to Mr. Holdred. 



37 



left by a, and adding the product to tliemimbrr above that which 

 is to be found ; and the. (juantities below the horizontal line, are 

 the sum of those above it : the same is to be understood in the 

 following modes :— 



No. 2. 



m 



P 



N 



9 

 r 



N + (j+r 



n 



Na 



= Q 

 = R 



:Q + R 



Na+ qa = S 



Na+ qa-^ra = T 



No. 3. 



Q 



or thus, according 

 to my notation. 



Q 



Q= Q + jPa 



„Q = jQ+2Pa 



Na+ n =iQ 



2Na+ qa + n =,fi 



■3Na.-^2qa + ra + 7i = sQ *P3Q=2Q + 3Pa 



It is evident that 3Q will be found the same, whichever of these 



methods of adding the numbers together is adopted ; or since the 



producing numbers in the right-hand column are Q, R, S, T, 



all the methods of addition will be shortly represented as follows : 



First method. 



Q 

 R 



S 

 T 



gQ^Q+R + S + T, 



Second method. 



,Q = Q+R 

 2Q = Q + R + S 

 3Q=0 + R + S-f-T, 



Third method. 



Q = Q 

 R 



,Q = Q + R 



oQ: 



:Q+R + S 



T 



3Q=Q + R + S+T 



•The first method is the form of addition ado})ted in my Essay 

 on Involution and Evolution ; the second, is that adopted in the 

 Postscript ; and the third is that adopted by Mr. Holdred, which 

 he calls an invention ; but since he ascribes to himself the merit 

 of having employed this peculiar mode of addition, it is but justice 

 to myself tt» apprize the reader that he has taken the hint from 

 what is everywhere done in my book on Involution and Evolution, 

 as may be seen in the first two lines of every operation from page 

 42 to page -IS inclusive. The method of proof which he pretends 

 to liHve discovered (page 47 of his Supplement) is also the very 

 same in principle as is found in that work of mine in the two lines 

 referred to. 1 should not have noticed these things, which to 



some 



