GEO 



C E O 



und the conic feclions; and thofe of flie tlinj, or cubic or- 

 der, will be the cubical and Neliaii parabolas, the ciffoid of 

 the ancients, &c. 



But the curve of the firft gender (bccaufe a right line 

 cannot be reckoned among the curves) is the fame with a 

 line of the fecond order ; and a curve of the fxoiid gender, 

 the fame with a line of the tlirrd order ; and a line of an in- 

 finiteiimal order is that which a right line may cut in in- 

 finite points ; as the fpiral, cycloid, the quadratrix, and 

 every line generated by the infinite revolutions of a ra- 

 dius. 



However, it is not the equation, but the defcription, that 

 makes the curve a geometrical one ; the circle is a geometri- 

 cal line, not becaiife it may be exprefled by an equation, but 

 bccaufe its defcription is a poftulate : and it is not the iimpli- 

 city of the equation, but the cafinefs of the defcription, 

 which is to determine tlie clioice of the lines for the con- 

 ftruclion of a problem. The equation that expreffes a 

 parabola is more fimple than that which expreffes a circle ; 

 and yet the circle, by reafon of its more iimple conllruC^ion, 

 is admitted before it. 



The circle and the conic feftions, if you regard the di- 

 menfions of the equations, are of the fame order ; and yet 

 the circle is not numbered with them in the conRruftion of 

 problems ; but by reafon of its fimple defcription is de- 

 preffed to a lower order ; 11/2. that of a right line ; 

 fo that it is not improper to exprefs that by a circle, 

 which may be expreffcd by a riglit line, but it is a fault 

 to conftrucl that by the conic feclions, which may be 

 conftrufted by a circle. 



Either, therefore, the law mufl be taken from thediaien- 

 fions of equations, as obferved in a circle, and fo the dirtiiic- 

 tion be taken away between plane and folid problems : 

 or the law mu!t be aU.owed not to be ftrictly obferved in lines 

 of fuperior kinds ; but that lome, by reaion of their more 

 fimple defcription may be preferred to otliers of the 

 fame order, and be numbered with lines of inferior or- 

 ders. 



In conflruftions that are equally geometrical, the moft 

 fimple are always to be preferred : this Izw is fo univer- 

 fal as to be without exception. But algebraic exprcffions 

 add nothing to the fimplicity of the conttruttiou ; the 

 bare defcriptions of the lines here are only to be confider- 

 ed ; and thefe alone were confidered by thofe geometri- 

 cians, who joined a ci-cle with a right line. And as 

 tlK-fe are eafy or hard, the condruction becomes eafy or 

 hai'd : and therefore it is foreign to the nature of the 

 thing, from anything elfe to eibablilh laws about eon !l ructions. 



Either, therefore, with the ancients, we muft exclude 

 all lines befide the circle, and perhaps the conic fections, out 

 of geometry ; or admit all according to tlie fimplicity of the 

 defcription : if the trochoid were admitted into geometry, 

 we might, by its means, divide an angle in any^-iven ratio ; 

 would you therefore blame thofe who wotild make ufe (jt 

 this fine to divide an angle in the ratio of one number to 

 another ; and contend, that this line was not defined by an 

 equation, but that you mull make life of fueh liiws as are de- 

 fined by equations ? 



If, when an angle were to be divided, for inllance, into 

 looi parts, we (hould be obliged to bri^ig a curve defined by 

 an equation of above a hundred dimeiifions to do the bufinefs ; 

 which nobody could deferibe, much lefs underlland ; and 

 fliould prefer this to the trochoid, which is a line well known, 

 and defcribed eafily by the motion of a wheel, or circle : 

 who would not fee the abfurdity .? 



Either, therefore, the trochoid is not to be admitted at 

 all in geometry; or elfe, in the conftrudlion of problems, it 



Vol. XV]'. 



is to be preferred to all lines of a more difficult defcription, 

 and the reafon is the fame for othrr curves. Hence, the tri- 

 fcttions of an angle by a Conchoid, which Archimedes, in 

 his Lemmas, and Pappus, in his CuUetlions, liavp preferred 

 to the invention of all others in this cafe, miift be allowed 

 to be good ; fince v.e mu(l either exclude all lines, befide 

 the circle and right line, out of geometry, or admit them 

 according to the fimplicity of their delcriptions ; in which 

 cafe the conchoid yields to none except the circle. Equa- 

 tions are expreflions of arithmetical computation, and pro- 

 perly have no place in geometr)', except fo far as quantities 

 truly geometrical (that is, lines, furfaces, folids, and pro- 

 portions) may be faid to be fome equal to others : multipli- 

 cations, divifions, and fuch fort of computations, are new- 

 ly received into geometry, and that apparently contrary to 

 the firft deiign of this fcicnce : for whoever confiders the 

 conttruftion of problems by a right line and a circle, found by 

 the firll geometricians, wiU eafily perceive that gcom.elry wa» 

 introduced that we might cxpeditioully avoid, by drawing 

 lilies, the tcdioufnefsof computation 



It ihculd f -em, then fore, that the two fcicnces ought 

 not to be confounded together : the ancients fo induftrioufiy 

 dillinguifhed them, that they never introduced arithmetical 

 terms into geometry ; and the moderns, by confounding 

 both, have loft a great deal of that fimplicity, in \.hichthe 

 elegance cf geometry principally confifts. Upon the whole, 

 that is arithmetically more fimple, which is determined by 

 more fimple equations ; but that is geometrically more fim- 

 ple whicli is deter. lined by the more fimple drnwir.g of 

 lines ; and in geometry, that ought to be ivckoned bell 

 which is geometrically moft fimple. 



Geometrical Zofuj, or P/nd; called alfo fimply locus. 

 See Locu.s. 



Geomktkic.vl Me^Uum. See MKDif>r. 

 Glo^iktrical Met/j'oil of l/ie y/nc}inls. It is to be ob- 

 ferved that the ancients eftabliflied the higher parts of 

 their geometry on the fame principles as the elements of that 

 fcience, by dcmonftrations of the fame kind ; and that they 

 feem to have been careful not to fuppofe any thing done, till 

 by a previous problem they had ilieu'u how it vxas to be per- 

 formed. Far lefs did they fuppofe any thing to be done that 

 cannot be conceived, as a line or feries to be actually conti- 

 nued to infinity, or a magnitude to be diminifiied till it be- 

 comes infinitely lefs than what it was. The elements into 

 which they refolved magnitudes were finite, and fuch as might 

 be conceived to be real. Unbounded liberties have been 

 introduced of late, by which geometry, which ought to be 

 perfectly clear, is filled \\ ith myfteries. See Maclaurin's 

 Fluxior.s, Intr. p. 39, feq. 



Geomeikk.\l Ojailum. .See Cl'U\ k, Evolute, and O."?- 



CLLUM. 



Gkomkthical Ptue, is a meafure confifling of five feet. 

 See Paci:, a^id Foot. 



GHJ.METltlCAi. Plan, in Architecture. See Plan. 



Gf.om ETHICAL Plane. See Plane. 



Geo.methical Progrejftim. See G cornel r'lccl PnoGReK.^ 

 siox. 



Geo.mktiuc.vl ProporUon, called alio abfolntely, and fim- 

 ply, froporl'ion, is a (imiUtude or identity of ratios. Sec 

 Ratio. 



Thus, if A be to Bj as C to D, they arc in geometrical 

 proportion : fo 8, 4, 30, and 15, are geometrical propor- 

 tionals. See Proi'oktion. 



Geometrk-.vl Solution of a problem, is when the pro- 

 blem is directly folvcd according to the ilritl principles and 

 rules of geometry, and bv lines that are truly geometri- 

 cal. 



P In 



