GEO
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GEO
The Rcafon is, that Nature is not abftraSed ; Mechani- cal Levers and Wheels are not Geometrical Lines and Circles, as they arc fuppofed to be : Our Tafle for Tunes is not the fame in all Men ; nor at all times in the fame Man : And as to Aftronomy, there is no perfecl Regular ty in the Motions of the Planets ; their Orbits do not feem reducible to any fix'd known Law, &c.
The Error.% therefore, we fall into in Aftronomy, Mu- fie, Mcchsnicks, and the other Sciences to which Geometry is applied, do not properly arife from Geometry, which is an infallible Science ; but from the falfe Mifapplication of it. Mallebranch, Recherche de la Ver.
Geom^tircal Lim\ or Curve, call'd siKoJ/gebraic Zinc, or Curve, is that wherein the Relation of the Abfciffes to the Semi-ordi nates may be exprefs'd by an Algebraick E- quation. See Algebraic Curve.
Thus, fuppofe in a Circle, 'Tab. Geometry, Fig. 51. AB— fl AP=j? P M==fs then will PB=:fl — #, and consequently, fince P M 3 = A P P B 5 y z z=a x—'x 2 . — Again, fuppofe in Fig.Si. PC — x, AC-flPM-^ then will MC*=PM\ that is a 2 — x'z=iy i . See Equation.
Geometrical Lines are diltinguifh'd into QaSes, Orders, or Genders, according to the Number of the Dimenfi- ons of the Eqijation that expreffes the Relation between the Ordinatcs and the Abfciffa: ; or which amounts to the lame, according to the Number of Points in which they may be cut by a Right Line.
Thus, a Line of the firjl Order will be only a Right Line: Thofe of the fecond, or guadratick Order, will be the Circle, and the Conick SeBions^ and thofe of the third, or Ctthick Order, will be the Cubical and Nclian Parabolas, the Ci/foid of the Antients, $5>c. See Circle, Conic Se~ Bid?, Parabola, Cissom, &c.
But a Curve of the fir ft Gender (becaufe a Right Line can't be rcckon'd among the Curves) is the fame with a Line of the lecond Order ; and a Curve of the fecond Gen- der, the fame with a Line of the third Order ; and a Line of an infimtefimal Order is that, which a Right Line may cut in infinite Points; as the Spiral, Cycloid, the Quadra- trix, and every Line generated by the infinite Revolutions of a Radius. See Line.
However, it is not the Equation, but the Defcription, that makes the Curve a Geometrical one: The Circle is a Gen- metrical Line, not becaufe it may be exprefled by an Equa- tion, but becaufe its Defcription is a Populate : And it is not the Simplicity of the Equation, but the Eafinefs of the Defcription, which is to determine the Choice of the Lines for the Construction of a Problem. The Equation, that expreffes a Parabola, is more fimple than that which expref- fes a Circle; and yet the Circle, by reafon of its more fim- ple Conft ruftion, is admitted before it.
The Circle, and the Conick Sections, if you regard the Bimenfion of the Equations, are of the fame Order; and yet the Circle is not numbered with them in the Conftru- ftion of Problems, but by reafon of its fimple Defcription is deprefs'd to a lower Order, viz. that of a Right Line 3 fo that it is not improper to cxprefs that by a Circle, which may be exprefs'd by a Right Line : But it is a Fault to con- ftruct that by the Conic Sections, which may be conftru&ed by a Circle.
Either, therefore the Law mufl be taken from the Di- mensions of Equations, as obfe-rved in a Circle, and lo the PUlinclicn be taken away between plane and lolid Problems : Or the Law mufl be allow'd not to be ftrictly obferved in Lines of fuperior Kinds; but that fome, by reafon of their more fimple Defcription, may be preferr'd to others of the fame Order, and be numbered with Lines of inferior Or- ders-
Iti Conirru&ions that are equally Geometrical, the moft fimple are always to be preferr'd : This Law is fo univer- sal as to be without Exception. But Algebraick Expressi- ons add nothing to the Simplicity of the Conltruclion ; the bare Deicription of the Lines here are only to be confidcr'd; ,and thefe alone were confidcr'd by thofe Geometricians, who joined a Circle -with a Right Line. And as thefe are eafy .or hard, the ConOruclion becomes eafy, or hard .- And there- fore it is foreign to the Nature of the Thing, from any thing elfe to eitablifh Laws about Conftrudtions. See Constru- ction.
Either, therefore with the Antients, we mud exclude all Lines befides the Circle, and perhaps the Conic Sections, out of Geometry ; or admit all according to the Simplicity of the Defcription: If the Trochoid were admitted into Geometry, we might, by its means, divide an Angle in any given Ratio : Would you therefore blame thofe, who ■would make ufe of this Line to divide an Angle in the Ra- tio of one Number to another, and contend that this Line was not defm'd by an Equation, but that you mutt make ufe of fuch Lines as arc derin'd by Equations? See Trans- cendental.
If, when an Angle were to be divided, for inftance, into jooi Parts, we mould be objig'd to bring a Curve defin'd
by an Equation of above an hundred Dimenfions to do the Bufinefs ; which nobody could, defcribe, much 1 lets under- ftand ; and mould 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 Tee the Abfurdiry ?
Either therefore the Trochoid is not to be admitted at all in Geometry, or elfe in the Con {truer ion of Problems, it i s to be preferr'd to all Lines of a more difficult Defcription : And the Rcafon is the fame for other Curves. "
Hence, the Trifect-ions of an Angle by a Conchoid, which Archimedes in his Lemma's, and Pappus in his Collections have preferr'd to the Invention of all others m this Cafe, mufl: be allow'd good, fincc we mufl either exclude all Lines, befide the Circle, and Right Line, out of Geometry, or ad- mit them according to the Simplicity of their Defcriptions; in which Cafe, the Conchoid yields to none, except the Circle.
Equations are Expressions of arithmetical Computation, and properly have no Place in Geometry, except as far as Quantities truly Geometrical (that is, Lines, Surfaces, So- lids, and Proportions) may be faid, to be fome equal to others: Multiplications, Divifions, and fuch fort of Com- putations are newly received into Geometry, and that ap- parently contrary to the firft Defign of this Science ; for whoever confiders the Conftruction of Problems by a Right Line, and a Circle found by the firft Geometricians, will eafily perceive, that Geometry was invented, that we might expeditioufly avoid by drawing Lines the Tedioufnefs of Computation.
It mould feem therefore, that the two Sciences ought not to be confounded : The Antients fo induftrioufly diftinguim'd them, that they never introduced arithmetical Terms into Geometry ; and the Moderns by confounding both, have loft a deal of that Simplicity, in which the Elegancy of Geo- metry principally confifts. Upon the whole, that is arith- metically more fimple which is determined by more Am- ple Equations ; but that is geometrically more fimple which is determined by the more fimple drawing of Lines ; and in Geometry, that ought to be reckoned belt, which is geo- metrically moft fimple.
Geometrical !Pto, fee Plane.
Geometrical Solution of a Problem, is when the Pro- blem is directly folved, according to the ftricr. Principles and Rules of Geometry ; and by Lines that are truly Geome- trical. See Problem, and Solution.
In this Senfe we fay, Geometrical Solution, in Contradi- ftindtion to a Mechanical, or Injlrumefital Solution, where the Problem is only folved by Rules and Compares. See Mechanical.
The fame Term we Jikewife ufe in oppofition to all in- direct,, and inadequate Solutions, as by infinite Series's, ^c. See Series.
We have no Geometrical Way, of finding the Quadra- ture of the Circle; the Duplicature of the Cube, or finding of two mean Proportionals : Mechanical ways, and others, by infinite Series's, we have. See Quadrature, Dupli- cature, and Proportional.
The Antients, 'Pap fin informs us, in vain endeavour'd at the Trifect-ion of an Angle, and the finding out of two mean Proportionals by a Right Line, and a Circle. After- wards they began to confider the Properties of fevcral other Lines, as the Conchoid, the Ciffoid, and the Conic Secti- ons, and by fome of thefe endeavour'd to folve thofe Pro- blems. At length, having more throughly examined the Matter, and the Conic Sections being received into Geome- try, they diltinguifh'd Geometrical Problems into three Kinds, viz.
i Q Into 'Plane ones, which deriving their Original from Lines on a Plane, may be folved by a Right Line, and a Circle. See Plane.
2 Solid ones, which were folved by Lines deriving their Original from the Confideration of a Solid, that is, of a Cone. See Solid.
3 W Linear ones, to the Solution of which were required Lines more compounded. See Linear.
According to this Diftincrion, we are not to folve ft] id Problems by other Lines than the Conic Sections; efpecially if no other Lines but Right ones, a Circle, and the Conic Sections mufl: be received into Geometry.
But the Moderns advancing much farther, have received into Geometry all Lines that can be exprefs'd by Equations- and have diltinguifh'd according to the Dimenfions of the Equations, thofe Lines into Kinds; and have made it a Law, not to conftruct. a Problem by a Line of a Superior kind, that may be conftn.16r.ed by one of an inferior one. See Geometrical Line.
Geometrical 'Proportion, call'd alfo abfolutely and Am- ply, Proportion ; is a Similitude or Identity of Ratio's* See Ratio.
Thus, if A be to B, as C to D, they are in Geometrical Pro- portion : So 8, 4, 30 and 15 are Geometrical Proportionals. See.PRopoE.TiON.
Geome-
