ON THE THEORY OF NUMBERS. 245 



19. Gauss's Second, Third, and Fifth Demonstrations. — The second de- 

 monstration (Disq. Arith. 262) depends on the theory of quadratic forms, 

 and will be referred to in its proper place in this Report. 



The third and fifth (which are in principle very similar to one another) 

 depend on much simpler considerations. 



A half -system of Residues for a prime modulus p is a system of \ (p — ] ) 

 numbers r, r 2 ... r,j(p_i), such that the/j — 1 numbers+r,, +r„.... +rj( p _i) 

 constitute a system of residues prime to p. We might take for the num- 

 bers r x r 2 &c, the even numbers less than p (as Eisenstein has done : see 

 Crelle's Journal, vol. xxviii. p. 246), but Gauss has preferred to take the 

 numbers 1, 2, 3 ...^ (p— 1). 



Let q be any number prime to p, and let k be the number of the numbers, 

 qr v qr v qr 3 .. . qr$( p -\), which are congruous, not to numbers in the series 

 r rj . .. r^( P _i), but to numbers in the series — r v — r 2 ,.. — ?}(?— 1> It 

 may be shown (by a method similar to that employed in Dirichlet's proof 



of Fermat's Theorem) th&tqi(P-V = (— 1)*, mod/?; so that/?) = (—1)*. 



Hence if q be a prime as well as p, and h! denote the number which replaces 

 k, when p and q are interchanged in the preceding considerations, we find 



It has, therefore, to be shown that k-\-k' = ± (p—1) (q—1), mod 2. The 

 way in which this is done is different in each of the two demonstrations, and 

 is a little complicated in each of them ; but by the aid of a diagram the con- 

 gruence may be demonstrated intuitively (compare Eisenstein : Crelle, xxviii. 

 p. 2t6). With a pair of axes Ox and Oy construct a system of unit-points 

 in a plane: only let no such points be constructed on the axes themselves. 

 If S be any geometrical figure, let (S) stand for the number of unit-points 

 contained inside it or on its contour, On Ox and Oy respectively take 

 OA=i<7, OB = ^9. Complete the parallelogram OACB, and draw its dia- 

 gonals, OQC, AQB. It is then easily seen that 



£=(QCA) - (QBO) 



A'=(QBC) - (QOA) 



A + A'=(ABC) - (AOB) 



= (OABC)-2(AOB) 



=(OABC), mod 2. 



But (OABC) = i (p-1) (q-l); therefore, finally, k + k! = ± (p-1) 

 (</— 1), mod 2. 



These demonstrations (the 1st, 3rd and 5th) introduce no heterogeneous 

 elements into the inquiry (the geometrical method of the preceding article 

 is to be regarded only as an abbreviation of an equivalent and purely arith- 

 metical process) ; they are based on the principles of the two theories with 

 which the Law of Reciprocity is most intimately connected, — those of the 

 residues of powers, and of quadratic congruences. The third, in particular, 

 appears to have commended itself above the rest to Gauss's judgment*. 



* " Sed omnes hie demonstrationes," (he is speaking, apparently, of the 1st, 2nd, 4th, and 

 6th,) "etiamsi respectu rigoris nihil desiderandum relinquere videantur, e principiis nimis 

 heterogeneis derivatas sunt ; prima forsan excepta, quae tamen per ratiocinia magis laboriosa 

 procedit, operationibusque prolixioribus premitnr. Demonstrationem itaque genuinam 

 hactenus baud aflfuisse non dubito pronunciare ; esto jam penes peritos judicium, an ea, 

 quam nuper detegere successit," (the 3rd,) " hoc nomine decorari mereatur."— Comm. Soc. 

 Gott. vol. xvi. p. 70. 



