On certain properties of square numbers and other quadratic forms, with a table by which all the old numbers up to 9211 may be resolved into not exceeding four square numbers

Abstract

In examining the properties of the triangular numbers 0, 1, 3, 6, 10, &c., the author observed that every triangular number was com­posed of four triangular numbers, viz. three times a triangular num­ber plus the one above it or below it; and he found that all the natural numbers in the interval between any two consecutive triangular numbers might be composed of four triangular numbers having the sum of their roots, or rather of the indices of their distances from the first term of the series constant, viz. the sum of the indices of the four triangular numbers which compose the first triangular number of the two .