no 



is 4 = li" and 



one third of this. Wherefore the volume 

 of one of these pyramids is equal to its base x — J — • 



Again, by dividing a cube into three irregularly-shaped 

 pyramids (Figure 28), as shown by these models, each 

 pyramid has the same base and height as the cube, but only a 

 third of its volume. As multiplying the base by the height 

 gives the volume of the cube, so multiplying its base by a 

 third of its height must give a third of its volume, that is, the 

 volume of one of the equal pyramids. 



I will not further occupy your time by ocular demonstration 

 that the formula holds good whatever the form of the 

 pyramid, but will conclude with a practical exhibition of the 

 utility of the system. 



When materials of any description, such as corn, or gravel, 

 or broken stone are formed into regular heaps for measure- 

 ment, they are usually put into a shape more or less like a 

 truncated pyramid. Suppose that these models (Figure 24) 

 be a heap of sand to be measured, and that the heap is 12 

 feet long at the bottom and 8 feet at the top ; 8 feet broad 

 at the bottom and 4 feet at the top ; and 3 feet high measured 

 perpendicularly: what is the volume of it ? 



One or other of two methods is usually employed to find 

 this out. The first is to multiply the mean length by the 

 mean breadth, and the product by the height. From the 

 dimensions we have assumed, the mean length is 10 feet and 

 the mean breadth 6 feet, and the height 3 feet. Therefore, 

 10 x 6 x 3 = 180 cubic feet for the volume of the heap. 



The other method is — Multiply the mean base by the 

 height. From the above dimensions the lower base is 12 X 8 

 = 96 square feet, and the upper 8 x 4 = 32 square feet. 



The mean base is therefore - — — 64 square feet, which, 



multiplied by the height, 3 feet, gives 192 cubic feet as the 

 volume of the heap. Both these cannot be right. Is either ? 

 Let us divide the heap into shapes that we know how to 

 measure. There is the central parallelepiped 8 feet long, 4 feet 

 broad, and 3 feet high. Its volume, therefore, is 8 X 4 X 3 

 = 96 cubic feet. There are two long prisms each 8 feet long, 

 2 feet wide, and 3 feet high, which being put together as in 

 Figure 19, make a parallelepiped whose volume is 8 X 2 X 3 

 =°48 cubic feet. There are two short prisms each 4 feet long, 

 2 feet broad, and 3 feet high, which also being placed together 

 form a parallelepiped whose volume is 4x2x3 = 24 cubic 

 feet. There are four pyramids, whose bases are each 2 feet by 

 2 feet == 4 square feet, and whose height is 3 feet. The 

 volume of each is therefore 4x| = 4 feet, or 16 cubic feet 



