Part 30 (2/2)

It is desirable to refer to the corresponding case in plane geoure is obtained by passing a plane through the parallelepiped parallel to a lateral edge The saonal plane of a parallelepiped

These two propositions are fundamental in the ular parallelepipeds are to each other as the products of their three dimensions_

This leads at once to the corollary that the voluular parallelepiped equals the product of its three dimensions, the fundamental law in the mensuration of all solids It is preceded by the proposition asserting that rectangular parallelepipeds having congruent bases are proportional to their altitudes This includes the incommensurable case, but this case may be omitted

The number of simple applications of this proposition is practically unlimited In all such cases it is advisable to take a considerable number of numerical exercises in order to fix in eometry furnishes a certain nu property of the rectangular parallelepiped, often called the rectangular solid:

If the edges are _a_, _b_, and _c_, and the diagonal is _d_, then (_a_/_d_)2 + (_b_/_d_)2 + (_c_/_d_)2 = 1 This property is easily proved by the Pythagorean Theorem, for _d_2 = _a_2 + _b_2 + _c_2, whence (_a_2 + _b_2 + _c_2) / _d_2 = 1

In case _c_ = 0, this reduces to the Pythagorean Theoreeometry

THEOREM _The volume of any parallelepiped is equal to the product of its base by its altitude_

This is one of the few propositions in Book VII where a e It is easy to make one out of pasteboard, or to cut one from wood If a wooden one is made, it is advisable to take an oblique parallelepiped and, by properly sawing it, to transfor three different solids

On account of its aard forure is sometimes called the Devil's Coffin, but it is a name that it would be well not to perpetuate

THEOREM _The volume of any prism is equal to the product of its base by its altitude_

This is also one of the basal propositions of solid geometry, and it has many applications in practical ive a sufficient list of proble numerical measure e of the pupils, the following problem is a type:

[Illustration]

If this represents the cross section of a railway eh, _b_ feet wide at the bottom, and _b'_ feet wide at the top, find the number of cubic feet in the embankment Find the volume if _l_ = 300, _h_ = 8, _b_ = 60, and _b'_ = 28

The ular parallelepiped and cube, was known to the ancients Euclid was not concerned with practical eometry appears in his ”Elements” We find, however, in the papyrus of Ahyptian builders, long before his tiular parallelepiped A before the Christian era, rules were known for the construction of altars, and a a cube with twice the voluiven cube (the ”duplication of the cube”) was attacked by many mathematicians The solution of this probleeoiven cube, then _e_3 is its volume and 2_e_3 is the volue of the required cube is _e_[3root]2 Now if _e_ is given, it is not possible with the straightedge and coh it is easy to construct one equal to _e_[sqrt]2

The study of the pyrains at this point In practical ular pyramid It is, however, a simple matter to consider the oblique pyra volumes we sometimes find these forular pyraht by the perimeter of its base_

This leads to the corollary concerning the lateral area of the frustuular pyramid may be considered as a frustum with the upper base zero, and the proposition as a special case under the corollary It is also possible, if we choose, to let the upper base of the frustues above that point, although this is too complicated forin the general idea of _pyramidal space_, the infinite space bounded on several sides by the lateral faces, of the pyra on two sides of the vertex

THEOREM _If a pyramid is cut by a plane parallel to the base:_

1 _The edges and altitude are divided proportionally_ 2 _The section is a polygon sious proposition of plane geoh the vertex so as to cut the base We shall then have the sides and altitude of the triangle divided proportionally, and of course the section will ment, and therefore it is si plane h the vertex, or it may cut the pyramidal space above the vertex In either case the proof is essentially the saular pyramid is equal to one third of the product of its base by its altitude, and this is also true of any pyramid_

This is stated as two theorems in all textbooks, and properly so It is explained to children who are studying arithmetic by means of a hollow pyramid and a hollow prism of equal base and equal altitude The pyrarain, and the contents is poured into the prism