GEOMETRY. 



Arcliiinedps's mcnrui-ation of a circle ; and tJi* commentary not to mi-ntion livinp^ writers, will aUrays be held in eft'eem 

 on Euclid, by I'rocliis, who lived under the empire of and veneration by thofL- that are devoted to the ftudy of 



geometry and mathematics. " " 



Anatlafius. 



The confequent inundation of ignorance and barbarifm 

 was unfavourable to geometry as well as to the other fci- 

 eiices ; and thofe few who applied themfelvcs to this fcience, 

 &c. were calumniated as majficians. However, in thofe 

 times of European darknefs, the Arabians were diltinguilhed 

 as the guardians and promoters of fcience ; and from the 

 ninth to the fourteenth century they produced many ailrono- 

 mers, geometers, geograj)hers, &c. from whom the mathe- 

 matical fciences were again received into Spain, Italy, and 



See Ei.EMKNTf;. 



The province of geometry is alraoft infinite : few of our 

 ideas but may be reprefented to the imagination by lines, 

 upon which they become of geometrical coiifideration : it 

 being geometry alone that makes comparifons, and finds the 

 relations of !ines. 



Ailronomy, mufic, mechanics, and, in a word, all the 

 fciences which confider things fufceptiblc of more and lefs, 

 /. e. all the prccife and accurate fciences, mar be referred to 

 geometry ; for all fpeculative truths confiding only in the 



other parts of Europe, fomewhat before the beginning of relations of things, and in the relations between thofe rela 

 the 1 5tli century. Some of the earllell writers after this tions, they may be all referred to lines. Confequencea re.-.y 

 period are Leonardus Pifenus, Lucas Paciolus or de Burgo, be drawn from them ; and thefe confequenccs, again, behijr 

 and otliers between 1400 a'l J 1500. After this period rendered fenfible by lines, they become permanent obieds 

 appeared many editions of Euclid, or commentaries upon which may be conllantly cxpofed to a rigorous attention and 

 his Elements; e.g. Orontius Fineus, in 1530, publilhcd a examination: and thus we have infmite opportunities both 

 commentary on the fix firll books ; as did James Peletarius of enquiring into their certainty, and purfuing them farther. 

 in 1557 ; and about the fame time Nicholas Tartaglia pub- The realon, for inftance, why we know fo diftinftly, and 

 liflied a com;nentary on the whole 15 books. We might mark fo precifely, the concords called oSa-oe, fj'lh, fourth, 

 here menlinn other editions or commentaries ; fuch are thofe &c; is that we have learnt to cxprcfs founds by lines, /. e. 

 of Commandine, Clavius, Billingdy, Scheubehus, Harli- by chords accurately divided ; and that we know, that the 

 nus, Dafypodius, Ramus, Herigon, Stevinus, Saville, Bar- chord, which founds ottave, is double of that with which 

 row, Tucquet, Dechnles, Furnier, Scarborough, Keill, it makes octave ; that the fifth is the fefquialterate ratio, or 

 Cann, Stone, and many others. (See Elkments ) as three to two ; and fo of the reft. The ear itfelf cannot 



At the revival of letters, there were few Europeans capable judge of founds with fuch a nice precifion ; its judgments 

 of tranllating and commenting on the works of the ancient are too faint, vague, and variable, to form a Icience. The 

 g-eomefrs; and geometry made little progrcfs till the time of fined beft-tutied ear cannot dillinguifh many of the differences 

 Des Cartes, who publilhed his Geometry in 1637. However, of founds; whence many muficians deny any fuch di.Terences • 

 not to mention all thofe wlio extended geometry beyond its as making this fenfe their judge. Some, forin'lance, admit 

 elementary parts, fuch as Theodofius in his Spherics, Se- no difference between an octave and three ditones: and 

 renus in his feftions of the cone and cylinder, Kepler in his others, none between the greater and leflertonc; the comma. 

 Nova Stereonietria, &c.; in 1635, Bonaventure Cavalerius, ^n which is the real difference, is infcnfible to them ; and much 

 Italian of the order of .lefuits, publilhed his " Geometry of more the fchifma, wliich is only half the comma. 

 Indivifiblcs ;" Torricelli his " Opei-a Geometrica ;" Viviani, It is only by reafon, then, that we learn, that the length 

 his "Divinationes Geometries," "ExercitatioMathematica," of the chord which makes the difference between certain 

 "De I^ocisSolidis," "De Maximi J et Minimis," &c.; Vieta, founds being divihble into feveral parts, there may be a 

 his "Effectio Geometrica," &c. ; Gregory St. Vincent, in great number of different founds contained therein, ufeful 

 1647, publilhed his treatife, entitled " Quadratura Circuli & in mufic, which yet the ear cannot dillinguilh. Whence it 



Hyperbi:' " '" ' ^ " '' "''' '^ '" " .1 . ■ 1 ■ 



and para 



could only have fucceeded'in that art by good luck, or force 

 of imagiivalion ; i.e. mufic would not ha-.e been any fcience 

 founded on incontellibie demonilrations : though we allow, 

 that the tunes compofed by force of genius and imagination, 

 are ufually more agreeable to the ear than thofe compofed 

 by rule. So, in mechanics, the htavinefs of a weight, and 



erbola," a work aboundmg with excellent theorems follows, that had it not been for arithmetic and geometry, we 

 paralogifms; and Pafcal, about the fame time, pub- iliouldhavehadnofuchthingasregularfixedmufic ; andthatwe 

 lifiied his treatife of the cycloid. Geometry, as far as it 1 ■ > • -■ .... 



was capable of deriving aid and improvement from the arith- 

 metic of infinites, was indebted to the labours of Fermiit, 

 Barrow, Waliia, Mercator, Brounker, .1. Gregory, Huy- 

 gens, and others, to whom we may add Newton and Leih- 

 (See Flu.xion's.) 



mtz. 



But fir ifaac Neu'ton contributed 



to the progrefs of pure geometry by his two treatifes, " De the dillance of the centre of that weight from the fulcrum, 



Quadratura Curvarum," and " Enumcratio Lincarum Tertii or point by which it is fuftained, being tufccptible of plus and 



Ordinis" (fee Ccuvk): and ftill farther by his incompara- minus, they may both be expreffed by lines: whence geonirtrr 



ble and immortal work, entitled " Philolopliiae Naturahs becomes applicable to this fcit nee; in virtue of which, infinite 



Principia Mathemalica," which will always be confidered difcoveries have been made, of the utmoft ufe in life, 



as the moft extenfive ar.d foccefsful application of geometry Geometrical lines and figures are not only proper to re- 



to phyfics. ^Ve cannot forbear tranlcribing in this place prcfent to the imagination the relations between magnitudes, 



the compliment paid to this author by the editors of the or between things fufceptiblc of more and lefs ; as fpaces. 



Encyclopedic, «ho, confidoring the various monuments of times, weights, motions, &c. but they may even reprefcnt 



the author's genius, and that he had made his principal thing.s which the mind can no otherwifc conceive, e. vr. the 



difcovery before the age of twenty-four, are tempted, they relations of incommenfurable magnitudes, 



fay, to fubfcribe to the words of Pope, that the fagacity of It muft be obferved, that this ufe of geometry amoijg the 



Newton ailonilhes even celeilial intelligences, and that they ancients was not llric'Uy fcientifical, as anion.'- us ; but rather 



contemplate him as a being occupying a kind of middle fymbolical : they did not argue, or deduce things and pro- 



Itation between man and themlclvcs ; or at leall they cannot pcrties unknown, from lines ; but reprefented or dei'r.catcd 



forbear exclaiming, homo hominl quid pi-iejlat ! what a dillance by them things that were known. In effccl, they were iKit 



doe 1 there fubi.ll between one man and another ! ufed as means or inllruments of difcovering, but as ima-res 



The modern geometers are innumerable ; and the names or characters, to preferve, or commiuiicate, tlie tiifcovcries 



•fCule.i, Maclauriuj R.Simfon, T.Stewart, T.bimpfun, &c. thjit were already made, 



Pz ••Tk« 



