being 
relinquished4 ,but no effort of life is 
lost regardless  as to whether the      
action is performed in one or another   
form4 ,the  totality of matter is       

perpetually being acted upon by the     
totality of  life4 ,every appulse of    
life against matter means an added push 
in the  direction of spiritualization4  
,the totality of such appulses of life  
against matter may seem infinitely small
 in the visible results which  they     
produce in the process of               
spiritualization2 but with each there   
is an eternal gain in that movement that
 shall end in the complete              
transmutation of materiality into       
spirituality4 ,this action of life in   
metamorphosing matter1 the nether pole  
of the great pair of opposites1  into   
spirituality1 its copolar factor1 in its
 outward1 visible effects1  is what we  
vaguely call evolution4 ,and such it is2
 for life is merely  unfolding that     
which it has enfolded4 ,matter1 having  
been involved as a  phase of kosmic     
involution1 is now being evolved4    ,in
 the genesis of the kosmos there appear 
to be three great undulations  in the   
universal current of life4 ,the first of

 these prepares the  field by depositing
 that elemental essence which is to     
become the  world-plasm2 the second     
precipitates the universum of           
materiality1  spatiality and            
intellectuality1 not as we now know     
them1 of course1  but as potencies2 the 
third great undulation in the current of
 life  effects the endowment of the     
world-plasm with those tendencies that  
are to build around themselves forms    
appropriate to their fulfillment4  ,this
 ensoulment of the world-plasm with     
tendencies and the consequent           
segmentation of it into separate forms  
by these tendencies constitute  the     
primary stages of that procedure of life
 which results ultimately  in the       
up-raisement of matter and its final    
exaltation into pure  spirit4 ,hence1   
the entire mass of materiality is       
besieged on all sides  by the sum-total 
of life and the former is being raised  
slowly and  irresistibly to heights that
 are immeasurably more sublime than its 

present degree of grossness4    ,it     
appears paradoxical1 therefore1 that    
life1 although in all respects  vastly  
superior to matter1 should become the   
apparent vassal of  materiality and give
 itself up to all the strict rules of   
imprisonment  which are imposed upon it 
by the properties and qualities which we
  observe in matter4 ,it seems so       
subject to every whim and fancy of      
matter that one is inclined to think    
that matter and not life is the  chief  
designer of universal destiny4 ,this is 
not a condition to be  wondered at so   
much1 for the reason that this apparent 
vassalage1 this  seeming enslavement of 
life by matter1 is due to that superior 
and most  marvelous adaptability of life
 which it enjoys in contradistinction to
  the relative unpliability of matter1  
and due also to the fact that life  is  
kinetic and matter1 being a mere deposit
 of life1 is static4 ,life is  mobility 
while matter is immobility and thus in  
possessing a greater  range of freedom  

is1 of course1 correspondingly superior2
 but in this  adaptation of itself to   
the labyrinthine cavities and           
multiformed  interstices in matter it   
exhibits but a seeming serfdom which is 
really  not a serfdom but a mastery4 ,it
 is as if a man had taken lumber1       
hardware and stone and built a house    
wherein he might dwell--life has  merely
 used matter1 molded and fashioned it so
 as to make for itself  a medium1 a     
dwelling-place wherein it operates1 not 
as a slave but as  a master possessing  
unlimited freedom of motility4 ,in the  
production  of a form life stamps upon  
it1 once for all time1 the path of its  
engendering action4 ,it leaves its      
finger-prints upon the mold which  it   
makes for itself4 ,so that if we would  
know where life has been  or where it is
 we should look for its finger-prints   
7organization72  we should observe the  
sinuosities which mark its pathway1     
remembering  always that it is life that
 has formed the intricacies and         

complexities  of the form into which it 
pours itself so accommodatingly in order
 that  it may raise that form1 develop  
and transmute it into something higher  
and better4    ,when we speak of form it
 must not be understood thereby that    
reference is made only to the gross     
physical form1 but to the entire  range 
of vital assumptions or vehicles which  
life ensouls for purposes  of           
manifestation4 ,this range we believe to
 cover the whole path of  kosmogenesis  
seriating from the densest to the most  
subtle4 ,our chief  concern1 however1 is
 the immediate effect which the totality
 of life's  operations will have upon   
humanity or the form which it ensouls as
  the human organism4 ,for it is        
impossible that humanity shall escape   
either the general or the specific      
results of the exalting power which     
life exerts over materiality and its    
appurtenances4 ,it is1 of course1       
impossible here to go into the various  
implications of this general  forward   

movement of the universum of materiality
 or even to outline  briefly the        
divergent lines of operation into which 
a satisfactory  exposition of this view 
would naturally lead4 ,and then to do so
 would  be inappropriate in a volume of 
this kind4 ,so we shall have to be      
content at this juncture to limit our   
study to a consideration of what  we    
believe to be some of the immediate     
indications of this vast and  most      
far-reaching phenomenon4    ,in the     
chapter on the ,genesis and ,nature of  
,space it is shown that  the material   
universe is engendered at the same time 
and by the same  movement or process as 
the universum of spatiality and         
intellectuality  and that as the passage
 from chaos to kosmos proceeds the      
function of  this movement is changed   
gradually from engenderment to          
exaltation  wherein materiality is      
transmuted into spirituality4 ,it is1 of
  course1 obvious that as materiality is
 exalted so are spatiality and          

intellectuality2 and that as the one    
becomes more and more refined1  capable 
of answering to higher and yet higher   
requirements so do  all the others4     
,for1 at work in all and through all of 
these1 is  the current of life which    
pervades them1 engendering1 sustaining  
and elevating as it proceeds4 ,so that  
as matter has evolved added             
characteristics and properties1 each    
answering to a given need  and arising  
out of the necessities inhering in the  
stage at which  it appeared1 so has the 
intellect evolved faculties to          
correspond  therewith4 ,in other words1 
the evolution of faculties for the      
expression of the human intellect has   
proceeded synchronously with  the       
evolution of material qualities4 ,and   
whenever a new faculty or an  additional
 scope of motility is achieved by       
humanity there is always  found a set of
 kosmic conditions which answers        
thereto4 ,the cardinal  principles of   
the doctrine of evolution are not1      

therefore1 adverse to  the conclusion   
that the organs of                      
sense-perception--hearing1 touch1       
sight1 taste and smell--have not been   
endowed upon the human race or  attained
 by it at one time2 but rather that each
 answering to a newly  acquired need and
 opening a wider scope of motility for  
the intellect  has been evolved         
separately and in due order4 ,it would  
also seem that  the quality of          
consciousness1 as it has been manifested
 in the various  stages of life through 
which it has passed1 and especially the 
mineral1  vegetal and lower animal1 has 
not always been of the same degree of   
efficiency4 ,nor has it enjoyed the same
 kind of freedom which it now  enjoys in
 the highly evolved genus homo4 ,it is  
equally apparent that  matter itself has
 not always been in possession of the   
same qualities  and characteristics     
which it now exhibits2 but that it1 too1
 has  gone through various stages of    
evolution bringing forward into each    

new stage the transmuted results of each
 preceding one as a basis  for further  
evolution and expansion4 ,the           
innumerable archological  evidences     
which support this view make it         
unnecessary to do more than  state the  
facts1 as they appear to be             
substantiated by indubitable            
testimonies4 ,furthermore1 it is        
believed that the outstanding           
implications of these phenomena will not
 be successfully controverted  by those 
who are disinclined to see such         
implications in the  evolutionary       
process4 ,in a previous chapter we have 
briefly sketched  the characteristics   
which mark the upspringing of a new     
faculty showing  how1 at first1 it      
appears as an abnormality which exhibits
 itself in  a very few individuals only1
 and that in a more or less indefinite  
manner2 and how later the number of     
individuals in which it appears         
gradually increases1 the definiteness of
 the faculty1 at the same time1         

appearing more marked2 then1 like a     
tidal wave1 it recurs in a still  larger
 number of persons until1 at last after 
a long period of time  usually several  
thousands of years1 it becomes universal
 exhibiting  itself in every individual 
and appearing as a hereditary           
characteristic  of the entire human     
race4 ,it is1 therefore1 not without    
assurance as to  the ultimate soundness 
of this view that we make the assertions
 which  follow this brief introduction4 
,it has already been stated that for a  
very obvious reason1 namely1  the       
satisfaction of the needs of our present
 humanity1 the intuition  is for the    
time dominated by the intellect and held
 in subjugation by  it so that all of   
man's external operations are governed  
and dictated  almost entirely by the    
intellectuality1 allowing the intuition 
only  rare moments when it can come to  
the fore at all4 ,this is the rule in   
the evolution of faculties and          
characteristics4 ,the higher faculty1   

although potentially present in every   
way1 is ever held in abeyance  while the
 lower is brought1 under the rigors of  
its own evolution1 to a  point where its
 joint operation with the higher may be 
executed with  the least possible       
friction and retardation as also with   
the greatest  possible cordination and  
coperation4 ,accordingly1               
notwithstanding the  fact that          
materiality must possess in potentiality
 all the qualities  which it will at any
 time reveal1 it is nevertheless        
necessary that  these qualities shall   
come forth gradually and in due order4  
,similarly1  humanity has come into     
possession of its various faculties of  
mind1 and  powers of physiological      
functions1 by insensible degrees1 the   
higher  always being held in abeyance   
until the lower is fully developed4     
,those faculties which are to bestow    
added powers1 additional freedom  and a 
greater scope of motility are the ones  
which appear later than  those which are

 truly primitive in character4 ,these   
facts have been  amply demonstrated by  
the science of embryology wherein it is 
shown  that ontogeny is a recapitulation
 of phylogeny4 ,that is1 the  history of
 the development of the individual is a 
recapitulation of  the development of   
the species4 ,thus the various stages of
 development  through which the human   
embryo passes while in utero are but a  
repetition of similar stages through    
which the entire human species has      
passed in its phylogenetic development4 
,wherefore1 it is certain that  humanity
 has not attained1 at one and the same  
time1 all the powers of  mind and body  
which it now possesses2 that the        
childhood of the human  race represented
 a time when it had but few faculties or
 organs of  sense-perception--indeed a  
time when the higher sense-organs of    
smell1  taste and sight were entirely   
lacking although residing in            
potentiality  therein4    ,it is        
undoubtedly true that the earth has     

passed through a similar  evolution with
 respect to its own material            
characteristics1 that  its childhood    
was1 in all points1 analogous to the    
childhood of  humanity2 that the air1   
earth and water were wholly absent1     
except in  potentiality1 during the     
nebulous youth of its genesis4 ,it is   
even  probable that there are at work   
to-day processes which in the future    
shall culminate in the evolution of     
newer1 higher and more complicately     
organized species of plants1 animals and
 minerals4 ,every year brings  fresh    
evidences that crystallize the          
conviction that the earth has been  the 
scene for the appearance of many strange
 orders of animal life4  ,fossiliferous 
strata are continually yielding         
incontestable testimonies  of changing  
flora and fauna4 ,we count the animal   
and vegetal life of  to-day as being    
more highly developed than that of any  
other previous  age1 and it is well that
 this is so1 for simplicity of          

organization and  primality of          
manifestation are always succeeded by   
complexity and a  greater scope of      
adaptability4    ,we have said that the 
whole of that movement of the intellect 
which  has brought forth the            
metageometrical creations of            
hyperspaces1 the  curvature of space and
 its manifoldness together with the     
entire  assemblage of mathetic          
contrivances are merely the early       
evidences of  the appearance in the     
human race of a new faculty1 a new      
medium of  perception whereby the       
,thinker shall acquire a still greater  
range of  motility than that now offered
 by the intellect4 ,attention has been  
called also to the fact that this       
phenomenon has been manifested not      
alone in the field of mathematics1 but  
in art1 religion1 politics  and also in 
science in which we have only to witness
 the marvelous  strides already made in 
the discovery of radio-active           
substances1  the ,,roentgen1            

,,becquerel1 ,,leonard and other kinds  
of rays4 ,it ,is quite  confidently     
believed that these forward movements in
 every branch of  intellectual pursuit1 
these combined efforts of the intellect1
 in  peering into the occult side of    
material things1 are in response to the 
evolutionary needs of the ,thinker1 and 
in addition1 are the evidences1  and    
shall in time be the cause1 of the      
development of an additional set  of    
faculties4 ,function1 or the performance
 of acts1 determines faculty  or the    
power of action and ultimately the organ
 itself4 ,thus the mere  wish to perform
 aroused by desire and vitalized by the 
will actually  terminates1 in the course
 of time1 in the genesis of a faculty1  
or the  power to perform4 ,the constant 
upreaching yearnings of the ,thinker    
through his intellect for greater       
freedom and a larger scope of action1   
the desire to peer into the mysteries of
 life and mind1 the infantile           
out-feelings of the mentality after some

 safer and surer basis for its  theory  
of knowledge cannot fail in producing   
not only the faculty or  power to       
satisfy these cravings but the very     
organ or medium by virtue  of which the 
satisfaction may be attained4    ,it is 
not strange that in mathematics the     
intellect should have found  first the  
clue to the existence of a higher sphere
 of intellectual  research wherein it   
might become the creator of the various 
entities  which peopled the new found   
domain2 it is not strange that the      
mathematician should1 in this instance1 
have assumed the role of  the prophet   
proclaiming by various mathetic         
contrivances 7although  unconsciously7  
that the human race is nearing that time
 when it shall  actually be able to     
function consciously in some higher     
sphere2  neither is it to be wondered at
 that the voice of the prophet is heard 
and respected throughout the earth2 for1
 indeed the mathematician is  a         
spokesman who1 as a rule1 is unmoved by 

sudden outbursts of passion  and        
ecstatic frenzies of emotions but calmly
 and dispassionately  verifies his      
conclusions1 tests them for consistency 
and having found  them to satisfy the   
most rigorous mathetic requirements     
hesitates not  to propound them4 ,for   
this cause humanity respects the        
mathematician1  and when he speaks      
listens to his voice4 ,it is well1 too1 
that this  is so2 for the history of    
mathematics is clearly the history of   
the  development of the intellect4 ,so  
exact a determinator of the quality  of 
intellectual efficiency is it that its  
reign may be said to be an  absolute    
monarchism whose lines of dominance     
extend to the minutest  desire or       
appetency4 ,it has always been the guide
 of the intellect1  going before1 as it 
were1 blazing the trail1 pushing back   
the frontiers  of the intellect's domain
 and clearing away the debris so that   
the  intellect with its retinue of      
servitors might have an easy path of    

progress4    ,mathematics1 however1 has 
not the aptitude to serve the intuition 
as  it serves the intellect4 ,so the    
path into which the intuition would     
lead humanity the mathematician1 because
 of his training and peculiar           
functions1 is unprepared to enter4 ,it  
is for this reason that when            
mathematics leads the intellect up to   
that point where it encounters  life it 
fails1 it becomes confused and its      
dictatorship becomes a  mockery1 its    
decrees remain unexecuted and futile4   
,in taking this view  we have certainly 
no desire to offend the mathematician or
 to detract  from the glory of his      
monarchistic rulership over the         
intellectual  progress of the race2 for1
 in truth1 mathematics is the diadem of 
gold  wherewith man has crowned his     
intellect4 ,yet it is well1 yea         
imperative  in the light of recent      
developments in the realm of hyperspace1
 that  a careful discrimination should  
be made as between the sphere of the    

intellect and that to which the         
intuition shall attain4    ,the         
intuition1 long held in abeyance until  
the intellect should be  fully crowned  
and reach the zenith of its powers1 is  
now coming to the  front4 ,it will be   
many centuries perhaps before it shall  
have grown to  such proportions as those
 already attained by the intellect2     
perhaps  a few thousand years may pass  
before the intuition shall have evolved 
to that point where it may labor as     
coadjutor to the intellect2 but         
undoubtedly the time will come when it1 
too1 shall reward the ,thinker's  labors
 with that which shall be more precious 
than the crown of gold  which the       
intellect has won4 ,then1 the intellect1
 grown old and  decrepit with years of  
reigning shall become dim and           
crystal-shaped and  finally pass into   
automatism or reflexive movements where 
without the  urge of volitional impulses
 it will perform with exactness1        
precision  and utter loyalty the tasks  

which it has learned so well to execute 
in the days of its forgotten glory4     
,mankind will then be free4 ,a new      
freedom1 wherein the erstwhile lightning
 flashes of intuition will  become fused
 into one glorious sheen of             
all-revealing light1 shall come  to men 
and thus the race resplendent will walk 
the earth enshrined in  the majesty of  
divine powers attained as a result of   
millions of years  of aspiration4       
,that there are supersensuous realms so 
far above the range of our  senses as to
 be entirely beyond their ken needs now 
no proof or  argument2 for the scientist
 has demonstrated1 by the invention of  
instruments of extreme delicacy and     
precision1 that such a world does       
really exist4 ,already we know of stars 
so distant that1 though light  traverses
 in the brief space of an hour six      
hundred million miles1  they might have 
ceased to shine before the pyramids were
 built and  yet be visible to us in the 
skies4 ,if the human eye were as        

sensitive  as the spectroscope many     
thousand tints and shades might be added
 to  the world of color2 if they        
possessed the magnifying powers of the  
microscope we should live in constant   
terror and awe of the monstrous         
entities that teem in the water which we
 drink and in the air which  we breathe2
 and if our ears could detect the       
microphonic vibrations  which register  
in the delicate apparatus of some       
microphones the dead1  vacuum-stillness 
of nature's great silences would appear 
as a babel of  voices by the seaside4   
,the sense of touch1 responding to the  
same range  of vibrations as the        
micrometer1 would reveal actually the   
interstices  between particles of the   
densest elements2 and gold1 silver1     
platinum  and mercury would seem but    
honeycombs of matter4 ,but1 to the      
forward-looking there is no element of  
absurdity in the expectation  that all  
these senses shall1 one day1 be able to 
dispense with the  artificial aid of    

physical apparatus and perform1 with    
even greater  precision and             
faithfulness1 the task which they now   
perform so crudely  and ineffectively4  
,there are without doubt vibrations of  
taste and  smell which are so far above 
the range of these senses that they have
  no effect upon them whatsoever4       
,notwithstanding the fact1 however1 that
  the galvanometer1 microscope1 the     
microphone1 the spectroscope and the    
telescope have extended thus the sphere 
of sense-knowledge there are  yet       
subtler vibrations to which these       
delicate instruments do not and  ought  
not be expected to respond4 ,but to say1
 as do many empiricists1  that since    
these phenomena cannot be detected by   
scientific instruments  they do not1    
therefore1 exist seems to be expecting  
too much of  material means as well as  
exposing oneself unnecessarily to       
criticism  on the grounds of extreme    
materialistic appetences4    ,there is  
indeed need of a more liberal attitude  

among men of science  towards the world 
of the unseen4 ,intolerance of the data 
which it  offers will for a time perhaps
 preserve the aloofness of scientific   
dogmatism inviolate but there will most 
surely come a reaction against  the     
dogmatism of science and men will seek  
freedom and attain it  despite their    
fetters4 ,sir ,,oliver ,,lodge1 in his  
book1 the ,survival  of ,man1#bi says3  
,man's outlook upon the universe is     
entering upon  a new phase4             
,simultaneously with the beginning of a 
revolutionary  increase in his powers of
 physical locomotion--which will soon be
  extended to a third dimension and no  
longer limited to a solid or  liquid    
surface--his power of reciprocal mental 
intercourse is also in  process of being
 enlarged2 for there are signs that it  
will some day be  no longer limited to  
contemporary denizens of earth1 but will
 permit a  utilization of knowledge and 
powers superior to his own1 even to the 
extent of ultimately attaining          

trustworthy information concerning other
  conditions of existence4      #bi ,see
 pp4 #cch1 #cda4    ,it is the author's 
good fortune that he has for a period   
extending over  several years been able 
to verify the conclusions which ,sir    
,,oliver  ,,lodge expresses in the      
above1 and thus to satisfy his own mind 
that the  process by which man's mental 
powers are being enlarged is indeed     
demonstrable by actual observation and  
experimental methods4    ,,lodge        
continues3      ,the boundary between   
the two states--the known and the       
unknown1    is still substantial1 but it
 is wearing thin in places1 and like    
excavators engaged in boring a tunnel   
from opposite ends amid the    roar of  
water and other noises1 we are beginning
 to hear now and    again the strokes of
 the pick-axes of our comrades on the   
other    side4    ,,camille             
,,flammarion#cj cites #dcf cases of     
psychic manifestations  examined by     
himself and which establish beyond any  

reasonable doubt  that there are certain
 perceptive faculties1 namely1          
clairvoyance and  clairaudience1 that   
crop out in certain individuals1        
sometimes in  abnormal conditions and   
sometimes normally1 the very unusual    
character  of which proves their        
rudimentary nature and the potency of   
their  maturescence in the humanity of  
the future4 ,among the cases cited by   
,,flammarion are #ahf instances of      
manifestations from the dying received  
by persons who were awake2 #gj cases    
were manifestations received by  persons
 asleep2 #eg were observations of direct
 transmission of thought  without the   
aid of sight1 hearing or touch or other 
physical means2 #di  were cases of sight
 at a distance or clairvoyance by       
persons awake1  in dreams or in         
somnambulism and #gd cases of           
premonitory dreams or  predictions of   
the future4 ,indeed1 there are few      
persons now living  who have not had    
similar experiences1 if not exactly like

 these1 of  the same nature4 ,these     
examples1 of course1 may be greatly     
multiplied  in every country in the     
world1 and it is unnecessary to         
enumerate  them further2 for1 when once 
the existence of such faculties has been
  demonstrated in persons1 either in a  
normal or an abnormal condition1  their 
presence can no longer be questioned by 
the fair-minded4 ,it is1  then1 only a  
question of evolution before they will  
appear in the  normal way and their     
universalization1 as transmissible      
characters1 be  an accomplished fact4   
,when we are brought face to face with  
this sort  of phenomenon which seems to 
be increasing rapidly the conclusion is 
inevitably forced upon us that since    
evolution must be a continuous  process 
and matter destined to yield higher and 
more refined powers and  humanity to    
come into a far more extensive scope of 
motility because  of the opening avenues
 of knowledge1 it is not impossible that
 these  acuter senses1 these new        

faculties are now existing in the human 
race in a rudimentary stage and are     
designed to become the universal        
possession of all4 ,that this is to be  
the almost immediate outcome  of the    
perpetual exalting power which life     
exercises not alone over  materiality   
but over human organs and faculties as  
well1 seems to be  the one big1         
outstanding implication of the          
evolutionary process4 ,the  presence of 
such functions as the ability to sense  
the invisible and  the inaudible1 to    
answer to vibrations far subtler than   
anything in the  scope of our external  
sense-organs1 certainly indicates the   
existence  of rudimentary faculties     
which make these functions possible4    
,back  of these vague1 indefinite       
functions1 back of every supernormal or 
abnormal manifestation of man's         
mentality and back of all that class of 
phenomena which take their rise out of  
supersensuous areas must lie1  in ever  
increasing potency1 faculties and       

organs1 however rudimentary1  which are 
the source of these manifestations4     
,life1 that ineluctable  agent of       
creation1 which is incessantly pushing  
outward the confines  of the intellect's
 scope of motility1 never wearying1     
never tiring  nor sleeping1 has long    
ago1 in the dim and distant past of     
man's  evolution1 laid the foundations2 
and in fact1 with one stroke of its     
creative hand1 has molded the organs    
which are to become the active  media of
 these new faculties4 ,and now1 these   
incipient demonstrations1  these        
infantile struggles which we see now and
 again outputting from  them1 are but   
the specializing processes through      
which1 in their later  development1     
these organs are proceeding4 ,these are 
the outward signs  which should tell us 
that life is breaking up these organs   
into special  parts1 assigning to each a
 certain division of labor and making of
  each a perfect cordinate of all the   
others4 ,it is1 by these very  dispread 

exhibitions1 cutting up1 specializing   
and by slow degrees  determining the    
function1 character and general tendence
 of the organs  of expression wherewith 
these manifestations shall be           
centralized and  put into effective     
operation4 ,in doing this1 it is but    
following its  accustomed procedure1 the
 procedure which it adopted when it     
produced  the eye1 the ear1 the heart   
and the spleen4 ,we shall1 therefore1   
gauge  our understanding of the purport 
and end of evolution2 in fact1 we  shall
 determine our exact intellectual       
comprehension of life itself1 by  the   
attitude which we adopt towards it and  
the mode of its appearance4  ,much      
depends1 accordingly1 upon the posture  
which we assume towards  life--whether  
we shall say the totality of life's     
creative powers has  been dissipated in 
the bringing of the human body to its   
present degree  of perfection2 whether  
we shall say that it is neither         
necessary nor  possible for life to     

produce other organs and faculties which
 shall  respond to the unseen world     
about us revealing its glories in a way 
far more perfect than do our external   
sense-organs reveal the wonders  of the 
world of sensation2 or1 whether we shall
 conclude from these  most palpable     
evidences that life has yet other powers
 and faculties  which it designs to     
bestow upon the human mind and other    
organs and  capabilities with which it  
shall endow the human body so that man1 
in his evolution1 shall be enabled to   
rise to still higher spheres  while yet 
incarnate4 ,there may be1 and           
undoubtedly are1 those who1  for various
 reasons prefer to take the former      
positions and there are  certainly those
 who like ,,lodge1 ,,flammarion1        
,,hudson1 ,,crookes and a host  of      
others1 preferring the latter view1     
would rather believe in the  strength of
 the great mass of corroborative        
testimonies that we are  even to-day in 
the midst of the matutinal hours of a   

newer1 a better  and a far more         
efficient era of human evolution than   
any through which  we have hitherto     
come4      #cj ,see ,unknown1 p4 #dhe1  
et4 seq4    ,already1 recent scientific 
investigations and the results obtained 
therefrom have begun to turn the        
attention of medical authorities to  the
 activities of two very small organs    
situated in the mid-brain and  known as 
the pineal gland and the pituitary body4
 ,these organs1  and especially the     
pineal gland hitherto supposed to be a  
vestige  of the past1 are now beginning 
to be recognized as rudimentary organs  
belonging to the future evolution of    
humanity4 ,dr4 ,,charles ,,de ,m4       
,,sajous1 who is an authority on the    
pituitary body1 believes that it  has no
 active internal secretions but is an   
epithelio-nervous organ  which controls1
 through nerves leading to the adrenals 
and thyroid  bone1 the processes of     
general oxygenation1 metabolism and     
nutrition4  ,little is known of the     

functions of the pineal gland except    
that it is  an ovoid1 reddish organ     
attached to the posterior cerebral      
commissure  projecting downward and     
backward between the anterior pair of   
the  corpora quadrigemina4 ,it is       
otherwise known as the conarium the     
pinus or epiphysis4 ,situated at the    
base of the brain1 it  is held in       
position by a fold of the pia mater     
while its base is  connected with the   
cerebrum by two pedicles4 ,it contains  
amylaceous  and gritty1 calcareous      
particles constituting the brain sand4  
,there  are1 however1 marked structural 
resemblances between the pineal  gland  
and the pituitary body and their        
formation is perhaps the  most          
interesting feature of the development  
of the thalamencephalon  or mid-brain4  
,the hypophysis cerebri or pituitary    
body is a small1  ovoid1 pale1 reddish  
mass varying in weight from five to ten 
grains  and situated at the basal       
extremity of the brain in a depression  

of  the cranium known as the sella      
turcica1 a configuration very much  like
 a ,turkish saddle in shape4 ,it is a   
composite1 ductless gland and  consists 
of two divisions1 an anterior and a     
posterior1 connected  by an intermedial 
portion--all of which are attached to   
the base of  the cerebrum by the        
infundibulum4 ,the anterior lobe is     
larger than  the posterior and very     
vascular1 springing in its development  
from  the buccal cavity of the embryo2  
the posterior lobe is situated in  a    
depression of the anterior and is a     
brain-process4 ,the pituitary  body     
itself is lodged in a cavity of the     
sphenoid bone called the  pituitary     
fossa4 ,this is a most remarkable       
position1 for the reason  that the      
sphenoid1 or wedge-shaped1 bone which   
lies at the base of  the skull          
articulates from behind with the        
occipital and in front with  the frontal
 and ethmoid bones and by lateral       
processes with the  frontal1 parietal   

and temporal bones4 ,from this position 
it binds  together all the bones of the 
cranium1 and moreover1 articulates with 
many bones of the face4 ,it is upon the 
upper surface of the sphenoid  bone     
which occupies such a prominent and     
commanding position in the  cranium1 in 
a deep depression1 that the pituitary   
gland is located4    ,each nasal chamber
 is lined by a mucous membrane called   
the pituitary  or ,schneiderian4 ,this  
membrane is prolonged into the meatuses 
and air  sinuses which open into the    
nasal chambers4 ,the pituitary membrane 
is  thick and soft and diminishes the   
size of the meatuses and air sinuses4   
,it is covered by a ciliated columnar   
epithelium and contains numerous        
racemose glands for the secretion of    
mucous or pituita4 ,it is also  vascular
 and the veins which ramify it have a   
plexiform or net-work  like arrangement4
 ,it divides into two membranes--a      
respiratory1 which  is concerned in     
breathing1 and an olfactory region4 ,the

 respiratory  region corresponds to the 
floor of the nose1 to the inferior      
turbinated  bone and to the lower third 
of the nasal septum4 ,the olfactory     
region  is the seat and distribution of 
the olfactory nerve and corresponds  to 
the base of the nose1 to the superior   
and middle turbinals and the  upper     
two-thirds of the nasal septum4         
,recent developments prove that this    
gland has a profound influence  over the
 animal economy4 ,it is believed by some
 that the pituitary  body actually      
destroys certain substances which have a
 toxic influence  on the nervous system2
 others believe that it secretes        
material  media for the proper action of
 the trophic or nutritive apparatus2    
still others believe that it influences 
blood-pressure4 ,it is known1  however1 
from experimentation1 that its removal  
in dogs1 cats1 mice  and guinea pigs    
causes a fall of temperature1 lassitude1
 muscular  twitchings1 dyspnoea or      
difficult breathing1 and even speedy    

death4  ,hypertrophy of the gland is    
directly associated with certain        
diseases1  such as giantism and         
acromegaly4 ,the latter is a disease    
which  causes a general enlargement of  
the bones of the head1 feet and hands1  
usually occurring between the ages of   
twenty and forty years1 and most        
frequently in females4 ,the fact that   
these diseases are so closely           
associated with a hypertrophic condition
 of the pituitary gland has led  to the 
conclusion that perhaps the giants or   
,cyclops of ancient times  were cases of
 giantism or acromegaly4 ,this view1    
while interesting  from the standpoint  
of the functions of the pituitary gland1
 is not  necessarily a correct one2 for 
the age of giants1 when men attained  to
 a much larger stature than at present1 
can be accounted for on  other grounds1 
namely2 that the early mesozoic man1 on 
account of his  having to live among    
animals1 trees and other vegetation of  
such huge  size1 had naturally to be    

fitted with a frame proportional to     
other  animals in order that he might   
successfully cope with his environing   
conditions4 ,nature thus wisely fitted  
him for the conditions which she  had   
prepared for the scenes of his life4    
,the facts adduced in the foregoing     
description are purely empirical  and   
may be verified by any who seek to      
establish their correctness or          
incorrectness4 ,but we are about to     
introduce a species of testimony  which 
while it may also be verified will not  
be found so easy of  verification as the
 above-mentioned physiological facts1   
and not by  the same means2 yet they are
 nevertheless deserving of a place here4
  ,it is the liberal attitude that we   
must adopt towards all phenomena1       
excluding none that give promise of the 
widening and deepening of  our knowledge
 and an explanation of much that has    
seemed heretofore  unaccountable4    ,we
 have noted how subtle is the physical  
connection between these two  bodies1   

the pineal gland and the pituitary body2
 we have seen how  profound is the      
effect which the latter has been        
demonstrated1 in a  measure1 to have    
over the entire bodily economy2 but     
there is even  other testimony to the   
effect that those gifted with the inner 
vision  can observe the pulsating aura  
in each body1 a movement which is not   
unlike the pulsations of the heart and  
which never ceases throughout  life4 ,in
 the development of clairvoyance it is  
known that this motion  becomes         
intensified1 the auric vibrations       
becoming stronger and more  pronounced4 
,the pituitary body is the energizer of 
the pineal  gland and1 as its pulsating 
arc rises more and more until it        
contacts  the pineal gland1 it awakens  
and arouses it into a renewed activity  
in much the same manner as current      
electricity excites nervous tissue4     
,when the pineal gland is thus aroused  
clairvoyant perception is  said to      
become possible4 ,these are facts which 

cannot be proved by  the materialistic  
man of science nor can they be          
demonstrated to the  layman who has to  
depend alone upon sense-deliveries for  
his knowledge4  ,this is true for the   
reason that1 in the first place1 it is  
necessary  that he shall either feel in 
his own mid-brain the energizing        
activity  of these two organs and have  
his entire nerve-body shaken from crown 
to toe by the down rushing currents of  
that subtle energy with which  the      
pituitary body floods it or be himself  
the perceiver of its  activities4       
,nevertheless attention is here called  
to these phenomena  and the conclusions 
drawn therefrom are offered as a means  
of denoting  the probable line of       
investigations which will establish the 
directions  which we should pursue and  
the source whence we shall find         
outcropping  the new faculties and their
 organs of expression4    ,we confess to
 a knowledge of the fact that men of    
empirical science  have long maintained 

a rather skeptical1 if not contemptuous1
 attitude  towards all these phenomena  
but it is also felt that there is far   
more  of discredit in their attitude    
than of credit2 for1 in so doing1 they  
have voluntarily adopted measures by    
means of which the knowledge  that they 
so eagerly seek is shut out from their  
attainment4 ,in vain1  then1 is appeal  
made to the intellect to remove the     
barriers which it  unconsciously        
interposes between itself and the goal  
of its pursuit2  in vain do we appeal to
 the materialist to give ear to         
testimony the  data of which cannot be  
made amenable to his knife and scalpel  
neither  to the microscope nor          
microphone2 in sheer vanity is he       
adjured to  look within--into the       
interior of life1 of mind and the things
 which  he handles with his             
instruments--for the answers to his     
queries1 for  the path which leads into 
the wake of life and consciousness4     
,because  his utter loyalty and devotion

 to the modus vivendi of the intellect  
will not permit this2 but1 after all1 it
 is not wholly wise to allure  him away 
unbetimes from his search after truth   
through superficialities  nor to        
inveigle him into giving up his         
tenacious prosecution of the  physically
 determinable4 ,we would not have it so2
 for1 perchance1 he1  too1 one fine day1
 in the quiet of his laboratory shall   
come upon  the data which may           
substantiate in his own mind the long   
settled  conclusions of the occultist   
who1 frequently and not without cause1  
grows impatient at the scientist's      
obstinate delay4 ,these two workers1    
the empiricist and the occultist1 must  
ultimately come together as             
collaborators--the one working upon the 
form1 the vehicle1 physical  matter and 
the other seeking to understand the     
life1 the interior  forces which        
produce1 the creative element4 ,they    
cannot remain always  aloof from one    
another2 for they1 too1 are as men      

digging a tunnel from  opposite ends4   
,finally1 the partition will break and  
thus will dawn a  new day for the       
knowledge of humanity and men will see  
the rationale1  the truth and good sense
 of coperation in this respect4    ,it  
can be said with confidence that        
whatever in the future may be  learned  
as to the physiologic functions of the  
pituitary body and  the pineal gland1 it
 suffices to know that it is life which 
they  express and that1 too1 in a far   
superior manner than any of the other   
sense organs4 ,the modus of these two   
glands differs in a very  marked way    
from that of the organs of sight1       
hearing1 taste1 smell and  feeling4 ,for
 these latter are designed for contact  
with the external1  objective world of  
sensations1 their growth and evolution  
being  dependent upon stimuli received  
from without while with the former the  
case is far different1 in fact1 just the
 opposite4 ,the mode of life  of the    
pituitary body and the pineal gland1    

instead of receiving  sustenance and    
impetus from external stimuli1 is rather
 dependent  upon impacts received from  
the ,thinker's own consciousness and    
made  to impinge upon them by an        
exclusively interior process4 ,situated 
in  the mid-brain1 safely secluded from 
all external interference1 they  are    
naturally limited to stimuli which come 
from within1 or it may  be said1 they   
are responsive to excitations that are  
more spiritual  than those which come   
through the external sense-organs4 ,if1 
as has  been said they control the      
internal processes of metabolism        
7anabolism  and katabolism71            
oxygenation1 nutrition1 and other       
important internal  movements1 none of  
which can be said to be under the       
control of the  intellect1 is it not1   
therefore1 justly assumed that their    
response  is directed towards stimuli   
which arise interiorly or upon a plane  
higher than the intellectual8 ,it is a  
matter of scientific knowledge  that    

those persons gifted with clairvoyance1 
and commonly known as  sensitives are   
far more responsive to nervous          
excitation than those  who are not so   
gifted4 ,this would seem to imply that1 
on account of  the superactivity of     
these two organs1 the entire nerve-body 
has1 in  consequence1 become more       
delicately and subtly organized4 ,they  
seem  to act as a switchboard for the   
regulation of the flow of the current   
of life through the body4 ,not only do  
they come more nearly to an  adequate   
expression of the physiologic function  
of life1 but1 as their  energization    
means an enlargement of the scope of    
perception by giving  the ,thinker's    
active consciousness access to hitherto 
unapproachable  realities and by        
penetrating the outer mask which life   
ensouls and also  laying bare a domain  
of unlimited knowledge the manifestation
 of which  is far more real than        
anything the senses can disclose1 it is 
evident  that they constitute1 in their 

collaborative functions1 a more highly  
adaptable medium for the expression of  
the ,thinker's consciousness4 ,and  if  
so1 for the kosmic consciousness which  
is the source of all forms of           
consciousness1 they furnish a           
specializing and adaptizing agency4     
,now1 in all those cases of             
inspirations1 revelations1 telepathic   
communications1 clairaudience1          
clairvoyance1 dreams1 visions1 etc41    
wherein the ,thinker is enabled to      
perceive facts and verities which  are  
then presented to his consciousness in a
 manner clearly without  the province of
 the common sense-organs1 it must be    
apparent that  these manifestations are 
apprehended by a perceptual mechanism   
which  is entirely independent of       
external sense presentations but which  
is  an interior and subtler form of     
psychic activity4 ,sounds which are     
heard by so-called sensitives and       
objects which are perceived by eyes     
that are keener than those organs said  

to have been evolved from the  medusa   
cannot be heard by other persons nor    
perceived by them in any  way4 ,thus it 
would seem that there are inner organs  
of perception which  respond to these   
finer vibrations and which enable the   
person so gifted  to apprehend them4    
,there are those who1 presumably basing 
their assertions upon actual            
observation and knowledge1 unqualifiedly
 assert that in order to  gain contact  
with the inner worlds all that remains  
to be done is the  awakening of the     
pituitary body and the pineal gland4    
,when this  is accomplished man will    
again possess the faculty of perception 
in  the higher worlds1 but on a grander 
scale than formerly 7when humanity  was 
in its infancy and exercised a lower    
form of psychic power only72  because it
 will be in connection with the         
voluntary nervous system1 and           
therefore1 under the control of the     
will4 ,through this inner perceptive    
faculty all avenues of knowledge will be

 opened to him and he will  have at his 
service a means of acquiring information
 compared with  which all other methods 
of investigation are but child's        
play4#ca  ,it is the lack of this       
ability to see1 with our physical eyes1 
as it  were1 by the ,roentgen rays1 to  
penetrate the inwardness of things  that
 has baffled and confounded men for so  
long a time and which has  eventually   
led certain mathematicians and others to
 conjecture such  strange1 and in many  
cases1 illogical possibilities for the  
denizens  of four-space4 ,this inability
 together with the desire to fathom the 
innermost complexities of solids and to 
handle1 albeit with unholy  hands1 the  
supersensuous1 the mysterious and the   
unapproachable  identity of             
things-in-themselves have induced the   
more zealous among  them to contrive    
some kind of hypothesis which would1 at 
least1 offer  an explanation of these   
phenomena4 ,it has driven them to       
wrestle  with metaphysical possibilities

 in a vain endeavor to grasp that       
which1 ignis-fatuus like1 ever evades   
their slightest intellectual  approach4 
,but why this prolonged struggle1 why   
this intellectual  maneuvering and      
sophistry8 ,we can calculate1 compute1  
excogitate1  says ,,paul ,,carus1#cb and
 describe all the characteristics of    
four-dimensional space1 so long as we   
remain in the realm of abstract  thought
 and do not venture to make use of our  
motility and execute  our plans in an   
actualized construction of motion2 but  
as soon as we  make an a priori         
construction of the scope of our        
motility1 we find  out the              
incompatibility of the whole scheme4    
,thus mathematicians are  forced to     
relinquish all hopes of transforming the
 world of life into  a sort of          
four-space dwelling place where         
everything is done according  to the    
laws of mathematics4 ,but whether they  
shall accept it or not  there is a      
wider1 truer and more rational view     

which recognizes all  metageometrical   
investigations1 as well as all kindred  
phenomena1 as  universal evidences      
indeed1 as the very causes which1 in the
 future  humanity1 will actually awaken 
and cause to be accelerated in their    
development these little inner          
sense-organs1 the pineal gland and the  
pituitary body1 whose perfect           
development promises to provide for  the
 ,thinker's consciousness an avenue of  
expression such as humanity  has        
possessed never before4 ,and too1 it is 
not without full knowledge  of the fact 
that it has been customary1 among       
certain scientists or  perhaps all of   
them1 to regard these bodies1 at least  
the pineal  gland1 as vestigal organs   
belonging to the past of human          
evolution1  that we make these          
assertions4 ,yet1 as man proceeds in the
 perfection  of mechanical science1 in  
the development of instruments of       
precision  that aid his external senses1
 responds more and more to the subtle   

vibrations teeming everywhere in the    
atmosphere about him1 and comes1  in the
 course of time1 naturally to possess a 
more sensitively keyed  nervous         
mechanism1 a finer body and higher      
spiritual aspirations1  there will be a 
corresponding widening of his scope of  
vision and the  attainment of larger    
powers of perception which must         
inevitably1 in the  very nature of      
things1 tend towards a deeper and truer 
knowledge4      #ca ,rosicrucian        
,cosmo-,conception1 p4 #dgg1 ,,max      
,,heindel4      #cb ,,foundations ,,of  
,,mathematics1 p4 #ij4    ,in view of   
the foregoing1 it is believed that the  
general results of  this pituitarial    
awakening which may be expected as      
humanity continues  to evolve should be 
seen in the marked effects which will be
 wrought  in the entire metabolistic    
area of the human body whereby a gradual
  intensification and sensitization of  
the whole neural mechanism  will raise  
the peculiar efficiency of all the      

senses whether purely  physiologic or   
psychic4 ,for there are undoubtedly     
notes so delicate in  their intensity   
that they transcend the grasp of the    
audital nerves2  scents and fragrances  
so subtle in their excelling purity that
 it is  beyond the powers of our present
 olfactorial contrivances to detect     
them2 colors and other external stimuli 
so sublimely supersensuous that  a      
nervous mechanism perhaps ten-fold more 
delicate and responsive than  ours is   
required to apprehend them4 ,all these1 
and more than at present  is            
conceivable1 will come1 with the aid of 
pituitarial stimulation1  within the    
purview of a more highly developed      
humanity of the future4  ,and because   
mathematics have led a movement into the
 very camp of  the                      
intellectuals--logic-bound and tethered 
by the severest rigors  of              
mathesis--whereby the intolerant        
intellect has been compelled1  by rules 
of its own making1 to recognize the     

existence of the  supersensuous1 and by 
looking into the glaring light of the   
sun of the  intuitable to gain strength 
of vision and boldness to press forward1
  a great and far-reaching service has  
been wrought for humanity4 ,and  in the 
tower of hyperspace mathematics have    
erected a monument to the  intellect    
which1 as long as the human race        
remains1 will mark the great  turning   
point in man's path to the highest life4
    ,what if it were possible that the  
scientist1 when he had carried          
instruments to their utmost precision   
and penetration1 should suddenly1  or   
otherwise1 be endowed with a            
clear-perceptivity of sight1 hearing    
and smell1 so that he could with his own
 powers of vision1 feeling and  hearing 
take up the task where the microscope1  
the microphone and the  micrometer left 
off and delve into depths far too       
unfathomable for his  appliances1       
perceiving the innermost realities of   
things and processes8  ,what if it were 

possible for him1 with these added      
powers1 to see and  examine without the 
aid of the magnifying lens the electron1
 the atom  and the molecule8 ,what if   
the cell1 the bacterium1 and other      
invisible  forms of life would then     
deliver up their secrets to his knowing 
mind8  ,what if he could sense with his 
own inner vision1 the ultra-violet and  
the infra-red rays2 what indeed1 if     
spirit itself1 the innermost sheath  of 
life1 should be visible and palpable to 
him and he could note the  internal     
processes1 the action and movements of  
the infinitesimals  of life8 ,think you 
not that such direct contact1 such      
immediate and  incontrovertible         
knowledge would be far superior to any  
advantage which  his manufactured       
devices now bestow8 ,it is even so4     
,thus will react upon man's perceptive  
apparatus the flood of light  which the 
awakened intuition will shed upon them  
and thus will man rise  higher1 driven  
on by the current of life with the mass 

of materiality1  to a point of complete 
spiritualization and take additional    
steps in  that direction which leads to 
,raja ,yoga or the ,royal ,union with   
the  divine life of the universe4       
,before this step is taken1 however1 and
 before the passage from  mechanics to  
biogenetics is made1 as made it must be1
 man must win a  complete mastery over  
matter4 ,but this he will do2 for more  
and more  he is learning to put all     
those forms of labor which are so       
exacting  as to leave him no time for   
the development of his higher powers    
into  the hands of machinery4 ,he will  
not be free until he has done this      
well-nigh completely4 ,this is the task 
of the intellect and with it  man must  
win his way to these higher faculties   
which are destined to  succeed the      
intellect whereupon he will be ushered  
out of a life bound  and restricted by  
mechanics to a life of unimaginable     
freedom1 the  intuitive life4    ,the   
outcome of these new faculties of       

perception and the development of  the  
intuition will be the springing up of a 
new species of art that1  turning away  
from appearances and sinking beneath or 
rising above1  superficialities1 will   
seek to portray in newly found colors1  
the  plastic essence of things so that  
we shall have an art which pertains  to 
the real1 superseding that which        
pertains to the phenomenal4  ,language  
and the need of it will pass away2 for  
man will have outgrown  the use of signs
 and symbols in his communion with his  
fellows and will  use the language of   
the intuition--direct and instantaneous 
cognition4  ,philosophy will be         
regenerated1 re-created4 ,speculation   
will give  way to truth and there shall 
be but one philosophy and that shall be 
the knowledge of the real4 ,mathematics1
 the royal insignia of the  intellectual
 life1 because it can deal only with    
immobilities1 with  segments and parts  
and has no aptitude for the continuous  
flow1 will  yield its kingdom to a      

higher form of kinetics which will serve
 the  intuitive faculty as mathematics  
now serve the intellect4 ,science will  
then be no longer empirical in its      
method2 but a system of direct and      
incontrovertible truths4 ,religion will 
rise to meet these changes which  will  
come in the path of human evolution2 and
 faith will surrender its  place to     
knowledge4 ,ethics1 recast in a new     
mold1 will deal with the  new aspect of 
man's relation to his fellowmen4 ,man1  
for whose highest  good these ultimate  
changes will come1 will be a new        
creature1 a higher  and better man2 and 
humanity shall evolve a new race4 ,there
 shall1  indeed1 be a new heaven and a  
new earth4        ,,the ,,end4          
,,bibliography        ,,american        
,,mathematical ,,monthly3      ,vol4    
,,viii1 p4 #afa1 ,p4 ,,barbarin1 for ,le
 ,mathematiche1       ,trans1 by ,,geo4 
,b4 ,,halsted4 ,see also pp4 #ca-ce4    
,vol4 ,,viii1 #aija1 p4 #bag1 ,suppl4   
,report on ,non-,euclidean ,geometry1   

,,geo4 ,,bruce ,,halsted4      ,vol4    
,,ix1 p4 #ei1 ,,geo4 ,b4 ,,halsted4     
,idem1 pp4 #aec-aei4      ,vol4 ,,xi1   
pp4 #he-hf1 ,simon's ,claim for ,gauss4 
,,american ,,mathematical ,,society     
,,bulletin1 ,lancaster1 ,pa43      ,vol4
 ,,ii1 series #ahie4      ,vol4 ,,xx1   
#aiad1 pp4 #dji-dab1 ,r4 ,c4            
,,archibald4      ,,annals ,,of         
,,mathematics1 #b;d series1 #aijb3      
,vol4 ,,xii1 ,f4 ,s4 ,,wood4            
,,astronomical ,,society ,,notices3     
,vol4 ,,xvi1 p4 #hj1 #ahef4             
,,ball',s ,,short ,,account ,,of ,,the  
,,history ,,of ,,mathematics4           
,,boston ,,colloquium1 for #aijc1 ,n4   
,y4 #aije1 ,forms of ,non-,euclidean    
,spaces1 ,vols4 ,i-,,iii1 pp4 #ca1 et   
seq41 #ahia4      ,,bulletin ,,new      
,,york ,,mathematical ,,society3        
,vol4 ,,ii1 pp4 #a-add4      ,vol4      
,,iii1 pp4 #a1 et seq41 #ahia1 ,g4 ,b4  
,,mcclintock4        ,,cambridge        
,,mathematical ,,journal3      ,vol4    
,,iv1 #ahde1 ,,cayley1 on ,analytical   

,geometry of n-,dimensions4             
,,canadian ,,institute ,,transactions3  
,vol4 ,,viii1 pp4 #cae-cci1 #aiji1      
,,august ,,kirschbaum4      ,,cayley1   
,a41 vide ,science1 ,vol4 ,,ix1 pp4     
#ei-fc3      ,amer4 ,math4 ,mo41 ,vol4  
,,ii1 pp4 #ajb-ajf4      ,amer4 ,math4  
,mo41 ,vol4 ,,vi1 pp4 #ei-fe4           
,review of ,reviews1 ,june1 #ahie1 pp4  
#fic-fid4      ,,class ,,book ,,of      
,,non-,,euclidean ,,geometry3      ,vide
 ,science1 ,,xiii1 pp4 #dfb-dfe4        
,,cornhill ,,magazine 7n4s473           
,vol4 ,,xxxvii1 pp4 #fdg-ffe1 ,c4 ,h4   
,,bryan1 on the ,fourth ,dimension4     
,,easy ,,non-,,euclid3      ,monist1    
,,xix1 pp4 #cii-djb4      ,,educational 
,,review3      ,november1 #aijb1 issue1 
pp4 #cdf1 et seq4      ,,euclid',s      
,,parallel-,,postulate1 by ,,john ,w4   
,,withers4        ,,flatland1 by ,a4    
,,square4      ,,foundations ,,of       
,,geometry3      ,science1 ,vol4 ,,vi1  
pp4 #dhg-dia4      ,,foundations ,,of   
,,mathematics1 by ,,paul ,,carus4       

,,fourth ,,dimension3      ,by ,,herman 
,,schubert1 ,c4 ,h4 ,,hinton1 ,,claude  
,,bragdon4      ,,fourth ,,dimension1   
,,simply ,,explained3      ,edited by   
,h4 ,p4 ,,manning4      ,,fourth        
,,dimension3      ,cornhill ,magazine1  
n4s41 ,vol4 ,,xxxvii1 pp4 #fdg-ffe1 by  
,c4 ,h4 ,,bryan4      ,,fourth          
,,dimension1 ,,the ,,riddle ,,of3       
,h4 ,a4 ,,bruce1 in ,scientific         
,american ,supplement1 ,vol4 ,,lxvi1 p4 
#adf4        ,,geometry1 ,,foundations  
,,of1 by ,o4 ,,veblen3      ,popular    
,science ,monthly1 ,vol4 ,,lxviii1 pp4  
#ba1 et seq4      ,,geometry1           
,,lobachevski',s3      ,theory of       
,parallels1 trans4 by ,,geo4 ,,bruce    
,,halsted4      ,,geometry1             
,,non-,,euclidean3      ,american       
,mathematical ,monthly1 ,vol4 ,,vii1 pp4
 #abc-acc4      ,,geometry1             
,,non-,,euclidean1 ,,report ,,on        
,,progress ,,in3      ,reprinted from   
,science1 n4 s41 ,vol4 ,x1 ,no4 #bea1   
pp4 #ede1       et seq41 by ,,geo4      

,,bruce ,,halsted4      ,,geometry1     
,,non-,,euclidean1 ,,thesis ,,on3       
,by ,m4 ,l4 ,,gerard4      ,,geometry1  
,,non-,,euclidean3      ,by ,,roberto   
,,bonola1 trans4 by ,h4 ,s4 ,,carslaw1  
,university of       ,sydney1 ,n4,s4,w4 
,,geometry1 ,,non-,,euclidean3          
,,julian ,,lowell ,,coolidge1 pp4 #df1  
et seq4      ,,geometry1                
,,non-,,euclidean3      ,by ,h4 ,p4     
,,manning4      ,,geometry1 ,space and1 
in ,light of ,physiologic1 ,psychologic 
and     ,physical ,inquiry1 vide ,open  
,court1 #aijf1 by ,e4 ,,mach4           
,,halsted1 ,,george ,,bruce3            
,cayley1 ,a41 ,biog4 of1 vide ,amer4    
,math4 ,mo41 ,vol4 ,,ii1 p4 #ajb4       
,four-fold ,space in ,two-fold ,time1   
vide ,science1 ,vol4 ,,xix1 p4 #cai4    
,hoel1 ,biog4 of1 ,american ,math4 ,mo41
 ,vol4 ,,ix1 pp4 #ii-aja4               
,lambert's ,non-,euclidean ,geometry1   
vide ,n4 ,y4 ,math4 ,soc4 ,bull41       
,vol4 ,,iii1 p4 #gi4      ,lobachevski1 
,n41 ,biog4 of1 vide ,amer4 ,math4 ,mo41

 ,vol4 ,,ii1         pp4 #acg-aci2 ,open
 ,court1 ,vol4 ,,xiii1 pp4 #daa-dae4    
,message of ,mathematics to ,natural and
 ,rational ,theology1 vide              
,hibbert's ,journal1 ,vol4 ,,vii1 #ahji4
      ,non-,euclidean ,geometry1        
,inevitable1 vide ,monist1 ,vol4 ,,iv1  
p4 #dhc4      ,non-,euclidean ,geometry1
 ,historical and ,expository1 ,american 
,mathematical ,monthly3        ,vol4 ,i1
 pp4 #gj-gb2 #aab-aae2 #adi-aeb2        
#ahh-aie2 #bbb-bbc2 #bei-bfj2           
#cja-cjc2 #cde-cdf2 #cgh-cgi2 #dba-dbc  
7#ahid74        ,vol4 ,,ii1 pp4 #aj2    
#db-dc2 #fg-fi2 #ajh-aji2 #add-ade2     
#aha-bad2         #bef-beg2 #cji-cac2   
#cdf-cdh 7#ahie74        ,vol4 ,,iii1   
pp4 #ac-ad2 #ce-cf2 #fg-fi2 #aji2       
#acb-acc 7#ahif74        ,vol4 ,,iv1 pp4
 #aj2 #gg-gi2 #aja-ajb2 #agj-aga2 #bjj2 
#bdg-bdi2 #bfi-bgj2         #cjg-cjh    
7#ahig74        ,vol4 ,v1 pp4 #a-b2     
#fg-fi2 #abg2 #bah2 #bij-bia 7#ahih74   
,review of ,reviews1 ,vol4 ,,ii1 #ahie1 
pp4 #fic-fid4      ,space1 ,new ,ideas  

,about1 ,popular ,science ,monthly1     
,vol4 ,,xi1 p4 #cfd4      ,,helmholtz1  
,h41 ,on ,origin and ,meaning of        
,geometric ,axioms3      ,vide ,mind1   
,part ,i1 ,no4 #c1 ,july1 #ahgf4 ,for   
replies and discussions       of the    
above1 vide ,nature1 ,,jevons1 ,vol4    
,,iv1 p4 #dha2 ,j4 ,l4 ,,tupper1        
,nature1 ,vol4 ,v1 p4 #bjb2 ,,helmholtz1
 in ,academy1 ,vol4 ,,iii1 p4 #eb2      
,mind1 ,part ,,ii1 ,april1 #ahgh4       
,,hermathena1 ,vol4 ,,xiii1 pp4 #dia1 et
 seq41 #aijf4      ,,higher ,,space1 ,a 
,,primer ,,of1 by ,,claude ,,bragdon4   
,,ingleby1 ,c4 ,m43      ,a             
,non-,mathematical ,criticism of a ,part
 of ,professor ,clifford's      ,lecture
 on ,curved ,space1 ,macmillan's        
,magazine1 ,october1 #ahgb2      also   
,nature1 ,vol4 ,,vii1 #ahgc4            
,,journal ,,of ,,mathematics1           
,,american3      ,vol4 ,i1 pp4 #bfb1    
#bgf1 #chd1 #chf2 ,,halsted 7#ahgh74    
,,jour4 ,,philos41 ,,psychol4 ,,and     
,,scientific ,,methods3      ,vol4 ,,vi1

 pp4 #fag1 et seq4        ,,kant1       
,,emmanuel1 trans4 by ,,fred4 ,,paulsen4
      ,,kant1 ,,emmanuel1 ,prolegomena1 
trans4 by ,j4 ,p4 ,,mahaffy and      ,j4
 ,h4 ,,bernard1 pp4 #ce-ci4             
,,lambert1 ,,biography ,,of3            
,american ,math4 ,mo41 ,vol4 ,,ii1 pp4  
#bji-baa4      ,,lie1 ,,sophus3         
,science1 ,vol4 ,,ix1 pp4 #ddg-ddh4     
,,lobachevski1 ,n43      ,open ,court1  
,vol4 ,,xii1 pp4 #daa-dae1      ,amer4  
,math4 ,mo41 ,vol4 ,,ii1 pp4 #acg-aci4  
,,london1 ,,edinburgh ,,and ,,dublin    
,,philosophical ,,magazine3      ,vol4  
,v1 series #f1 p4 #eei4                 
,,mathematics1 ,,foundations ,,of3      
,,paul ,,carus4      ,,mathematical     
,,papers ,,of ,,chicago ,,congress1 p4  
#ib4      ,,mathematical ,,papers1 pp4  
#ee-ga4      ,,mathematics 7a brochure73
      ,by ,c4 ,j4 ,,keyser1 ,adrian     
,professor of ,mathematics1 ,columbia   
,univ4      ,,mathematics1 ,,higher3    
,edited by ,,mansfield ,,merriam and    
,,robert ,s4 ,,wood1 #aijb1 pp4 #ejh1   

et seq4      ,,mathematics1 ,,spiritual 
,,significance ,,of3      ,c4 ,j4       
,,keyser1 reprinted from ,columbia      
,univ4 ,quar41 ,december1 #aiaa4        
,,memorial ,,volume ,,of ,,scientific   
,,papers3      ,pub4 by ,univ4 of ,st4  
,andrews1 pp4 #c-de1 ,d4 ,m4 ,y4        
,,somerville4      ,,mind 7,magazine73  
,vol4 ,i1 #ahgf1 pp4 #cjc1 et seq4 ,j4  
,n4 ,p4 ,,land4      ,vol4 ,,ii1 #ahgg1 
p4 #dj1 ,j4 ,n4 ,p4 ,,land4      ,vol4  
,,ix1 pp4 #ebb1 et seq4 #aijj4          
,,monist 7,magazine73      ,vol4 ,,ii1  
pp4 #ej1 #cba1 ,,paul ,,carus4          
,vol4 ,,iv1 #ahid1 ,articles by ,g4 ,b4 
,,halsted and ,j4 ,,delboeuf4      ,vol4
 ,,vi1 #ahie1 ,,herman ,,schubert4      
,vol4 ,,vii1 ,g4 ,b4 ,,halsted1 on      
,saccheri4      ,vol4 ,,xiii1 pp4       
#hj-ajb2 #bah-bcd4      ,vol4 ,,xiv1    
,article by ,e4 ,,mach4      ,vol4 ,,xv1
 p4 #abg1 ,,paul ,,carus4      ,vol4    
,,xvi1 #aijf1 p4 #dcc1 ,,david ,p4      
,,abbott4      ,idem1 #aijf1 p4 #fe1 ,c4
 ,j4 ,,keyser1 on ,mathematical         

,emancipations4        ,,nature         
7,magazine73      ,vol4 ,,viii1 pp4 #h1 
#ad-ag1 #cf-cg 7#ahgc74      ,vol4 ,,xv1
 pp4 #ecc-ecg 7#ahgg74      ,vol4 ,,xlv1
 ,p4 #djd 7#ahia74      ,,newcomb',s    
,,philosophy ,,of ,,hyperspace3         
,science1 ,vol4 ,,vii1 p4 #bab4         
,,new ,,york ,,math4 ,,soc4 ,,bulletin3 
,vol4 ,,iii1 p4 #gi 7#ahic71 ,f4 ,s4    
,,wood on ,forms of ,non-,euclidean     
,space4        ,,open ,,court           
7,magazine73      ,vol4 ,,xvi1 pp4      
#eac-eba4      ,vide ,december1 #aiac   
edition1 p4 #geg4        ,,philosophy   
,,of ,,mathematics1 by ,,paul ,,carus4  
,,philosophical ,,review3      ,vol4 ,v1
 #ahif1 p4 #age1 ,f4 ,c4 ,s4 ,,schiller4
      ,idem1 p4 #ceg1 ,j4 ,h4 ,,hyslop4 
,vol4 ,,vii1 p4 #fae4      ,vol4 ,x1 pp4
 #aac1 #bbi1 #cad1 #cge1 #dhh1 #egi1    
#ehc4      ,vol4 ,,xii1 p4 #dic4        
,,popular ,,science ,,monthly3      ,no4
 #ff 7#aije71 pp4 #fci-fdf4      ,vol4  
,,lxvii1 pp4 #fci1 et seq4      ,vol4   
,,lxviii1 pp4 #ba1 et seq4      ,vol4   

,,lxxviii4 p4 #eed 7#aiaa71 ,,samuel    
,,moffat ,,weyer4      ,,popular        
,,astronomy3      ,vols4 ,,vii and      
,,viii 7#aijj71 ,g4 ,b4 ,,halsted4      
,vol4 ,,xviii1 pp4 #db1 et seq4 7#aiaj71
 ,w4 ,h4 ,,pickering4      ,vol4 ,,lxxv1
 p4 #agi1 ,w4 ,w4 ,,payne1 on           
,attraction and ,figure of the          
,earth4      ,vide ,no4 #hd 7#aija71 pp4
 #ahg-aij4      ,,popularization1 ,,the1
 ,,of ,,non-,,euclidean ,,geometry3     
,amer4 ,math4 ,mo41 ,vol4 ,,viii1 pp4   
#ca-ce4      ,,psychological ,,bulletin3
      ,vol4 ,,viii1 p4 #bb 7#aiaa74     
,,queen',s ,,quarterly3      ,vol4 ,,xx1
 pp4 #dca-ddf1 ,a4 ,l4 ,,clark 7#aiac74 
,,revue ,,generale ,,des ,,sciences3    
,no4 #bc1 trans4 ,nature1 ,vol4 ,,xlv1  
#ahib4      ,,royal ,,astronomical      
,,society ,,proceedings3      ,vol4     
,,xvi1 pp4 #hj-he4      ,,royal         
,,society ,,proceedings1 ,,london3      
,obituary ,notices of ,deceased         
,fellows1 ,vol4 ,,xvi1 p4 #fi           
7#ahfg-fh74      ,,royal ,,society ,,of 

,n4 ,s4 ,w4 ,,journal ,,and             
,,proceedings3      ,vol4 ,,xxxv1 p4    
#bdc 7#aijb71 ,g4 ,h4 ,,knibbs4         
,,science 7,magazine73      ,vol4 ,,iii 
7new series71 ,no4 #eh1 #ahif4          
,vol4 ,,vi1 pp4 #dhg-dia 7#ahig74       
,vol4 ,,vii1 pp4 #a-g1 ,philosophy of   
,hyperspace1 ,s4 ,,newcomb4      ,idem1 
p4 #hfa 7#ahih74      ,vol4 ,,xiii1 pp4 
#dfb-dfe4      ,vol4 ,,xiv 7new series71
 pp4 #gaa-gab1 ,g4 ,b4 ,,halsted4       
,vol4 ,,xix1 pp4 #dja-dac4      ,vol4   
,,xx1 pp4 #cec-def1 ,,josiah ,,royce4   
,vol4 ,,xxii1 pp4 #bgj-bga4      ,vol4  
,,xxxv1 ,no4 #ijf 7#aiab74      ,vol4   
,,xxxvii1 pp4 #hhe1 et seq41 ,c4 ,j4    
,,keyser1 on ,concerning the            
,figure of the ,universe of ,space4     
,,science ,,et ,l',,hypothese1 ,r4      
,,poincar4      ,,scientific ,,american 
,,supplement3      ,vol4 ,,lxvi1 pp4    
#ad-af 7#aijh74      ,,scientific       
,,romances1 ,c4 ,h4 ,,hinton4           
,,thought1 ,,new ,,era ,,of1 ,c4 ,h4    
,,hinton4        ,,westminster ,,review3

      ,vol4 ,,clii1 pp4 #fge1 et seq4   
,,zollner1 ,j4 ,c4 ,f41 ,on ,space of   
,four ,dimensions3      ,quar4 ,jour4 of
 ,science1 ,vol4 ,,viii 7#ahgh71 pp4    
#bbg-bcg4            ,,index        ,a  
,abeyance1 higher faculties held in1    
until lower are fully developed1 #ccd   
-- intuition kept in1 #cch1 ff          
,ability1 to sense the inaudible and the
 invisible1 #cdb      ,absolute1 no room
 for the1 #aja      ,abstract thought1  
invigorating power of1 #cc1 #dj         
,abstractions1 realizing1 #add1 #bid    
,absurdity1 analysis inevitably ends in 
an1 #cai-bj      ,acrobatics1 mathetic1 
#adf      ,acromegaly1 #cdg             
,action1 automatic1 of the intellect1   
#bec-d      -- engendering1 of life on  
the form1 #ccj      ,activities1 space1 
as physical and chemical phenomena1     
#bbi-cj      ,actualities of the        
physical world pushed over into the     
conceptual1 #aai      ,actuality        
imperceptible to the intellect1 #abf    
,adaptability of life1 #ccj      -- --  

mathematical laws1 #cg-ci               
,adaptation1 purpose of1 #cbg-h         
,adaptations made by an organism1 #afb  
,additional freedom bestowed by higher  
faculties1 #ccd      ,affinity1 zones   
of1 #bb1 #abd      ,agency1             
interpretative1 intellect as an1 #aff-g 
,agent1 ego as an1 of the ,thinker1     
#bdc1 ff      -- engendering1 life as   
an1 #cca      -- of creation1 life as   
an1 #cdc      ,ahmes1 an ,egyptian      
priest1 #dd      ,alchemy of            
psychogenesis1 #bbe      ,alcohols1     
eight different1 from one formula1 #aee 
,algebraic quantities and space1 #abe-g 
,allowability of the rules of logic1    
#afc      ,aloofness1 the1 of scientific
 dogmatism1 #cdj      ,alphabet of      
space-genesis1 #bcg      -- the         
geometric1 #aic      ,amenability of    
mind to the laws of evolution1 #bh-ca   
,analogies1 use of1 to popularize the   
four-space1 #abh-i      ,analogy1       
difference between two processes        
illustrated by1 #gd      ,analysis      

incapable of dealing with life1 #cai    
-- the manifold1 the fiat of1 #gg       
,analyst and the manufacture of the     
space-manifold1 #gg      -- disregards  
the conformity of the nature of things1 
#hd      ,analytics and the mechanical  
origin of the universe1 #dj-a      --   
perisophical nature of1 #cbj1 ff      --
 the four-space1 a curiosity of1 #dj    
,anchorage1 fourth dimension denied1    
#bfe      ,angularity of consciousness1 
#abj      ,answer of the senses to new  
needs1 #ccb      -- the1 of             
consciousness to realism1 #ig           
,anti-,euclidean geometry1 #fi          
,apotheosis of the definition1 #gh      
,apparatus1 man's perceptive1 #ceg      
,apparent vassalage of life1 #cbi-cj    
,appearance1 dynamic1 space as1 #ajd-g  
,appetences1 materialistic1 of          
empiricists1 #cdj      ,appulses of life
 against matter1 #cbh                   
,appurtenances of materiality1 #cca     
,a priori1 #ahe1 ff      -- --          
knowledge1 #aaj-e      ,aprioriness1    

#aab1 #aaf      ,apriority1 the         
principle of1 #aac1 #bbc                
,arbitrariness of the common mensurative
 quantities1 #da      ,archeological    
evidences1 #ccb      ,archimedes1 #gj   
,aristotle1 #cbb      ,arrangement of   
the contents of the kosmos1 #cag-h      
,art and the criterion of truth1 #cbc   
-- -- -- higher consciousness1 #aha     
-- evidences of new faculties in the    
field of1 #cce      ,assemblage of      
mathetic contrivances1 #cce             
,assumptions1 category of1 #ahi      -- 
procedure based upon1 disadvantages of1 
#afc      ,at-one-ment1 #bbd1 #beb1 #bgj
      -- defined as the end of          
evolutionary activity1 #a               
,attachment of the consciousness to the 
order of becoming1 #cje                 
,attainment1 the difficulty of1 by the  
intellectual method1 #bjf      -- of    
space-consciousness1 #bbe               
,attitude1 the need of a more liberal1  
#cdj      -- towards things spatial1    
#cjf      ,audital nerves1 notes that   

transcend the grasp of1 #cee      ,aura1
 pulsating1 #cdh      ,automatism of the
 intellect1 #bec1 ff2 #cch      ,avenues
 of knowledge opened to inner perceptive
 faculties1 #cec      ,awakening of the 
faculty of awareness in a new domain1   
#ij      -- -- -- pituitary body and the
 pineal gland1 #cec      -- pituitarial1
 general results of1 #cee-h             
,awareness1 a determinant of conception1
 #abj      -- as a gauge of the         
existence of things1 #afa-bjb      --   
degrees of1 #afe      -- hyperspace1 a  
symbol of a more extensive1 #ahj      --
 progress in1 #aga      -- ,thinker's   
sphere of1 #bgd1 #bhc        ,b         
,baltzer1 #ff      ,barrier1 the1 to the
 ,thinker's certitude1 #ahf-h           
,barriers1 freedom determined by absence
 of1 #bhj      ,becoming1 endless       
labyrinth of1 #bbj      -- the kosmos1  
in a state of1 #bfe      -- the order   
of1 #cje      ,becomings1 infinity of1  
#bcd      ,becquerel rays1 #ccf         
,being1 identification of consciousness 

with1 #bje      -- kathekotic1 #baj     
,being1 the interiority of1 #bij      --
 the numericity of1 #cji      -- the    
range of1 co-extensive with reality1    
#afh      ,beings1 discarnate1 #aed     
,beltrami1 ,eugenio1 #a-b1 #fc-e1 #ff-g 
-- -- and the manifold1 #cag      -- -- 
and the pseudosphere1 #fd-e             
,besiegement of matter by life1 #cbi-cj 
,bessel1 on geometry as incomplete1 #eh 
,bewilderment1 new realities cause1     
#afi-gj      -- of the mind1 #ahb       
,biogenetics1 passage from mechanics to1
 #ceg      -- turning to1 from          
mechanics1 #cbb      ,birth1 as a fourth
 dimensional process1 #aei      ,blind  
men and the elephant1 #beg      ,bolyai1
 ,janos1 #b-c1 #df1 #fj1 #ff1 #hc1 #hg  
,boundaries of a hypercube1 #acf-h      
,brain1 the physics of1 #bib            
,brain-consciousness1 #ahd1 #ahf      --
 standards of1 #aij      ,brotherhood1  
and the tendencies of the ,thinker's    
consciousness1 #cja      -- the keynote 
of the intuition1 #beb      ,bulwarks1  

the formal1 of geometry1 #gg-i        ,c
      ,carpet1 the sensible world as a1 
#aif      ,carus1 ,paul1 on             
four-dimensional space1 #bed      -- -- 
on metageometricians1 #ajb      -- -- on
 space1 #aaj      -- -- on the          
representation of the tesseract1 #acf-g 
,cayley1 #c-d1 #ed-e1 #ff-g             
,cell-activity as a performance a       
priori1 #aad      ,cell-colony1 #bej1   
#cja      ,cell-consciousness as an     
aposterioristic phenomenon1 #ahe        
,centers1 pyknotic1 #bcb                
,certainty1 geometric1 the basis of1    
#gg-h      -- of mathetic conclusions1  
provisional1 #cg      ,chaogenic period 
of involution1 #bjh1 ff      ,chaogeny1 
#bca1 #bcc      -- the evolution of     
chaos into order1 #d      -- the        
laboratory of1 #aic                     
,chaomorphogeny1 defined1 #d1 #bca-b    
,chaos1 as egg-plasm1 #bfe      --      
duration of1 #bfh      -- movement of   
life in1 #cjg      -- shores of1 #cjg   
,chaos-geometricity1 #cjg      --       

-intellectuality1 #cjg      --          
-materiality1 #cjg      -- -spatiality1 
#cjg      -- -,theos-,kosmos1 #bae1     
#bcc-d1 #bcg      -- -- -- a triglyph   
symbolizing kathekos1 #aa-b             
,character of the universe1 fixed by    
consciousness1 #afb                     
,characteristics1 changes in the1 of    
reality1 #afi      -- fourth            
dimensional1 of the ether1 #aeg      -- 
minds of similar1 fall into zones of    
affinity1 #abd      -- of non-,euclidean
 space1 #gb-c      ,chasm1 kosmic1      
between the real and the ideal1 #ajg    
,chemists1 speculative1 and the ourth   
dimension1 #aee      ,chrism1 sacred1 of
 creative mentality1 #ia      ,circle1  
significance of process of squaring1    
#aji      ,circle1 the symbology of1    
#cji      ,circuits1 closed1 our        
interests as1 #afg      ,clairaudience1 
#ceb      ,clairvoyance1 #ceb           
,clairvoyant perception1 #cdi           
,clause1 enabling1 of metageometry1     
#gf-g      ,clavius1 ,christoph1 #df1   

#eb1 #hc      ,clifford1 #ff      ,code1
 psychic1 for systematizing cognitions1 
#aij-a      ,cognition1 instantaneous1  
organ of1 #beh      -- intimacies of    
direct1 #bje      -- the method of1     
#ahf-h      ,cognitions1 intuitive1     
#ade1 #aib-c      -- psychic code for   
systematizing1 #aij-a                   
,commensurable quality1 dimension as1   
#adj      ,communal consciousness1 #bgj1
 ff      ,communalization and the       
intuition1 #cja      ,conarium1 the1    
#cde      ,conceivability and evolution 
of mind1 #bd      -- ultimate range of1 
#bgh      ,conceiving1 power of1 derived
 from sense-experience1 #bf             
,concept1 as a shadow1 #aai      -- the 
hyperspace1 gradual rise of1 #bg-h      
,conception and awareness1 #abj      -- 
every1 based upon prior experience1 #be 
-- freedom1 the1 of the mind from1 #bh  
,conceptions1 symbolic1 dependence      
placed upon1 #adg      ,concepts1 and   
the ,thinker1 #bdd      -- intuitograms 
as1 #bdh      -- quality of1 dependent  

upon sense-experience1 #bf              
,concepts1 the perception of1 #bee      
,conceptual1 related to the objective1  
#bia      -- the1 and mathematicians1   
#ga      ,conceptualization1 the act of1
 #d      -- the power of1 dependent upon
 evolution of mind1 #be                 
,conclusions1 mathetic1 complex         
dependence of1 #cg      ,concrete hewn  
into shape by the intellect1 #bie       
,congruency1 the1 between intellect and 
the universe1 #cai1 ff      ,congruity1 
the1 between concepts and objects1 #abj 
,connection1 the1 traced out between    
reality and object1 #abf                
,co-originality of things1 #cjc         
,consciousness and kosmos1 graph        
representing1 #bga      -- -- the       
character of the sensible world1 #aff   
-- -- -- wake of life1 #bje      -- --  
time1 #bbd      -- angularity of1 #abj  
-- a priori1 #ahe      -- a replica of1 
and judgments1 #bje      -- as a scale1 
#afe      -- -- barrier to ultimate     
knowledge1 #bjg      -- -- determinant  

of dimension1 #ade      -- -- life1 #bie
      -- -- variable quantity1 #aga     
-- attachment of1 to the order of       
becoming1 #cje      -- dawn of1 and     
dimension1 #agi      -- degrees of1 #afb
      -- deprived of the validity of    
notions1 #cad      -- evolution of1 #bej
      -- expansion of1 #afd1 #bgb       
-- extended1 #hi-ij      -- focus of1   
#bij      ,consciousness and the planes 
of the kosmos1 #afc      -- fragmentary 
view of1 #bfj      -- freedom of1 in the
 genus homo1 #ccb      -- habitation of1
 #bgh      -- higher1 and art1 #aha     
-- -- planes of1 #agh      -- history   
of1 #bih-i      -- identification of1   
with objects1 #ahi-ij      --           
intuitional1 #bee      -- keyed to the  
entire range of reality1 #afg-h      -- 
kinds of1 #bgj1 ff      -- kosmic1      
,elysian fields of1 #bce      -- of the 
primitive man1 #bec      -- organ of1   
#ahd1 ff2 #ij      -- psychics of1 #bib 
-- raised from the sensuous to the      
conceptual1 #gb      -- recipient of    

truths from dual sources1 #bf      --   
sensible world as instrument of1 #aii   
-- triple presentation of notions of    
space to1 #gb      -- turned inward1    
#bhc      -- unification of the states  
of1 #aib      -- youthful1 of mankind1  
#abb      ,consequences1 the science of1
 #he      ,considerations1 mathematical1
 and reality1 #abh      ,consistency1   
kosmic1 #agd      -- self1 of           
hyperspatial hypotheses1 #fg      -- the
 criterion of geometry1 #hf      -- --  
inconsistence of1 #agc      -- --       
metageometrician curtailed by1 #he      
,construction1 idealized1 cannot be     
objectified1 #add1 #bgf      -- ideal1  
the meaning of1 #d      -- mental1      
trafficking in1 #aed                    
,consubstantiality of intellectuality   
and spatiality1 #cca1 ff                
,continuity of the psychic plasm1 #bfj  
,contrivances1 mathetic1 the passing of1
 #bda      ,convergence of parallel     
lines1 #cae      ,convolutions1 duplex1 
#adf      ,cordinates1 as spatial       

determinants1 #bjc      -- of           
hyperspace1 #id1 #ig1 #ii1 #aae-f       
-- systems of1 #cja1 ff      -- three1  
#acb1 #bcb      ,copolarity of ideas and
 objects1 #abg      ,corpora            
quadrigemina1 #cde      ,corpuscular    
orbits of particles1 #aeb      ,cosmos1 
significance of1 #d      ,creation1     
impregnated screen of1 #bcb             
,creations1 the1 of the intellect1 #agc 
,criterion of geometry as consistency   
and convenience1 #hf      -- -- truth1  
#cbc-d      ,crookes1 #cdd      ,cube1  
as succession of planes1 #adh      --   
generation of the1 #add      --         
illustrated1 #acc      -- the           
generating1 #acd      ,curiosities1     
analytic1 energy spent in elaborating   
misappropriated1 #ci      ,current1     
electric1 as component in the fourth    
dimension1 #aec      ,curvature of      
space1 the doctrine of1 formulated by   
,riemann1 #e      -- -- -- valid element
 in the notion of1 #cje      -- the     
measure of1 #fa      ,curved space and  

metageometricians1 #caf        ,d       
,d',alembert1 note1 #ea      ,dante1    
#aha      ,darwin1 #aha      ,day1 the  
,great ,kosmic1 #bac      ,de ,tilly1   
#ff      ,death1 as fourth dimensional  
unity1 #aei      -- of the intellect1   
#aie      ,decrement of the diametrical 
ratio to the circumference of circle1   
#cab      ,deeps1 fearful1 of kosmic    
mind1 #ga1 #ia      ,definition1        
apotheosis of1 #gg-h      -- as an      
arbitrary determination1 #ajj      --   
the deification of1 #cg      ,degrees of
 realism1 #afd-e      ,deiform1 basic   
idea of the1 #e      ,demarkation       
between reality and phantasy1 the line  
of1 #agc      ,deposits of life1 #agc   
,descartes1 #cbb      ,designs1 cut out 
in materiality by life1 #bfd            
,details1 the power to dispose of       
innumerable1 #cjc      ,determinations1 
geometrical1 the necessity of1 inheres  
in logical     deductions1 #gh      --  
the factors of conscious1 #afb          
,determinative period of mental         

development1 #ca      ,development of   
the intuitive faculty1 future1 #ahh     
,diacritics of life1 #bhf               
,diameter1 the ratio of1 to the         
circumference1 #caa-b      ,difference  
between concept and thing1 #aba-b       
-- between mathematical and perceptual  
space1 #abj      -- between the ideal   
and the actual is dynamic1 #ajh         
,differential1 among minds1 #abd        
,difficulty of imagining the fourth     
dimension1 #ij      -- of propagating an
 intuition1 #cae      ,difficulties in  
the acquisition of real knowledge1 #cbe 
-- of hyperspace1 the logical1 #aci-da  
,dilemma of metageometricians1 #ig-h    
,dimension1 a distinct stage in         
psychogenesis1 #bg      -- -- system of 
space measurement1 #e      -- and       
analysis1 #afc      -- -- ,d',alembert1 
note1 #ea      -- -- the action of a    
tartrate in1 #aef      -- as an         
arbitrary contrivance1 #bfb      -- --  
-- assemblage of elements1 #ic      --  
-- direction1 #adj      -- -- extent1   

#ih      -- -- lying near the surface of
 things1 #aei-fj      -- -- space1 #if  
-- current definitions of1 #ih-ajj      
-- denied legitimate anchorage1 #bfe    
-- determined by consciousness1 #ade    
-- does not explain spiritualism1 #aee  
-- evolution of the fourth1 #dd      -- 
fourth1 direction of1 #acd      -- as   
jack of all trades1 #aef      --        
impossible to actualize1 #abd      --   
key1 to non-,euclidean geometry1 #ij    
-- movement of a plane into1 #adg-h     
-- no motion of material masses into1   
#abe      -- not a priori1 #aaf      -- 
proof of the existence of the fourth1   
#acf      ,dimensionality1 and the      
intellect1 #bjj      -- as dependent    
upon the will of the investigator1 #ie  
,dimensionality1 conception of1 purely  
conventional1 #ib      -- logical       
difficulties which beset1 #ig      -- of
 perceptual space1 #gb      -- -- space1
 the four1 #adb-c      ,dimensions1     
four1 no basis in consciousness for1    
#agb      -- piled upon one another1    

#cjj      -- vanity1 the1 of segmenting 
space into many1 #bii      ,direction as
 dimension1 #ii      ,disappointment    
with the sensible world1 #abj-a         
,discoveries1 metageometrical1 as       
excrescences of mental unfoldment1 #aca 
-- never the result of methodic         
reflection1 #bi-cj      ,disorder1 the  
edict of1 and space1 #bcj               
,disorderliness1 the fringe of1 #bgg    
,dispossessal of the intellect by the   
intuition1 #aie      ,dissimilarities1  
besetment of1 #bia      ,distinctions1  
between conceptual and perceptual space1
 fundamental1 #gc-e      ,diversity1 a  
result of the fragmentative tendency of 
life1 #f      -- as transfinite         
quantity1 #db      ,divinity1 prize of1 
won by the intellect1 #dc      ,duadic  
phase of evolution1 #baj-a      ,duodim1
 defined1 #f2 #abh1 #abi1 #acd1 #agg    
-- consciousness1 #afc      ,duopyknon1 
#bab1 #bac      -- the meaning of1 #f   
,duopyknosis1 #bjh-aj1 #bah      -- as a
 stage in the evolution of space1 #f    

,duration and space1 #bbd      --       
eternal1 #bcd      ,dynamic appearance1 
space as1 #ajd1 #ajg      ,dynamism1 of 
the intellect1 functional1 #cjc         
,e      ,earth1 the nebulous youth of1  
#cce      -- a new1 #ceh      ,east     
effloresces as ,west1 #cjf      ,ego1   
the1 #bgj      -- and percepts1 #bdd    
-- as sovereign1 #bdd      -- ,thinker's
 dependence upon1 #bde      -- --       
treatment of1 #bdf      ,egopsyche1 #bfe
      -- as an agency of                
self-consciousness1 #beh      --        
defined1 #g      ,egos1 compared to     
choppy sea1 #bef-g      ,egypt1         
birth-place of geometry1 #dd            
,eisenlohr1 #dd      ,elaborative period
 of the non-,euclidean geometry1 #ff-gj 
-- -- of mental development1 #ca-cb     
,element of evolution1 basal1 #bei      
,elements of the non-,euclidean         
geometry1 #gi-hj      ,elephant and     
blind men1 #beg      ,ellis1 ,wilmot    
,e41 on ether as four-dimensional1      
#aeg-i      ,embrace of direct          

cognition1 #bje      ,engenderment of   
space1 #bfj      ,ensoulment of the     
world-plasm1 #cbi      ,entities1       
hyperspace1 and the phenomenal world1   
#abh-acj      ,enumeration1 reformation 
of the system of1 #ch      ,enveilment  
of consciousness1 #bgc-d                
,environment1 artificial and natural1   
#afb      ,epiphysis1 the1 #cde         
,equidistantial1 described by           
hyperspatial perpendicular1 #hj         
,essence1 elemental1 as world-plasm1    
#cbi      ,ether as possessing fourth   
dimensional characteristics1 #aeg       
,ethics recast in a new mold1 #ceh      
,euclid1 #df1 #gj1 #hc1 #bfc      -- and
 the parallel-postulate1 #de      --    
never-dying elements of1 #ec            
,evidences1 mathematical1 exemplifies   
intellectual evolution1 x      -- the   
vanity of fragmentary1 #bjd             
,evolution1 a continuous process1 #cbg  
-- and the norms of reality1 #age       
-- basal element of1 #bei      --       
commencement of1 #bcb-c      --         

intellectual1 forward movement of1 #ahd 
-- kosmic1 vicissitudes of1 #bae      --
 laws of1 govern mind1 #bh      --      
mental1 results of1 #abb      -- of     
faculties1 synchronous with evolution of
 matter1 #ccb      -- -- material       
characteristics of the earth1 #cce      
-- -- organs1 time required for1 #ahh   
-- preparation of the field of1 #bah    
,exaltation of matter into spirit1 #cbi 
,examples of new perceptive evidences1  
#cda-c      ,existence on a higher      
plane1 states of1 #afb      ,experience1
 corroborative testimony of1 denied     
hyperspace1 #bfc      -- prior1 and     
conception1 #be      ,experiences1      
spatial1 systematization of1 #gh        
,extension1 space as an unbounded1 #fa  
,extra-spatiality1 degraded into        
spatiality1 #bfa      ,extravaganza1    
mathematical1 and the fourth dimension1 
#aef      ,eyes1 as ,roentgen rays1 #cec
        ,f      ,fact-mass1 #bhi        
,factors of conscious determinations1   
#afb      -- four1 of the criterion of  

truth1 #cbd-e      ,facts as facets of  
truth1 #bhd      -- attempts to reform1 
#cjd      -- logic as symbolism of1 #bhg
      -- -- the modeler of1 #bhh        
,faculties1 dual derivation of1 #afb    
-- evolution of1 synchronous with       
evolution of material     qualities1    
#ccb1 ff      -- extended1 #bci      -- 
foreshadowed by the hyperdimensional1   
#aca      -- higher1 man must win his   
way to1 #ceg      -- new faculties      
evidenced by four-space1 #bd-bi      -- 
rudimentary nature of1 #cda1 ff      -- 
the source of1 #cdi      ,faculty and   
the intellect1 #bdg1 ff      -- as      
transmissible character1 #bea      --   
determined by function1 #ccf      --    
greatly extended1 #aag      -- higher   
than the intellect1 #abf      --        
,i-making1 self-consciousness as1 #bdc  
-- intuitive1 #ahe      -- of perception
 in higher worlds1 #cec      -- --      
awareness1 the awakening of1 #hi-ia     
-- overshadowed by the intellect1 #ahh  
-- of perception in higher worlds1 #cec 

-- rudimentary condition of1 #aib1 ff   
,faculty1 outcropping of1 #bdi-ea       
,failure of efforts to justify the      
objective existence of     four-space1  
#abe-f      ,failures at solving the    
parallel-postulate1 outcome of1 #dh1 #hc
      ,faith1 dispossessed by knowledge1
 #ceh      ,fay1 mathetic1 #afj         
,fechner1 #ci      ,finity and          
unboundedness of space1 #gf             
,flammarion1 ,camille1 #cda-b1 #cdd     
,flexity1 as property of hyperspace1 #fc
      ,flexure1 space as a1 #cje        
,fluxional1 between sense objects and   
ideal representations1 #abb-c           
,fluxion1 psychic1 as difference between
 memory image and     object1 #g-h2 #abb
      ,focus of consciousness1 #afc1    
#bij      -- -- the intellect1 #caj     
,fohat1 and the creation of morphons1   
#bai      -- as creative energy1 #bac   
-- ,creator1 #h      ,fohatic energy1   
#bbf      ,form1 as vehicle of life1    
#ccj      -- definition of1 #h      --  
driven back over the path of its        

genesis1 #cbg      -- pure1 cannot exist
 in nature1 #bid      -- the idealty of1
 #aaj      -- the universe not a pure1  
#ajh      ,formative period of mental   
development1 #ca      -- -- of          
non-,euclidean geometry1 #ee            
,formula1 eight different alcohols from 
one1 #aee      ,four-dimensionality1    
justification of1 #agf      ,four-space1
 a curiosity of analytics1 #dj      --  
and ,riemann1 #bg      -- as a          
divertisement1 #age      --             
consciousness does not act in the1 #agb 
-- existence of1 denied1 #aga-b      -- 
movement of matter in1 #aeg      --     
reality of1 glibly proclaimed1 #aed     
-- the1 #h1 #bdj      -- -- denizens of1
 #cec      -- -- domain of1 #aed      --
 -- study of1 #abd      -- use of       
analogies to popularize1 #abh           
,fourth dimension1 analogical reasoning 
of1 #agg-i      -- as a transcendental  
problem1 #adj      -- -- an attitude of 
the intellect1 #bjj      -- electric    
current as a component in1 #aec      -- 

imaginability of1 #ajf-g      -- ,simon 
,newcomb on1 #abe      ,fragmentariness 
of the intellectual method1 #afd        
,fragmentary view of the universe1 #bfj 
,fragmentation1 tendency to1 #bif       
-- harks back to cell-division1 #cja    
,freedom1 a new1 #cch      -- determined
 by absence of bonds1 #bhj      --      
mathematical1 #fj      -- mental1 #ff-g 
-- now dawning for the mind1 #cb      --
 of consciousness in genus homo1 #ccb   
-- regal freedom of the mind1 #aah      
-- three degrees of1 #abe      --       
unrealizable1 for the ,thinker1 #bee    
,fringe1 chaogenetic1 #cjh1 #cji      --
 of disorderliness1 #beg      -- --     
kathekosity1 #bbi      ,frischauf1 #ff  
,full1 the universe as a1 #caj          
,function determines faculty1 #ccf      
,functions back of latent faculties1    
indefinite1 #cdc      -- cellular and   
histologic1 #bec      ,functioning1     
instinctive1 of the intellect1 #bhi     
,fundamentals1 totality of1 kosmic1 #bfe
        ,g      ,gamut of realism1 #afi 

,gauge1 awareness as a1 #afa-bjb        
,gauss1 ,charles ,frederick1 #h-i1      
#ef-h1 #ei1 #agd      -- -- -- as       
formulator of the non-,euclidean        
geometry1 #eg      ,geminos of ,rhodes1 
#de1 #eb      ,generability1 as property
 of space1 #fb      -- of hyperspace1   
,keyser on1 #add      -- -- space by    
lines1 #adc      ,generation1 the1 of   
the hypercube1 #acd      -- of the      
hypertetrahedron1 #ace-f      ,genesis1 
of space1 #baa1 #bbg1 ff      -- -- the 
earth1 its nebulous youth1 #cce      -- 
-- -- form1 #cbg      -- -- -- sensible 
world1 #afg      ,genus homo1 freedom of
 consciousness in1 #ccb                 
,geometricity1 #bff      ,geometries1   
non-,euclidean1 based upon a negation of
 the latent     geometrism1 #bfb-c      
-- three possible1 #ed      ,geometrism1
 engenderment of1 #bfb      --          
established by life1 #bfd      -- kosmic
 geometrism1 establishment of1 #bca     
-- latent1 #beg      -- native          
geometrism of space1 #bfb      --       

rediscovered by the intellect1 #i-aj    
,geometrism1 the basis of1 #bcg1 #bfa   
,geometry1 anti-,euclidean1 #fi      -- 
and the study of magnitudes in space1   
#bjc      -- artificial1 #bfb-d      -- 
a two-fold1 #ei      -- breakdown of1   
#bff      -- determinative period of the
 non-,euclidean1 #fa-d      -- diverse  
systems of1 #cbc      -- ,euclid's      
,elements of1 #ec-d      -- formal      
bulwarks of1 #gg-i      -- ,imaginary   
,geometry1 ,the1 #fj      -- natural    
geometry1 #bfa      -- ,plato and the   
divine geometry1 #aic      -- possible  
systems of1 #agd      -- radical essence
 of pure1 #he      ,geometry1           
non-,euclidean1 at variance with the    
parallel-postulate1 #hc-e      -- --    
based upon a misconception1 #ia      -- 
-- determined by qualitative            
differences1 #adj      -- -- elaborative
 period of1 #ff-gj      -- -- first     
published treatise on1 #fj      -- --   
formative period of1 #ee      -- -- the 
final issue of1 #hh-i      -- -- growth 

and development of1 #ib      -- --      
invalidation of1 #agd      -- -- key to1
 #ij      -- ,lambert's ,non-,euclidean 
geometry1 note1 #ea      -- --          
popularization of1 #ff-g      -- --     
,schweikart's treatise on1 #ei      --  
-- self-consistence of1 #eg      -- --  
some elements of1 #gi-hj      -- --     
superperceptual knowledge of1 #gb       
,geometry1 non-,legendrean1 #gj      -- 
symbolic1 and commensurable quality1    
#ade      ,gerling1 as correspondent of 
,gauss1 #eh      ,germ-plasm1 continuity
 of1 #bfj      ,giantism1 #cdg          
,glimpse1 a1 of the reality of space1   
#cad      ,glorification of the flesh1  
#bbg      ,groups1 transformation1      
discovery of1 #cj      ,guide-posts to a
 new domain1 #bgc1 #bgg      ,gulf1     
interposed between manifestation and    
non-manifestation1 #bjf        ,h       
,halley1 memoir of1 mastered by ,la     
,grange1 #ej      ,halstead1 ,g4 ,b41   
note1 #di1 #ff2 note1 #gi      ,heindel1
 ,max1 note1 #cec      ,helmholtz1 #ci1 

#ff      ,hewer1 the1 of the concrete1  
the intellect as1 #bie      ,hinton1 ,c4
 ,h41 #aed      -- on the fourth        
dimension1 #aec      ,history1 the1 of  
mind1 three great epochs of1 #ca-cb     
,homogeneity1 the1 of realism1 #afe     
,hoel1 ,j41 #fj1 #ff      ,hudson1 #cdd 
,humanity and the exalting power of     
life1 #cca1 ff      ,hypercube1         
boundaries of1 and mirrors1 #acf-i      
-- the generation of1 #acd1 #ade        
,hyperdimensional1 as a prophecy of new 
faculties1 #aca      ,hyperspace1 #age  
-- a monument to the intellect1 #cef    
-- and the involved procedure of        
arriving at a recognition of its        
relations1 #gc      -- and the passage  
thither1 #bic1 ff      ,hyperspace as a 
figurative mountain-peak1 #bjj      --  
-- -- movement1 #dd-fh      -- -- an    
all-powerful something1 #afj      -- -- 
-- idealized construction1 #aj      --  
-- evidence of new faculties1 #bd       
-- -- ignes fatuii1 #aed      -- --     
illusion1 #bjb      -- a symbol of      

higher consciousness1 #ahj-d      --    
concept of1 as an evolutionary quantity1
 #dd-e      -- confounded with real     
space1 #ia      -- creation of1 #cce    
-- denied the corroborative testimony of
 experience1 #bfc      -- direction of1 
baffling to mathematicians1 #adf      --
 discovery of1 a sign of mental         
evolution1 #ahj      -- domain of1 a    
fairy-land1 #bci      -- magnitudes of1 
the non-sensuous1 #gb      -- mysterious
 hiding place in1 #abi      -- the      
logical difficulties of1 #aci-ada       
-- the six pillars of1 #fb-c            
,hyperspatiality as the toys of         
childhood1 #bda      ,hypertetrahedron1 
the1 #ace      ,hypertrophy of the      
pituitary body1 #cdg      ,hypervolume1 
the1 #acd      ,hypothesis1 a           
superfoetated1 #ada      -- four-space1 
utility of1 #abi2 #aed      ,hypotheses1
 admissibility of1 #aah      --         
incompatibility of the non-,euclidean1  
#fg2 #ga      -- ,riemannian1 #hj       
-- solution of1 ,bolyai1 #ca            

,hyslop1 ,james ,h41 on the logical     
difficulties of hyperspace1 #aci-ada    
,i      ,icosahedron1 examination of    
the1 #bhe      ,ideal1 and the real1    
kosmic chasm between1 #ajg      --      
perceptual value of1 #bge1 ff           
,ideas and words as symbols1 #abf-g     
-- ,malebranche on1 #bbb      -- realism
 of1 #bd-ce      -- the symbolism of1   
#bje      ,identification of            
consciousness with being1 #bje-f      --
 with the objects of study1 #ahi-ij     
,identity of things-in-themselves1 #cec 
,ignes fatuii and hyperspaces1 #aed     
,illustration of plane-rotation1        
#adh-aej      -- -- the tesseract1 #acc 
,images1 totality of1 recoils upon us1  
#afg      ,imaginability of the fourth  
dimension1 #ij1 #ajf-g                  
,imagination1 premises of1 the          
mathematical1 #adf      ,impossibility  
of plane-rotation1 the structural1      
#aea-c      ,impressions1 the symbolism 
of1 neurographic1 #ahf      ,impulse1   
the satisfaction of the original        

creative1 #caj-a                        
,incomprehensibility of reality to the  
intellect1 #abf      ,incongruity1 life 
estranged by a radical1 #cbe            
,individual as space1 #bbc              
,ineptitude of intellectual             
determinations for vitality1 #cad       
,infinite1 interpreted in the terms of  
the finite1 #hb      ,infinitesimals of 
unity1 numbers as1 #da      ,infinity as
 a process1 #aji      -- of becomings1  
#bcd      ,infinity of parallels through
 a given point1 #gj      -- -- space1 a 
capital illusion1 #aie      -- the      
concept of1 #bgg      -- -- innate dread
 of1 #ajc      -- -- relativity of1 #aid
      ,influence of abstract thought1   
#cc      -- -- ,kant on the             
non-,euclidean geometry1 #di-ej      -- 
-- ,la ,grange1 #ea-b      -- -- the    
intellect1 #cae      ,infundibulum1 the1
 #cde      ,inner organs of perception1 
#ceb      ,innermost1 the1 realities of 
things1 #cef      ,insouciance of the   
geometer1 #if1 #bid                     

,instant-exposure and intellect1 #caa   
,instrument1 intellect likened to a     
color-bearing1 #cac      -- for the     
measurement of the passage of space1    
#big      -- of consciousness1 the      
sensible world as1 #aii      -- -- life1
 form as an1 #cbh      ,integers1 as    
fractional parts of unity1 #da          
,intellect and its domination of the    
intuition1 #ccc      -- -- -- final     
union with the space-mind1 #aid      -- 
-- -- topography1 #cab      -- --       
spatiality1 #bfc      -- -- the deposits
 of life1 #agc      -- -- -- designs cut
 by life in materiality1 #bfd      -- --
 -- dictum of ,sensationalists1 #bf     
-- -- -- instant-exposure1 #caa      -- 
-- -- intuitive faculty1 #bdg1 ff       
-- -- -- prize of divinity1 #dc      -- 
as a color-bearing instrument1 #cac-d   
-- -- -- searchlight1 #afh              
,intellect as a fashioner of phenomena1 
#aii1 ff      -- -- hewer of the        
concrete1 #bie      -- -- sole          
interpretative agency1 #aff-g      --   

automatism of1 #bec1 #cch      -- cannot
 seize life1 #bhb      -- crowned by a  
diadem of gold1 #cch      -- dominated  
by the intuition1 #bej      -- fashioned
 for matter only1 #bca      -- follows  
in the grooves of logic1 #bid      --   
hyperspace as a monument to1 #cef       
-- in the field of vitality1 ix      -- 
its aptitude for starts and stops1 #bib1
 #cjb      -- -- instinctive tendency to
 fragmentate1 #bif      -- quality1     
determined by mathematics1 #ccg      -- 
makes for individuality1 #beb      --   
misses the ceaselessness of life1 #bja  
-- modus vivendi of1 its influence upon 
knowledge1 #ahd      -- the constitution
 of1 #bia      -- -- cut and mode of1   
#afg      -- -- focus of1 #caj      --  
-- illusion of1 #bdf      -- --         
illusionary character of1 #cbc      --  
-- incomprehensibility of reality to1   
#abf      -- -- instinctive functioning 
of1 #bhi      -- -- judgments of1 #cjb  
-- -- moods of1 #bjb      -- --         
predominating tendency in1 #cbj-a       

-- -- scientific tendence of1 #afe-f    
-- -- struggle of1 against dispossessal1
 #aie      -- unsuited to deal with     
reality1 #cbb1 ff      ,intellectuality 
and reality1 #cjd      -- -- spatiality1
 consubstantial1 #cca                   
,intellectuality as co-extensive with   
spatiality and materiality1 #bcf      --
 the source of1 #bfj      -- -- ,thinker
 makes his own1 #bdb      ,intelligence 
and automatism1 #bec1 ff      -- the    
,thinker as a pure1 #bdc      --        
transfinite intelligence and the degrees
 of realism1 #afd      ,intent1 the     
kosmic1 of the intellect1 #cbf          
,interests1 the sphere of our1 as closed
 circuits1 #afg      ,interior1 the     
great1 #bij      ,interiority of being1 
#bij      ,interpretation1 the standards
 of1 vary as consciousness varies1 #aga 
,interstices of materiality1 #bfd       
,intuition and brotherhood1 #beb      --
 -- communalization1 #cja      -- -- the
 riddle of spatiality1 #cbe      -- as  
dispossessor of the intellect1 #aie     

-- cannot be served by mathematics1 #ccg
      -- dominated by the intellect1    
#ccc      -- held in abeyance1 #cch1 ff 
-- its domination of the intellect1 #bej
      -- the development of a spatial1  
not absurd1 #ade      -- -- need of a   
sympathetic attitude towards1 #bdi      
-- -- results of the development of1    
#ceg      ,intuitional consciousness1   
#bee      -- the superiority of1 over   
the rational1 #ahg      ,intuitions and 
the lead of life1 #aia      -- -- --    
,thinker1 #bg      -- free1 mobile and  
formless1 #aff      -- the              
conceptualization of1 #bdh      -- --   
humility of1 #afe      -- -- nature of1 
#ahe      ,intuitograms1 as concepts1   
#bdh      ,intuitograph as means of     
contacting the egopsychic consciousness 
by     the ,thinker1 #aj      -- as     
super-concepts1 #bee      -- the        
difficulty of transmitting1 #cae        
,invariability1 the vaunted1 of the laws
 of mathesis1 #cg      ,invariants1     
psychological1 #bd      ,investigations1

 metageometrical1 and the new           
sense-organs1 #ced      ,involution1 as 
antithesis of evolution1 #aj-aa      -- 
kosmic involution1 #bbf      -- of      
matter1 #cbh      -- the movement of1   
#bai      -- -- seven stages of1 #bjh-ab
      ,ions1 creation of1 #bai      --  
magnitude of a hydrogen ion1 #bbe       
,j      ,judgments a priori1 ,kant on1  
#he-f      -- and the faculty a priori1 
#aij-a      -- -- zones of affinity1    
#abd      -- based upon a replica of    
consciousness1 #bje      -- no          
trustworthy1 can be predicated upon     
fragmentary knowledge1 #bhb      -- of  
the intellect1 #cjb      -- the         
lessening of error in1 #bef      -- --  
more complex the more at variance with  
the nature of things1 #ge      -- --    
synthesis of1 #beg-h      -- valid      
judgments long delayed1 #agj-a          
,judicative power of mathematics1 #ahj  
,justification for a multi-dimensional  
quality in space1 #bfb      -- of       
four-dimensionality1 #agf               

,justification of sense-deliveries by   
one another1 #gf      -- -- the         
existence of the fourth dimension1 #abe 
,k      ,kant1 #he1 #aha1 #ahb1 #cbb    
-- and the faculty of thinking1 #bfa    
-- -- -- idea of space1 #cbb-c      --  
influence of1 on the hyperspace         
movement1 #di-ej      -- on the nature  
of things1 #aai      -- -- space as an  
intuition1 #aae      ,kathekos1 #bcc-d  
-- as chaos1 #bff      -- -- symbol of  
,chaos-,theos-,kosmos1 #aa-ab      --   
symbology of1 #bcd      ,kathekosis1    
note1 #bbg      ,kathekosity1 fringe of1
 #bbi      -- rock-bound coast of1 #bfg 
-- significance of1 #ab      ,kathekotic
 consciousness1 #bgb      -- period1    
#bji      ,key to the mysteries of      
nature1 the fourth dimension as1 #aca   
,keyser1 ,cassius ,jackson1 and freedom 
of the mind1 #cc      -- -- -- on       
attitude of metageometricians1 #ga      
-- -- -- -- dimensionality1 #id-e       
-- -- -- -- four-dimensionality of      
space1 #adb-c      -- -- -- --          

generability of hyperspace1 #add        
,klein1 ,felix1 #ab1 #ed-e1 #ff-g       
,knowledge1 all1 relative1 #aja      -- 
barrier to the certitude of the         
,thinker's1 #ahf-g      -- fabric of1   
#aif      -- hypothetical nature of1    
#ahi1 ff      -- immeasurable realm of1 
laid bare by the telescope1 #bih        
,knowledge1 mathematical1 apriority of1 
questioned1 #cg      -- nature of the   
non-,euclidean1 superperceptual1 #gb    
-- real1 difficulties of acquiring1 #cbe
      -- related to the stream of life  
and to the shore of materiality1 #cbc   
-- relative1 degrees exist for1 #afd    
-- sphere of1 #ahd      --              
systematization of1 #abf      --        
ultimate1 consciousness as a barrier to1
 #bjg      -- unification of1 #bef      
,kosmogenesis1 the latent geometrism of1
 #bfd      -- -- scope of1 #bcg         
,kosmometer1 the1 #big      ,kosmos1 and
 consciousness1 graph representing1 #bga
      -- arrangements of the contents   
of1 #cag-h      -- in a state of        

becoming1 #bfe      -- magnitude of1    
#cjh      -- moods of1 #caa      --     
space as the consistence of1 #bci       
-- see ,cosmos1 #ab        ,l           
,labor1 division of1 between the        
tuitional and the intuitional           
faculties1 #aic-d      -- mathematical  
labors1 significance of1 #agf      ,la  
,grange1 ,joseph ,louis1 #ab1 #ci1      
#ej-c1 #cba1 #cbb      -- -- -- -- and  
the parallel-postulate1 #ea-b           
,lambert1 ,john1 #eh      -- -- and the 
theory of parallels1 #dh-i              
,language1 the passing of1 #ceg         
,legendre1 #gj      ,leibnitz' dictum1  
#bf      ,leonard rays1 #ccf            
,license1 mathematic1 permissibility of1
 #ch      ,lie1 ,sophus1 #ab1 #fg       
-- -- and transformation groups1 #cj    
,life1 analysis incapable of dealing    
with1 #cai      -- and consciousness as 
one1 #bbd      -- -- form rooted in     
pyknosis1 #bad1 #cbh      -- -- the     
fourth dimension1 #agb      -- -- --    
inaptitude of mathematics for1 #agi     

-- -- -- power to create1 #big      --  
as agent of creation1 #cdc      -- --   
creative agent1 #bfd      -- --         
expression1 #cjc      -- -- vassal of   
materiality1 #cbi-cj      -- causative  
agencies in prolonging1 #abc      --    
current of1 as engendering element1 #cca
      -- deposits of1 and the intellect1
 #agc      -- estranged by a radical    
incongruity1 #cbe      -- exalting power
 of1 and humanity1 #cca1 ff2 #cdb       
-- exhibition of its remains1 #bhb      
-- flow of1 #bfe1 #big      -- form as  
an instrument of1 #cbh      --          
indescribable signs of1 #bai      --    
infinitive action of1 #cbg      --      
intellect has no aptitude for1 #bca     
-- intuitive1 values of1 x      --      
larger life of the ,thinker1 #aif       
-- lead of1 followed by intuition1 #aia 
-- limits of1 fixed by consciousness1   
#aih1 ff      -- most solid facts of1 as
 shadows1 #aie      -- motility of1     
admits of endless variations1 #ce       
,life1 movement of chaos in1 #cjg       

-- passage of1 #bja      -- passage     
through spatiality1 #bhh      -- power  
to manipulate1 #cjc      -- recurrent   
movement of1 #cjg      -- totality of   
egoic1 #bdc      -- undulations in the  
current of1 #cbi      -- uniqueness of1 
#da      -- wake of1 and consciousness1 
#bje1 #bhh      ,life-cycle1            
evolutionary results of the1 #bei       
,life-stream1 the1 #bgh      ,light1    
polarized1 and the fourth dimension1    
#aef      -- consciousness as a         
spreading1 #afh      ,light-years and   
space1 #bgh      ,limits1 the sphere of1
 consciousness as1 #afc      ,line1 as  
generating element1 #add      -- the    
straight1 a curved1 #gf      ,lineage of
 every principle runs back to           
monopyknosis1 #bac      ,lines1         
perpendicularity of1 in four space1 #acj
      ,lobachevski1 #ee1 #fj1 #ff1 #hg  
,lodge1 ,sir ,oliver1 #cdj-a2 #cdd      
,logic as architect1 #aah      --       
conventional forms of1 #bfb      -- data
 of1 #bhf      -- intellect follows in  

the grooves laid out by1 #bid      --   
miracle power of1 over facts1 #bhh-i    
-- rules of1 allowability under1 #afc   
-- -- -- the game of1 #afd      -- as   
symbolism of facts1 #bhg      ,logos and
 the limits of space1 #aic      -- being
 of1 #bah      -- body of being of1 #bad
      -- consciousness of1 #bjd      -- 
creative1 #ab1 #bac      -- the1 #baf-g 
,lorenz1 #hc      ,luther1 ,martin1 #aha
        ,m      ,magnitudes1 geometry1 a
 study of1 #bjc      -- non-sensuous1 of
 hyperspace1 #gb      ,makrokosmic      
consciousness1 #bgj      ,malebranche1  
,n41 #bbb2 see note      ,manifestation 
and non-manifestation1 #bjf             
,manifold1 finite1 though unbounded1 #gj
      -- manufactured by the analyst1   
#gg      -- the1 #fa-b1 #agb            
,manifoldness1 analytical               
manifoldnesses1 as mental excitants1    
#hi-ij      -- as a conventional        
construction1 #bfb      -- -- an        
intuition1 #cah      -- of space1 as a  
near-truth1 #cag      ,manning1 #hb1    

#aeg      -- and maneuvers in the fourth
 dimension1 #aeb      ,manvantara1 #bac1
 #bbh-i1 #bcb      -- as evolution and  
involution combined1 #ac      ,mask1    
logic as a1 #bhh-i      ,mass-termini1  
of lines1 #bcf      ,mastery of life    
over matter1 #ccj      -- of the        
sensible world1 #aif-g      ,materiality
 as a deposit of life1 #bie      -- --  
consubstantial with spatiality1 #cca1 ff
      -- becoming spatialized1 #bfa     
-- characteristics of1 #bfj      --     
engendered by kosmic mind1 #bfa      -- 
engenderment of1 #bid      --           
interpenetrative with spirituality1 #bcf
      -- interstices of1 #bfd      --   
neglect of1 #cbj      ,materiality1     
shore of1 #cbc      -- transmuted into  
spirituality1 #cbh      ,mathematicians 
and the definition1 #cg      -- -- --   
limitations of consciousness1 #agh      
-- -- -- phenomenal world1 #aah-i       
-- as prophets1 #ccf      -- respected  
by humanity1 #ccg      -- the gods of   
mathesis1 #hd      ,mathematics1 a      

determinant of the quality of the       
intellect1 vii      -- and the criterion
 of truth1 #cbc      -- -- -- kosmic    
intent of the intellect1 #cbf      -- as
 symbology1 #ahd      -- ,euclidean1    
#adj      -- fails when it encounters   
life1 #ccg      -- its inaptitude for   
life1 #agi      -- -- kingdom yielded to
 kinetics1 #ceh      -- possesses no    
judicative power over life1 #ahj      --
 the orthodoxy of1 #cd      ,mathesis1  
and conceptional space1 #bjc      --    
conduct of the intellect in the field   
of1 vii      -- definition of1 #ac      
-- domain of1 as origin of fourth       
dimension1 #acj      -- gods of1 #hd    
-- marvelous domain of1 #aah      --    
realm of1 not submissive to laws of     
sensible space1 #cd      -- things of1  
as emblems of kosmic forces1 #bci       
-- world of1 #fg1 #abg      ,matter and 
mind1 syncretization of1 #bii      --   
appulses of life against1 #cbh      --  
as a deposit of life1 #ccj      --      
honeycombs of1 #cci      ,matter        

metamorphosed by life1 #cbh      -- mind
 as wedded to1 #bdi      -- movement of1
 in four-space1 #aeg      -- qualities  
of1 and faculties1 #ccb      -- seven   
planes of1 #bab      -- spiritualization
 of1 as end of evolution1 #cbg      --  
totality of1 acted upon by totality of  
life1 #cbh      -- unity of1 with space1
 #bfj      -- universum of1 #bfa        
,matutinal ceremonials of creation1 #bba
      ,measurability1 as property of    
space1 #fb-c      ,measurement of       
hyperspaces1 the science of1 #gc        
,measurements1 space determined by the  
number of1 #fa      -- systems of space1
 #ia      ,measurer of space1           
n-dimensionality as a1 #bif-g           
,mecanique ,analytique1 #ej-a1 #cba     
,mechanics of matter1 the passage from1 
to the dynamics of life1 #cbf      -- to
 biogenetics1 the passage from #ceg     
-- the turning from1 to biogenetics1    
#cbb1 ff      ,mechanism1 the doctrine  
of1 due to analytics1 #dj               
,memories1 stored in the omnipsyche1    

#beh      ,men as gods1 #bhi            
,mental evolution1 determinative period 
of1 #ca      -- -- elaborative period   
of1 #ca-cb      -- -- formative period  
of1 #ca      ,mentality1 inner mysteries
 of creative1 #ia      -- infantile     
out-feelings of the1 #ccf      -- the   
principle of1 #baj1 ff                  
,mentalities1 the adaptation of         
phenomena to1 #ahd      ,mentograph1    
the1 #aba      -- the basis of          
intellectual consciousness1 #ac         
,mesozoic man and his environments1     
#cdg-h      ,metageometrical            
investigations and the new sense-organs1
 #ced      ,metageometricians and       
hyperspace1 #bch      -- -- proof of    
rotation about a plane1 #adi-aea      --
 -- the curved space1 #caf      -- -- --
 fourth dimension1 #ade      -- -- --   
key to the mysteries of nature1 #aca    
-- baffled by the direction of          
hyperspace1 #adf      -- dilemma of1    
#ig-h      -- eschew sense-data1 #gc-d  
-- ,keyser on the attitude of1 #ga      

-- perceptual obliquity of1 #ajb      --
 -- seed thought of1 #ahb      -- turned
 to idealized constructions1 #bfc       
,metageometry1 as a stepping stone1 #bch
      -- defined1 #ac      -- ,riemann1 
the father of1 #fc      -- the fabric   
of1 #ff      ,metamorphosis of matter by
 life1 #cbh      -- of the monopyknon1  
#bad      ,metamorphotic stage of       
space-genesis1 #bbh      ,metaphysician1
 the preserves of1 usurped by           
mathematicians1 #ada      ,meta-self as 
medium of kosmic consciousness1 #ac1    
#bag      ,methods of the ego1 #cdd     
-- scientific1 and the ample explication
 of phenomena1 #aed      ,mikrokosmic   
consciousness1 #bgj      ,mind1         
amenability of1 to the evolutionary     
movement1 #bh-cb      -- and freedom    
from conception1 #bh      -- -- -- --   
problems of physical existence1 #cb     
-- -- the new freedom1 #cc      -- as   
vehicle of life1 #bhe-f      -- --      
wedded to matter1 #bdi      -- coevalism
 of1 and space1 #bdj      --            

consubstantial with space1 #bci-dj      
-- divine1 of the kosmos1 #agf      --  
evolution of1 and conceivability1 #bd   
-- of the duodim1 #abh      -- powers   
of1 not attained simultaneously1 #ccd   
-- profound influence of hyperspace on  
the1 #ia      -- space as progenitor of1
 #bbd      -- three great epochs of the 
history of1 #cj-ca      -- tuitional1   
the limitations of1 #aib-c      -- the  
unity of1 with space1 #bbd              
,mind-principle1 the quintopyknon as the
 basis of1 #bba1 ff      ,mirrors1      
three1 and the hypercube1 #acf-h        
,mobility1 life as1 #bfd      ,molecule1
 the four-dimensional1 ,newcomb on1 #aei
      ,monadic phase of evolution1      
#baj-a      ,moneron and man1 the gulf  
between1 #bh      ,monopyknon1 the1     
#bac1 #baf      ,monopyknosis1 defined1 
#ag1 #bjh-aj1 #bah      ,monstrosities1 
mathetic1 #bjb      ,moods of the       
kosmos1 #caa      ,morphogeny1 #ac1     
#bcc-d      ,morphons1 the creation of1 
#bai      ,motility1 the scope of1 #aca 

,movement1 forward1 in knowledge1 #ccf  
-- hyperspace as an evolutionary1 #dd-fh
      -- -- influence of ,kant on1      
#di-ej      -- of life in chaos1 #cjg   
,multi-dimensional quality of space1    
justification for1 #bfb                 
,multiplication of hypotheses1 #aah     
,mutuality1 intuition as promoter of1   
#cja1 ff      ,mysteries1 inner1 of     
creative mentality1 #ia      -- kosmic1 
low-lying plains of1 #bci      --- of   
life and mind1 #cbf      -- -- the      
sciences and the four-space operator1   
#aca      ,mystery1 kosmic1 the         
revelation of1 foreshadowed in manifold1
 #gg      -- of reaction1 #afg-h      --
 -- space decreases as copes of         
consciousness increases1 #bgc1 ff       
,n      n-,cordinates1 #bjc             
n-,dimensionality1 as a phase of        
geometry1 #cjj      -- -- property of   
intellectuality1 #bfi      -- -- quality
 of conceptual space1 #ad      --       
irreconcilable with perceptual space1   
#aef      -- no justification for1 #bce 

-- not surprising that the intellect    
fell upon1 #bif1 ff      -- predicates  
concerning1 #aeh      n-,space1         
imperceptibility of1 #id                
,nasir-,eddin1 #df1 #ec      ,natural   
symbols1 ideas as1 #abf      ,nature1   
representative1 of things1 #aai      -- 
the vacuum-stillness of1 #cci           
,naught to unity1 consciousness         
registers from1 #afe      ,near-truth1  
#cjj1 ff      -- as variation from      
criterion of truth1 #cbd      -- based  
upon partial knowledge1 #ad      --     
space-curvature as a1 #cjf      --      
space-manifold as a1 #cah               
,nebulosity1 #bfg      ,necessity1 as a 
bulwark of geometry1 #gg-h              
,nerve-body1 the1 #cdi      ,neurogram1 
#ad      ,neurograph1 the ,thinker's    
scrutiny of1 #aij      ,neurographical  
communications1 #bdd      ,newcomb1     
,simon1 #ad-e1 #ff      -- -- and the   
four-dimensional molecule1 #aei      -- 
-- on the fourth dimension1 #abe        
,non-,euclideans and the meaning of     

dimension1 #ig      -- -- things as they
 actually are1 #ga      -- dilemma of1  
#hf      -- meaning of the term1 #fi    
-- results obtained by1 #hh             
,non-manifestation1 as antithesis of    
manifestation1 #ajf      -- and         
manifestation1 #bjf      ,non-methodic  
advancement of human consciousness1 #bii
      ,non-spatiality1 cut off from     
spatiality1 #bff      ,norm1 as a value 
assigned by the intellect1 #afg      -- 
consciousness as a1 #ae2 #afa-bjb       
-- of life1 found by synthesis1 #cjj    
-- -- the natural geometry1 #bfc        
,note1 the dominant1 of communality1    
#cja      ,notions of space1 triple     
presentations to the consciousness1 #gb 
,number1 as a phase of kosmogenesis1    
#cji      -- numbers as infinitesimals  
of unity1 #da      ,numericity of being1
 #cji        ,o      ,objects and ideas1
 #abf-g      -- passage of1 into the    
fourth dimension1 #acj      -- of study1
 identification of consciousness with1  
#ahi      ,obliquity1 perceptual1 of    

metageometricians1 #ajb                 
,obscurantism1 mathetic1 #aed           
,occultist and the scientist1 #cej      
,omnipsyche1 #bfe      -- as agency of  
kosmic consciousness1 #beh      -- --   
neglected factor of evolution1 #bei     
-- defined1 #ae      ,one-space         
represented by a line1 #acd             
,ontogeny recapitulates phylogeny1 #bg1 
#ccd      ,order1 kosmic1 the difference
 between1 and hyperspace1 #adc      --  
mathematical1 discovered by the         
intellect1 #bfa      -- the fiat of1    
#bjh      -- the totality of1 #bfe      
,orderliness1 #bff      ,organ1         
determined by function to be performed1 
#ccf      ,organs broken up by life into
 special parts1 #cdc      -- the        
evolution of1 #ahh      ,orthodoxy of   
mathematics1 #cd      ,outcome of new   
adjustments1 #bef      ,out-feelings of 
the ,thinker1 infantile1 #ccf           
,outlook1 the spiritualization of our   
mental1 #dj        ,p      ,papyrus1 a  
hieratic1 #dd      ,parallel-postulate  

and the surface of a sphere1 #gj      --
 as basis of non-,euclidean geometry1   
#ed      -- -- stated by ,euclid1 #ee   
-- -- -- -- ,lobachevski1 #ee      -- --
 -- -- ,manning1 #hb      -- failures to
 demonstrate1 #dh1 #hb-c      -- in the 
,elements of ,euclid1 #af      --       
,saccheri's proof of1 #dg               
,parallels1 convergence of1 #hj      -- 
infinity of1 through a given point1 #gj 
-- meet at infinity1 #hh      -- theory 
of1 #dg-i      ,passage from mechanics  
to biogenetics1 #ceg      -- from       
mechanics to dynamics1 #cbf      -- from
 three-space to four-space1 #bib-c      
-- of mathetic contrivances1 #bda       
-- of space1 #af1 #big      ,pathway of 
life1 sinuosities in1 #ccj              
,patterns of the intellect1 ready-made1 
#aff      ,pentagrammaton of space1 #bcg
      ,perceptibility in the ,thinker1  
the faculty of1 #ahh      ,perception1  
domain of1 and the sensible world1 #ai  
-- inner organs of1 #ceb      --        
replaced by conception1 #bee            

,percepts1 and the ego1 #bdd            
,period1 chaomorphogenic1 #bjh1 ff      
-- determinative1 #fa-e      --         
kathekotic1 #baj      ,perisophical     
nature of analytics1 #cbj1 ff           
,perisophism1 a1 #cjj                   
,permissibility of mathetic license1    
unlimited1 #ch      ,perpendicularity1  
#acb1 #agg      -- of lines in the      
four-space1 #acj      ,phantasy and     
reality1 line of demarkation between1   
#agc      -- the world of1 #adf         
,phantom-ideal1 the1 #bgg      ,phase of
 the world age1 evolutionary1 #bji      
,phenomena1 efforts to explain1 on the  
basis of the fourth dimension1 #abi     
-- fashioned by the intellect1 #aii     
-- fragmentary interpretation of1 #cjb  
-- mind's method of apprehending1 #ahf  
-- physical phenomena amply explained by
 scientific methods1 #aed               
,phenomenal1 the inverse of realism1    
#bgh      ,philosophy and the criterion 
of truth1 #cbc-d      -- regeneration   
of1 #ceh      -- systems of1 and zones  

of affinity1 #abd      ,phosphorescence1
 dim1 added to the unilluminated pool of
     sense-consciousness by the         
intuition1 #bf      ,phylogeny and      
ontogeny1 #ccd      -- represented in   
ontogeny1 #bg      ,physical as         
embodiment of all possibilities1 #bbg   
-- phenomena as space-activities1 #bbi  
,physicality1 the principle of1 #baj1 ff
      ,physics of the brain1 #bib       
,pi proportion1 significance of1 #caa-b 
,pickering on space-curvature1 #bgi     
,pineal gland1 energized by the         
pituitary body1 #cdh-i      -- -- the1  
#cdd1 ff      -- -- not a vestigal      
organ1 #ced      ,pituitarial awakening1
 general results of1 #cee-h             
,pituitary body and the ,thinker1 #cea  
-- -- as an organ1 #cdd-ceb      -- --  
,c4 ,de ,m4 ,sajous on1 #cde      -- -- 
hypertrophy of1 #cdg      -- -- location
 of1 #bdf      -- -- the1 #cdd1 ff      
,plane1 as generating element1 #add     
-- movement of1 into the fourth         
dimension1 #adg      -- realism appears 

to be divided into1 #afd      --        
rotation about a1 #acj1 #adf-aea      --
 the seven planes of matter1 #bab       
,planes1 a cube as a succession of1     
#adh-i      ,plasm1 kosmic1 strivings   
of1 #bae      -- psychic1 #bfj      --  
the differentiation of1 #bad            
,plato1 #cbb      -- and the divine     
geometry1 #aic1 #bcg      -- -- --      
shadow consciousness1 #bha      -- on   
,god as geometrizer1 #aha      -- --    
ideas1 #cd      ,play1 sensuous1 of the 
intellect1 #bie      ,plenum1 space as  
a1 #bcj      -- the universe as a1 #ajg 
,poincair1 #ij      ,point1 an infinity 
of parallels through a given1 #gj       
-- as generating element1 #add      --  
position of1 #acb      -- succession of 
points1 a line as1 #adh      ,polarity  
between concrete and abstract1 #bid-e   
,ponderability1 as property of          
hyperspace1 #fc      ,popularization of 
the fourth dimension1 #abh      -- of   
the non-,euclidean geometry1 #ff-g      
,possibilities in the world of          

hyperspace1 #abi      -- of four-space1 
the marvelous1 #acj                     
,postulate-systems1 the multiplication  
of1 #he      ,powers of mind1 not       
attained at one and the same time1 #ccd 
,pralaya1 #bac      -- as kosmic        
quietude1 #ag      ,predicates1 all     
mathematical1 not justified by the      
phenomenal1 #abh      ,principle1       
ensouling1 #bad-e      ,printing press1 
the invention of1 and the laws of       
psychogenesis1 #cj      ,problem1 the   
fourth dimension as a transcendental1   
#adj      ,procedure1 the involved1 of  
arriving at hyperspace notions1 #gc     
,process1 evolution a continuous1 #cbg  
-- space as a dynamic1 #cjg             
,proclus1 #hc      ,profundities1       
kosmic1 and hyperspace1 #aef            
,progression eastward terminates at the 
west1 #cjf      ,proof of rotation about
 a plane1 #adf-aea      ,propagation of 
intuitographs1 difficulty of1 #cae      
,propositions1 geometric1 subjunctive   
quality of1 #ch      ,pseudosphere and  

the shape of space1 #gj      -- as basis
 of ,beltrami's calculations1 #ag       
-- the nature of1 #fd-e      ,psychics  
of consciousness1 #bib      --          
transmuted1 #bfj      ,psychogenesis1   
alchemy of1 #bbe      -- kosmic1 #bgb   
-- laws of1 and the printing press1 #cj1
 #aaf      ,psychogenesis1 mind's place 
in1 #abc      -- outgrown phase of1 #cjb
      -- outstanding facts of1 #aca     
,psychogeny1 defined1 #ag      ,ptolemy1
 #de1 #eb1 #hc      ,publicist1 the     
mathematical1 and the duodim1 #abh      
-- the non-,euclidean1 #adf      ,pyknon
 as a kosmic principle1 #bab      -- -- 
basis of space-genesis1 #ag             
,pyknosis1 #bab      -- and involution1 
#baj1 #bca      -- as a metamorphotic   
process1 #bad      -- the seven         
processes of1 #ag1 #bid        ,q       
,quality1 dimension as commensurable1   
#adj      ,quantities1 algebraic1 cannot
 represent space1 #abe-g      ,quantity1
 consciousness as a tri-space1 #agb     
,quartodim1 #ah1 #acd                   

,quartodimensionality1 #bib             
,quartopyknosis1 #ag1 #bjh-aj           
,quartopyknotic1 principle as basis of  
spirituality1 #bbh      ,quintopyknon1  
#bbj      ,quintopyknosis and the a     
priori1 #bbc      -- meaning of1 #ag1   
#bjh-aj      ,quintopyknotic principle  
as basis of mentality1 #bbh        ,r   
,race1 humanity to evolve a new1 #ceh   
-- resplendent1 #cch      ,rajah-,tamas1
 #bbj      ,rajah ,yoga1 #ceg           
,rational1 valueless when unsanctioned  
by the intuition1 #ahg      ,reaction1  
as a mystery1 #afg-h      ,real1        
,thinker as a part of the1 #bja         
,realism and its degrees1 #afd-e1 #bgi  
-- -- the ,thinker1 #cae      -- as     
life1 #bhf      -- consciousness as     
touch-stone of1 #afh      -- found in   
the direction inverse to the phenomenal 
world1 #bgh      -- homogeneity of1 #afe
      -- illusionary nature of1 #cf     
-- infinite gamut of1 #cf      -- its   
degrees and the states of consciousness1
 #aig      -- new scope of1 bewilders   

the mind1 #agj-a      -- not of         
mathematical import1 #bij      -- of    
concepts and the sensible world1 #dj    
-- -- the domain of mathesis1 #cc-d     
-- psychological quality of1 #bhe       
-- the thread of1 #afi      -- three    
dimensional scope of1 #agb              
,realities1 abstractionizing1 #add1 #bid
      -- incomprehensible to the        
intellect1 #abf      -- innermost1 of   
things1 #cef      -- lesser give way to 
greater1 #aie      -- natural symbolism 
of1 #abh      -- our1 non-existent to   
beings on spirit-levels1 #ce      --    
shaped upon conventional models1 #bid   
-- supersensuous1 #ahj      ,reality1 as
 life and consciousness1 #agc      --   
barrier to the cognition of1 #ahi1 ff1  
#bgc      -- comprehension of1 #afe1    
#cjc      -- current of1 #cbb      --   
defined as life1 #ah      -- flow of1   
#cai      -- flowing stream of1 #abd    
-- myriad ways of presenting itself1    
#bhg      -- naked contact with1 #aic   
-- not an inscrutable quantity1 #aid    

-- obscured1 #cjb      -- of the        
four-space glibly asserted1 #aed      --
 thread of1 #bjj      -- universum of1  
#abg      ,realms1 supersensuous1 #cch  
,re-becoming1 life and consciousness as 
a1 #beg      ,reciprocity of            
consciousness and realism1 #aig-h       
-- of the manifest and the unmanifest1  
#bag      ,recoil of images upon us1    
#afg-h      ,reflexive development of   
the intellect1 #bed      ,relativity of 
all knowledge1 #aja      ,religion and  
realism1 #agj      -- the changes in1   
#ceh      ,remains of life exhibited to 
the intellect1 #bhc      ,remaking of   
moods by the intellect1 #bja            
,replica of consciousness as basis of   
judgments1 #bje      ,representation1   
sensuous1 compared with a shadow1       
#aai-abj      ,research1 improvement in 
the methods of1 #abe      ,reservoir1   
psychic1 of evolution1 #bfj             
,residuum1 the unexplained1 #aeg-h      
,revelational impressions1 #bi          
,riddle of spatiality and the intuition1

 #cbe      ,riemann1 ,g4 ,f4 ,b41 #ah1  
#ai1 #ed1 #ff-g1 #hg      -- and a      
limited space1 #ajh-i      -- -- the    
curvature of space1 #e      -- -- --    
determinative period1 #fe      -- --    
finite space1 #ajc      -- -- --        
four-space1 #bg      -- -- -- manifold1 
#cag      -- -- -- as inventor of the   
manifold1 #hf      -- on the bases of   
geometry1 #fa-d      ,roentgen rays1    
#ccf      ,rotation about a line1 #adg-h
      -- about a plane1 #adf-aea      --
 -- -- -- illustrated1 #adh-ej      --  
intra-corpuscular1 #aeb                 
,rudimentary organs1 vague functions    
lying back of1 #cdc      ,rule in the   
evolution of faculties1 #ccc        ,s  
,saccheri1 ,girolamo1 #dg-h1 #eb1 #hc   
,sajous1 ,c4 ,de ,m41 on the pituitary  
body1 #cde      ,schematism1 the        
suitability of the present1 #cbj-a      
,schweikart1 ,ferdinand ,karl1 #ai1 #ff1
 #fi      -- on the non-,euclidean      
geometry1 #eh-fj      ,science ,absolute
 of ,space1 #fj      ,science and       

reality1 #agj-a      -- no longer       
empirical1 #ceh      -- skeptical       
attitude of1 #cdi      ,scientist1      
method of the1 opposed to the analyst1  
#hd-e      ,scopographic impressions1   
#ai1 #aij      ,screen1 the impregnated1
 of creation1 #bcb      ,sea1 choppy1   
and egos1 #bef-g      ,search1 the path 
of1 for spatial understanding1 #bgh     
,seb1 the god1 #bae      ,sects1        
coplanar1 #gi      ,self and the        
immensity of space1 #bbc      -- of the 
universe1 #bag      -- the kosmic1 #bgh 
,self-consciousness1 #bgj      -- and   
space1 #bbc      -- as ,i-making        
faculty1 #bdc      -- determination of1 
#cjb      ,self-consistency1 #agc       
,sella turcica1 the1 #cde               
,semi-,euclidean geometry1 the1 #gj     
,sensationalists on the intellect1 #bf  
,sense-data1 spurned by the             
metageometrician1 #gc                   
,sense-delivery1 one1 justified by the  
other1 #gf      -- symbolism of1 #bjc-d 
,sense-knowledge1 the sphere of1        

extended1 #cci-dj      ,senses1 the     
answer of1 to new needs1 #ccb           
,sensible world1 as domain of           
perception1 #ai      ,sensographic      
impressions1 as perceptions1 #ai        
,senso-mechanisms of the intellect1 #afe
      ,sentience1 the principle of1     
#baj1 ff      ,septopyknon1 the1 #bbg   
,septopyknosis1 as kosmic materiality1  
#ah1 #bjh-aj      ,septopyknotic1 the1  
principle1 #bbh      ,seven planes of   
matter1 #bab      ,sextopyknon1 the1    
#bji      ,sextopyknosis1 defined as    
kosmic sensibility1 #ah1 #bjh-aj        
,sextopyknotic1 as emotional principle1 
#bbh      ,shadow likened to a sense    
presentation1 #aai      ,shadowgraphs1  
#bha      ,silences1 nature's great1    
#cci      ,similitude of agent and      
principle1 #bdc-f      ,sinuosities in  
the pathway of life1 #ccj               
,socrates1 #cbb      ,solitariness of   
intellectual testimony1 #gf      ,space 
and edict of disorder1 #bcj      -- --  
four-dimensional entities1 #abh-acj     

,space and quartopyknons1 #bai-bj       
-- -- self-consciousness1 #bbc      --  
-- the idealized construction1 #add     
-- as a dynamic process1 #baf      -- --
 all-mother1 #bbi-cj      -- -- an      
assemblage of spheres1 #adc      -- --  
becoming1 #aie      -- -- boundary of   
another space1 #di-ej      -- --        
consistence of the kosmos1 #bci      -- 
-- curved1 #afd      -- -- dynamic      
appearance1 #bbh1 #cjg      -- -- eldest
 born of kathekosis1 #bae      -- --    
engendered product of life1 #cjh1 ff    
-- -- evolution1 #bbf1 ff      -- --    
finite extension1 #fb1 #ajf      -- --  
generable quantity1 #ajc      -- --     
infinite continuum1 #gj      -- --      
intuitional concept1 #bbe      -- --    
kosmic order1 #bfe1 #bfg      -- --     
nether pole of non-manifestation1 #bjg-h
      -- -- path of life1 #bic-d      --
 -- pseudosphere1 #gj      -- -- sheer  
roominess1 #aae      -- -- system of    
cordinates1 #ii      -- -- unbounded    
extension1 #fa      -- best study of1   

the consciousness1 #bfi      --         
conceptual1 as basis of non-,euclidean  
geometry1 #gb      -- -- fundamentally  
distinct from perceptual1 #gb-ge      --
 elliptical1 #fg      -- engenderment   
of1 #bfj      -- essential nature of1   
#bab      -- fabric of1 the1 lends      
itself to measurement1 #ii-ajj      --  
four-1 #h      -- fractionalized space1 
#bif      -- foundations of1 #bba       
-- four-dimensional1 #da      -- ,paul  
,carus on1 #ced      -- fourth dimension
 of1 efforts at making it thinkable1    
#abe      -- generability of1 by lines1 
#adc      -- genesis of1 #bbg1 ff       
-- geometric1 purely formal             
construction1 #ge-f      -- limits of   
and the ,creative ,logos1 #aic      --  
manifoldness of1 #gg      -- mind and1  
the coevalism of1 #bdj      -- mystery  
of1 #bgh1 ff      -- nature of1 answered
 to by mind1 #aih-i      -- nether pole 
of non-manifestation1 #bjg-h      --    
non-,euclidean1 the characteristics of1 
#gb-c      -- not a pure form1 #aaj     

-- passage of1 #af      -- path of      
search for1 must be ,thinker-ward1 #bgh 
-- perceptual1 irreconcilable with the  
fourth dimension1 #aef      -- possible 
curvature of1 and the non-,euclidean    
geometry1 #gf-g      -- primeval1 and   
tridimensionality1 #bcf      -- problem 
of devising a1 #hg-h      --            
psychological nature of1 #cje      --   
real1 confounded with hyperspace1 #ia   
-- ,riemann on the curvature of1 #e     
-- the unity of1 #bbd      -- -- -- --  
with matter1 #bfj                       
,space-activities1 as chemism and       
physicism1 #bbi-cj      ,space-center1  
the1 #cjf-g      ,space-consciousness1  
#bbe      -- as a direct process1 #bbd  
-- mergence of1 with the individual     
consciousness1 #aie                     
,space-curvature1 as an arbitrary       
construction1 #bfb      -- as           
near-truth1 #cjf      -- nature of1 #bgi
      -- no need for1 #bcg      --      
translation of1 in terms of the         
intellect1 #cae      ,space-forms1 the  

construction of1 #gd                    
,space-genesis1 #bj1 #bcb1 #bcc      -- 
alphabet of1 #bcg      -- completion of1
 #bbf      -- norm of1 #agd      --     
pyknon as a basis of1 #ag      --       
symbolism of1 #bjh1 ff1 #bce            
,space-manifold1 the1 #bgi              
,space-measurement1 and reality1        
,euclid's system of1 #if      -- systems
 of1 #bja      ,space-mind and          
archeological evidences1 #aic      -- --
 knowledge1 #ahg      -- -- the         
,thinker1 #bec1 #bgb      -- attainment 
of1 #bbe1 #bca      -- granaries of1    
#bce      -- the realization of1 #bch   
,space-process1 the1 #cje               
,space-realities1 hyperspace as a       
stepping stone to1 #bch      ,spark1 the
 omnipsyche as a1 #beh      ,spatiality 
and the intellect1 #bfc-d      --       
consubstantial with materiality1 #bj1   
#cca1 ff      -- cut-off from           
non-spatiality1 #bff      --            
interpenetrative with materiality1 #bcf 
-- nonconformity of1 with logic1 #afd   

-- riddle of1 and the intuition1 #cbe   
--1 rise of1 #ahj      -- ,thinker's    
outlook in1 #caf      -- world of       
nascent1 #bcb      ,spencer1 ,herbert1  
#cbb      ,spinoza1 #cbb                
,spiritism1 the phenomena of1 #aed      
,spiritualists1 the claim of1 regarding 
the fourth dimension1 #aed-e            
,spirituality1 materiality transmuted   
into1 #cbh      -- principle of1 #baj1  
ff      -- seeds of1 #bba               
,spiritualization of the flesh1 #bbg    
-- -- man's mental outlook1 #dj      -- 
-- matter1 the end of evolution1 #cbg   
,squares1 the form of1 in the           
non-,euclidean geometry1 #ei      ,st4  
,marie1 ,flye1 #ff      ,stage of       
pyknosis1 #bjh1 ff      --1             
quartopyknotic1 #bai1 ff      ,stanley1 
,hiram ,m41 on space as dynamism1 #ajd-g
      ,states1 the two1 of              
consciousness2 one unaware of the other1
 #aib      ,straight1 a1 determined by  
two points1 #gi      ,stuff1            
etherealized four-dimensional1 #afj     

,sum1 angular1 #gi      -- triangular1  
less than two right angles1 #ei         
,superconcepts1 as intuitographs1 #bee  
,superconceptual1 as the intuitional1   
#bj      ,superperception and automatism
 of the intellect1 #bj                  
,superperceptual1 the1 #gd              
,superunodim1 the1 #agg                 
,swedenborg1 #ahj1 #ahd      ,symbol1   
psychic1 and the brain1 #aij      --    
words as symbols of ideas1 #abf-g       
,symbolism of life1 #bhf      -- --     
neurographical impressions1 #ahf      --
 -- realities1 #abh      -- --          
sense-deliveries1 #bjd      -- --       
space-genesis1 #bjh1 ff      ,symbology1
 depicting planes of consciousness1 #bgj
      -- of mathematical knowledge1     
#bjc1 ff      ,synchronous evolution of 
faculties and properties of matter1 #ccb
      ,syncretization of mind and       
matter1 #bii      ,synthesis and the    
norm of reality1 #cjj      ,synthetic   
school of geometry1 #ef        ,t       
,tactographic impressions1 #aij         

,tartrate and a component in the fourth 
dimension1 #aef-g      ,task of mental  
evolution1 the supreme1 #afi            
,taurinus1 #ef-g1 #fi      ,telescope1  
its effect upon consciousness1 #bih     
,tendence1 intellect as scientific1     
#afe-f      ,tendencies1 destined to    
flower as faculties1 #hi-ia1 #ccb       
,tendency to fragmentate1 the intellect 
and1 #bif      ,tesseract as represented
 by ,carus1 #acf-h      -- illustrated1 
#acc      -- the1 #bj1 #ade      -- the 
elements of1 #ace      -- the four      
cordinates of1 #acb      ,testimony of  
the intellect1 incapable of comparison1 
#gf      ,tetragrammaton1 the1 #cjh     
,tetrahedron1 the1 #ace                 
,thalamencephalon1 the1 #cde            
,thales1 #dd      ,theory of knowledge1 
#bij      ,thingness of objects and the 
impossibility of comprehending1 #abb    
,things1 the dual nature of1 #aai       
-- -- value of1 #afa                    
,things-in-themselves1 the identity of1 
#cec      ,thinkability of the fourth   

dimension1 #abe      ,thinker as a pure 
intelligence1 #ahd1 #bdh      -- --     
engenderer of the intellect1 #bia       
-- -- principal to the ego1 #bdc1 ff    
-- -- six times removed from the        
sensible world1 #bja      -- -- the     
source of the intuition1 #bf      -- and
 concepts1 #bdd      -- -- his approach 
to the kosmic mind1 #bjb1 #bbc      --  
-- -- larger life1 #aif      -- -- --   
treatment of the ego1 #bdf      -- -- --
 of sense-impressions1 #gd-e      -- -- 
realism1 #cae      -- -- space1 #aic    
-- -- a segment of reality1 #aga      --
 the1 and the amplitude of the degrees  
of realism1 #aih1 #bgj      -- and the  
barriers of consciousness1 #bjg      -- 
-- -- brotherhood of man1 #cja      --  
-- -- degrees of realism1 #age      --  
-- -- manipulation of concepts1 #bee    
-- -- -- nature of space1 #bfh      --  
-- -- neurograph1 #aij      -- -- -- new
 faculty1 #cce      -- -- -- perceptive 
faculty1 #ahh      -- -- -- pituitary   
body1 #cea      -- defined as the       

spiritual man1 #ba      -- his          
consciousness and the ether1 #bhj       
-- -- dependence upon the ego1 #bde     
-- -- method of contacting the external 
world1 #ahi-aia      -- -- outlook upon 
the universe of spatiality1 #bdb        
,thinker1 his schematism of cognitive   
powers1 #bjf      -- -- sphere of       
awareness1 #bhc      -- the difference  
between1 and the intellectuality1 #bef  
-- -- evolutionary needs of1 #ccf       
-- -- upreaching yearnings of1 #ccf     
,thinker-ward1 space must be sought in a
 direction1 #bgh      ,thinking1        
abstract1 elevating influence of1 #cc   
,thought1 prolonged abstract1 benefits  
of1 #bdi      ,time1 as an aspect of    
consciousness1 #bbd      -- divested of 
timeliness1 #bgi      ,todhunter1 #df   
,tool1 fashioning1 life as a1 #bfd      
,topography of the intellect1 #cab      
,touchstone1 consciousness as a1 #afh   
,tracery of connection between ideas and
 objects1 #abf      ,trafficking in     
mental constructions1 #aed              

,transcendental1 the realm of1 and      
mathematicians1 #adb      ,transfinite  
as a limit1 #abb1 #baf                  
,transfinity1 #ba-b      ,tree-reality1 
#abf      ,triadic phase of evolution1  
#baj-a      ,triangles1 angular sum of1 
#gh1 #hh      ,tridim1 the1 #bb1 #abi1  
#acd1 #adg1 #agg      ,tridimension1    
the1 #adg      ,tridimensionality and   
primeval space1 #bcf      -- and the    
space-mind1 #aic      -- a quality of   
perceptual space1 #bb      -- mastery of
 the phenomena of1 #agb      -- the     
sufficiency of1 #aih      ,tripyknon1   
the1 #bab1 #bac      ,tripyknosis1 #ag1 
#bjh-aj1 #bai      ,truth compared with 
facts1 #bhh      -- criterion of1 #e1   
#cbc-d      -- facets of1 #bhd      --  
kinds of1 #cbd      -- logic does not   
illuminate1 #bhg-h      ,tuitive1 the1  
and the intuitive faculties1 #aib-c     
,u      ,ultima ,thule1 the1 #bjg       
,unbounded1 the1 as a finite extension1 
#gj      ,undulations1 three1 in the    
current of life1 #cbi      ,unfoldment1 

mental1 and metageometrical discoveries1
 #aca      ,uni-circle1 the1 #cjh       
,unification of all knowledge1 #bef     
,uniqueness of real space1 #ie          
,unitariness of all existences1 #bda    
,unity1 as end of analysis1 #db      -- 
death as a fourth dimensional1 #aei     
-- kosmic1 integers as fractional parts 
of1 #da      -- of mind and space1 #bbd1
 #bcj      -- the new realization of1   
#bbe      ,universality1 geometric1     
based upon the formal character of      
assumptions1 #gg-h      ,universe1 a    
glorified1 #bbf1 #bfh      -- as a full1
 #cjh      -- -- -- plenum1 #ajg      --
 and the seven planes of matter1 #bab   
-- character of1 fixed by consciousness1
 #afb      -- limited and conditioned1  
#abg      -- not a pure form1 #ajh      
-- the perfected1 and the circle1 #caj  
-- -- theory of the mechanical origin   
of1 due to analytics1 #dj      -- unity 
in the1 #cja-b      ,universum of       
appearance1 #ahg      -- of life and    
consciousness1 #beg      ,universum of  

reality1 #abg      -- -- space1 #bfi    
,unknowable as a symbol1 #aid      --   
the darkness which shuts out the1 #bjg  
,unodim1 the1 #abi1 #acd1 #agf-g      --
 -- consciousness1 #afc      -- defined1
 #bb      ,up-raisement of matter1 #cbi 
,upspringing of a new faculty1 #ccc     
,v      ,vacuum-stillness of nature1    
#cci      ,validity of mathematical     
conclusions1 #aja      ,value assigned  
by the intellect to the sensible world1 
#afg      -- assumptional1 vitiating    
influence of1 #afc      -- -- of the    
ideal1 #bge      ,vanity of fragmentary 
evidences1 #bjd      -- of intellectual 
method1 #cbe1 ff      -- -- segmenting  
space into many dimensions1 #bii        
,vassal1 life as the1 of materiality1   
#cbi      ,vehicle of life1 form as1    
#ccj      -- -- -- mind as1 #bhf        
,vicissitudes of kosmic evolution1 #bae 
,vision1 the inner1 #cef        ,w      
,wachter1 #fi      ,wallis1 ,john1 #df1 
#hc      ,weissman1 note1 #bfj          
,words1 as symbols of ideas1 #abf-g1    

#bjd      ,world and the child mind1    
#aba      -- as instrument of           
consciousness1 #aii1 #bih      -- fabric
 of1 and geometrism1 #bfa      --       
impossibility of objectifying the fourth
 dimension in the     perceptual1 #abd  
-- of phantasy1 conditions of1          
identified with sensible realm1 #adf    
-- phenomenal1 and hyperspace entities1 
#abh-acj      -- sensible1 as a carpet1 
#aif      -- the domain of a perception1
 #ai      -- -- genesis of1 #afg        
,world-plasm1 kathekotic1 #bfg      --  
as elemental essence1 #cbi        ,y    
,yoga1 ,rajah1 #ceg      ,youth of the  
earth1 the nebulous1 #cce        ,z     
,zollner1 and the claims of the         
spiritualists1 #aed      ,zones of      
,affinity1 #bb1 #abd      ,zoometer1    
the1 #big            ,transcriber's     
,notes      ,variations in spelling1    
punctuation and hyphenation have been   
retained    except in obvious cases of  
typographical errors4 ,inconsistencies  
between the text and index have been    

resolved in favour of the text4         
,duplication of the sub title 7,the     
,mystery of ,space7 on consecutive      
pages has been removed4      ,italics   
are shown thus italic4      ,in the five
 cases where illustrations could be     
realistically rendered    using only    
text symbols this has been done4