GEOMETRY. 



" The Egyptians," Gale obferves, " ufed geometrical 



figures, not only to exprefs the generations, mutations, and 

 dcftriiftions of bodies ; but the manner, attributes, &c. of 

 the fpirit of the univerfe, who, diftufing himfelf from the 

 centre of his unity, through infinite concentric circles, per- 

 vades all bodies, and fills all fpacc. But of all other figures 

 they mo fl afFefted the circle and triangle; the firl\, as 

 being the moll perfeft, finiple, capacious, &e. of all figures : 

 whence Hermes borrowed it to reprefent the divine nature; 

 defining God to be an intelleftiral cii'cle or fphere, wliote 

 centre is everywhere, and circumftrence nowhere." See 

 Kirch. CEdip. ^Egyptiac. aixl Gale Phil. General, lib. ii. 

 cap. 2. 



The ancient geometry was confined to very narrovv' bounds, 

 in comparifon of the modcrii. It only extended to right 

 lines- and curves of the firft order, or conic feftions ; 

 ■whereas into the modern geometry new lines of infinitely 

 more, and higher orders are introduced. 



Geometry is commonly divided into four parts, or 

 brandies ; planimetry, altimctry, longimetry, and itereome- 

 try ; which fee refpeitively. 



Geometry, again, is dillinguiflied into thcorctii-al or fpccu- 

 lat'nv, and prdilical. 



The firll contemplates the properties of continuity ; and 

 dcmonftrates the truth of general propofitions, called theo- 

 rems. 



■ The fecond applies thofe fpeculations and theorems to par- 

 ticular ufes in the folution of problems. 



Geometry, fpecuLlivc, again may be diftinguifhed into 

 tlcmentary and fui/inte. 



Geometry, eL'tnentary or common, is that employed in the 

 eonfidcration of right lines, and plane furfaccs, and folids 

 generated from them. 



Geometry, higher, or fublime, is that employed in the 

 confideration of curve lines, conic feftions, and bodies 

 fiormed of them. 



The writers who have cultivated and improved geometry 

 may be diftinguifhed into elementary, prattical, and thole of 

 the fublimer geometry. 



The principal writers of elements, fee enumerated under 

 Elements. 



Thofe of the higher geometry are Archimedes, in his 

 books De Sphasra, Cylindro, and Circuli Dimenfione ; as 

 alfo De Spiralibus, Conoidibus, Sphaeroidibiis, De Quadra- 

 tura Parabols, and Arenarius : Kepler, in his Stercometria 

 Nova ; Cavalerius, in his Geometria IndivifibiUum ; and 

 Torricellius, De Solidis Sph;erahbus ; Pappus Alexan- 

 drinjis, in Colleftionibus Mathematicis ; Paulus Guldinus, 

 in his Mechanics and Statics ; Barrow, in his Lcdliones Geo- 

 metrica; ; Huygens, De Circuli Magnitudine ; Bullialdus, 

 De Lineis Spiralibus ; Schooten, in his Exercitationes Ma- 

 thematics ; De Billy, De Proportione Harmonica ; Lalo- 

 ▼era, De Cycioide, For. Erneft. Com. ab Haibenflein, in 

 Diatonic Circulorum ; Vivjani, in Exercit. Mathcinat. de 

 Eorniutione & Menfura Fornicum ; Bap. Palma, in Geomet. 

 Exercitation. and Apoll. PergKus, De Seftionc Rationis. 



For praftical geometry, the fulleft and completeft trea- 

 tifes are thofe of Mallet, written in French, but without the 

 demonftrations ; and thofe of Sehwentcr and Cantzlerus, 

 both in High Dutch. In this clafs are like wife to be 

 ranked Clavius's, Tacquet's, and Ozanam's Praitical Geo- 

 netries ; De la Hire's Ecole des Arpenteurs ; Reinholdus's 

 Gcodjctia ; Hartman Bcyers's Stereometria ; Voigtel's 

 Geometria Subterranea ; all in High Dutch : Hulfius, Ga- 

 llleus, Goldmannus, Scheffelt, and Ozanam, on the Scftor. 

 &c. &c. &c. 



Xlw fcience of geometry is founded on certain axvoms, or 



felf-evident truths (fee Axiom) ; it is introduced by defini- 

 tions of the various objefts which it contemplates, and the 

 properties of wliich it invelligates and demonitrates, fuch as 

 points, lines, angles, figures, furfaces, and folids : — lines 

 again are confidered as llraight or curved; and in their re- 

 lation to one another, either as inclined or parallel, or as per- 

 pendicular : — angles as riglit, obhqUe, acute, obtufe, external, 

 vertical, &c. : — figures, with regard to their various boun- 

 daries, as triangles, which are, in refpetl: of their fides, 

 equil ifertil, ifofceks, and Icalene, and in reference to their 

 an»'les, rio;ht-anglcd, obtule-angled, and acate-angled ; as 

 quadrilaterals, which comprcliend the panillelogram, in- 

 cluding the reftangle and Iquaie, the rliombus and rhom- 

 boid, and the trapezium and trapezoid ; as multilaterals or 

 polygons, comprehending the pentagon, hexagon, heptagon, 

 &c. ; and as circles : — and as folids, including a prifm, 

 parallckpipcdon, cube, pyramid, cylinder, cone, fphere, and 

 the frallum of either of the latter. We ihall not here at- 

 tempt to compile a complete fyilem of geometry, as it 

 would occupv too many of our pages, in a work from its 

 nature protracted and enlarged to a very great extent ; and 

 this is the lefs necefiary, becaufe the reader will find under 

 the titles above enumerated, and others naturally connected 

 witli and derived from them, the moil efiential and important 

 principles of geometry, together with the operations that 

 are founded upon them ; and becaufe any perfon who is de- 

 firous of acquainting himielf with the icience of geometry, 

 in its whole extent and application, wilUiave recourfe to one 

 or other of thofe numerous treatifes, in a more enlarged or 

 more compendious form, which may be eafily procured. 

 The Elements of Euclid by Dr. R. Simpfon occur firfl to 

 our recolleftion, and deferve particular recommendation ; 

 but the objeft of the geometrical iludent may be fatis- 

 fatlorily attained by T. Simpfon's Geometry, or by the trea- 

 tifes of Emerfon, Hutton, Bonnycaille, Leilie, Slc. &c. 



But as the analytic method of treating geometrical quef- 

 tions is lefs generally known, and as complete treatifes on 

 this fubjeft are only to be found in fcu-cign works, we have 

 been induced to devote a confiderable fpace to this part of 

 the fcience ; the following treatife is chiefly comjiiled from 

 the " Feuilles d' Analyfe'' by Monge, wliieh were puMiflied in 

 feparate portions for the ufe of the polytechnic fchool, and 

 afterwards collected in a quarto volume. A more elemen- 

 tary work has lately been publidied by Garnier in oftavo, 

 to which the reader is referred. 



Geomjjtry, Analytic. — Method of defining the pofition 

 cf a point in a plane. 



A point M (y/n.2A;y7j, PlateMWl.fg. I.) is defined by re- 

 ferring its pofition to two lines, as A Y, A X, generally at 

 right angles to each other, but they may be inclined at any 

 given angle. 



If M Q be drawn perpendicular to A Y, and M P per- 

 pendicular to A X, then O M, M P, are called the co-or- 

 dinates of the point M ; the diftance of the point from 

 A Y is ufually denoted by .t:, and its dillance from A X 

 by V. 



The point of interfeftion of the two lines AY, AX \% 

 called the origin of the CK-ordinates, and the lines A Y, 

 A X, produced each way io Y' and X', are called axes. 



If the diflance of the point M from tliefe axes is given, 

 ■u/'s. M Q = <7, M P ^ ^, then x — a, y — l\% the equation 

 to the point M. 



But if the point M be fituated in any other of the a'.ijles, 

 the fign of « and h will vary, and thefe variations are governed 

 by the fame rules, as tlie fines and cofines in in<orio- 

 metry. 



For inftance, if the point M is fituated in the angle 

 J Y -A. X, 



