I. Continuation of the subject of a paper read Dec. 22, 1853, the supplement to which was read Jan. 12, 1854, by Sir Frederick Pollock, &c.; with a proof of Fermat's first and second theorems of the polygonal numbers, viz. that every odd number is composed of four square numbers or less, and of three triangular numbers or less

The object of this paper is in the first instance to prove the truth of a theorem stated in the supplement to a former paper, viz. “that every odd number can be divided into four squares (zero being considered an even square) the algebraic sum of whose roots (in some form or other) will equal 1, 3, 5, 7, &c. up to the greatest possible sum of the roots.” The paper also contains a proof, that if every odd number 2n + 1 can be divided into four square numbers, the algebraic sum of whose roots is equal to 1, then any numbern is composed of not exceeding three triangular numbers.