20 



OF SPACE. 



The diagonals are paraJlel, because the lines in which 



they terminate are parallel and equal, and every line and 



C angle of the one prism is equal to the 



corresponding line and angle of the other 



prism ; consequently the prisms are 



^^ equal. Thus AB=CD, AE=CF, DE= 



BF, the angle EAB=DCK, EAH=:GCF, 



and BAH=DCG. 



178. Theorem. Prisms are to each oilier 

 in the joint ratio of their bases and their 

 heights. 



Triangular prisms are in the same ratio as the paral- 

 lelepipeds on bases twice as great, of which they are the 

 halves; and all prisms may be divided into triangular 

 prisms, by planes passing through lines similarly drawn 

 on their ends, and they will be equal together to the 

 half of a parallelepiped on a basis twice as great ; conse- 

 quently two such prisms are in the same ratio as the pa- 

 rallelepipeds. 



179. Theorem. All solids of which the 

 opposite surfaces are planes, and the sides 

 such that a straight line may be drawn in 

 them from any point of the circumference 

 of the ends parallel to a given line, are to 

 each other in the joint ratio of their bases 

 and their heights. 



For if they are terminated by rectilinear figures, the 

 solids are prisms ; and if they are terminated by curvilinear 

 figures, they will always be greater than prismatic figures, 

 of which the bases are inscribed polygons, and less than 

 figures of which the bases are circumscribed polygons; and 

 if the proposition be denied, it will always be possible to 

 inscribe a prism in one of the solids which shall be greater 

 than any solid bearing to the other solid a ratio assignably 

 less than the ratio determined by the proposition, and to 

 circumscribe 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 incremen t of the solid, 

 which is cut off by planes determin- 

 ing the increment of the height, sup- 

 pose 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 anothtr solid be similarly described on the 

 lower surface of the segment 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 in- 

 crement coincides 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 figure parallel to the 

 basis, is proportional to the square of its distance from the 

 plane of the vertex. 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 assignable quan- 

 tity,consequentlyeach section is as the square of any homo- 

 logous line belonging to it, or, by the properties of similar 

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

 from the plane of the vertex. If then the area of the base 

 be a, the whole height b, and the distance of any sec- 

 tion from the plane of the vertex x, the area of the section 



will be — .a, and the fluxion of the solid ttx'x, of which 

 bo bb 



a 

 the fluent is i — x', and when xzzb, the content is lax, 

 bb 



which is one third of the content of the whole prismatic or 

 cylindroidal solid. Hence a pyramid is one third of the cir- 

 cumscribing prism, and a cone one third of the circum- 

 scribing cylinder. 



182. Theorem. The fluxion of any solid 

 is equal to the parallelepiped of which the 

 base is equal to the section of the solid, and 

 the height to the fluxion of its height. 



For every part of a solid may be considered as touching 

 some pyramidoidal solid, and having the same fluxion : 



