40 INTRODUCTION. 



scribe a prism less than any solid bearing a ratio assignably greater. 

 Such solids may not improperly be called cylindroids. 



180. Theorem. The fluxion of any solid, described 

 by the revolution of an indefinite line, passing" through a 

 vertex, and moving round any figure in a plane, is equal 

 to the prismatic or cylindroidal solid, of which the base is 

 the section parallel to the given plane, and the height the 

 fluxion of the height. 



In any increment of the solid, which is cut 

 oiF by planes determining the increment of the 

 height, suppose a prismatic or cylindroidal solid 

 to be inscribed, of which the base is equal to 

 the upper surface of the segment, and the sides 

 such that a line may always be drawn in them parallel to a given 

 line passing through the vertex and the basis of the solid : and let 

 another solid be similarly described on the lower surface of the seg- 

 ment as a basis : then it is obvious that the increment is always 

 greater than the inscribed solid, and less than the circumscribed; 

 and that when the increment is diminished without limit, its two sur- 

 faces are ultimately in the ratio of equality, and the increment coin- 

 cides with the cylindroid described on its basis. Such solids may 

 be termed in general pyramidoidal. 



181. Theorem. All pyramidoidal solids are equal to 

 one third of the circumscribing prismatic or cylindroidal 

 solids of the same height. 



The area of each section of such a tigure, parallel to the basis, is 

 proportional to the square of its distance from the plane of the ver- 

 tex. For each section is either a polygon similar to the basis, or it 

 may have polygons inscribed and circumscribed, which are similar 

 to polygons inscribed and circumscribed in and round the basis, and 

 which may differ less from each other in magnitude than any assign- 

 able quantity, consequently each section is as the square of any ho- 

 mologous line belonging to it, or, by the properties of similar tri- 

 angles, as the square of the distance from the vertex, or from the 

 plane of the vertjex. If, then, -the area of the base be a, the whole 

 height bf and the distance of any section from the plane of the vertex 



