OF SPACE. 39 



178. Theorem. Similar parallelepipeds are in the 

 triplicate ratio of their homologous sides. 



For the joint ratio of the bases and heights is the same as the tri- 

 plicate ratio of tlie sides. 



177. Theorem. A plane, passing through the dia- 

 gonals of two opposite sides of a parallelepiped, divides it 

 into two equal prisms. 



The diagonals are parallel, because the lines in which they tenul- 

 nate are parallel and equal, and every line and 

 angle of the one prism is equal to the correspond- 

 ing line and angle of the other prism ; consequent- 

 ly the prisms are equal. Thus ABirCD, AE=CF, 

 DE=BF, the angle EAB=:DCF, EAH=:GCF, 

 and BAH=:DCG. 



178. Theorem. Prisms are to each other in the joint 

 ratio of their bases and their heights. 



Triangular prisms are in the same ratio as the parallelepipeds 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 tlieir 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 parallelepipeds, 



179. Theorem. All solids, of which the opposite sur- 

 faces 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 in- 

 scribed polygons, and less than figures of which the bases are cir- 

 cumscribed 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 

 \e&% than the ratio determined by the proposition, and to circum- 



