1797-J 
which, multiplied into itfelf, produces the given 
term, and a mark fet over a term cannot give 
exiftence to a term which was not in exiftence. 
Ex nihilo nihil fit, —a is no number at all, 
and 4/—a is nothing at all. 
It may be afked, whence comes it to pafs that 
mathematicians of great name have, for fo 
great a length of time, permitted thete imagi- 
nary quantities to occupy moft of their attention? 
Jt might be anfwered, that they were fallible, 
like the philofophers who contended and rea- 
foned for the Ptolemaic fyftem; and when we 
read of fuch exceflive abfurdities as are daily 
permitted by our courts of law, or fwallowed 
down by whole nations, we muft not be fur- 
prifed that mathematicians, whofe {ole bufinefs 
is truth, fthould fometimes fall into fimilar errors 
to thofe of the greateft part of mankind. I ad- 
mire Newton, when, from an apple’s fall, he 
inveftigated the laws of gravity, or from a 
child’s bubble, difentangled the rays of light ; 
but, © non audeo dicere de tali viro,” he ap- 
pears to me * incredibiliter repuerafcere,’” when 
he was flinging about his impoffible roots, and 
not to have been fo well employed as Scipio 
and Lelius in their aural amufements. 
Another of your correfpondents afks this 
queftion :—‘** Are all infinite additions equal ?” 
The queftion is eafily anfwered. Additions may 
be carried on for ever, but the fum is, at any 
period, finite. We may fuppofe the act, that 
a man it employed to add two to itfelf, for ever 
making as many additions in the day as you 
pleate, and, at the end of » additions, the fum 
will be 27. If, in the fame manner, another 
adds four to itfelf, at the end of the fame 
time the fum will be-4”, and the firft fum 
will be to the latter {um as 27:4”, or as 
4:23 that is, the latter fum will be double 
the former, It matters not what number 7 is; 
whether the two men have been employed a 
thoufand million of times, the: period elapfed 
fince the firit form of this earth, or one day, 
the fums are to’ each other as 1:2. EH one 
man, in the above period, makes m, whilft the 
other is making 7 additions, the fums are to 
each other 2m:4n, or as m:27, a finite ratia, 
which may be varied at pleafure. 
I remember, at Cambridge, we ufed to bandy 
about in the f{chools the terms infinite ratio, 
infinite quantity, infinitefmal quantity; but fuch 
terms were well enough to make a frefhman ftare 
.and puzzle a moderator; and I with, with all 
my heart, that no other falfe reafoning pafled 
current in that learned feminary. ‘Thus, from 
your correfpondent’s queltious, I can prove, ac- 
Mathematical Correfpondence. 
207 
cording to the “ argumentum ad fophos,” that 
an infinite fam may be infinitely greater than 
another infinite fum, For, whilft our men 
above were employed in their additions, fuppofe 
another to amufe himfelf with the arithmetical 
~ progreffion 1, 2, 3, 4, &c. to x terms. Confe- 
quently, the fum of his feries will be to the fum 
h 4 nol 
of the firft mentioned feries as »w—— 3 2m 2 
2 
m-+-1:4 3 and as our Cambridge fophifts tell 
us that z-1 may be infinitely greater than 
four, my pofition is proved. This, to be furey 
is infinite nonfenfe, but may, for ought { know, 
have its ufe. At the entrance into the ancient 
{chools was prefixed sdsic and yeorelerilog evoilury 
and a young man may be called upon to find the 
{quare root of —1, or the‘laft term but one of 
an infinite feries, before he is initiated into the 
myfteries of Dr, Hey’s leCtures. 
Puito-Cosa. 

To the Editor of the Moathly Magazine. | 
STR, 
HE fubftance of ithe query propofed by your 
correfpondent Pui ariTaMus, appears to 
be a contradi¢tion in itfelf ; for, to require the 
fum of a feries, continued ad infinitum, necef- 
farily fuppofes a limit to infinity ; which is ab« 
furd, and contrary to the ideas we asfix to the 
term. 
Indeed, the doétrine of infinity is fo very 
abftruie, that the commoly received Opinions 
concerning it appear to me paradoxical. very 
much doubt, whether a fair explanation can be 
given to the affertion, that every magnitude is 
infinitely divifible—how can a particle of matter 
be divided inta an infinite number of parts, 
fince the number of component parts, let them 
be never io imal], muft be, in the aggregate, 
equal to the given particle, and therefore finite 2 
I am, your conftant reader, 
Port/mouth, Sept. 7, 1797. E. H. 

New MATHEMATICAL QUESTION. 

Question XAXV.—By Mr. Pirul Newten, 
ADMITTING two globes of fine gold to be 
each 30 inches in diameter; what difference 
muft there be between the thicknefs of the 
fhells, fo that oxe may juft fwim in rain-water, 
and the other in air; the denfities of th three 
fubftances, gold, water, and alr being as 
19640, and 1000, and 1h 3 

, & The Biographical Memoir of the late Mr. Wright, of Derby, could nat he prepared in 
time for infertion in the prefent Number. I: will certainly appear, in rbe neat. : 
Ee2 
ORIGINAL 
