GEOMETEICAL AXIOMS. 21 



an axiom that through any three points in space, not 

 lying in a straight line, a plane may be drawn, i.e. a 

 surface which will wholly include every straight line 

 joining any two of its points. Another axiom, about 

 which there has been much discussion, affirms that 

 through a point lying without a straight line only one 

 straight line can be drawn parallel to the first ; two 

 straight lines that lie in the same plane and never 

 meet, however far they may be produced, being called 

 parallel. There are also axioms that determine the 

 number of dimensions of space and its surfaces, lines 

 and points, showing how they are continuous ; as in 

 the propositions, that a solid is bounded by a surface, 

 a surface by a line and a line by a point, that the 

 point is indivisible, that by the movement of a point 

 a line is described, by that of a line a line or a surface, 

 by that of a surface a surface or a solid, but by the 

 movement of a solid a solid and nothing else is 

 described. 



Now what is the origin of such propositions, un- 

 questionably true yet incapable of proof in a science 

 where everything else is reasoned conclusion? Are 

 they inherited from the divine source of our reason 

 as the idealistic philosophers think, or is it only that 

 the ingenuity of mathematicians has hitherto not been 

 penetrating enough to find the proof? Every new 

 votary, coming with fresh zeal to geometry, naturally 



