SQTJAEES INTO WHICH ODD NUMBERS MAY BE DIVIDED. 
59 
There are certain numbers, which, added to the terms of the series in its ordinary 
state, convert it into a series whose diiferences have been added “ imerso ordineT 
If the number of terms be 3, add 
If 4, add 
If 5, add 
If 6, add 
If 7, add 
If 8, add 
0 4 0 
0 8 8 0 
0 12 16 12 0 
0 16 24 24 16 0 
0 20 32 36 32 20 0 
24 40 48 48 40 24 0 
&c. «&c. 
The law of the formation of these numbers is obvious. 
These numbers apply to both series, and to any consecutive terms in either ; that is, 
[e. f ] 0, 12, 16, 12, 0, added to any 5 consecutive terms of either series, converts them 
into 5 terms whose differences have been added ^'inverso ordine;’' and what is still more 
remarkable, the middle roots of the first eight (or indeed n) terms having their differ- 
ences added “ inverso ordine" are the middle roots which answer for any eight [or n\ 
consecutive terms, whose differences have been added “ inverso ordine ” through the 
unlimited extent of the whole series. Thus if the first 8 terms be formed with the 
differences added “ 
inverso ordine,'" 
1 
5 
13 25 
41 61 
85 
113 
add 
0 
24 
40 48 
48 40 
24 
0 
^29 
^53 
’73 
^89 
^^101 
^ n 09 
0,0, 0,1 
0,2, 4, 3 
-2,2, 6, 3 
-1,0, 6, 6 
-2, 0,6, 7 - 
-5, 2, 6, 6 
—5, 2, 4, 8 — 
7,0, 0,8 
+2, 0, 0,5 
-1, 0,6,4 
+ 1, 2,2,8 
0,2,2,9 - 
-4, 0,6, 7 
-3, 0,0,10 
+ 1,0, 4, 6 
-2, 0,4, 9 
+2,0,0,7 -1,0,0,10 
the indices and the roots of the squares will be as above. 
Now take the 8 consecutive terms, beginning with 181 : — 
181 221 265 313 365 421 481 545 
add 0 24 40 48 48 40 24 0 
19181 ^^245 ^^305 ^^361 ^’413 ^®461 ^^505 ^«545 
-9,0,0,10 -9,2,4,12 -11,2,6,12 -10,0,6,15 -11,0,6,16 -14,2,6,15 -14,2,4,17 -16,0,0,1 
-7,0,0,14 -10,0,6,13 - 8,2,2,17 - 9,2,2,18 -13,0,6,16 -12,0,0,19 
- 8,0,4,15 -11,0,4,18 
- 7,0,0,16 -10,0,0,19 
the middle roots are the same for both. 
These last matters add weight to the suggestion already made, that the properties of 
numbers referred to are connected with the “ mysterious and abstruse ” properties 
alluded to by Feemat, as enabling him to prove the theorem he announced of the 
polygonal numbers. 
