ments of three parallel lines not situated in the same plane, are parallel.

Cor. 3. Parallel planes are everywhere equally distant. (481) [Let r, s, t, be perpendicular to M, N.]

B

Cor. 4. Two angles, having their sides parallel and open- (482) ing in the same direction, are equal, and their planes are parallel. For, let the sides AB, AC, of the angle A, be parallel respectively to the sides ab, ac, of the angle a, and open in the same direction; draw Bb, Cc, parallel to Aa, then will the quadrilaterals, Ab, Ac, Bc, be parallelograms, and the sides of the triangles BAC, bac, Fig. 1102. severally equal,— .'. ▲ A = a; but Aa Bb = Cc, .. (480) the plane BAC will be parallel to the plane bac.

=

Cor. 5. A Dihedral angle, or the angle which one plane (483) makes with another, is measured by the inclination of two perpendiculars drawn through the same point in its edge, one in each side.

For, make A a perpendicular to the planes BAC, bac, then will Aa be perpendicular to AB, AC, ab, ac, and the dihedral angle BAac will be measured by the plane angle BAC≈ bac; from which it follows that the point A, through which the perpendiculars AB, AC, are drawn, may be taken anywhere in the edge, Aa, of the dihedral angle.

Cor. 6. The segments of lines intercepted by parallel (484) planes are proportional.

For, let ABC, abc, be any lines whatever, piercing the parallel planes, P, P, P, in the points A, B, C, a, b, c; and through B draw mBn parallel to abc, piercing the planes in m, n. Since A, B, C, m, n, are in the same plane, and mA parallel to Cn, we have

AB: BC= mB : Bn 11 ab: bc.

A

B

Fig. 1103.

PROPOSITION IV.

If a line passing through a fixed point revolve in any (485) manner so as constantly to intersect two parallel planes, the figures thus described will be similar.

SIMILAR SOLIDS.

For, let VaA, VxX, VyY,..., be positions of the revolving line passing through the fixed point, V, and piercing the parallel planes in A, a, X, x, Y, y, ...; drop the perpendicular VpP, piercing the planes in P, p, and join PA, PX, pa, px. The radii vectores PX, px, have the constant ratio Vp: Vp, and APX apx; .. (426) the plane figures

< =

AXY, ..., αxy, ..., are similar.

X

Fig. 111.

249

Definition. The solid (V, AXY ...) is denominated a Cone, when the perimeter AXY... is wholly curvilinear, and it becomes a Pyramid when AXY ... is made up of straight lines. The cone is circular when its base, AXY is a circle, and right if the dicular fall in the centre of the base.

perpen

Cor. 1. In similar cones [the pyramid is to be included] (486) the altitudes, radii vectores, like chords and generating lines for corresponding positions, are proportional; and the bases are as the squares of these lines.

Thus, VP : Vp = PX : px

and

=

chordXY: chdry = VX : Vx, base(AXY...): bs(axy ...) = (PX): (px)=....

Cor. 2. If the vertex, V, be carried to an infinite distance, (487) the lines Aa, Xx, Yy, will become parallel, and the figure axy will = AXY .... Under these conditions the solid (AXY

... 9

axy ...) is denominated, a Cylinder when the perimeter AXY ... is a curve, and a Prism when AXY... is a polygon; and these magnitudes are said to be right or oblique according as the sides are perpendicular to or inclined to the bases. The cylinder and prism are also distinguished by their bases. When the base is a parallelogram it is obvious that all the other faces will be parallelograms and those opposite to each other equal, in which case the prism is called a Parallelopipedon; and the Cube is a right parallelopipedon of equal faces.

SECTION SECOND.

Surfaces of Solids.

PROPOSITION I.

The surface of a Polyhedron, that is, any solid bounded (488) by planes, may be found by computing the areas of its several faces.

PROPOSITION II.

The convex surface of a Right Circular Cone is meas- (489) ured by its slant height multiplied into the semicircumference of its base.

Let y be the convex surface included between any two positions of its Generatrix, l, and r the intercepted portion of the circular base; then, since y is obviously a continuous function of x, giving to y, x, the vanishing increments [k], [h], we have, (311), (148),

=

Fig. 112.

yzcircumference (circumference of base). Q. E. D. Cor. 1. The circular conical sector is measured by its (490) slant height multiplied into half its base. y = lx.

Cor. 2. The frustrumental surface of the right circular (491) cone is measured by its slant height multiplied into the half sum of its bases.

For let VAB, Vab, be conical sectors having the same vertical angle, V, and, consequently ABba the frustrumental surface in question, we have

*No constant is to be added, since y and x vanish together.

SURFACES OF REVOLUTION.

ABba = VAB - Vab = VA • ‡i̟AB – Va • ‡ab = (Va+aA) • AB- Va tab

251

Cor. 3. The convex surface of the Right Circular Cylin- (492) der, is equal to its height, multiplied into the circumference of its base; for ab becomes = AB.

If a continuous curve referred to rectangular coördi- (493) nates, revolve about the axis of abscissas, the derivative of the surface thus generated, regarded as a function of the corresponding abscissa, will be equal to the circumference described by the ordinate multiplied into the square root of unity increased by the square of the derivative of the ordinate also regarded as a function of the abscissa.

For let the surface Z be described by the revolution of any continuous curve, z, around the axis, x, M, m, k, h, the vanishing increments

and

3

M

of

then

but (491),

9

M = m[Tу+(y+k)],

= (h2 + k2)* • π(2y+k) ;

[N] _ (^* + **) *. • 2y = 2πy (1 + [ * ]')*
(h2 k2)* •«

=

h

Z'z_r2 = 2πy(1+y';_s»). Q. E. D.

h

Fig. 113.

It follows that, in order to determine the surface generated by a particular curve, we have only to eliminate y and y' by aid of its equation, and to return to the function.

PROPOSITION IV.

A Spherical Zone is equal to its altitude multiplied (494) into the circumference of the sphere.

Let Z be a zone generated by the arc, z, of a circle revolving about a diameter, which we will assume as the axis of x, the origin being at the centre and the radius = r. We have

where there is no constant to be added if we make the surface begin at the axis of y. Now let Z, Z1, Z2, T2, be corresponding values of Z and x; we have

zone(Z2- Z1) = zone Z2- zone Z1 = (x-x1)• T 2r. Q. E. D. zoneŽ1 π •

2

Cor. 1. A spherical zone is equal to the convex surface (495) of the circumscribing cylinder, described by the revolution of a rectangle with a radius equal to that of the sphere.

Surface described by S = sur. described by C.

C S

Fig. 114.

Cor. 2. The surface of the sphere is equal to the convex (496) surface of the circumscribing cylinder,

Fig. 1142.

Cor. 3. The surfaces of spheres are to each other as the (497) squares of their radii [r], or diameters [2r], or circumferences [π • 2r].