ISl 1.] on his Elements cf-the Arithmetic of Infinites. 
cipher p'-efixed is infinitely less than th$ 
same senes without tlie cipher;—-liiat 
—that 2-}-l 13 not the 
same as 1-^2 that 4 — 3 is greater than 
1—1 . 1 
3^2;—that-is equal to that 
' 2 — 1—1 ^ 3 
the series &c. is to the se¬ 
ries 1-}-2-4-3-^4, as 1-|~1 to 1, but 
not as 2 to 1; thougli, I say, I cannot for 
one moment suppose tliat any thing 
which I can have said can convince such 
a man; yet 1 trust I have succeeded in 
my object, which was not so much to 
convince you as to satisfy others, that the 
mathematical sciences do not abound in 
those foolish conceits, glaring absurdi¬ 
ties, quirks, quibbles, and paradoxes, 
which are every where to be met with in 
your “ True Arithmetic,” and which are 
delivered with such a parade of ostenta¬ 
tion ; with such airs of Self-importance; 
and with such marked contempt of all 
‘tncdern mathemal'iciam, even Newton and 
not excepted, as miglit lead those 
wiio are unacquainted with tliese sci¬ 
ences, to fnrin the most unfavourable con¬ 
clusion, not only rc'pcctmg the evidence 
o? their principles, but also respecting 
their nature and tendency. Such, Sir, 
was my oljoct, and this object I flatter 
myself j[ have accomplished. 
i know imt^hetlier you will consider 
these observations worthy of nolice ; be 
this as it may, I feel thoroughly assured, 
that however much I may have failed in 
conviticing you of the fallacy and absur¬ 
dity of your “ True Arithmetic,” I have 
fully satisfied others on this point. I 
sha.t therefore liave little inclination to 
•resume the subject; for conceiving that I 
have fairly beaten and vanquished you 
with your own weapons; broken your 
rusty sword ; captured your general; and 
dispersed your army; I feel no anxiety as 
to any effiu'ts which you may hereafter 
he able to make. Should yon, there¬ 
fore, once more rally your forces, I shall 
most probably leave you in quiet pos¬ 
session of tlie small portion of territory 
which you now occupy, agd shall content 
myself with smiling at the puny efforts 
which you may make to destroy the vali¬ 
dity, beauty, and accuracy of the mathe¬ 
matical sciences, defended as those 
sciences are by truth, reason, and argu¬ 
ment. 
That you may not think, Sir, that I 
have looked no further into your book 
than the 8th proposition, allow me to 
conclude my observations vvith the foi- 
lowing extracts. At page 26 is the fol¬ 
lowing remark, For infinite 
collected number can no otherwise sub¬ 
sist than casually, oraccoidiug to the in¬ 
finite in power, of whicir mode of sub¬ 
sistence these expressions are obvious 
images.” At prop. 21 it is asserted, that 
“ The difference between ^ and 1-|- 
t^c. is 1;” though at prop. 
• 1 . 
2 It IS said that -—- is equal to 
1 +1-1-1, &c,—Prop. 22,13 repree 
sent the difference between —~— and 1 
2—i ® 
in infinite series of whole numbers.^’ To 
this prop, is added the foilow-ing curious 
corollary, “ In like manner the difference 
r 1 1 1 
between-— and - between-- and 
0—1 2 4—1 
-} and so on, may be shewn in infinite 
series of whole numbers: and thus a* 
Plato says of justice in a republic, und in 
the human soul zee shall evidently see, m 
it zoere, in large what is not so obvious in 
SJiiall lettersP I knovv not, Sir, what 
affinity there is between “ Justice in a 
Republic” and the fraction-; 
or between 
the “ Flurnan Soul” and the fractioa 
and I really cannot help thinking that 
your illustration would have been much 
more readily comprehended, at least by 
your English readers, if you had said 
that a surloin of beef be more evi¬ 
dently seen while whole than when dis^ 
iribuled. 
As, in your reply, you defied me to 
prove that the hi«t term of an infinite 
series multiplied by the number of terms 
was not equal to the sum of the series, 
so, Sir, I cannot finish these observations 
without defying you to prove an assertion 
contained in the second corollary to 
your eleventh proposition, which is thus 
expressed: “Hence, also, the assertion 
of modern mathematicians, that the sum 
of any number of terms of the arithme¬ 
tical series of odd numbers, 1,3, 5, 7, 9, 
&c. is equal to the square of that number 
isfahed* Now, Mr. Taylor, if you caU 
point out that number of terms to which, 
if the series 1, 3, 3, 7, 9, Ikc. be carried, 
the sum obtained by adding together all 
the terms of the said series, shall not be 
equal to the square of tlie said ivumber 
of terms, Izvill concede to you every thing 
which 
