original Proposition in the Theory of Numbers. 457 



numbers." He assumes it as the basis upon which his own 

 theorems respecting prime numbers are founded, and to which, 

 of consequence, the disquisitions of Gauss are equally to be 

 referred. 



Euler's demonstration, which has been generally adopted 

 by other writers, is exceedingly simple and convincing, but 

 does not appear capable of being extended to meet the case 

 of composite divisors, without a very complicated and operose 

 process. Such a process is sketched in the concluding para- 

 graph of Part IV. chap. ii. of Legendre's Theorie ; but the 

 writer does not appear to have actually gone through it, nor 

 am I aware that any one else has. 



A different demonstration, not less simple, was given by 

 Mr. Ivory in Leybourn's Repository; and being engaged at 

 the time in a research, which caused the want of a more en- 

 larged theorem to be urgently felt, I was glad to perceive thai 

 the new demonstration was more tractable than Euler's, and 

 might easily be made to include the general case. The result 

 of my attempt is given in the Annals of Philosophy for Fe- 

 bruary 1826. 



The mode of demonstration, however, is little less artificial 

 than that of Euler, and did not supply a ready transit to the 

 distinct cases which occur in its practical application. But by 

 closely observing the indications which arise in actual prac- 

 tice, I soon afterwards found myself in the track of a demon- 

 stration of the most satisfactory description ; simple, convin- 

 cing, and naturally adapting itself to all the requisite practical 

 uses. It is this which I now wish to submit to mathemati- 

 cians ; and I trust that the Theorem itself, which is here 

 proved, will not be regarded as a mere extension of Fermat's, 

 but recognised as the primordial idea from which his disco- 

 very was a fragment fortuitously detached. 



Theorem. — If N, P, are prime to each other and c in- 

 dicates the number of integers less than P and prime to it, 



N c — 1 • 



* i is an integer. 



N tt -R 

 Dem. 1. — Let us consider the general formula 



P 



an integer; where R a is the remainder left after dividing 



N M by P: — R is prime to P ; for had they a common factor, 

 it would divide IP+R a , that is N"; and so P and N" would 

 have a common factor ; which is against the hypothesis 

 (A). Hence R M is an integer ^P, and prime to it. 



Third Series. Vol. 1 1. No. 69. Nov. 1837. 3 N 



