GEOMETRY. 431 
and managed by them.’* Mathematicians indeed abridge 
their computations by tlie fuppoiltion of infinites ; but 
when they pretend to treat tliem on a level with finite 
quantities, they are fometimes led into fuch dodtrincs as 
verify the obfervation of tliis judicious author. To 
mention an inftance or two : the progrenion of the num¬ 
bers I, 2, 3, 4, 5, &c. in their natural order, is fuppofed 
to be continued to infinity, till by the continual addition 
of units an infinite number is produced, which is con¬ 
ceived to be the termination of this feries. This infinite 
number-is fuppofed to be fiill capable of augmentation 
and dimfcnution; and yet it is faid, “ that it is neither iiv 
creafed nor diminiflied by the addition or fubtrailion of 
the fame units from rvhich it was fuppofed to be gene¬ 
rated.” In a progreflion of this kind, the number of 
terms is always equal to the laft or greateft teiiu, and 
is finite when the laft term is finite.' If the number of 
terms be fuppofed infinite, the laft term cannot be finite; 
and yet it is laid, “ tliat in fuch a progreflion continued 
to infinity there is an infinite number of finite terms.” 
It is evident, that no finite number can become infinite 
by the addition of unit, or of any other finite number; 
and yet “ a greateft finite fqiiare number is fuppofed in 
Inch a progreflion, the next to which (though it exceed 
that finite number by an unit only) is I'uppofed infinite.” 
From thefe fuppofitions it is inferred, “ that in fuch a 
progreflion continued to infinity there are finite numbers 
whole fquares become infinite;” though it feems very 
evident, that a finite number taken any finite number of 
times can never produce more than a finite number. 
We may perceive from thefe inftances, that it is not by 
founding the higlier geometry on the doctrine of infinites 
we can propofe to avoid the apparent inconliftencies that 
have been objedted lo it; and fince an excellent author, 
who has always diftinguiihed himfelf as a clear and acute 
writer, has had no better fuccefs in eftablifliing it on 
thefe principles, it is better for us to avoid them. 
Thefe I'uppolitions however may be of ufe, when em¬ 
ployed with caution, for abridging computations in tlie 
jnveftigaiion of theorems, or even for proving them 
where a Icrifpulous exaftnefs is not required ; and we 
would not be underftood to affirm, that the methods of 
indivifibles and infinitcfimals, by whicli fo many uncon- 
tefted truths have been dil'covered, are without a foun¬ 
dation. We acknowledge further, that there is fome- 
tiiing marvellous in tlie doclriiie of infinites, that is apt 
to pleafe and tranfport us ; and that the method of infi- 
nitefimals has been profecuted of late with an acutenefs 
and fubtlety not to be paralleled in any other feience. 
But geometry is heft eftablillied on clear and plain prin¬ 
ciples; and tliefe fpeculations are ever obnoxious to 
fome difficulties. If tlie greateft accuracy has been al¬ 
ways required in tliis fcience, in reafoning concerning 
finite quantities, we apprehend that geometricians can¬ 
not be too fcriipulous in admitting or treating of infi¬ 
nites, of which our ideas are fo imperfect. Philofophy 
probably will always have its myfteries. But thefe^lre 
to be avoided in geometry : and we ought to guard 
againll abating from its ftridliiefs and evidence, the ra¬ 
ther, that an abfurd pliilofophy is the natural produ6l 
of a vitiated geometry. 
It is juft at the fame time to acknowledge, that they 
who firft carried geometry beyond its ancient limits, and 
they who have fince enlarged it, have done great fer- 
vice,, by deferibing plainly the methods which they 
found fo advantageous for this purpofe (though they 
might appear exceptionable in fome rel'peits), that 
others might proceed with the fame facility to improve 
it. Some of them have been I'o cautious as to verify 
their difeoveries by demonftrations in the ftriClell form ; 
and others were able to have done this, had they not 
chofe rather to employ their time in extending the 
fcience. At firft, the variation from the ancient me¬ 
thod was not fo confiderable, but that it was eafy to have 
rccourle to it, when it Ihould be thought iiecelfary for 
VoL. VIII. Islo. 514/ 
the fatisfaftion of fuch as required a fcrupuloiis exadl- 
nefs. The geometricians in tlie mean time made great 
improvements. They had the accurate method and ex¬ 
amples of Archimedes before them, by wliich they might 
try their difeoveries. Thefe lerved to keep them from 
error, and the new methods facilitated their progrefs. 
Thus their views enlarged ; and problems, that appeared 
at firft fight of an infuperatile difficulty, were afterwards 
refolved, and came at length to be delpiled as too fim- 
ple and eafy. The meiifuration of parabolas, hyperbo¬ 
las, fpirals of all the higher orders, and of the famous 
cycloid, w'ere ainongft the earlieft produftions of this 
period ; fome of which feeni to have been difeovered by 
leveral geometricians almoft at the fame time. It is not 
necelTary for our purpofe to deferibe more particularly 
what difeoyeries were’ made by Torricelli, MelTrs. de 
Fermat, and de Roberval, Gregory a Sto. Vincentie, &c, 
by whom the theorems of Archimedes were continued., 
and applied to the menfuration of various figures. 
The Aritkmetka Infinitonm of Dr. Wallis was the fuHeft 
treatife of this kind that appeared before the invention 
of the method of fluxions. Archimedes had confidered 
tlie funis of the terms in an arithmetical progreflTion, and 
of their fquares only, (or rather the limits of thefe fuma 
deferibed above,) thefe being fufficient for the menfu¬ 
ration of the figures he had examined. Dr. Wallis treats- 
this fubjedf in a very general manner, and afligns like 
limits for the funis of any powers of the terms, whether 
the exponents be integers or frattions, pofitive or nega¬ 
tive., Having difeovered one general theorem that in¬ 
cludes all of this kind, he then compounded new pro- 
greflions from various aggregates of thefe terms, and ' 
enquired into the I'ums of the powers of thefe terms, by 
which he was enabled to meafure accurately, or by ap¬ 
proximation, the areas of figures without number. But 
he compofed this treatife (as he tells us) before he had 
examined the writings of Archimedes, and he propofes 
his theorems and demonftrations in a lefs accurate form. 
He fuppofes the progreflions to be continued to infinity, 
and inveftigates, by a kind of indu 61 ;ion, the proportion 
of the fum of the powers to the product that would, 
arife by taking the greateft power as often as there are 
terms. His demonftrations, and fome of his expreffions, 
(as when he fpeaks of quantities more than infinite,) 
have been excepted againft. But it was not very diffi¬ 
cult to demonftrate the greateft part of his propofitions 
in a ftri< 5 ter method ; and this was efte6i:ed afterwards by 
himfelf and others In various inftances. He chofe to 
deferibe plainly a metliod which he had found very 
commodious for difeovering- new theorems; and it mult, 
be owned, that this valuable treatife contributed to pro¬ 
duce the great improvements which foon followed tlfter. 
A like apology may be made for others who have pro¬ 
moted. his doctrine fince his time, but have not given 
us rigid demonftrations. In general, it muft be owned, 
that if the late difeoveries were deduced at lengtli, in 
the very fame metliod in which the ancients demonftratecl 
their theorems, the life of man could hardly be fufficient 
for confidering tliem all: fo that a general and concife 
method, equivalent to tlieir’s in accuracy and evidence, 
that comprehends innumerable theorems in a few gene¬ 
ral views, may well be efteemed a valuable invention. 
Cavalerius was I'enflble of tlie difficulties, as Well 
the advantages, that attended his method. He fpeaks 
as if he forefaw tliat it lliould be afterwards delivered 
in an unexceptionable form, that might I'atisfy the moft 
fcrupuloiis geometrician ; and leaves this Gordwi knot., 
as he exprefles himfelf, to fome Alexander. Its form, 
indeed, was foon altered, and many improvements were; 
made by the mathematicians who prol'ecuted it fince his 
time, that deferve to be mentioned with efteem. But 
the method ftill remained liable to fome exceptions, and 
was thought to be lefs perfect than.that of the ancients 
on feveral grounds. 
Sir Ilaac Newtoa accomplifired what Cavalerius wiflicd 
5 P 
