Cor. 1. Any two similar prisms are to each other as the cubes of their homologous edges.

For, since the prisms are similar, their bases are similar polygons (D. 16); and these similar polygons may each be divided into the same number of similar triangles, similarly placed (B. IV., P. XXVI.); therefore, each 'prism may be divided into the same number of triangular prisms, having their faces similar and like placed; consequently, the triangular prisms are similar (D. 16). But these triangular prisms are to each other as the cubes of their homologous edges, and being like parts of the polygonal prisms, the polygonal prisms themselves are to each other as the cubes of their homologous edges.

Cor. 2. Similar prisms are to each other as the cubes of their altitudes, or as the cubes of any other homologous lines.

PROPOSITION XX. THEOREM.

Similar pyramids are to each other as the cubes of their homologous edges.

Let S-ABCDE, and S-abcde, be two similar pyramids, so placed that their homologous angles at the vertex shall coincide, and let AB and ab be

any two homologous edges: then will the pyramids be to each other as the cubes of AB and ab.

For, the face SAB, being similar to Sab, the edge AB is parallel to to the edge ab, and the face SBC being similar to Sbc, the edge BC is parallel to bc; hence, the planes of the bases are parallel (B. VI., P. XIII.).

E

B

Draw SO perpendicular to the base ABCDE; it will

also be perpendicular to the base abcde. plane at the point o then will SO be to So, as SA is to Sa (P. III.), or as AB is to ab; hence,

Let it pierce that

base ABCDE : base abcde :: AB2: ab2.

Multiplying these proportions, term by term, we have,

ᎪᏴ

base ABCDE × 1SO : base abcde So :: AB3: ab.

But, base ABCDE × 480 is equal to the volume of the pyramid S-ABCDE, and base abcde × So is equal to the volume of the pyramid S-abcde; hence,

pyramid S-ABCDE : pyramid S-abcde :: AB3 : ab3;

which was to be proved.

Cor.

Similar pyramids are to each other as the cubes of their altitudes, or as the cubes of any other homologous lines.

If we denote the volume of any prism by V, its base by B, and its altitude by H, we shall have (P. XIV.),

If we denote the volume of any pyramid by V, its base by B, and its altitude by H, we have (P. XVII.),

If we denote the volume of the frustum of any pyramid by V, its lower base by B, its upper base by b, and its altitude by H, we shall have (P. XVIII., C.),

V =

}(B + b + √B × b) × H (3.)

A REGULAR POLYEDRON is one whose faces are all equal regular polygons.

There are five regular polyedrons, namely:

1. The TETRAEDRON, or regular pyramid-a polyedron bounded by four equal equilateral triangles.

2. The HEXAEDRON, or cube-a polyedron bounded by six equal squares.

3. The OCTAEDRON-a polyedron bounded by eight equal equilateral triangles.

4. The DODECAEDRON-a polyedron bounded by twelve equal and regular pentagons.

5. The ICOSAEDRON-a polyedron bounded by twenty equal equilateral triangles.

In the Tetraedron, the triangles are grouped about the polyedral angles in sets of three, in the Octaedron they are grouped in sets of four, and in the Icosaedron they are grouped in sets of five. Now, a greater number of equilateral triangles cannot be grouped so as to form a salient polyedral angle; for, if they could, the sum of the plane angles formed by the edges would be equal to, or greater than, four right angles, which is impossible (B. VI., P. XX.).

In the Hexaedron, the squares are grouped about the polyedral angles in sets of three. Now, a greater number of squares cannot be grouped so as to form a salient polyedral angle; for the same reason as before.

In the Dodecaedron, the regular pentagons are grouped about the polyedral angles in sets of three, and for the same reason as before, they cannot be grouped in any greater number, so as to form a salient polyedral angle.

Furthermore, no other regular polygons can be grouped so as to form a salient polyedral angle; therefore, Only five regular polyedrons can be formed.

14

BOOK VIII.

THE CYLINDER, THE CONE, AND THE SPHERE.

DEFINITIONS.

1. A CYLINDER is a volume which may be generated by a rectangle revolving about one of its sides as an axis.

E

P

H

N

K

Thus, if the rectangle ABCD be turned about the side AB, as an axis, it will generate the cylinder FGCQ-P. The fixed line AB is called the axis of the cylinder; the curved surface generated by the side CD, opposite the axis, is called the convex surface of the cylinder; the equal circles FGCQ, and EHDP, generated by the remaining sides BC and AD, are called bases of the cylinder; and the perpendicular distance between the planes of the bases, is called the altitude of the cylinder.

MK

The line DC, which generates the convex surface, is, in any position, called an element of the surface; the elements are all perpendicular to the planes of the bases, and any one of them is equal to the altitude of the cylinder.

Any line of the generating rectangle ABCD, as IK, which is perpendicular to the axis, will generate a circle whose plane is perpendicular to the axis, and which is equal to either base: hence, any section of a cylinder by a plane perpendicular to the axis, is a circle equal to either base. Any section, FCDE, made by a plane through the axis, is a rectangle double the generating rectangle.