244 



in which last series the differences are reversed or added " inverse 

 ordine." The appropriate roots of these numbers are 



35 7 9 11 13 15 17 



5 37 65 89 109 125 137 145 



-1,0,0,2 -1,2,4,4 -3,2,6,4 -2,0,6,7 -3,0,6,8 -6,2,6,7 -6,2,4,9 -8,0,0,9 



+ 1,0,0,6 -2,0,6,5 0,2,2,9 -1,2,2,10 -5,0,6,8 -4,0,0,11 



0,0,4,7 -3,0,4,10 



+ 1,0,0,8 -2,0,0,11 



which may be immediately obtained from the former series, the 

 middle roots being the same ; and the exterior roots, one of them 

 one less, the other one more. In this way any consecutive 8 terms, 

 with the differences reversed, may be each divided into 4 squares 

 throughout the whole series. 



And the same is true of 4 terms, 5 terms, or any number of terms. 

 If 3 terms have the differences reversed, the numbers added are 



040 



If 4 terms 8 8 



If 5 terms 12 16 12 



If 6 terms 16 24 24 16 



If 7 terms 20 32 36 32 20 



&c. &c. &c. 



The law under which these numbers are formed is obvious enough. 

 The same numbers exactly are to be added to the other series (1,3, 



9, 19, &c.) to produce the same result. 



012345 6 7 



If the 2 series be blended together, thus 1 1 3 5 9 13 19 25, 

 &c., the differences will be 2, 2, 4, 4, 6, 6, 8, 8, &c. ; and if an odd 

 number of terms be taken (so as to begin and end with a number 

 from the same series), and the differences be added inverse ordine, a 

 similar result occurs. Take 1 1 terms. 



1 3 5 9 13 19 25 33 41 51 61, and add the differences " in- 

 verso ordine" the series becomes, with its indices and roots, 



12345 



I 11 21 29 37 



0,0,0,1 +1,0,1,3 -1,0,4,2 0,2,3,4 +1,0,0,6 6 



-1,2,4,4 43 (middle term.) 



II 10 9 8 7 -3,3,4,3 

 61 59 57 53 49 



-5,0,0,6 -3,0,1,7 -4,0,4,5 -2,2,3,6 0,0,0,7 



-2,2,4,5 



