ESSAY-REVIEWS 



HIGHWAYS AND BYWAYS IN THE THEORY OP NUMBERS, 



by L. J. MoRDELL (Manchester College of Technology) : on The History 

 of the Theory of Numbers, vol. ii, Diophantine Analysis, by Leonard 

 Eugene Dickson, Professor of Mathematics in the University of 

 Chicago. [Pp. XXV + 803.] (Washington: Carnegie Institution of 

 Washington, 1920.) 



All mathematicians interested in the Theory of Numbers, and this means 

 sooner or later most pure mathematicians, "K-ill welcome vol. ii of Prof. 

 Dickson's " Chronological Histo^\^" It notes practically everything written 

 on the subject, sums up the results of a paper in a few hnes, and might serve 

 as a model of orderly arrangement. 



This history adds considerably to the increasing debt of mathematicians 

 to America, and is a real necessity in their hbraries. Writers ■will hereafter 

 frequently refer to it rather than give a large list of references — a tendency 

 already noticed with respect to vol. i. He vdU be a rash investigator, indeed, 

 who does not consult it. The energy and untiring patience devoted to his 

 task by Prof. Dickson \vi\l save his readers endless trouble. He has placed 

 them under the deepest obhgations, and indeed has set an example that may 

 well be followed in other branches of mathematics. The results in the Theory 

 of Numbers are now being made so accessible that the next thirty years 

 should see even more progress than the preceding period, during which 

 numerous theorems of outstanding importance have been proved. 



Vol. ii, dealing with such a large number of famous questions, in many 

 of which mathematicians of the present day are displacing considerable 

 interest, and some of which are kno-wTi even to the la^Tnan, will soon convince 

 the reader that the Theory- of Nmnbers is still of supreme importance. It 

 is concerned -w-ith the solution of diophantine equations — that is, to find the 

 rational values oi x, y, 2 . . . satisfying the equation 



f {x, y, 2 . . .) = m (i) 



where / denotes a pohmomial in ;r, >', ^ . . . and in is given. The term 

 "diophantine equation" (so called after Diophantus, who flourished about 

 A.D. 250) has been frequently used when integer values only are required 

 for the unknowns, the term indeterminate, when rational solutions are 

 desired ; but writers have not always adhered to this distinction. It is 

 unimportant in discussing the general equation (i), but, for a non-homogeneous 

 equation, the distinction between rational and integer solutions usually is of 

 considerable importance. 



We shall now indicate the relation betn-een the study of the integer 

 solutions of equation (i) and some branches of the Theory of Numbers, as 

 given in this book. When / is a binary quadratic, the solution of 



ax^ -f 2bxy 4- cy^ = m 



contains practically the elements of the Theory of Numbers as developed by 

 Gauss. When / is a quadratic form in n variables, it gives rise to the arith- 

 metical theory of the general quadratic form as developed by Gauss, Eisenstein, 



647 



