326 BEPOET — 1875. 



being 454 ; and hence that every number greater than 454 can be expressed 

 as a sum of at most 7 cubes ; and further, that every number greater than 

 2183 can be expressed as a sum of at most 6 cubes. A small subsidiary 

 table (p. 276) shows that the number of numbers requiring 6 cubes gradually 

 diminishes — ?. g. between 12'' and 13* there are seventy-five such numbers, 

 but between 13^ and 14* only sixty-four such numbers ; and the author 

 conjectui-es " that for numbers beyond a certain limit every number can be 

 expressed as a sum of at most 5 cubes." 



42. For the decomposition of a number into biquadrates we have 

 Bretschneider, " Tafeln fiir die Zerlegung der Zahlen bis 4100 in Bi- 

 quadrate." CreUe, t. xlvi. (1853), pp. 3-23. 

 Table I. gives the decompositions, thus : — 



viz. 696=6. l'-fl.2*-l-2.3*+2.4S&c. 



And Table II. enumerates the numbers which are sums of at least 2, 3, 4 

 .... 19 biquadrates ; and there is at the end a summary showing for 

 the first 4100 numbers how many numbers there are of these several 

 forms respectively : 28 numbei-s are each of them a sum of 2 biquadrates, 

 75 a sum of 3, .... 7 a sum of 19 biquadrates. The seven numbers, eaoh of 

 them a sum of 19 biquadrates, are 79, 159, 239, 319, 399, 479, 559. 



Art. 6. [F. 16. Binary, Ternary, ^-c. quadratic and higher fonns.'] 



43. Euler worked with the quadratic forms rr.r'+c?/' (p and q integers), 

 particularly in regard to the forms of the divisors of such numbers. It will 

 be suificient to refer to his memoir : — 



Euler, " Theoremata cii'ca divisores numerorum in hac forma pa^ + gj- con- 

 tentorum" (Op. Arith. Coll. pp. ,35-61, 1744), containing fifty-nine theorems, 

 exhibiting in a quasi-tabular form the linear forms of the divisors of such 

 numbers. As a specimen : — 



"Theorema 13. Xumerorum in hac forma ff--f-76 F contentorum divisores 

 primi omnes sunt vel 2, vel 7, vel in una sex formularum 



28m + l, 28m-|-ll, 



28m -I- 9, 28rr + 15, 



28m + 25, 28m -,-23, 



seu in una harum trium 14m -f 1, 



14m-f9, 



14m -f 11 



sunt contenti;" viz. the forms are the three litn + l, 14m4-9, 14wi-f 11. 

 But Euler did not consider, or if at all very slightly, the trinomial forms 



ax'^+hxy + cy^ nor attempt the theory of the reduction of such forms. This 



was first done by Lagrange in the memoir 



Lagrange, ' Me'm. de BerUn,' 1773. And the theory is reproduced 

 Legendre, ' Theorie des Nombres.' Paris, 1st edit. 1798 ; 2nd edit. 1808, 



§ 8, " Reduction de la formule Ly' + Myz+'Mz- a I'expression la plus simple" 



(2nd ed. pp. 61-67). 



44. But the classification of quadratic forms, as established by Legendre, is 

 defective as not taking account of the distinction between proper and im- 

 proper equivalence ; and the ulterior theory as to orders and genera, and the 



