ON THE CONE.

629. DEF. A Conical surface is a surface generated by a moving straight line continually touching a given curve and passing through a fixed point not in the plane of the curve. Thus the surface generated by the mov

ing line A A' continually touching the curve ABCD, and passing through the fixed point S, is a conical surface.

630. DEF. The moving line is called the Generatrix; the curve which directs the motion of the generatrix is called the Directrix; the generatrix, in any position, is called an Element of the surface.

631. DEF. A conical surface generated by an indefinite straight line consists of two portions, called Nappes, one the Lower, the other the Upper Nappe.

Α'

B

S

D

A

632. DEF. A Cone is a solid bounded by a conical surface and a plane.

face.

633. DEF. The Lateral surface of a cone is its conical sur

634. DEF. The Base of a cone is its plane surface.

635. DEF. The Vertex of a cone is the fixed point through which all the elements pass.

636. DEF. The Altitude of a cone is the perpendicular distance between its vertex and the plane of its base.

637. DEF. The Axis of a cone is the straight line joining its vertex and the centre of its base.

638. DEF. A Section of a cone is a plane figure whose boundary is the intersection of its plane with the surface of the

cone.

639. DEF. A Right section of a cone is a section perpendicular to the axis.

640. DEF. A Circular cone is a cone whose base is a circle. 641. DEF. A Right cone is a cone whose axis is perpendicular to its base. The axis of a right cone is equal to its altitude.

642. DEF. An Oblique cone is a cone whose axis is oblique to its base. The axis of an oblique cone is greater than its altitude.

643. DEF. A Cone of Revolution is a cone generated by the revolution of a right triangle about one of its perpendicular sides as an axis.

The side about which the triangle revolves is the axis of the cone; the other perpendicular generates the base, the hypotenuse generates the conical surface. Any position of the hypotenuse is an element, and any element is called the slant height.

644. DEF. Similar cones of revolution are cones generated by the revolution of similar right triangles about homologous perpendicular sides.

645. DEF. A Truncated cone is the portion of a cone included between the base and a section cutting all the elements. 646. DEF. A Frustum of a cone is a truncated cone in which the cutting section is parallel to the base.

647. DEF. The base of the cone is called the Lower base of the frustum, and the parallel section the Upper base.

648. DEF. The Altitude of a frustum is the perpendicular distance between the planes of its bases.

649. DEF. The Lateral surface of a frustum is the portion of the lateral surface of the cone included between the bases of the frustum.

650. DEF. The Slant height of a frustum of a cone of revolution is the portion of any element of the cone included between the bases.

651. DEF. A Tangent line to a cone is a line having only one point in common with the surface.

652. DEF. A Tangent plane to a cone is a plane embracing an element of the cone without cutting the surface. The element embraced by the tangent plane is called the Element of Contact.

653. DEF. A pyramid is inscribed in a cone when its lateral edges are elements of the cone and its base is inscribed in the base of the cone.

654. DEF. A pyramid is circumscribed about a cone when its lateral faces are tangent to the cone and its base is circumscribed about the base of the cone.

PROPOSITION XXXII. THEOREM.

655. Every section of a cone made by a plane passing through its vertex is a triangle.

Let SBD be a section of the cone S-ABC through the vertex S.

We are to prove the section SBD a triangle.

The straight lines joining S with B and D are elements of

the surface.

They also lie in the cutting plane,

(for their extremities lie in the plane).

§ 630

Hence, they are the intersections of the conical surface with the plane of the section.

BD is also a straight line,

(the intersection of two planes is a straight line).

... the section SBD is a A.

§ 446

Q. E. D.

656. Every section of a circular cone made by a plane parallel to the base is a circle.

A

S

C

C

B

Let the section a bc of the circular cone S-ABC be parallel to the base.

Let O be the centre of the base, and let o be the point in which the axis SO pierces the plane of the || section.

Through SO and any number of elements, SA, SB, etc., pass planes cutting the base in the radii OA, O B, etc.,

and the section a b c

Now oa and ob are

in the straight lines o a, ob, etc.

respectively to OA and O B, § 465

(the intersections of two || planes by a third plane are lines).

..the A Soa and Sob are similar respectively to the

ASOA and SO B,

and their homologous sides give the proportion

$279

That is, all the straight lines drawn from o to the perimeter

of the section are equal.

... the section a bc is a O.

Q. E. D.

657. COROLLARY. The axis of a circular cone passes through the centres of all the sections which are parallel to the base.

PROPOSITION XXXIV. THEOREM.

658. The lateral area of a cone of revolution is equal to one-half the product of the circumference of its base by the slant height.

Let A-E FGH K be a cone generated by the revolution of the right triangle AOE about AO as an axis, and let S denote its lateral area, С the circumference of its base and L its slant height.

Inscribe on the base any regular polygon E F G H K,

and upon this polygon as a base construct the regular pyramid A-E F G H K inscribed in the cone.

Denote the lateral area of this pyramid by s, the perimeter of its base by p, its slant height by 1,

(the lateral area of a regular pyramid is equal to one-half the product of the perimeter of its base by the slant height).

Now, let the number of the lateral faces of the inscribed

pyramid be indefinitely increased,