ON THE THEORY OF NUMBERS. 229 



imperfectly represented in those celebrated treatises. The arithmetical 

 memoirs of Gauss himself, subsequent to the publication of the ' Disquisi- 

 tiones Arithmeticae ;' those of Cauchy, Jacobi, Lejeune Dirichlet, Eisen- 

 stein, Poinsot, and, among still living mathematicians, of MM. Kummer, 

 Kronecker, and Hermite, have served to simplify as well as to extend the 

 science. From the labours of these and other eminent writers, the Theory 

 of Numbers has acquired a great and increasing claim to the attention of 

 mathematicians. It is equally remarkable for the number and importance 

 of its results, for the precision and rigorousness of its demonstrations, for 

 the variety of its methods, for the intimate relations between truths 

 apparently isolated which it sometimes discloses, and for the numerous 

 applications of which it is susceptible in other parts of analysis. " The 

 higher arithmetic," observes Gauss*, confessedly the great master of the 

 science, " presents us with an inexhaustible store of interesting truths, — of 

 truths, too, which are not isolated, but stand in a close internal connexion, and 

 between which, as our knowledge increases, we are continually discovering 

 new and sometimes wholly unexpected ties. A great part of its theories 

 derives an additional charm from the peculiarity that important propositions, 

 with the impress of simplicity upon them, are often easily discoverable by 

 induction, and yet are of so profound a character that we cannot find their 

 demonstration till after many vain attempts; and even then, when we do 

 succeed, it is often by some tedious and artificial process, while the simpler 

 methods may long remain concealed." 



2. It is the object of the present report to exhibit an outline of the 

 results of these later investigations, and to trace (so far as is possible) 

 their connexion with one another and with earlier researches. An attempt 

 will also occasionally be made to point out the lacunce which still exist 

 in the arithmetical theories that come before us ; and to indicate those 

 regions of inquiry in which there seems most hope of accessions to our 

 present knowledge. In order, however, to render this report intelligible 

 to persons who have not occupied themselves specially with the Theory of 

 Numbers, it will be occasionally necessary to introduce a brief and summary 

 indication of principles and results which are to be found in the works of 

 Gauss and Legendre. It is hardly necessary to add that we must confine 

 ourselves to what we may term the great "highways of the science; and that 

 we must wholly pass by many outlying researches of great interest and im- 

 portance, as we propose rather to exhibit in a clear light the most funda- 

 mental and indispensable theories, than to embarrass the treatment of a 

 subject, already sufficiently complex, with a multitude of details, which, 

 however important in themselves, are not essential to the comprehension of 

 the whole. 



3. There, are two principal branches of the higher arithmetic : — the Theory 

 of Congruences, and the Theory of Homogeneous Forms. The first of 

 these theories relates to the solution of indeterminate equations, of the form 



a n x n + a n _ l x n ~ 1 + +a l x+a =Py, 



in which a n a n _ x . . . o^ o and P are given integral numbers, and x and y 

 are numbers which it is required to determine. The second relates to the 

 solution of indeterminate equations of the form 



r \x x x 2 . . . x m ) = j.\l, 



in which M denotes a given integral number, and F a homogeneous function 



* Preface to Eisenstein's ' Mathematische Abhandlungen,' Berlin, 1847. 



