240 



into 1 2 + l a + ! 2 + 6 2 , and then the equal roots are discovered. It is 

 proved from the known properties of numbers that this property of 

 having 2 roots whose difference will be 0, 1, 2, 3, &c., as far as is 

 possible, belongs to all odd numbers. A new symbol is then sug- 

 gested to represent the division of a number into 4 squares, such 

 that 2 of the roots will have a given difference, and these are made 

 the exterior roots ; the number or figure denoting the difference is 

 placed above on the left hand: thus 4 25 denotes 0, 0, 3, 4 or 

 -2, 1, 4, 2 ; '25 denotes 1, 2, 4, 2. 



The general theorem is this : If any odd number of odd numbers 

 be in arithmetical progression (4 being the common difference), as 

 9, 13, 17, 21, 25, then if the common difference be assumed as the 

 index of the difference of roots to the middle term, and the higher 

 terms in the series have as indices (4+1), (4 + 2), &c. in succession, 

 and the lower terms have as indices (4 1), (4 2), &c., the series 

 with its indices will be 



23456 

 9 13 17 21 25 



and if the terms less than the middle term be divided into 4 squares 

 with exterior roots having the differences indicated by their respect- 

 ive indices thus, 



234 



9 13 17 



0,1,2,2 -1,2,2,2 -2,0,3,2 



then the terms greater than the middle term will have this relation 

 to the terms less than the middle term ; the terms equidistant from 

 the middle term will have their middle roots the same, and the dif- 

 ferences of the exterior roots will increase ; those nearest the middle 

 term will have a difference of 1, the next 2, and so on, thus: 



23456 

 9 13 17 21 25 



0,1,2,2 -1,2,2,2 -2,0,3,2 -2,2,2,3 -2,1,2,4 



An algebraic proof is then given as to a series whose middle 

 term is n and common difference p ; and as n may be odd or even, 

 and ^7 also, and the index of differences may be minus as well as plus, 

 the theorem applies frequently to even numbers, but not universally. 

 The following example is given of the theorem applied to 1 7 terms 

 of a series whose first term is 25, and common difference 1 : 



