# 1911 Encyclopædia Britannica/Cylinder

**CYLINDER** (Gr. κύλινδρος, from κυλίνδειν, to roll). A
cylindrical surface, or briefly a cylinder, is the surface traced
out by a line, named the generatrix, which moves parallel to
itself and always passes through the circumference of a curve,
named the directrix; the name cylinder is also given to the solid
contained between such a surface and two parallel planes which
intersect a generatrix. A “right cylinder” is the solid traced
out by a rectangle which revolves about one of its sides, or the
curved surface of this solid; the surface may also be defined as
the locus of a line which passes through the circumference of a
circle, and is always perpendicular to the plane of the circle. If the moving line be not perpendicular to the plane of the circle, but moves parallel to itself, and always passes through the circumference, it traces an “oblique cylinder.” The “axis” of a circular cylinder is the line joining the centres of two circular sections; it is the line through the centre of the directrix parallel to the generators. The characteristic property of all cylindrical surfaces is that the tangent planes are parallel to the axis. They are “developable” surfaces, *i.e.* they can be applied to a plane surface without crinkling or tearing (see Surface).

Any section of a cylinder which contains the axis is termed a “principal section”; in the case of the solids this section is a rectangle; in the case of the surfaces, two parallel straight lines. A section of the right cylinder parallel to the base is obviously a circle; any other section, excepting those limited by two generators, is an ellipse. This last proposition may be stated in the form:—“The orthogonal projection of a circle is an ellipse”; and it permits the ready deduction of many properties of the ellipse from the circle. The section of an oblique cylinder by a plane perpendicular to the principal section, and inclined to the axis at the same angle as the base, is named the “subcontrary section,” and is always a circle; any other section is an ellipse.

The mensuration of the cylinder was worked out by Archimedes,
who showed that the volume of any cylinder was equal
to the product of the area of the base into the height of the solid,
and that the area of the curved surface was equal to that of a
rectangle having its sides equal to the circumference of the base,
and to the height of the solid. If the base be a circle of radius
*r*, and the height *h*, the volume is π*r*^{2}*h* and the area of the curved
surface 2π*rh*. Archimedes also deduced relations between the
sphere (*q.v.*) and cone (*q.v.*) and the circumscribing cylinder.

The name “cylindroid” has been given to two different surfaces. Thus it is a cylinder having equal and parallel elliptical
bases; *i.e.* the surface traced out by an ellipse moving parallel
to itself so that every point passes along a straight line, or by a
line moving parallel to itself and always passing through the
circumference of a fixed ellipse. The name was also given by
Arthur Cayley to the conoidal cubic surface which has for its
equation *z*(*x*^{2} + *y*^{2}) = 2*mxy*; every point on this surface lies on
the line given by the intersection of the planes *y* = *x* tan θ,
*z* = *m* sin 2θ, for by eliminating θ we obtain the equation to the
surface.