1866.] 



alluded to by Fermat, 



119 



Having explained the construction of the square and the indices which 

 belong to the series of which it is composed, I propose to point out the 

 properties which discover the roots of the squares into which the odd 

 numbers (which are found in the different parts of the square) may be 

 divided. 



If the first odd number in the square be of the form (4 ^; + 2) . w + 1 (or 

 2 w+1, 6 n + l, 10 n-\-ly 14 n+\y &c.), n being of any value whatever. 



the {n—p)ih. term will be 



these are the roots 



(the number itself would be 2 ?i^ + 2;?^ + 2^ + 1) ; and as the index of the 

 nth. term is n + {n—\)y or 2^—1, the index of the {n—p)i\v will be 

 2 n— (2^+1), and the algebraic sum of the roots may be made equal to 

 the index ; it is therefore a term in a diagonal series [that the j?)th 

 term is 2 w^ + 2^j^ + 2^+ 1, will appear by finding in the usual way the 

 («— ^)th term of a series whose first term is (4 p + 2). w+ 1, and the 

 terms of which increase by 4, 8, 12, IG, &c. . . . Aii] ; but as two 

 of the roots are equal, n, it is the first term of a vertical series descend- 

 ing, thus: P+^,Pi n 



(n-2), w + 2 

 p+\,Py 0^-3), (n + 3). 

 I call this term (A)* and indicate it by that letter, and from this term 

 the roots of many others may be derived (which are indicated by other 

 letters connected with A in an invariable manner), whose sq^uares will 

 compose the number that belongs to that term. For example, counting 

 2^ + 1 squares backwards from A is a term I have distinguished as W ; 

 the roots of the number belonging to it are 



These roots may, of course, be either positive or negative ; arranging 

 them thus, — p + 1, —p^ n, —4^ + 2, we have the algebraic sum of their 

 roots equal to 2 7i— 6^ + 3, which is the index of the square in which W 

 is found going backwards from A, the index of which is 2 n~2p-\-l (as 

 already stated) ; immediately adjoining W are two squares which I have 

 called M and N respectively. 



M is the term next before W, and is composed of the roots 



n-2p + 2, n-2p + 2, 3^ + 2,^+1 



N is the term next after W, and consists of the roots 



n — 2 p, n— 2p, 3^+1,2) 



Each of these will traverse the square diagonally, the algebraic sums of 

 * See Diagrams Nos. 2, 3, and 4. 



