his Geschichte der Mathematik (1796), was not an inspiring teacher. At the age of nineteen Gauss discovered a method of inscribing in a circle a regular polygon of seventeen sides, and this success encouraged him to pursue mathematics. He worked quite independently of his teachers, and while a student at Göttingen made several of his greatest discoveries. Higher arithmetic was his favourite study. Among his small circle of intimate friends was Wolfgang Bolyai. After completing
his course he returned to Brunswick. In 1798 and 1799 he repaired to the university at Helmstädt to consult the library, and there made the acquaintance of Pfaff, a mathematician of much power. In 1807 the Emperor of Russia offered Gauss a chair in the Academy at St. Petersburg, but by the advice of the astronomer Olbers, who desired to secure him as director
of a proposed new observatory at Göttingen, he declined the offer, and accepted the place at Göttingen. Gauss had a marked objection to a mathematical chair, and preferred the post of astronomer, that he might give all his time to science. He spent his life in Göttingen in the midst of continuous work. In 1828 he went to Berlin to attend a meeting of scientists, but after this he never again left Göttingen, except in 1854, when a railroad was opened between Göttingen and Hanover. He had a strong will, and his character showed a curious mixture of self-conscious dignity and child-like simplicity. He was little communicative, and at times morose.
A new epoch in the theory of numbers dates from the publication of his Disquisitiones Arithmeticæ, Leipzig, 1801. The beginning of this work dates back as far as 1795. Some of its results had been previously given by Lagrange and Euler, but
were reached independently by Gauss, who had gone deeply into the subject before he became acquainted with the writings of his great predecessors. The Disquisitiones Arithmeticæ
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 Arithmeticæ, treating of