was already in print when Legendre's Théorie des Nombres appeared. The great law of quadratic reciprocity, given in the fourth section of Gauss' work, a law which involves the whole theory of quadratic residues, was discovered by him by induction before he was eighteen, and was proved by him one year later. Afterwards he learned that Euler had imperfectly enunciated that theorem, and that Legendre had attempted to prove it, but met with apparently insuperable difficulties. In the fifth section Gauss gave a second proof of this "gem" of higher arithmetic. In 1808 followed a third and fourth demonstration; in 1817, a fifth and sixth. No wonder that he felt a personal attachment to this theorem. Proofs were given also by Jacobi, Eisenstein, Liouville, Lebesgue, A. Genocchi, Kummer, M. A. Stern, Chr. Zeller, Kronecker, Bouniakowsky, E. Schering, J. Petersen, Voigt, E. Busche, and Th. Pepin.[48] The solution of the problem of the representation of numbers by binary quadratic forms is one of the great achievements of Gauss. He created a new algorithm by introducing the theory of congruences. The fourth section of the Disquisitiones Arithmeticœ, treating of congruences of the second degree, and the fifth section, treating of quadratic forms, were, until the time of Jacobi, passed over with universal neglect, but they have since been the starting-point of a long series of important researches. The seventh or last section, developing the theory of the division of the circle, was received from the start with deserved enthusiasm, and has since been repeatedly elaborated for students. A standard work on Kreistheilung was published in 1872 by Paul Bachmann, then of Breslau. Gauss had planned an eighth section, which was omitted to lessen the expense of publication. His papers on the theory of numbers were not all included in his great treatise. Some of them were published for the first time after his death in his collected works (1863–1871). He wrote two memoirs on
