CYLINDER, (from the Greek av)ardetv, to roll) a solid formed by moving a straight line in the periphery of a circle, parallel to another straight line, which passes through the centre, and which makes any given angle with the plane of the circle. until the line come again into its first position.
The surface described by the moving line from the circle to any indefinite extension, is called a cylindric gui:fitce ; the straight line which passes through the centre of the circle is called the aris of the cylinder, and the circle the base of the cylinder.
If the axis be at right angles to the plane of the base, the cylinder is called a rigid cylinder ; but it' at oblique angles, then it is called an Oligne cylinder.
Euclid confines his definition only to a right cylinder, and defines it to be a solid formed by revolving a rectangle round one of its sides.
It is evident from this definition, that all sections passing through the axis, or parallel to the axis, have their opposite sides parallel ; viz., those which are formed by the cutting plane and the cylindric surface. From the definition here given, the following consequences may be drawn, as being ton obvious to require any formal demonstration. If a plane, parallel to another plane, drawn along the axis of a cylinder, touch the periphery of the base, the plane so posited will touch the surface of the cylinder, and will meet it in a line parallel to the axis ; but if such plane cut the plane of the base of the cylinder within its periphery, it will cut the cylindric surface in two parallel lines, and the common section of the plane and the cylinder will be a parallelogram, and, lastly, the common section of a plane, parallel to the base, with the cylindrical surface, is a circle with its centre in the axis. Let us now consider the property of a section which
will meet the plane of the base of the cylinder, but which will neither pass along the axis, nor be parallel thereto.
Figure 1.—Let A 11 1. n be a section of the cylinder through the axis, cutting the section proposed, and let the cutting plane meet the plane of the base AFEDNO ill RS: through the centre D of the base, draw the diameter A n, at right angles to a s. Let the plane be drawn through A 11 and the axis n of the cylinder, meeting the cutting plane in the line NI II no I. T. and the surface of the cylinder, in A AI and 1; L. Through and any other point, G, in m L draw Q a and I P parallel to s; through a Q and D a, draw the plane K Q o F, and through I P draw the plane Tr N E, parallel to the plane a Q o F, cutting A n at c ; and because a Q and i P are parallel to it s, the planes aoor and i P N n, are also parallel to it s ; therefore F o and E N. are respectively parallel to a Q and i P; conse quently. the figures aQC,F and i P N E, are parallelograms, and therefore a Q is equal to F o, and I P equal to E bemuse E N is parallel to n s, it is at right angles to A D. and hence it is plain that i r is bisected by DI L. Now the triangle A NI T, has the side A T cut into several parts by the intermediate points a, c, and AI T cut into other parts at the intermediate points a, 0, L, by lines parallel to the side A AI, and therefore the parts A D, D c, c 11, are respectively in the same ratio with the parts Dt x, n G. a L ; these being premised, we have therefore Therefore Ac X an ::A10 X GL