109 



figure of a solid projected upon paper. This is especially the 

 case when solids of one form have to be divided into solids of 

 other forms for purposes of measurement or of demonstration. 



You will readily see, without my occupying time by actual 

 demonstration, how, by forming rectangular parallelepipeds 

 with these cubic inches, we can prove that multiplying their 

 length by their breadth will give their base, and multiplying 

 this base by their height will give their volume, in a manner 

 analogous to that employed with square inches to form and 

 measure rectangles : — how we can thus illustrate that cubic 

 measures are the result of the double multiplication of lineal 

 measures, just as square measures are the result of their single 

 multiplication : — how we can show the meaning of the terms 

 cube and cube root as we did that of square and square root: — 

 how we can extend the rule for measuring rectangular parallele- 

 pipeds to all parallelepipeds, by dividing rectangular ones into 

 two equal prisms as in these models (Figure 19), and then 

 arranging the prisms as in Figure 20, to form a sloping 

 parallelepiped, wherein it is evident that the base, height, and 

 volume of the solid are unchanged however much its shape 

 may be : — and how, by this division of parallelepipeds into 

 prisms we may prove that the volume of the prism is equal to 

 one of its rectangular sides as base multiplied by half its height. 



As all planes bounded by straight lines may be divided into 

 triangles, so all solids bounded by such planes may be divided 

 into pyramids. Therefore if we can measure pyramids we can 

 measure all such solids. That the formula for measuring 

 pyramids — volume — base x J height, is correct may be 

 proved by concrete demonstrations of various kinds, as, for 

 instance, this one based on the beautiful one imagined by blind 

 Dr. Sanderson, that is before alluded to. 



Let us take the two cubes represented by these models 

 (Figures 21 and 22) and say they are each three inches in 

 length, breadth, and height. They are consequently equal to 

 each other, each having a volume of 27 cubic inches, and their 

 basis are equal to each other, being each 3" x 3" ■— 9 square 

 inches. If one be now divided into six equal pyramids 

 (Figure 21), and the other into six equal layers (each a 

 parallelepiped) as in Figure 22, it is evident that each pyramid 

 is equal to each layer, for each is respectively the sixth part of 



equal cubes, that is, each has a volume equal *■ 



to 



6 



= 41 



cubic inches. This is evidently the volume of each layer, for 

 the height of each is -J = J an inch, and the base 9" x $ — 4^. 

 But as the base of the pyramid is also 9" it must also be 

 multiplied by i" to produce its volume of 4J inches. _ As the 

 height of two pyramids is equal to that of the cube, this height 



