436 
Observe here again, sir, and you will 
find that your objection vanishes as soon 
@s itis examined. You say, I do not 
in these instances obtain the remainder 
"+ 1—1-++-1—1+41—1, &c. True sir, 
but what if I obtain a remainder equal to 
it! Have you any objection to this? 
Now mark, in every subtraction, if it is 
tly made, the remainder added to 
what is subtracted is equal to the sub- 
trahend, by my second postulate, which 
you say you admit. Consequently sir, 
to I—14 14-41, &c. let —14-9—14 
I—1, &c. be added, and the sum is 1; 
and ‘the like conclusion is true in the 
_ other instances. But if this be the case 
—1+2—1+-1—1, &e. is equal to.1—1-++ 
I—itit—1, &e. Por if toi—i+1—1 
H1i—i, &e. -1—1-+-1—141—1, &c: 
be added, and the sum is 1; and if also 
to 1—1+4-1—i1+1—1, &c. —1-+2—1 
+1—1, &c. be added, and the sum is 
also 1. E think you'will not dény, Mr. 
Saint, that 1—1+4-1--1, &c. and —1+42 
—1+-1—1, &c. must be’ equal to each 
other. Now, if it clearly appears from 
all this, that such expressions as 1—1,; 
1—2-+4-1, &c. are not equivalent to 0, 
and yet are not quantities, is there any 
absurdity in asserting that they are ana- 
Iogous to points at the extremities of 
Ines, which are something belonging to, 
without being lines; and therefore that 
these »expressions are something helong- 
ing to number, without being number? 
» Why you exult so much at my having 
by a very obvious deduction shown the 
truth of my method of finding the last 
term of an infinite series, I cannot con- 
eeive. For in the eighth proposition, I 
have demonstrated the truth of ‘this 
universally, and I chose previously to 
elucidate it by induction in the third 
Proposition, from the facility with which 
such induction may be made. My 
eighth proposition, therefore, is as follows: 
“ In every series of terms in arithmetical 
or geometrical progression, or in any 
Progression in which the terms mutually 
exceed each other, the last term is equal 
to the first term, added to the second 
term, diminished 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 ifthe num- 
ber of terms be infinite, the last term 
is equal to the series multiplied by 
By as 
- Demonstration: 
“ Let the terms, whatever the series 
may be, be represented by a, b, c,d, e, 
then a--b—a-}+-c—b-+-d—c-+-e—d = e, 
Mr. Taylor in reply to Mr. Saint, 
[June 1, 
a i : 
+b—2 . 
, +c—b 
+d—e 
-he—d 
= Se ° 
But if the number of terms be infinite, 
viz. if the series be a+-b-+-c-+-d+e+f-+-g, 
&c. ad infin. then this series multiplied 
by 1—1, will be = a-+-b—a-+-c—b+d 
—c-+-e—d+f—e-+g—t, &c.” Q.E.D- 
Now, Sir, what becomes of your exul- 
tation; and how came you to be guilty 
of so unpardonable an omission, as not 
even to mention this proposition? You 
have, however, been guilty of a greater 
and more unpardonable omission than 
even this. For having. granted that 
the number of terms in an infinite series 
; and alsa 
1 
cannot be greater than q 
that my method in proposition 3, of ob- 
taining the last term of an infinite series 
is Just; you have wholly neglected to 
notice the necessary consequence of this 
concession, which is, the complete sub 
version of the leading propositions in Dr. 
Wallis’s Arithmetic of Infinites, as E 
have abundantly shown in the treatise 
under discussion. Thus in the infinite se- 
ries 04-14-2454, &c. the last orgreat- 
est term is O+1+1-+-1-+41, &c. and the 
number of terms is 1-+-1+-1+1-+41, &c. 
and O+4-1-++1-+1+-1, &c. multiplied by 
1+1+1+-1, &c. produces 0-+-1+42-4+3 
+4+5, &c. Thus too in the series 
0-+-1-++-4-4-90-+16-+425, &c.; the last 
term is O--1454547+9+4 11, &e. and 
the number of terms is 1-4-1+-1-++1, &c. 
and the last tern: multiplied by the nums 
her of terms is equal to 0-+-1-++-4-++-9-++16, 
&c. Thus again, in the series 0-148 
+274644125, &e. ; the last term is O-- 
1+-7 +.19-+487-+461, &c, and the number 
of terms is 1-++-1--14-1, &c. and the last 
term multiplied by the number of terms, 
produces 0--14-8-+27-+-641125, &c. 
And so in other instances which are enu- 
merated in prop. 3. Hence, as I infer 
in corol. 4, to prop. 8. “ In every infinite 
series whether fractional or integral, the 
terms of which have an uninterrupted 
conunuity, the Jast term multiphed by 
the number of terms wiil be equal to the- 
sum of the series. Now if this, Sir, be 
admitted to be true, and I defy you, or 
any mathematician, to show that it ig 
not, the following propositions of Dre. 
Wallis, are evidently false. ‘*In the 
arithmetical series O4-1+-2-+8-+4-4, &c. 
ifthe last term be multiplied into the 
, nombee 
