1811.) 
which they may be. answered, demand 
on my part, that.the reply to them should 
be written with at least, equal severity ; 
yet [ will not so far degrade myself in the 
confutation of them as to act the part of 
a Reviewer. 
In your own language therefore, “ Now: 
sir to-the point.” My three first postu- 
lates, you say, “ You readily grant ;” but 
you are averse to assent to my fourth 
postulate, which, as you say, runs thus, 
“That to multiply one number, or one 
series of nuinbers, by another, is the 
same thing as to add either of those num- 
bers, or series of numbers, to itself, as 
often as there are units in the other.” 
You add, ‘¢ Now to say nothing of the 
‘ absurdity of calling this a postulate, which 
is, in reality, a definition, I do not be- 
lieve that it conveys even your own 
meaning, for surely you will not say that. 
3, multiplied by 2, isthe sameas 3 added 
twiee to itsell—for 3 added once to itself 
makes 6, and if added twice to itself it 
will make 9; and I cannot think, sir, that 
you meant to say that 3, multiplied by 2, 
is equal to 9,” I have only to say, in 
answer, that if I am in an error in this 
instance, your own favourite moderns 
have, unhappily for me, led me into it, 
And the first cause of my error was. Wol- 
fius, who, in his Algebra, p. 2, says, 
§*When unity is contained as oft in-one 
number, as another in a third, the two 
numbers are called) factors. or co-effi- 
¢iénts, and the third is the. product, 
arising from the one drawn into, or mul. 
tiplied by the other, and és no other than 
adding a number to itself, as often as 
there are units in the other; but it is 
done sooner by multiplication.” Now 
that I should be wrong is not at all won- 
derful, ‘but it seems that even that great 
modern mathematician Wolfius, is-also 
wrong according to Mr. Saint. And 
perhaps also, sir, you may he of opinion, 
that a? for instance, is not the second 
power of a, but the sirst. power of it, for 
a?, you may say is the first multiplication 
of a by itself. I bowever, agree with 
modern mathematicians, that. 6 multi- 
plied by 2 is the same thing as adding 6 
to itself twice, or 2 to itself six times, and 
that a is the first power of a, and a? the 
second power of it. You add, ‘* Now if 
you had te multiply the series, 14-1--1-+-1 
&c. ad infinitum by 1—1, since you 
have asserted in the corollaries to your 
first proposition, that 1—1 is that which 
is neither quantity nor. nothing, but 
which is something, belonging to.number 
without being nembery you would 
he 
/ 
Mr, Taylor in reply to Mr. Saint; 
435 
thus have to add the infinite. series 
14+1+-1-+1, &e. to itself as many times 
as are denoted by that which is neither 
quantity nor nothing, but which is some= 
thing belonging io number without being 
number.” Observe sir, with what fae 
cility this objection may be answered, 
According to the above citation from 
Wolfius, the multiplication of two terms 
is equivalent to the addition of one term 
to itself, as often as there are units in the 
other. Now as there are no units in 
1—1 it being an infinitesimal, and there 
are in 1-+-1+-1-4, &c. it will be the same 
thing to add 1—41 to itself, as many 
times as there are units in the infinite 
series 14+1+1-+1, &c. as to multiply 
1+1+1+1, &c. by 1—1. And so it, 
evidently is according to my theory, For 
Tsay, and have demonstrated that 1—1 
added to itself infinitely is in the aggre- 
gate equal to 1, though in the distributed 
form 1—1+41—141—1, &c. it is only 
equal to wd, 
1-1" 
In your next objection, you think that 
you have great matter for triumph. 
As a demonstration that the series 
1—1-+-1—1+-1—1, &c. produced from 
1 
1-1 
the expansion of —— is equal t 
RRS Ph gaat, SARE SO 
I said 
From 1 
Subtract I—141—141—1, &e. 
The remainder “+1—141—141—1, &e. 
To this you object, ‘ thatif instead of 
placing the subtrahend 1 over the first 
term of the second line, I had put it over 
any of the succeeding terms in the same 
line, as in the following instances, 1 
should not have obtained the remainder 
*+1—1+-1—1+41—-1, &c. as, may be 
seen on Inspection. 
From 1, 
Subtraet 1—1+1—1-+41, &e. 
Remainder. is —1-+2—1-1—1, &c. 
From 
1 
Subtract 1—1-+4-1—i-11, &c, 
Remainder is 11. -+-1—1, &c. 
From 1 
Subtract 1—i-+-1—1-41, &e. 
Remainder is —1-+-1—1-+-2—1, &c. 
Thave not inserted your second ine 
Stance, because it is not intelligible, 
Owing perhaps to errors of the press, and 
I have corrected an error in your ‘third 
instance, as. you will easily. see, which 
also. was perhaps an error of the press, 
tr , Observe 
