sections. These prisms may be said to be circumscribed about the pyramid 0-ABC.

Also using the triangular sections as upper bases, construct prisms whose lateral edges shall be parallel to 40, and whose altitudes shall be the distance between the parallel sections. This set of prisms may be said to be inscribed in the pyramid

0-ABC.

For every circumscribed prism there is an equivalent inscribed prism, except for the circumscribed prism having ABC for its lower base, for which there is no equivalent inscribed prism.

The difference between the sum of the circumscribed prisms and the sum of the inscribed prisms is prism M.

By increasing the number of divisions into which 4S is divided, the divisions can be made as sinall as we please, and the volume of prism M can be made as small as we please, although not equal to zero.

The difference between the sum of the circumscribed prisms and the sum of the inscribed prisms can therefore be made as small as we please, but not equal to zero.

The volume of the pyramid 0-ABC differs from the sum of the circumscribed prisms or the sum of the inscribed prisms by less than they differ from each other.

Consequently the difference between the volume of the pyramid and either the sum of the circumscribed prisms or the sum of the inscribed prisms can be made less than any assignable quantity, but not equal to zero.

Therefore the volume of the pyramid is the limit of the sum of the circumscribed prisms or of the inscribed prisms as their number is indefinitely increased.

Q.E.D.

964. Triangular pyramids having equal altitudes and equivalent bases are equivalent.

Let the pyramids O-ABC and R-DEF have equivalent bases and a common altitude AS.

44

B

To Prove O-ABC and R-DEF equivalent.

Proof.

Divide the altitude AS into a number of equal parts, and through the points of division pass planes parallel to the plane of the bases.

The corresponding sections made by these parallel planes are equivalent. (?)

Inscribe in each pyramid a series of prisms having the triangular sections as upper bases, and the distance between the sections as their altitudes.

The corresponding prisms of the two pyramids are equivalent. (?)

The sum of the prisms inscribed in 0-ABC is equivalent to the sum of the prisms inscribed in R-DEF.

If the number of divisions into which AS is divided is indefinitely increased, the sum of the prisms inscribed in 0-ABC approaches the volume of 0-ABC as its limit, and the sum of the prisms inscribed in R-DEF approaches the volume of R-DEF as its limit. (?)

Since these variable sums are always equal, their limits are equal. Consequently vol. 0-ABC= vol. R—DEF. (?) Q.E.D. SANDERS' GEOM. - 20

PROPOSITION XVIII. THEOREM

965. The volume of a triangular pyramid is equal to one third the product of its base and altitude.

E

B

Let O-ABC be any triangular pyramid.

To Prove the volume of O-ABC and altitude.

=

the product of its base

Proof. Construct on the base ABC the triangular prism BD, with its lateral edges AD and CE each equal and parallel to OB. The prism BD is made up of the triangular pyramid 0-ABC and the quadrangular pyramid O-ACED.

Pass a plane through OC and OD, dividing the quadrangular pyramid O-ACED into two triangular pyramids 0-ACD and

O-CED.

Pyramids 0-ACD and O-CED are equivalent. (?)

Pyramid O-CED may be read C-ODE.

Pyramids C-ODE and O-ABC are equivalent.

..the three triangular pyramids composing the prism BD are equivalent.

O-ABC is equal to one third of the prism BD.

O-ABC= the product of its base and altitude. (?) Q.E.D.

966. EXERCISE.

Find the altitude of a triangular pyramid whose

volume is 50 cu. in. and whose base is 12 sq. in.

967. EXERCISE.

The volume of a parallelopiped is m cu. in. Find the altitude of an equivalent pyramid whose base is one of the triangles into which the base of the parallelopiped is divided by its diagonals.

PROPOSITION XIX. THEOREM

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

B

Let O-ABCDE be any pyramid.

To Prove the volume of O-ABCDE = the product of its base and altitude.

Suggestion. Divide the pyramid into triangular pyramids and apply § 965.

969. COROLLARY I. The volumes of pyramids are to each other as the products of their bases and altitudes; if their bases are equivalent, the pyramids are to each other as their altitudes; and if their altitudes are equal, the pyramids are to each other as their bases.

970. COROLLARY II. The volume of any polyhedron may be found by dividing it up into triangular pyramids, computing their volumes separately, and taking the sum of their volumes.

971. EXERCISE. The altitude of a pyramid is 8 ft. and its base is a regular pentagon each side of which is 4 ft. Find the volume of the pyramid. [§ 754.]

PROPOSITION XX. THEOREM

972. The volume of the frustum of a pyramid is equal to the sum of its bases and a mean proportional between its bases, multiplied by one third of its altitude.

Let XS be a frustum of the pyramid 0-XY; and let B represent the area of the lower base, b the area of the upper base, and a the altitude of the frustum.

To Prove

vol. XS = a(B+ b + √ B x b).

Proof. Let m represent the altitude of the pyramid o-RS, and v represent the volume of the frustum

XS.

R

The volume of the frustum XS is equal to the difference between the volumes of the pyramids o-XY and 0-RS.

Substitute this value of m in (1).

v = {} B × a + {} (√B+ √b)a√b.

v = { B× a + a√ Bxb+b xa.

v =

973. COROLLARY.

a(B+b+√Bxb).

Q.E.D.

The frustum of a pyramid is equivalent to the sum of three pyramids having a common altitude equal to that of the frustum, and whose bases are the upper and lower bases of the frustum and a mean proportional between them.