122 4 /?’. Saint's Rejoinder to Mr, Taylor, [Sept, 1, 
respectively ciestroyeH by tho^e in tlie 
Inwer^ there can he no resulting term or 
letter. 
Y 'll continue your reply by accusing 
me of a “ still more unpardonable omis¬ 
sion" than that of this splendid and fa¬ 
mous eighth proposition, for you say that 
“ Having gran'^ed rb.at the number of 
terms in an infinite series cannot be 
greater than and also that my me¬ 
thod in proposition 3 of obtaining the last 
term of an infinite series is just, you have 
wholly neglected to notice the necessary 
consequence of this concession, \^’hich is 
the complete subversion of the leading 
urnposinons of Dr Vvailis’s Arithmetic 
of Infinites, as I have abundantly shewn 
in the treatise under discussion. Thus 
in the infinite series &c. 
the last orgreatest term is O-pl-pl-pl-f-l, 
&:c. and the number of terms is 
&c." Now all tiiis, Mr. 
Tavlor, I readily grant, but I nevertbele‘s 
deny your conclusion, namely, that ‘'The 
last term multiplied by the number of 
terms produces the sum of the series." — 
Nay I assert that the conclusion drawn 
from your own principles is precisely that 
of Dr. Wallis, which i'?, that “ In ihe 
aritfimetical series 0-pl-j-2-f-3-}-4, &c. 
if the last term be multiplied into the 
numl'er of terms, the product will be 
double the sum of all tiie series.” In 
proof of this assertion I must beg of you 
to attend to the following multiplication. 
&c. 
Nc. 
0-i-i+i-fi-H 
O-j-1 -i~ 1 — 5“ 1 “I" 1 
04-14-1+1-fl 
_ Q+l-f 1+1 + 1 
0+1+2+3+4+4+3+2+1 
You must wilfully blind your eyes, Mr. 
Taylor, not to see in this product the 
truth of Dr. Wallis’s conclusion and the 
fallacy of your own. For you cannot 
Vail to perceive that this product consists of 
double the natural series 0+1+2+3+4, 
that is to say double th.e series of whicii 
your multiplicand 0+1+i+l+l, &c. is 
the last term, and ymur'^ multiplier 
l+l + l+l+l, &c. the number f terms. 
If you are startled at this conclusion, let 
me advise you, Sir, to multiply six terms 
by six; seven by seven; eight hy eight; 
and so on as far as you please. You 
will find the results respectively 0+1+ 
2.+3+4+5+5+4+3+2+] ; — 0+1+ 
*4~S T 4''^-4“^"1"^'T^”I“-“}~2+2+1 '-X-, 
0+1+2+3+ 4 -f 5+6+7+7+6+5+4 
3+2+1 i—that is t.* -av, you will find 
each product to be double the Mim ot the 
series, agreeably to the coi elusion of Dr. 
Wallis. Now, Sir, whatever number of 
terms n there may be in vour multipli* 
ca%:l, since you must have the same num¬ 
ber of terms n in your mul ipliei, you 
will obtain a series of this fomi 0+1+2 
+3+4, . • • • . to n+' +ii — l4~o—24" 
n — 3.&c. to 1; — therefore, rea¬ 
soning by the method or Induction which 
you i)ave employed in the demonstration 
of your 3d proposition, and which you 
have defended in vonr reply to my letter, 
when n the number of terms is infinite 
you will still obtain for the product double 
the sum of the infinite series 0+1+2+3 
+4, o:c. And now, Sir, I think you 
must feel yourself vanquished with your 
own weapons. What think you now of 
tlie glorious discovery to which you lay 
such strong and frequent claiiijs? Think 
you not. Sir, that I had other reasons for 
omitting to notice this discovery than a 
“conviction of its truth”? And am I 
not warranted, Sir, after such a display of 
error and imbecility, to adojit your own 
words in the corollary to your oth prop, 
changing only Dr. Wallis {ov IMr. Taylor, 
which will then stand thus, “ Hence, as 
tlie w hole of tlie Arithmetic of Infinites 
of Jlr. Taylor is founded-on the above 
false proposition, no part ofthat arithmetic 
is to be considered as demonsfrutive; and 
such conclusions in it as may happen to 
be true are nor legitimately deduced. 
In tlie conclusion of your reply you 
inform me in what manner you obtained 
tlie remainder 1—1 in subtracting 1 + 1 
from 2; and you ask, “Is not the sub¬ 
traction actually made?” I answer, if it 
be, whar then becomes of your proposi¬ 
tion? For if the subtraction of 1 
from 1 gives 1—1 how is it, Mr. Taylor, 
that “ numbers connected together by a 
netiative sign are different from the same 
numbers when actually subtracted and ex^ 
pressed by one number T' 
I think, Sir, I have now noticed every 
article in your reply, and though I cannot 
flatter myself w ith the hope that in these 
ocservations I have used any arguments 
that wn'il appear convincing to the man 
who maintains that 6 multiplied by 2 is 
the same thing as adding C twice to it¬ 
self;—that J+1 is not equal to 2; that 
0-4-1 
1—1 IS not equal to 0:—that -is less 
1—1 
^^'30 +^4—that an infinite series with a 
