^418 
G E O M 
metria Indivifibilium ; Torricelli’s Opera Geometrica ; 
Viviani, in his Divinationes Geometrica:, Exercitatio 
Ma. hemanca, De Locis Solidis, De Maximis & Minimis, 
'&c.; Vieta, in his Efteftio Geometrica, Supplement. 
Geometriae, Seftiones Angulares, Refponfiim ad Pro- 
blema, Apollonius Gallus, See. ; Gregory St. Vincent’s 
■Quadratura Circuli; Fermat’s Varia Opera Mathema- 
tica; Dr. Barrow’s Le^tiones Geometricas ; Balliald de 
Lineis Spiralibus; Cavaleritis ; Schooten and Gregory’s 
Txercitationes Geometricae, and Gregory’s ParsUiiiver- 
falis, See. ; De Billy’s treatife De Proportione Harmo- 
‘nica; La LovCra's Geometria veterum promota; Slu- 
fiiis’s Mefolabium, Problemata Solida, &c. ; Wallis, in 
his treatifes De Cycloide, CifToide, See. De Proportioni- 
btis, De Seiitionibus Conicis, Arithmetica Infinitorum, 
De Centro Gravitatis, De Setlionibus Angularibus, De 
Angulo Contaftus, Cuno-Cuneus, &c. &c. ; Hugo De 
Omerique, in his Analyhs Geometrica ; Pafeal on the 
Cycloid; Step. Angeli’s Problemata Geometrica ; Alex. 
Anderfon’s Suppl. Apollonii Redivivi, Variorum Pro- 
blematum Pradtice, See.-, Baronius’sGeomet. Pi'ob. &c.; 
Guido Grand! Geometr. Demonftr. &c.; Ghetaldi Apol¬ 
lonius Redivivus, &c.; Ludolph van Colen oraCeuIen, 
•de Circulo et Adferiptis, See. ; Snell’s Apollonius Ba- 
tavus, Cyclometricus, &c.; Herberftein’s Diotome Cir- 
. culorum ; Palma’s Exercit. in Geometriam ;• Gtildini 
Centro-Baryca ; with feveral others equally eminent, of 
more modern date, as Dr. Rob. Simfon, Dr. Mat. Stew¬ 
art, Mr. Tho. Simpfon, &c. Since the introdudlion of 
the New Geometry, or the Geometry of Curve Lines, 
as exprelTed by algebraical equations, in this part of 
geometry, the following names, among .many others, 
are more cfpecially to be relpedfed, viz. Des Cartes, 
Schooten, Newton, Maclaurin, Brackenridge, Cramer, 
Cotes, \Varing, &c. But in Pradlical Geometry, the 
chief writers are, Beyer, Kepler, Ramus, Clavius, Mal¬ 
let, Tacquet, Ozanam, Wolfius, Gregory, with innu¬ 
merable others. 
Geometry is diftinguiflied into Theoretical or Specu¬ 
lative, and Pradlical. Theoretical or Speculative Geometry, 
treats of the various properties and relations in magni¬ 
tudes, demonftrating the theorems, &c. And Pra&ical 
Geometry, is that which applies thofe fpeculations and 
theorems to particular ufes in the I’olution of problems, 
and in the mealurements in the ordinary concerns of 
life.—Speculative Geometry again may be divided into 
Elementary and Sublime : Elementary or Common Geometry, 
is that which is employed in the confideration of right 
lines and plane furfaces, with the folids generated from 
them. And the Higher or Sublime Geometry, is that which 
is called “ the method of indivijibles, or injinitejimals 
and is employed in the conlideration of curve lines, conic 
feftions, and the bodies formed of them. This part has 
■ been chiefly cultivated by the aflillance of the improved 
Hate of algebrg, and the modern analyfis of fluxions. 
Geometry is hence valued for its extenfive ufefulnefs; 
but it has been moft admired for its evidence ; mathe¬ 
matical demonrtration being fuch as has been always 
fuppofed to put an end to difpute, leaving no place for 
doubt or cavil. It acquired this high character by the 
great care of the old writers, who admitted no princi¬ 
ples but a few felf-evident truths, and no demonftrations 
but fuch as were accurately deduced from them. The 
fcience being now vaftly enlarged, and applied with luc- 
cel's to philofophy and the arts, it is of greater import¬ 
ance than ever that its evidence be prelerved perfedf. 
But it has been objedted on feveral occaljpns, that the 
modern improvements have been eftablilhed for the moll 
part upon new and exceptionable maxims, of too ab- 
flrufea nature to deferve a place amonglt the plain prin- 
ciples-of the ancient geometry: and fome have proceeded 
jTo far as to impute falle reafoning to thofe authors who 
have contributed molt to the late difeoveries, and have 
at the fame time been molt cautious in their manner of 
deferibing them. 
E T R Y. 
I41 the method of indivifibles, lines were conceived to 
be made up of points, furfaces of lines, and folids of I'ur- 
faces ; and fuch fuppofitions have been employed by 
feveral ingenious men for proving the old theorems, and 
difeovering new ones, in a brief and eafy manner. But 
as this do 6 trine was inconfiftent with the Uriel principles 
of geometry, fo it foon appeared that there was fome 
danger of its leading them into falfe conclufions : there¬ 
fore others, in the place of indivifible, fubflituted infi¬ 
nitely fmall divifible elements, of which tliey fuppofed 
all magnitudes to be formed; and thus endeavoured to 
retain, and improve, the advantages that w'ere derived 
from the‘former method for the advancement of geome¬ 
try. After thefe came to be reliflied, an infinite fcale 
of infinites and infinitefimals (afeending and defeending 
always by infinite Heps) was imagined and propofed to- 
be received into geometry, as of the greatell ufe for pe¬ 
netrating into its abftrufe parts. Some have argued for 
quantities more than infinite; and others fora kind of 
quantities that are laid to be neither finite nor infinite,, 
but of an intermediate and indeterminate nature. 
This way of confidering what is called the fublime 
part of geometry has fo far prevailed, that it is gene¬ 
rally known by no lefs a title than the Science or Geometry 
of Infinites. Thefe terms imply fomething lofty, but 
myllerious; the contemplation of which may be I'uf- 
pedled to amaze and perplex, rather than fatisfy or en¬ 
lighten the underllanding, in the profecution of this 
fcience ; and, while it feems greatly to elevate geometry,, 
may poflibly lelfen its true and real excellency, which- 
chiefly confills in its pcrfpicuity and perfedl evidence 
for we may be apt to reft in an obfeure and imperfeiit 
knowledge of fo abftrufe a dodtrine, as better fuited to 
its nature, inftead of feeking for that clear and full view 
we ought to have of geometrical truth ; and to this we 
may aferibe the inclination which has appeared of late 
for introducing myfteries into a fcience wherein there 
ought to be none. 
There were fome, however, who diflilced the making 
much ufe of infinites and infinitefimals in geometry. Of* 
this number was fir Ifaac Newton (whofe caution was 
almoll as diftinguithing a part of liis charatler as his 
invention), efpecially after he faw that tliis.liberty was 
growing to fo great a height. When the certainty of 
any part of geometry is brought into queftion, the moft. 
efl'eftual way to fet the truth in a full light, and to pre¬ 
vent difputes, is to deduce it from axioms or firft prin¬ 
ciples of unexceptionable evidence, by demonftrations 
of the ftricleft kind, after the manner of the ancient geo¬ 
metricians. This is our defign in the follou’ing treatil'e. 
But, before we proceed, it may be of ufe to confider the 
fteps by which the ancients were able, in feveral inftances, 
from the menfuration of right-lined figures, to judge of 
fuch as were bounded by curve lines; for, as tliey did not 
allow themfelves to refolve curvilineal figures into rec¬ 
tilineal elements, it is worth while to examine by what 
art they could make a tranfition from the tine to the 
other : and as they were at great pains to finifti their de¬ 
monftrations in the moft perfect manner, fo by following 
their example, as much as poftible, in demonftrating a 
method fo much more general than their’s, we may beft 
guard againft exceptions and cavils, and vary lefs from 
the original foundations of geometry. 
They found, that fimilar triangles are to each other 
in the duplicate ratio of their homologous fides ; and, 
by refolving fimilar polygons into fimilar triangles, the 
fame propofition was extended to thefe polygons alfo. 
But when they came to compare curvilineal figures, 
tliat cannot be refolved into reftilineal parts, this me¬ 
thod failed. Circles are the only curvilineal plane 
figures conlidered in the elements of geometry." If they 
could have allowed themfelves to have conlidered thefe 
as fimilar polygons of an infinite number of fides, (as 
fome have done who pretend to abridge their detnon- 
llrationsj) after proving tjut any fimilar polygons in- 
feribed 
