III. On some remarkable relations which obtain among the roots of the four squares into which a number may be divided, as compared with the corresponding roots of certain other numbers

Abstract

The property of numbers, which is the subject of this paper, first presented itself to my attention in the case of the odd squares 1, 9, 25, 49, &c. (2 n ∓ 1) 2 ; any two adjoining odd squares may be divided (each of them) into 4 square numbers, whose roots will have this remarkable relation to each other: two of them will be identically the same; and as to the other two, one of them will be 2 less, and the other will be 2 more than the roots of the preceding or subsequent odd square; for example, 25 and 49 may be divided into squares, the roots of which being placed below, will appear thus: — 25 49 so 49 81 -2, 1, 4, 2 -4, 1, 4,4 0, 2, 3, 6 -2, 2, 3, 8 or thus 0, 0, 3, 4 -2, 0, 3, 6. In comparing the roots of the adjoining odd squares, 2 roots (placed in the middle) are the same; of the others, one is 2 more, the other 2 less than the corresponding roots of the other.