J 8 i 1. ] on his Elements of the Arithmetic of Infinites^ 
121 
sufficient cause for exultation ; allow me 
to repeat that cause. You had exulted 
in your preface tiiat'your discovery “ af¬ 
forded a splendid instance of the absur¬ 
dity which may attend reasoning by in¬ 
duction from parts to wholes, or from 
wlioles to parts, when the wholes are 
themselves infinite,and yet so early as 
in your third proposition I found you 
“ JXeusoning by induction from parts to 
tcholes, when the wholes are themselves in- 
Jinitef now surely, Sir, it was allowable 
to stop here to exult at your sudden and. 
open violation of your owm precept, par¬ 
ticularly as you have omitted no oppor¬ 
tunity in your “True Arithmetic,'^ not 
only to exult at what you are pleased to 
call the errors of modem mathematicians^ 
but also to speak of those mathematicians 
themselves in a manner neither respectful 
nor decorous, and in terms which generally 
imply a certain bloated self-sufficiency, 
(not to say insufferable arrogance) which 
is rarely found to be the concomitant of 
science and knowledge. 
You go on by accusing me of an “ un¬ 
pardonable omission" in not even men- 
tioning your eighth proposition; permit 
me therefore, Sir, in my own justification 
.frankly to state to you the causes for that 
■omission. Having pointed out, most 
clearly as I conceived, as many errors, 
absurdities, and contradictions of your 
■postulates and leading propositions as ap¬ 
peared to nie abundantly sufficient to 
■convince any unprejudiced mind of the 
falsehood of ybur “ True Arithmetic," I 
did not think it necessary, neither did I 
wish, to follow you through the whole 
work, minutely stating every blunder and 
absurdity; every inflated proposition and 
empty demonstration; or every insignifi¬ 
cant sneer and pointless sarcasm at tiie 
■modern mathematicians. Nor did I think, 
Sir, it would be candid, generous, or even 
manly, after having, as I conceived, van¬ 
quished the enemy, to pursue him to r/erz/A; 
to allow him naquarter', or to exhibit him 
in all the cruel pomp and slow parade of a 
Roman triumph. No, Sir, conceiving 
that in my attack 1 had broken through 
the front line of the enemy; disconcerted 
Ins whole army; and entirely frustrated 
his designs, 1 wished rather to imitate 
the conduct of a British iiero, and to de¬ 
mist from the warfare the moment 1 per¬ 
suaded myself its object; was accom¬ 
plished. You however, having rallied 
your forces, in the language of defiance 
now dare me to the battle. As I am 
fully prepared for action I accept your 
♦ 
challenge, and will immediately attack 
your army of Invincibles, headed as it is 
by that unconquerable genera! your eighth 
proposition. Now then. Sir, laying aside 
ail figure of speech, let me request you 
once more to read attentively the enun¬ 
ciation of this famous proposition'—fof 
wliich purpose, and that 1 may the bet* 
ter animadvert upon it, allow me to put 
it down in your owii words. “In every 
series of terms in aritlimetical or geome¬ 
trical progression, dr in any progression 
in which the terms mutually exceed each 
other, the last term is equal to the fiisj 
term, added to the second term.,dimi¬ 
nished by the first, added to the. third 
term diminished by the second, added to 
the fourth term diminished by the third, 
and so on. And if the number of terms 
be infinite the last term is equal to the se¬ 
ries multiplied by 1—1." Now., Sir, 
wlien you have duly considered, this enun¬ 
ciation, let me ask you whether it mean^ 
any thing more or less than — If 
from any series of terms all the terins ex¬ 
cept the last be taken axvay, the last term 
only will remain ; say, Mr. Taylor,'does 
your boasted proposition amount to any 
thing else than this truism. No, Sir, to 
use your own expression, I will defy you 
to prove that it does. Yet this glorious 
truth! this important proposition is fol¬ 
lowed by what you are pleased to dignify 
by the appellation of o. demonstration, and 
which Consists in nothing more than put¬ 
ting ilown a series of letters with the sign 
“k or plus before them, and the same se¬ 
nes of letters except the last with the 
Sign — or minus^ and then shewing that 
since the positive and negative terms de¬ 
stroy each other, the last term or letter 
will be left alone: thus confirming my 
statement as to the purport (f your pro¬ 
position. The latter part of this yonr 
proposition however I deny, namely, 
that “If the number of terms be infiniie 
the last term is equal to the series mul¬ 
tiplied by 1 — 1." For ^f the series 
a-kb-kc-f-d-j-e, &c. be multiplied by 
1—3 1 ,as follows: 
' a-|-b4-c-|-d-ke, &c. 
1—1 
a-j-h-j-c-j—d“|~e, Nc. 
—a — b — c—<1 — e, ike. 
it is manifest that every positive term 
will have its corresponding negative one, 
and that this must necessarily be the case 
whatever be the number of terms in the 
series; and consequently since all the 
terms m the upper line of the product are 
Q ^ respectively 
