56. Scholium II. The student may derive some aid in comprehending the preceding discussion of the regular polyedrons by constructing models of them, which he can do in a very simple manner, and at the same time with great accuracy, as follows.

Draw on card-board the following diagrams; cut them out entire, and at the lines separating adjacent polygons cut the card-board half through; the figures will then readily bend into the form of the respective surfaces, and can be retained in that form by gluing the edges.

EXERCISES ON BOOK VII.

THEOREMS.

1. THE volume of a triangular prism is equal to the product of the area of a lateral face by one-half the perpendicular distance of that face from the opposite edge.

2. The lateral surface of a pyramid is greater than the base. Suggestion. Join the projection of the vertex on the base with the corners of the base.

3. At any point in the base of a regular pyramid a perpendicular to the base is erected which intersects the several lateral faces of the pyramid, or these faces produced. Prove that the sum of the distances of the points of intersection from the base is constant.

Suggestion. The distances in question are proportional to the distances of the foot of the perpendicular from the sides of the base, and these distances have a constant sum. (v. V., Exercise 16.)

5. In a tetraedron, the planes passed through the three lateral edges and the middle points of the edges of the base intersect in

a straight line.

Suggestion. The intersections of the planes with the base are medial lines of the base. Therefore they intersect in the line joining the vertex with the point of intersection of the medial lines of the base.

6. The lines joining each vertex of a tetraedron with the point of intersection of the medial lines of the opposite face all meet in a point, which divides each line in the ratio 1:4. Note. This point is the centre of gravity

of the tetraedron.

Suggestion. If AF and DG are two of the lines in question, they must intersect, since they both lie in the plane passed through AD and the middle point E of the opposite edge. Moreover, since EF ED and EG EA (I., Exercise 38),

GF is parallel to AD and is equal to §AD.
Whence HF HA and GH = HD.

B

A

D

E

The lines through C and B will also each cut off of AF. Hence the four lines have a common intersection.

7. The straight lines joining the middle points of the opposite edges of a tetraedron all pass through the centre of gravity of the tetraedron, and are bisected by the centre of gravity. (v. III., Exercise 7.)

8. The plane which bisects a diedral angle of a tetraedron divides the opposite edge into segments which are proportional to the areas of the adjacent faces.

Suggestion. Consider the volumes of the two parts into which the tetraedron is divided.

9. If a, b, c, d, are the perpendiculars from the vertices of a tetraedron upon the opposite faces, and a', b', c', d', the perpendiculars from any point within the tetraedron upon the same faces respectively, then

Suggestion. Join the point in question with the vertices of the tetraedron, and compare the volumes of the four tetraedrons thus obtained with the volume of the given tetraedron.

10. The altitude of a regular tetraedron is equal to the sum of the four perpendiculars let fall from any point within it upon the four faces.

11. Any lateral face of a prism is less than the sum of the other lateral faces. (v. Proposition II.)

PROBLEMS.

12. Given three indefinite straight lines in space which do not intersect, to construct a parallelopiped which shall have three of its edges on these lines. (v. VI., Exercise 8.)

13. Within a given tetraedron, to find a point such that planes passed through this point and the edges of the tetraedron shall divide the tetraedron into four equivalent tetraedrons. (v. Exercise 6.)

BOOK VIII.

THE THREE ROUND BODIES.

1. Or the various solids bounded by curved surfaces, but three are treated of in Elementary Geometry,—namely, the cylinder, the cone, and the sphere, which are called the THREE

ROUND BODIES.

THE CYLINDER.

2. Definitions. A cylindrical surface is a curved surface generated by a moving straight line which continually touches a given curve, and in all of its positions is parallel to a given fixed straight line not in the plane of the curve.

Thus, if the straight line Aa moves so as continually to

touch the given curve ABCD, and so that in any of its positions, as Bb, Cc, Dd, etc., it is parallel to a given fixed straight line Mm, the surface ABCDdcba is a cylindrical surface. If the moving line is of indefinite length, a surface of indefinite extent is generated.

M

B C

α

b c

d

The moving line is called the generatrix; the curve which it touches is called the directrix. Any straight line in the surface, as Bb, which represents one of the positions of the generatrix, is called an element of the surface.

To draw an element through any given point of a cylindrical surface, it is sufficient to draw a line through the point parallel to the given fixed straight line, or parallel to ar element (I., Postulate II.).