316 
you called the remainder 0 or nothing, or 
rather dot or *, to which you annexed 
the other terms of your second, line, or 
number to be subtracted, with their 
signs changed, agreeably to the common 
rule for the subtraction of algebraic 
quantities. Now, surely, as the author 
of an Elementary Treatise, you oughi to 
have previously demonstrated the grounds 
-of this method of subtraction: passing 
over, however, this unpardonable omis- 
sion, I would ask, why the first term of 
your second line should be actually sub- 
_tracted from the subtrahend, rather than 
put down after that subtrahend with its 
siva changed, in like manner as all the 
terms after the first in that line are put 
down? in which case, instead of the dot 
or * in the remainder, you would have 
had {—1; to this, perhaps, you will 
answer that you would still have had the 
series 1—1+1—111—1, &c. fora re- 
mainder, which I also readily admit; but 
what, let me ask, would have been the 
result of the second part of your demon- 
stration, where you attempt to shew that 
1—2-+-1 is an infinitesimal; if, instead 
ef actuad/y subtracting the first term of 
your second line from the same term of 
your first, you had only put the former 
down with its sign changed after the 
latter 2 Would you not, in this case, 
instead of > 1—2-+-1-++-1—2+1-11—2, 
&c. have obtained 1—1-+-1—2+1+-1— 
24+-1+1—2, &c. fora remainder? and 
how then; Sir, would you have sewn 
that this latter series consisted of your 
boasted infinitesiinal 1—2+-1 ?—Again, 
if in the first exainple, instead of placing 
the subtrahend 1 over the first term of 
the second line, you had put it over any 
of the succeeding terms in the same line, 
as in the following instances, you could 
not have obtained the remainder 
~-1—1+1—1, &c. as may be seen on 
anspection = F 
From 1 
Subtract 1—1+1—1+1, &c. 
* Remainder is —1+-2—14-1—1, &c. 
ees, 
. From : 1 
» Subtract 1—1-+-1—1, &c. 
Remainder is —1--i+1—1+1, &c. 
From 11 
Subtract 1—1+1—1+1, &e. 
Remainder is —1+i-+1—1, &c. 
’ From 1 
. Subtract 1—1-+-1—1-+41, &c. 
Remainder is —141—1+2—1, &c. 
Remarks on Taylor’s Elements 
[May 1, 
thus an infinite number of remainders 
might be obtained from the infinite 
variety of positions in which the subtra- 
hend might be placed, and any one of 
these I will affirm to be as corréctly the re- 
mainder as the one you have above given ¢ 
and what indifferent person would not 
consider-my affirmation as of equal weight 
with yours, till you have demonstrated 
that to obtain the ¢rue remainder it is 
absolutely necessary that the subtrahend 
should be placed over the first term of 
the series to be subtracted. The re- 
mainder in your second example might 
be varied in a similar manner by putting 
the first term of the second line uncer 
the second, third, fourth, term, &c, of 
the first line; but you would not then 
obtain for a remainder a series which 
would be constituted of a repetition of 
your infinitesimal 1—2-+1; unless, 
therefore, you can demonstrate that-the 
true remainder can only be obtained by 
that particular position in which you 
have thought proper to place the subtra- 
hend and series to be subtracted, your 
fundamental proposition is, to use your 
own language, false, and the superstruc- 
ture which you have raised upon it in- 
stantly falls to the ground; or I should 
rather have said the temple erected by 
Wallis and Newton, which you have in | 
vain attempted to demolish, still stands 
firm, unshaken, and immutable, upon 
the eternal and adamantine rock of 
science and truth, 
In your corollaries to this proposition, 
you are pleased to assert that the expres. 
sions 1—1, 1—2-+4+1, &c. are “ nenher 
quantities, nor’ nothings ;” that they are 
not quantities I am ready to allow, as 
numbers are rather the measures or re- 
presentatives of quantities, than quanti- 
ues themselves: but that they are noé 
nothing I deny, and I will defy you to 
prove that they are something. The in- 
genious Bishop Berkeley very shrewdly 
asked, * Whether evanescent incre- 
ments might not be called the ghosts of 
departed quantities:” what then, may L 
ask, shall your non-quantities be called, 
which are something yet neither quantity 
nor nothing 2 Surely these can only be 
the shadows of the ghosts of departed 
nothings ! 
Your second proposition is thus enun- 
ciated: ‘* There cannot-be a greater 
number of terms in any infinite series 
1 5 Dhtge 
than <a which is equal to 14-1-++- 1-1-1, 
&c. ad infinitum,” This enunciation 
ought. 
