242 report — 1859. 



The equation I - 1 I -l=( — 1) may be expressed in words by- 



saying that " if p and q be two primes, the quadratic character of p in regard 

 to q is the same as the quadratic character of q in regard to p ; except both 

 p and q be of the form in + 3, in which case the two characters are opposite 

 instead of identical." 



Gauss, who attributes the first enunciation of this theorem to Legendre, 

 while he justly claims the first demonstration of it for himself*, appears to 

 have considered that Euler was unacquainted with the theorem, at least in 

 its simple form. (See Disq. Arith. Art. 151.) Nevertheless, we find in the 

 ' Opnscula Analytica' of Euler, vol. i. p. 64>, a memoirf the concluding para- 

 graph of which contains a general and very elegant theorem, from which the 

 Law of Reciprocity is immediately deducible, and which is, vice versa, 

 deducible from that law. But Euler (loc. cit.} expressly observes that the 

 theorem is undemonstrated ; and this would seem to be the only place in which 

 he mentions it in connexion with the theory of the Residues of Powers ; 

 though in other researches he has frequently developed results which are 

 consequences of the theorem, and which relate to the linear forms of the 

 divisors of quadratic formulae. But here also his conclusions repose on 

 induction only; though in one memoir he seems to have imagined (for his 

 language is not very precise) that he had obtained a satisfactory demonstra- 

 tion. The theorem, in a form precisely equivalent to that in which we have 

 cited it, was first given by Legendre, in a Memoir contained in the ' Histoire 

 de l'Academie des Sciences ' for 1785. (See pp. 516, 517.) But the demon- 

 stration with which he has accompanied it is invalid for several reasons. (See 

 Gauss, Disq. Arith. Art. 151, 296, 297, and the Additamenta.) 



17. Jacobi's extension of Legendre s Symbol. — The symbol ( - j, the intro- 

 duction of which has greatly contributed to simplify the theories of the higher 

 arithmetic, does not appear in the Memoir just referred to. It first occurs 

 in the ' Essai sur la Theorie des Nombres;' the first edition of which ap- 

 peared at Paris in 1798, and the second in 1808. 



Jacobi, in a note communicated to the Academy of Berlin in 1837 %> has 

 extended the notation of Legendre. If P z= p i p 2 P3 • • • • where p l p 2 p 3 denote 



(equal or unequal) uneven prime numbers, Jacobi defines the symbol I- \ 

 by the equation 



GHsNSNs)- 



and observes that we then have the equations 



*(P-1)(Q-1). 



/p\ i(J. J -i;i<4-'i/r)\ 



* " Pro primo lmjus elegantissimi Theorematis inventore ill. Legendre absque dubio 

 habendus est, postquarn longe antea surami geometrse Euler et Lagrange plures ejus casus 

 speciales jam per inductionem detexerant. . j . . In ipsum theorema proprio marte incideram 

 anno 1795, dum omnium, quae in arithmetica sublimiori jam elaborata fuerant, penitus 

 ignarus, et a subsidiis literariis omnino prseclusus essem. Sed per integrum annum me tor- 

 sit, operamque enixissimam effugit," etc. — Comm. Soc. Gbtt. vol. xvi. p. 69. 



f Observationes circa divisionem quadratorum per nnmeros primos (Comment. Arith. 

 vol. i.p. 477). 



t Ueber die Kreistheilung und ihre Anwendung auf die Zahlentheorie. See the Monats- 

 Bericht of the Berlin Academy, vol. ii. p. 127 (Oct. 16, 1857), or Crelle's Journal, vol. xxx. 

 p. 166. 



