III. On Fermat’s theorem of the polygonal numbers, with supplement

Abstract

This paper (with its Supplement) proposes a proof of the first two theorems of Fermat, relating to the polygonal numbers, viz. that every number is composed of not exceeding three triangular numbers, and not exceeding four square numbers. And this is done by a method entirely new, founded on the properties of the triangular numbers and the square numbers, and the relation they bear to each other, and on the expansion of an algebraical expression of three members into a line , a square , and a cube , so as to obtain every possible value of the whole expression; and throughout the proof every number or term in a series (except in the Table) is expressed by the roots of the squares that compose it, and the roots only are dealt with, and not the numbers or the squares that compose them; a Table is constructed from the triangular numbers, thus (see opposite page). Mode of constructing the Table. The series of triangular numbers is in the centre of the Table. Below that series the adjoining terms are united, and they form the square numbers 1, 4, 9, &c.; the next adjoining terms are united, and they form the next row, and so on.