The consequent inundation of ignorance and barbarism was tinfli•ou•able to geometry, as well as to theother sciences; and those who applied themselves to this science, or indeed to any In of learning incomprehensible by the vulgar. were calumniated as magicians. In those times of European darkness, the Arabians were distinguished as the guardians and promoters of science ; and from the ninth to the timrtecuth century they prodneed many astronomers, geometers, geographers, &e., from whom the mathematical sciences were again received into Spain, Italy, and other parts of Europe, somewhat before the beginning of the fifteenth century. Some of the earliest writers after this period, are Leonardus Pisenus, Lucas Paciolus or de Bingo. and others who flourished between 1400 and '1500. After this period appeared many editions of Euclid, or eonmentaries upon his Elements; Orontius Fineus, in 1530. published a commentary on the six first books; as did James Peletarins in 1557 ; and about the same time, Nkolas Tartaglia pub lished a commentary on the whole fifteen books. We might also mention other editions or commentaries; such as those of C'ommandine, l layius Seheubelins, I I ail inns, Dasypodills. Ramus, Ilerigon. Stevinus, Saville, Barrow, Tucquet, I )echales, Furrier, Scarborough, Keil I, Cann, Stone, and ninny others.
At the revival of letters, there were few Europeans capable of translating and commenting on the works of the ancient geometers; and geometry made consequently but little pro gress till the time of Des Cartes, who published Iris Geometry in 1037. however, not to mention all those who extended geometry beyond its elementary parts. such as Theodosius, in his ,,,oherics, Serenus, in his Sections of the Cone and Cylinder ; Kepler, in his _Ara Stereometria, &e.; in 1635, Bonaventure Cavalerins, an Italian, of the order of Jesuits, published his Geometry of ; Torricelli, his Opera Geometric(' ; Vivinni, his Divinationes Geomet•ice•, Exerci !alio Mathew: lien, De Loris Soli/is, De ilaximis et Minh/11s, &e. ; V e t a, EPctio Geometric', &e. ; Gregory St. Vincent, in 1647, published his treatise. intitled Quadra tura Cirenli et Hyperbola, a work 'hounding with excellent theorems and paralogisms; and Pascal. about the same time, published his Treutise of the Cycloid. Geometry. as far as it was capable of deriving aid and improvement from the arition etic4,1 in finites, was indebted to the labours of Fermat, Barony, Wallis. Jlereatur, Brounker. .1. Gregory, gees, and others ; to NI hum we may add Newton and Leibnitz. 13ut Sir Isaac Newton cont.' Hotted to the progress of pure geometry by his two treatises. be Quudratnra Curvarum, and Enumeratio Linearnm Thrill Ordinis : and still farther by his ineouqurable sod immortal work. honied, Philosophia ...Vaturatis Prineipia Jlathematica, which will always be considered as the most extensive and successfid application of geometry to physics.
The modern Geometers are innumerable ; and the names of ( Simpson, T. Stewart, T. Simpson. &e. not to mention living writers, will alway s be held in esteem and by those who are devoted to the study of geometry. and mathematics.
The province of geometry is almost infinite: few of our ideas but what may he represented to the imagination by lines, Which they become of geometrical consideration: it being geoinetry alone that makes comparisons and finds the of lines.
Architecture, mechanics, astronomy, music, and in a word, all the sciences which consider things susceptible of more and less, i. e. all the precise and accurate sciences, may be re ferred to geometry ; for all speculative truths consisting only in the relations of things, and in the relations between those relations, they may be all referred to lines. Consequences may be drawn from them ; and these consequences, again, being rendered sensible by lines, become permanent objects, which may be constantly exposed to a rigorous attention and examination : thus atrording to us infinite opportunities both of inquiring into their certainty, and pursuing them farther.
Geometrical lines and figures are not only proper to represent to the imagination the relations between magni tudes, or between things susceptible of more and less ; as spaces, times, weights, notions, &c., but they may even represent things which the mind can no otherwise conceive, for example, the relations of incommensurable magnitudes.
it must be observed, that this use of geometry among the ancients was not strictly scientilical, as among us; hut rather Syl b01 : they did not argue, or deduce things and pro perties unknown, from lines, but represented or delineated by them things that were known. In effect, they were not used as means or instruments of discovering, but as images or characters, to preserve, or communicate, the discoveries already made.
The ancient geometry was confined to very narrow bounds, compared with the modern. It only extended to right lines and curves of the first order, or conic sections; whereas in modern geometry new lines, of infinitely more and higher orders, are introduced.
Geometry is commonly divided into four parts, or branches; ALTIMETRY, STEREOMETRY, P1.ANIMETRY, and LONGLMETRY. See those word:.
It is again distinguished into theoretical or speculative, and practical. The first contemplates the properties of con tinuity ; and deimmstrates the truth of general propositions, called theorems. The second applies those speculations and theorems to particular uses in the solution of problems. Speculative geometry, again, may be distinguished into elenientarg and sublime. '1'lle former is that employed in the consideration of right lines and plane surfaces, and solids generated from them. The higher Or sublime geometry is that employed in the consideration of curve lines, conic sec tions, and bodies formed of them.