1811] 
ought certainly to have terminated with 
& 1 
the fraction = 
2 and it should have 
been the object of the demonstration to 
prove that 7 
series 14-14-14-1, &c. whereas, by ad- 
ding to this part of the enunciation the 
words “ which is equal to 141-4-1+-1, 
&c. ad infinitum,” the proposition is 
rendered identical, and means neither 
more nor less than that there cunnot be 
@ greater number of terms in any infi- 
mite series, than un infinite number 
of terms! or that the number of 
terms in an infinite series is infinite ! 
Wow, as it would’ be the height of folly 
for one moment to dispute the truth of 
this assertion, I will not dwell on the 
demonstration’ which you have been 
pleased tv give to this notable truth, but 
will only ask you, by what method you 
1141, 
ae equal to the infinite 
—1 
rather than 
. a 
obtained 
“f 
s* you must 
wee) 
pA 
fox the sum of 1 and 
have obtained by first expres- 
3 ) 1 1 
sing the sum of 1, and Ear by Tot, 
and then by adding the product of the 
integer and denominator to the nume- 
rator, and placing their sum over the 
denominator, agreeably to the common 
rule for reducing a mixed number to an 
iniproper fraction ; but, suppose you had 
‘written the integer 1 before the fraction 
1 
~ : thus 14+—, and reduced this 
mixed number as above, would you 
domaclol4, 
not have obtained Fy ot aed 
&c. instead of So 14-21-41, 
&c.? Will you say that these resulting 
series 14-14-1-+-1, &c. and 1-+-2+41+1, 
&c. are equal, are the same, are identical ? 
Ifnot, ought you not to have proved by 
way of Lemma, previously to your enter- 
ing upon the “Elements of the True 
Arithmetic,” and as an indispensable 
requisite to understand even the first 
roposition of your work, that the num- 
Ber denoted by a, added to,the number 
denoted by b, is not the same as the 
number denoted by b, added to the num- 
Ber denoted by a, or that a+-b is not the 
Mowtury Mas. No, 212.; 
' P ° 
of the True Arithmetic of Infinites. 
$17 
same with b4-a? I must therefore, sir, 
push this question, why did you adopt 
the position of the integer 1 after the 
: J 1 
fraction 7 
j rather than before it? I 
anticipate your answer in these words, 
“because this position, and this only, 
would produce the resalt which I have 
obtained.” How necessary then was it, 
sir, L again repeat, that you should have 
previously proyed, that, in the addition 
of numbers, a particular regard to their 
position was essentially necessary to 
obtain, I will not say their correct sum, 
for that I deny, but the conclusions which 
you have deduced ia your propositions. 
In your third proposition from among 
many curious specimens of your reason. 
ing, I will select the following. For 
1--1+-1+-1, &c. ad infin. is evidently 
equal to the last term of the series 
1243-14, &c. ad infin. For the sum 
of two of these terms, beginning from the, 
first term, viz. 1-+-1 is equal to the se« 
cond term of the series 14-24-34, &c. 
The sum of three of the terms, beginning 
from the first term, viz. 1-1-1, is 
equal to 3, or the third term of the ses 
ries. The sum of four of the terms is 
equal to 4, or the fourth term of the 
series, and so on; and therefore the sum 
of the infinite series 14141441, &c, 
will be equal to the last term of the 
series 14-2+3-+4, &c.” I mean, not, 
sir, to dispute the justness of this infere 
ence, but can it be possible that you - 
should have deduced sach conclusions 
from such premises? You, who only a 
few pages before, in your preface, were 
vaunting in these words, “I rejoice to 
find as the result of this discovery, that it 
affords a most splendid instance ef the 
absurdity which may attend reasoning by 
induction from parts to wholes, or Srom 
wholes to parts, when the wholes are thems 
selves infinite?” Have you not here reas 
soned by induction from “parts to 
wholes,” when the wholes are themselves 
infinite? And may it not be perempto- 
rily demanded of you, “first to cast out 
the beam which is in thine own eye, thas 
thou mayest see clearly to cast out the 
mote of thy brother’s eye,” 
In your corollary to this proposition 
0) : 
you say, “ And 4 less than 
1 oe aa bpeiy 
7 by To” here again, sir, I 
will not stop to dispute your conclu. 
25 
sit, 
