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A very simple instance will ilhistrate the matter. 

 If from a point outside a circle we draw a straight line 

 to touch the curve, the distance between the starting- 

 point and the point of contact has certain geometrical 

 properties. If the starting point be shifted nearer and 

 nearer to the circle the distance in question becomes 

 shorter, and ultimately vanishes. But as soon as the 

 point passes to the interior of the circle the notion of 

 a tangent and distance to the point of contact cease 

 to have any meaning; and the same anomalous con- 

 dition of things prevails as long as the point remains 

 in the interior. But if the point be shifted still further 

 until it emerges on the other side, the tangent and its 

 properties resume their reality ; and are as intelligible as 

 before. Now the process whereby we have passed from 

 the possible to the impossible, and again repassed to the 

 possible (namely the shifting of the starting point) is a 

 perfectly continuous one, while the conditions of the 

 problem as stated above have abruptly changed. If, 

 however, we replace the idea of a line touching by that 

 of a line cutting the circle, and the distance of the point 

 of contact by the distances at which the line is inter- 

 cepted by the curve, it will easily be seen that the 

 latter includes the former as a limiting case, when the 

 cutting line is turned about the starting point until it 

 coincides with the tangent itself. And further, that the 

 two intercepts have a perfectly distinct and intelligible 

 meaning whether the point be outside or inside the area. 

 The only difference is that in the first case the intercepts 

 are measured in the same direction ; in the latter in 

 opposite directions. 



The foregoing instance has shown one purpose which 

 these imaginaries may serve, viz., as marks indicating a 

 limit to a particular condition of things, to the applica- 



