136 SCIENCE AND METHOD. 



few more than is strictly necessary, the harm is not 

 great. The essential thing is to learn to reason 

 exactly with the axioms once admitted. Uncle 

 Sarcey, who loved to repeat himself, often said that 

 the audience at a theatre willingly accepts all the 

 postulates imposed at the start, but that once the 

 curtain has gone up it becomes inexorable on the 

 score of logic. Well, it is just the same in mathe- 

 matics. 



For the circle we can start with the compass. The 

 pupils will readily recognize the curve drawn. We 

 shall then point out to them that the distance of the 

 two points of the instrument remains constant, that 

 one of these points is fixed and the other movable, 

 and we shall thus be led naturally to the logical 

 definition. 



The definition of a plane implies an axiom, and 

 we must not attempt to conceal the fact. Take a 

 drawing-board and point out how a movable ruler 

 can be applied constantly to the board, and that 

 while still retaining three degrees of freedom. We 

 should compare this with the cylinder and the cone, 

 surfaces to which a straight line cannot be applied 

 unless we allow it only two degrees of freedom. 

 Then we should take three drawing-boards, and we 

 should show first that they can slide while still re- 

 maining in contact with one another, and that with 

 three degrees of freedom. And lastly, in order to 

 distinguish the plane from the sphere, that two of 

 these boards that can be applied to a third can also 

 be applied to one another. 



Perhaps you will be surprised at this constant use 

 of movable instruments. It is not a rough artifice, 



