34 INTRODUCTION. 



157. Definition. A cone is a solid described by Ibe 

 revolution of an indefinite right line passing through a ver- 

 tex, 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 catting each other 

 are in one plane. 



For a plane passing through one of them may be supposed to re- 

 volve on it as an axis until it meet some point of the other; and tlien 

 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 (62), and must, therefore, be the common section of 

 the planes. 



161. Theorem. A straight line, making right angles 

 with two other lines at the point of their intersection, is at 

 right angles to the plane passing through those lines. 



B Let AB be perpendicular to CD and EF 



intersecting each other in A : take AC at 

 pleasure, and make ACrzADzzAEnAF ; 

 draw through A any line GH, and join 

 CE,DF; then the triangles ADH, ACG 

 are equal and equiangular, AHnAG and 

 DH=:CG; but since the triangles CBE, 

 DBF, are equal, and equiangular, the angles BCG and BDH are 

 equal, and the triangle BCGzrBDH, BGnBH, and the triangles 

 ABG, ABH, are equal and equiangular : consequently the angle 

 BAGzzBAH, and both are right angles: and the same may be 

 proved of any other line passing through A ; tlierefore AB is perpen- 

 dioular to the plane passing through CD and EF (145). 



