GEO
C Hi )
GEO
The fecond applies thofe Speculations and Theorems to particular Ufes in the Solution ofProblems. Sec Problem. Speculative Geometry, again, may be diftinguiih'd into Elementary, and the Sublimer.
Elementary, or Common Geometry, Is that employed in the consideration of Right Lines, and plain Surfaces, and Solids generated therefrom. See Plain, &c.
The Higher, or Sublimer Geometry is that employ 'd in the consideration of Curve Lines, Conic Sections, and Bodies form'd thereof. See Curve, &c.
Herodotus, lib. II. and Strabo, lib. XPII. affert the Egyp- tians to have been the firft Inventors of Geometry ; and the annual Inundations of the Nile to have been theOccafion: For that River bearing away all the Bounds and Land- marks of Mens EfSares, and covering the whole Face of the Country, the People, fay they, were obliged to diStinguifh their Lands by the consideration of their Figure, and Quan- tity ; and thus by Experience and Habit, form'd themfelves a Method, or Art, which was the Origin of Geometry. — -A farther Contemplation of the Draughts or Figures of Fields thus laid down and plotted in Proportion, might naturally enough lead them to the Difcovery of fomc ot their excel- lent and wonderful Properties 5 which Speculation continu- ally improving, the Art became gradually improved, as it continues to do to this Day. jfofephus, however, feems to attribute the Invention to the Hebrews : And others among the Antients make Mercury the Inventor. Polid. Virgil, de Invent. Rer.L.l. C. 18.
The Province of Geometry is almoft infinite: Few of our Ideas, but may be reprefented to the Imagination by Lines, upon which they Straight become of Geometrical Considera- tion ; it being Geometry alone that makes Comparifons, and finds the Relations, of Lines. See Line.
Aftronomy, Mufic, Mechanicks, and, in a Word, all the Sciences which confider Things fufceptible of more, and lefsj i. e. all the prccife and accurate Sciences, may be refer'd to Geometry : For all Speculative Truths only confifting in the Relations of Things, and in the Relations between thoSe Re- lations, they may be all referred to Lines. Confequenccs may be drawn from them ; and thefe Confequcnces, again, being render'd fenfible by Lines, they become permanent Objects, constantly expofed to a rigorous Attention and Exa- mination : And thus we have infinite Opportunities both of enquiring into their Certainty, and purfuing them further. See Art, and Science.
The Reafon, for inftance, why we know fo diftinctly, and mark fo precifely, the Concords call'd Octave, Fifth, Fourth, f$c. is, that we have learnt to exprefs Sounds by Lines, i. e. by Chords accurately divided 5 and that we know that the Chord, which founds Octave, is double of that which it makes Octave withal 5 that the Fifth is in the fofauialterata Ratio, or as three to two ; and io of the reft.
The Ear it felf cannot judge of Sounds with fuch Pre- cision; its Judgments are too faint, vague, and variable to form a Science. The fineSt, befi tuned Ear, cannot diftin- guiSli many of the Differences of Sounds ; whence many Muficians deny any fuch Differences ■ as making their Senfe their Judge. Some, for inltance, admit no Difference be- tween an Octave and Three Ditones : And others, none be- tween the greater and leffer Tone; So that the Comma, which is the real Difference, is infenfible to them 5 and much more the Schifma, which is only half the Comma.
'Tis only by Reafon, then, that we learn, that the Length of Chord which makes the Difference between certain Sounds, being divifible into feveral Parts, there may be a great Num- ber of different Sounds contained therein, ul'eful in Mufic, which yet the Ear cannot diftinguifh. Whence it follows, that had it not been for Arithmetic and Geometry, we had had no fuch thing as regular, fix'd Mufic ; and that we could only have fucceeded in that Science by good Luck, or Force of Imagination, i. e. Mufic would not nave been any Science founded on incontestable Demonstrations : Tho' we allow that the Tunes compofed by Force of Genius and Imagination, are ufually more agreeable to the Ear, than thofe compofed by Rule. See Sound, Tune, Gravity, Concord, £<?£■
So, in Mechanicks, the Heavinefsofa Weight, and the Diftance of the Centre of that Weight from the Fulcrum, or Point it is fuftained by, being fufceptible of plus, and minus they may both be exprefs'd by Lines 5 whence Geo- metry becomes applicable hereto 5 in virtue whereof, in- finite Difcoveries nave been made, of the utmoft ufe in Life. See Balance, Stilyard, £f>c.
Geometrical Lines and Figures, are not only proper to re- prefent to the Imagination the Relations between Magni- tudes, or between Things fufceptible of more and lefs 5 as Spaces, Times, Weights, Motions, ££c. but they may even reprefent Things which the Mind can no otherwife con- ceive ; e.gr. the Relations of incomenfurable Magnitudes. Sec Incommensurable.
We do not, however, pretend, that all Subjects Men may have occafion to inquire into, can be exprefs'd by Lines.
There are many not reducible to any fuch Rule : Thus, the Knowlege of an infinitely powerful, infinitely juft GOD, ofi whom all Things depend, aud who would have all his Crea- tures execute his Orders to become capable of being hap- py, isthe Principle of all Morality, from which a thotifand undeniable Confequcnces may be drawn, and yet neither the Principle, nor the Confequences can be exprefs'd by Lines, or Figures. Mallebranche Recherche de la Verite, 5f. II. Indeed, the antient Egyptians, we read, ufed to exprefs all their Philofophical, and Theological Notions by Geome- trical Lines. In their Relearches into the Reafuri of Things; they obferv'd, that GOD, and Nature afreet Perpendicu- lars, Parallels, Circles, Triangles, Squares, and harmoiii- cal Proportions 5 which engaged the Priefts and Philofopbers to reprefent the Divine and Natural Operations by fuch Fi- gures : In which they were followed by Pythagoras, Plato^ &c. Whence that Saying of Boethius, Nullum 'Divinoruni fcientiam iysa^irelfay attingere poffe. See Platonic, Py- thagorean, See:
But it muft be obferv'd, that this Ufe oiGeometry among the Antients was not Strictly Scientifical, as among us 4. but rather Symbolical: They did not argue, or deduce Things and Properties unknown from them; but reprefented or de- lineated Things that were known. In effect, they were not ufed as Means or Instruments of difcovering, but Images of Characters, to preferve, or communicate the Difcoveries made. See Symbol, and Hieroglyphic.
" The Egyptians, (Gale obferves) ufed Geometrical Fi- " gures, not only to exprefs the Generations, Mutations; " and Destructions of Bodies ; but the Manner, Attributes,- " &c. of the Spirit of the Univerfe, who diffufing himfelf " from the Centre of his Unity, thro' infinite concentric " Circles, pervades all Bodies, and fills all Space. But of " all other Figures they moSt affected the Circle, and Tri- " angle 5 the firff, as being the molt perfect, Simple, capa- " cious, &c. of all Figures : Whence Hermes borrowed it to " reprefent the Divine Nature ; defining God to be an in* " tellectual Circle or Sphere, whole Centre is every where, 11 and Circumference no where." See Kirch. Oedip. JEgy- ptiac. and Gale Phil. General. Lib. I. c. II.
The Antient Geometry was confined to very narrow Bounds^ in com pari Ton of the. Modern. It only extended to Right Lines and Curves of* the firft Order, or Conic Sections j whereas into the modern Geometry new Lines of infinitely more and higher Orders are introduced. See Curve.
The Writers, who have cultivated and improved Geome* try, may be diftinguifh'd into Elementary, Practical, and thofe of the fublimer Geometry.
The principal Writers of Elements , fee enumerated un- der the Article Elements.
Thofe of the Higher Geometry are Archimedes, in hi* Books de Sphara, Cylindro, and Circuit 1)ir>ienfione 3 as alio de Spiralibus, Convidibus, Spheroi dibits, de ^uadratura 'Parabola'; and Arenarius-. Kepler, in his Stereometric Nova $ Cavalcrius, in his Geometria ludivifibilhtm ; and 1'orricellius, de Solidis Sphsralibus 5 Pappus Alcxandrimis^ in Colletlionibus Mathematicis ; Paulus Guldinus, in his Me- chanicks and Staticks j Barrow, in his LcBiones Geometri- es ; Huygens, de Circuit Magnitudinc ; Sulliatdits, de Lh nets Spiralibus ; Schcoten, in his HExercitationes Mathema- tics ; de 'Billy, de c Proportione Harmonica 5 Lalovera, de Cycloide ; Fer. Ernefi. Com. ab Herbenftei/^ in Z)iateme Ctrculoritm 5 Viviani, in Exercit. Mat'hemat. de Forma- tione, and Menfura Fomicum ; Bap. Pahna, in Gecmet.Ex- ercitation. and A poll. Pergmis^ de Sefiione Rationis.
The Writers on the Seatons of the Cone, and Sphere ; fee under Conicks, and Sphericks.
For 'Practical Geometry, the fulleft and compleateft Trea- tifes are thofe of Mallet, written in French ; but without the Demonstrations: And thofe .of Schtventer, and Cantzlerus % both in High-Z)mch.— In this Clafs are likewife to be rank'.j Clavius\ Tacquet\ and Ozmam's Practical Geometries ;' iDe la Hire's Ecole des Arpenteurs ; Reinholdus's Gccde/ia- 7 Hartma?i Beyer's Stcreometria j Vo'igtel\ Geometria Sub- terranca, all in High-Dutch ; Huljiits, Gallileus, Goldman- nits, Scheffelt, and O&anctm, on the Setlor.—
GEOMETRICAL, fomething that has a relation to Geo- metry. See Geometry,
Thus we Say, a Geometrical Method, a Geometrical Ge- nius, a Geometrical Strictnefs, a Geometrical Ccntf-ruction, and Geometrical Demonftration, &c. See Demonstra- tion, &c.
Geometry it felf leads us into Errors : After once reducing, a thing to Geometrical Consideration, and finding that it an- fwers pretty exactly, we purfue the View, are pleas'd with the Certainty and Agreeablenefs of the DemOnilrationE - 7 and apply the Geometry further and further, till we outrun Nature. Hence it is, that all Machines do not fucceed : That all Compositions of Mufic, wherein the Concords are the moSt rigidly obferv'd, are not agreeable : That the moft exact AStronomical Computations do not always foretel the precife Time and Quantity of an Eclipfe.
