420 
GEOMETRY. 
■precautions of Arcliimedes came to be confidered as a 
clieck upon geometricig,ns, that ferved only to retard 
their progrefs. Therefore, inftead of his aflignable finite 
figures, indivifible or infinitely fmall elements ■were fub- 
feituted ; and thefe being imagined indefinite, or infinite, 
in number, their fiim was luppofed to coincide with tlie 
curvilincal area, or folid. 
It was, however, with caution that thefe fuppofitions 
were at firlt employed in geometry by Cavalerius, the in¬ 
genious author of the method of indivifibles, and by 
Olliers. Ke difeovered a method, which he found to be 
of a very extenfive ufe, and of an eafy application, for 
meafming or comparing planes and folids; and would 
not deprive the public of fo valuable an invention. In 
propofing it, lie ftrove to avoid the fuppofing magni¬ 
tude to confifl of indivifible parts, and to abfiract from 
the contemplation of infinity ; but he acknowledged, 
that there remained fome difficulties in this matter which 
he was not able to refolve. Therefore he fubjoined more 
unexceptionable demonfirations to thofe he had deduced 
from his own principles ; and the difputes which enfued 
(tlie fir/'t of any moment that were known between geo¬ 
metricians) juftified his precautions. Afterwards, infi¬ 
nitely fmall elements were fubftituted in jtlace of his 
indivifibles; and various improvements w'ere made in 
this dotfrine. The method of Archimedes, however, 
was often kept in view, and frequently appealed to as 
the fiireft tefi of every new invention. The*Ji armony 
betwixt the conclufions that arofe from the old and new 
methods contributed not a little to the credit which the 
latter at firff acquired ; till being more and more re- 
lifiied, they came at length to be generally admitted on 
their own evidence, and feerned to merit lb favourable 
a reception, by the great advantages that were derived 
from them for rcfolving the mofi: difficult problems, and 
demonffrating the mofi general theories, in a brief and 
eafy luanner. 
But when the principles and ftridi method of the an¬ 
cients, which had hitherto preferred the evidence of 
this fcience entire, were fo far abandoned, it was diffi¬ 
cult for the geometricians to determine where they fliould 
flop. After they had indulged themfelves in admitting 
quantities, of various kinds, that were not affignable, in 
fuppofing fuel) things to be done as could not be poffibly 
effedied (againfl the conflant pradlice of the ancients), 
and had involved themfelves in the mazes of infinity ; 
it was not eafy for them to avoid perplexity, and fome- 
times error, or to fix the bounds to thefe liberties -when 
they were once introduced. Curves were not only con¬ 
fidered as polygons of an infinite number of infinitely 
little fides, and their differences deduced from tlie dif¬ 
ferent angles that were fuppofed to be formed by thefe 
fides ; but infinites and infinitefimals were admitted of 
infinite orders, every operation in geometry and arith¬ 
metic applied to them witlr the fame freedom as to finite 
real quantities, and fuppofitions of this nature multi¬ 
plied, till the higher parts of geometry (as they were 
mofi commonly deferibed) appeimed full of myileries. 
From geometry the.,infinites and infinitefimals palled 
into philofophy, carrying with them the obfeurity and 
perplexity that cannot fail to accompany them. An 
adtual divifion, as well as a divifibility, of matter in infi¬ 
nitum^ is admitted by fome. Fiitiids are imagined con- 
fifting of infinitely finall particles, which are compofed 
ttipmfclves of others infinitely lefs ; and tfiis fubclivifion 
is fuppofed to be continued without end. Vortices are 
propof’ed, for folving the phenomena of nature, of inde¬ 
finite or infinite degrees, in imitation of the infinitefimals 
in geometry ; that, when any higher order is found in- 
fufiicient for this purpofe, or attended with an infuper- 
able difliculty, a lower order may preferve fo favourite 
a fcheme. Nature is confined in her operations to adt 
by infinitely finall fteps. Bodies of a perfedt Irardnefs 
are rejedted, and the old doebrine of atoms treated as 
imaginary, becaufe in their adtions and collifions they 
might pafs at once from motion'to refi, or from reft to 
motion, in violation of this law. Thus the dodlrine of 
infinites is interwoven with our fpeculations in geometry 
and nature. Suppofitions, that were propol'ed at firft 
diffidently, as of ufe for difeovering new theorems ii) 
this fcience with the greater facility, and were fuftered 
only on that account, have been indulged, till it has be¬ 
come crowded with objedfs of an abftrufe nature, which 
tend to perplex it and the other fciences that have a de¬ 
pendence upon it. 
They who have made ufe of infinites and infinitefi. 
mals with the greatefl liberty, have not agreed as to the 
truth and reality they would aferibe to them. The ce¬ 
lebrated Mr. Leibnitz owns them to be no more titan 
fidtions. Others place them on a level with finite quan¬ 
tities, and endeavour to demonllrate their reality from 
magnitudes being fufceptible of augmentation and dimi. 
nutioii without end,* from tlie properties of tlte prom-ef- 
fions of numbers that may be continued at pleafure, and 
from the infinity which fome geometricians have aferibed 
to the hyperbolic area. But in thefe arguments they 
feem to fuppofe the infinity which they would demonfiratc. 
It was a-principle of the ancient geometricians, that 
any given line may be produced, and its parts fubdi- 
vided, at pleafure: but they never Ifippofed it to be pro¬ 
duced till it fliould become infinitely great ; or to be 
fubdivided, till its parts fliould become infinitely fmall. 
It does not necelfarily follow, that, becaufe any given 
right line may be continued further, it can be produced 
till it become actually infinite, or that we are able to 
conceive fuch a line to be deferibed, fo as to admit it in 
geometry. In general, magnitude is capable of beino- 
increafed without end ; that is, no term or limit can be 
afligned or luppofed beyond which it may not be con¬ 
ceived to be further increafed. But from this it cannot 
be inferred, that we are able to conceive or fuppofe 
magnitude to be really infinite ; or, if we are able to 
join infinity to any fuppofed idea of a determinate quan¬ 
tity, and to reafon concerning magnitude aifually infi¬ 
nite, it is not furely with that perfpicuity that is re¬ 
quired in geometry. In the fame manner, no magnitude 
can be conceived fo finall, but a lefs than it may be 
fuppofed; but we are not therefore able to conceive a 
quantity infinitely fmall. A given magnitude may be 
fuppofed to be divided into any affignable number of 
parts; but it cannot therefore be conceived to be divided 
into a number of parts greater than what is affignable. 
The parts of a given line may be fuppofed to be conti¬ 
nually bifedbed till they become lefs than any line that 
is propofed ; and this is fufficient for completing the 
demonfirations of the ancients. But it is acknowledged 
by thofe who have treated the dodlrine of infinites in the 
fulleft manner, that “ there is fomething inconceivable 
in fuppofing an infinitely great or infinitely fmall num¬ 
ber or figure to be produced or generated; and that the 
puifage from finite to-infinite is obfeure and incompre- 
henfible:” and therefore it is better for us, in treating 
of fo (Iridl a fcience as geometry, to abftradb from thefe 
fuppofitions. The abftrufe confequenees that have been 
deduced from them by ingenious men, may the rather 
induce us to beware of admitting them as neceffary 
principles in this fcience, and to adhere to its ancient 
principles. 
Mr, Locke, who wrote his-excellent Effay, “ that we 
might difeover how far the powers of the underftanding 
reach, to what things they are in-any-degree.proportion¬ 
ate, and where they fail us,” obferves, “ that whilft 
meii'talk and difpute bf infinite magnitudes, as if they 
had as complete and pofitive ideas of them as they have 
of the names they ufe for them, tor as they have of a 
yard, or an hour, or an-y other determinate quantity, it 
is no wonder if the incomprehenfible nature of the thing 
they difeourfe of, or reafon about,deads them into per¬ 
plexities and contradiitions; and their'minds be over¬ 
laid by an object too large-and mighty to bedurveyed 
a and 
