633 



Commutative. (1) I say that a X 1 1 X a. The theorem is 

 obvious for a = 1. We can verify analytically that if it is true for 

 a = a, it will be true foi a = a + 1. 



(2) I say that a X b ~ b X a. The theorem has just been proved 

 for b = 1. We can verify analytically that if it be true for b = (3 it 

 will be true for & = ft + 1. 



This monotonous series of reasonings may now be laid aside; but 

 their very monotony brings vividly to light the process, which is uni- 

 form, and is met again at every step. The process is proof by recur- 

 rence. We first show that a theorem is true for n 1; we then show 

 that if it is true for n 1 it is true for n, and we conclude that it is 

 true for all integers. We have now seen how it may be used for the 

 proof of the rules of addition and multiplication that is to say, 

 for the rules of the algebraical calculus. This calculus is an instru- 

 ment of transformation which lends itself to many more different 

 combinations than the simple syllogism; but it is still a purely an- 

 alytical instrument, and is incapable of teaching us anything new. 

 If mathematics had no other instrument, it would immediately be ar- 

 rested in its development; but it has recourse anew to the same 

 process i.e., to reasoning by recurrence, and it can continue its 

 forward march. Then if we look carefully, we find this mode of 

 reasoning at every step, either under the simple form which we have 

 just given to it, or under a more or less modified form. It is there- 

 fore mathematical reasoning par excellence, and we must examine it 

 closer. 



The essential characteristic of reasoning by recurrence is that it 

 contains, condensed, so to speak, in a single formula, an infinite num- 

 ber of syllogisms. We shall see this more clearly if we enunci- 

 ate the syllogisms one after another. They follow one another, if one 

 may use the expression, in a cascade. The following are the hypo- 

 thetical syllogisms: The theorem is true of the number 1. Now, if it 

 is true of 1, it is true of 2; therefore it is true of 2. Now, if it is 

 true of 2, it is true of 3; hence it is true of 3, and so on. We see that 

 the conclusion of each syllogism serves as the minor of its successor. 

 Further, the majors of all our syllogisms may be reduced to a single 

 form. If the theorem is true of n 1, it is true of n. 



We see, then, that in reasoning by recurrence we confine ourselves to 

 the enunciation of the minor of the first syllogism, and the general 

 formula which contains as particular cases all the majors. This unend- 

 ing series of syllogisms is thus reduced to a phrase of a few lines. 



It is now easy to understand why every particular consequence of 

 a theorem may, as I have above explained, be verified by purely an- 

 alytical processes. If, instead of proving that our theorem is true for 

 all numbers we only wish to show that it is true for 

 the number 6 for instance, it will be enough to establish the first five 



