NUMBER AND MAGNITUDE 631 



2+2=4 can be verified because it is particular. Each individual enun- 

 ciation in mathematics may be always verified in the same way. But 

 if mathematics could be reduced to a series of such verifications it 

 would not be a science. A chess-player, for instance, does not create 

 a science by winning a piece. There is no science but the science of the 

 general. It may even be said that the object of the exact sciences is to 

 dispense with these direct verifications. 



Let us now see the geometer at work, and try to surprise some of 

 his methods. The task is not without difficulty; it is not enough to 

 open a book at random and to analyze any proof we may come across. 

 First of all, geometry must be excluded, or the question becomes 

 complicated by difficult problems relating to the role of the postulates, 

 the nature and the origin of the idea of space. For analogous rea- 

 sons we cannot avail ourselves of the infinitesimal calculus. We must 

 seek mathematical thought where it has remained pure i.e., in Arith- 

 metic. But we still have to choose ; in the higher parts of the theory 

 of numbers the primitive mathematical ideas have already undergone 

 so profound an elaboration that it becomes difficult to analyze them. 



It is therefore at the beginning of Arithmetic that we must expect 

 to find the explanation we seek; but it happens that it is precisely in 

 the proofs of the most elementary theorems that the authors of classic 

 treatises have displayed the least precision and rigor. We may not 

 impute this to them as a crime; they have obeyed a necessity. Begin- 

 ners are not prepared for real mathematical rigor; they would see in 

 it nothing but empty, tedious subtleties. It would be waste of time to 

 try to make them more exacting; they have to pass rapidly and without 

 stopping over the road which was trodden slowly by the founders of 

 the science. 



Why is so long a preparation necessary to habituate oneself to this 

 perfect rigor, which it would seem should naturally be imposed on 

 all minds ? This is a logical and psychological problem which is well 

 worthy of study. But we shall not dwell on it; it is foreign to our 

 subject. All I wish to insist on is, that we shall fail in our purpose 

 unless we reconstruct the proofs of the elementary theorems, and give 

 them, not the rough form in which they are left so as not to weary 

 the beginner, but the form which will satisfy the skilled geometer. 



Definition of Addition 



I assume that the operation x-\-l has been defined; it consists in 

 adding the number 1 to a given number x. Whatever may be said of 

 this definition, it does not enter into the subsequent reasoning. 



We now have to define the operation oH-o, which consists in adding 

 the number a to any given number x. Suppose that we have defined 

 the operation x + (a 1) ; the operation x + a will be defined by 



