3st.) 
_ 4—1 by the infinite series 14-11-11, 
&c. would be to add that to itself which 
is “ neither quantity nor nothing” an 
infinite number of times; and this sam 
_ teing equal to the former (unless indeed 
you deny that 2 multiplied by 3 is the 
— same with 3 multiplied by 2, or, more 
generally, that a multiplied by & is the 
same with 6 multiplicd by @) you would 
have an infinite number added to itself 
“neither quantity wor nothing” times, 
equal to ‘neither quantity nor nothing” 
added to itself an infinite number of 
times!—I know not, Mr. Taylor, what 
you may-think of this, but [ will tell you 
| most freely that I think it to be infinite 
~ monsense ! And Iwas nat a little asto- 
__nished to meet with this “ splendid in- 
Stance of absurdity,” to use your own 
~ language, in the very outset of a work 
in which you most modestly observe that 
| * The rambling and precipitate genius of 
modern mathematicians, eager to arrive 
at some conclusion which may be applicable 
_ to practical purposes, neglects thut rigid 
Vatcurucy of demonstration, which may be 
alled the impregnable fortress of the ma- 
thematical science, and for which the we- 
“nius of ancient mathematicians was so 
re-eminently distinguished.” : 
But I pass, Sir, from your postulates 
9 your first proposition, the enunciation 
id demonstration of which I will here 
nt down at length, as affording a fair 
specimen of the accuracy of your logic. 
«¢ Proposition 1, 
an infinitesimal, or infinitely 
¥ 
a 
4111s 
a 
3 1 
all’ part of the fraction Ty and’ an 
lite se f 1—1 Hi : 
I ries of 1—1 is equal to 1" 
in like manner, also, 1—2-++-1 is an in- 
initely small part of 
11 + 9 — 11 4 21 1 91 1 Sc 
a 11 
infin. and an infinite series of 1—2-+-1 
11-24 —1-b9— 11, &c. 
&e. 
a+ —— ad infin, and an 
mite series of 1—2 is equal to 
Be ie tio, Thus, too, 1—3 
ey ' 1— 9-9. 9_—9.. Ba, 
€ infinitesimal of- 4-8. ied 
ep 1-3 —3—S—3, Ke. 
‘tl , and so, of 
T-1 
of the True Arithmetic of Infinites. 
S15 
From 1° 
Subtract 1—141—141—141-1, &c, 
Remainder *+1—141—14+1—141, &c. 
“ But by the second postulate the re- 
mainder added to what is subtracted is 
egual to the subtrahend, Hence the so- 
ries 1—1+1—1+1—1, &c. added to 
1—1-+-1—1+1—1, &c. is equal to’ 
1. ‘The series 1—14-1—1+1—1, &c. 
is therefore equal to 
a 
and conse-: 
1.1 
quently 1—1 is an infinitesimal. For 
it cannot be O, since an infinite series of 
@, added to an infinite series of 0, can 
never be equal to 4, 
“Tn like manner, 
If from 1—i—1+2—1—14 + 2m I—1, &eL 
Subtract 1—2+141—94+ j{41—944, &c. 
Remaind. - 1—2+1+41—v-+141—2, ac. 
and thercfure 1—2-L1 is an infinitesimal ; 
and so of the rest. 
“Corol. 1. Hence such expressions 
as 1—1i, 1—2-+4+-1, 1—2, &e. are neither 
quantities nor nothings, but they are 
something belonging to number, without 
being nunrber ; just as a point, which is 
the extremity of a line, is something be-, 
longing to, without being a line. 
*“Corol. 2. Ifence, likewise suclz 
expressions when they are considered as. 
paris of infinite series, are not to be 
taken separate from the terms by which 
they are expressed, viz. 1—1, for instance. 
is not to be considered as a subtraction: 
of 1 from 1; for, in this case, it woul, 
be 0. Nor is 1-2 to be considered as 
a subtraction of 2 from 4; since it wauld’ 
then be —1i. But these expressions are, 
always to be considered in connexion 
with the nuinbers by which they are 
formed. 
“< Coro]. 8, Hence, the series which, 
are called by modern mathematicians 
neutral and diverging series, are .erro= 
neously so called, for they are in reality: 
convering series.” 
In this proposition, Sir, you begin by 
aflirming that 1—1 is an znfinitesinal, 
without having previously defined what 
constitutes an infinitesimal; perhaps, 
however, the qualifying words “ infinitely 
small part” which follow were designed 
to supply this deficiency. Your demone 
stration, I presume, begins at the word 
“Vrom ;’—ifso, let me ask you by what 
means you obtained the remainder 
ve 1 11 + 4 — 141-1, hc? © 
Your answer rust certainly be, that you 
actually subtracted the first term of the 
second line, or nun ber to be subtracted 
from the first term (and here only tern) 
6! the first ine or subtrahend, and that 
you 
