Cylinder

CYLINDER. In its oldest mathematical sense, the space swept out by a rectangle (fig. I) rotating round one side as axis. It is from the Greek kylindros, roller, from kylindein, to roll. The side (called generatrix, g) parallel to the axis (a) traces the curved cylindric surface; the other two (each equal to r) trace the circu lar bases of such a right circular cylinder. The nearest-lying gen eralization supposes g not perpendicular to the parallel base planes, but oblique, inclined at a fixed angle (like a pencil in writ ing), and tracing in the planes equal circles about the ends of the oblique axis as centres (fig. 2).

perpendicular distance between the planes is called the cylinder's altitude (h). The area of each base is clearly the product of base and altitude (h) is the cylin der's volume its curved surface (which suppose rolled out on a tangent plane) is the parallelogram of its edge (g) and the circumference of its base, i.e., 27f rh. A sector of the cylinder has a volume and curved surface proportional to the sector's angle. Subtracting from this volume that of the triangular prism of the axial sector-planes and the plane through parallel chords of the sector in the two bases, we obtain the volume of the cylinder seg ment cut off by a plane parallel to the axis, the formula being easily found by trigonometry.

Archimedes.

The metrical relations of cone, hemisphere and cylinder of the same base and height were especially studied by Archimedes of Syracuse (c. 225 B.c.), who showed (in Book I of his Sphere and Cylinder) their volumes to be as I : 2: 3; and the surface of the sphere to equal the curved surface of the (right) circumscribing cylinder ; i.e., two-thirds of its whole surface. These relations were deemed by Archimedes so important and beautiful that he expressed the wish that the sphere-cylinder figure be engraved on his tomb, a wish fulfilled at command of Mar cellus and furnishing a mark by which the quaestor Cicero iden tified it in 75 B.e., after nearly a century and a half.

Later View.—The more modern mind regards rather the cylin dric surface as the cylinder itself (compare the cone, q.v.), and defines it in full generality as the path of a right line (g) moving without turning, i.e., always parallel to itself or some fixed line or direction. To make such a surface definite, we must prescribe some directrix (d), generally a curve which the generatrix (g) shall describe passing through its points (fig. 3). Obviously such a surface is developable; that is, it may be imagined as flattened or rolled out smooth, without stretching, tearing or crinkling, on a tangent plane which evidently touches the surface full length along an element in any one of its positions. In case the directrix be an ellipse, the surface has been called "cylindroid," and may be defined as the path of an ellipse moving always parallel to itself, its centre always on a fixed line or axis. The same name is also applied to Cayley's conoidal cubic surface traced by the inter section of two moving planes y = x tan 0, z =m sin 2 0, whence, on eliminating the parameter 0, there results the equation of the surface, z = 2mxy. Plainly any cylindric surface may be vividly conceived as a straight tube traced by a directrix moving always straight and keeping always parallel to itself.

Sections.

If any directrix has a centre, the line through it, along which the centre moves parallel to g, is called the axis. Any plane through the axis makes a "principal section," namely, two opposite parallels, elements of the surface, which, with two paral lel sections of this plane by two parallel planes through the cylin der, form a parallelogram. This parallelogram becomes a rectangle if the two planes be perpendicular to the generatrix g. In a right circular cylinder such a perpendicular plane cuts it in a circle, but any plane oblique to the axis, in an ellipse, which is thus seen to be the parallel projection of a circle on an inclined plane, a fact lead ing directly to many properties of that curve (see ELLIPSE). If the circular cylinder be not right but oblique, then the plane sec tions across it are in general elliptic, but become circular for planes parallel to the directrix circle, and also in sections "sub contrary," i.e., of planes perpendicular to the principal section and inclined to the axis equally, but oppositely, to the directrix plane. The same condition occurs in oblique cones (see CONE).

A Limiting Case of the Cone.

The cylinder may be regarded as a cone whose vertex has withdrawn to infinity (Desargues, 1639). Accordingly all oblique sections of the right circular cylinder yield ellipses, but those parallel to an element (g) or the axis yield a pair of parallels, a limiting case of the parabola. Still further turned, the plane again cuts through, giving ellipses. But the same two parallels may be held for a limiting case of the bZ = 1 which for b=oo becomes x= an by perbola straightened out into a pair of right lines tangent at the :, 2 vertices. The conjugate, hyperbola, x2 d2-2 = —1, on the same supposition (b= oo) becomes a pair of imaginary lines, x= ±a V —I. (See CONIC SECTIONS and CO-ORDINATES.) (`V. B. SM.)