OF SPAClf. 



17 



the radius and a line equal to the circum- 

 ference. 



Suppose the circle to be described by the revoUition of 

 the radius : the elementary triangle to which the fluxion of 

 the circle is proportional (l4l), is equal to the contempo- 

 raneous increment of the rectangle, of which the base is 

 equal to the circumference, and the height to half the radius : 

 consequently the whole areas are equal (47). 



144. Theorem. The circumferences of 

 circles are in the ratio of their diameters. 



Supposing the circles to be concentric, and to be de- 

 scribed by the revolution of different points of the same 

 right line, the ratio of the fluxions, and consequently that 

 of the whole circumferences,' will be the ratio of the 

 radii, or of the diameters (47). 



Scholium. The diameter of a circle is to its circum- 

 ference nearly as 7 to 22, and more nearly as 113 : 355, or 

 1 : 3.14159205359 ; hence the radius is equal to 57.29578° 

 =3437.74S7'=:2oa«04.8"; and, the radius being unity, 

 lO=:.017453293, l'=:.000290888, and l"=.OO0OO4848. 



145. Definition. A straight line is 

 perpendicular to a plane, when it is perpen- 

 dicular to every straight line meeting it in 

 that plane. 



146. Definition. A plane is perpen- 

 dicular to a plane, when all the straight lines 

 drawn in one of the planes perpendicular to 

 the common section, are perpendicular to 

 the other. 



147. Definition. The inclination of a 

 straight line to a plane is the angle contained 

 by that line, and another straight line drawn 

 from its intersection with the plane to the in- 

 tersection of a perpendicular let fall from 

 any point of the line upon the plane. 



148. Definition. The inclination of 

 two planes is the inclination of two lines, one 

 in each plane, perpendicular to the common 

 section. 



149. Definition. Parallel planes are 

 such as never meet, although indefinitely 

 produced. 



150. Definition. A solid angle is made 



VOL. II. 



biy the meeting of two or more plane angles, 

 in different planes. 



151. Definition. Similar solid figures 

 are such as have all parts of their surfaces si- 

 milar and similarly placed : and which have 

 all their sections, in similar directions, re- 

 spectively similar. 



152. Definition. A pyramid is a solid 

 contained by a plane basis and other planes 

 meeting in a point. 



153. Definition. A prism is a solid 

 contained by planes of which two that are 

 opposite, are equal, similar, and parallel, and 

 all the rest parallelograms. 



154. Definition. A cube is a solid 

 contained by six equal squares. 



155. Definition. A solid of revolu- 

 tion is that which is described by the revolu- 

 tion of any figure round a fixed axis. 



156. Definition. A sphere is described 

 by the revolution of a semicircle on its dia- 

 meter as an axis. 



157. Definition. A cone is a solid 

 described by the revolution of an indefinite 

 right line passing through a vertex and 

 moving round a circular basis. 



158. Definition. A cylinder is a solid 

 described by the revolution of a right angled 

 parallelogram about one side. 



159. Theorem. Two straight lines cut- 

 ting each other are in one plane. 



For a plane passing through one of them may be sup- 

 posed to revolve on it as an axis until it meet some point of 

 the other ; and then the second line will be wholly in the 

 plane (62). 



160. Theorem. If two planes cut each, 



other, their section is a straight line. 



For the straight line joining any two points of the section 

 must be in each plane (92), and must therefore be the 

 common section of the planes. 



161. Theorem. A straight hne, making 

 right angles with two other lines at the point 



D 



