760. Exercise.-Cut a cube by a plane so that the section shall be a regular hexagon.

761. Exercise.—Pass a plane through a given straight line which shall divide a given triangular prism into two equivalent truncated prisms.

762. Exercise.-Construct a parallelopiped of which three edges lie upon three given straight lines in space.

763. Exercise.-Pass a plane through a given point which shall divide a given regular tetraedron into two equal parts.

PROBLEMS FOR COMPUTATION

764. (1.) A rectangular block of marble is I m. 9 dcm. long, 9 dcm. 6 cm. broad, and 8 dcm. 9 cm. thick. What is its weight, if a cubic meter weighs 2675 kg.?

(2.) A barn with a gable roof is 60 ft. long, 30 ft. broad; the height from the floor to the eaves is 25 ft., to the gable 32 ft. Find its contents.

(3.) The area of the base of a right prism is 12 sq. in., its total area is 295 sq. in.; the base is a regular hexagon. What is the volume?

(4.) The great pyramid is estimated to have cost ten dollars a cubic yard, and three dollars besides for each square yard of surface; in this estimate the lateral faces are considered to be planes. The altitude of the pyramid is 488 ft., its base is 764 ft. square. What was its cost?

(5.) Express the volume of a cube in terms of the length of a diagonal.

(6.) What is the ratio of an edge of a cube to that of a regular tetraedron of the same volume?

(7.) The area of the lower base of a frustum of a pyramid is 100 sq. cm., of the upper base 30 sq. cm., and the altitude of the frustum is 5 dcm. What would be the altitude of the complete pyramid?

(8.) What is the volume of a frustum of a regular triangular pyramid, if its slant height is 3.5 ft., a side of the lower base 4 ft., of the upper base 1.5 ft.?

(9.) The total surface of a regular tetraedron is 400 sq. ft. What is its volume?

(10.) The area of a face of a regular octaedron is 1 sq. ft. What is its volume?

(11.) What is the ratio of the lateral area of a regular tetraedron to the lateral area of a prism constructed upon the same base and having one of its lateral edges coincident with an edge of the tetraedron?

(12.) Find the volume of a truncated triangular prism, if the sides of a right section are respectively 2.416, 3.213, 1.963 in., and its lateral edges are 7.645, 6.633, 2.742 in.

GEOMETRY OF SPACE

BOOK VIII

THE CYLINDER

765. Def.-A curved line, or curve, is a line no part of which is straight.

The curve may or may not lie entirely in one plane. An example of the first kind is the circumference of a circle; an example of the second kind is a curve like a corkscrew.

766. Def.-A cylindrical surface is a surface generated by a moving straight line which continually intersects a given fixed curve and is constantly parallel to a given fixed straight line.

Thus, if the straight line AB moves so as continually to intersect the curve AC and remains parallel to the line PQ, the surface generated, ABDC, is a cylindrical surface.

767. Defs.-The moving line is called the generatrix; the fixed curve is called the directrix.

Any one position of the generatrix, as EF, is called an element of the surface.

768. Remark. The generatrix is usually supposed to be indefinite in extent, so that the surface generated is also of indefinite extent.

The directrix may be any curve whatever. But for the student who has not studied the appendix the proofs are rigorous only when the directrix is considered to be the circumference of a circle.

PROPOSITION I. THEOREM

769. The sections of a closed cylindrical surface made by two parallel planes cutting the elements are equal.

GIVEN the closed cylindrical surface RS cut by two parallel planes, not parallel to the elements, in the sections TS and RU.

TO PROVE

Let A, C, and E be any three points in the perimeter of the upper section, and AB, CD, and EF the corresponding

elements; B, D, and F being the points where these elements meet the perimeter of the lower section.

Through AB and CD pass a plane. Pass another through AB and EF.

Then AC is parallel to BD and AE to BF.
Hence AC-BD and AE=BF.

The angles CAE and DBF are also equal.

§ 544

$118 $ 557

If, therefore, the planes of the two sections be superposed so that BD shall coincide with AC, F will fall on E.

Now, if we suppose AC to be fixed and the point E to describe the perimeter of the upper section, then F will describe the perimeter of the lower section.

But in the superposed position of the sections F would always coincide with E.

Hence the perimeters of the two sections would coincide throughout. Therefore the sections are equal.

Q. E. D.

YO. Defs.-A cylinder is a solid bounded by a closed cylindrical surface and two parallel planes.

The cylindrical surface is called the lateral surface, and the equal sections formed by the parallel planes the bases of the cylinder.