Upon the base CFH and upon the sections as lower bases construct prisms having their lateral edges parallel to VC and their altitudes equal to h.

This set of prisms may be said to be circumscribed about the pyramid.

Also with the sections as upper bases construct prisms having their lateral edges parallel to VC and their altitudes equal to h.

This set of prisms may be said to be inscribed in the pyramid.

The first circumscribed prism (beginning at the top) is equivalent to the first inscribed prism, the second circumscribed to the second inscribed, and so on until the last circumscribed remains. 8677

Hence the sum of the inscribed prisms differs from the sum of the circumscribed by the lower circumscribed prism P-CFH.

But the pyramid is intermediate between the total inscribed and the total circumscribed prisms. Ax. 10

Therefore the difference between the pyramid and either

of these totals is less than the difference between the totals themselves, i. e., less than the lower circumscribed prism.

But the volume of this prism is the product of its base and altitude, and since its altitude can be indefinitely diminished, while its base remains the same, its volume can be made as small as we please. $187 That is, the total of the inscribed prisms, or the total of the circumscribed prisms, can be made to differ from the pyramid by less than any assigned volume.

But they can never become equal to the pyramid. Ax. 10 Therefore the volume of the pyramid is their common

limit.

Q. E. D.

PROPOSITION XXI.

THEOREM

702. Two triangular pyramids having equal altitudes and equivalent bases are equivalent.

GIVEN the triangular pyramids A-BCD and A'-B'C'D' having equivalent bases BCD and B'C'D' in the same plane and having a common altitude BF.

TO PROVE

the pyramids are equivalent.

Divide BF into any number of equal parts and denote one of these parts by h.

Through the points of division pass planes parallel to the bases and cutting the two pyramids.

The corresponding sections made by these planes in the two pyramids will be equivalent. $698

Inscribe in each pyramid a series of prisms having the sections as upper bases and having the common altitude h. The corresponding prisms, having equal altitudes and equivalent bases, will be equivalent.

$677 Therefore the total volume (or S) of the prisms inscribed in A-BCD will equal the total volume (or S') of the prisms inscribed in A'-B'C'D'.

Now suppose the number of divisions of the altitude BF to be indefinitely increased.

Then S will approach the volume of the pyramid A-BCD as a limit, and S' will approach the volume of the pyramid A'-B'C'D' as a limit.

$ 701

Since the variables S and S' are always equal to each other, their limits are equal.

$186

That is, the volumes of the pyramids are equal.

Q. F. D

PROPOSITION XXII. THEOREM

703. The volume of a triangular pyramid is one-third the product of its base and altitude.

TO PROVE its volume is one-third its base ABC by its altitude.

Construct a triangular prism having ABC for its base and its lateral edges equal and parallel to BS.

Taking away the triangular pyramid S-ABC from the prism, we have left the quadrangular pyramid S-DACE. Divide the latter by the plane SDC into two triangular pyramids S-DAC and S-DCE.

These pyramids have equal bases, the triangles DCA and DCE. $ 116 They have equal altitudes, the perpendicular from the common vertex S upon the common plane of their bases.

Therefore they are equivalent.

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It can also be shown that the pyramids S-ABC and S-DAC, regarded as having the common vertex C, have equal bases and equal altitudes.

Hence these two pyramids are equivalent.

Hence all three are equivalent.

Therefore the pyramid S-ABC is one-third of the prism. But the volume of the prism is the product of its base and altitude.

And the pyramid has the same base and altitude.

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Hence the volume of the pyramid is one-third the product of its base and altitude.

Q. E. D.

704. The volume of any pyramid is equal to one-third the product of its base and altitude.

GIVEN
TO PROVE

the pyramid O-ABCDE, whose altitude is OZ.