423 
G E O M 
fignable ; and ws may well be allowed to hefitate at 
fiich a fuppolition in fti’ift geometry. Any-finite fpace 
being propofed, the hyperbolic area (terminated by the 
curve, tlie afymptote, and a given ordinate) will exceed 
it by producing the curve and afymptote to an affign- 
able diftance ; and there is no afiignable limit in this, as 
in fome other cafes, which the area,may not furpafs in 
magnitude. See the article Fluxions. Therefore it 
is faid, that this area would be infinite, if the curve 
and afymptote could be infinitely produced. But no 
argument for admitting magnitude atfually infinite can 
be deduced from this, which does not more eafily appear 
from hence, that, a parallelogram of a given height 
would be infinite if it could have an infinite bafe; from 
which it cannot be inferred that fuch a bale or paral- 
Jclogram can abtually exift. It is often laid, that a 
rettangle of a given height on an imaginary bafe, as the 
analyfts fpeak, is imaginary : but we cannot thence in¬ 
fer, that an imaginary line or redtangle can exift. It is 
not, however, our intention to maintain the impoffibility 
of infinite magnitude ; but to fbew, that fuch doctrines 
are not neceflary confequences of the received principles 
of tills fcience, and not very proper to be admitted as 
the groundwork of the high geometry. 
In the fame treatife (fedf. 196,) a proof is offered, to 
Blew that, in the infinite feries of numbers proceeding 
in their natural order, there are finite numbers whofe 
fquares become infinite, which are called indetermi¬ 
nable, and are fuppofed to occupy the obfeure paflage 
from the numbers that are afiignable to thofe that are 
infinite. A greatell finite fquare is luppoled in this 
progrefiion, and reprefented by nn\ all that precede it 
are finite, and all that follow after it are luppofcd infi¬ 
nite. The numbers in this progreffion between n and nn, 
being lefs than nn, are finite ; but being greater than n, 
their fquares are greater than nn, and therefore, by the 
fuppofition, are infinite. But how can we admit the 
fuppofition of a greatefi: finite fquare number, fuch as is 
here exprefied by nn'> The number nn, being finite, is 
not the next to it in the progrefiion (which exceeds it 
by unit only) alfo finite > Should we allow that a finite 
number becomes infinite by adding unit to it, or even 
by fquaring it, how lhail we diftinguifh finite from infi¬ 
nite ? We commonly conceive finite magnitude to be 
afiignable, or to be limited by fuch as are afiignable, 
and to be fufceptible of further augmentation : and 
therefore infinite magnitude would feem. to imply, 
either that which exceeds all afiignable magnitude, or 
that which cannot admit of any further augmentation ; 
thefe being direblly oppofite to what we mofl: clearly 
conceive of finite magnitude. But neither of thefe con- 
ftitute the idea of infinite magnitude, as it mult be un¬ 
derflood in that treatife. The former is applicable to 
thofe numbers which the author calls finite and indeter¬ 
minable; which, being fuppofed to produce infinite 
Iquares, mull therefore exceed all afiignable numbers 
whofe fquares are affignable and finite. The latter is 
aferibed to that infinite only which he calls metaphysical, 
and excludes from geometry. -We are at a lofs to form 
a diflindl'idea even of finite itfelf as it is here underflood ; 
and it would feem, that the more art and ingenuity is 
employed in penetrating into the dodlrine of infinites, 
it becomes the more abllrufe. 
A proof is offered a pojleriori, (fe£t. 393,) to fhev/, 
that there are finite fractions in the feries, i, f, i, 
&c. whofe fquares become infinitely little in the feries 
41) zV’ Tbe ftim of the firft feries corre- 
fponds witli tlie area included betwixt the common hy¬ 
perbola and its afymptote; and is faid to be infinite 
when the feries is fuppofed to be continued to infinity. 
The fum of the latter feries correfponds with the area 
of an hyperbola of a higher order, and is faid to be 
finite, even when the feries is fuppofed to be continued 
to infinity j. bec.aufe there is a limit which tliis fum can 
E T R Y. 
neverequal, to which Itowever it continually approaches, 
as we have already deferibed. This being allowed, it 
is fuppofed farther, that there is an infinite number of 
finite terms in the firft progrefiion ; and it is thence de- 
monflrated, that there are finite fradlions in the firft fe¬ 
ries whofe fquares become infinitely little in the fecond, 
thus : “ If it fliould be pretended, that all the finite 
terms in the firft feries have their fquares finite in the 
fecond, there would be an infinite number of finite terms 
in the fecond as well as in the firft ; and the Anns of 
botli would be infinite ; fo that the contrary of an un¬ 
doubted truth, that is univerfalLy received, v/ould be 
demonftrated.” If we could allow that tltere is an in¬ 
finite number of finite terms in tlie firft feries, this ar¬ 
gument might have fome weight. But this is a fuppp- 
lition we cannot admit. For the denominator of any. 
fradfion in the firft feries is always equal to tlie number 
of terms from the beginning, and muft be fuppofed infi¬ 
nite when the number of terms is fuppofed infinite; 
but a fradlion that has unit for its numerator, and is 
fuppofed to have an infinite number for its denominator, 
cannot be fuppofed finite, but infinitely little ; fo that 
we cannot fuppofe an infinite number of terms in the 
firft feries to be finite. It is often faid in this treatife, 
that there is an infinite number of finite terms in the na¬ 
tural feries i, 2, 3, 4, 5, &c. continued to infinity. But 
we are at a lofs to conceive how this can be admitted ; 
fince, in any fuch progrefiion, the lafl or greatefi term 
is always equal to the number of terms from the begin¬ 
ning, and cannot be fuppofed finite v/hen the number of 
terms is fuppofed infinite. There is an afiignable limit 
which the fum of the terms of the fecond feries never 
amounts to ; but there is no afiignable limit which the 
fum of the firft feries may not furpafs ; and the fum of 
the terms of the firft is-greater than the fum of the cor- 
refponding terms of the fecond, in a ratio that, by con¬ 
tinuing the terms, may exceed any afiignable ratio of a 
greater magnitude to a lefs : and as this is eafily un- 
derllood and demonftrated, fo there is no neceflity for 
having recourfe to fuch abftrufe principles in order to 
account for it. 
It is of no life to cite authorities on this fubjedl, but 
as they may juftify us in ellablilhing fo noble a part of 
geometry for avoiding principles that are fo much con- 
tefled. What Ariftotle taught of infinite magnitude isw'ell 
known. Mr. Leibnitz, who for obvious reafons cannot 
be fufpecled of any prejudiceagainfl the dodlrine of infi¬ 
nites, exprelfes himlelf thus : On s'embaraj/'e dans les feries 
des nombres qui vonl a I'infini. On congoit un dernier terme, un 
nombre infini, ou infiniment petit ; mais tout cela ne font que des ■ 
fiElions. Tout nombre ef fini & qjfignable, toiitc ligne L'ef de 
mcme, Elllii de Theodicee, dife. prelim, fedl. 70. 
We have fubjoined thefe remarks, in order to ftiov/ 
why we have not followed an author who has merited 
fo well of mathematicians, and who on every other oc- 
cafion has been juftly applauded for his clear and diftindl 
way of explaining the abftrufe geometry. They who 
treated of infinites before him proceeded, as he oblerves,. 
with a timoroufnefs which the contemplation of fuch 
an objedl naturally infpires : Quand on y etoit arrive (fays 
he), on s'arrefoit avec une efpecc d'ejfroy & de fainte horrent^; 
—on regardoit Tinfini comme un mijicre qu'il falloit refpeBer, 
& qu'il n'etoit pas permis d'approfondir. They Itopped 
when they Came to infinity with a fort of holy dread, 
and refpedled it as an incomprehenfible myflery. He 
adventures farther, in order to difeover the fourcc, and 
penetrate into the firft principles, of geometrical truth. 
Infinity, according to him, is the great trunk from which 
its various branches are derived, and to \yhich they all 
lead. In this great purfuit he difplays infinite and finite 
with a freedom that puts us in mind of the ancient poet 
and his gods, whom he reprefents with the paflions of 
men, and mingles in their battles. We doubt not, that, 
if i\ full and perfect account of all that is moft. profound 
in 
