FREE BOOKS

Author's List




PREV.   NEXT  
|<   104   105   106   107   108   109   110   111   112   113   114   115   116   117   118   119   120   121   122   123   124   125   126   127   128  
129   130   131   132   133   134   135   136   137   138   139   140   141   142   143   144   145   146   147   148   149   150   151   152   153   >>   >|  
en from this source of error not having been sufficiently determined or appreciated that the lamentable wreck of the United States ship Huron off the coast of Hatteras occurred some twenty years ago. X THE FAIRYLAND OF GEOMETRY If the reader were asked in what branch of science the imagination is confined within the strictest limits, he would, I fancy, reply that it must be that of mathematics. The pursuer of this science deals only with problems requiring the most exact statements and the most rigorous reasoning. In all other fields of thought more or less room for play may be allowed to the imagination, but here it is fettered by iron rules, expressed in the most rigid logical form, from which no deviation can be allowed. We are told by philosophers that absolute certainty is unattainable in all ordinary human affairs, the only field in which it is reached being that of geometric demonstration. And yet geometry itself has its fairyland--a land in which the imagination, while adhering to the forms of the strictest demonstration, roams farther than it ever did in the dreams of Grimm or Andersen. One thing which gives this field its strictly mathematical character is that it was discovered and explored in the search after something to supply an actual want of mathematical science, and was incited by this want rather than by any desire to give play to fancy. Geometricians have always sought to found their science on the most logical basis possible, and thus have carefully and critically inquired into its foundations. The new geometry which has thus arisen is of two closely related yet distinct forms. One of these is called NON-EUCLIDIAN, because Euclid's axiom of parallels, which we shall presently explain, is ignored. In the other form space is assumed to have one or more dimensions in addition to the three to which the space we actually inhabit is confined. As we go beyond the limits set by Euclid in adding a fourth dimension to space, this last branch as well as the other is often designated non-Euclidian. But the more common term is hypergeometry, which, though belonging more especially to space of more than three dimensions, is also sometimes applied to any geometric system which transcends our ordinary ideas. In all geometric reasoning some propositions are necessarily taken for granted. These are called axioms, and are commonly regarded as self-evident. Yet their vital principle is not so much tha
PREV.   NEXT  
|<   104   105   106   107   108   109   110   111   112   113   114   115   116   117   118   119   120   121   122   123   124   125   126   127   128  
129   130   131   132   133   134   135   136   137   138   139   140   141   142   143   144   145   146   147   148   149   150   151   152   153   >>   >|  



Top keywords:
science
 

imagination

 

geometric

 
geometry
 

reasoning

 

demonstration

 

Euclid

 

logical

 

called

 

ordinary


dimensions

 
allowed
 

branch

 
mathematical
 
confined
 

strictest

 

limits

 

actual

 

related

 

closely


desire

 

distinct

 

EUCLIDIAN

 

Geometricians

 

incited

 
arisen
 

inquired

 

critically

 

carefully

 

foundations


sought

 

explain

 
designated
 

propositions

 

necessarily

 

fourth

 

dimension

 

Euclidian

 

belonging

 

applied


hypergeometry
 
transcends
 

common

 

adding

 

evident

 
assumed
 

system

 
parallels
 
presently
 

regarded