

GEOMETRICAL INSTRUMENTS AND MODELS. 33 



lines, and points which exist in space, being, in technical phrase- 

 ology, the " loci " obtained by imposing one, two, or three 

 restrictive conditions upon these indeterminates. As often hap- 

 pens in similar cases, the mode of representation thus introduced 

 is capable of being extended so as to apply to other objects or 

 conceptions beside that for which it was first employed; and 

 thus mathematicians have been led to consider complexes of more 

 than three indeterminates, or, again, complexes not possessing the 

 properties which we have enumerated as characteristic of space. 

 This is the origin of such phrases as "a space of four dimen- 

 sions," or of such assertions as " it is conceivable that a space 

 may not be exactly similar to itself at all its points." These 

 speculations are perhaps not calculated directly to promote our 

 knowledge of the space in which we live and move, and to which 

 they seem entirely inapplicable ; but they have had the effect of 

 advancing our knowledge of the relations of quantity, and have 

 thus had an indirect, but not unimportant, influence upon the 

 recent progress of geometrical science. 



So great has been the influence of the Cartesian mode of repre- 

 sentation upon geometrical speculation that it has perhaps, to a cer- 

 tain extent, and in certain cases, unduly led away the minds of geo- 

 metricians from that direct intuition of space upon which geometry 

 must after all be founded. And there can be no doubt that an 

 Exhibition of models such as those included in the present Cata- 

 logue is calculated to render a great service to geometrical science 

 by calling attention to the concrete shapes of objects, which are 

 too apt, even in the mind of the serious student, to exist only as 

 conceptions very imperfectly realised. 



We may for the purposes of this introduction adopt a threefold 

 classification of the properties of space, as being either, i. Pro- 

 perties of Situation ; or, 2. Graphical Properties ; or, 3. Metrical 

 Properties. Of each of these three classes of properties we shall 

 here say a few words to illustrate their importance and meaning. 



i. The Properties of Situation of a figure in space are those 



D 2 



