OF SPACE. 



37 



A M O 



H 



D B 



The opposite sides of all the figures are parallel, because they are 

 the sections of one plane with two parallel planes (168) : the corres- 

 ponding sides of two opposite planes being, for the same reason, pa- 

 rallel to each other, contain equal angles (165), and they are also 

 equal, as being the opposite sides of parallelograms ; consequently 

 the opposite figures are the doubles of equal triangles, and are, there- 

 fore, equal parallelograms. 



171. Theorem. If a prism be divided by a plane, pa- 

 rallel to its two opposite surfaces, its segments will be to 

 each other as the segments of any of the divided surfaces 

 or lines. 



Let the prism AB be divided by 

 the plane CDE parallel to AFG and 

 BHI. Find FK a common mea- 

 sure of FD and DB(119), make KL 

 =FK, and let the planes KMN, LOP be parallel to AFG; then the 

 prisms AK, ML may be showii to be contained by similar and equal 

 figures similarly situated, in the same manner as it is shown of paral- 

 lelepipeds, and there is no imaginable difference between these 

 prisms: they are therefore equal; and the prism AD is the same 

 multiple of AK that FD is of FK, and AB the same multiple of AK 

 thatFB isofFK,or AD ; AKzrFD : FK, and AB : AK=:FB : FK, 

 whence AD ; ABzzFD ; FB, and the prisms are in the same ratio as 

 the segments of the line FB, or of the parallelogram GB (27), 



If the segments are incommensurable, they are still in the same 

 ratio, for it may be shown that the ratio of the prisms is neither 

 greater nor less than that of the lines. 



172. Theorem. Parallelepipeds on the same base, 

 and contained between the same planes, are equal. 



The parallelepiped AB is equal 

 to CD standing on the same base 

 BC, and terminated by the plane 

 A ED. For each is equal to the 

 parallelepiped EF ; since the trian- 

 gular prism GB is similar and equal 

 to the triangular prism HC, and 

 deducting these from the solid HCI, 



