422 
GEOMETRY. 
for, by inventing the method of fluxions, and propofing 
it in a way that admits- of ftridl demonftration, which 
requires the fuppofition of no quantities but fuch as asre 
finite, and eafily conceived. The computations in this 
method are the fame as in the method of infinitefimals ; 
but it is founded on accurate principles, agreeable to 
tlie ancient geometry. In it, tlie premilfes and conclu- 
fions are equally accurate ; no quantities are rejected as 
infinitely fmall, and no part of a curve is fuppofed to 
coincide with a right line. The excellency ot this me¬ 
thod has'not been fo fully defcribed, or fo generally at¬ 
tended to, as it feems to deferve; and it has been lome- 
times reprefented as on a level in all thefe refpedfs with 
the method of infinitefimals. Our defign has been to 
fhow its advantages in a clearer and fuller light, and to 
promote the objedt of the great inventor, by ellablifli- 
ing the higher geometry on plain principles, perfectly 
confiftent with each other, and with thofe of the ancient 
geometricians. 
In a late treatife afcribed to a celebrated French au¬ 
thor, juftly efteemed for his various writings, feveral 
arguments are propofed, for admitting magnitude ac¬ 
tually infinite ; not that kind which has no limits, com¬ 
prehends all, and can receive no addition, which he 
calls inctaphyfical ; but that which he defines to be greater 
than any finite magnitude, which he diftinguifiies from 
the former, and calls geometrical. PuiJ'que la grandeur ejl 
Jufceptible d’augmentation fans fin, on la pent concevoir ou Jup- 
pofer augmentee' vne infinite de fois, c^ejl-a-dire, qu'elle Jera 
devenue infinie. Et, en ejfet, il efi impqjjible que la grandeur 
Jufceptible d’augmentation fans fin foit dans le meme cas que fi 
eile n'en etoit pas Jufceptible Jans Jin. Or, fi elk ne Vetoit 
pas, elle demeureroit toujours finie ; done etant Jufceptible d'aug¬ 
mentation fans fin, elle pent ne demeurer pas toujours finie, ou, 
ce qui efi le meme, devenir infinie. Elem. de la Geom. de 
ITnfini, 83.—Becaufe magnitude is fufceptible of aug¬ 
mentation without end, the author concludes, that we 
may fuppofe it augmented an infinite number of times. 
But, by being fufceptible of augmentation withotit end, 
we underfiand only, that no magnitude can be afllgned 
or conceived fo great but it may be fuppofed to receive 
further augmentation, and that a greater than it may 
Hill be alfigned or conceived. We eafily conceive that 
a finite magnitude may become greater and greater witii- 
out end, or that no termination or limit can be alfigned 
of the increafe which it may admit: but we do not 
therefore clearly conceive magnitude increafed an infi¬ 
nite number of times. Mr. Locke acknowledges, that 
we eaiily form an idea of the infinity of number-, to tlie 
end of whofe addition there is no approach: but he 
difiinguilhes betwixt this and the idea of an infinite num¬ 
ber ; and fubjoins,' tiiat, how clear foever our idea of 
the infinity of number may be, there is nothing more 
evident than the abfurdity of the aftual idea of an infi¬ 
nite number. 
The latter part of the argument amounts to this: 
“ It is impoflible that magnitude, being fufceptible of 
augmentation without end, can be in the fame cafe as if 
it v/as not fufceptible of augmentation without end. 
But if it was not fufceptible of augmentation without 
end, it would remain always finite. Therefore, fince it 
is fufceptible of augmentation without end, the con¬ 
trary mull be allowed : that is, it may not always re¬ 
main finite, or it may become infinite.” The force of 
which argument feems to be taken ofi', by confidering, 
that, if magnitude was not fufceptible of augmentation 
without end, it tvould not only remain ahyays finite, but 
there would necellarily be a term, limit, or degree, of 
magnitude, which could never be exceeded, or there 
might be a greatefi: magnitude. And, by allowing tliat 
there is no fuch term or limit, magnitude is not fup, 
poled to be in the fame cafe as if it were not fufceptible 
■of augmentation without end, though we Ihould refufc 
that it may become infinite. What is oppolite to tlie 
fuppofing magnitude fufceptible of augmentation with- 
a 
out end, is not the fuppofing It always finite, (for finite 
magnitude is capable of being increafed without end,) 
but the fuppofing it fufceptible of no augmentation 
at all, or of an augmentation that has a limit or end. 
The feriesof numbers, i, 2, 3, 4, &c, in their natural 
order, may be continued without end ; and it is faid, 
that “we never come nearer the end of the progrellion, 
liow great foever tlie number may be to which we ar¬ 
rive ; which is a charadter that cannot belong to a feries 
of a finite number of terras. Therefore this natural fe¬ 
ries has an infinite number of terms.” And it is added, 
that “ though we can go over a finite number of terms 
only, yet all the terms of this infinite progrellion are 
equally real.” But if we can conceive this feries to 
have any end, it feems to be evident, that we mull ap¬ 
proach to this end as we proceed frotja the beginning to¬ 
wards it; and tliat, while we advance, the diftance of 
any term from the end mult decreafe (whether this dif, 
tance be called finite or infinite) by the fame quantity 
as tlie difiance from any fubfequent term decreafes, or 
tile dillance from the beginning of the feries increafes. 
If vve cannot conceive the feries to have an end, then 
we can have no idea of its lall term. If we. fuppofe 
this feries to be continued to infinity, it would indeed 
be abfurd, after fuch a fuppofition, to fay that the num¬ 
ber of its terms is finite. But, in treating this fcience 
ftridtly, it may, perhaps, be better to avoid this fuppo¬ 
fition. Foj' if it is only a finite number of terms we can 
clearly conceive, liow fhall we judge of the reality of 
the rell ? or wherein lliall vve place the reality of thole 
which it is impoflible for us to aflign ? of which two 
kinds are faid to be in this fame feries, each infinite in 
number j the firft of whicli are faid to be finite, but in¬ 
determinable ; the latter, actually infinite. 
The argument from the infinity of the hyperbolic 
area is much infilfed on. “The hyperbolic area {Elm. 
de la Geom. de I'Infin. pref.) is as really infinite, as a de¬ 
termined parabolic area is two-thirds of the circum- 
Icribed paralle.'ogram. It is trifling to fay, that the 
one can be actually defcribed, and the other cannot. 
Geometry is entirely intelledtual, and independent of 
the adlual defeription and exifience of the figures whofe 
properties it difeovers. All that is conceived necelfary 
in it has tlie I'eality which it fuppofes in its object. 
Therefore the infinite which it demonftrates is as real as 
that which is finite,” &c. And the learned author, af¬ 
ter infifiing on this fubjeift, concludes, that, “not to 
receive infinity as it is here reprefented, with all its ne- 
cefl'ary confequences, is to rejedt a geometrical demon- 
Itration ; and that he who rejedts one, ought to rejedb 
them all.” But though the adtual defeription of the 
figures which are confidered in geometry be not necel- 
lary, yet it is requifite that we fliould be able clearly to 
conceive that they may cxifi and a diftindf idea of the 
manner how they may be fuppol’ed to be defcribed or 
generated is necefl'ary, that they may have a place in 
this fcience. Principles that are propofed as of the 
mofl: extenfive ufe, and as the foundation of all the 
luhjime geometry, ought to be clear and unexception¬ 
able. It this fcience is entirely intelledlual, or if the 
reality of its objedts is to be confidered as having a de¬ 
pendence on their being conceived by the mind, it would 
feem, that there muft be a difference betwixt tlie reality 
ot finite aflignable lines orjmmbers, and the reality we 
can aferibe to infinite lines or numbers, which are not 
affignable, and cannot be fuppoted to be produced or 
generated but in a manner that is allowed to be incon¬ 
ceivable. As for what is faid of the parabolic and hy¬ 
perbolic areas, we can conceive any portion of .the pa¬ 
rabola to be accurately defcribed, and its area to be de¬ 
termined, though no exact figure of this kind fhould 
ever exiff. We can alfo conceive, that the hyperbola 
and its afymptote may be produced to any aflignable 
diftance: but we do not fo clearly conceive that they 
may be produced to a difiance greater than what is all 
fignable j 
