x.\-\i.] LIMITS OF SCIENTIFIC METHOD. 767 



foundations from which we started, and they are, so far as 

 I can see, the foundations which we can never quit without 

 tottering. Scientific Method must begin and end with the 

 laws of thought, but it does not follow that it will save us 

 from encountering inexplicable, and at least apparently 

 contradictory results. The nature of continuous quantity 

 leads us into extreme difficulties. Any finite space is 

 composed of an infinite number of infinitely small spaces, 

 each of which, again, is cojnposed of an infinite number of 

 spaces of a second order of smallness ; these spaces of the 

 second order are composed, again, of infinitely small 

 spaces of the third order. Even these spaces of the third 

 order are not absolute geometrical points answering to 

 Euclid's definition of a point, as position without mag- 

 nitude. Go on as far as we will, in the subdivision of 

 continuous quantity, yet we never get down to the ab- 

 solute point. Thus scientific method leads us to the 

 inevitable conception of an infinite series of successive 

 orders of infinitely small quantities. If so, there is nothing 

 impossible in the existence of a myriad universes within 

 the compass of a needle's point, each with its stellar sys- 

 tems, and its suns and planets, in number and variety 

 unlimited. Science does nothing to reduce tlie number 

 of strange things that we may believe. When fairly 

 ])ursued it makes absurd drafts upon our powers of com- 

 prehension and belief 



Some of the most precise and beautiful theorems in 

 mathematical science seem to me to involve apparent con- 

 tradiction. Can we imagine that a point moving along a 

 perfectly straight line towards the west would ever get 

 round to the east and come back again, having performed, 

 as it were, a circuit through infinite S})ace, yet without 

 ever diverging from a ])erfectly straight direction ? Yet 

 this is what happens to the intersecting point of two 

 straight lines in tlie same plane, when one line revolves. 

 The same ])aradox is exhiljited in the hyperbola regarded 

 as an infinite ellipse, one extremity of which has passed to 

 an infinite distance and come back in the opposite direction. 

 A vaiyiiig quantity may change its sign by ])assing either 

 tlirough zero or through infinity. In the latter case there 

 must be one intermediate value of the variable for which 

 the variant is indifferently negative infinity and positive 



