OF SPACE. 



19 



every parallelepiped are equal and equian- 

 gular parallelograms. 



The opposite sides of all the figures are parallel, because 

 they are the sections of one plane with two parallel planes 

 (168): the corresponding sides of two opposite planes be- 

 in", for the same reason, parallel 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 therefore 

 equal parallelograms. 



171. Theorem. If a prism be divided 

 by a plane parallel to its two opposite sur- 

 faces, its segments will be to each other as 

 the segments of any of the divided surfaces 

 or lines. 



A. 



G 



M O 



H 



■ r ^ =-r 1 Let the prism AB 



,Nf\,.Ei\ .Ei,\ -lA be divided by the plane 



F K L D ' B CDE parallel to AFG 



and BHI. Find FK a common measure of FD and DB 

 (119), make KL=:FK, and let the planes KMN, LOP be 

 parallel to AFG ; then the prisms AK, ML may b& shown 

 to be contained by similar and equal figures similarly situ- 

 ated, in the same manner as it is shown of parallelepipeds, 

 and there is no imaginable difference between these prisms : 

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

 multiple of AK that FDisof FK, and AB the same multiple 

 of AK that FB is of FK, or AD : AK=FD : FK, and AB . 

 AK=:FB : FK, whence AD : AB=FD : 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. Theokem. Parallelepipeds on the 

 same base and contained between the same 

 planes, are equal. 



D The parallelepiped AB 



is equal to CD standing on 

 the same base BC, and 

 terminated by the plane 

 AED. For each is equal 

 jj to the parallelepiped EF ; 

 since the triangular prism 

 GB is similar and equal 

 IQ the triangular prism HC, and deducting these from 

 the solid HCI, the remainders AB and EF are equal. 



And in the same manner it may be shown that CD=:EF ; 

 therefore AB=CD. 



173. Theorem. Parallelepipeds on equal 

 bases and of the same height are equal. 



Each parallelepiped is equal 

 to the erect parallelepiped on 

 the same base. Let one of j 

 these be so placed that the 

 plane of one of the sides AB 

 may coincide with the plane BC of the other parallelepiped 

 CD, and that EBC may be a straight line. Then producing 

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

 will be equal to CD (172). Now completing the parallel- 

 epiped 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 

 therefore equal, and the parallelepipeds AF and BH are 

 equal, and AFz;CD. 



174. Theorem. Parallelepipeds 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) ; consequently 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 third 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 b being 

 the bases, c andd the heights, e,J, and g the three parallel- 

 epipeds, a -.b-.-.e: g, and c.d:: g -.f; ac : Id^Ze-.f. 



Scholium. Hence is derived tlie common mode of 

 finding the content of a solid, by multiplying the nume- 

 rical representatives of its length, breadth and height, and 

 thus comparing it with the cubic unit of the measure, - 



176. Theorem. Similar parallelepi|3eds 

 are in the triplicate ratio of, their homologous 

 sides. 



For the joint ratio of the bases and heights is the same 

 as the triplicate ratio of the sides. 



177. Theorem. A plane pjissing through 

 the diagonals of two opposite sides of a pa- 

 rallelepiped, divides it into two equal prisms. 



