3S INTRODUCTION. 



the remainders AB and EF are equal. And in the sani€ manner tt 

 may be shown that CDizEF ; therefore ABr:CD. 



173. Theorem. Parallelepipeds on equal bases, and 

 of the same height, are equal. 



^ P Each parallelepiped is equal to the 



p^ ^^^~^\ €!rect parallelepiped on the same base, 



llll^ plane of one of the sides AB may coin- 



A ^ cide with the plane BC of the other 



parallelepiped CD, and that EBC may be a straight line. Then pro- 

 ducing FB, and making CG parallel to it, the parallelepiped BH 

 will be equal to CD (172). Now, completing the parallelepiped IK, 

 as the parallelogram CF is to EF, so is KI to AF (171) ; and as CF 

 to BG, so is KI to BH, but EF is equal to the base of AF, and BG 

 to the base of CD, they are tlierefore equal, and the parallelepipeds 

 AF and BH are equal, and AF=:CD. 



174. Theorem. Paralellepipeds, of the same height, 

 are to each other as their bases. 



For one of them is equal to a parallelepiped of the same height on 

 an equal base which forms a single parallelogram with the base of 

 the other; and this is to the other in the ratio of the bases (171); con- 

 sequently the first two are in the same ratio. 



175. Theorem. Parallelepipeds are to each other in 

 the joint ratio of their bases and their heights. 



For one of them is to a tliird parallelepiped of the same height with 

 itself, but on the basis of the second, in the ratio of the bases, and the 

 third is to the second in the ratio of the heights, consequently the first 

 is to the second in the joint ratio of the bases and the heights. Thus, 

 a and h being the bases, c and rfthe heights, e,y, and g the three pa^ 

 rallelepipeds, a',h\\e ', g^ and c \ d\\^ \f', ac ; bdzze [f. 



Scholium. Hence is derived the common mode of finding the 

 content of a solid, by multiplying the numerical representatives of its 

 length, breadth, and height, and thus comparing it with the cubic 

 unit of the measure. 



