1866.] 



alluded to by Fermat, 



117 



If therefore the roots of the square, into which any odd numher may be 

 divided, bej?, q,n, n-{-\, and the number be increased by 4, 8, 12, &c. 

 . . . {An), the mih term in the series will be composed of squares whose 

 roots will be^, qyU — (m — 1), + ; two of the roots will be constant, the 

 others will vary, and their differences in the successive terms will be 1, 3, 

 5, 7, &c. (2^—1); and if you discover the roots of any term in the series, 

 you can find the roots of all the terms. 



The second series resembles the first in having two roots constant, and 

 two variable ; the differences between the variable roots are, in the first 

 term 0, in the second term 2, in the third term 4, &c., and the indices of 

 the terms are therefore 0, 2, 4, 6, 8, 10, &c., which are placed vertically 

 by the side of the square. For if two numbers are equal, as ?^, and one 

 of them be increased by 1, and the other diminished by 1, and the process 

 be continued, the result will be 



'2^1' 



n n ^ 



n—ljU+l I If these be treated as 

 n — 2,n-^2 y roots, the sums of<J 2n^ + S 

 n—3,n-^3 \ their squares will be I 2n^+l8 

 &c. &c. J t,&c. &c. 



The sums of the squares increase by 2, 6, 10, 14 ... . (4n—2), and the suc- 

 cessive differences of the variable roots are 0, 2, 4, 6, &c. ; and if the roots 

 of the squares into which any odd number may be divided be p, n, n, 

 and the number be increased by 2, 6, 10, &c., the roots of the mth term 

 will be p, q, n—(m — \), n-\-{m — 1), and if the roots of any one term be 

 known, the roots of all the others may be found. 



The small figures in the upper right-hand corner of each division or 

 small square are the indices of the third set of series. In this set all 

 the roots are variable. The character of the first set of series is, that 

 two roots in every term differ by an odd number; the character of 

 the second set of series is, that two roots in every term differ by an 

 even number ; but in the third set of series, the algebraic sum of all the 

 roots of the squares into which the successive terms may be divided is 

 successively 1, 3, 5, 7, 9, &c. (an odd number) : the sum of the roots of 

 the squares into which an odd number can be divided cannot be an even 

 number. 



The following Table will explain in what manner the series is formed 

 from the roots of the squares into which any odd number may be divided, 

 so as to make the algebraic sum equal to 1. I have preferred to use 

 figures instead of algebraic symbols, as being more readily understood and 

 more easily dealt with ; but the result is the same whatever figures or 

 symbols may be used. The series begins from the centre. 



Let — 7, —3, 2, 9, which are the roots of the squares into which the 

 odd number 143 may be divided, be placed in the centre, and let the posi- 

 tive roots be increased downwards and decreased upwards, and the nega- 



M 2 



