PROPOSITION XXXI. PROBLEM.

626. Through a given point to pass a plane tangent to a given cylinder.

CASE I. When the given point is in the curved surface of the cylinder.

Let A C be a given cylinder, and let the given point be a point in the element A A'.

It is required to pass a plane tangent to the cylinder and embracing the element A A'.

Draw the radius O A, and A T tangent to the base;

and pass a plane RT' through A A' and A T.

The plane RT is the plane required.

For, through any point P in this plane, not in the ele ment A A',

pass a plane to the base, intersecting the cylinder in the MN,

and the plane R T' in MP.

From the centre of the O MN draw QM.

MP and MQ are respectively to A T and A O,

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(the intersections of turo || planes by a third plane are || lines);

..ZPMQ=TAO,

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(two not in the same plane, having their sides and lying in the same

direction, are equal).

.. PM is tangent to the O M N at M.

.. P lies without the OM N,

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to it.

and hence without the cylinder.

.. the plane RT does not cut the cylinder, and is tangent

CASE II.

When the given point is without the cylinder.

Let P be the given point.

It is required to pass a plane through P tangent to the cylinder.

Through P draw the line PT to the elements of the cylinder,

meeting the plane of the base at T.

From 7 draw TA and TC tangents to the base.
Through P T and the tangent TA pass a plane R T'.

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Cons.

the plane R T', passing through PT and the point A will contain the element A A',

(two lines lie in the same plane).

And, since R T' also contains the tangent A T,

it is a tangent plane to the cylinder.

In like manner, the plane T'S', passed through PT and the tangent line TC,

is a tangent plane to the cylinder.

Q. E. F.

627. COROLLARY 1. The intersection of two tangent planes to a cylinder is parallel to the elements of the cylinder.

628. COR. 2. Any straight line drawn in a tangent plane, and cutting the element of contact, is tangent to the cylinder.

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

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 alti

tude.

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.

S

B

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.

$630

They also lie in the cutting plane,

(for their extremities lie in the plane).

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.

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Q. E. D.