MATHEMATICAL DISCOVERY. 47 



do not understand mathematics? If the science 

 invokes only the rules of logic, those accepted by 

 all well-formed minds, if its evidence is founded on 

 principles that are common to all men, and that none 

 but a madman would attempt to deny, how does it 

 happen that there are so many people who are 

 entirely impervious to it? 



There is nothing mysterious in the fact that every 

 one is not capable of discovery. That every one 

 should not be able to retain a demonstration he has 

 once learnt is still comprehensible. But what does 

 seem most surprising, when we consider it, is that 

 any one should be unable to understand a mathe- 

 matical argument at the very moment it is stated to 

 him. And yet those who can only follow the argu- 

 ment with difficulty are in a majority ; this is incon- 

 testable, and the experience of teachers of secondary 

 education will certainly not contradict me. 



And still further, how is error possible in mathe- 

 matics ? A healthy intellect should not be guilty 

 of any error in logic, and yet there are very keen 

 minds which will not make a false step in a short 

 argument such as those we have to make in the 

 ordinary actions of life, which yet are incapable of 

 following or repeating without error the demonstra- 

 tions of mathematics which are longer, but which 

 are, after all, only accumulations of short arguments 

 exactly analogous to those they make so easily. Is it 

 necessary to add that mathematicians themselves are 

 not infallible? 



The answer appears to me obvious. Imagine a 

 long series of syllogisms in which the conclusions of 

 those that precede form the premises of those that 



