GEOMETRY. 327 



CHAPTER HI. 



GEOMETRY. 



GEOMETRY, is that branch of Mathematics which treats of the 

 measurement of space, and the properties of lines, surfaces, and 

 solids. The name is derived from the Greek, y^, the earth, or land, 

 and petpov, a measure : and this science was thus designated from its 

 early application to the measurement of land. Under this branch, 

 we would also include the modern Descriptive Geometry ; for rea- 

 sons already given in the introduction to the present department. 

 The science of Geometry, is one of the most beautiful, as it is also 

 one of the most useful, among the exact sciences ; and, from its 

 frequent applications, in all the arts of construction, it is a branch 

 which we think should be studied by every mechanic, if not gene- 

 rally introduced into our common schools. 



The origin of Geometry, is ascribed by some writers to the Hin- 

 doos : but by others, as Herodotus, to the Egyptians, who employed 

 it in retracing their landmarks, after each subsidence of the Nile. It 

 was introduced into Greece by Thales, and his pupil, Pythagoras, 

 both of whom travelled in Egypt. Thales discovered that all angles 

 inscribed in a semi-circle are right angles ; and Pythagoras, besides 

 noticing the five regular solids, discovered that the square on the 

 hypothenuse of a right angled triangle is equal to the sum of those 

 on the two sides. Hippocrates of Chios, by the quadrature of his 

 famous lunulse, was the first to discover the exact area of a curvili- 

 near figure ; 450 B. C. Eudoxus, the friend of Plato, found the 

 measure of the pyramid and cone ; and Archimedes of Syracuse, that 

 of the sphere and its circumscribed cylinder, which were sculptured 

 on his tomb. This involved the quadrature of the circle, towards 

 which Archimedes gave the first approximation. The two famous 

 problems, in the Platonic School, of the trisection of an angle, and 

 the duplication of the cube, led to the invention of geometrical loci : 

 and the spiral of Conon, the quadratrix of Dinostratus, the conchoid 

 of Nicomedes, and the cissoid of Diocles, are curves having reference 

 to these problems. 



Among the best ancient works on Geometry, were the Mathema- 

 tical Collections of Pappus; and Euclid's Elements; which were 

 first translated from the Arabic into Latin, by Adhelard, an English 

 monk of the 12th century. The Arabians seem to have made no 

 advances in Geometry ; though Mahomet of Bagdad wrote an 

 original work on Mensuration. Gerbert, already mentioned in the 

 introduction to this department, also wrote a treatise on Mensuration; 

 and the first printed treatise on Algebra, by Paccioli, already referred 

 to, related in part to Geometry. Van Ceulen of Cologne, who died 

 in 1610, calculated the ratio of the circumference to the diameter of 

 a circle as far as to 36 places of figures : and Albert Girard, another 

 Fleming, first found the area of a spherical triangle. Descartes first 



