"however it is beautiful and surprising that the prime numbers p for which m is a residue are precisely those which belong to certain arithmetic progressions of increment 4m; for the others m is a non-residue"

Fascinating. At first I was confused because I thought he was referring to the law of reciprocity. But it's actually a different law:

  m = 3
  = not a square mod 5. (reciprocal)
  = not a square mod 7. (not reciprocal)
  = 5² mod 11.          (not reciprocal)
  = 4² mod 13.          (reciprocal)

  Add 4*3 = 12:

  = not a square mod 17 (reciprocal)
  = not a square mod 19 (not reciprocal)
  = 7² mod 23.          (not reciprocal)

  Add 4*3 = 12:

  = not a square mod 29 (reciprocal)
  = not a square mod 31 (not reciprocal)