3 6 SCIENTIFIC APPARATUS. 



which exist irrespectively of the magnitude and even of the shape 

 of its parts, depending solely on the connection of the parts, and 

 on their situation with reference to one another. As neither the 

 term " properties of situation," nor the description which we have 

 just given of these properties, can be regarded as conveying a 

 distinct image, a few very simple examples of what is meant may 

 not be out of place. 



If we draw, upon any surface such as a plane, two closed con- 

 tours of however complicated an outline, it is quite possible that 

 they may never meet one another, or that they meet in one or more 

 points, and do not traverse one another. But if they traverse one 

 another at all, they must do so an even number of times ; i.e. 

 twice, or four, or six times, &c. The truth of this proposition will 

 be easily admitted, and it will be seen that, to understand the 

 assertion made, we require no conception of magnitude, nor even 

 the conception of the straight line or plane. All that we require is 

 the idea of a continuous closed curve, and of a surface upon which 

 it is drawn. 



Again : conceive of two bodies, one a hollow sphere, the other 

 a hollow anchor ring, and let a person imagine himself placed 

 successively in the interior of each of these two hollow bodies. 

 The two closed spaces in which he will thus successively find him- 

 self differ from one another at least in one remarkable respect. 

 There is but one way of travelling from one point A inside the 

 sphere, to another point B, also inside it : we might, of course, 

 trace any number of routes we please from A to B, but all these 

 routes are really reducible to one and the same route ; and an 

 elastic thread connecting A and B might be stretched so as to 

 assume the shape of any one of them. But now take two points 

 A and B inside the hollow anchor ring, and it. will be seen at once 

 that there are two different ways, irreducible to one another, of 

 travelling from A to B. We have thus before us an example of a 

 singly connected space (the interior of the sphere), and a doubly 

 connected space (the interior of the anchor ring). The distinc- 



