G E O :M E T R Y. 
41!) 
fcrlbed in circles are in the duplicate ratio of tlie dia¬ 
meters, they woulcf Iiave imni diately extended this to 
the circles tliemfelves ; and would have confidercd 'he 
fecond propofition of the twelfrh book of the Elements 
as an eafy corollary from the nrfl. But there is ground 
to think that they would not have admitted a demon- 
ftration of this kind. It was a fundamental principle 
with them, that the difference of any two unequal quan¬ 
tities, by which the greater exceeds the lefs, may be 
added toltfelf till it ffiall exceed any propofed finite quan¬ 
tity of the fame kind : and tliat they founded their pVo- 
pofitions concerning curvilineal figures upon this prin¬ 
ciple in a particular manner, is evident from the demon- 
firations, and from the exprefs declaration of Archi- 
-medes, who acknowledges it to be the foundation upon 
which he eftablifhed his own difcoveries, and cites it as 
■alfumed by the ancients in demonftrating all their pro- 
pofitions of this kind.. But this prEincipIe feems to be 
inconfiftent with the admitting of an infinitely little 
quantity or difference, which, added to itfelf any num¬ 
ber of times, is never fuppofed-to become equal to any 
finite quantity whatfoever. 
They proceeded therefore in another manner, lefs di- 
redt indeed, but perfedlly evident. They found, that 
the inferibed fimilar polygons, by increafmg the number 
of their fides, continually approached to the areas of tlte 
circles ; fo that the decreafing differences betwixt each 
circle and its inferibed polygon, by ffill further and fur¬ 
ther divifions of the circalar arches which the lides of 
tlie polygons fubtend, could become lefs than any quan¬ 
tity that can be alligned ; and that all this while the 
lintilar polygons obferved the fame conftant invariable 
.proportion to each other, viz. that of the fquares of the 
diameters of the circles. Upon this they founded a de- 
monffration, tliat the proportion of the circles them- 
lelves could be no other than that fame invariable ratio 
of the fimilar inferibed polygons ; of which we ffiall give 
a brief abffradt, that it may appear in what manner tliey 
• were able, in this inffance, and fome others of the lame 
■ nature, to form a demonitration of the proportions of 
curvilineal figures, from what they had already difeo- 
■ vered of redtilineal ones.. And that the.general reaion- 
ing by wlvich tliey demonftrated all their theorems of 
this kind may more eafily appear, we lhall reprefent the 
circles and polygons by right lines, in the lame manner 
as all magnitudes are exprelfed in the fifth book of tlie 
. Elements. 
Suppofe the right - p „ o . n 
lines AB and AD ^ Q EyD 
to reprefent the two ' ‘ 
areas of the circles that are compared together ; and let 
A P, A Q, reprefent any two fimilar polygons inferibed 
in thefe circles. By further continual fubdivifions of 
the circular arches which the fides of the polygons fub¬ 
tend, the areas of the polygons increafe, and may ap- 
■ proach to the circles A.B and AD fo as to difler from 
them by lefs than any aflignable meafure; the triangle 
which is fubdudled. from each fegment at every new lub- 
divifion being always greater than the half of the feg- 
ment. The polygons inferibed in the two circles, as 
they increafe, are ever in the fame conftant proportion 
to each other.; and. this invariable ratio of thele poly¬ 
gons muft alfo be the ratio of the circles tliemfelves. 
For, if it is not, let the ratio of the polygons A P and 
A Q to each other be, in the firft place, the fame as the 
ratio of the circle A B to any magnitude A E lefs than 
the circle A D ; fuppofe the fubdivifions of the arches 
of the circle AD to be continued till the difference be¬ 
twixt the circle and inferibed polygon become lefs than 
E D, fo that the polygon may be reprefented by A q, 
greater than AE ; and let Ap reprefent a polygon in- 
icribed in the circle A.B, fimilar to the polygon A q. 
Then, fince A P is to A Q as A B is to A E l>y the fupl 
pofition, and theipolygon Ap is to the fimilar polygon 
A j as A P is to A Q; it .follows, that AB is to A E as 
Kp is to A y; and that the circle A B being greater than 
A p, a polygon inferibed in it, AE, mult be greater than 
A q. But A y is fuppofed to be greater titan A E ; and 
thefe being repugnant, it follows, tluit the polygon A P 
is not to the polygon A Q as tlte circle A B is to any 
magnitude (as A E) lefs than tlie circle A D. For tlie 
fame reafon, A Q is not to A P as A ID is to any magni¬ 
tude (asAF) lefs than AB. From which it follows, 
that we cannot fup- , o is- n 
pofe A P to be to AQ ^ 
as A B is to any mag- p q 
nitiule A e greater 
than A D ; becaufe if we take A F to A B as A D is to 
A r, A F will be lefs than A B, and A P will be to A Q 
as A F lefs than A B to A D ; againft what has been de¬ 
monftrated. It follows, therefore, that AP is not to 
A Q as A B is to any magnitude greater or lefs than A D; 
but tliat the ratio of the circles A B and A D to each 
otlier, muft be the fame as the invariable ratio of the 
fimilar polygons AP and AQ inferibed in them, which 
is the duplicate of the ratio of their diameters. 
In the fame manner the ancients have demonftrated, • 
tliat pyramids of the fame height are to each other as 
their bafes ; that fpheres are as the cubes of their dia¬ 
meters, and that a cone is tlie third part of a cylinder 
on the fame bafe and of the fame height. In general, 
it appears from this demonftration, that when two vari¬ 
able quantities, A P and A Q, which always are in an 
invariable ratio to each other, approach at the fame 
time to two determined quantities, A B and A D, fo that 
they may differ lefs from them than by any aflignable 
meafure, the ratio of thefe limits A B and A D'muft be 
the fame as the invariable ratio of the quantities A P 
and AQ: and this maybe confidered as the raoftfimple 
and fundamental propofition in this dotcrine, by which 
we are enabled to compare curvilineal fpaces in fome ot 
the more fimple cafes.—This general principle;may ferve 
for deTnonftrating many other propofitions, befides the 
elementary theorems already mentioned. 
By many otlier obfervations and demonftrations, Ar- 
chimedes lias highly diftinguifiied himfelf amongft tlie 
geometricians, and has done the greateft honour to this 
part of learning. Fie has not, .however, efcaped the 
cenfures of fome writers, who being unable to reconcile 
their own conceits with his demonltiations, have repre¬ 
fented liim as in an error, and miilcadiiig matliematlcians 
by iiis atithority. But tliough mathematicians may be 
grateful, autliority has not any place in this fcience ; 
and no geometrician ever pretended, from the higheli 
veneration for Archimedes, fir Ifaac Newton, or others, 
to reft on their judgment in a matter of geometrical de¬ 
monftration. The purfult of general and eafy metliods 
may have induced Ibme to make tife ot exceptionable 
principles; and the vaft extent which the fcience has of - 
late acquired, may have occalioned their propqling in¬ 
complete demonftrations. They may have alfo iome- 
times fallen into miftakes: but it will be found difficult 
to aflign one falfe propofition that has been ever ge.ne- 
rally received by geometricians; and it is hardly poi- 
fible, that accufations of this nature can be more mif- 
placed. 
In what Archimedes had demonftrated of the limits 
of figures and progreftions, there were valuable hints to¬ 
wards a general method of confidering curvilineal figures; 
fo as to fubjedl them to raenfuration by an exadt qua- - 
drature, an approximation, or by comparing them with 
others of a more fimple kind. Such methods have been 
propofed of late in various forms, and upon ditferent 
principles. The firft eflays were deduced from a care¬ 
ful attention to his fteps. But that his method might 
be more eafily extended, its old foundation was aban¬ 
doned, and fuppofitions were propofed which he had 
avoided. It i.as thought unnecelfary to conceive the 
figures circumferibed or inferibed in the curvilineal area, 
or folid, as being always allignable and finite; and the 
precautions 
