96 HISTORY OF THE THEORY OF NUMBERS. [CHAP, n 



mations to a fraction by means of a series of fractions m/n, the error being 

 < h/ri 2 . He gave a slight modification of Dirichlet's 257 method. 



H. Minkowski 268 proved that if = ax + @y and 77 = yx + fy have 

 any real coefficients of determinant a d fiy = 1 and if , "no are any given 

 real numbers, there exist integers x, y for which ( )(i7 170) = i- 

 In particular, if a is irrational and b is not an integer, there are integers x, 

 y for which (y ax b}(x c) < f ; the case c = gives a better 

 approximation than Hermite's, 261 since 1/4 < V2/27. 



E. Cahen 269 discussed the approximate solution in integers of a system 

 of linear equations. 



E. Borel 270 proved that if a, b, M are any given real numbers, integers 

 x, y, z, numerically < M , can be assigned such that 



ax + by + z < -- Va 2 + 6 2 + 1, 



6 being independent of a, b, M (but not found). Again, intervals (A n , B n ) 

 can be found such that A n and B n increase indefinitely with n and such 

 that, if a is any irrational number between and 1, integers p n , q n exist 

 for which 



Pn 



q n 



a 



A n < q n < n n . 



At least one of three successive convergents to a satisfies the first inequality 

 [cf. Hurwitz 263 ]. 



Minkowski 271 proved that if a is real we can choose integers x, y such 

 that x/y a \ < Ify 2 and deduced the existence of solutions of sx ry = 1 

 if s, r are relatively prime integers. He gave a new proof, suggested by 

 D. Hilbert, 2710 of his 264 theorem on n real linear forms. He discussed (pp. 

 47-58) the maximum value of the minimum of p -f v p , where and 

 77 are real linear forms. He treated (pp. 68-82) the equivalence and mini- 

 mum of three linear forms , 77, f, and gave theorems on the values of their 

 sum or product. 



B. Levi 272 proved Minkowski's 264 thoerem, and for the limit case in 

 which no integers, not all zero, make each |/ t - < 1, proved his result 

 that then at least one of the /; has integral coefficients. 



288 Math. Annalen, 54, 1901, 91-124, see pp. 108, 116 (Ges. Abhandl., I, 320); French transl., 

 Ann. de l'6cole normale sup., (3), 13, 1896, 45. For an account of Minkowski's investi- 

 gations, see Verhandl. des dritten intern. Math. Congresses Heidelberg, 1905, 164. 

 Proof by J. Uspenskij, Applications of continuous parameters in the theory of numbers, 

 St. Petersburg, 1910; cf. Jahrb. Fortschritte Math., 1910, 252. 



269 Bull. Soc. Math. France, 30, 1902, 234-242. He also made additions to the subject in 



his article in the Encyclop6die des Sc. Math., 1906, tome I, vol. Ill, 89-97. 



270 Jour, de Math., (5), 9, 1903, 329-375; Comptes Rendus Paris, 163, 1916, 596-8. Lecons 



sur la thdorie de la croissancc, 1910, 143-154. Cf. A. Denjoy, Bull. Soc. Math, de 

 France, 39, 1911, 175-222. 



271 Diophantische Approximationen, Leipzig, 1907, 1-19, 28. 



2710 Cf. J. Sommer, Vorlesungen iiber Zahlentheorie, 1907, 65-72; French transl. by A. Levy, 

 1911. 



272 Rendiconti Circolo Mat. Palermo, 31, 1911, 318-340. 



